Текст
                    ACOUSTIC FIELDS
AND WAVES
IN SOLIDS
VOLUME II


1. FIELD EQUATIONS A. Acoustic Field dp 3 VT=^-F Vsv=-S B. Electromagftetic Field -V x E = ЭВ 3D If **H--+J. + J. B.1 Quasistatic Approximation E = -V0> V • D = , C. Constitutive Relations S =d-E + sE:T D = eT -E +d:T or T = -e-E + c^:S D = • E + e.S D. Power Densitj p = pv B = jjl-H- Jc = a • E P(,) = -v(/) • Ш + E(/) x H(/) PaTK = ReK_v* • T + E x H*) In the quasistatic approximation E(/) x Н(/)^Ф(0-^О(г) E x H* ^ Ф(/Ы))* 2. FIELD OPERATORS IN RECTANGULAR COORDINATES 0 -djdz d/dy V X d/dz -djdy д/дх ■д/дх djdx О 0 0 d/dz д/ду О д/ду 0 djdz О д/Вх О О д/dz djdy д/дх О 3. COORDINATE TRANSFORMATIONS 'д/дх О О о д/dz д/ду д/дх О [/] = [а][г] [а] = Jizx azy azz_ A. Stiffness and С mplianc Matrices [c] = [M]lcJM] [л'] = Г a2 XX a2 xz "I a2 vv < A. Qyiflzy avza azx&xx azyaxy azza _&хх@ух &ХУ&УУ 2aZydz О О " д/ду 0 О д/dz д/dz 3/3// О д/дх &yy&zz "Т" &yz&zy uyx&zz "Т" Gyz&zx &y\№zx ^yx^zv ! Qxiflzz "Ь &XZ&ZV &xz@zx Q$vP-zy ^xiflzx \ &xy&yz axzayy axzayx axx®yz &хх@уу axyayx_ f/V] is obtained from [M] by shifting the factors 2 into the lower left-hand submatnx. U. Piezoelectric Matrices C. Permittivity Matrix
ACOUSTIC FIELDS AND WAVES IN SOLIDS VOLUME II B. A. AULD Senior Research Associate in the W. W. Hansen Laboratories of Physics and Lecturer in Applied Physics Stanford University Л WILEY INTERSCIENCE PUBLICATION JOHN WILEY & SONS New York London Sydney Toronto
CONTENTS VOLUME 1 1 Particle Displacement and Strain 1 2 Stress and the Dynamical Equations 33 3 Elastic Properties of Solids 57 4 Acoustics and Electromagnetism 101 5 Power Flow and Energy Balance 135 6 Acoustic Plane Waves in Isotropic Solids 163 7 Acoustic Plane Waves in Anisotropic Solids 191 Й Piezoelectricity 265 Appendix 1 Cylindrical and Spherical Coordinates 349 Appendix 2 Properties of Materials 357 Appendix 3 Acoustic Plane Wave Properties 383 Index 411 VOLUME II 9 Reflection and Refraction 1 10 Acoustic Waveguides 63 11 Acoustic Resonators 221 12 Perturbation Theory 271 13 Variational Techniques 333 Appendix 4 Rayleigh Wave Properties 375 ItiMiography 395 Index 397
Copyright © 1973, by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. No part of this book may be reproduced by any means, nor transmitted, nor translated into a machine language without the written permission of the publisher. Library of Congress Cataloging in Publication Data Auld, Bertram Alexander, 1922- Acoustic fields and waves in solids "A Wiley-Interscience publication." Bibliography: p. 1. Solids Acoustic properties 2. Sound-waves Industrial applications 3. Elastic waves 4. Wave- motion, Theory of. I. Title. QC176.8 A3A84 534'.22 72 8926 ISBN 0-471-03701-х (V. 2) ISBN 0-471 03702-8 (Set) Printed in the United States of America 10 987654321 PREFACE This book has developed from a lecture course on mechanical waves and vibrations in solids, for first and second year graduate students. It is intended to present, in a manner congenial to the disciplines of Applied Physics and Tlectrical Engineering, a coherent treatment of the subject, starting from fundamentals and proceeding to applications. In Volume I acoustic field theory was developed step-by-step from the basic principles of mechanics and electricity. The present volume applies this theory to a variety of scattering, waveguide and resonator problems. As in the previous volume, the material is organized along the lines used in graduate level electromagnetic theory texts. This approach is of particular importance in connection with acoustic waveguide theory, where recent advances in thin film technology and waveguide transducer design have encouraged the application of microwave electromagnetic concepts to problems in acoustics. Generally speaking, acoustic field problems are significantly more difficult than electromagnetic problems, and approximation methods must often be used to obtain solutions. For this reason, chapters on two powerful approximation procedures (perturbation theory and variational techniques have been included). These chapters contain many examples chosen to demonstrate the procedures and to present practical information for the applications engineer. It has not been possible to treat the topics of diffraction, amplification and nonlinear acoustics in this volume. However, some recent references on these important and rapidly developing areas of acoustics theory are presented in the bibliography. The examples and end-of-chapter problems are meant lor classroom use and have been selected to illustrate both concepts and problem solving methods in a progressive manner. Appendix 4 gi%es a tabulation of the Rayleigh surface wave properties needed for the transducer calculations described in Chapter 10 and the perturbation calculations in t luptcr 12. Stanford, California B. A. Auld
Chapter 9 REFLECTION AND REFRACTION Л. WAVE SCATTERING AT BOUNDARIES 1 H. ISOTROPIC SOLIDS 2 C. ANISOTROPIC SOLIDS 6 IV ACOUSTIC FRESNEL EQUATIONS FOR ISOTROPIC SOLIDS 21 K. ACOUSTIC FRESNEL EQUATIONS FOR ANISOTROPIC 38 SOLIDS PROBLEMS 57 К Г FERENCES 61 Л. WAVE SCATTERING AT BOUNDARIES In Volume 1 the fundamentals of acoustic field theory were developed *tep-by-step from the basic principles of mechanics and electricity. Volume II Starts from the field equations given in symbolic form on the front cover papers, and applies the theory to a variety of acoustic boundary value problems. The cover papers also list rectangular coordinate representations of the field operators, transformation properties of the constitutive parameters, and a number of useful identities. With this information on hand, the experienced reader should be able to proceed without constantly referring back to Volume T. The simplest, and one of the most important, boundary value problems in ttlectromagnetism and acoustics is the scattering of a uniform plane wave incident upon a plane boundary between two different media. In the next chapter it will be seen that many different waveguide configurations can be analyzed by using solutions to this simple scattering problem. A brief introduction to the subject was given in Chapter 4 of Volume I, which considered the case of a wave incident normally on a plane boundary. This chapter will deal with the more complicated case of oblique incidence.
2 REFLECTION AND REI RACTION When a plane wave impinges on an interlace between two different media, it is necessary that certain boundary conditions be satisfied at the interface. Because these conditions cannot be satisfied by the incident wave alone, it is necessary to include a certain number of reflected waves in the first medium and transmitted waves in the second medium. If the incident wave travels normal to the interface, the reflected and transmitted waves are also normal to the interface; this was the case treated in Example 3 of Chapter 4. For an obliquely incident wave the scattered waves travel in different directions. This change in direction of the transmitted waves is called refraction The character of the transmitted (or refracted) waves depends very strongly on the nature of the second medium. When there is only one wave velocity for each propagation direction (electromagnetic waves in an isotropic solid), there is only one refracted wave direction. For electromagnetic waves in an anisotropic solid and for acoustic waves in an isotropic solid, there may be two wave velocities for each propagation direction and two refracted ware directions may occur (birefringence). Acoustic waves in an anisotropic medium may have three wave velocities for each propagation direction and three refracted ware directions may occur (trirefringence). In elcctromagnetism, propagation directions of the plane waves scattered at a plane boundary are given by Snell's Law and the amplitudes of the scattered waves are given by the Fresnel Equations. This chapter will develop and examine corresponding relationships for acoustic wave scattering in isotropic, anisotropic, and piezoelectric solids. B. ISOTROPIC SOLIDS B.l Snell's Law Consider a plane boundary, in the xz plane, between media with different acoustic properties (Fig. 9.1). The particle velocity and traction force must be continuous at all points on the boundary, v = v' T-n-T'-n. (9.1) This means that the fields on both sides of the boundary must have the same functional dependence on x and z at the boundary plane. Since plane wave fields are described by wave functions the incident and scattered waves must all have the same component of к tangential to the boundary. This constraint, illustrated in Fig. 9.1 for an incident wave with kx = 0, is the basis for deriving Snell's Law. B. ISOTROPIC SOLIDS 3 FIGURE 9.1. Acoustic plane wave scattering at a plane boundary between two isotropic media. The derivation is performed most efficiently by using the slowness surfacet and reasoning geometrically. Figure 9.2 shows constructions for both the electromagnetic and acoustic cases, assuming an incident wave which propagates in the yz plane. In an isotropic medium the electromagnetic wave velocity is independent of the propagation and polarization directions, and the slowness surface is a single sphere. From Fig. 9.2(a), continuity of k. at the boundary then leads directly to the isotropic electromagnetic Snell's Law ft) . fl CO . „ V) . c, ,,. — sin 6, = — sin 6n = — sin GT. (9.2) К v. Ve The transmitted wave is bent (or refracted) away from the direction of the incident wave. Acoustic media, on the other hand, are birefringent even in the isotropic case (Example 1 of Chapter 6 in Volume I) and there are several scattered waves with the same value of kUm as the incident wave. From Fig. 9.2(b) and (c) the isotropic acoustic Snell's Law for either shear or longitudinal incidence is therefore sin 0, = — sin 0 — — sin 6, = — sin 6'„ (9.3) t The slowness surface gives ihc magnitude of kw as a function of its direction.
(a) Electromagnetic <b) Acoustic, (с) Acoustic, longitudinal shear incidence incidence FIGURE 9.2. Derivation of Snell's Law relations from the slowness surface. with Or, = в1щ = os elu = on = et o:es = e; o'Tl = e;. Scattering at a boundary between birefringent media produces two transmitted (or refracted) waves. B.2 Critical Angles The geometrical constructions used for deriving the Snell relations do not always give real intersections with the slowness surface for all of the scattered waves. An example is shown in Fig. 9.3(a) for the case of an electromagnetic wave. The physical significance of this situation may be deduced from the dispersion relation for the transmitted wave, Since fcrz = fc2 > 7^ . B. ISOTROPIC SOLIDS 5 the у component of kT is pure imaginary, The plus sign is chosen to satisfy the physical requirement that the transmitted field e—ikzze\kTv\v must approach zero as у -> —со. The incident wave experiences total internal reflection, and the transmitted wave is called evanescent (Fig. 9.4). This effect occurs for any angles of incidence greater than the critical value (defined in Fig. 9.3(b)). That is, (0„). = sin-'^. (9.4) Real values of the critical angle occur only when V, < V'f. For acoustic waves several critical angles may occur because of the greater complexity of the slowness surface, and they may appear in both reflection and transmission. It is seen from Fig. 9.3(b) that the critical incidence angle occurs when the transmitted wave travels parallel to the boundary. For shear wave incidence and the relative acoustic velocities shown in Fig. 9.5 there are three critical angles. These correspond, respectively, to propagation of the reflected longitudinal wave, the transmitted longitudinal wave, or the
6 REFLECTION AND REFRACTION у Totally reflected incident wave vr Evanescent \ wave \ FIGURE 9.4. Electromagnetic field distribution along the у direction under conditions of total internal reflection in the upper medium. transmitted shear wave parallel to the boundary. As in the electromagnetic case, the field of a scattered wave becomes evanescent when the incidence angle exceeds the critical value for that particular scattered wave. The number of critical angles for other combinations of incident wave type and relative acoustic velocities are listed in Table 9.1. Values of the critical angles are easily obtained from the slowness surface constructions. C. ANISOTROPIC SOLIDS C.l Snell's Law For anisotropic media, where the wave vector amplitude varies in a complicated manner with angle of propagation, an analytic statement of Snell's Law is difficult to obtain. However, the scattering angles can still be obtained graphically from the slowness surface. The electromagnetic case is now birefringpnt and there are two reflected and two transmitted waves, just as in the isotropic acoustic case. In the anisotropic acoustic case the two waves of shear type may be nondegenerate, and two extra scattering angles are now possible (Fig. 9.6). Anisotropic acoustic media are trirefringent and produce three transmitted (or refracted) waves. (a) s n (ecr)T, = v„iVi (b) sin (ecr)R, = vsiv, (c) sm (e„)T„ = vs/v; FIGURE 9.5. Critical angles for an incident shear wave (l^'/K, > 1). TABLE 9.1. Critical Angles for Acoustic Plane Wave Scattering at a Plane Boundary Between Isotropic Solids Shear Wave Incidence v» > v'i > v'a 1 critical angle V'\ > К > К 2 critical angles V[ > Vg > Vs 3 critical angles Longitudinal Wave Incidence V[ > V'i > Vg No critical angle V{ > Vt> ys' 1 critical angle V[ > Vs > Vj 2 critical angles
FIGURE 9.6. Scattering of a quasilongitudinal plane wave at a boundary between anisotropic media, showing construction of the scattering angles from the slowness surfaces. Figures 9.7 and 9.8 illustrate some of the unusual features of anisotropic scattering that arise from nonparallclism of the wave vector к and the group velocity Vs. The illustrations refer to a cubic crystal with propagation in a cube face, the yz plane, and with a free boundary lying along the rz plane. Since the second medium is vacuum, only reflected waves occur. In Example 4 of Chapter 7 in Volume I it was shown that there is one pure shear wave polarized normal to the cube face, and quasishear and quasilongitudinal waves polarized in the cube face. The pure shear wave has only two stress components TTy and Tx2, while the quasishear and quasilongitudinal waves have stress components Tvv, Тгг, and Tvs. Consequently, the quasishear and quasilongitudinal waves arc not coupled to the pure shear wave by the free surface boundary conditions, Tyx = Tvu = Туг = 0. (9.5) С. ANISOTROPIC SOLIDS 9 In Fig. 9.8 both the I and R quasishear waves have positive values of kx, but negative values of (Vg),. Under these conditions the ray vectors (or energy How directions) and the wave fronts (or surfaces of constant phase) have the relative orientations shown in Fig. 9.9. Г.2 Critical Angles Another consequence of elastic anisotropy is that the critical angle phenomenon is much more complicated than for isotropic media. As the wave vector angle 0, of the incident quasishear mode in Fig 9.8 is increased, the reflected quasilongitudinal wave vector kH, eventually becomes parallel to the z axis at A (Fig. 9.10a). Beyond this angle, the quasilongitudinal wave is vranescent and decays exponentially away from the boundary. As the incidence angle is further increased, a second critical angle occurs when к reaches point В in Fig. 9.10(b). Beyond this point a noneianescent reflection again appears. In this case both reflected waves lie on the quasisheai bianch. That is to say, the evanescent wave in Fig. 9.10a changes from quasilongitudinal to quasishear as it shifts from A to B. The wave vector of the second reflected quasishear wave is directed downward, but the energy flow is upward (I l£ 9.11).
FIGURE 9.8. Quasishear scattering at an |010]-ori- cnted free boundarj of a cubic crystal. Incidence is in the (100) plane. Reflected Incident quasishear quasishear FIG1Щ Ь 9 9 Ray vectors (solid arrows) and wave fronts (dashed lines) corresponding to Figure 9.8. C. ANISOTROPIC SOLIDS 11 FIGURE 9.10(a). Quasishear scattering at an Г010]-oriented free boundary of a cubic crystal, with (100) plane propagation. Condition for the existence of an evanescent quasilongitudinal wave ( .3 Conical Refraction К lias been seen that anisotropy of the acoustic slowness surface can lead to some unusual and interesting wave scattering effects at a plane boundary Ixrtween two media. Л case of special interest occurs when the boundary is normal to a three-fold crystal symmetry axis in the second medium, and the incident wave impinges normally on the boundary. The cube diagonal direction of a cubic crystal ([111] in Fig. 9.12a) is one example of this kind of symmetry axis. Section I of Chapter 7 in Volume I listed some of the general characteristics of acoustic plane waves propagating along a three-fold symmetry axis. The waves arc cither pure transverse or pure longitudinal, and the transverse (or shear) waves are degenerate. Also, and this is a most important point, (lie shear slowness curves cross the symmetry axis at an angle. Figure 9.12b illustrates this behavior for the [111] direction in a cubic crystal, where one i>l llic shear curves corresponds to a [I T0]-polarized pure shear wave and the
12 REFLECTION AND REFRACTION FIGCJ R E 9.10(b). Quasi-shear scattering at a 1010]-oriented free boundary, with (100) plane propagation. Incidence beyond the second critical angle. other solution reduces to a pure shear wave polarized at right angles to the plane containing [HO] and [111]. According to Sections G and H of Chapter 7 in Volume I, the energy (or group) velocity is always normal to the slowness surface. This means that the energy velocities of the shear waves propagating along [111 ] in Fig. 9.12b are both deflected away from k. The group velocity of the [lT0]-polarized shear wave is deflected toward the [001] axis and the group velocity of the First reflected quasishear FIGURE 9.11. Ray vectors (solid arrows) and wave fronts (dashed lines) corresponding to Figure 9.10(b). (c) riGURH 9.12. Group velocity deflections for pure shear waves travelling along the threefold symmetry axis [111] in a cubic crystal
14 REFLECTION AND REFRACTION other shear wave is deflected toward the [ПО] axis. There are, however, still further complications. Figure 9.12b represents slowness curves for waves propagating in the plane defined by the [110] and [0011 crystal directions, but it is clear from the cubic symmetry of the crystal that identical curves will be obtained for propagation in any plane that passes through a cube edge and a cube face diagonal. Slowness curves for the plane passing through [100] and [011] therefore appear as shown in Fig. 9.12c. The shear waves propagating along [111] now have different polarizations than those given in Fig. 9.12b. In itself, there is nothing unusual about this; degenerate shear waves can always be combined to produce arbitrary polarization directions. The unusual feature of the present situation is that the group velocity directions are also different than they were in Fig. 9.12b. For the [Oil]-polarized pure shear wave in Fig. 9.12c the group velocity is deflected toward [100] and the other shear wave is deflected toward [011 ]. Slowness curves for the plane passing through [010] and [101] also have the same shape as in Fig. 9.12b and c, giving two additional polarization and energy velocity directions for the pure shear waves traveling along [111]. This kind of behavior always occurs when shear waves propagate along a three-fold crystal symmetry axis. The detailed behavior of the waves is best illustrated by finding wave solutions for a specific example and then calculating the power density (or Poynting) vector P given by ID on the front cover papers. EXAMPLE 1. Acoustic Plane Wave Propagation Along the U1T] Direction in a Cubic Crystal. A rotated coordinate system appropriate to this problem is shown in Fig. 9.12, and the stiffness matrix referred to these coordinates is given by (3.46) in Volume Г. It is most convenient to write the strain-displacement relation in terms of stiffness rather than compliance. If a complex waveform e*<wt-'«"> is assumed, the acoustic field equations then take the form ev T V • г — n = с: Vsv 9 dt dt -;7сГ5. = iwpiv (a) /га T"i- = -ik{c"3vz. + c«tv) (d) -'7c TV = iWpVy. (b) 1(0 7Л- = —ik(c'[3vz. - см»*») (e) -,7c TT = iwpvz~ (c) |"«)Г3- = —ikc33vz- (f) /ш Г4- = —ikctiv.y~ сто = (h) iwTB. = ikc"svv~ (i) (9.6) for the nonpiezoelectric case. Following the examples of Chapter 4 in Volume I, these equations are grouped according to the variables they contain. From (c) and (f) —ikT3~ io)pvt~ iioTr = ikc'3iv,~, (9.7) C. ANISOTROPIC SOLIDS 15 and elimination of 7"3- gives <pw* - c33k-)uz- = 0. (9.8) This is a pure longitudinal wave, with dispersion relation pc,,2 = c^fc2 (9.9) and extra stress components 7> = - - fa, <9-10> со Гг = (9.11) from (9.5d) and (9.5e). One of the pure shear wave solutions is obtained from (b) and (g) -ikTr = itopvr (9.12) ,7„Г4. = -ikcu-vr, (9.13) and the dispersion relatio is p<»2 = cl№. (9.H) There is an extra stress component 7V = -r;5<y, (9.15) to from (9.6i). The remaining equations give the second pure shear wave solution. From (a) and (h), -ikTs- = /oiptv (9.16) иоТъ. = -/ArrlW, (9.17) and the dispersion relation is the same as (9.14). Extra stress components, ю to are obtained from (d) and (e). The group velocity Vs is always parallel to the power density vector P given by I .D on the front cover papers. For the nonpiezoelectric case v* - T P Consider, for example, the //"-polarized shear wave. This wave has field components ,v, Tt. = 7>.-, 7V = Tu.r., from (9.13) and (9.15). Accordingly, components of
16 REFLECTION AND REFRACTION (b) HLiURt 9.13. Poynting vector directions for (/"-polarized and "-polarized pure shear waves propagating along the " (or [111]) direction in Fig. 9.12a. C. ANISOTROPIC SOLIDS 17 x" FIGURE 9.14. Cone of Poynting vector directions swept out by rotating the polarization direction of a pure shear wave propagating along the " axis in Fig. 9.12a. Tr = Tr-X., Г2. = Try, and Г5- = Tx.z-. Components оГР arc therefore О ' * T ' * T ^ " 12 = — ~ Vi~l i~y — — - IV ' Х~У — Cl!i I I Pv- = - \ vt-Trr = 0 (9.21) 1 1 к as shown in Fig. 9.13b. A little thought will show that these polarization and power llow directions are consistent with the results obtained from the slowness curves in Fig. 9.12b. To calculate the power flow direction for a shear wave with arbitrary polarization, linear combinations of the .r" polarized and .'/"-polarized solutions arc formed and then used to calculate the complex Poynting vector. The particle velocity field v = x"v cos x + S"v sin x (9.22) is polarized a( an angle x with the x" axis (Fig. 9.14) and the corresponding stress Pare /"»- =~2"*W=0 (9.20) Pz- = — — Г?-Г,-г- = — - IV J"^г- = — C« |ty]2. The polarization and power flow directions are therefore as shown in Fig. 9.13a. For the ж"-polarized shear wave, the field components in (9.16) to (9.19) are vx~,
18 REFLECTION AND REFRACTION field components arc к Tx-X- = - - cKv cos x Ty-y = ^ c'^v cos x к Ту-г,- = t£p sin z (9-23) A: 7VS- = - - c^e cos * A: 7>„. = - c'^v sin z. Components of the Poynting vector are thus P*- = - \ uf~Ti-x- = j - c^, № cos 2* *V- = - J «£■ *Vv = - i ~ c"& \tf sin 2* (9.24) 1 4 I At L L Oi These components reduce to (9.21) and (9.20), respectively, for z = 0andz = W2; and the angle between P and i" is the same for all polarization directions As the polarization angle is rotated in the ж y" plane, the power flow angle rotates twice as fast in the other direction. The power flow must rotate twice as fast as the polarization, because the polarization is time-harmonic and a reversal of the polarization direction cannot change the power flow direction. Example 1 has shown that the power flow direction for a shear wave traveling along the [111] direction in a cubic crystal rotates about the cone shown in Fig. 9.14 as the polarization angle is rotated. This behavior occurs in all cases of shear wave propagation along a threefold crystal symmetry axis. Л rather remarkable refraction effect results when an acoustic wave travels into a medium of this kind. Suppose that a shear wave is normally incident upon the boundary between an isotropic medium and a cubic medium with the [111] axis normal to the boundary (Fig. 9.15). According to the boundary conditions r \ — \ in (9.1), the transmitted wave in the second medium must have the same shear polarization as the incident wave. When the polarization of the incident wave is rotated, the power flow direction of the transmitted (or refracted) wave in the second medium rotates around a cone of refracted wave direction . C. ANISOTROPIC SOI IDS 19 Cubic medium FIGURE 9.15. Internal conical refraction of a shear wave incident on the boundary between an isotropic medium and a cubic medium. In electromagnetism this phenomenon is called internal conical refraction, and the same term is used in acoustics. Experimental observation of the effect is difficult because one cannot realize a true uniform plane wave in practice and must work with a beam of finite diameter. Such a beam is equivalent to a distribution of uniform plane waves with wave vectors к distributed about the beam axis. Since each plane wave refracts in a slightly different way, the power transmitted into the anisotropic medium is "smeared out" over the cone of refraction. Figure 9.16 shows this effect schematically for the acoustic case. The acoustic intensity, indicated by shading in the figure, is seen to peak at the correct point on the cone but it is spread out over the entire conical surface. Internal conical refraction of electromagnetic waves was confirmed experimentally more than 80 years ago, but the acoustic effect was not observed until the experiments of Papadakis on NaCI and CaCo3 in 1963 and those of McSkimin and Bond on quartz in 1965. Conical refraction into a cone of elliptical cross section has also been reportcd.f Figure 7.10 in Volume T shows that, for wave propagation in a cube Bice, there are two points ф and ® with the same group velocity but different wave vector directions (normal to the ray surface). Tf one examines the ray curves for different plane sections passing through the [100] axis, one finds that there is a complete cone of wave vector directions corresponding to the I Reference 12 at the end of the chapter.
20 REFLECTION AND REFRACTION "[110] Axes in sample General configuration FIGURE 9 16 "Smearing out" of the conical refraction effect for a finite diameter acoustic beam in NaCl. (After Papadakis). group velocity at ф and Q). This is similar to the cone of power flow (or group velocity) directions corresponding to the wave vector к in Fig. 9.15, and leads to a phenomenon called external conical refraction. Suppose that acoustic energy is traveling along the [100] direction in a cubic crystal and impinges at normal incidence upon a boundary with an isotropic medium. Because of the boundary conditions (9.1) the acoustic field external to the cubic crystal must contain the same transverse wave vector components as the internal field. The latter, however, can have any value of к on the cone of D. ISOTROPIC FRESNEL EQUATIONS 21 wave vector directions passing through ф and ® in Fig. 7.10; the external wave vector must therefore lie on an external refraction cone. Although this phenomenon is well known in optics it has not yet been observed experimentally in acoustics. D. ACOUSTIC FRESNEL EQUATIONS FOR ISOTROPIC SOLIDS The preceding sections have dealt with Snell's Law, which describes the propagation directions of scattered waves at a boundary. To describe the scattering process completely it is necessary to have additional equations which relate the scattered wave amplitudes to the amplitude of the incident wave. These correspond to the Fresnel equations in electromagnetism. This bection is concerned with isotropic media, where the acoustic Fresnel equations take the simplest form because plane waves are either pure shear or pure longitudinal. I) I Horizontally Polarized Shear (SH) Wave Incidence The case of horizontally polarized shear wave incidence is the most straightforward. If propagation is in the yz plane (Fig. 9.17), the incident wave has v, = xAe ik "г у vT =xB'e~ik>t RGU RE 9.17. Reflection and refraction of a horizontally polarized shear (SH) wave at a plane boundary between isotropic media.
22 REFLECTION AND REFRACTION acoustic field components ( )г = £44 = _ (Mi Си/4в-*х- (9.25) ,„ . _ c« 5(^)r _ (Ui --an ко су (о The boundary conditions (9.1) can therefore be satisfied by reflected and transmitted waves of the same polarization; that is, (9.26) (9.27) (».)„ = Be**' (От = B-e^1. According to Snelfs Law the scattering angles are such that (A)i = (к,)и = (Kh = fc,; and the boundary equations at у = 0 are therefore A + B= B' - ^4((AM - = ~ - W o> «) where the common exponential e lk*z has been dropped. These equations are solved directly for the reflection coefficient Ts and the transmission coefficient Г' p __ * _ z»cos e» ~~ Z*cos e» (9 98) s ~ a ~ zs cos os + z; cos 0; p, _ 6^ _ 2Z,cos 0, (0 29) * a zs cos es + z; cos 0; where Zs = (pc«)"2 and Zs' = (p'cU)1* are the characteristic shear wave impedances for the upper and lower media. The angle of incidence 6S and the angle of transmission 6's must satisfy the condition si" 0, = К (9 30) sin 0's V'/ from bneirs Law. The reflection and transmission coefficients vary with the incidence angle 6S in a manner that depends on the velocity ratio V'JVS and the impedance ratio Z'JZS of the two media. If Ks'/F6 < 1 there can be no critical angle for the transmitted shear wave (Table 9.1), and the scattering coefficients vary smoothly with 0S. Since B't < 0S when Ks'/Ks < I, the reflection coefficient Г, D ISOTROPIC FRESNEL EQUATIONS 23 for the case Z'sjZa < 1 always goes to zero at some value of incidence angle 0S (Fig. 9.18a). This does not happen when V'JVS < 1 and Z'JZ, > I. Tf V'JVS > 1, as in Fig. 9.5, a critical angle does occur (Fig. 9.18b). Beyond this angle there is total reflection, and the wave in the second medium is evanescent. Since 6's > 0S when V'sjVs > I, the reflection coefficient has a zero only when z;/Zs > I. D.2 Vertically Polarized Shear (SV) Wave Incidence In this case there is no coupling to the horizontally polarized shear waves, but both reflected and transmitted longitudinal waves are excited (Fig. 9.19). 0 10 20 30 40 50 60 70 80 90 es (a) FIGURE 9.18. Scattering coefficients for a horizontally polarized shear (SH) wave at a boundary between yttrium aluminum garnet and fused silica. In this and following figures, only the magnitude is shown when the scattering coefficient is complex, (a) Incident wave in the yttrium aluminum garnet (F,'/Fs < 1 and Z'JZ, < I)
24 REFLECTION AND REFRACTION 0 10 20 30 40 50 60 70 80 90 <b) FIGURE 9.18b. (b) Incident wave in the fused silica <V'JVS > 1 andZ'JZ, > 1) The particle velocity fields shown in the figure arc v,, = - X X k'* л _-Лт«-г vT5 = —— ese (9.31) D. ISOTROPIC FRESNEL EQUATIONS 25 (9.32) FIGURE 9.19. Reflection and refraction of a vertically polarized shear (SV) wave at a plane boundary between isotropic media. Only two stress components arc relevant to the boundary conditions (9.1), that is, _ c12 дюг cu dvv UO OZ Ш cxv iw\cz Су I From (9.31) and Fig. 9.19, continuity of r„ and vz at у = 0 gives the boundary value equations (A - в,) sm e, + в, cos o, = e; sin о- - e; cos 0; (9.зз) and (/Is + Bs) cos 0, + Вг sin 0г - В.; cos 0; + В; sin 0;, (9.34) where the common exponential factor p-'fc*z has boon dropped Continuity conditions for ГЬК and Tv. are obtained from (9.32). That is, (As + BsXcKks — clYks sin 0S) cos 6S + B,(c12/cz sin 0, + СцЛ^сов* 0,) = В'£с[2кг - с;,k; sin 0.;) cos 0; + BKsfc, sin 0; + cnk't cos2 0;) (9.35) у
26 REFLECTION AND REFRACTION given in Section 6.B of Volume 1, and Snelfs Law, k, = ks sin 0„ = k; sin 6's = fc, sin 0[ = kj sin 0;, where k, = — , etc. It is also convenient to use the relation Л + 2/j. cos2 0j = (Я + 2^) cos 20s, derived from Snell's Law and И = >* + 2И These substitutions lead to the set of scattering equations As sin 0S = -B, cos 0, - B/cos 0,' + S5 sin 0S + Bs' sin 0; (a) /4S cos 0S -= —B, sin 0, + Bt' sin 0г' - fls cos 0S + B's cos 0^ (b) T ■ -AgfjLk.sm 20, - -BIX + 2//>fctcos20s + ВЦЛ' + 2(j.')k\cos20^ + B^ks sin 20s - B's(i'k's sin 20; (c) -Astxkscos 20s = -B^k,sin 20t - B'^'k\ sin 20[ - Bsfiks cos 20s - B>% cos 20^. (d) (9.37) D. ISOTROPIC TRESNLL EQUATIONS 27 From (9.37) the reflection and transmission coefficients are 1 Is A, Д 1 Is л Д r> _ _ AJL* Д Г' B's ~ A* Д' ~ Д (9.38) Here Д is the determinant of the coefficients on the right-hand side of (9.37) iintl Д,8 is the determinant obtained by replacing the coefficients of BL with those of As, etc. It is convenient to simplify the secular determinant by iciiioving factors sin 0г, sin 0J, sin 0„ sin Q\ from columns I to 4 respectively and a factor kz from rows 3 and 4. This gives Л — к* sin 0, sin 0\ sin0S sin 0\ —col©( —cotfl] 1 1 —cotO, co\.6't (2 sin2 0,-1) —(A' + 2/,')(2 sin2 (»; - I) W + 2/() — -—j 2« Lot 0, -2ft col 6, sin20, sinHlt —2ft col 0, — 2fi cot 0'( —/j(2 — esc2 б,) fi'(2 — esc2 0\ (9.39) Because of the complexity of these results, a general discussion of SV wave maltering is not possible. Critical angles may be calculated from Snell's Law, and some qualitative information about the scattering amplitudes can hi4 deduced from this. To proceed further, numerical computation must be пмч1. Figure 9.20 gives as an example the scattering properties of an SV wave 11I 1111 interface between fused silica and yttrium aluminum garnet (assumed яи)1м)р1с) and (Л, - Bs)cH(kz sin 0e - fc, cos2 в,) + бгс44(кг + к, sin 0,) cos 0, = Вуи(кя sin 6; - К cos2 0S) - B'lC'^kz + k\ sin 6\) cos 0|. (9.36) Equations (9.33)-(9.36) can be simplified by using the isotropic elastic relations cn — c\2 + 2c41 с.,, = A
28 REFLECTION AND REFRACTION D.3 Longitudinal (P) Wave Incidence In this case only the incident wave is changed in Fig. 9.19 and the scattered waves appear as shown in Fig. 9.21. Since the scattered wave polarizations and angles are the same in both figures, only the terms on the left hand side of (9.37) need to be changed. These terms become -AfCosdf (a) A, sin 0, (b) (9.40) AS). + 2ц)к, cos 20, (c) —Affik, sin 20{, (d) (a) Fl GU RE 9.20. Scattering coefficients for a vertically polarized shear (SV) wave incident on a boundary between fused silica and yttrium aluminum garnet, (a) Reflection coefficients. D. ISOTROPIC FRESNEL EQUATIONS 29 0 10 20 30 40 50 60 70 80 90 (b) FIGURE 9.20b. (b) Transmission coefficients. liud the reflection and transmission coefficients are found to be A, д _Д[- A д Bs д.. Аг д E* _Д1. Аг д (9.41)
30 REFLECTION AND REFRACTION у FIGURE 9.21. Reflection and refraction of a longitudinal (P) wave at a plane boundary between isotropic media. where the secular determinant Д is again given by (9.39). In Fig. 9.22 the scattering properties of a P wave are shown for a boundary between fused silica and yttrium iron garnet (assumed isotropic). D.4 Reflection at a Free Boundary For the important practical problem of reflection at a stress-free boundary the above results become relatively simple. This is a fortunate result because it permits quick and easy solutions to some of the waveguide problems discussed in Chapter 10. The analysis for SH shear wave incidence is applied to the free boundary case by setting c'u = 0 in (9.28). This gives total reflection with zero phase angle, \\ 1. (9.42) By contrast, an SV wave or a P wave is not totally reflected at a free boundary (Fig. 9.23a and b). Each incident wave scatters partly into a reflected wave of the other type. The laws governing the scattering of an SV wave are D. ISOTROPIC FRESNEL EQUATIONS 31 10 20 30 40 50 60 70 80 90 в, (a) FIGURE 9.22. Scattering coefficients for a longitudinal (P) wave incident on a boundary between fused silica and yttrium aluminum garnet, (a) Reflection coefficients. obtained by setting the transmitted amplitudes B't and B's equal to zero in (9.37c and d), and then solving for B{ and Bs. This gives Г = ' - 'S A —pks sin 20, [iks sin 20, —/гк, cos 20, — /гк,. cos 20, -(Л + 2[i)kl cos 20, [iks sin 20, —fikt sin 20, — fiks cos 20, i ml where 2(VJVS) sin 20, cos 20, sin 20, sin 20, + (VjVf cos2 20, sin 20, sin 20, - (F,/F,)2 cos2 20, Г.. = - sin 20, sin 20, + (VJV,)* cos2 20, sin 0, _ Vi sin 0, ~ V, (9.43) (9.44)
32 REFLECTION AND REFRACTION 61 (b) FIGURE 9.22b. (b) Transmission coefficients. Similarly, for longitudinal wave incidence r _ sm 20, sin 20, - (VJV,? cos2 20, " sin 26, sin 20, + (VtjVsf cos2 20, 2(Г,/К,) sin 20, cos 20, sin 20, sin 20, + {VJVf cos2 20, Figures 9.24 and 9.25 show typical curves of these reflection coefficients as a function of the angle of incidence. As anticipated from Table 9.1 there is no critical angle for the case of longitudinal wave incidence. Another feature of interest is that, in both cases, the incident wave is totally scattered into the wave of the other type at two values of the incidence angle. This phenomenon, which has some similarity to the Brewster angle effect in SV wave P wave cos 0, I , = ks COS 0, -Voltage" £,= -sm29s "Current" /, = (vY)s cot 0, (Та), 1 cos 0, ft = hi cos в/ 4/oltage"£,= -(M\-^\ - "Current" /;= (vy)i п, = sir 20, щ = cos 20s (c) I IGURE 9.23. Reflection at a free boundary, (a) Vertical shear (SV) incidence, (b) Longitudinal (P) incidence, (c) Equivalent circuit. (After Oliner)
34 REFLECTION AND REFRACTION -0.; -0 0 10 20 30 40 50 60 70 80 90 FIGURE 9.24(a). Reflection of a vertically polarized shear (SV) wave at a stress-free boundary in yttrium aluminum garnet (assumed isotropic). 20 1.8 D. ISOTROPIC FRESNEL EQUATIONS 35 1.6 - 1.4 - 1.2 - 0 10 20 30 40 50 60 70 80 90 FIGURE 9.24(b). Reflection of a vertically polarized shear (SV) wave at a stress-free boundary in fused silica. wave (9.42), an incident shear wave with arbitrary polarization will scatter into an elliptically polarized reflected wave. I here arc two important and useful algebraic relations connecting the reflection coefficients (9.43)-(9.46) The first one, Г„ = Г„, (9.47) is obvious from inspection. It can also be easily shown that rf,+rjsFs,= I. (9.48) I his relation will be very useful for some of the derivations in Chapter 10. Reflection coefficients at a free boundary can also be calculated from an rrt|iiivalciit transmission line circuit (Fig. 9 23c) Equivalent "voltages' and i—i—Г optics, has been used experimentally to convert longitudinal waves into shear waves. It might appear at first from the values of the reflection coefficients that power conservation is violated at these points. This apparent anomaly arises from the manner of defining the wave amplitudes. The power divides between the reflected SV wave and the reflected P wave in the ratio Г" 1 — Г2 for Fig. 9.24, and the power divides between the reflected P wave and the reflected SV wave in the ratio 1 — Г2 for Fig. 9.25. In Fig. 9.24 the totally reflected SV wave (0S > 0cr) undergoes a phase shift upon reflection. Since the phase shift is always zero for an SH 20 18 16 1.4 1.2 1. 08 06 0.1 02
36 REFLECTION AND REFRACTION £ C^yy)y sin 26s i, = ("Л + 2[i sin2 0, (9.49) 10 20 30 <ЗД 50 60 70 80 90 FIGURE 9.25(a). Reflection of a longitudinal (P) wave at a stress-free boundary in yttrium aluminum garnet (assumed isotropic). D. ISOTROPIC FRESNEL EQUATIONS 37 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1.0 0 10 20 30 40 50 60 70 80 90 ei FIGURE 9.25(b). Reflection of a longitudinal (P) wave at a stress-free boundary in fused silica. Consider the case of an incident P wave. This is represented by an incident wave on the P transmission line in the figure. The reflected longitudinal wave i* calculated by ordinary electrical transmission line theory, using the input impedance ZIN seen looking into the right-hand transformer. Since the reflection coefficients in (9.43) to (9.46) are defined in terms of the particle velocity field, analogous quantities in the transmission line analogue are the current reflection coefficients. For the P transmission line I I I I I I 1 Г / / / / / / - / / I I L "currents" for the SV and P waves are defined in terms of the stress component Tvy and the particle velocity component vy\ that is, Vtf (Tvvh
40 REFLECTION AND REFRACTION ky CO Pure shear""""--- eJ 1 / ** \ 1 со > т Pure shear""""-4 к у 01 1 /Z axis 1 ) Ы FIGURE 9.26. Scattering of horizontal shear waves at a plane boundary between hexagonal media, with propagation in the meridian plane. In the first medium the incident wave has fields (t.V)r = Аеш at x = 0, and the three possible reflected waves have fields (M = BeUoi = Ce*"* (7j/z)a (Тгх)ц = ЖрСм)"2/"' (956) E. ANISOTROPIC FRFSNFI EQUATIONS 41 Incident z*-polarized shear wave FIGURE 9.27. S after ng a b u dary between hexagonal media with different crystal axis orientations. lit the same plane. In the isotropic problem treated in Section D it was possible to assume that the reflected wave is polarized in the same plane as the incident wave, but this is not always true for anisotropic problems. The three possible transmitted waves in the second medium of Fig. 9.27 arc easily expressed in terms of the crystal axes X'Y'Z'; that is, (Tx'x'h OvOt (ТУ'х')т ("z'b (TVa')t Eetcot -G(p'c'uy'*e"°1. (9.57) To satisfy the boundary conditions (9.54) it is necessary to refer all field components to the same coordinate system. A matrix description of the transformation Irom the crystal axes X' Y'Z' to the coordinate axes in Fig. 9.27 is [«] 0 cos v sin у 0 —sin ip cos y>_
42 REFLECTION AND REFRACTION and the transmitted fields therefore transform according to the relations+ = 'a' ■» « vY> cos у — "z' s'n У "г = Vy' sin v + pz' cos v т f grx = TX x' r ' yx = rr v' cos у Tz'x' s'n V T 1 г Y' sin у + Tz'x' cos у (9.58) The boundary condition equations at v = 0 are therefore »я. В = E 00 к : С = ^cos у — G sin у (b) „г: /1 + D = Fsin у + С cos у (с) 7W: «(Pr„)* -Е<№* <d> Tw: C(Pce6)1/2 - -F(p'4)' 2 cos у f C^'c^'2 sin у (e) Тгх:-А(рс44р2 + DOu)1'* = -F(p'c;6)1,2siny - GO't^cos у (0- From (a) and (d) it follows that В - E = 0, and the remaining four equations can be solved for C, D, F, and G in terms of the incident wave amplitude A. The reflection coefficients at a; = 0 are then found to be (9.59) С (ZZZ.' - ZvZ'y) sin 2y А Д (9.60) p _ (z„ + z;xz, - z;) sin2 у + cz, + z;xzs - z;) cos2 у (Q61) гг /1 A where Z, = (pr™)1'- Zs = (pc«)-/2 ^ = ( and A=(ZV+ Z'z){Zz + Z'y) sin2 у + (Zs + z;XZ, + Z;) cos2 y, (9.62) and the transmission coefficients are _F 2Z,(Z;+Z,) sin у (9 63) 1 s /1 Д = C=2Z,(Z; +Z„)cosy *z А Д t The three stress components ж, ?лц zx that appear in the boundary conditions (9.54) are components of the traction force on the boundary. Since this is a vector quantity, it transforms in exactly the same way as v. The stress transformation can thus be written down directly in this case, where X — x. E. ANISOTROPIC TRESNFL EQLATTONS 43 These results show that there is a scattering of the incident shear wave into Kneeled and transmitted shear waves with orthogonal polarization. Since the two shear polarizations travel with different phase velocities the polarizations of the total reflected and transmitted fields vary with distance from the boundary. This coupling into other polarizations at the boundary is somewhat analogous to the coupling between SV and P waves at a boundary between isotropic solids (Figs. 9.19 and 9.21). An important distinction, however, is thai polarization coupling in the isotropic case disappears when the waves are normally incident on the boundary, 'flic example treated here shows that this is not always true for anisotropic problems. Reflection into the orthogonally polarized shear wave, described by (9.60), disap pears only when у = 0 or л- 2 in Fig. 9.27. When у = -л 2 1 0 from (9.64) and ihe transmitted wave is linearly polarised parallel to the incident polarization. The fttnsmitted and incident polarizations are again parallel when у = 0. K.2 Piezoelectric Media In piezoelectric media, wave scattering analysis is further complicated by coupling between the acoustic and electromagnetic field equations. The acoustic field equations (LA on the front cover papers) have three uniform plane wave solutions for each propagation direction, and the electromagnetic held equations (LB on the front cover papers) have two solutions. These solutions are coupled together through the piezoelectric constitutive relations, j'iv ng five coupled wave solutions for each propagation direction. Because ihese are hybrid waves, with both acoustic and electromagnetic field components the boundary conditions v = v' T • ft - Г • ft (9 65) ft x E — ft x E' n x H = й x II must be satisfied at boundaries between piezoelectric solids or between a piezoelectric solid and a nonpiezoclectnc solid Since there are five different types of plane waves in piezoelectric media, scattering problems may involve up to five reflected waves and five transmitted waves (Fig. 9.28a). I here are therefore ten unknown wave amplitudes p be determined from the ten linear equations obtained by expressing (9.65) in component form. Snell's Law relations are obtained from the same kind of slowness curve constructions that were used in Fig. 9.2. For an exact analysis, slowness curves must be obtained from the coupled wave theory picscntcd in Section F of Chapter 8 in Volume I: that is, slowness curves are needed for the quasiclcctromagnctic, quasiacoustic and stiffened acoustic wave types. "I he incident wave may be of any type Figure 9.29(a) illustrates Hie case of a quasiclcctromagnetic incident wave Because of the large
44 REFLECTION AND REFRACTION Boundary conditions v = v' nxE = nxE' T-ii = T'-ft AxH=ftxH (a) Exact solution-10 boundary conditions, 10 unknown amplitudes. Boundary conditions v v' Ф = Ф' T-fi = T'-ft T)-fi=D'-n (b) Quasistatic approximation-8 bounda у conditions 8 unknown ampl tudes FIGURE 9.28. Wave scattering at a plane boundary between piezoelectric media. difference in scale of the slowness curves, the quasiacoustic and stiffened acoustic waves in this situation propagate very nearly normal to the boundary. An example of quasiacoustic or stiffened acoustic wave incidence is shown in Fig. 9.29b. In this case the quasielectromagnetic scattered waves are evanescent, unless the incidence angle is very close to zero. For scattering problems of practical interest, the condition shown in Fig. 9.29b usually exists. That is, 2 К ** » ft>V.v. (9.66) F. ANISOTROPIC FRESNEL EQUATIONS 45 Quas acoustic or stiffened acoustic waves Quasielectromagnetic waves Quasielectromagnetic waves Quasiacoustic or stiffened acoustic waves MGURE 9.29(a). Snell's law construction for an incident quasielectromagnetic wave at a piezoelectric interface. The quasielectromagnetic curves are not to scale. They should be approximately 10 4 times the quasiacoustic or stiffened acoustic curves in size. where cfj and efs are typical constitutive parameters. In such cases it is |u-imissiblc to use the quasistatic approximation (l.B.l on the front cover papers), where E - -V<D н = о, (967) (iuI to replace the exact boundary conditions (9.65) with the quasistatic boundary conditions, v = v' T • n = T' • Й Ф = Ф' D-fi = D'-ii. (9.68)
46 REFLFCTION AND REFRACTION Quasiacoustic or stiffened acoustic waves Quasielectromagnetic waves Quasielectromagnetic waves Quasiacoustic or stiffened ac stic waves FIGURE 9.29(b). Snell's law construction for an incident quasiacoustic or quasiacoustic wave at a piezoelectric interface. The quasielectromagnetic waves are not to scale. They should be approximately 10 1 times the quasiacoustic or stiffened acoustic curves in size. In component form (9.68) gives only eight linear equations, rather than ten, and can therefore accommodate only eight scattered waves. However, it will be seen in the following examples that the quasistatic approximation gives only four wave solutions for each medium. The required balance between the number of boundary conditions and the number of scattered wave amplitudes is thereby preserved bXAMPLE 4. Consider a cubic medium belonging to one of the piezoelectric classes (23 and 43m). A plane, mechanically free boundary is assumed normal to the X crystal axis (Fig. 9.30). To conform with the convention established earlier in the chapter, the у coordinate axis is taken to be normal to the boundary and the E. ANISOTROPIC FRESNEL EQUATIONS 47 X.y Incident stiffened acoustic shear wave A* Reflected stiffened | acoustic shear iJ5 Cubic piezoelectric >L medium N. vR® Л ^ / 1 i Reflected evanescent / if wave Bf / v/s/////7s///s//e/s/ss//*'//,7///. :w Nt ^btimiiilumWHiiH j -V/.,/./-уу/,;/,;/, у Vacuum || Transmitted evanescent wave 1 FIGURb 9.30 Sc tteri ■ : tiff d u ti hear wave at a stress-free boundary of a cubic piezoelectric medium. incident wave is assumed to propagate in the yz coordinate plane. The medium below the boundary is vacuum. Under these conditions the boundary is said to be ulectrically free. For this problem, the quasistatic electrical boundary conditions Ф = Ф' (9.69) ure the same as in (9.68), but only one of the mechanical houndary conditions, T - n = 0, (9.70) is required. This gives only five boundary conditions, but a balance still exists between the number of boundary conditions and the number of scattered wave amplitudes. The lower medium in Fig. 9.30 is a vacuum, and does not support acoustic pves. Accordingly, the maximum number of scattered waves is five rather than ■glit. In the quasistatic approximation the acoustic fields associated with plane waves in E pie/oeleciric medium are obtained by solving the stiffened ChristofTel equation *yiK \<-kl + т^гт; J '«J*J = P°>** (9-70 where I = \lx + y/s + il. is (i unit vector in the wave propagation direction, and the corresponding clcctiic poicutials are calculated from . I d.eaJ,,)
48 REFLECTION AND REFRACllON In (9.71) and (9.72) the matrix lu is evaluated by making the substitutions a г dz in the matrix representation of V$ given on the front cover papers. The matrix liK is then obtained by transposing.t For waves propagating in the yz plane in Fig. 9.30, (9.71) takes the form ce e'v + 2с5В/Л 0 0 fc2 with and (9.72) is с{\1\ + c£j; (eg + c£)V, l55 '14 T S V .2 1 2eJfllyl. vx Ф = t Chapter 8 of Volume I gives derivations of (9.71) and (9.72), which appear as (8.147) and (8.146), and also applies the equations to some specific examples. E. ANISOTROPIC FRESNEL EQUATIONS 49 The potential equation shows that only the -r-polarized particle velocity is coupled to the electric potential. The incident wave, therefore, will be taken to be an x- polarized wave,t v, =\Aseto"-^I\ (9.73) which satisfies the dispersion relation c£(kl + kl) + 4 ^ JfrP. P1"2 (9-7*J and has an electrical potential U» e« k\ In the vacuum region (у < 0 in Fig. 9.30) the electric potential satisfies Laplace's equation ?2Ф = 0. (9.76) The transmitted wave function is therefore Ф, = В'се'ш kr'\ (9.77) vvith 'I о satisfy the boundary conditions at у — 0 in Fig. 9.30, the incident wave and all of the scattered waves must have the same г component of к (Snell's Law condition). If the propagation angle of the incident wave is 0S, substitution of (fr„)i = -(Wi cot es, in I о (9.74) yields pto2 sin2 8S (kz)\ = - . (9.79) cu + 4 ~7T sin2 °* cos'2 °s XX Since (fc2)T in (9.78) must be the same as (fcf )r, (ky)is imaginary. This means that the transmitted wave is evanescent, as in Fig. 9.4. The sign of (ky\ is chosen so that 'I' ► 0 when .y —>- — со. That is (Ars>r = iik^. (9.80) The reflected waves in the piezoelectric solid in Fig. 9.30 must satisfy the dispersion relation (9.74), subject to the Snell's Law condition (kjt, = (кг\. I his gives . „A ч , (фн(к% С«((*Л + ».),) + ^ ^ + ^ - I I in waves poleri/cd in the >/z plane, the scaitering problem is nonpiezoelectric.
=cot26s (9.82) 50 REFLECTION AND REFRACTION / ЭФ.Д E. ANISOTROPIC FRESNEL EQUATIONS 51 /Vs for the other waves are obtained from the first constitutive equation. There are jiikt two stress components (Тхг, Txy), and only reenters into the boundary condition (9.70). This problem, then, has only three boundary condition equations. I liese arc i ii cos 0S „ cE cos 0, A.-^-p (I + 2**) = Bs - (I +2K2) icJi cos 20, Ф: ^ sin 20, g„sin20, 2^ (1 + 4Q'« — p— - — Bs—;—^— — Я —5- —; h В (b) «>„: (9.87) exi sin 6S cos 2f?s exi sin 0S cos 20s ' ~ t ■ - -B- РГ 2eri sin 0S (1 + 2K2) <oe0 sin 0, в< V. 4^ e ¥„ (c) where mid The amplitude of the incident shear wave in Fig. 9.30 is As, and Яч, Bf, В' arc ampli- Imles of the scattered shear and evanescent waves. Solutions for the scattering Ouoilicients are I- " As ~ A ' в, д, г« = л = "f (9-88) Г' =В: = ^ " As A wticrc Л is the determinant of the coefficients on the right-hand side of (9.87) and Л., is the determinant obtained by replacing the coefficients of Bs with those of A„, |tc Modification of this analysis to allow for other clectiical boundary conditions in I lie second medium is relatively simple. For the special case of a short circuit PflttriCfil boundary condition (Ф — 0), (9.87c) is not required and the problem Iw-i nines trivial. The electric potential in the second medium is zero (B'e = 0); and which can be rearranged as - (1 + ^.sin2 Л cot2 6>3 = 0 (9.81) by using (9.79). It is easily verified that (9.81) has two solutions ®.--(,+3sH The first of these is clearly a reflected wave of the same type as the incident wave and is described by (9.73) and (9.75), with the substitutions As — (kv)K -► — (kv)I (^z)ll ~** (^z)t- The second solution has a negative value of (k2)H and is therefore a reflected evanescent wave (Fig. 9.30), with №Лс = -'(1 + ^sin2 4 V-)i- <9-84> If the particle velocity of this wave is vltc = kB/^Kf*, (9.85) the corresponding electric potential is ф = (*Лс(*Л . ,<„«- tRe.r) (9 86) from (9.72) and the Snell's Law condition To satisfy the boundary conditions (9.69) and (9.70) one must calculate the fields D and T from the piezoelectric constitutive relations D es-(-VO)+e:S T = -e-( VO)+cB:S given on the front cover papers. Only the у component of D is required for the boundary conditions. For the transmitted evanescent wave, this is simply
52 REFLECTION AND REFRACTION Incident stiffened acoustic shear wave As Hexagonal (6mm) medium Vacuum Reflected stiffened acoustic shear wave Bs Л Reflected evanescent JJ wave Br mmmmY.z ij Transmitted evanescent wave B'„ FIGURE 9.31. Scattering of a stiffened acoustic shear wave at a stress-free boundary of a hexagonal (6mm) medium. (9.87a), (9.87b) can be satisfied by taking Bc=0 B„ = As. In this case there is no evanescent reflected wave inside the solid. EXAMPLE 5. A piezoelectric scattering problem with somewhat different characteristics has the same physical configuration as Example 4, but the cubic crystal is now replaced by a hexagonal (6mm) crystal, oriented with X parallel to у and Г parallel to z (Fig. 9.31). In the x,y,z coordinate system, the constitutive matrices take the form (9.89) l33 c13 C13 0 0 o- ей cK CV1 0 0 0 «g 4 411 0 0 0 0 0 0 c& 0 0 0 0 0 0 0 0 0 0 0 0 CK E. ANISOTROPIC FRESNEL EQUATIONS 53 ez3 ezi ezi 0 0 0 0 0 0 0 0 eXb 0 0 0 0 e.\s 0 For waves traveling in the yz plane, the stiffened Christoffel equation (9.71) is 0 0 k- A'A" En , E,2 , E , Aw i 0 (4 + O'J* egll + eftl From (9.72), the electric potential is fir. Ф = par (9.90) (9.91) xx As in Example 4, the only piezoclcctrically active solution is the ^--polarized shear wave; and this solution is chosen for the incident wave, Vj = х/и/'<и' -кгг). (9.92) T he dispersion relation is therefore tind the corresponding potential is _ As exa ,t k].r) (9.93) Ф. (9.94) xx The wave transmitted into the vacuum is the same as in the previous example, Ф.( B'reHwt kJr) (9.95) with (kzh = №Л (fc„)i' = /(*-_),•
54 REFLECTION AND REFRACTION If the horizontally polarized shear wave (9.92) is incident at an angle 6S in Fig. 9.31 (&z)i = к sin 0K, and о (k% = sin2 0„ (9.96) E +е_хц_ from (9.93). Reflected waves inside the crystal must satisfy the dispersion relation (9.93), subject to the Snell's Law condition (^z)k = (^z)f This gives (*v)n = C*=)/ cot f?s. (9.97) In this case the stiffened ChristofTel equation provides only one reflected wave solution—a shear wave of the same type as the incident wave. There are, however, three field components (7"ст, Ф, Dv) to be matched at the boundary; and a second reflected wave is needed. This additional solution can be found by returning to the quasistatic piezoelectric equations ю»Л*(/^/,)Ф = -wkHl^JrJvj, (9.99) from which the stiffened ChristofTel equation was derived/!" In the present problem (9.98) and (9.99) reduce to («■Ж + k\) - Pco*K = -iWA.5(Aj + *|)Ф (a) + cgfcf - p^f, + (<:£ + c^kjev. = 0 (b) (9 ]oo) (cf2 + C^)k,J<,Vv + (cgkt + cfikl - pW*)vz = 0 (c) и?*хх<к* + к1)Ф * -1«*л-5(*; + *!)»«- (d) The relevant equations are (a) and (d). If (d) is solved for Ф and the result is substituted into (a), the dispersion relation (9.93) is obtained. It can be seen, however, that another solution to (a) and (d) is* kl+kl = 0 u, = 0 (9.101) Ф^ 0. This provides the reflected evanescent wave Фцв B,,*>'<°" Wiie = '№z)i that is needed to satisfy the boundary conditions in Fig. 9.31. (9.102) t These equations appear as (8.142) and (8.143) in Chapter 8 of Volume I % Although this solution has no particle motion, it does have a stress ticlu (Problem 8). E. ANISOTROPIC FRESNEL EQUATIONS 55 Incident evanescent wave Ac Hexagonal (6mm) I medium I ^\^\4444\W4\\\W444\W\\W^vl Vacuum 0» I Reflected stiffened shear wave Bs I ij Reflected evanescent wave Be >ii)i))i)i)iiniw<t>M.\\^w^\^-> Y.z T it Transmitted evanescent wave B'e FIGURE 9.32. Scattering of an incident evanescent wave at a stress-free boundary of a hexagonal (6mm) medium. From the above solutions, the three boundary equations are ■ I lie piezoelectrically stiffened shear velocity, As is the amplitude of the incident tlmiv wave, and flv, fie, B'e are the scattered wave amplitudes. The scattering coefficients Г„ = BJA„ 1« = W •n«" found by solving these equations simultaneously. In some cases it is necessary to find the scattering coefficients when an evanescent wiive of the type defined by (9.101) is incident on the boundary—that is, the wave ■Irc.iys exponentially toward the boundary (Fig. 9.32). Boundary equations for this X.y
56 REFLECTION AND REFRACTION problem arc obtained by making the substitutions lue\s cos 0. — A, —-" sin 0, (9.104) As : s ' A" 0 - -4x"« in parts (a), (b), (c) of (9.103). The modified equations arc then solved for the scattering coefficients ■1 ye = Bgf At, vee = в,/л Г' = B'JA,. ее e' e An important special case arises when a short-circuit electrical boundary condition (Ф =0) is applied at у = 0 in Fig. 9.31. The boundary condition on Dv is no longer required, and (9.103) reduces to —-— cos в, = В cje vb cos0s + Д..--Г5 sin es (.a) (9.10 B„ - В.. lb) since B'e = 0. In this case xx Am Д Д (9.106) where д _ _ \ <.v v ' УДГ-У (9.107) is the determinant of the right-hand side of (9.105) and Дет is the determinant obtained by replacing the coefficients of B„ with those of A„ etc. 11 should be noted that in this example, by contrast with Example 4, the reflected evanescent wave still remains when the short-circuit electrical boundary condition is applied. PROBLEMS 57 Ae —^ sin 0S = Bs~ лл cos 0, + BK —~ sine, (a) (9.108) A. = - тЦг~ В, - Br (b) when an electrical short circuit is imposed at the boundary; and the scattering coefficients are л .p _ ->s<? A« - Д (9.109) Г — д • where the determinants A£e, Д„ arc obtained by replacing the coefficients of В and B, in Л with those of Ac in (9.108). PROBLEMS 1. Calculate complex Poynting vectors for plane waves propagating along the Z-axis in sapphire (Problem 4, Chapter 3 of Volume 1). Compare with the lingle between P and к calculated from Fig. 3.8 in Appendix 3 of Volume 1. 2. Find the complex Poynting vectors for plane waves propagating along llie Z-axis in materials belonging to the trigonal crystal classes 3m and 32 (Problem 4, Chapter 8 of Volume I). Calculate the change in direction of P due to the piezoelectric effect in lithium niobate and quartz. X Using the relations derived in Section B.2, calculate the critical angles relevant to Figs. 9.20 and 9.22. Locate these angles on the figures. 4. The formula (7.115) for [Z£] in Chapter 7 of Volume I applies to isotropic, as well as anisotropic, materials. Derive the boundary condition equations (9.27) and (9.37) by using the impedance concept. 5. Use the impedance concept to find reflection coefficients for the configuration, shown at the top of page 58. Solve the problem for incident waves Of both shear and compressional types. <». I he two media in Problem 5 are separated by a slab of nonpiezoclectric cubic material with axes oriented as shown in the figure. Use the transformation law for impedance (Problem 25, Chapter 7 in Volume 1) to derive design equations for shear and compressional wave quarter-wave matching trans- loimcis (Problem 12 in Chapter 6). Boundary equations for the incident evanescent wave problem become
58 REFLECTION AND REFRACTION Incident wave Hexagonal (non piezoelectric) Cubic (nonpiezoelectric) Problem 5 7. A longitudinal wave in an isotropic medium is normally incident on the boundary of a nonpiezoelectric cubic medium with axes oriented as shown. Use the particle velocity polarizations derived in Problem 8, Chapter 7 of Volume I to find all transmitted and reflected wave amplitudes. Assuming that the cubic medium is gallium arsenide and neglecting the piezoelectric Hexagonal (nonpiezoelectric) Cubic (nonpiezoelectric) Cubic (nonpiezoelectric) Problem 6 effect, use Fig. 7.3 to estimate the energy flow directions for the scattered waves. Make a sketch showing wavefronts and energy flow directions for the various waves. 1» Isotropic Cubic (nonpiezoelectric) Z, [0011 Problem 7 PROBLEMS 59 8. Show that the xy stress components associated with the incident and Krlleeted shear waves in Fig. 9.31 are and that the xy stress component associated with the reflected evanescent wave [defined by (9.101) and (9.102)] is (TOT)Kc=-Be^eX6sin 0/,"->. 4 Solve the boundary condition equations (9.103) for the scattering coefficients Tgs = BJA„ Г„ = BJAS, V„ = B'JA,. 10. Verify that the fields пх=2А^'cos o>y}Vs T„ = 2 Mc* + e-' sin wyjVt ioi у > 0, and Ф - 2iA\ex& i№l r -r—e cos a>yJ V„ MeA-J 2iAeeXa У>0 У<0 D - 0, at all values of у Hiilisfy the boundary conditions for Txv, Ф, Dy at у = 0 in Fig. 9.31. Show that the solution of Problem 9 reduces to this result when 0, — 0, except for an mldiiive constant in the potential function. II. I he quasistatic piezoelectric equations used to obtain (9.100) in Example 5 were derived from the original piezoelectric field equations in Chapter 8 of Volume I by making the following substitutions for partial derivatives h respect to the spatial variables, d_ dy 3z = -ifc, = -iky = — ik. lunations (9.100) may be adapted to fields with arbitrary functional dependence on y, z by resubstituting the partial derivatives djdy and djde for
CHAPTER 10 ACOUSTIC WAVEGUIDES A. GUIDED WAVES 63 II. METHODS OF ANALYSIS 66 C. FREE ISOTROPIC PLATE 73 I) ISOTROPIC PLATE ON AN ISOTROPIC HALF SPACE 94 I FREE ISOTROPIC CYLINDER 104 |. ISOTROPIC RECTANGULAR STRIP 114 ti. M1CROSOUND WAVEGUIDES 118 II ANISOTROPIC WAVEGUIDES 128 I PIEZOELECTRIC WAVEGUIDES 134 I. RECIPROCITY RELATIONS AND MODE ORTHOGONALITY 151 K. EXCITATION OF WAVEGUIDE MODES 161 I INPUT IMMITTANCE OF WAVEGUIDE TRANSDUCERS 163 M. TRANSMISSION LINE MODEL FOR ACOUSTIC WAVEGUIDES 177 N WAVEGUIDE SCATTERING PROBLEMS 190 <> GROUP VELOCITY AND ENERGY VELOCITY 199 PROBLEMS 207 КI I ERENCES 214 A. G1IDED WAVES Chapter 9 was concerned with scattering of acoustic plane waves at a boundary between different media. This interaction is of primary importance in problems where the dimensions of a solid body are large compared with the acoustic wavelength, and its volume is not completely occupied by the waves. 1'ioblcms of this kind are analogous to "optical" problems in electromagnet- ism, where there is also only an occasional interaction of the waves with the boundary of the medium. The scattering solutions presented in the previous 63
60 REFLECTION AND REFRACTION — ik„ and —ik.. Prove that there arc then two general solutions to (9.100a) and [9-I00d], one defined by кое xx and the other by vx = 0 Use these differential equations to solve the scattering problem for an a-polarizcd shear wave normally incident on the boundary in Fig. 9.31, and show that the solution obtained is equivalent to the one given in Problem 10. (Note that the electric field Ey must go to zero at у — — со, unless there is an electrical charge layer at that point.) 12. A perfectly conducting plane with potential Ф = 0 is placed at у = —h in Fig. 9.31. Use the differential equations in Problem 11 to solve the scattering problem for an .^-polarized shear wave normally incident on the boundary у = 0. Show (hat this reduces, when h = 0, to the solution determined by (9.105). 13. Show that solutions of the form given in Problems 10 and 11 apply to any case where a pure mode (v either parallel or perpendicular to k) is normally incident on the boundary between a piezoelectric medium and an unbounded vacuum. 14. Problem 13 is modified by replacing the unbounded vacuum with a piezoelectric medium that supports a normally propagating pure mode with the same polarization as the incident wave. Show how the solutions found in Problem 13 arc modified in this case. 15. Consider an A"-oricnted interface between a cubic piezoelectric material and vacuum (Fig. 9.30). Use the exact plane wave solutions given in Example 4, Chapter 8 of Volume I and the exact boundary conditions (9.65) to find the scattered waves produced by normal incidence of a F-polarized quasi- acoustic wave. Repeat for the Z-polarized quasiacoustic wave and the X- polarized purely acoustic wave. 16. Consider a solid-to-vacuum interface oriented normal to the [110] direction of a cubic piezoelectric medium. Using the exact plane wave solutions obtained in Example 5, Chapter 8 of Volume 1, solve the normal- incidence scattering problem for incident waves of purely acoustic, quasi- acoustic, and stiffened acoustic types. REFERENCES 61 REEI RENTES SnelVs Law and the Fresnel Equations 1. L. M. Brekhovskikh, Waves in Layered Media, Academic Press, 1960. 2. W. M. Ewing, W. S. Jardetsky, and F. Press, Elastic Waves in Layered Media, Ch. 2, 3, McGraw-Hill, New York, 1957. 3. E. G. H. Lean and H. J. Shaw, "Efficient Microwave Shear-Wave Generation by Mode Conversion," Appl. Phys. Lett. 9, 372-374 (1966). I. W. P. Mason, Physical Acoustics and the Properties of Solids, pp. 22 32, van Nostrand, New York, 1958. 5. M. J. P. Musgrave, "The Propagation of Elastic Waves in Crystals and Anisotropic Media," Reports on Progress in Ph . it 22, pp. 74-96 (1959). ft. G. Nadeau, Introduction to Elasticity, pp. 241 258, Holt, Rinehart and Winston, New York, 1964. 7. S. Ramo, J. R. Whinncry, and T. van Duzer, Fields and Waves in Communication Electronics, pp. 342-365, Wiley, New York, 1965. 8. M. J. P. Musgrave, Crystal Acoustics, Ch 11, Holden-Day, San Francisco, 1970. H.i. E. G. Henneke II, "Reflection-Refraction of a Stress Wave at a Plane Boundary between Anisotropic Media", J. Acous. Soc. Am. 51, 210-217 (1972). Conical Refraction 9. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, pp. 324-329. Pcrgamon, New York, 1960. III. H. J. McSkimin and W. L. Bond, "Conical Refraction of Transverse Ultrasonic Waves in Quartz," /. Acous. Soc. Amer. 39, 499 505 (1966). II. E. P. Papadakis, "Ultrasonic Internal Conical Refraction in Rocksalt and Calcite," /. Appl. Phys. 34, 2168 2171 (1963). \1. K. S. Aleksandrov and Т. V. Ryzhova, "Internal Conical Refraction of Elastic Waves in Ammonium Dihydrogen Phosphate," Soviet Physics—Crystallography 9, 298-300 (1964). ti. P. C. Waterman, "Orientation Dependence of Elastic Waves in Single Crystals", Phys. Ree. 113, 1240-1253 (1959). Lquii alent Netw ork Methods I'I A. A. Oliner, "Microwave Network Methods for Guided Elastic Waves," IEEE Trans, on Microwave Theory and Techniques MTT-I7, 812-826 (1969). Piezoelectric Media H I Rurdati, G. Barzilai, and G. Gerosa, "Elastic Wave Excitation in Piezoelectric Slabs," IEEE Trans, on Sonicsand Ultrasonics, SU-15, 193 202 (1968).
66 ACOUSTIC WAVEGUIDES discussed in Section M, and application to the problem of waveguide scattering is considered in Section N. The chapter concludes with a section on an important relationship between the power flow, stored energy, and group velocity of a guided wave. B. METHODS OF ANAIYSIS The simplest of all acoustic waveguide structures, an unbounded plate with strcss-frcc surfaces will be used to illustrate the three most commonly applied methods for waveguide analysis. B.l Potential Theory The most general method for solving isotropic waveguide problems makes use of a representation of the particle velocity field in terms of a scalar and a vector potential. For isotropic media, which are always nonpiezoelectric, the left-hand side of the Christolfel equation (9.71) takes the form 4Уз- ^xlv k'2\ cm + (cn - c4i) ^>Jx ^У^У ^V^Z LL I J. LI Matrix multiplication of the term on the right gives ~ljlxi'r + fa + 1л\У Jz(Lrx + fa + lzl\) Consequently, the Christoffel equation can be written as r„A-2v + feu - c44)k(k • v) = orpy (10.1) for an isotropic medium. This governs plane wave solutions with harmonic time variation. To obtain the general equation for plane wave solutions, the substitutions V ^- -ik, 3/3/-*-iou, are inverted. This gives 32v r«VS + ('c,i-«,,)V(V.T) dt2 (10.2) or cnvrV • v) - c44V x V x v = p — , at (10.3) B. METHODS OF ANALYSIS 67 where the vector identity V x V x A = V(V • A) - V2A (10.4) has been used to rearrange terms. Solutions of (10.3) are obtained by expressing v in terms of a scalar potential and a vector potentialf v-v"(I> + Vx4' (10.5) Substitution into (10.3) gives v(cu¥4» - я|^ф) - V x (cAi\ xVx*+^T)=0, (10.6) since V-V x ¥ = 0 V x Yii> = 0. For the second term, the quantity in brackets is set equal to the gradient of an arbitrary function/. That is c14Vx \ xW + P^ = Vf. Application of the identity (10.4) converts this to V(V .«Г -/) - V2Y + —2^,W = 0, V f 3r where VH - (c,ulp)1"2. Since/ is arbitrary, it can always be chosen to cancel V • 41 in the first term on the left. The vector potential 4? can thus be taken as a solution to the vector wave equation This means that there are no necessary constraints on the divergence V • *F. Problems treated in this chapter and the next will, however, use the zero divergence condition (V • ¥ = 0). The first term in (10.6) is made zero by simply requiring that the scalar potential Ф satisfy the scalar wave equation oc,, 1 32Ф „ ™ " Ц ^ - 0, (Ю.8) where V\ = (cjpf*. |- In piezoelectric problems the symbol Ф is also used to represent the scalar electrical poleniial, Chapter 8 in Volume I. This docs not, however, lead to any ambiguities. Mechanical potentials arc applied only to isotropic problems, and these are always nonpie/oelcctric.
64 ACOUSTIC WAVEGUIDES chapter all involved a plane boundary but, by using the same "ray" concepts that have been developed in electromagnetic optics, they may be applied point-by-point to curved boundaries - provided the radius of curvature is much larger than a wavelength. This chapter deals with the behavior of acoustic waves in the immediate vicinity of material boundaries, where the boundaries have the effect of guiding waves along their surfaces. Guided waves have energy flow mainly along the direction of the guiding configuration, or waveguide. In applied electromagnetism, where waveguides have been widely used for a number of decades, a variety of waveguide components—attenuators, phase shifters, directional couplers, etc.,- have been developed for the purpose of controlling and interconnecting guided electromagnetic waves. By contrast, applications of guided acoustic waves have primarily exploited the dispersive characteristics of waveguide propagation in delay line devices. It is only recently that advances in thin film technology and the development of efficient waveguide transducers have made it possible to consider applying the more sophisticated electromagnetic waveguide concepts and devices to acoustic systems. In analyzing waveguide problems the aim is to find solutions to the field equations and the boundary conditions in which the energy flow is along the boundary. The treatment here will be restricted to straight waveguides of uniform cross sectional shape; guided wave solutions are assumed to be proportional to e~""\ where the waveguide is aligned along the 2 axis and ft is called the propagation constant of the wave. As in electromagnetism, solutions for straight uniform structures apply to a good approximation in guiding systems which are slowly curved in direction (Fig. 10.1) or slowly tapered in cross section. Generally speaking, acoustic field problems are significantly more difficult to solve than are electromagnetic problems, and the basic characteristics of the solutions themselves are often more complicated. These differences, which arise from the differing nature of acoustic and electromagnetic fields, have already been demonstrated in Chapter 9. Nevertheless, acoustic and electromagnetic guided wave problems also have many strong similarities. These similarities are especially evident when the field equations are expressed in the symbolic notation given on the front cover pages, and will be exploited toward the end of this chapter by applying electromagnetic microwave methods to acoustic waveguide excitation and scattering problems. Useful similarities between electromagnetism and acoustics also exist at the stage of finding wave solutions for specific waveguide structures. General analytical methods are much the same in the two cases, and some solutions are exactly analogous in all respects. Where this is not so, common features still exist and are helpful in providing physical understanding and in suggesting approximation procedures. A. GUIDED WAVES 65 (a) (b) FIGURE 10.1 (a) Wave guiding in two dimensions, (b) Wave guiding in three dimensions. Every waveguide structure is capable of supporting an infinite number of different guided wave solutions, called )\ areguide modes, each with a different piopagation constant (i and a different field distribution. The first nine tectums in this chapter develop and review modal properties of the most important acoustic waveguides. Only the lossless case will be considered here. Approximate methods for treating lossy waveguides, as well as other problems of a more complex nature, are discussed in Chapter 12. Waveguide modes provide a natural basis for analyzing waveguide excitation and tcattcring problems, and have been widely used for this purpose in electromagnetism. Modal analysis begins by expressing the total field in a waveguide as a superposition of all the modes supported by the structure. Substitution of this mode expansion into the field equations leads to a set of mode amplitude equations that closely resemble electrical transmission line equations • ml can therefore be used to construct a transmission line circuit model for the waveguide problem. Acoustic waveguides may be treated 111 exactly the same way The formal procedure for setting up acoustic mode expansions is detailed 111 Sections J and K. this formalism is then applied to waveguide li nisduccrs in Section L. Principles of transmission line modeling are
68 ACOUSTIC WAVEGUIDES In the free isotropic plate waveguide shown in Fig. 10.2 propagation is along the 2 direction and the fields are assumed to be uniform in the j- direction. The strain component, S^, is therefore zero and the free boundary conditions take the form T3,v = c^Sj^ — 0 T„ = cuSyv + cl2Szz - 0 (10.9) Tzv — c^.tSzv = 0 at у = ±6/2. These can be expressed in terms of the potentials by using (10.5) and the definition of strain given by the second acoustic field equation on the front cover papers. For simple geometries such as this, the wave equations for the potentials are solved by separation of variables, assuming harmonic time dependence; and the arbitrary constants in the solutions arc specified by the boundary conditions (10.9). As the general solutions for this geometry are more easily obtained by the transverse resonance method, only one specific type of potential theory solution will be considered. These arc solutions for which the scalar potential Ф vanishes. For fields varying as <-'<"'' 'iz), (10.7) is .y= +6/2 Free boundary (y) у//////////уЖ//лу///////////, — 6/2 Vpree boundary n «= 1 mode «ж* ///////AY/'//, У /free boundary у = + 6/2 / -J. Ыу) ■= -6/2 \ ^Free boundary T ■ 0 n'O mode Txz(y)\ у//////////,у/.Ш/:'///////////. -//////////хул V////'////////,y, FIGURE 10 2. Particle velocity and stress fields for the tvto lowest order SI I modes of a free isotropic plate waveguide. B. METHODS Ob ANVLVSIS 69 dvr д ( д „ \ at у = ±b}2. This can be satisfied by taking either С = D = 0 in (10 10) and к1х=пф (n = 1,2, 3,...), or £= F= 0 in (10.10) and kls = 0. 1 he dispersion relation is then F = 0 (/; odd) E = 0 (/; even) and the particle velocity, strain, and stress fields have the transverse variations shown in Fig. 10.2 These are called the SH (horizontal shear) modes of an isotropic plate. 11.2 Superposition of Partial Waves The potential theory method is limited in that it is not applicable to anisotropic problems. These problems arc very difficult in general and no universal and has general solutions XVX = (A cos k,j/ + В sin ktsy)e iPz Vv = (C cos ktsy + D sin ktsy)e~i0x (10.10) xVt = (E cos klsy + F sin ktlty)e with kf, + f = af/Vi The form of the boundary conditions (10.9) suggests that a simple solution may exist when only the strain components SXj) and Szv are present. Since the fields vary only with у and z, this requires vu = vz = 0. In terms of the vector potential this means that --ti = o --тх = о, by which can be satisfied by taking 4'a. = (). The only nonvanishing strain component in (10.9) is then Sxy, and
70 ACOUSTIC WAVEGUIDES FIGU RE 10.3. Slowness curves for the partial wave solutions in a piezoelectric plate waveguide. The electromagnetic slowness curves are verj small compared with the acoustic curves and are not shown to scale. B. METHODS OF ANALYSIS 71 M-omvt because they reduce to electric potentials satisfying Laplace's equation when the piezoelectric constants arc allowed to go to zero. Finally, all partial waves must be superimposed with amplitudes chosen to satisfy the acoustic and electrical boundary conditions at the waveguide boundaries. The partial wave solutions are obtained from (9.98) and (9.99), expressed symbolically asj V • cK: Vsv = -pio2\ - iwV • e • УФ (10.12) and co2V • (c's - V<1)) = - V • с: V,v, (10.13) by assuming solutions proportional to e •ikiv+>'iz\ д trial value of \\V = /?/«> is first assumed, and (10.12), (10.13) are then solved for the transverse wave vector component kjui, the velocity field v, and the potential Ф for each partial wave. In a piezoelectric medium there arc 8 such partial wave solutions (Fig. 10.3), which must generally be obtained by numerical computation. For the free piezoelectric plate problem these 8 partial waves are matched to the 2 electrostatic waves outside the plate by means of 3 acoustic boundary conditions and 2 electrical boundary conditions at each boundary. This leads to a 10 x 10 characteristic determinant, which must vanish for any value of \\V = /3/cj corresponding to a modal solution. Iteration by means of numerical computation is used to obtain values of V which reduce this boundary value determinant to zero. From these values, dispersion curves w(P) are obtained for the various modes of the waveguide. B.3 Transverse Resonance For isotropic media and planar geometries the superposition of partial waves method becomes sufficiently simple that it can be carried out analytically, rather than numerically. The analysis can be further simplified by making use of the transverse resonance principle, which has been used to great advantage in electromagnctism. This principle is based on a realization of the fact that, for waveguides with lossless boundaries, the mode solutions are traveling waves along the waveguide axis and resonant standing waves in the transverse direction. Because of this, the partial waves propagating transversely downward (Figure 10.4) must have the correct propagation angles and wave vectors so that they reconstruct themselves after reflection from the lower and upper boundaries in succession. The problem of SH mode propagation on an isotropic plate is used to illustrate the method. Figure 10 4 shows the Iransversely resonating horizontal shear wave pattern for this case. It was seen in Section D.4 of Chapter 9 I Or l.aplace Wanes (Reference 79 at the end of the chapter). \ Sec Problem 11 in Chapler 9. method of analysis is available. One method (superposition of partial waves) can, however, be used for any planar problem, such as an anisotropic free plate waveguide or an anisotropic plate over an anisotropic substrate. In this approach all possible plane wave solutions for the media involved are first obtained. The wave vectors are then oriented so that all waves have the same wave vector component, p\ along the waveguide propagation direction z (Fig. 10.3). This usually requires imaginary wave vector components normal to 2 for some of the partial waves. The figure shows a piezoelectric plate problem, under conditions where the quasistatic approximation is applicable. It was seen in Examples 4 and 5 of Chapter 9 lhat evanescent partial waves always exist under these circumstances. These may be termed electrostatic
72 acoustic; waveguides F1GU RE 10.4. Partial wave pattern for transverse resonance analysis of SH wave propagation on an isotropic plate with free boundaries. that a shear wave with horizontal polarization scatters only into itself at a free boundary. The transverse resonance condition then states that the partial waves must experience a phase shift of some integral multiple of 2тт during each round trip from у — —b\2 back to the same point. This fact is all that is required to obtain the SH mode solutions. It is, however, desirable at this point to develop a symmetry principle which will greatly simplify the discussion of Lamb waves in the next section. It can be seen in Fig. 10.2 that the velocity field distributions have even symmetry with respect to reflection in the xz plane for и even, and odd symmetry forn odd. This separation into even and odd symmetries is characteristic of modes on structures with a plane of symmetry. Because of this, the reflected partial wave amplitude in Fig. 10.4 can differ from the incident partial wave amplitude at у = 0 only by a factor ±1. Reflection in the xz plane interchanges the incident and reflected waves, and this must leave the field pattern completely unchanged for symmetric modes or changed by a sign reversal for antisymmetric modes. The outstanding advantage of the transverse resonance procedure is that the boundary value problem is solved once and for all, to obtain the reflection coefficients; these reflection coefficients can then be used directly to solve a variety of guided wave problems. According to (9.42) the velocity field reflection coefficient for a horizontal shear wave at a free boundary is 1. If the incident and reflected waves in Fig 10 4 are vx\ - Ae l{-k"y "z) and С. FREE ISOTROPIC PLAIE 73 the symmetry condition then requires that В = ±A. The reflection condition at у = —h\2 demands that *x\\ — 1 j-l> or ±A\e~i<~k"b/2"lz) = /4e~'<s's6 2|/,г) From this, the resonance condition is eik"b - ±1, and fc^-T1. (Ю.14) The waveguide dispersion relation (10.11) then follows directly from the partial wave dispersion relation k'L + /32 = (w/Vsf. (10.15) In this presentation the transverse resonance calculation has been formulated in terms of the reflection coefficient. It can also be presented in terms of impedance, using the equivalent network model of Fig. 9.23c. In either case I he essence of the technique is that it uses the results of a boundary value problem, solved once and for all, to obtain solutions to a variety of other problems. The example chosen here is somewhat trivial. A clearer picture of I he economy of effort provided by the method will be given by the treatment of Lamb waves in the next section. (. FREE ISOTROPIC PLATE fhe unbounded plate waveguide is an idealized, rather than a truly physically-realizable, structure but it is a good approximation to a number of practical configurations. Historically, the isotropic plate was the first acoustic guided wave system to be completely analyzed, and it is of fundamental importance because it illustrates the general character of guided acoustic waves in a relatively simple manner. One mode family of this structure (the SH modes) has alicady been considcicd in Section В and will be treated somewhat more fully in this seclion. The other modes supported by the structure will also be developed here. Propagation along the z az.is will be assumed, as in Section B.
74 ACOUSTIC WAVEGUIDES C.l SH Modes Тл the previous section the dispersion relation for these waves was obtained by both potential theory and transverse resonance methods. The physical meaning of this relation, (10.11), is easily pictured by using the graphical construction of Fig. 10.5. The two arrows on the diagram represent the incident and reflected partial waves in Fig. 10.4. Tl was seen in the transverse resonance analysis that each of these partial waves has K = P- For the incident partial wave the ;;/, or transverse component, of the wave vector is —Onrjb) and for the reflected wave it is +(mrjb). By plotting these wave vector components on the slowness curve for the isotropic medium, which is simply a circle, one can easily visualize their behavior as the frequency is varied. If the frequency is increased, the angle of partial wave incidence в steadily increases, and at very high frequencies the partial waves propagate essentially along z. Under these conditions Pico « IIV,. For decreasing frequencies, the incidence angle 0 decreases and becomes w V FIGURE 10.5. Slowness diagram for SH modes on an isotropic plate of thickness b. C. FREE ISOTROPIC PL ME 75 >- Non propagating mode FIGURE 10.6. Wave functions for propagating and nonpropagating SH waveguide modes •его when П7Г 1 ho ~ V, ' At this point b and, since the partial waves are reflecting back and forth directly across the guide, there is no variation of the waveguide field along z. For m < ojck the partial waves no longer lie on the slowness curve. Since k,s is always determined by (10.14) it follows from (10.15) tha: /9 must be imaginary. This means that the waveguide fields, which are proportional to the propagating wave function e for w > v>cn, are proportional to the exponential (or nonpropagating) wave function e '**z for ш < шсп. In the second case the guided wave decreases exponentially with distance from the source that excites it (Fig. 10.6). Under these conditions the waveguide mode does not transport energy along the plate. The frequency wCM at which the mode changes from propagating to nonpropagating is called the cutoff frequency. Dispersion curves showing this behavior are given in Fig. 10.7. Field distributions for the SH modes were obtained in part 1 of Section В by using potential theory. They may also be calculated from the transverse lcsonance method, by combining the partial wave fields. For waves traveling
in the positive and negative z directions, ( with and O,, = cos + fc/2)]e^»* (Тхг)п = T ^ cos Г"77 (» + Ь/2)1<г*" (Ю.16) <u L fc J (Te).--^«n[^(» + b/2)V-. Л„ = [(«/ K,)2 - W)2]1/2 (" = 0, 1,2, 3,. . .) "i - (тЩГ \ bm / J where Zs = c„/K, is the plane shear wave impedance. (10.17) C.2 Lamb Waves From the transverse resonance analysis of SH waves it might seem plausible that other modes of the plate structure could be constructed by taking C. FREE ISOTROPIC PLATE 77 у F1GURL 10.8. Partial wave pattern for transverse resonance analysis of uncoupled SV wave propagation on an isotropic plate with mixed boundary conditions, Tm — 0, т„ = 0 at у = ±bj2. vertically polarized shear partial waves or longitudinal partial waves in the transverse resonance patterns of Figs. 10.8 and 10.9. For a plate with free boundaries such SV (vertical shear) and P (pressure) waves do not exist individually, but are coupled. This is seen very simply from the reflection coefficients (9.43) to (9.46), which show that vertically polarized shear and longitudinal waves are coupled at a free boundary. A partial wave pattern for these polarizations must therefore appear as in Fig. 10.10. The wave vectors of the shear and longitudinal partial waves must all have the same component ft along the z axis, and the shear and longitudinal partial waves consequently propagate at different angles to the z axis for these modes, which arc called Lamb wave (Fig 10 J 1 Dispersion Relations. In this problem the transverse resonance condition requires that the incident shear and longitudinal partial waves (traveling downward in Fig. 10.10) must reconstruct themselves after successive reflection from the lower and upper faces of the plate. This condition can be у FIG URE 10.9. Partial wave pattern for transverse resonance analysis of uncoupled P wave propagation on an isotropic plate with mixed boundary conditions, Tyy = 0, iv = 0 at у = ±bj2.
78 ACOUSTIC WAVEGUIDES .v Incident and reflected longitudinal partial waves Incident and reflected vertically polarized shear partial waves HGURE 10.10. Partial wave pattern for transverse resonance analysis of Lamb wave propagation on an isotropic plate with free boundaries. established analytically, without solving the boundary value problem, by using the reflection coefficients (9.43) to (9.46). Analysis is greatly simplified by using the symmetry principle established in part 3 of Section B. The modes are either symmetric or antisymmetric with respect to reflection in the xz plane, and the reflected partial wave amplitudes at у — 0 differ from the incident wave amplitudes at the same point only by a factor ±1. If particle velocity fields of the incident partial waves are proportional to A,e-lktVT, A,e and those of the reflection partial waves arc proportional to > FIGURE 10.11. Wave vector diagrams for lamb waves on an isotropic plate of thickness b. Imaginary transverse wave vector components are shown by dashed lines. C. FREE ISOTROPIC PLATE 79 the symmetry condition requires that Bt = ±Л, В, = ±AS. The reflection coefficients at the lower boundary {y = — A/2) in Fig. 10.10 impose the further condition -А^к„Ы2.- ' A le-,kM'2r -Г.1 _Ase-ik^\ ±r„ = where kt, and ku are the magnitudes of the transverse wave vector components for the longitudinal and shear partial waves. For modal solutions the characteristic determinant of this equation must vanish. With use of the reflection coefficient relation (9 48), the characteristic equation becomes sin (ktl + kts)bj2 sin (kn - ku)bj2 The right-hand side is expanded and multiplied top and bottom by cos knbj2 cos ktshj2, giving tan klsbj2 _ 1 + Г„ tan fc„6/2 ~ 1 - Ги for the symmetric solutions and tan fct,,ft/2 _ 1 - Г„ tan kHb/2 ~~ 1 + Г„ for the antisymmetric solutions. Finally, by expressing the partial wave incidence and reflection angles in terms of ktl, kts, and fj these are reduced to the Rayleigh-Lamh frequency equations tan klsbj2 tan knbj2 for the symmetric solutions and tan ktsbl2 (fcj.-/*■)- (10.18) (10.19) tan knbj2 for the antisymmetric solutions. In (10.18) and (10.19) the transverse wave vector components are related to w and the propagation factor ft by (.10.20)
80 ACOUSTIC WAVEGUIDES and fi\ (10.21) Dispersion relations for the symmetric and antisymmetric solutions can then be obtained by solving (10.18)—(10.21) for <o as a function of f>. The dispersion curves (Fig. 10.12) are complicated, and it is fortunate that they can be interpreted in a rather simple physical way. Tt was noted, in connection with Figs. 10.8 and 10.9, that simple SV and P-type guided waves do not exist on a plate with free boundaries. Such waves do, however, occur for the mixed boundary conditions Tm - - 0, t, = 0 shown in the li urcs Tin is casil confirmed by ferring to (9.3 ) and (9.40), with the transmitted wave amplitudes li\ and B's equal to zero. For shear wave incidence the mixed boundary conditions staled here require only that (9.37b) and (9.37c) be satisfied. It is obvious by inspection that these conditions are met when 7tvs FIGURE 10.12. Lamb wave dispersion curves for the lower order modes of an isotropic free plate with VJ Vs = 1.9056. C. FREE ISOTROPIC PLATE 81 and B« Г.. = — = — I That is, there is no coupling of an incident vertical shear wave to a longitudinal reflected wave and it is totally reflected with a velocity reflection coefficient of —1. Similarly from (9.40) an incident longitudinal wave is found to be totally reflected into a longitudinal wave, again with a velocity reflection coefficient of —1. Because of these simple reflection conditions, SV and P guided waves (Figs. 10.8 and 10.9) do exist for these mixed boundary conditions (or their duals, Tyr = 0, and vy = 0). They have simple solutions and dispersion relations of the kind obtained for the SH waves (Fig. 10.7). With free boundary conditions, the vertical shear and longitudinal particle motions are coupled at the boundaries, and the Lamb waves (Fig. 10.10) therefore have the nature of coupled SV and P waves. The phenomenon of wave coupling is important in many branches of engineering and physics, and it is well known that the dispersion curves for the uncoupled waves are split at their crossover points when coupling is introduced.t Far from crossover the coupled wave dispersion curves nearly coincide with those of the uncoupled waves. This behavior is illustrated in Fig. 10.13 where dispersion curves for the uncoupled modes are shown by dashed and solid lines, and the symmetric mode dispersion curves from Fig. 10.12 are superimposed. Because the coupling is very strong, the coupled waves exhibit large departures from the uncoupled curves. One of the clearest examples of coupled mode behavior is the lowest order symmetric Lamb wave, which begins on the m=0 P mode curve and then shifts over to the n = \ SV mode at the first crossover point. Owing to the strength of the coupling, this particular mode does not remain on then — I SV curve and is eventually depressed below the /;=0 SV curve. All of the other symmetric modes approach SV curves asymptotically as fib becomes very large. Similar behavior is displayed by the antisymmetric Lamb waves; all except the lowest order mode approach SV curves when fib becomes large. The special nature of the two lowest order modes has an important physical consequence that will be explored fully in the next subsection. The concept of waveguide mode cutoff was introduced in the subsection on SH modes, where it was shown that these modes have a real value of fi above the cutoff frequency and an imaginary value of fi below the cutoff frequency. This was illustrated by the dispersion curves shown in Fig. 10.7. t Sec Problem 2 ai ihe end of the chapter.
82 ACOUSTIC WAVEGUIDES 4 6 8 10 12 fib FIGURE 10.13. A comparison of the symmetric I amb waves with uncoupled SV and P modes for an isotropic plate. Lamb waves exhibit much more complicated behavior than the SH modes. The propagation constant ft is always real for propagating modes but is usually complex for nonpropagating modes. To show the complete dispersion curves one must therefore use a three-dimensional plot, as in Fig. 10.14. In the regions where ft is real the curves will be recognized as the first three symmetric Lamb modes in Figs. 10.12 and 10.13. The method of mode indexing is related to the mode orthogonality relations developed in Section J and will be explained at that point. Examination of these curves shows that there are four cutoff points (А, А', В, B') where ft changes from real to imaginary or complex. At Л and A' cutoff occurs for ft = 0, as in the case of SH modes. Points В and B', on the other hand, have finite values of ft. С FREE ISOTROPIC PLATE 83 0b Imaginary FIGURE 10.14. Dispersion curves for the three lowest order symmetric Lamb waves on a free isotropic plate. (After Mindlin). Each continuous solid curve (such as L3—JL3) corresponds to a wave earning power or decreasing exponentially in the + direction, while each continuous dashed curve (such as L 3-L 3) corresponds to a wave carrying power or decreasing exponcntialK th d eti Th th d f h ing mode indices is explained in Part 3 of Section J. I'icld Distributions. As in the case of SH modes, field distributions can be obtained by combining the partial wave fields. Only the particle velocity field components are given here. These are Д2 _ b2 vv = ~kn sm кцу cos kubj2 - -———^cos клЪ}2 sin kuy 2kts = ТФ\ cos kny cos klsbj2 - P *«'cos Ac„6/2 cos kuy L 2ft2 (a) (b) (10.22)
84 ACOUSTIC WAVEGUIDES for the symmetric modes and ft2 — ^7* A:,,cos kny sin ktllbj2 + —— sin k^b/l cos kisy vz = Tip sin kny sin klsbj2 2*„ ? ~ kj 2ff ipz sin /ct(fc/2 sin kuy , iPz (a) (b) (10.23) for the antisymmetric modes. The transverse wave numbers ktl and fc,s are given by (10.20) and (10.21). These expressions are rather complicated and a physical description of the field patterns is best obtained by looking at some limiting cases. It is of interest first to consider /3 = 0, that is, the points where the dispersion curves cross the w axis in Fig. 10.12. The Rayleigh-Lamb frequency equations then reduce to tan kubj2 tan fc„b/2 - 0 for the symmetric waves and tan klsbj2 ■ oo tan k„bj2 for the antisymmetric waves. The symmetric waves therefore have or whereas or k,Jb = Ntt (N = 0,2,4,. -■) knb = Л'тг (N = 1,3,5,. ■•), KJb = Ntt (Л' = 1,3,5,. -) M = Ntt (N = 0, 2,4, . •■) (10.24) (10.25) (10.26) (10.27) for the antisymmetric waves. Substitution of these results into the field expression shows, with the use of (10.24) and (10.25), that the particle velocity fields at fi = 0 are as illustrated in Fig. 10.15. In both the symmetric and antisymmetric sequences there is an alternation between pure shear and pure longitudinal types of transverse standing waves. At small values of /3 the particle displacement field patterns have the appearance shown in Fig. 10.16. For comparison, the particle displacement fields are also given for the two lowest order SH modes. In the symmetric modes the boundaries nf the plate periodically dilate and contract; these modes arc, therefore, often called dilatational.]' The antisymmetric t They are also often called longitudinal wave* and arc usually designated by L„. The term longitudinal will not be used here, to avoid confusion with longitudinal plane waves. C. FREE ISOTROPIC PLATE 85 WWW У/ащШУ УУУУУУУ ws//y///y/y//*wy//r*wi'/«,yMv. 'УЯУ"У>ЧЯ"—wr—y - I / 4 ',,,^У/Ууу/МШУУУУШУ#УУУМ'/У/у, ш/уууш-уу&щ-'мьу-тж тшуу>у,уум\фхш/,>у>,ш. p Symmetric or dilatational waves ry9±.:y>v>-yyy- ШУУУУУУУУУУУуУ/У/УУуу^УУУ/УУУУУУУУШ ^w^\\\\\\\\\\\<\\n\\\\4N\4^^^ Antisymmetric or flexural waves FIGURE 10.15. Lamb wave particle velocity field distributions at (I = 0. The four lowest order dilatational and flexural modes arc shown. modes are called flexural because of the periodic flexing motion of the boundaries. An interesting field pattern occurs when со Ум. = v72 К (10.28) in Fig. 10.12. Under these conditions ^ = ~72~?' v * y* which means that the SV partial waves are traveling at 45° to the z axis in Fig. 10.10. According to (9.43), there is no coupling to the P wave in this Case; and the Lamb waves arc therefore pure SV waves (Fig. 10.17). These arc called Lame ware solutions. The character of the field patterns changes strikingly as fi increases. When o>lfi > Vt in Fig. 10.12 the wave vectors of the partial waves appear as shown in Fig. 10.11 a. Both ktl and kls are real, and the longitudinal and shear parts of the Lamb wave (ields vary trigonometrically with y. At („//У = vl the longitudinal part of the field is uniform in the у direction lor \\ < vijfi < Vx the partial wave vectors are as shown in Fig. 10.1 lb.
Lamb Waves FIGURE 10.16. Field distributions for the lowest order modes on an isotropic plate with free boundaries (/? f=a 0). у У = 6/2 у = - b/2 FIGURF. 10.17 Partial wave pattern for Lame wave propagation At a 45 angle of incidence there is no coupling of the SV waves with the P waves. C. FREE ISOTROPIC PLATE 87 FIGURE 10.18. Dispersion curves for the fundamental modes Lx and F| in a free isotropic plate. The longitudinal part of the field is now confined near the boundaries of the plate and decays exponentially into the interior. Tn Fig. 10.12 all waves except the fundamental symmetric and antisymmetric modes and Fi) remain above the line «>//? = F"K, which is approached in the limit fjb — со. From (10.18), (10.19), (10.22), and (10.23) the limiting field patterns for all but the two fundamental modes are found to be predominantly a shear motion, tV~sinM" ~ 1)V-*-- for L„(«>1) h 2тг(2и — 3)?/ .щ, r r- i ^ i\ Uj,~cos—1 — e|!PZ for F„(n > 1), lb (10.29) plus a longitudinal motion tightly localized at the boundaries. These are essentially pure SV waves. For the fundamental mode L,, mjfj becomes less than Vs as /3 increases (Fig. 10.12), and cojfj is always less than Vs for F,. The partial wave vectors in this situation are shown in Fig. 10.1 Ic. For mode F, and fJ-^O the transverse wave vector components become so large that il is immaterial whether they arc real or imaginary. At large values of fj, however, both the Lx and F, modes become tightly bound to the surfaces of the plate and their velocities approach degeneracy (Fig. 10.18). These surface
88 АООТ. STIC WVVFGUIDFS waves arc sufficiently important to warrant a more detailed and separate discussion in the following subsection. C.3 Rayleigh Waves Dispersion Relation. Figure 10.18 shows that со IЯ for both the Ц and FY Lamb waves approaches a constant value Vu < Vs, Vt, as ffh becomes large. Since «•-т-т and kn and kis become large imaginary quantities in this limit. The mode fields are therefore very tightly bound to the surfaces and the interior of the plate is essentially undisturbed (Fig. 10.19). In the limit as /5Л > со the symmetric and antisymmetric waves Li and F, become exactly degenerate, and the field patterns can be combined to form a surface wave on either the upper or lower boundary as shown schematically in Fig. 10.20. The thickness of the plate is of no consequence in this situation, and the surface waves are then valid solutions for a half-space (or infinite substrate) with a free boundary surface. These solutions are called Rayleigh surface waves. Other kinds of surface waves will be encountered in some of the following sections. In parts (1) and (2) of this section the transverse resonance method was used to find the SH modes and Lamb wave solutions for an isotropic plate. FIGURE 10.19. Partial displacement field patterns for modes L] and E, for (Ih * со. C. FREE ISOTROPIC PLATE 89 ■v///////////*tt«///////////. ALl = e-'/M = e" (a) A. = с-'/Звг ''/////'//'/Лжи*,*;', AFl = — е-'Вчг . (b) FIGURE 10.20. Combination of L, and h\ modes to form surface, or Rayleigh, waves on the upper and lower boundaries, in the limit fib >- со. ALi and AFl are the amplitudes of the L, and bx fields. Partial waves were assumed to reflect back and forth between the boundaries Of the plate, and dispersion relations were obtained by imposing a resonance condition on this transverse standing wave pattern. The same technique can he used to find the dispersion relation for a Rayleigh surface wave propagating on an isotropic half-space. It has been seen that an SV wave or a P wave incident on a free boundary scatters into reflected waves of both types (Fig. 10.21a and b), with reflection coefficients Г,„ Tss, I1,,, Vsl given by (9.43) to (9.46). For the Lamb wave problem, reflections were assumed at both boundaries of the plate. The Rayleigh wave problem, by contrast, has only one reflecting boundary. In this case there are no incident waves arriving from the depth of the substrate (A, — At = 0), but the reflected waves Bs = VSSAS + VflAt B, = VlsAs + VHAt, (10.30) must still exist (Fig. 10.21c). This obviously requires that the reflection coefficients l\s, l\s, Y,„ Г , all become infinite. That is, the denominator of I lie reflection coefficient formulas must be zero. The transverse resonance condition for Rayleigh waves is therefore sin 26s sin 20, + (VtjV,f cos2 26„ = 0. (10.31) '1 he reflected waves now arc both evanescent, and must decay exponentially into the depth of the substrate. Consequently, 0 3 )
(с) sin 20s sin 20, - (У,/!/)2 cos2 20, ~ sin 20, sin 20, + (V,}Vsf cos2 20, 2(F,/FS) sin 20, cos 20, ' sin 20, sin 20, + (F,/!/)2 cos2 20s FIGURE 10.2E Transverse resonance analysis of Rayleigh wave propagation. The z components of к for all of the incident and reflected waves arc equal to />Е, the Rayleigh wave propagation constant. F\i — F„ sin 20v C. FREE ISOTROPIC PEATE 91 FIGURE 10.22. Isotropic Rayleigh wave velocity KR as a function of the bulk shear wave velocity Vs and the bulk longitudinal wave velocity V,. and sin fj, = ft) sin 0, = ft. К ft) eos 0. — — i cos 0, = —i The characteristic equation (10.31) may now be rewritten as fc = (4 + ^)2- (Ю.ЗЗ) After squaring and using the relations fa - 4 = (colV,f fa - «2, = («,/!/)*. from (10.20) and (10.21), (10.33) becomes {2pt - (to/I/)2}' " 16ft№ - WVf}{Bl - (co/Vf) = 0. This is multiplied out and divided by /5tj(oj/F,)2, giving where ft. is the Rayleigh wave velocity. An allowable solution for FR/F, must be real and positive, and only one such solution exists. This is given in Fig. 10.22 as a function of the shear and longitudinal velocities in the substrate. Since this exact solution for VnjVs
92 ACOUSTIC WAVEGUIDES Propagation 0 о i) Propagation (b) FIGURE 10.23. (a) Particle velocity and (b) particle displacement field distributions for a Rayleigh surface wave traveling to the right on an isotropic substrate. Ла = 2я7/7ц. t A range — 1 to + 1/2 is theoretically possible (Problem 4 in Chapter 6 of Volume 1) but is not found experimentally. C. FREE ISOTROPIC PLA'lE 93 illustrates the resulting field patternf v = ± i/Sufe-"* - 2a"K" е-и*)в^« (10.36) vz = аАё-*иу - 2PR , e-'Atr*'*', (10.37) Pii + *ts for the case of a wave traveling to the right. As in the Lamb wave solutions, the particle motion is elliptically polarized in the yz plane. In the figure it can be seen the the Rayleigh wave motion is retrograde near the surface and reverses its sense at depths greater than approximately one-fifth of a wavelength. The major axes of the ellipses are normal to the surface and the aspect ratio varies with depth. Coupled Surface Waves. In an infinitely thick plate the surface waves on the upper and lower boundaries do not interact. For a plate with finite, but large, thickness the field pattern of a surface wave on the upper boundary has a small residual amplitude at the lower boundary and vice versa. These small residual amplitudes cause a continuous coupling between the two surface waves, which can no longer be treated separately. The proper wave solutions for the coupled surface wave problem are the fundamental symmetric and antisymmetric Lamb waves L, and Fx (Fig. 10.19). It was noted in the section on Lamb wave dispersion relations that, in general, coupling between two waves induces a separation or splitting of their dispersion curves. Increased coupling of the surface waves on the upper and lower faces of the plate is therefore the cause of the increased separation of the Lx and Fj dispersion curves with decreasing fib in Fig. 10.18. In the infinitely thick plate the Lt and F, modes have the same phase velocity, ft t = ArE = fin, and the field patterns in Fig. 10.20a and b propagate without change. For a plate of finite thickness, L, and Fj have different phase velocities If a surface wave disturbance is excited on the upper boundary at z = 0 the Lamb wave amplitudes and AF must be equal and in phase at that point, just as in Fig. 10.20a. The fields excited are therefore described by ALl = A exp (-ifiLz) ЛК] = A cxp {-ifiFz). I Other forms appear in the references, but they are easily converted by using (10.33). is not easy to calculate, an approximate solution is often used instead, namely, Vu 0.87 + 1.12«t — = , (10 35) К i + cr where 1 - 2(VsIV,f 2(1 - [VJVtf)' called Poisson's ratio, ranges from 0 to J for actual materials.! The figure shows that the approximate expression is accurate to better than 0.5 %. Field Distribution. The particle velocity field for a Rayleigh wave is found by combining fields of the longitudinal and shear partial waves. Figure 10.23
94 4COUSTIC \\A\EGUIDES By contrast with the infinite plate case, however, the field pattern cannot persist as it propagates. Because fjbi and /?F( arc unequal, A,^ and vlFj continually shift in relative phase as the waves travel. After a propagation distance /, such that AF will be in phase opposition to Abt, and the pattern will correspond to a surface wave on the lower boundary (Fig. 10.20b). Propagation through a distance 21 will again return Alt and /1F] to an in-phasc condition and the surface wave energy will all reappear at the upper boundary. Because of this "beating" between the two nondegenerate waveguide modes, energy will continue to transfer periodically back and forth between the upper and lower faces of the plate. This behavior, which is characteristic of coupled wave systems, will be encountered again in part 3 of Section T. The length of one spatial period of the energy transfer process, called the beat wavelength, is A6 = 2w . (10.38) PFl ~ Ры Energy is completely transferred from one face to the other in one-half beat wavelength. As the coupling is increased by reducing the plate thickness b, the difference between /3Fi and /3Lj becomes larger and the energy transfers from the top of the plate to the bottom in a shorter distance.t D. ISOTROPIC PLA1E ON AN ISOTROPIC HALF SPACE Historically, the first investigations of guided wave propagation in elastic media were stimulated by the problem of seismic shock propagation in the earth's crust. Because the radius of the earth is much larger than the wavelength of seismic disturbances, this problem can be simplified by considering the earth's surface to be the top of an infinite half space. It was on the basis of this model that Rayleigh waves were first predicted and then observed experimentally. The structure of the earth is, however, more complicated than this, and other types of seismic waves can also exist. At the simplest level, one must allow for the fact that the earth's crust has different clastic properties than does the underlying material. In the planar model this situation is represented by an isotropic plate rigidly bonded to an isotropic half space (or substrate) having different material properties. Additional complications, in the form of multiple layers of different materials, are often required for models of scismological problems. Only the case of a single layer will be treated here, however. Problems of this kind, originally of interest only to seismologists, have in recent years taken on new importance ■j For furlher details see Reference 10 in Chapter 6, Volume 1. D. ISOTROPIC PLATE ON AN ISOTROPIC HALF SPACF 95 Reflected partial wave ^^.^^-^ Transmitted partial wave ^^S^S^S^S^S^ Incident partial wave FIGURE, 10.24. Partial wave pattern for transverse resonance analysis of Love wave propagation on an isotropic plate over an isotropic half space. with regard to high frequency acoustic surface wave devices for electronic signal processing. Wave propagation on an infinite half space differs in a very fundamental way from propagation on a plate of finite thickness. In both cases the problem is two-dimensional; that is, the fields in Figs. 10.4 and 10.24, for example, are uniform with respect to the x direction. The free plate, however, is closed on both sides by totally reflecting boundaries normal to the у axis, while the plate on an infinite half-space is open on one side. In the second case there is a possibility of radiation or energy leakage out of the wave into the half space, and it will be seen below that such leaky wave solutions do, in fact, exist. It has already been seen in Section С tha't leaky waves do not occur in the free plate problem. In seismology and electronics, unbounded structures do not, of course, actually occur in practice; nevertheless, the open structure is often the most convenient and suitable analytical model to use for many problems. 11.1 Love Waves The simplest solutions to the problem of a plate on a half space are Love waves. These arc shear waves polarized parallel to the plate boundaries and reduced to the SH waves of part 1 Section С when the mass density or stiffness
96 ACOUSTIC WAVEGUIDES Vr B'e ik'"b!i Zs cos 6, + Z; cos 0; £ *«* By rearranging terms and evaluating cos 0f and cos 0^ in terms of wave vector components, this can be expressed as V'7'к' i tan A- fa — . When k'ts is a real number the transmitted wave in Fig. 10.24 carries energy away from the plate and the solution is a leaky wave. Since solutions which trap and guide the energy are of greater interest the substitution к Is where a.',, is the transverse decay constant of the acoustic field in the half space, is made. The transverse resonance condition is then tan ktJb = • (10.42) D. ISOTROPIC PLVTE ON AN ISOTROPIC HALF SPACE 97 This is to be solved simultaneously with kjs = (a>IVsf — ft* (10.43) a|; = p'2 - UojV'tf. (10.44) Equation (10.44) shows that trapping can occur only when V\ > Vs. The reason for this is pictured graphically in Fig. 10.25, which shows slowness diagrams for the partial waves in trapped and leaky Love wave regions. Problems of this kind are often encountered in electromagnetism and graphical methods developed for their solution are also useful here. Equations (10.43) and (10.44) are first combined by eliminating /3, (10 45) Graphical plots are then made of (10.42) and (10.45) versus coordinates a'tl, k'fs, and simultaneous solutions are obtained from the intersection points (Fig. 10.26). For a given parameter ratio V'S7.'JVSZS (10.42) can be calculated once and for all, and (10.45) is a simple straight line which shifts along the aj2 axis with increasing frequency at a rate determined by the shear velocities in the two media. In Fig. 10.26 each branch of (10.42) corresponds to a separate Love mode. At m = 0 there is an intersection at к}2 = 0 for/; = 0 only. All other modes tire therefore leaky waves. As frequency is increased, each mode in succession makes a transition at a)2 = 0 into a trapped wave. These transitions occur at о.Гп = (пп)2 . (10.46) and Bl = {птт? *J . (10.47) For in-* cc the transverse wave numbers in the plate approach лтг/2, and MJfi approaches Vs. The dispersion curves (Fig. 10.27) are obtained by using (111.43) to evaluate В(ы) from the solutions k„(w) of Fig. 10.26. At high frequencies these curves all approach <u//3 = F, in the same manner as the SI I waves. There are no cutoff frequencies. Below a certain transition iicquency each mode become leaky. D.2 Generalized Lamb Waves These solutions are polarized in the vertical plane and bear the same relation lo I amb waves as Love waves do to SH waves. They are, however, much more complicated in their behavior. of the underlying half space goes to zero. The partial wave pattern for this type of wave is shown in Fig. 10.24. Following the transverse resonance procedure used for SH waves in part 3 of Section В the particle velocity fields of the incident, reflected, and transmitted waves are taken to be cm - Be-M"'J+Pz) From (9.42) the horizontal shear reflection coefficient at the free boundary у — b\2 is Г, = 1. Since в,, is the reflected wave at the upper boundary, у = Ы. = *1 = i (10.39) vrR he At the interface у = — b\2 the scattering coefficients are, from (9.28) and (9.29), z,< cos o, — z; cos о; _ = ^ щ Zs cos 6S + Z'f cos 6's vrl and 2Z.cose, _ .v, (Ю.41) z, cos o„ + z; cos Transverse resonance requires that the two reflection conditions (10.39) and (10.40) be satisfied simultaneously. That is, zs cos o, - z; cos e; e-1*"*
cd с О о со О о Э1-» + I + с' I cd" с Г-н I О cd cd о о cd ^ „ .л СО ° с cd Гч1 о + cd о cd О cd с + cd О) Г-Г cd с cd О О I cd О О J3 FIGURE 10.27. Love wave dispersion curves. у Longitudinal partial waves Shear partial waves HGURE 10.28. Partial wave pattern for generalized Lamb waves on an isotropic plate over an isotropic half space. 10
Plate kz _ k-2 _ § S bstr te Plate — 1 K H OJ ~~ w к (Jj 1 1 1 У " Substrate (b) TIGURC 10.25. Slowness diagrams for partial waves, in the (a) trapped and (h) leaky wave regions. Part (b) does not strictly represent a wave solution because leaky waves arc attenuated and must therefore have complex values of ft. D. ISOTROPIC PLATE ON AN ISOTROPIC H\LF SPACE 99 10 20 30 40 50 riGURE 10.26. Graphical construction of Love wave dispersion curves (V'^Z'JVSZS = 1, h — 1). Note that branches with klx tan klls negative are excluded, since x'ts in (10.42) must always be positive. The partial wave pattern is as shown in Fig. 10.28. Scattering relations at Hie interface у = —bj2 are very complicated (Sections D.2 and 3 in Chapter 9), and it does not seem advantageous to use them directly for calculating dispersion relations. Instead, the analysis may be performed by returning lo the scattering equations (9.37) and (9.40). The partial wave amplitudes in the plate (At, Л„ Bt, Д) must satisfy stress-free surface boundary conditions at у — bjl. Since A, and As are amplitudes of the reflected waves at this surface, the condition is TV ~B,elk,lb"r Д giktshf'2 В il-|S&/2 _ ; Using this equation As and A, can be eliminated from (9.37) and (9.40). II can then be shown from (9.47) and (9.48) that (9.37) and (9.40) both lead to the same set of four linear equations for B,, 7J„ B[. Bs. The determinant of this set of equations gives the characteristic equation for generalized Lamb waves.
102 ACOUSTIC WAVrGUIDLS Dispersion relations are obtained by solving (10.49) simultaneously with the auxiliary conditions (10.50) «;2 = [3- - {oiivf <; = f - Wf. To carry the problem beyond this point requires numerical computation. The details, which are thoroughly covered in references at the end of this chapter, will not be discussed here. As in the case of Love waves, the characteristics of generalized Lamb waves depend strongly on the ratio of the shear velocity V',. in the substrate to the shear velocity Vs in the plate. If the two velocities are appreciably different and Кя » V's there is only one generalized Lamb wave solution. This reduces to a Rayleigh wave on the substrate when fib -> 0 and exists only over a range of fib for which e>IP < V,. When V„« V's there is an infinite number of solutions which fall into two families of modes, often called the M, series and the M2 series. These reduce to the symmetric and antisymmetric Lamb waves, respectively, when the density or stiffness of the half space goes to zero. The fundamental modes of the two series (Mn, M2I) have especially interesting properties. For plate thicknesses approaching zero (fib -*■ 0) the Mu mode approaches a Rayleigh type surface wave on the substrate (Fig. 10.29a), while the higher order Tvl, modes and all the M2 modes are leaky waves. As the plate thickness is increased, the first additional mode to become trapped is the M2, mode, which is called the Sezawa wave in this region (Fig. 10.29b). For very high frequencies or thick plates (fJb > oc) the Mu mode approaches a Rayleigh t pe surface wave on the upper boundary of the plate l 0 9c). All higher modes (M,„ M2i, /' > 1) degenerate into essentially vertically polarized shear waves in the plate when^A-^ со. That is, they behave like the Lamb waves of a free thick plate. The behavior of mode M2i depends critically on the relative material parameters of the plate and the substrate. If V's and Vs are appreciably different this mode also approaches a shear wave in the plate when fib—>*oo; but for certain special combinations of material parameters with V's Ve it becomes a bound, or surface-type, wave at the plate-substrate interface. Waves of this type may also exist when Vs «a V, and Ks > V's. In this case the Rayleigh wave solution for the uuplated substrate (fib — 0) becomes a bound wave at the interface (with toffJ < V'j in the limit as fib -*■ со. General conditions for the existence of these interface waves arc discussed in the following subsection. D. ISOTROPIC PI.ATE ON AN ISOTROPIC HALF SPACE 103 MH Mode (a) /36- (b) /36 ss 0 ;;.,»*...v'.».., —j— -llfel-1- »*- —mm 1L Shear plate wave (VS«V') (с) /36-Э-оо Stonely wave (Vs ~ V-) FIGURE 10.29. Schematic velocity field distributions for the dominant modes Mu and M2, of an isotropic plate of thickness /> over an isotropic half space. The shear velocity in the plate Ks less than the shear velocity in the substrate V'. D.3 Stoneley Waves For certain specific combinations of material parameters, mode M21 becomes n surface wave tightly bound to the plate-substrate interface when fib ->- do. These waves arc called Stoneley waves. The field distribution is entirely composed of two partial waves decaying away from the surface in each medium (Fig. 10.30). Numerical computation is required for evaluation of the propagation velocity Vb and the field distributions; but some general slatements can be made about the allowed range of velocities and also about the ranges of material parameters within which Stoneley waves can exist. It has been shown that the Stoneley velocit) must lie between the velocity of Rayleigh waves and shear waves in the denser medium, that is, V'K < Fs < V't, (10 51)
104 ACOUSTIC WAVEGUIDES Stonely wavelength Tungsten Pa ic e velocity normal to the interface \ Particle velocity ^ parallel to the interface У у FIGURE 10.30. Particle velocity field components for a Stonelcy wave at an interface between polycrys- tallinc (isotropic) aluminum and tungsten media (after Farnell and \dlcr) where p > p. Limits on the values of c41/c« and pip' for which Stoneley waves can exist are given in Fig. 10.31. The figure shows plots of these curves for two acoustic velocity ratios which bracket the typical range for real materials (V,]VS^ 2). The solutions are very sensitive to the ratio VJVS in the more dense of the two media but are only slightly influenced by VjV, in the less dense medium. Table 10.1 shows pairs of isotropic (that is, poly- crystaline) metallic solids which support Stoneley waves."!" The figures tabulated give the ratio of the Stoneley velocity to the lower of the two shear velocities. E. FREE ISOTROPIC CYLINDER Up to this point, only planar structures have been considered and the guiding effect is confined to one dimension, as in Fig. 10.1a. For guidance in two dimensions (Fig. 10.1b) it is necessary to use a three-dimensional structure such as the cylindrical isotropic rod waveguide with stress-free boundaries t It should be noted that Table 10.1 does not include the case shown in Fig. 10.30. This discrepancy is due to the use of slightly different values of material constant by the ai thors, and it serves as a vivid illustration of the sensitivity of Stoneley wave solutions to the values of these parameters. а в > S D IV0 P3)S Э KO'O P3)s UlUinfEJllQ oojUfV uais§unj. PJPIN lunisaugrji^ joddo^j «">00|--OOOOOl/->M>00№ со о On -sr r— r 1 Гч1 о со — cs m — —' OO — ON О Г- О О CO Г] О m (4 гч гч r-i r*i r-1 r*-i rs п m rl ON ON d NO CO ON NO ON О rl on on о r- CO no on on on on on on d d d d О ON Г- On ON On Г On On rl i-~ r- CO ^ On On On On odd "Л 1Л in oo rl ON NO CO —< r- ГП О Г- .—■ *N m n r*i о ON On d On On On On on d On On О I - ON Aiouiuuy NO ON ON d ON On On On О lunujiuniy С 3 ■—) '■5 ON on d с E Г-] CO r— CO NO On On ON On On On d d О О a — 5 .э .3 •S « с u с -о «J — с о о £ ■- га .с 5 - to с с 1 о _ о га д U U со со О т О О
J06 ACOUSTIC WAVEGUIDES p (b) FIGURE 10.3F Range of existence of Stoneley waves. Solutions exist any where within the shaded regions (after Scholte). (a) VJ V„ = V[\ V[ = V3. (b) vjy, = viiv; = cc. (Fig. 10.32). In this case it is most convenient to apply the potential theory method (part 1, Section B). For time-harmonic fields with frequency w] (10.8) and (10.7) have cylindrical coordinate representations 11(гЭФ)+1^ + ^=_-!ф Oo.52) дЛ Ьг1 г2 дф~ dz2 V, and (10.53) E. bRIT ISOTROPIC CYI [NDER 107 EIG URE 10.32. Free circular rod waveguide. where rdr\ dr) f дф2 dz2 For waveguide problems, the appropriate solutions are of the form Ф(г, ф)е'ш'рх) 4f(r, ф)е'ш~рг). The standard separated variable solution to the scalar potential equation (10.52) is then РФ\ cos рф\ Ф = ARri(k„r)\ Icos рф) (10.54) 4 + where A is an arbitrary constant and Rv is a Bessel function. In Fig. 10.32 the potential must remain finite at r — 0, and Bessel functions of the first kind arc used; that is, „ .,. . . ,. ЯДА„г) = /ДА',/). It is easily checked by direct substitution (Problem 7) that the rector potential equation (10.53) has two independent divergenceless solutions and M = V x zY V. N = — V x M w where x¥ satisfies the scalar potential equation It follows that has the same form as (10.54), but with Rp(kur) = JAktj) kl + * = W. (10.55) (10.56)
108 ACOUSTIC WAVEGUIDES A general solution for the vector potential is then 4? - BM + CN fcos рф рф — cos рф . sin рф M = I —r \ kisr { cos рф . sin рф. iPz sin рф cos рф. (10.57) f—cos рф\\ v Sin рф ) I рф where primes indicate a derivative with respect to the argument. Substitution of (10.54) and (10.57) into v - V<D + V x 4/ gives the general particle velocity field in cylindrical coordinates. That is, Aktlj'v{knr) + №KiKr) + - C./„(fc,sr)l(COS P%~ipz r J (sin p<^> J - AJP(kar) + y£ BJv(klsr) + ktsCJ^(ktM~Sin РФ .r ktssr J{ cos рф , (10.58) v, - l-ipAJJLkur) - k,MJ.klsr)] cos рф sin рф According to Appendix 1 in Volume T the strain field in cylindrical coordinates has components 1 dv_r но dr Sz<t> ~ io> \r г оф I io> dz 2u»\dz гдф) 1 idvr dv\ 2im\dz dr) 1 /1 dvT di-ф _ сф\ 2ico\r дф dr rl (10.59) and these are converted to abbreviated subscript notation by the relations Si = s s4 = 2SZ^ s2 — Зфф s5 = 2STZ (10.60) s3 = szz Se = 2Sr0. E. FRFF ISOIROPIC GYL1NDI R 109 Stress components are calculated from (10.59), using (10.60) and T = c:S. For the free cylinder, the boundary conditions T„ = 7\ = 0 Trz =TS = 0 (10.61) ттф = ть = о must be applied at r = a. In the case of an isotropic medium the stiffness matrix has the same form in cylindrical coordinates as it does in rectangular coordinates (Appendix 1 Volume I) and the boundary equations (10.61) take the form 1 ко Tr 1 ко 1 дф. (10.62) дф Substitution of the particle velocity field (10.58) leads to the three equations in А, В, С \ J ********* ( —.'**>) ЪГ,кпф„а) K-lP)j'v(ktla) iBp (10.63)
110 ACOUSTIC W4VEGUIDFS The characteristic equation is obtained by setting the determinant of these equations equal to zero. Solving this problem in the general case is a formidable task, but it has been thoroughly analyzed in the references at the end of the chapter and the general propagation characteristics arc well understood. These characteristics are most easily summarized by considering first the azimuthally symmetric modes (p = 0). E.l Azimuthally Symmetric Modes For p = 0, С is decoupled from A and В in (10.63), and the characteristic equation is 2cH\8ktJl{ktsa) -c,«(k-tl + B-)J0(kHci) +2сык*,31(кпа) Тфкптаа) (kl - ДУЙМ) 0 (10.64) In (10.63) the constant С can have a nonvanishing value only when kfMkt.a) + — Jo(ktsa) = 0. (10.65) a Since A and В can be set independently equal to zero, this gives (from (10.58)), a set of solutions vr - vz = 0 еф = Ck.Xik^e^ = -CktJ^k^e (10.66) In these modes, the torsional modes, the particle velocity is entirely azimuthal (Fig. 10.33). It vanishes al r — 0 and alternates in sign with increasing r. The designation T(llj has been proposed for these modes, where 0 is the value of p, and q designates the solutions of (10.65) in order of increasing kls. A solution q = 0 exists, in which kts — 0 and гф r. For this mode, each cross section of the rod rotates rigidly about the axis (Problem 9). Propagation is at the bulk shear velocity Vs — (c.,„/p)1/2 and extends down to zero frequency, where the motion is a rigid rotation of the entire rod. The torsional modes are therefore very similar to the SH modes of the free plate (part 1 of Section C) both in the simplicity of the solutions and in the mode characteristics themselves. Two other families of azimuthally symmetric modes are obtained by constructing a two-by-two determinant from the upper left-hand corner of (10.63) and equating to zero. This gives the Pochhanimerfrequency equation. F1GURF 10.33. Particle velocity distributions for the three lowest order members of the torsional mode familv w The constant С must now be zero; but, in general, both A and В are non- vanishing. For the special case /3 — 0, however, the determinant becomes diagonal and solutions arc simply A - С = 0; Jy(kud) = 0, (10.67) which gives pr = = 0 from (10.58); and В = С = 0; -c]2J0(fc„«) + 2см.П(кии) = 0, (10.68) which gives юф, гг — 0. In the solutions (10.67) the particle motion is entirely axial and in (10.68) it is entirely radial (Fig. 10.34). When /1^0 these two motions are coupled, but С is still zero. T he particles therefore move in radial planes. These modes arc usually referred to as longitudinal or dilatational, the philosophy of this terminology being the same as for the symmetric Lamb modes of the free plate (Fig. 10.15). That is, the particle velocity fields have a radial component which does not vary
112 ACOUSTIC WAVEGUIDES 1-02 "o2 FIGURE 10.34. Schematic particle displacement patterns for the lowest order dilatational modes of a circular rod at P = 0. azimuthally, and the free boundary alternately dilates and contracts. It has been proposed that this set of modes should be subclassified, according to their axial and radial motions at /3 = 0, and designated as L^ and (Fig. 10.34). This would correspond to a subclassification of the symmetric Lamb waves in Fig. 10.15 as either axial or transverse types. Numerically calculated dispersion curves for the L^ and Rtv modes have the same general appearance as curves for the symmetric Lamb modes, where it was seen that the behavior could be interpreted in terms of coupling between guided wave solutions to a mixed boundary value problem. This approach is also instructive in the circular rod problem and will be discussed below. The lowest order mode propagates down to zero frequency and approaches the Rayleigh velocity VH (part 3 of Section C) as pa -> oo. E.2 Azimuthally Varying Modes Modes with p = I are usually termed Jlexural modes, the reason being that radial motions at opposite points on the circumference move in phase opposition and the boundaries consequently execute a flexing motion, as in antisymmetric Lamb waves (Fig. 10.16). Modes with p > 2 are often called higher order flexural modes. Just as in the p = 0 case discussed above, the matrix in (10.63) becomes diagonal when /8 = 0 and the modes can be classified as T , LM, and Rm according to their behavior at p = 0. E. FREE ISOTROPIC CYLINDER 113 In general, the azimuthally varying modes have particle velocity fields with r, фу and z components. Dispersion curves have the same general appearance as the azimuthally symmetric modes. For p = 1 the lowest order mode propagates down to zero frequency and approaches the Rayleigh velocity Vn as pa -»- со. The Rayleigh velocity limit still applies to the lowest order modes with p > 1, but these do not propagate down to zero frequency. E.3 Uncoupled P and SA Modes It was seen in part 2 of Section С that the characteristics of Lamb waves could be usefully interpreted in terms of uncoupled SV and P waves (Figs. 10.8 and 10.9). These are simple wave types which are solutions to the plate problem with mixed boundary conditions. Introduction of complete stress- free boundary conditions leads to coupling of these modes and explains the characteristic coupled wave appearance of the Lamb wave dispersion curves. This suggests that the same approach might be usefully applied to the circular rod waveguide. For the aximuthally symmetric case (p = 0), the uncoupled modes are easily obtained for the mixed boundary conditions tV = 0 (10.69) at r = a. The torsional modes T0o, discussed above, already satisfy these boundary conditions. From (10.58) two other classes of solution are easily found to be oT ktlJ,(ktlr)e *• vz = ipJ0(kllr)e-"!!! (10.70) 4k,fi) = o, and vr=-ipUklsr)e*»* ». = -ktMkt^e-1"* (10.71) Jt(fctsc) = 0. At (I = 0 the particle velocities become purely radial and purely axial, that is, radial standing waves of pressure (P) and axially polarized shear (SA) types, respectively. By analogy with the Lamb wave problem, the uncoupled waves for the cylindrical rod may therefore be designated as P waves (in (10.70)) and SA waves (in (10.71)). E.4 Cylindrical Rayleigh Waves Another cylindrical coordinate problem of considerable interest is that of wave propagation along ф. Solutions corresponding to (10.58), but with
114 ACOUSTIC WAVEGU1DFS direction FIGURE 10.35. Configuration for cylindrical Rayleigh wave propagation. е'"ф angle functions rather than sines and cosines and with /3 = 0, arc matched to stress-free boundary conditions at r — а. A wedge geometry (Fig. 10.35) is selected, to allow nonintegral values of the azimuthal wave number. One solution of the characteristic equation has the property that it reduces to the Rayleigh surface wave solution (part 3 of Section C) when a-+ <x>. For finite radius this solution, called a cylindrical Rayleigh ware, travels at a slightly higher velocity than the Rayleigh wave on a plane boundary (Fig. 10.36). F. ISOTROPIC RECTANGULAR STRIP In rectangular Cartesian coordinates, general separated variable solutions to the potential equations (10.7) and (10.8) are easily obtained. Since the FIGURE 10.36 Propagation velocity for cylindrical Rayleigh waves. A is the azimuthal wavelength. (After \ iktorov) cii 2c,14 2(сц + P44) I he wave becomes leaky at the point where the curve stops at the left. F. ISOTROPIC RECTANGULAR STRIP 115 ox p- +ilF dy where general solutions for4;v and V, arc T, = (A cos kxx + В sin кгх){С cos kyy + D sin kyy)e~i0z = (t cos krx + F sin kxx)(G cos kyy + И sin kyy)e~ipz, with (10.72) (10.73) kl+k The scalar potential solution is simply Ф = (J cos kx% + К sin kxx)(L cos kyy + M sin kyy)e~ with (10.74) k* + k\ + ft 1.1 Free Strip In view of the simplicity of these results it seems paradoxical that an analytical solution has not yet been found for the free rectangular strip (Fig. 10.37), while a solution does exist for the cylindrical problem (Fig. 10.32). The boundary conditions shown in the figure cannot, in fact, be satisfied with the potential functions (10.72) and (10.74), and the problem has been attacked TxX — TXy ~ гхг = о TIGURE 10.37. Tree rectangular isotropic strip waveguide. divergence of the vector potential W ts arbitrary, it may be taken as zero For a potential of the form 4?(x, у)е^\ the z component of the potential can then be calculated from the other two components; that is,
116 ACOUSTIC WAVFGUIDFS by approximation methods suitable for the thiii strip case (b « a). These make use of approximate boundary conditions on some of the surfaces.! Usually, exact boundary conditions are applied on the broad faces of the waveguide Tvv = Г„„ = T„ = 0 at у = 0, b and approximate boundary conditions Txx = 0 at x = 0, a are applied to the narrow faces. It can be shown that the neglected stress components (Txv and Tx.) on the narrow faces go to zero as the thickness b goes to zero. There arc also some exact solutions known for special conditions. At the Lame velocity (part 2 of Section C), the Lamb wave solutions reduce to pure SV waves (Fig. 10.17) which satisfy stress-free boundary conditions at all surfaces of a rectangular strip. Exact solutions have also been obtained at special frequencies and for special width-to-thickness ratios. F.2 Uncoupled Modes In view of the difficulty in obtaining exact solutions to the rectangular strip waveguide it appears desirable to apply to this problem the coupled wave formulation which proved to be so useful in the Lamb wave problem. As in the plate and cylindrical rod waveguides the uncoupled modes are generated by applying mixed boundary conditions. In the rectangular strip it is convenient to choose these to be vanishing normal particle velocity and tangential traction force (Fig. 10.38). It is easy to show that these boundary conditions are satisfied by (10.74) with A! — M = 0 and kx = nmja, kv = nnjb, where m and « are integers. The particle velocity field is then ТПП . mnx Птту — sin cos —- a a b тттх . птту — cos sin b a 0 (10.75) тттх птту _tpz vz - —ip cos cos e a b and the dispersion relation is f = ИК,)* - (^J - (^J. (10.76) f Reference 41 at the end of the chapter. F. ISOTROPIC RECTANGULAR STRIP 117 vx = 0 Txy = T„ = 0 vx = 0 Txy=Txz = Q FIGURE 10.38. Mixed boundary' conditions defining uncoupled modes for the rectangular isotropic strip waveguide Two other classes of solutions are obtained from the vector potential given by (10.72) and (10.73). These are (v)(f) тттх птту ..„„ sin cos —- e~tfiz a b p2 + (nnrjaf mnx . птту i\ -" cos sin —- e p" 'P ah птт mnx nny itt, vz = cos cos —- e p\ b a b obtained by taking В = С = 0 and lF, = 0 in (10.73), and p2 + (nnjbf . mnx nny Vx= : sin cos —- e p ip a b (?)(t) iP mnx . птту ,ft. cos sm —- e fz mn mnx nny ,„„ vz= cos cos — e'Pz. (10.77) (10.78) oFlained by taking lF, = 0 and F _ H = 0 in (10.73). Both of these classes of modes have the dispersion relation P2 = HKf (10.79)
118 ACOUSTIC WAVEGUIDES For m = 0 and [i — 0 these modes become very simple Equation (10.75) reduces to a pure P wave reflecting between the upper and lower boundaries, (10.77) reduces to a pure SV wave, and (10.78) becomes identically zero. With n = 0 and /7=0 (10.75) and (10.78) become, respectively, P and SV waves reflecting between the side boundaries, and (10.77) vanishes. F.3 Rectangular Strip on a Substrate In most waveguide applications it is important that there should be only a single propagating mode in the frequency band of interest; all other modes should be cut off. Otherwise, mode interference effects and multiple delay times will cause distortion of the transmitted signal. This poses a practical problem in the case of strip waveguides. The analysis of SH modes in Section В showed that the number of propagating modes increases as the signal frequency is increased and decreases as the transverse dimension of the waveguide is decreased. This means that the transverse dimension must be decreased when the signal band is shifted to higher frequencies, in order to preserve single mode operation. All types of waveguides share this property. Consequently, single-mode strip waveguides for operation at very high frequencies are too thin to be self-supporting and must be mounted on a substrate (Fig. 10 39a). Thi practical consideration further complicates the already difficult free strip problem. Approximate numerical methods have, however, been applied to the problem,+ and Fig. I0.39b shows typical phase velocity versus frequency curves for the four lowest order modes. The terms symmetric and antisymmetric refer to the field distribution in a horizontal plane. Modes described as Love-type become the Love waves of Section D.l when fl->cc, and the Rayleigh-typc modes become the generalized Lamb waves of Section D.2 under the same conditions. It should be noted that only the 1st Symmetric Raylcigh-type mode propagates for atb{V't < 0.9. G. MICROSOLND WAVEGUIDES The waveguide in Fig. 10.39 is one of a general class of waveguides that may be used to guide acoustic energy along a curved path on the surface of a solid body Cither planar or curved surfaces may be considered; the only constraint on the shape of the surface and the path of the guide along the surface is that the radius of curvature should be much larger than the wavelength (Л — 2тт}р) of the guided wave—typically, at least 10 times as large. Because these are open waveguides, too sharp a curvature causes the acoustic energy to radiate, t С. С Tu and G. W. Farnell, "On the Hcxural Mode of Ridge Guides for Elastic Surface Waves," Electronics Letters 8, pp. 68 70 (1972). G. MICROSOLND WAV FGLIIDES 119 p, 41, c4a Substrate shear velocity = v,' (a) 0.2 0.4 0)6 V/ FIGURE 10.39. Rectangular strip waveguide on a substrate, (a) Ba-sic configuration, (b) Typical velocity versus frequency curves for the lowest order modes. (After lu and farnell) or leak, from the guided wave*. Many waveguide structures in this class [inicrosound waveguides) have been investigated (Fig. 10.40). One of the earliest studies dealt with the ridge guide% shown in (a), which is closely related to Fig. 10.39, but differs in having a substrate of the same material t—Птс same effect was noted in Fig. 10.36. I Finite-clement computations have recently been reported for this structure. See R. IJuriiclge and F. J. Sabma, "Theoretical Computations on Ridge Acoustic Surface Waves Using the Finile-elcmcnt Method," Lleclromcs Letters!, pp. 720-722 (1971); P. R. l.agassc, "Analysis of a nispersionfrcc Guide for Elastic Waves," electronics LettersS, pp. 372-373* i |tl"7*)l * '
120 ACOUSTIC WAVEGUIDES (a) (W (c) № (e) « (E) FIGURE 10.40. Microsouud waveguides, (a) Ridge Guide [48, 52, 55]. (b) Channel Guide [48]. (c) Topographic guiding with a low ridge; arrows on and near the ridge indicate a greater freedom from constraint, and therefore a lower phase velocity, in the ridge [48]. (d) "Slow-on-fast" stripe guide. Stripe may be a mass-loading film on the s t te r it ma b a low densit Iec- trical conductor on a piezoelectric substrate [46,47,49, 53]. (e) "Fast-on-slow" stripe guide, (f) Guide with diffused- or growii-in material that has a low surface wave velocity, (g) Sandwich guide with a low-velocity center section [48]. (After R. M. White) as the strip. An advantage of this arrangement is that fabrication is simpler; however, use of two different materials provides tighter confinement of the acoustic field within the waveguide. The channel guide in (b) is an obvious structural modification of the ridge guide. Ridge and channel structures are examples of topographic guides. The guiding action arises from a geometrical deformation of the substrate surface, without use of a second material. G. MICROSOl ND WAVEGUIDES 121 Another type of microsound waveguide is based on the concept of reducing the Rayleigh wave velocity underneath the guiding structure. It has already been shown in the analysis of Love waves (part I of Section D) that waves can be trapped within a region with a lower phase velocity than its surroundings. Outside this region the fields decay exponentially to zero. This is a general principle of wave propagation and can be applied to both surface and volume waves. Parts c, d, e, f, and g of Fig. 10.40 show several kinds of microsound waveguide based on this principle. These may be categorized as structures which change the Rayleigh wave velocity by perturbing the surface boundary conditions (c, d, e) and structures which alter the elastic properties of the substrate. Of these waveguide types, the stripe guide and the slot guide have been subject to the most thorough study and only these two will be considered further. Superficially, the stripe guide in Fig. 10.40 and the strip-on-substrate guide in Fig. 10.39 appear to be identical; and, geometrically, they are. The distinction lies in the fact that the plating is only a small perturbation in the case of a stripe guide and the acoustic energy therefore resides almost entirely in the substrate; the guide in Fig. 10.39, on the other hand, has most of its stored energy in the rectangular strip, which must therefore be much thicker than for the stripe guide. In stripe and slot waveguides the guiding structure is defined by a pattern of thin film plating on the surface of the substrate. The purpose of this plating is to perturb the velocity of the Rayleigh wave propagating underneath it. For energy trapping within the guide it is necessary that the Rayleigh velocity be lower under the stripe in Fig. 10.40d and the slot in Fig. 10.40e than it is elsewhere. This means that the stripe plating must slow down the Rayleigh wave, while the slot plating must speed it up. Because of the complexity of the exact characteristic equation (10.49) for the plate-on-a- snbstrate problem it is desirable to use an approximate solution for the thin plating situation. A detailed treatment оГ the approximation will be given in Section B.l of Chapter 12 and only the end result will be given here. The effect of plating on the Rayleigh wave velocity can be predicted from the »ign of the quantity! I/ /1 _ (V'IV',f\,/2 L = — — ( yvj 'M , (10.80) where primes refer to the substrate medium. When L > 0 the Rayleigh wave velocity is increased by the plating, and when L < 0 it is decreased. It docs not appear possible to obtain an exact solution of the stripe and tlot waveguide problems. One method of analyzing the problem is to obtain I (Inference 20 at the end of ihe chapter.
122 ACOUSTIC YVAYTGUIDES field solutions in Regions I, II, and III in Fig. 10.41, using the approximations described above for the plated regions, and then to satisfy boundary conditions for one or more acoustic potential components along the lines у = 0, x = ±«/2- In Regions I, II, and III the acoustic fields must all vary with z according to the same factor e'fi. This condition is satisfied by superposing partial waves, as in the Lamb wave and Love wave analysis (part 2 of Section С and part 1 of Section D). In this case, however, the partial waves are Rayleigh surface waves (Fig. 10.42) rather than plane waves. Unperturbed Rayleigh waves are used in the unplated regions and perturbed solutions in the plated regions. Slowness diagrams corresponding to Fig. 10.25 are shown in Fig. 10.43. For field confinement near the slot or stripe it is clear that the partial wave velocity in Regions II and III must be greater than in Region 1. This wave trapping condition, which is the same as for 1 Region Ш (b) FIGURE 10 41. Stripe waveguide, (a) "Slow- on-fast" configuration, (b) "Fast-on-slow" configuration. G. MICROSOUND WAVEGUIDES 123 FIGURE 10.42. Pattern of partial Rayleigh waves for the analysis of stripe waveguide. Imaginary wave vector components are designated by dashed lines. Ixive waves, requires that the quantity L in (10.80) be less than zero for the stripe configuration (a) and greater than zero for the slot configuration (b). Since the variation of the field quantities with у will be different for the perturbed and unperturbed Rayleigh waves, it is not possible to match boundary conditions everywhere on the interfaces 1, II and I, III by using only these partial wave solutions. This is the reason the boundary conditions arc matched only on the lines у = 0, x — ±«/2 in Fig. 10.41. Boundary conditions for the plated region are ignored completely, on the grounds that the plating is very thin. In this way, approximate characteristic equations arc found to be of the form tan kaa\2 = ± (—L1 . (10.81) Here, the transverse wave number in the central region, ka, and the transverse decay constant in the outer regions, <x(TT must satisfy relations 3*+ k-fi = HVnl)2 (10.82) ft2 ~ «m - HVnnf- (Ю.83) 7.\ and Zxl are effective impedances for Rayleigh waves in the inner and outer regions. Because of the similarity of (10.81) to the characteristic equation (10.42) foLLove waves, the same graphical procedure can be used for calculating the piopagation characteristics. An infinite set of modes exists, each one being a tupped mode above a certain transition frequency and a leaky mode below that frequency. As in the Love wave case, the dominant mode propagates down to zero frequency. Calculated velocity curves and experimental results
(b) "Fast on-slow" configuration FIGURE 10.43. Surface wave slowness diagrams for trapped microsound waves. G. MICROSOUND WAVEGUIDES 125 0 0.1 0.2 0.3 0.4 FIGURE 10.44. Propagation velocity Vof the lowesl order symmetric mode for (he stripe configuration (gold on fused silica). V's is the substrate shear velocity. (After Ticrsten) for the two lowest order modes are shown in Figs. 10.44 to 10.47. These show that the slot configuration has the advantage of being less dispersive than the stripe configuration. Furthermore, in this configuration the energy !s concentrated primarily in an unplatcd region and the attenuation would therefore be expected to be less than for the stripe guide if the plating material Is lossy, as it always is experimentally. Г wo other formulations of the stripe and slot waveguide problems are given in references 46 and 110 at the end of the chapter. In the Adkins and Hughes treatment, an exact numerical solution for the Rayleigh wave velocity in the plated region is fitted to a power scries expansion in tobjV tnid somewhat different boundary conditions are used at ц = 0, a = ±« 2. The second method, due to Olincr el al., uses waveguide mode theory to (Hkulale the Rayleigh wave reflection coefficient at the edge of the plating (Section N). All three calculations agree quite well with each other and with experimental results. VV hen two stripe or slot guides are run close to each other, they interact through the exponentially decaying fields in Regions If and III of Fig. 10.41. The situation is analogous to coupling between Rayleigh waves on the upper
О 01 0.2 0.3 0.4 соб/V FIGURE 10.45. Propagation velocity V of the lowest order antisymmetric mode for the stripe configuration (gold on fused silica). V's is the substrate shear velocity. (\fter Tiersten) о o.i о.г и j 0 4 o,5 ubIV FIGURE 10 46 Propagation velocity V of the lowest order symmetric mode for the slot configuration (aluminum on T 40 glass). V's is the substrate shear velocity (After Tiersten) G. MtCROSOLIND WAVFGUIDLS 127 1.000 i 0.950 0.914 0.900 1 Г A um num- T-40 Glass 0.1 0.2 0.3 oib/V 0.4 0.5 E1GURE 10.47. Propagation velocity V of the lowest order antisymmetric mode for the slot configuration (aluminum on T-40 glass). V's is the substrate shear velocity. (After Tiersten) and lower faces of a plate (part 3 of Section C). If" the guides are not too close logclher, coupling is weak and the modes on the individual guides are essentially unperturbed. Гп this situation, the two lowest order coupled wave solutions for the double guide system have field distributions that are closely described by even and odd excitation, respectively, of unperturbed modes on the two guides. Excitation of just one of the two guides is then described by an appropriate superposition of the even and odd coupled waves, as in 'Fig. 10.20. Since these coupled waves have slightly different propagation l.iclors (/3+ and p ), the initially excited fic d pattern cannot pers'st as it propagates. Energy transfers back and forth between the two guides with a spatial period defined by the heat wavelength (10.84) IP- - P-\ The coupling length, or distance required for complete transfer of energy from one guide to the other, is equal to one-half beat wave length. This coupling pi maple has been used as the basis for a number of electromagnetic waveguide devices (directional couplers, power dividers, etc.) and it also allows similar devices to be realized acoustically. In the weak coupl nc lim t comp cte energy transfer from one guide to the other is possible, but the coupling Iciiglh is excessively large. Increasing the coupling induces a larger difference between /?. and (i in (10.84), thereby reducing the coupling length, but it also
128 ACOUSTIC WAVEGUIDES means that the modes of the individual waveguides are no longer unperturbed (L. R. Adkins and A. J. Hughes, "Investigations of Surface Wave Directional Couplers," IEEE Trans. SU-19, 45 58 (1972)). H. ANISOTROPIC WAVEGUIDE Up to this point, only isotropic structures have been considered. These are adequate for some purposes, but high frequency waveguides require the use of single crystal materials in order to reduce attenuation. Because of the anisotropy that is introduced in this way, the simple potential equations (10.7) and (10.8) no longer apply and cither superposition of partial wares or the transverse resonance method must be used to analyze single crystal waveguides. The problems that can be treated are therefore, almost without exception, restricted to planar geometries. Since anisotropic plane wave solutions arc themselves usually obtained only by numerical computation, these problems are necessarily more complicated than isotropic problems. Solutions for some cases can be found analytically: but, more commonly, numerical computation must be used to find the partial waves and also to satisfy the boundary conditions. Both kinds of problem will be discussed in this section, which considers only nonpiczoelectric waveguides. Piezoelectric structures are treated in Section 1. H.l Free Anisotropic Plate Analysis of the free isotropic plate waveguide in Section С showed that the modal (iclds are polarized either parallel to the faces of the plate (SH modes) or normal to the faces of the plate (Lamb waves). In the anisotropic case this occurs only for certain crystal symmetries and plate orientations. EXAMPLE i. Consider a nonpiezoelectric hexagonal plate (crystal classes 6jm, 6lmmm) with the Z axis oriented at an angle у with the plate (Fig. 10.48). According to Appendix 3 in Volume I there exist pure shear partial waves which arc polarized normal to the YZ crystal plane and satisfy the dispersion relation (<W'4 + <~м*4 - p"j2) = 0. Transformation into wave vector components along the coordinate axes у and z gives к у = к у cos у — kz sin у к% — kv sin if + kz cos у and kl(ceccas2 у + e„ sin2 y) + &f(coe sin2 у + (ц cos2 у) kjtzipt* - f44) sin 2y - p,o- = 0. (10.85) у II. ANISOTROPIC WAVEGUIDES 129 у = 6/2 FIGURE 10.48. SH wave propagation on a nonpiczoelectric hexagonal plate with free boundaries. The slowness diagram for this problem (Fig. 10.49) shows that the incident partial wave and the reflected partial wave have different values of ka. Superposition of these waves gives a particle velocity field vx = (Ae + В ,км*)е ,Рг. (10.86) lo obtain the stress held relevant to the boundary conditions it is useful to transform the strain components 10.87) ITGURE 10.49. Slowness diagram corresponding to Figure 10.48.
130 ACOUSTIC WAVEGUIDES into the crystal axis system Y, 7.. This gives Sxy = Sxycos у — .S.j.sin v1 Szx = i'j.ysin у + S,x cos v, and the resulting stresses are T.xr = ceK(S„cos v - 52a.sin y) Tzx = c^tS^sin у + -S^cos y). Transformation back to the coordinate axes then gives (c • — с ) = (cee cos2 v- + сц sin2 vVbj, 1,6 44 sin 2y Л'-j.. (.10.88) The stress-free boundary conditions require Txy =0 at I/ = ±/>/2. (10.89) This leads, in general, to a complicated transcendental equation which must be solved simultaneously with (10.85), after setting k, = ft. The problem is simple only when у = 0 or ttJ2. For у = 0 A'lu = —klx and (10.89) requires that S„ = -ik^{Ae"<i," + ft>"V) = 0 at.'/ = ±bj2. These conditions are satisfied if A = —B and From (10.85), the dispersion relation is + См/З* - p«>a = 0: (10.901 and the dispersion curves have the same general form as for the isotropic plate (Fig. 10.7). I Л characteristic equation for Lamb wave solutions on the crystal plate shown in Fig. 10.48 can also be derived analytically. For crystal axes oil arbitrary orientation numerical methods are required. This is more typical I of anisotropic plate problems. 112 Ravleigh and Pseudosurface Waves on an Anisotropic Half Space The anisotropic waveguide problem which has received the most attention is Rayleigh wave propagation on an anisotropic substrate. For the isotropic^ H ANISOTROPIC WAVEGUIDES 131 problem, it was seen in part 3 of Section С that the Rayleigh surface waves are made up of two partial waves, both having real decay constants into the substrate. On anisotropic substrates these simple solutions occur only for certain special orientations and propagation directions. Generally, three partial waves are required, the decay constants arc complex, and the particle motion no longer lies in a plane normal to the surface. Solutions of this kind are sometimes called generalized Rayleigh wares. Another feature of anisotropic surface wave propagation is that a "leaky" surface wave sometimes exists for certain ranges of propagation directions. This wave contains a bulk type of partial wave, which radiates or leaks energy into the substrate. The velocity is greater than that of the slowest quasishear bulk wave traveling in the same direction. At certain critical propagation directions the leaky surface wave becomes totally trapped at the surface and is called a pseudosurface wave; at this point the Rayleigh wave reduces to a pure bulk shear wave which travels along the boundary and itself satisfies slrcss-frce boundary conditions. Pseudosurface wave solutions require both numerical computation of the partial waves and numerical solution of the boundary value equations. Figures 10.50 10.52 show some typical examples for cubic crystals. Figure 10.50 applies to the (001) plane of germanium. This is a crystal with anisotropy factor A < 1; that is, the fast shear wave S2 propagating in a cube face (Part B.2 of Appendix 3 in Volume I) is a pure mode with polarization normal to the face. Along the [100] and [110] directions the slow shear wave Si is a pure mode polariaed in the plane of the cube face. It therefore satisfies the stress-free boundary condition at points in the figure. The ordinary surface wave approaches the Sx wave as the [110] axis is approached and becomes identical with it when the axis is reached. At the same point the pseudo- surface wave becomes nonradiating, with two partial wave components and an elliptical particle motion at the surface. The same type of field distribution occurs at the point 0 = 0 for the ordinary surface wave (marked x in the figure). Figure 10.51 shows surface wave velocity characteristics for a free surface *long the (TTO) plane of KCI, which also has an anisotropy factor A < 1. In this case the pure shear wave S, has the fastest velocity along the [110] direction. Along this direction the pseudosurface wave reduces to a non- radiating solution with two partial wave components and elliptical particle motion at the surface (marked x in the figure). At points marked the slow shear wave Sx itself satisfies stress-free boundary conditions on the free surface. In Fig. 10.52 the free surface is along the (111) plane for silicon. This has ни anisotropy factor A > 1 but the shape of the curves docs not depend upon Ihis parameter. At an angle (J = 30= with the [П0] axis a different type of
132 ACOUSTIC WAVEGUIDES 3600 3500 3200 _ 3100 3000 2900 2800 2700 1 (001] I [0101 Propagation Fast bulk /Dure shear wave 1100) Slow bulk quasishear wave Pseudosurface wave 50" I1O0I 1110] FIGURE 10-50. Pseudosurface wave propagation in the (001) plane of germanium. (After Lim and Farnell) pseudosurface wave behavior occurs. The normal surface wave retains its character and, in fact, reduces to a solution with only two partial wave components (marked x). The pseudosurface wave, on the other hand, becomes a bulk wave which satisfies the free boundary conditions. This wave has its group velocity directed along the boundary but its propagation vector tilted into the substrate. The phase velocity marked by the triangle is computed along the free surface. That is, it is greater than the bulk wave to which it corresponds. This particular bulk wave velocity does not appear on the figure because it corresponds to a wave vector к at an angle to the surface, while the angle в refers to bulk waves with к lying in the free surface. H. ANISOTROPIC WAVEGUIDES 133 11.3 Anisotropic Plate on an Anisotropic Half Space Tor certain crystal symmetries and orientations the guided modes for this geometry separate into horizontally and vertically polarized families just its in the isotropic case, and the solutions can be described as Love waves, Utmb waves, Rayleigh wanes, and Stoneley waves. Under these conditions the Love wave solutions can be obtained analytically, but the other wave types nrc very complicated and generally require numerical analysis. Some problems of this kind arc treated in the references at the end of this chapter. Figure 10.53 shows the range of parameters for which Stoneley waves can exist at an interface between two cubic media. The curves are of the same general shape as for the isotropic case but, because of the anisotropy, they now depend upon the propagation angle 0. [Hoi1 '
134 ACOUSTIC WAVEGUIDES I. PIEZOELECTRIC WAVEGUIDES The addition of piezoelectricity complicates still further the already-difficult anisotropic waveguide problem. Numerical computation is usually required; but simple examples do exist where the solutions can be obtained analytically by applying cither the superposition of partial wares or the transverse resonance methods. Illustrations of both methods will be given. 1.1 Free Piezoelectric Plate As in the nonpiezoelectric case, the piezoelectric plate problem can be treated analytically only for plate symmetries and orientations where the mode* separate into horizontally and vertically polarized families. 0.0 1.0 2.0 3.0 4.0 0.0 1.0 2 0 3.0 4.0 40 3.0 - 1.0 1 0 Unprimcd Piimcd (Cv, 0.595 0.595 0.0 1.0 0.595 0.0 1.0 0.0 0.489 0.0 0.489 0.489 0.516 0.516 0.0 1.0 0516 0.0 1.0 0.0 0.295 0.0 0.295 0.295 1 1 e = 45' 1 i 1 3.0 I К it) КI 10.53. Range of existence of Stonclcy waves at an interface between cubic media. Stoneley waves exist in the shaded regions Both media have the Mm* crystal axis directions, with the Z (or [001]) axis normal to the interface. Wave propagation is at an angle 0 with respect to the X (or [100]) axis. Stiffness i onslaiit ratio c\i = 0»/<-ц. С[г = сУс'ц, etc. in с maintained loiislaul. Values used for calculating the curves are given in the (able.
136 ACOLSTIC WAVEGUIDES EXAMPLE 2. The hexagonal plate geometry shown in Fig. 10.54 will be used to demonstrate some general properties of piezoelectric plate waves. In contrast with Example I, the plate now belongs о the iezoelectr с crystal cla s 6mm and e Z crystal axis is oriented normal to the page. Electrical short-circuit boundary conditions are applied at the surfaces of the plate, which are assumed to be mechanically free, and the thickness of the plate is sufficiently small that the quasistatic approximation can be used. Partial wave solutions appropriate to this problem were obtained in Example 5 of Chapter 9. It was seen there that the problem is isotropic in the yz plane and that the partial fields polarized in the yz plane have no interaction with the electric field. Vertically polarized modes of this structure are therefore identical with the Lamb waves treated in part 2 о Section C. Horizontal particle motions on the other hand, do interact with the electric field, and two kinds of piezoclcctrically active partial] waves exist. One of these (with dispersion relation (9.93)) is simply a piezoelectrically stiffened shear wave. The other, which has no particle motion but does have horizontally polarized shear stresses 7"a.„and Txz, follows the dispersion relation (9.101) and has an evanescent (or exponential) field variation in the у direction. In part 2 of Section В this second kind of wave was called an electrostatic wave because it reduces to an electric potential satisfying Laplace's equation when the piezoelectric coupling goes to zero. For these horizontally polarized waves, the scattering coefficients at a stress-free shor -circuit boundary have been calculated in xa pie 5, Chapter 9. I i g these coefficients, the transverse resonance method is formulated just as it was for the Lamb wave problem (part 2 of Section C), and the solutions obtained are found to have striking similarities with Lamb waves. If the particle velocities of the incident (downward traveling) and reflected (upward traveling) stiffened acoustic waves arc proportional to /V-''k*T-f, Sse-'k»E'r (10.91) and the electric potentials of the electrostatic waves are proportional to Aee "<«'"r, В,,е-'кл-', (10.92) FIGURE 10 54 SIi wa>e propagation on a hexagonal (6mm) piezoelectric plate with short circuit electrical boundary conditions. I PIEZOELECTRIC WAVEGUIDES 137 symmetry requires that Вг = ±AS Й, = ±A„. The reflection coefficients at the lower boundary of the plate impose the further condition гг T ~ Аге1к,Л- 1" Г Using (9.106) and (9.109) and taking kte nd to be tan к , tanh Щ2 = for the symmetric solutions, and tan кф\2 tanh Щ2 ~ i/5, the characteristic equations are P ^44 e А" X kfs 4-s T for the antisymmetric solutions. Dispersion curves are obtained by solving these simultaneously with + k% = (<»lf'f where V1/2 (10.93) (10.94) (10.95) If44 + A A is the stiffened shear wave velocity. As suggested by the similarity of (10.93) and (10.94) with the Rayleigh-Lamb frequency relations (10.18) and (10.19) the curves have a coupled wave form (Figs. 10.55 and 10.56). In this case, however, the coupling is between stiffened acoustic waves and electrostatic waves. Strong inter- nclions therefore occur only when p is imaginary. Field distributions for the plate wave solutions governed by the dispersion Nations (10.93) and (10.94) in Example 2 have a stiffened shear wave part described by partial waves (10.91) and an electrostatic wave part described by partial waves (10.92). The electrostatic field always has an imaginary trans- veise wave vector component and is therefore bound to the surfaces, decaying exponentially into the center of the plate. The horizontal shear wave field, on the oilier hand, usually has a trigonometric transverse variation as in the isotropic case (part I, Section C). For large values of jib, however, Figs. 10.55 and 10.56 tihow that tuff} for the fundamental symmetric and antisymmetric solutions hppioaohes n value less than IAccording to (10.95) ktf then becomes imaginary and the horizontal shear wave part of the field is also bound to the
Imaginary Rea FIGURE 10.55. Dispersion curves for the symmetric SH modes on an X-cut hexagonal (6mm) plate, with propagation normal to the Z axis. (The dotted curves on the left-hand side of the figure represent uncoupled acoustic and electrostatic waves.) (After Blcnstcin) J vT' asymptote tor modes 3, 5, 7 etc. Z~i 1 _~ 10 9~ 1.—-''' 5 3 M Л л -; / i / ! Л A; J./ M ' i t i t '■ f\\ i i i i i i asymptote for mode 1 fib 10 98765432 10123456789 10 Imaginary Real FIGURE 10.56. Dispersion curves for the antisymmetric Sll modes on an Jf-cut hexagonal (6mm) plate, with propagation normal to theZ axis. (The dotted curves on the left-hand side of the figure represent uncoupled acoustic and electrostatic waves.) (After Bleustein). I. PIEZOELECTRIC WAVEGUIDES 139 / X AMPLE 3. A hexagonal (6mm) substrate has its surface oriented normal to ihe X crystal axis. Wave propagation along Y is assumed (Fig. 10.57). Surface scattering coefficients for horizontally polarized waves were calculated in Example 5 of Chapter 9 and, for the special case of an electrical short circuit at the boundary, have already been used in Example 2 of this chapter. If the incident shear and clcclro- siatic waves have amplitudes A and A,,, respectively, the scattered amplitudes arc B^ I ssA, -Ь E,F/tt, Be ~ i\,As + TeeAc (10.96) в', - r'PSAs + V'fcAr. I Wave solutions of the same general type arc found in nonpiczoclcctnc media when the Reformation potential is included in the analysis. Yu. V. Oulyaev, Appl. Phys. l-ett. 20, pp 2IS 217 (1972). I Cylindrical Oleuslein Culvacv waves are discussed in Reference H6 at the end of this I iMpicr. plate surfaces. This situation, where the two dominant modes become tightly bound to the surfaces and approach phase velocity degeneracy with each other, is trongly eminiscent of the behavior of Lamb waves on an isotropic plate; and the same argument can be used to show that these surface waves are valid solutions for the half space or infinite substrate problem. The following subsection will discuss these horizontally polarized surface waves in detail and compare them with Rayleigh waves. 1.2 Rayleigh and Bleustein-Guljaev Waves on a Piezoelectric Substrate In the isotropic case it was seen that there is an intimate relationship between the dominant Lamb wave solutions on a thick plate and Rayleigh waves on an infinite isotropic substrate (Part 3, Section C). The preceding subsection r s just shown that an almost identical situation occurs in the case of SH modes on a piezoelectric plate. Here the dominant SH mode solutions are ielated to a horizontally polarized surface wave (the Bleustein-Gulyaev wave) on an infinite piezoelectric substrate.! This relationship can be used to define lhe Bleustcin-Gulyacv wave as a limiting case of plate waves; but, just as in (he Rayleigh wave case, it is much simpler to apply the transverse resonance method directly to the finite substrate geometry. In the Rayleigh problem it was noted that SV and P waves incident on a free surface scatter into reflected waves of both types. At the surface of an infinite substrate there can be no incident waves but the reflected waves arc still present (Fig. 10.21c); this means that the scattering coefficients must go to infinity. In the Bleustein- Gulyaev case one is concerned with scattering of horizontally polarized waves at the surface of a piezoelectric substrate; and the existence of the wave again demands that the relevant scattering coefficients become inlin te 'I he analysis will be illustrated by a specific example. J
140 ACOUSTIC WAVEGUIDES Vacuum Hexagonal (6mm) medium (b) Va u Hexagonal (6mm) medium (c) ||{ГяА5 + ГиАе)Л1А=0 (rtsAs + 1 seAe)A л _q FFGURE 10.57. Transverse resonance analysis of Bleustcin- Gulyaev wave propagation. The z components of к for all of the incident and scattered waves are equal to /JjG, the Bleustein- Gulyaev wave propagation constant. Transverse resonance requires that these remain finite when As = Ae = 0 (Eig. 10.57c). Therefore, Г„, Vse, Г,„ 1» must all go to infinity. In other words the denominator Д in the reflection coefficient formulas must be zero. Consider the simplest case, in which an electrical short-circuit boundary condition is applied at у — 0. From (9.107) whcie (10.97) I. PIEZOELECTRIC WAVEGUIDES 141 cos 0, + i-^- sin 0S = 0. (10.98) Both reflected waves must now be evanescent, and Ic vy_v / 0C f.... It follows, then, that sm 0,. = cos 0„ = in (10.98), which becomes Bug. (10.99) After squaring and using the relation one obtains the dispersion relation (1/Ю2 (5bo = -f 71 <»2 (10.100) I -I Uleustein-Gulyaev wave propagation is very sensitive to electrical boundary conditions at the substrate surface. This can be demonstrated by applying free, rather than short circuit, electrical boundary conditions to the surface. In this case the relleetion coefficients are found by solving the boundary equation (9.103) in Ex- aii pie 5, Chapter 9. The denominator of the reflection coefficient formulas is equal id the determinant of the right-hand side of (9.103) Setting this equal to zero gives the dispersion relation aiv,r Phe 7 j ^7 \2Ш (10 101) rather than (10.100). For the hexagonal (6mm) substrate considered in Example 3, SV and P w.ives arc not piezoelectrically active and, furthermore, the problem is i iiinplclcly isotropic in the yz plane. Consequently, the scattering problem is the stiffened shear wave velocity. The transverse resonance condition is therefore
142 ACOUSTIC WAVEGUIDES 'Perfect conductor ^tangential= °) ' Perfect open circuit (^normal = °) Free space (b) Piezoelectric medium щ Surface wave propagation FIGURE 10.58. Commonly used electrical boundary conditions for piezoelectncally active Rayleigh surface waves. for SV and P waves is exactly the same as for the isotropic case, and the Rayleigh wave solution is the same as in Part 3 of Section C. In this example two surface waves exist—a Rayleigh wave polarized in a plane normal to thm surface and a Bleustein-Gulyaev wave polarized parallel to the surfacel For substrates with general crystal symmetry and orientation, two surface wave solutions may still exist under certain conditions, but the solutions are much more complicated.")" It was seen in Part 2 of Section H that Rayleigll waves on anisotropic, but nonpiezoelectric, substrates arc not always polarized in a plane normal to the surface and that pseudosurface waves occur for certain propagation directions. This behavior may also occur in the piezol electric case. On piezoelectric substrates, Rayleigh waves may be piczol electrically active and may, therefore, also be sensitive to the electrical boundary conditions (Fig. 10.58).* These problems must, in general, be solved numerically by superposition of partial fields. Four partial wave solutions are required—three correspondinl to stiffened acoustic waves and the fourth to the electrostatic wave. Numerous calculations for various substrate materials and orientations have Ьсеи performed, and a tabulation of surface wave velocities for various substratel t See. С С Tseng, "Piezoelectric Surface Waves in Cubic and Orlhorhombic Crystals."* Appl. Phys. Letr.\6, p. 253 (1970), and G. Kocrbcr and R. F. Vogel, "Generalised Bleul stein Modes." IEEE Trans. SU-19, 3 8 (1972). J Annular Rayleigh waves have also been siudied for certain kinds of piezoelectric sub-l stratcs (С. K. Day and G. G. Koerber, "Annular Piezoelectric Surface Waves," lEEETrtm.s. SU-19, 461^66 (1972). I. PirZOELECTRIC W WKGUIDrS 143 *nd electrical boundary conditions is given in Reference 72 and also in Appendix 4. Typical curves of surface wave velocity versus propagation direction arc shown in Figs. 10.59-10.61 for short circuit (h = 0 in Fig. 10.58a) and free (h = со) electrical boundary conditions. At V = 50° in the Л'-cut crystal, the change in velocity due to the short circuit at the surface is small. This suggests that the tangential electric field is weak. Figures 10.62 [and 10.63 confirm this conclusion and show that the electrical potential distribution is little changed by the short circuit. At 0 — 100" in the Л'-cut crystal the velocity change is large. This corresponds to a large tangential electric field at the boundary (Figs. 10.64 and 10.65). In selecting crystal orientations for interdigital surface wave transducers (Section L), curves of lhis kind have proved to be very useful. Velocity curves have also been calculated for open-circuit boundary conditions (Fig. 10.58b). As will be seen below, these two sets of curves can also be used to calculate coupling between piezoelectric surface vvaves on adjacent substrates. 38001 1 ■—I 1 1 33001. I I I 0 50 100 150 Direction of propagation (в) FIGURE 10.59. Surface wave propagation on Л'-cut lithium niobate, with the electrical boundary conditions of Fig. 10.58(a). (After Campbell and Jones)
144 ACOUSTIC WAVEGUIDES 3800 S 3500 3300 50 100 Direction of propagation (в) 50 FIGURE 10.60. Surface wave propagation on У-cut lithium niobate, with the electrical boundary conditions of Fig. 10.58(a). (After Campbell and Jones) 1.3 Coupled Waves on Adjacent Piezoelectric Substrates It was shown in Part 3 of Section С that mechanical coupling between Rayleigh waves on the upper and lower surfaces of a plate can be analyzed in terms of the symmetric and antisymmetric coupled modes of the structure, and coupling between microsound waveguides was considered from the same point of view in Section G The same appro ch an b u cd for analysis of electrical coupling between Rayleigh waves on adjacent and identical piezoelectric substrates (Fig. 10.66 on page 150). Since the symmetric mode has £ШГ1Ш] = 0 at the center line of the gap and the antisymmetric mode has ^t:m«entmi = 0, the symmetric velocity V++ and the antisymmetric velocity Vj__ can be calculated from boundary conditions in Figs. 10.66a and b, I PIEZOELECIRK WA\ TGUIOFS 145 'МММ Z cut 37001 1—1 !_J l M 1 i 1 i I i i i I i 0 20 40 60 80 100 120 140 160 180 Direct on of piopagation (в) FIGURE 10.61. Surface wave propagation on Z-cut lithium niobate, with the electrical boundary' conditions of Fig. 10.58(a). (After Campbell and Jones) respectively. The beat wavelength, from (10.38), is then (10.102) 1.4 Stoneley and Maerfeld-Tournois Waves It was noted in part 3 of Section H that waves supported by an anisotropic pbtc on an anisotropic half space can, for symmetrical crystal orientations, be identified with Love waves, Lamb waves, Rayleigh waves, and Stoneley waves. A similar comment can be made about the piezoelectric problem. In ihis case, however, one must include both Bleustein-Gulyaev waves and Rayleigh waves, and there exists another extra wave solution (the Maerfeld- Tournois ware) thai bears the same relation to the Stoneley wave as the
146 ACOUSTIC WAV EGLTD S -01X x 10"3 m/sec FIGURE 10.62. Electrical potential and particle displacement fields for surface wave propagation at в = 50° on unmetal ized Jf-cut lithium niobatc. (After Campbell and Jones) Bleustein-Gulyaev wave does to the Rayleigh wave. That is to say, the Maerfeld-Tournois wave is a horizontally polarized wave bound to the boundl ary between two piezoelectric solids. It may also exist at the boundarj between a piezoelectric solid and a nonpiezoelectric solid, and it reduces to the Bleustein-Gulyaev wave as a special case. EXAMPLE 4. Consider two hexagonal (6mm) solids that arc rigidly bondcJ together at an interface normal to the Л'crystal axis in both media. The Y crystal axes in the two media are parallel and propagation is along this direction, which is designated as coordinate avis % (Fig 10 67, page 150) If the plane wave scattcringl coefficicntsat the interface arc not alreadyknown, one must start from the beginning! and solve the boundary value problem by superposing partial wave solutions. In thd present problem, these partial waves have already been found in Example 5,j Chapter 9. For Maerfeld-Tournois waves one is interested only in the solutions withl I. PIEZOEf FCTR1C WAVEGUIDES 147 0.22 0.20 - 0.18 0.16 - 0.14 - oT^ 0.12 -- 0.10 - - 0.08 0.06 0.04 0.02 -wXx 10~Jm/sec FIGURE 10.63. Electrical potential and particle displacement fields for surface wave propagation at 0 50° on metallized Л'-cut lithium niobate. (After Campbell and Jones) particle velocity polarized parallel to the Z crystal axis (that is, the x coordinate axis in Fig. 10.67). In the example cited there was only one plane wave with v polarized along x; that is, the stiffened SH wave governed by the dispersion relation (4.93). To satisfy boundary conditions at the interface, the evanescent (or electio- itatic) wave (9.101) must also be included. If all partial wave fields are assumed to decay away from the interface, kys — ia,JS in the lower medium and k'vs = /Vs in the upper medium. The boundary conditions 1 * = li T T' ф = Ф' A, = К
148 ACOUSTIC WAVEGUIDES FIGURE 10.64. Electrical potential and particle displacement fields for surface wave propagation at 0 = 100" on unmetallized X-cul lithium niobate (After Campbell and Jones) at у = 0 give a set of four linear homogeneous equations in the four partial wave amplitudes, and a Maerfeld-Tournois wave solution exists only if the determinant of this set of equations is zero. This gives the dispersion relation i i (10.103) where unprimed quantities refer to the lower medium in the figure and primed quantities to the upper medium. When the upper medium is taken to be vacuum (lcf$ = (^-5)' = 0 and (e^- Y)' = «о) (10.103) reduces to (10.101) for a Bleustein-Gulyaev wave at a free electrical boundary. Bleustein-Gulyaev waves at a short circuit boundary (10.100) are obtained by taking (c£)' = (e Y6)' = 0 and (<|-^)' = «\ The dispersion relation (10.103) still applies when the upper medium in Fij_ 10 67 is rotated so that the У crystal axis lies in the —z direction. In this case (сД)' = c„, {е'ххУ = €xx< an^ (eXbY = —e vs, and the dispersion relation becomes identical with (10.100) for a Bleustein-Gulyaev wave at a short circuit boundary. I. PIEZOEIFCTRIC WAVEGUIDES 149 70 -oiX x 10_Jm/sec FIGURE 10.65. Electrical potential and particle displacement fields for surface wave propagation at 0 = 100 on metallized X-cut lithium niobate. (After Campbell and Jones) As in the Stoneley wave case (Part 3 of Section D) conditions for existence of a Maerfeld-Tournois wave at the interface between two solids are rather difficult to fulfill. If V's > Vs, (10.103) has a real solution for cu/A.ut °nly when < K- The exact condition for a solution is t 1-Ы+ЫЫ-ч\ * W (ШЛ04) where a V * *xx + «Д-А/ таг) 1 Reference 85 at the end of the chapter
Free _y^Enormal - 0 h space h (a) Symmetric coupled mode
152 ACOUSTIC WAVEGUIDES be assumed that the set of acoustic waveguide mode functions is complete for all practical purposes. Proof of orthogonality is, however, essential. This requires, first of all, a derivation of two general acoustic field theorems—the real reciprocity relation and the complex reciprocity relation. J.l Real Reciprocity Relation Both this and the complex reciprocity relation apply to the general piezoelectric case, governed by the electromagnetic and acoustic field equations listed on the front cover papers ав — V x E = — dt 3D V x II = — + Js at dp V-T = V--F dt and the constitutive relations D = er • E + d : T S = d • E + sB : T. These equations may be written completely and conveniently in "matrix" form as 0 V 0 0 0 0 0 0 0 -Vx 0 0 Vx 0 H E_ a dt 0 0 0 .E 0 s: 0 0 u.- 0 0 d : 0 H + —F 0 0 L J, (10.106) To derive the real reciprocity relation, one takes the fields to be time-i harmonic (v -•- v(.r, y, z)eiat, etc.); that is, d\dt /о in (10.106). As in all the previous discussion, the constitutive parameter^ p, sK, etc. are time-independent. Two field solutions are assumed: v,, T,, E1; H, driven by sources J„, and h\, and v2, T2, E2, 112 driven by sources Js2 and F2. Solution "1" is written into (10.106) and the "matrix" scalar J. RECIPROCITY RELATIONS AND MODE ORTHOGONALITY 153 T H, _0 -d: 0 .e-T Et_ + yt-F1 + Et-3a. (10.107) Subscripts 1 and 2 are then interchanged and the result is subtracted from (10.107). After invoking the identities V-(ExH) = HVxE-E-VxH V.(t.T) = ».(V-T) + T:V> from the back cover papers and noting that the matrix array of constitutive parameters in (10.107) is symmetric, one obtains the real reciprocity relation V" • Oi • T2 - v2 • Tf + Ex x H2 - E2 x H,) = v2 • F, - vt • F2 + E2 - Jsl - Et • Jrt. (10.108) hi the nonpiezoelectric case this separates into Lorentz's reciprocity relation! lor electromagnetic fields and Lamb's reciprocity relation* for acoustic fields. 1 Since the quasistatic approximation is applicable to most problems of practical interest, it is convenient to re-cxpress the cross-product terms of 110.108) in this form by using the identities V • (-V<1> x H) = H ■ V x ( — УФ) + \7Ф • V x H = ¥Ф • V х Н and V . ФО = ФУ • D + УФ • D. « ombining these terms with the electrical source terms then gives V • К • T2 - v, - T, + O/icoDJ - Os(/wD,)] = v2 • F, Vl • F2 + Ф^/юр.и) - 0,(iftj/t>„) (10.109) 1 II. A. Lorcntz, Amst. Akad. van Wetensch. 4, p. 176 (1895-96). (Ill amb, Proc. London Math. Soc. 19, p. 144 (1889). product is taken with the "row vector" [-v2 T, -H, E,], giving -t.-(V7-T1) + Ts:V.v1 + H,-V xEi + E.-V xH, [v2 T2 H2 Е2]Г-0 0 0 0
1S4 ACOUSTIC WAVEGUIDES where the electrical sources are now charges rather than currents. An integral version of this quasistatic reciprocity relation was first stated by Lewis.f J.2 Complex Reciprocity Relation In this case the solutions are allowed to be nonperiodic functions of time. That is, v,(*, ij, z, t) = 2 vi»0, У. s)e""''etc. where the frequency components <яр arc completely arbitrary. As a matter of notational convenience the derivative djdt will be retained, rather than introducing a factor ioiv for each term in the summation. Solution "1" is written into (10.106) and the "matrix" scalar product is taken with the complex conjugate of solution "2," giving v? • (V • Tj) + T* : V,vt — H* • V x E, + Ef - V x H\ '№ T* H* Ef] 0 0 : s 0 0 — v* • Ft + E* • J,,. (10.110) A second equation, obtained by complex conjugation and interchanging subscripts, is added to this; and the identities from the back cover papers are again applied, as in deriving (10.108). Tn this case the sum can be reduced to V • {-v? • Tj - V! • T* + Ef x EE + E x H? 0 0 \ 0 :d- d dt T, 0 Hi 0 • e -T_ -Et_ '[v? Tf H* Ef] dt 0 0 :s 0 0 :d 0 0 -p. 0 -d: 0 I, -Et_ + (v| • Ft + v, • F.f) - (E2* • J5l + E, • J*) (10.111) t Reference 30 in Chapter 11. I RECIPROCITY RELATIONS AND MODF ORTHOGONALITY 155 when the constitutive matrices are real and symmetric, so that - d* e„ = €* Pu * = Ри- (10.112) In a medium with elastic and dielectric losses, the constitutive matrices are bymmclric but no longer pure real, and the complex reciprocity relation does not apply.f In the quasistatic approximation (10.111) becomes [v2* T* -УФ.*] p о 0 : s :E 0 ■ d : 0 d T + (v2* • Fa + Vl • F*) + Ф* % + Ф, ^ dt dt (10.113) .1.3 Waveguide Mode Orthogonalitj To derive orthogonality relations for waveguide modes, all field quantities are assumed to vary as e"°', and djdt therefore becomes /<•>. The source terms in the quasistatic reciprocity relations (10.109) or (10 113) arc set equal to xero, F1 = F2 = 0 Рл = Pc2 = 0- t Note, however, that the real reciprocity relation is still valid for lossy media, Por fields varying as euot, lossless media that exhibit rotary activity or Faraday rotation have complex constitutive matrices that satisfy relations (10.112). In such cases the complex reciprocity ■ elation is still applicable, but the real reciprocity relation is not. (See, for example, A. Ci.Gurevich, Ferritesat Microwave Frequencies, pp. 128 132, Consultants Bureau, New York, 1963. Also Problem 21 at the end of this chapter.) It can also be shown that ilic reciprocity relations are not applicable to systems that contain both piezoelectric and pir/oniagneiic (or biased magnetostrictive) coupling media (see E. M. McMillan, "Violation ftf ilic Reciprocity Theorem in Linear Passive Electromechanical Systems," J. Acous. Soc. timer. 18, 344 347 (1946)).
156 ACOUSTIC WAVEGUIDES Solutions "1" and "2" are then taken to be free modes with propagation factors @m and /9И respectively, (10.114) Vl = e iPmZvJx, y), etc, v, = e~ip"*vn(x, y), etc. Different orthogonality relations, applicable to different kinds of problems, can be obtained from the two different reciprocity relations. However, only those obtained from the complex reciprocity relation (10.113) will be considered here. They apply, therefore, only to lossless waveguides. Orthogonality derivations may be carried out for waveguide structures of arbitrary shape (Problem 25), but it has been seen in the preceding sections that waveguides of practical interest arc usually geometrically simple. For this reason, and to display the analytical details in the simplest and clearest manner, proof of mode orthogonality will be given only for layered waveguide structures (Fig. 10.68), where the material media may have arbitrary aniso- tropy and inhomogeneity, provided the properties do not vary along the Electrical boundary Me amcal boundaries Electrical boundary FIGURE 10.68. General layered waveguide structure. The waveguide cross section is infinite in the a direction and the fields are uniform along .i. .1. RECIPROCITY RELATIONS AND MODE ORTHOGONALITY 157 cross sccfri on (10.119) In (10.119) the integral is performed over the entire waveguide cross section, from i/ = —Л to у — b + h, but the acoustic terms contribute only over the range у — 0, b. If the acoustic boundary conditions in (10.118) arc cither stress-free or iij'iil, T • у = 0 or v = 0, at у - 0,b, iiiul I lie electrical boundary conditions are either short-circuit or open i in.nit, Ф = 0 or D • у = 0 at у = —A, h + h, * coordinate. For such structures the modal field distributions are independent of ж, and (10.114) is т, = «*ЧМ etc. <,0-,,5) Under these conditions the complex reciprocity relation (10.113) reduces to V{ }=f-{ + }-y = 0, (10.116) oz ay with { } = {-v* • T, - v, • TJ + ФЯнвОО + Ф^/шЦ,)*}. After substitution of (10.115). this becomes <</',„ - P*n){~< • t,„ - vm • t* + 4>t(i<ol>m) + Ф„(1о>В„)*} - = I" {-▼: ■ t„ - v,„ - t* + OK'«DJ + QJiitoDJ*} ■ ye рп*,г, ду (10.117) winch is then integrated with respect to у across the waveguide. The term on I he right-hand side leaves simply the values of { } at the transverse boundaries of the waveguide; and, since the acoustic medium in Fig. 10.68 occupies only part of the waveguide cross section, (10.117) is converted to hp* - nwmn = {-< • тт - v„,. t:j . я-* + + Фт(шВ„)*} . у] -ь~," (10.118) where К* = l~ j {-< ■ Tm - vm • t* + <K(io,DJ + Фт(иоОп)*} - z dy.
156 ACOUSTIC WAVEGUIDES Solutions "1" and "2 are then taken to be free modes with propagation factors /?„, and Д„ respectively, (10.114) Vl = e iPmZvJx, y), etc. v2 = е**~*чн(х, y), etc. Different orthogonality relations, applicable to different kinds of problems, can be obtained from the two different reciprocity relations. However, only those obtained from the complex reciprocity relation (10.113) will be considered here. They apply, therefore, only to lossless waveguides. Orthogonality derivations may be carried out for waveguide structures of arbitrary shape (Problem 25), but it has been seen in the preceding sections that waveguides of practical interest arc usually geometrically simple. For this reason, and to display the analytical details in the simplest and clearest manner, proof of mode orthogonality will be given only for layered waveguide structures (Fig. 10.68), where the material media may have arbitrary aniso- tropy and inhomogeneity, provided the properties do not vary along the Electrical boundary Me arucal boundaries Electrical boundary" FIGURE 10.68. General layered waveguide structure. The waveguide cross section is infinite in the a direction and the fields are uniform along i. .1. RECIPROCITY RELATIONS AND MODE ORTHOGONALITY 157 cross (10.119) lii (10.119) the integral is performed over the entire waveguide cross section, from у = —h to у = b + h, but the acoustic terms contribute only over the Oinge у = 0, h. If the acoustic boundary conditions in (10.118) arc cither stress-free or 11| (I, T • у = 0 or v = 0, at у - 0, b, ninl (he electrical boundary conditions are either short-circuit or open < limit, Ф = 0 or D • у = 0 at у = —A, h + h, .r coordinate. For such structures the modal field distributions are independent оГ л:, and (10.114) is v, = e->p-\n(y), etc. v2 = e-f»"*vn(y), etc- (,°-,15) Under these conditions the complex reciprocity relation (10.113) reduces to V'{ }=f-{ + }-y = 0, (10.116) os ay with { } = {-v* • Tj — vt • T* + ФК/coD,) + Ф^йоВД*}. Alter substitution of (10.115). this becomes V.tL - P*n){~< • T,„ - vm - T* + Ф*п(шВт) + Фт(коОа)*} • te-^"'" ду (10.117) which is then integrated with respect to у across the waveguide. The term on I lie right-hand side leaves simply the values of { } at the transverse boundaries of the waveguide; and, since the acoustic medium in Fig. 10.68 occupies only part of the waveguide cross section, (10.117) is converted to HP* ~ №ПРтя = {-v* • Tm - v,„ • T*} • y]-J + {Ф*п(шВт) + Фт(|«.Би)*} • y]Z%" (10.118) where f'mn = l~ j {-< ■ Tm - vm • T* + Ф*п0о,От) + Фт(шОУ} - z dy.
158 ACOUSTIC WAV ECUIDES the right-hand side of (10.118) is zcro.t Consequently, >W,„ ~ />t)P«,„ = 0: and an orthogonality relation for the waveguide modes is Pm„ = 0 pm * p*r (10.120) In the first part of this chapter it was seen that waves on isotropic waveguides always occur in pairs having equal and opposite propagation factors [I. Figure 10.14 illustrated this property for some of the symmetric Lamb modes. It will be seen in Section M that this kind of mode pairing also occurs in the most general piezoelectric waveguides. Mode labeling is arranged so that one member of each mode pair carries energy or decays exponentially in the +z direction, while the other carries energy or decays in the opposite direction. This is indicated by allowing the mode subscripts m and n to assume positive and negative values. That is, m=±M (10.121) n — ±Л', where M, N arc positive integers and P 1/ = -Pm (10.122) Р л = ~Py Positive subscripts (M and Л) refer to waves carrying energy or decaying in the +: direction, while negative subscripts (— M and —N) refer to energy How or field decay in the opposite direction. The mode indexing used in Fig. 10.14 can now be explained. Note that the curve from A' to В is labeled with a different index than the curve to the right of B. At any given frequency ©j it is seen that p_3 ^ p* in these two regions. The corresponding waves arc therefore orthogonal, according to (10.120), and must be indexed differently. Although p and the phase velocity K„ = «>//? arc positive in region A'B, the group velocity Vg = doldp (Section O) is negative. This is called a backward wave mode and, following the convention of assigning the sign of the subscript on the basis of power How direction, is labeled with a negative subscript. For the nonpropagating regions (complex or imaginary p) the subscript sign is determined by the direction of energy decay. In this way one arrives at a set of continuous curves, each labeled with a single mode index as shown in Fig. 10.14. For propagating modes, p„, and p„ are real. It follows, then, from (10.120) that F„„, - 0, /71 Ф и (pm, p„ real). (10.123) t "this is irtie, in general, for any kind of loss-less boundary conditions (Problem 20). I RECIPROCITY RELATIONS AND MODE ORTHOGONALITY 159 ! -v*m • I „ - т. ■ T* + Ф1 m(m,Om) + ФИ(|ИВ „,)*} • z dy (10 128) Nonpropagating (or cutoff) modes have pTs that are either pure imaginary or complex, and a distinction must be made between the two cases. When P,n, Pn are pure imaginary Pm = — Pm = P m Pt " ~Pn = P-n> from (10.122), and the orthogonality relation becomes P„,„ =0 m Ф —n (/?„„ pn pure imaginary). (10.124) I f pm and pn are complex Pm = #,„■. m' ^ ±m P* = p„; П' ф ±П and the orthogonality relation (10.120) becomes Pmn = 0 m ^ n' (pm, pn complex) (10.125) where n is neither +m nor —m. All three cases can be illustrated by the L3 dispersion curve in Fig. 10.14. Above A, and from A' to B', p3 is pure real and (10.123) applies. Between A' and A, p{, is pure imaginary and P* — —p-i = Р-ъ о (10.124) applies. Below B' Pt = P-* and (10.125) apphes. It is seen from the remarks above that there is always some value of л for which Pmn is nonzero. This has an important physical interpretation. For propagating modes P„„„ - \ J {-v* • T,„ + Фш0о>Ом)*} -idy (10.126) crnss section is nonzero (from (10.123)) and I.D on the front cover papers identifies this as the average power (low in the +z direction, per unit waveguide width nlong .v. According to this identification and the convention for positive and negative mode subscripts, p *t. м = Flfu (propagating modes). (10.127) I or nonpropagating modes the interpretation is simplest when pm is pure imaginary. In this case
160 ACOUSTIC WAVEGUIDES is nonzero (from (10.124)) and P-m.m = P*, m (Sm pure imaginary). (10.129) By definition, a nonpropagating (or cutoff) mode cannot by itself transport energy along a waveguide (Fig. 10.69a). If, however, a lossy load is placed on the waveguide at a finite distance from a source exciting the cutoff mode (Fig. 10.69b) the average power absorbed by the termination must be transmitted in some manner from the source to the load. For cutoff modes with imaginary B's this average power flow is carried by the cross-product terms between the fields of the positive-decaying mode M and the reflected mode — M. The average power flow is calculated by substituting v = Ал1уЛ1{у)е-*»> + A_rf M{y)e-if-MZ and corresponding expressions for T, Ф, D into the real part of the complex power formula. Using (10.124) and (10.128), this is then reduced to (Payg) z = 2<%* (АмА1ыРм_ m\ (10.130) which gives a physical interpretation of the integral in (10.128). Power transfer by cutoff waveguide modes is seen from (10.130) to be critically dependent on the relative phase angle of the reflected wave. Accordingly, the existence of a reflected wave does not always imply an average power flow.l For nonpropagating modes with complex values of/9, the integrals P В —в* have a similar interpretation. In this case power is transported by pairs of modes with the indices m and n' in (10.125). Source Source Load (b) FIGURE 10.69. Mechanism of power transfer in a cutoff waveguide, (a) Infinite waveguide;—no power transfer, (b) Waveguide terminated with an absorbing load—power transferred through interaction of the incident and reflected waves. K. EXCITATION OF WAVEGUIDE MODES 161 K. EXCITATION OF WAVEGUIDE MODES One of the most important applications of modal analysis is for analyzing waveguide excitation by sources placed either on the surface of an acoustic waveguide or within its volume. The principles of these calculations will be developed in this section and then used in the following section to evaluate the input electrical immittance of a waveguide transducer. The starting point of the derivation is, again, the complex reciprocity relation (10.113). As in the orthogonality derivation, all field quantities are assumed to vary as еш' and only layered geometries (Fig. 10.68) are considered. In this case, however, it is necessary to retain the source terms F1; F2 and prl, pe2, and (10.116) is replaced by T{ }-* + ^{ }-y = v*'l'\ + »i-FJ+**('«>P,i'l+*i('«'Prt)*. oz oy (10.131) where { } is defined as before and the right-hand side of the equation is a function of у and z only. The waveguide may be excited by volume sources on the right-hand side of (10.131), by traction force sources T • у and velocity sources v at the acoustic boundaries in Fig. 10.68, and by potential sources Ф and surface charge sources D ■ у at the electrical boundaries. It is assumed that the excited field in the waveguide can be represented by the mode expansion v{y, z) = 2 ajz)vm(y) ■m T(.y, г) • z = J fi„,(2)Tm(!/) • 2 Ф(.У, г) - 2 ат(г)Фт(у) m m These arc the only field components appearing in the orthogonality relation (10.120). All other components can be evaluated from these, since Vsv S = — Uo E = -УФ, mid the stress and electric displacement fields are given by
162 ACOUSTIC WAVFGLIDES L. INPU'l 1MMITTANCF Ob WAVEGUIDE TRANSDUCERS 163 L. INPUT IMMITTANCL OF WAVEGUIDE TRANSDUCERS Section К gave a method for finding the waveguide fields excited by arbitrary distributions of mechanical and electrical sources. This theory provides the basis for dealing with a wide variety of waveguide transducer problems. Analysis of the thin disk piezoelectric transducer in Chapter 7 of Volume I showed that it is not sufficient to calculate the amplitude of the radiated acoustic wave in terms of the electric current (or voltage) applied at the input terminals of a transducer. The reason is that the acoustic fields react back on the electrical source and, in this way, affect the level of excitation. This back reaction always occurs when the electrical source has a finite electrical immittance. The degree of back reaction on, or loading of, the source depends on the relationship between the source immittance and the input immittance of the transducer. Calculation of the input immittance is therefore of crucial importance in transducer analysis. These considerations applied equally well to piezoelectric waveguide transducers and also, with mechanical immittance substituted for electrical immittance, to mechanical waveguide transducers. Most experimental investigations of waveguide transducers have been concerned with Raylcigh surface waves, and an enormous variety of different excitation techniques have been explored. Successfully operated Rayleigh and inicrosound waveguide transducers of different kinds are shown in Fig. 10.70. At the present time, the interdigital electrode array (Part VII in the figure) is the most widely used transducer and is chosen in the following example as an illustration of modal analysis.! motion i FIGURE 10.70. Types of Rayleigh wave and microsound waveguide transducers that have been successfully operated experimentally. (I-IX After Reference 94, X After Reference 96, XI After Reference 105, XII After Reference 100, XIII After Reference 104, XIV After Reference 51, XV After Reference 109.) I. Transduction by dm en point or line on surface. Pickup or drive transducer used in exploratory studies. Alternatively metal stylus might contact bare surface and have transducing crystal at its upper end. \ kaylcigli wave excitation problems are also analyzed by means of radiation theory in an minute half space. see, for example. reference 15 at the end of the chapter. To evaluate the mode amplitudes am(z) in (10.132), solutions "1" and "2" in (10.131) are taken to be П = v(?/, z), etc., from (10.132), and v2 = e^"zyn(y), etc., with F2 = 0, pe2 = 0. Integration across the waveguide then gives f" 1 4am{z)Pmne*»* + {-v„* . T - v . T*K*S* • y|-0l oz m + (OfibD) + Ф(/»В.)*}е«««. гь гч-ь+h = чу"* • F{y, z)dy+\ <bt<**\w>p.{y, г)) dy, (10.133) jo -й wherePmn has been defined in (10.119) and F(y, z), pe(y, s) are the prescribed acoustic and electric volume source distributions. According to the orthogonality relation (10.120) the summation on the left-hand side of (10.133) has only a single nonzero term. If я is a propagating mode (/9„ real), one has 4РИП (J; + a J?) - fJLz) + fvn(z), (10.134) where .Ш = К(У) ■ Ш z) + y(y, z). T*(?/)} • y|U - {0*(j,)(JcoD(y,s)) + 0(y,g)(io>Dn)*} - у|ГТ (10-135) is the forcing function due to surface sources and гь [y=b+h f,M = <{У) • Ш *) + Ф*„(у)(коРе(У, z)) dy (10.136) jo jv—h is the forcing function due to volume sources. Tf the waveguide modes have stress-free (Tn • у = 0) and short-circuit boundary conditions (Ф„ = 0), the surface excitation function (10.135) shows that excitation is completely prescribed by the applied traction forces and potentials at the surface. When the modal boundary conditions are changed, corresponding changes must be made in the surface excitation specifications. The mode amplitude equations (10.134) are seen to be exactly analogous to the normal mode equations introduced in Chapter 6 of Volume T and can he solved hy using the techniques illustrated by Examples 3 and 4 in Chapter 6 and Examples 12 and 13 in Chapter 8. In the next section these methods will be applied to waveguide transducer problems.
Compressional wave transducer Shear wave transducer \ Nonpiezoelectric substrate Thin piezoelectric film Piezoelectric block Thin liquid film \^Nonpiezoelectric substrate Array of -interdigital electrodes Nonpiezoelectric \ Array of substrate N» interdigital electrodes (c) Bulk wave transducer III Comb structure FIG U RE 10.70 (Continued). П Piezoelcctricallу driven surface-wavetrans- ducers. (a) Bulk wave transducer crystals driving surface, (b) Thin piezoelectric film on surface. Interdigital electrode array beneath or on top of piezoelectric film, or single-phase electrode array and ground plane on opposite side of film may be used [101, 102, 107]. (c) Separate piezoelectric block set on electrode array deposited on substrate with thin liquid film. III. Wedge (left) and comb (right) transducers. Angle of wedge is set so wavelength of bulk wave measured along contact surface is approximately equal to surface wavelength A. Absorber attenuates reflected bulk waves. Wedge transducers can be designed to have high conversion efficiency [97, 98]. Suface of liquid ^Transmitting transducer (a) Incident bulk wave (a) Specimen >~ Support Transm tting transducer Receiving transducer IV Guiding stripe Receiving transducer Solid Liquid Asymmetrical grooves Waveguide ««4 Conducting stripes Incident bulk wave Surface wave Piezoelectric l«0 FIGURE 10.70 (Continued}. IV. Transduction at liquid-solid boundary, (a) Scheme used to measure surface wave velocity. Pressure wave from drive transducer is less strongly reflected at some angle of incidence 0, given by I) = sin 1 (^liquid/t'amfscf) at which the incident pressure wave is converted to a surface wave, (b) Scheme used to measure attenuation of surface waves along a guiding stripe. Point of measurement varies as depth of immersion of solid changes. V. Transduction by bulk-surface mode conversion, (a) Conversion from incident bulk wave to surface waves at symmetrical grooves cut into surface. Groove period equals surface wavelength and wavelength of bulk wave projected onto surface plane, (b) Conversion at asymmetrical grooves from incident surface wave to downward propagating bulk wave and then to surface wave propagating on lower surface of plate. The asymmetrical cut causes waves to emerge in preferred directions shown, (c) Conversion from bulk longitudinal to surface wave on surface waveguide, (d) Mode conversion at conducting stripe1, on piezoelectric substrate. 165
Electrode array i~J>' Piezoelectric plate n la e Piezoelectric crystal (a) (b) X/4 Signal generator -i -i Matching network w VII FIGURE 10.70 (Continued). VI. Single-phase electrode array transducers, (a) Basic array structure showing array of constant pitch and ground plane on piezoelectric plate, (b) Approximate distribution of electric fields in plate at one instant of time. VII. Intcrdigital electrode array transducers, (a) Arrav of constant period on piezoelectric crystal, (b) Sketch showing approximate distribution of electric fields in crystal at one instant of time, (c) Unidirectional transducer. The passive array on the left acts as a reflector. Movable Absorber v Probe Signal generator Piezoelectric Detector Nonpiezoelectric plate supporting an. / Г _ electrode array \ // I—— \ / i A Signal generator Absorber Pi I block with electrode array Detector Piezoelectric (b) VIII Piezoelectric Negative electrode г -i Positive e ectrode FIGURE 10.70 (Continued). VIII. Coupling to piezoelectric fields at surface, (a) Probing with small flexible tip attached to coaxial cable. Absorber attenuates wave launched to left, (b) Coupling to electrode array on nonpiezoelectric plate and to a second piezoelectric substrate, both separated from Ihe bottom piezoelectric. IX. Piezoelectric solid driven by moving domain of high electric field in Gunn effect oscillator. X. Intcrdigital grating array [96]. 167
Acoustic propagation Matching network -60° Phase shifter XII FIGURE 10.70 (Continued). XI. Unidirectional operation with use of a grounded serpentine electrode [105, 108]. XII. Multiphase electrode array used for unidirectional transduction [100]. FIGURE 10.70 (Continued). XIII Multichannel surface wave directional couplers have been constructed by depositing unconnected metal fingers on the substrate surface [50]. Operation is based on the beat wavelength concept discussed in part (3) of Section С and part (3) of Section I. One-way transduction has been realized by bending a directional coupler of this kind around an ordinary intcrdigital transducer [104]. XIV Excitation of the lowest order antisymmetric mode on a ridge waveguide by means of (a) direct piezoelectric conversion and (b) bulk wave conversion [51]. XV, Wedge excitation of stripe waveguide, l'his is a two-dimensional analogue of the wedge transducer in II and is capable of equally high conversion efficiency [109].
1 о ACOUSTIC WAVEGUIDES EXAMPLE 5. Input Admittance of the Interdigital Transducer. An array of uniformly spaced metallic fingers is deposited on the surface of the piezoelectric substrate, and voltage excitation of opposite polarity is applied to alternate fingers (Fig. 10.71). The electric fields applied to the substrate in this manner produce a spatially periodic distribution of piezoelectric stress and, if the spacing of the fingers is chosen to conform to the wavelength of the Rayleigh wave, this stress generates constructive Rayleigh wave radiation in both forward and backward directions. In applying the waveguide excitation formalism summarized by (10.134) to (10.136), it is convenient to assume that the waveguide boundaries are at the upper and lower surfaces of the substrate. That is, h = 0 in Fig. 10.68. The Rayleigh waves, of course, satisfy stress-free mechanical boundary conditions at the upper surface у = 0 and the lower surface у = h can be ignored in the usual experimental arrangement, where the substrate is many wavelengths thick. Furthermore, it is useful to assume that the electrical boundary conditions at у = 0 arc open-circuit.t This condition will be indicated by a superscript <x> appended to the Rayleigh wave fields; that is, D»+(0)=D^(0)=0 where the + and — subscripts designate Rayleigh waves traveling in the +z and —z directions. Since the interdigital transducer applies only an electrical excitation at the substrate surface, the volume source term in (10.134) is zero and the surface source term has only an electrical part. According to (10.135) this is -0&<P)C'»D(P,s))-? for the positive-traveling Rayleigh wave and -Ф£!<Р)(й!Д><0, '))•* for the negative-traveling Rayleigh wave. If Pnn for the positive Rayleigh wave is set equal to Pu, the Rayleigh wave power per unit width along x, then Pnn is —PK for the negative Rayleigh wave. Similarly, /i„ is PR for the positive wave and — £H for the negative wave. The mode amplitude equation (10.134) therefore becomes {ct + /ftl)"Rl-(Z) = ~ 47^ (Фй+(°)0'ю1>(0, z))\ . у (10.137) «,< (*) = ~ {«>if (0)(/wD(0, *))} • У (Ю.138) Is"*) ф This commonly used approximation in piezoelectric boundary value problems is justifiable for materials such as lithium niobate that have a large dielectric permittivity (see part 1 of Section В in Chapter 12). L. INPUT 1MMIT1ANCE OF WAVEGUIDE TRANSDUCERS 171 for positive and negative traveling waves respectively. If the ends of the transducer structure are z = ±/, the positive wave is zero at the end z = —I and the negative wave is zero at the end z = +/. Direct integration of the mode amplitude equations, as in Chapter 6 of Volume I, therefore gives «R+(z) = e-iptlz г l { }E+ • у dC, z> +/, (10.139) (10.140) where the bracketed quantities { }n and{ }n arcdefinedby(10.137)and(10.138). The Rayleigh wave amplitudes are thus 1Ф,°° I I C+l Ы = 4^fjJ , e'H" V'oDJO, Q) dt 4Г,х J е-*^ (iwDJO, Q)di (10.141) (10.142) Before evaluating (10.141) and (10.142) it is necessary to relate the quantities under the integrals to the electrical terminal quantities I and V in Fig. 10.71. The I 1С. U RE 10.71. Interdigital Rayleigh wave transducer.
О 0.2 0.4 0.6 0.8 1.0 w/L (b) FIGURE 10.72. (a) Fourier amplitude coefficients in (10.143) and (b) Capacitance Cs per ringer pair in (10.144) as a function of finger width to spacing ratio w\L in Figure 10.71. (After Engan) L. INPUT IMMITTANCE OF WAVEGUIDE TRANSDUCERS 173 acoustic response of the substrate is neglected when calculating the relationship of A/(0» z) to the applied voltage V. This is called the weak-coupling approximation, and best agreement with experiments is obta ned by using the zero-stress permittiv- lt eT for this electrostatic field problem For an interdigital array of infante length, Dy has been found to have the Fourier series representation Dy(0,z) = J e^/l^sinO + I)-г, (10.143) 71—0 ^ where irVPn(2k* - 1) A; cos (1 - н./£)], the P„'s are Legendre polynomials of the first kind, and К is the complete elliptic integral of the first kind. Amplitudes of the first five Fourier amplitudes are given in Fig 10.72a. The same calculation also provides another quantity needed in evaluating the transducer admittance; that is, the capacitance per unit width along x for a single pair of transducer fingers (Fig. 10.72b), K(k) c' = <- + '-,^ra- (,0J441 If end effects are neglected, (10.143) may be applied to transducers offinite length. In this case (10.141) and (10.142) involve only elementary integrals of the kind valuated in Example 4 of Chapter 6 in Volun e I, ё~г^ sin (In + 1) -f dl = -ite'V*1 I sin I /SR + y-^-V - e-i(2«+i>(rj/i) \ i !_ I. (Ю.145) Near the fundamental resonance of the transducer, where L яа Au/2, ill Fig. 10.71 and
1 4 ACOUSTIC WAVEGUIDES only the second term in (10.145), with n =0, makes a significant contribution. The positive-wave amplitude (10.141) is therefore K+| = \AX\ h»4 sin 2; - (10.146) where Л =7 V ' L K(s 1 - k2) ' from (10.143). If /V is the number of finger pairs in the transducer, / ЛХ. Also «r( to - (on) plxL — тт = ■ , <"o where <o„ is the fundamental resonant frequency. When these substitutions are made in (10.146) /W(cu — <o0) sin —— ш0 *K,-l*-0 K(yJ\ - k2) Nir((v — «)„) to0 4Pn (10.147) Since |Ф£_|„ о in (10.142) is always equal to |Ф£,|„.-0,Т the negative-traveling wave amplitude is l«n-l = l«n+l- A simple power balance calculation may now be used to find the input conductance of the transducer. The average power supplied by the source in Fig. 10.71 is *ау=К7«1И'. (,0J48) where Ga is the input conductance of the transducer per unit width along x. Since the transducer itself is lossless, this must equal the radiated acoustic power per unit width, PAV = ( l«R+l2 + l«R-IVn = 2 |eR. \2PR- (10 149) If (10.148) and (10.149) are equated and the mode amplitudes are substituted from the preceding paragraph, the input conductance per unit width along x is found, finally, to be Nir(m — wn) Ca = Go(«'0) sin Nir'tn — o>0) (10.150) t See Part 1 of Section M. L. INPUT IMM1TTANCE OF WAVEGUIDE TRANSDUCERS 175 with n I \ IK) 2У>/ эт \1^)K+1«0 4A. This is the "radiation" conductance of the transducer and represents the radiated power for unit applied voltage. Its form is characteristic of end-fire antenna performance, with a mid band conductance proportional to /V2 and a frequency bandwidth proportional to 1//V (Fig. 10.73a). The measured input conductance for an actual transducer is shown for comparison in Fig. 10.73b. In the weak-coupling approximation, the potential | Ф,™ |„ 0 in (10.147) can be related to the free boundary potential Фк \y 0 used in the perturbation calculations of part 1 in Section 12.B. From (12.36) T o=f4^4<j>R.,u). FIGURE 10.73. Frequency response of the input admittance for an intcrdigital Rayleigh wave transducer. (Vfter W. R. Smith, ct. al.)
176 ACOUSTIC WAVEGUIDES If this is substituted into (10.150). the midband conductance is / * \2\Ayn (10.151) where №1 (from the Ingebrigtsen perturbation formula (12.38)) is a commonly accepted measure of piezoelectric surface wave coupling. Usually, the midband conductance is expressed in terms of the total static capacitance of the transducer, Cr = NCS, (»0.152) where Cs is given by (10.144). In this case, К(к)К(\/ I - A:2) (10.153) For a transducer with equal finger widths and spacings A: = I/л/2, from (10.143), and K(k)=K(\ll — кг) = 1.854 in (10.153). The midband conductance is then G„(4) = 2.87(onGrV — (10.154) For comparison, a quasiempirical treatment of the same problem, in which the surface wave transducer is modeled by a series of bulk wave transducers,! gives C>;n) = 2.55F4CT/V (10.155) where Fis an empirically determined factor. Experimentally, F has been found to be in the range I ± 0.2 for lithium niobate substrates of various orientations. When ш ^ w0, the transducer input admittance contains a radiation susccptancc term Bjco) in addition to the radiation conductance Ga(o>) and the static susceplance wCT. This additional term cannot be evaluated easily by the method described here but the frequency dependence of Ba(o>) may, however, be obtained from Smith's bulk wave model, and it is found to agree closely with the experimental results (Fig 10 73h) An alternative approach is to use Kino's variational formula% for transducer admittance t Reference 103 at the end of the chapter. X Reference 95 at the end of the chapter. M. TRANSMISSION LINE MODEL 177 (Section D of Chapter 13), ico f Ф(0, z)D(0, z)-ydz Y = -±i . (10.156) This method, which is not limited to the weak-coupling approximation, reproduces (10.153) and also gives an explicit formula for Ва(м). A practical disadvantage of the intcrdigital transducer analyzed in Example 5 is that it radiates equally in both directions. This means that 50% of the acoustic power is ost (see part Vll(a) of Fig. 10.70). Not all Rayleigh wave transducers have this disadvantage. The wedge transducer, for example, in part 111 of Fig. 10.70 radiates only to the right and the grating in part V(b) also has some directivity. Parts VII (с), XI, XII, and XIII of the figure show modified interdigital structures that radiate acoustic waves only to the right. The transducers shown in VII(c) and XI significantly reduce the transducer bandwidth, and require either additional electrical components or increased precision in the fabrication techniques. In XII the bandwidth is not reduced, but additional fabrication techniques are required. The structure shown in XIII, by contrast, neither reduces the bandwidth nor requires special techniques and extra components. M. TRANSMISSION LINE MODEL FOR ACOUSTIC WAVEGUIDES The strong similarity of the mode amplitude equations (10.134) to the normal mode formulation of electrical transmission line equations has already been noted at the end of Section K. Using this analogy, it is possible to define equivalent transmission line currents and impedances for the waveguide modes. This is not a particularly advantageous procedure for waveguide excitation problems, where the mode amplitude formulation is easier to use, but it will be seen in the next section that transmission line equivalent circuits arc especially useful for scattering problems. M.I Positive- and Negative-Traveling Guided Waves In Chapter 6 of Volume I it was seen that there is an intimate relationship between the positive- and negative-traveling wave amplitudes (or normal modes) and the voltage and current in an electrical transmission line. This lclationship provides the basis for transmission line modeling of acoustic waveguides. In general, a transmission line model can be constructed only when waveguide modes can be grouped in pairs (labeled M and M, or N mid —N, in (10.121)) with equal and opposite propagation factors It will
178 ACOUSTIC WAVEGUIDES be shown, first of all, that this grouping occurs in all straight, uniform piezoelectric waveguides.! General Piezoelectric Waveguides. Modal solutions for the waveguide in Fig. 10.74 must satisfy the quasistatic equations V-T=r V.v = sE:— + d- д (-V<P) (10.157) dt dt V • D — V • (-eT • V<1> + d : T) = 0, obtained from LA, LB. I, and l.C on the front cover papers. These equations are subject to appropriate mechanical and electrical boundary conditions on the boundary C. To be specific it will be assumed that these boundary conditions are TN =0 or v = 0 Ф = 0 or D.JN-0, (10.158) although more general lossless boundary conditions might also be used. For steady state time-harmonic fields, v(r, 0 - v„(r)e"0', etc.; Boundary contour С HGURC 10.74. General piezoelectric waveguide with a common mechanical and electrical boundary C. f Nonrcciprocal systems, such as ferromagnetic or semiconducting waveguides in a do magnetic field, are specifically excluded from this discussion. M. TRANSMISSION LINE MODEL 179 the field equations (10.157) become v • Тш = Uopvt0 Vsv„, = "<>(rs' : T„, - d • VOJ (10.159) V.D(O = V.(-er^w + d:Tw) = 0, and a general positive-traveling mode solution to these equations takes the form vM = ?.„(*, я)«Г"а" тш = TjK*, i/)e Dw = D4(.r,!j)e If the frequency transformation to —*■ — со is applied to (10.159), it is found by direct substitution that the corresponding transformation of (10.160) is v —*• v = —v to T —to to 1 > T — г фю - ф_., = фт О» > D _„ = D„. (10.161) When the transformed time factor tr1"' is included, the transformed solution, I = Tu(*, У)е~'ш Ф = фф>1/)е (10.162) D = D„(.<:, y)e-iilaU^\ also satisfies the original time-dependent field equations (10.157). For a propagating mode (fiM real), complex conjugation of (10.162) gives -Vjr(.r, ууш+^г) *S,t>, уУ1"**-'"1 (Ю.163) D_w(ar, y)eti,ot^z). Since the equations (10.157) arc pure real, this is also a valid solution and represents a negative-traveling mode with time dependence e""' and a propaga- lioii factor — (5Л1. Piezoelectric waveguide modes, therefore, always occur in pairs (M and MYf with propagation constants ±fiif; and, for propagating modes, the I 1 he proof given here is applicable onlv to propagating modes (рл real), but the statement i»n be generalized by analytically continuing the solutions as functions of frequency from piop.igaling regions to nonpropagating regions.
180 ACOUSTIC WAVEGUIDES fields of these mode pairs satisfy relations *-м(х> y) = -v*n(a;, У) т_ис*,) = П(*,*) EXAMPLE 6. In С and D of Appendix 4. fields of a +Z-propagating Rayleigh wave at the surface of a - У-cut electrically free lithium niobate substrate are given as pi/2 pi/2 " (ФвЬ-с pi/2 R = 2.625 x 10-* w1;2/90' X=a - 1.777 x 10 *iov*/V = 14.50 o> 1/2 /95° (Дк^Г° = 36.79 x 10-la «гЛ/-85°. pjv. Z. . According to (10.164), the surface fields for a —Z-propagating Rayleigh wave are (P"^° = -2.625 x 10 6 с»"2/-90° "it = -1.777 x 10-fi <Wo° ^f? = 14.50,0 m /-95° (г>к,^Г" = 36.79 x 10-la оЛ« /85°. £^ Reflection-Symmetric Waveguides. If the waveguide in Fig. 10.74 is isotropic, or if the material medium has a crystal symmetry plane normal to the propagation direction, the problem is invariant with respect to a reflection in the xy plane z^-z (10.165)1 In this case, proof of the mode-pairing property is much simpler, and the relations between positive- and negative-traveling wave fields are mordj stringent. When a waveguide structure is symmetric with respect to the transformation I (10.165), any modal solution transformed according to z—>- —z still satisfies! all the field equations and boundary conditions of the problem. Consequently J M. TRANSMISSION LINT MODEL 181 1 0 0" [«] = 0 1 0 .0 0 -1. and the field quantities in (10.160) transform according to the following matrix equations iV] = WW [Г] = [M][T] ф' = Ф [/)'] = [a][D]. The transform of solution (10.160) is therefore ' My — v. D -D Vs. ! mxj- Myy Mzz My г -Г Mxz мху (10.166) В hese fields represent a negative-traveling mode with propagation factor \—P.\r, and should therefore be labeled — M. This shows that modes in fcflection-symmetric waveguides occur in pairs with equal and opposite propagation factors and have field components satisfying the relations v~Mx — vMx< T — T * Mxx 1 Mxx> T-Mvz T Myz* v~My = v3fu> Т-итуу — TpIt T-Mxz = —T, mz — V Mz T-Mzz — ^Mzz Mxz* Т-мху = TM.x, (10.167) D Ф-.1/ = Фд* D-nf,= -Dliz- -my — "my Uy contrast with the general case, (10.167) applies to both propagating and nonpropagating modes. Since (10.164) applies to both general and reflection-symmetric systems, additional constraints on the field components of a reflection-symmetric the transformed field is a modal solution traveling in the opposite direction. For the transformation (10.165) the matrix [a] given on the front cover papers is
182 ACOUSTIC WAVEGUIDES EXAMPLE 7. Lithium niobate belongs to the trigonal crystal class 3m, and the plane normal to the Z crystal axis is not a reflection symmetry plane. The Rayleigh wave surface components in Example 6 do not, therefore, satisfy the conditions in (10.167). On a — K-cut lithium niobate substrate, the X-axis direction is normal to a crystal symmetry plane; an A'-propagating Rayleigh wave should therefore exhibit all he reflection symmetry properties listed in (10.167). From part С of Appendix 4, the particle velocity field for a +X-propagating Rayleigh wave is 0.7298 x 10"° «>1'* /-90° 3.150 x 10-° ro1'2 /90° 2.219 x 10-6wl/2 /0'. Surface fields for a negative-traveling wave can always be calculated from (10.164). The —X-propagating Rayleigh wave therefore has a particle velocity field 0.7298 x 10-fi ш1/2 /90° 3.150 x 10"° cu1 2 /-90° 2.219 x 10 0 <о1/г /о°, in accord with (10.167). Similarly, the potential and the electrical displacement are consistent with both (10.164) and (10.167). (cU*)g-0 pi, 2 ' JR. ('"lly)i/ f) pi 2 (vTlz)y-0 pi'2 ' К (p Hy)yi 0 p1 ,2 ' 11 (vU-)y О ' II M. TRANSMISSION LINE MODEL 183 M.2 Waveguide "Voltage," "Current," and "Impedance" From (10.121) and (10.132) the waveguide fields v, T • z, Ф, Dz can be written in terms of normal mode amplitudes as У, z) = 2,(a+„(z)vM(x, y) + a v(z)y v(x, y)) TO, y, z) • г = 2 (a, M(2)T m(x, y). £ + a_.v(2)T_JU(ir, //). 2) Ф(*. ?y, г) = J (я+;И(г)Ф 1f(r, ?/) + «_„(г)Ф л1(х, у)) О0168) г/, z) = 2(«кгг(2)Оиг(Ж, ?/) + a_u(z)D_M.(x, у)), м In (10.168) the normal mode amplitudes a v and a_w satisfy differential equations (10.134), which have the same general form as the normal mode transmission line equations introduced in Chapter 6 of Volume I. Therefore, waveguide "voltages" V4(z) and "currents" fu{z) defined by the relations ' r:^+;i« со..») а-ы — ум — AvAv will automatically satisfy the voltage-current "transmission line" equations dVM if( 7 т . w = -»AjtZu/A/ + Vsst dz Zu These "voltages" and "currents" are not physical quantities, but simply provide a useful means for describing the z variation of the waveguide fields. Substitution of (10.169) into (10.168) gives v = |(КдК*)»3?Се, У) + Tу)) Т " 2 = 2 {Vu{*№\*. И) ■ * + /«(гУГЭДж, .V) ■ 2) Ф = ЦУ„(г)Ф%>(х, у) + 1л[(г)Ф^(х, у)) (10.171) \\ here D* = I (VK(z)D%i(x, у) + I3l(z)D%{x, у)) = vjj + v u, etc. (10.172) me the parts of the fields associated with "voltage" and v'/r = Zu(vM - y_.„), etc. (10.173) guide may be obtained by combining (10.164) with (10.167). For example, (10.164) requires that v-mjc — V-Mu — v Mz — ~v*lz> Ф-м = Ф*/- This is consistent with (10.167) only if v4x, vMv are pure imaginary vMz, Фз/ are pure real. Accordingly, the particle velocity field in a reflection-symmetric waveguide is always elliptically polarized with a major or minor axis along the z direction, and the sense of polarization rotation reverses with the propagation direction. A similar conclusion applies to the electric displacement field.
184 ACOUSTIC WAVEGUIDFS are the parts of the fields associated with "current." Because of the non- physical nature of V3T and /jU the equivalent transmission line "impedance" Zu in these equations can be specified rather arbitrarily and may therefore be chosen in different ways to suit particular problems. For general waveguide structures the positive- and negative-traveling J fields are related by (10.164), and the "voltage" and "current" parts of the field, vV = (v* ~ v|r) T5,-2 = (TJZ + Tjr).4, etc. v<7) Tjj • 1 - Z3tCT„ - T*f) • 2, etc., (10.174J usually contain all field components. Reflection-symmetric waveguides, on the other hand, are much simpler. Since V-Mx — Mx -Mv u3Iy Mz Mz in (10.167), the "voltage" part of the particle velocity field (10.172) has onbJ x and?/ components, while the "current" part (10.173) has only as component! For reflection-symmetric waveguides the mode functions appearing in (10.17Ш are therefore 1 if 2vMx . 0 . 0 0 ®W = 2Ф.И-. in = Z м г = M « = Z 0 0 IT. Mxz IT Myz 0 (10.1751 ф((; = о Z 2D Mz- M.3 Waveguide Power Flow According to Chapter 5 in volume I the complex power flow in an electricae transmission line is given by W-^Vl*. (10 176) A true and complete representation of a piezoelectric waveguide by a set or transmission lines should therefore describe complex power flow in the M. TRANSMISSION LINE MODEL 185 I waveguide by a sum of terms like (10.176); that is, W = ZiVufc. (10.177) at This requirement is, in fact, not often satisfied. However, the usefulness of the transmission line model is not seriously impaired by this limitation. A I more important requirement is that Pavg - &cW = »c 2' I VMI%. (10.178) м This also is not satisfied in general, but it does apply to many problems of practical interest. I For simplicity, the question of waveguide power flow will be discussed Only for reflection-symmetric waveguides. In this case the field relationships (10.167) lead to an important simplification of the mode orthogonality relations. For two positive-traveling modes (in = W and n = Л') the orthogonality relation'!' is ) {-"Ufa - vZ-Ju,. ~ ^JMzz + ®„0«>DXz)* l ггпкч ■ Ul'l Ion - vMxT%xz - vMyT%z - vMJl-„ + Ф*(иоВЛГг)} dS = 0, Pu*P%\ (10.179) unci for one positive- and one negative-traveling mode (m = M and и = —N) ill is I J" {-«&Лг« - + v^T„Zz ~ Фл,(1"«мОл-,)* m-f I Ion + + v3IvT*yz - v3TsT%zz + Ф^(к*Ол/г)} dS = 0, (10-180) When these equations are added, one obtains ( {-v***Tv„ ~ ckTUyz - vSIzT*zz + Ф,*(Ы).Ш)} dS = 0, *' 14 nMrl ■ Ktl) Рлг*(Р£)\ (Ю.181) which is a general orthogonality relation for reflection-symmetric wave- indcs. According to l.D on the front cover papers the average power flow I Ik-ic ilie orthogonality relation (10.120) has been generalised to include waveguides of in I'ilfluy cross sectional shape (Problem 25).
186 ACOUSTIC WAVEGUIDES along z is cross st'ction If this is rewritten as Pavo = St' J К~"Л: - vv T* - vtTzz + Ф (i«.DJ*) rfS, (ТОНН section substitution of (10.171) and (10.175) leads to the expression Payg = 2 4ZA.(JK3//£) (-i>Mv„ - »ЬЛ*. М.Л" J - »*.TSr„ + <S>b(iu>DMJ)*dS (10.182) for average power flow in a reflection-symmetric waveguide. When /?jlf, /?lV are pure real or pure imaginary, the integral in (10.182) is zero unless M = N\ (from (10.181)). This means that the condition (10.178) is satisfied for &Щ modes with pure real or pure imaginary propagation factors, provided the field distributions are suitably normalized. For modes with complex propaga-j tion factors, on the other hand, the integral in (10.182) takes on nonzero values only for certain combinations M ^ N. This means that power is transported only by pairs of modes having different subscripts, as wal anticipated in the last paragraph of Section J. M.4 Mode Functions for Nonpiezoelectric Waveguides with Reflection Symmetry For reflection symmetric waveguides the fields in (10.175) split into сопи plctely independent "voltage" and "current" parts. The notation can thus nu simplified by introducing new symbols for the "voltage" part of the ficlJ and the "current" part of the field.f For nonpiezoelectric waveguidesj till "voltage" part of the field for the A/lh mode is described by the column vector G u(z, .V, z) »2lAx> &z) If' Z) (10.1 Щ t References 112 and 113 at the end of the chapter. I Piezoelectric reflection-symmetric waveguides are discussed in Reference 111. M. TRANSMISSION LINE MODEL 187 Qu('. y,z) = (10.184) and the "current" part of the field by Tv„(x, y, z) TMvz(x, У у z) Bach of these fields is regarded as the product of the "voltage" or "current" amplitude with an appropriately defined rector mode function. That is, gm = Km(z)Em(x, y) where the vector mode functions are defined as (10.185) gjtfs Члгх (10.186) The "voltage" VM in (10.185) is chosen to have units of stress and the current IM has units of velocity. The impedance ZM in (10.169), which is ni|ual to VMjIM, must have units of stress/velocity. For consistency with < 10.183) to (10.185) gЛ1г and qMz must therefore be dimensionless quantities. Components gMx and gMv have the dimensions of 1/ZW, while qMx and qMv have the dimensions o(Z3I. In this notation the orthogonality relation (10.181) may be stated succinctly us I (10.187) section where the dot indicates a matrix scalar product. The "impedance" ZM is an nrbitrary parameter; but it is often convenient to choose cup Tt (10.188) 1 his gives a real impedance for propagating modes (jiM real) and an imagin- iny (or complex) impedance for cutoff modes. Average power flow will be ■Ven by (10.178) if all modes present in the summation have cither real or imaginary fTs and the modal field distributions are normalized so that the
188 ACOUSTIC WAVEGUIDES nonzero integrals in (10.182) are equal to unity. This condition is satisfied by applying the normalization J 4M'Z$fdS=-\ (10.189) croes section to the vector mode functions (10.186). For each specific problem, the vector mode functions are obtained by substituting field solutions for positive- traveling waves into (10.183) and (10.184), choosing a value for the "impedance" ZM, and imposing the normalization condition (10.189). This procedure is best illustrated with a specific example. EXAMPLE 8. According to part 1 of Section C, the SH modes of a free isotropic plate have just three field components vx, Txz, and Txy. Only the first two of these enter into the vector mode functions. For a positive-traveling wave ('*).!/ = cos ГМет It (T«)„ = -cos (3 + A/2) Mi (у + m hi = [(ш/ E"s)2 - (МфП>\ and the equations in (10.185) are Mtt cos —r- (v + hl2)e~,pMz b Г ~g\1x(>/)~ 0 = Fm(z) 0 _0_ 0 cos — 0/ + bWe-Px* b CO 0 0 Ямх(У) 0 . 0 . The "voltage" must therefore have the form (10.190) (10.191) (10.192) where V0yr is the "voltage" amplitude corresponding to the fields (10.190), and the* "current" is If the "impedance" is chosen to be top Рм (10.193) (10.194) M. TRANSMISSION LINF MODEL 189 it follows from (10.191) that M-rt cos Рись cos Accordingly, (У + A/2) (У + A/2) ЧмхКуУ 1 Мп рС.ц Mir = - i^cos— [(// + Л/2)], (10.195) (10.196) where K0;W is to be evaluated by imposing the normalization condition (10.189). That is. Г +6/2 л -6/2 Чм ■ g*f dy = <jMxgMx dy J-6/2 J—b/2 Substitution from (10.195) and (10.196) gives "« Г "/2 о ГАГ" -r cos2 — (у + 6/2) iJfl J -6/2 L b = -I. Pcit (10.197) dy -2, M=0 2 *Wf which requires \l/2 |EWl (10.198) (10.199) I he phase of УоЛ1 in (10.199) is arbitrary. If it is taken to be 0°, the normalized vector mode functions for the SH modes of a free isotropic plate are Ъм = (cos p^fr +A/2)JJ< \ LA JM(2Pc44)I«, M^0| г 0 / _0 Since (10.200) (10.201) I c°s \-J-(y + A/2) cos — (j/ + Л/2) U/ = 0 (10.202) lot Л/ rV, these obviously satisfy the orthogonality condition (10.187).
190 ACOUSTIC WAVEGUIDES VMM. P. с (a) П \////,V/A (b) Very thin slots p,c T&-~* ГД' V 1*' V У'"'" wit. 61 P, с (0 FIGURE 10.75. Examples of collinear, isotropic acoustic waveguide junctions. N. WAVEGUIDE SCATTERING PROBLEMS The analysis of scattering in acoustic waveguides is a difficult and relatively unexplored subject, and only a brief introduction can be given here. The simplest problems of this kind arc junctions of two collinear waveguides (Fig. 10.75), where the guides may have different material properties, different geometrical shapes, or may be separated by a planar discontinuity. An incident waveguide mode is directed toward the junction from one side, and one seeks to find the amplitudes of all the reflected and transmitted waves. As in the plane wave scattering problems of Chapter 9, the scattered wave amplitudes are calculated by applying boundary conditions T"„z L"z (io2o3> ф = ф D ■ z = D' ■ £ at the interface between the two guides. A sufficient number of reflected waves in the first guide and transmitted waves in the second must be assumed to permit solution of these equations. Two simple examples of nonpiezo- electric problems will serve to demonstrate how these calculations are performed and to illustrate some general features of the scattering process. EXAMPLE 9. Consider first a junction of two isotropic plate waveguides wilh the same thickness b but different material properties (Fig. 10.75a) The Mih SH mode N. WAVEGUIDE SCATTERING PROBLFMS 191 in the left-hand waveguide is assumed to be incident on the junction. That is (vx) = Av cos -у (У + b/2) ГМтт It cos — (y + hj2) (10.204) e "м Рм = [(«V F,)2 - (МфП^. At the plane of the junction it is necessary that vx and Txz be continuous. These boundary conditions can be satisfied by assuming a reflected SHj,f mode in the left- hand guide 'M-n Юн = Влг cos (у + 6/2) с*'*8 (Jxzhi = вм cos OJ and a transmitted SH.lf mode in the right-hand guide, ~M; ((A*)T = BM cos b — 0/ +/3/2) ;uide, (y + bjl)\e-ifl'^ (10.205) ,_' , „, Рмсц [М- \l xzh — —Им cos ~(y +bl2)\e-^s (10.206) At z = 0 the boundary condition equations (10.203) then give vx: AM+BM=B'M (AM - BM) P'mc'u , 'I hese equations are solved for the particle velocity reflection coefficient ("x(°))i AM or Рмсм РмСц PmQ** fhrcu Рмсм + P'mc'u ' (10.207) nml the transmission coefficient Рмс'ла РмСц со со (10.208)
192 ACOUSTIC Vv'AVEGUTDF.S Stress reflection and transmission coefficients arc, correspondingly, 1 lA',T , (10.209) - (Тт(0))ц = (Г,(0))л, ^ • Comparison with Example 3(b) of Chapter 4 in Volume I suggests that PMc4Jw and Р'лгс'м1<о can be interpreted as acoustic impedances of the Mth SH modes in the two waveguides. This is, however, not the only way the problem can be formulated. An alternative is provided by the transmission line model of part (4) inSection M. Vector mode functions for the problem have already been given in (10.200) and (10.201), and (using (10.185)) the boundary condition equations in (10.203) can be written as / 2 \1/2 ГЛЬ- 1 I 2 \1/2 ГМтт 1 I v Kj'(0)UJcos br{y+H= Кл/(0)Ыcos It*+bm\ I Г„: -/дг(и)(2рс4;)"гсо5^0/ + A/2)] = -/u(0)(2PV;4)1'2cos^0/ + A/2)] (10.210)] This shows that ™o> = (^J4<o> '"<0) - Ш"7^0' '.i In other words, the junction behaves like a transformer of turns ratio connected between transmission lines with characteristic impedances to p 1 40 p £|7 as illustrated in Fig. 10.76. The input impedance looking into the junction from the left is therefore = Wit, (Ю.213) and the voltage reflection coefficient in Fig. 10.76 is JJ^^-z*. ,10.2M) (KF(v>hr (Kw(0))i ZIK+ZV N. WAVEGUIDE SCATTERING PROBLEMS 193 7 ' _ W Air — I I 2 = 0 г = 0 FIGURE 10.76. Equivalent circuit for SHJf mode propagation through the junction shown in Figure 10.75a. Substitution of impedance values from the previous equations converts this to <*iW.v (10.215) which is the same as (10.207) because (^»(Q))it _ fe(Q)V (Ku(0)h ОлДО)), from (10.183) and (10.185). P.XAMPLE 10. The junction problems illustrated by Fig. 10.75b and с arc much more difficult. If the waveguides in (c) are isotropic and only the SFI modes .ue considered, the junction boundary conditions at z = 0 are (a) vjll, 0) = vjy, 0) T„(y, 0) = T'jg, 0) T*(y, 0) = T'xz(y, 0)} d\2 < \y\ < A/2 (b). (10.216) hu this problem, boundary conditions (a) must be satisfied over part of the junction cross section and boundary conditions (b) over the remainder. Referring to I sample 9, one easily sees that these conditions cannot be satisfied by taking a .single reflected wave and a single transmitted wave. Suppose, for instance, that the incident wave from the left is a (SH)U mode and that there is a single reflected mode of the same type. According to the equation for Г„ in (10.210), boundary condition (10.216(b)) can be satisfied only if Ли(0) - /v(0) = 0; but this equality also imposes the same boundary condition over the remainder til the waveguide cross section, which is not consistent with the boundary condition (10.216(a)). To satisfy both of the conditions (10.216(a)) and (10.216(b)) it is JV:1
194 ACOUSTIC WAVEGUIDES necessary to assume an infinite series of reflected and transmitted modes. That is to say, the junction acts as a waveguide mode converter. Similar remarks apply to the junction illustrated in Fig. 10.75(b). In Fig. 10.75(c) the waveguides on both sides of the junction arc identical. It will be assumed that only the fundamental (M = 0) SH mode is propagating, all other SH modes are below cutoff. That is M > I (10.217) If the (SH)0 wave is incident on the junction from the left, the total field in the left-hand waveguide may be written as QU, *) = vx(y, z)' 0 0 0 0 Fo(z)800y) + £ K„(*)ev<!,) (10.218) Л/>1 ««ОЧоМ + 2Мг)Ч^); (Ю.219) and the total field in the right-hand guide is G'd/, ) Q'(2/,*) 0 0 0 0 W 1 /o(^o0/) + 2 'Ji0*b/C") (Ю.221) ЛГ 1 because the left-hand and right-hand waveguides, being identical, have the same vector mode functions gJf, qM. These are given in explicit form by (10.200) and (10.201), but it will be more convenient to retain the general formulation until the end of the calculation. According to the two parts of (10.216), the boundary condition TMfe/,0) = tUv,o) applies over the entire waveguide cross section. This is equivalent to Q(y, 0) Q'(?/, 0) (10 222) in (10.219) and (10.221); and one can therefore write Q(y,0) /u(0)qovy) + 2 W0)qtf(y) = f'o(0)qo(l/) + £ /jf (0)<Ц,0/). (10.223) M XT 1 N. WAVEGUIDE SCATTERING PROBLEMS 195 where use has been made of the fact that trz(y, 0), and therefore Q0/, 0), is zero for \y\ > d\1 (boundary condition (10.216(b))). Another relation may be obtained from the particle velocity boundary condition in (10.216(a)). The applies only for \y\ < d/2 and is equivalent to G(y, 0) = G'(y, 0) Ы < d/2 in (10.218) and (10.220). That is, F0(0)gu0/) + У Vv(0)Zm(!/) <(0Ш'/) + 2 У'мШмМ- (Ю-226) л/Ti м i Because the reflected modes with M ^ 1 in the left-hand guide all travel in the —z direction F„(0) = -rv(0)ZM; (10.227) but Fu(0) = Im(0)Zm (10.228) for the transmitted waves. Using (10.225), (10.227) and (10 228), one can rearrange (10.226) in the form rd 2 (Fu(0) - У«(0)ЫУ) ="2 2 Zlfg w(y) gw0/) ■ QU , 0) dy', (10 229) M 1 J -d/2 where the "impedances" of the higher order modes are Wp (lip Zn = = i -ртгг-^ in , M * 0 (10.230) Pi The "currents" in (10.223) may now be calculated in terms of the aperture field Q(y, 0) by using the orthogonality relation (10.187). To do this, (10.223) is multiplied by g*(i/) and integrated over the waveguide cross section. That is g;Vfe/) • QO/'. 0) dy Ш g* ■ Чо dy + 2 s% ■ 4w dy' J-b/2 J-b/2 If .1 J-b 2 J l! M -1 J-1/2 where a dummy integration variable у has been introduced. From the orthogonality condition (10.187) satisfied by the vector mode functions gv and qM, there is only one nonzero "current" term for each choice of N. According to (10.189), this gives rdii 7„(0) = f'0(0) = - g*(y') • Q(?/', 0) dy' (10.224) J—it 2 and 4,(0) = /лг(0) = - P "g*, (У) • Qfy', 0) dy', (10.225)
196 ACOUSTIC WAVEGUIDES Finally, (10.229) is multiplied by Q*(>i, 0) and integrated from -d\2 to d]2. The result is then rearranged as d/2 ^o(O) - Fu(0) 2 %ZM jj (0*(y", 0) • gu(</"»(Q0/, 0) ■ gW)) <*/ «ЙГ* |j%(./,0)-g*(?/)^/' (10.231) using (10.224). Expression (10.231) has a very simple and useful physical interpretation. According to (10.224) and (10.231), the 0th mode "current" is continuous at the junction but the "voltage" is discontinuous. This means that the junction behaves like a series impedance Zs = iXs inserted in a transmission line of characteristic impedance cop Zq = ~7T~ = pVs, Pa as shown in Fig. 10.77. The value of this series impedance is given by (10.230) and (10.231). Because the higher order modes are below cutoff, they die away exponentially in both directions and, at appreciable distances from the junction, only the ft, Z0 = ft, 4 y////////////////////////////rf////////%\ fy///*/////////.--У////// ''V////A '' 1 SH0 mode fields Higher mode fields v._ FIGURE 10.77. Equivalent circuit for SH0 mode transmission through the junction shown in Figure 10.75(c). The SHy mode fields are governed by the equivalent circuit. N. WAVEGUIDE SCATTERING PROBLEMS 197 incident, reflected, and transmitted SH„ waves remain. The relative magnitudes of these SHn waves can be calculated from the equivalent circuit in Fig. 10.77. Substitution of the vector mode functions (10.200), (10.201), and the mode impedances (10.230) into the series impedance expression (10.231) gives Zs .X„ — = ,— = Afi0 rf/2 vr?i"i я Ш"(У°- 0) c°s[t" 0/' + fc/2>]cos[^ + hl2)\dy'df where (10.232) Pa = ЧК and the complex conjugates have been dropped because the mode functions are all pure real. To calculate ZJZ0 it is necessary to know the stress distribution in the aperture of the junction; that is T^(:v,0), \y <dj2. This is a very difficult problem to solve rigorously, and it is fortunate that (10.232) is a variational expression. In essence, this means that an accurate value of ZJZ0 can be obtained by using an approximate trial function for the stress field Txz(y, 0) in the plane of the junction. Variational techniques provide a powerful method for attacking otherwise intractible problems, and will be discussed at some length in Chapter 13. The present problem is an exact analogue of one of the classical problems in electromagnetic waveguide theory (namely, the capacitive diaphragm in a parallel plate or rectangular waveguide (Fig. 10.78)). As seen in the figure, the equivalent circuit for the diaphragm is a shunt admittance connected across a transmission line with characteristic impedance corresponding to the fundamental waveguide tnode. The normalized shunt susceptance BJY0 for the electromagnetic problem, which is the same as the normalized series reactance XJZ0 for the acoustic problem, has been evaluated to a high degree of accuracy. Full details of the calculation are given in advanced electromagnetism texts,t and only the final resuli will be quoted here. If all of the higher order modes are very far below cutoff, the term (ro/Fs)2 I See, for example, R. E. Collin, Field Theory of Guided Waves, Gh8, McGraw-Hill, New York. 1960.
198 ACOUSTIC WAVEGUIDFS Parallel plate f waveguide Very thin capacitive diaphragm F (a) Waveguide configuration . В 1ъ Yd (b) Equivalent circuit FIGURE 10.78. Electromagnetic analogue of the acoustic junction problem in Fig. 10.75c. in Гл/ can be ignored, and — = i In esc — , (10.233) where я, SUn p is the wavelength of the SH0 mode. This result is compared in Fig. 10.79 with a more accurate solution obtained by using a better approximation to Глт. The acoustic waveguide junction in Fig. 10.75b is also analogous to a standardj electromagnetic problem. In this case a transformer is required, in addition to the series reactance, to account for the different geometries of the two waveguides. Examples 9 and 10 have treated only the simplest free plate modes (the! SH family); analysis becomes very much more complicated when LamlJ modes are considered. These have two components in each of the vectol mode functions. Furthermore, since the cutoff modes have complex valuesJ off}, orthogonality and normalization conditions are more complicated than! for SH modes. Little work has yet been done on Lamb wave scattering problems, but an even more difficult case has recently been studied in an approximate manner.! This is the problem of Rayleigh wave scattering at a junction between a plated and an unplated substrate (Fig. 10.80). An indication of the importance t Reference 110 at the end of the chapter. O. CROUP VELOCITY AND ENERGY VELOCITY 199 5 4 3 2 - 1 1 1 -., 1 1 1 1 _L 1 Exact *j 1 - / f- f i 1 1 1 1 / — / Eq. (10.233) - / l l l 0.2 0.4 0.6 0.8 1.0 b_ Xsh0 FIGURE 10.79. Normalized series reactance XJZ0 in Figure 10.77, for d\b = 1/2. (After Montgomery, Dicke, and Purcell). of this kind of junction has already been given in the treatment of stripe- and slot-type microsound waveguides. To analyse these waveguides accurately it is necessary to evaluate the reflection coefficient for obliquely incident waves. The transmission line model described in part 4 of Section M can be adapted to this problem, and it is found that the junction can be represented approximately by the equivalent circuit in Fig. 10.80. This calculation neglects the effect of nonsurface wave modes excited at the junction and also assumes that there is a negligible difference between the Rayleigh wave impedances and field distributions in the plated and unplated regions. For the cases in which the Rayleigh wave is incident in the unplated region and the plating is sufficiently thick there is an appreciable amount of scattering into the lowest order Love wave (Part 1 of Section D) in the plated region. Calculated reflection coefficients for this situation arc given in Fig. 10.81. O. GROUP VELOCITY AND ENERGY VELOCITY I l Chapter 7 of Volume I the group velocity Ve of a wave was defined as the propagation velocity of a modulation envelope applied to the wave, and was
Free surface Gold Plating/^' Fused quartz Modified Rayleigh wave nL :1 Love wave Love 30.0 60.0 Incidence angle, в 30.0 60.0 Incidence angle, в FIGURE 10 80 Equivalent circuit representation of Rayleigh wave scattering at the boundary between plated and unplated substrate regions. (After Ol.ner, Berto... and Li). и a "a. с з •x3 с a a "a. ♦J a •o с 3 о a ё 3 с a a S S с о - и a ■t С ш л с ос — >• I- и с "я --^ О N 5 з I- о- 3 2 с ■ о о "о — ,ьп ей " о $ 201
202 ACOUSTIC WAVEGUIDES calculated for the case of a uniform plane wave. If the modulation envelope is one-dimensional, V. = ; (Ю.234) ok in the more general case of a three-dimensional modulation envelope, „ dm . dw do) nn1K\ Vv = x—- + y- !-£-—. (10.235) dkx ok,j dkz Expressions for the group velocity of a guided wave arc calculated in exactly the same way. If the wave is guided in three dimensions (Fig. 10. lb), only a one-dimensional modulation can be applied—the field variation across the waveguide cross section is completely specified for each type of guided wave. The group velocity is then К = . (Ю.236) dp Dispersion relations for uniform plane waves have a linear relationship between to and k, and the group velocity in (10.234) and (10.235) is therefore independent of frequency. This is not usually true for guided waves. The dispersion relations obtained in this chapter generally express m as a nonlinear function of p. An illustration of this is given by the dispersion curves for the SH modes of an isotropic plate (Fig. 10.7). Only the curve for the fundamental mode (я = 0) is a straight line. For all other modes Va = dtojdp differs from the phase velocity Vs = vo\p, and both velocities are functions of frequency. This property (called frequency dispersion) is of great importance in most practical applications. If the signals to be transmitted cover a band of frequencies, signal distortion resu ts when the propagation velocity i frequency-dependent. In communications applications, where accurate reproduction of a signal is essential, frequency dispersion is to be avoided. Other applications, such as radar and sonar systems, require transformation (or processing) of the signals, and in such cases an appropriate form of frequency dispersion is often deliberately introduced into the system. Guided acoustic waves are sometimes used for this purpose. EXAMPLE 11. Because guided wave dispersion relations are generally presented in the implicit form fi(w. P) = 0, (10.23 the group velocity is most easily calculated by using implicit differentiation; that is, O. GROLP VELOCITY AND ENERGY VELOCITY 203 from part I of Section C, one may use «(/», « = - ^ - Pi (Ю.239) for the nth SH mode. Accordingly, V„ = -- ZmfV, mjVs At the cutoff frequency («и = (птт/Ь)Ух) Уе=0, as is readily apparent from the slope of the dispersion curves (Fig. 10.7). (.10.240) For many types of guided waves, dispersion curves can be obtained only by numerical computation. In these cases the general characteristic equation of the system may be used for (10.237); the value of p corresponding to a particular mode is substituted into (10.238) after the differentiation. In the case of Lamb waves, functions Q(w, p) for the symmetric and antisymmetric solutions are obtained by substituting (10.20) and (10.21) into (10.18) and (10.19). The derivatives diljdp and 3Q/3co in (10.238) can be taken analytically and may then be evaluated numerically for particular modes of interest. When a wave is guided in only two dimensions (Fig. 10.1a) the modulation envelope may be two-dimensional. If the phase propagation is allowed to Itcive an arbitrary direction in the xz plane.f the method followed in Chapter 7 of Volume I may be used to obtain an expression „ dm _ do v* = x^ + z^r (1024l) °Px OP г for the group velocity. With a dispersion relation presented in the implicit fonn I2(w, px, pt) = 0 (10.242) Ihis is most conveniently evaluated by implicit differentiation; that is, V*=-^4r- (Ю.243) Ал in the plane wave case, (10.242) defines a slowness curve for the waves; v^ii in (10.243) and therefore the group velocity Vg are always normal to litis slowness curve (Fig. 10.82). The direction of the group velocity can thus I 11 in 1 is, the guided wave fields vary as e
204 ACOUSTIC WAVEG IDES FIGURE 10.82. Relationships between the group velocity Vg, the propagation vector (J and the slowness curve. The magnitude of p/co is 1IV„(0). be easily deduced by inspecting the curve. Guided wave propagation characteristics are usually presented in the form of numerical plots of рЬаЯ velocity V„ — w//3 as a function of the phase propagation direction 0 (seM for example, Figs. 10.59 to 10.61). If в = tan-' pjp„ the normal to the slowness curve (defined by 1/^,(0)) makes an angle — tan -i dVJdO (10.24 with the z axis (Fig. 10.82). This is the group velocity direction. The orthogonality relations derived in part 3 of Section J were obtained choosing solutions "1" and "2" in the complex reciprocity relation (10.113) to be the fields of different waveguide modes at the same frequency ы. An important relation between group velocity, power flow, and stored energy is derived by choosing solutions "1" and "2" in the complex reciprocity relation to be fields of the same waveguide mode at different frequencies ы and o> + fifiM That is, vi = ^eO-yJx, y), etc. Ma+SaH„-i(Pm'. tfim)zt у,. — e Ьт(х, У) + 8vn,(x, y)), etc. (10.245) O. GROUP VELOCITY AND ENERGY VELOCITY 205 Here, dfim and dvm, etc., are changes due to the frequency shift 8co, and /3m is assumed to be real (propagating mode). Substitution of (10.245) into the left-hand side of (10.113) gives '" ^«{-т? ■ Тц - v, • Tf + ФГОЪВО + Ф,(/(й) + 6fl))D,)*} - г + {-v* . Т, - тц • Т* + OJCiwDO + U\(i(w + 3w)D2)*}, (10.246) where da- dy This is integrated over the cross section of the waveguide in Fig. 10.74, and Gauss's theorem is used to convert the second term in (10.246) into a line integral around the waveguide boundary C. For any set of lossless boundary conditions (Problem 20), this line integral is zero. The integral of (10.246) over the waveguide cross section then reduces to i&Rmute j 2{-y*-TM + <bJivJDJ*}-tdxdyt (10.247) cross section when second-order terms in <5 are ignored. Similarly, substitution of (10.245) into the right-hand side of (10.113) and integration over the cross section gives I da j (pv,* • ym + T •(: s* : Tm - : d • УФт) cross section - УФ * (■ d : Tm - • er • W>J) d* dy = dw j (pv* • vm + T * : Sm - УФ* • Dm) Лс dy. (10.248) cross Section Wince (10.248) is equal to (10.247), and is therefore a real quantity, the group Velocity is 2Ste Г {-v* • T„, + Фв(/а>Б J*} • z d* dy '^-"(sT-) = r • (10249) vam j J>m • v* + T,„ : S * - \Фт ■ D*) dz dy According to the complex Poynting Theorem of Section 8.G in Volume T, the numerator in (10.249) is 4 times the average power flow PAV, and the denominator is 4 times the average stored energy UAV per unit length of the waveguide. The waveguide group velocity V„ and the waveguide energy
206 ACOUSTIC WAVEGUIDES velocity Vt are therefore equal, Кй = Ve = —. 0°-25°) U^AV just as they were for uniform plane waves. Relation (10.249) can be further simplified by applying the complex Poynting Theorem to a length dz of the waveguide and substituting fields of the mth waveguide mode. The surface integral in Poynting's Theorem is independent of 2 (for a propagating mode) and goes to zero at the waveguide boundaries; the integral over the closed surface is therefore zero. For a freely propagating mode there are no sources (F = pe = 0), and therefore where This means that the average stored energy per unit length of waveguide in (10.250) is given by UAV = 2f Zym*yZdxdy, (10.251) Js 4 which avoids laborious calculations of the stress field and the electric potential. Using (10.250), it is then possible to calculate the average waveguide power flow from the particle velocity field alone; that is, Рлу= V.\ Eym-<dxdy, (10.252) Js 2 This result can also be applied to structures which guide in only two dimensions (Fig .10.1a), provided the group velocity is collinear with the phase velocity. EXAMPLE 13. For one-dimensional waveguides (such as the isotropic plate) power flow and stored energy should be evaluated for unit length in a direction normal to the propagation direction. For the SH plate modes of Part 1 Section С and UAV = J j cos* \j (?/ + */2)J d,j = , (10.254) PROBLEMS 207 from (10.16). According to (10.250) the group velocity is then This agrees with (10.240), which was obtained directly from the dispersion re -a on. PROBLEMS 1. Derive the Rayleigh-Lamb frequency equations (10.18) and (10.19), and verify the field distributions (10.22) and (10.23). 2. Find solutions of the form vt = А1ехШ-*х) for the coupled differential equations \rjr or! dz" giving a dispersion relation m = fik) and a particle displacement ratio »i _ A{k) v, B(k) for each solution. Show that the coupled dispersion curves "split" at points where the uncoupled (K = 0) curves cross each other, and verify that the coupled solutions differ appreciably from the uncoupled solutions only in the neighborhood of a crossover point. According to (9.43), an SV wave incident at 45° to a free boundary scatters only into a reflected SV wave. When the SV components of a Lamb wave satisfy this condition, the wave is called a Lame wave [see (10.28)]. Show that the first terms within the brackets in (10.22) and (10.23) go to zero under these conditions. 4. When to//3 — У1л/2, Lamb wave solutions contain only longitudinal partial waves. Prove this statement, using the free-surface scattering constants derived in Chapter 9, and verify that the second terms in the field distribution equations (10.22) and (10.23) go to zero under these conditions.
208 ACOUSTIC WAVEGUIDbS 5. Derive the v field distribution equations (10.36) and (10.37) for Rayleigh waves, and show that these correspond to Fig. 10.23. Find the associated T field equations. 6. Use Fig. 10.26 to show that Love mode fields arc tightly confined to the plate at high frequencies, even when V'a -=ss V„. 7. Show by direct substitution that M = V x zip N = (F>))V x M have zero divergence and are solutions to the vector wave equation V2*= -fej* when Derive the potential functions given in (10.57). 8. Derive (10.58) and the boundary condition equations (10.62). 9. Use the standard Bessel function identities 1 d_ JJy) = JVM ydy у" ~ y"n and y-j-Wy)] = -PJ„(y) + yJv-&) dy to demonstrate that (10.65) is equivalent to J2(kua) = 0, which has roots re = kua - 0, 5.136, 8.418, etc.; and show that the dispersion relations for torsional modes are therefore «=(fJ-(l5- I For tv = 0, but the corresponding v field in (10.66) is zero. Verify that the solution ru = Ore «""V» PROBLrMS 209 corresponding to tq = 0 is obtained by substituting the vector potential into (10.53). 10. Starting from (10.58), derive the characteristic equation for waveguide modes in a circular cylinder with rigid boundaries. Find the torsional mode field distributions and dispersion relations for this structure. 11. Prove that the Lame waves considered in (10.28) and in Problem 3 have Txx — Tyx — T,x = 0, where z is the propagation direction and a: is parallel to the surfaces of the plate, and verify that they satisfy all the boundary conditions for a free rectangular strip waveguide (Fig. 10.37) with arbitrary width a. Show that these strip waveguide solutions are valid only at the discrete frequencies given by n = 1,2, 3 Do the P-type plate waves considered in Problem 4 also provide exact solutions for the rectangular strip waveguide? 12. Only г-propagating SH and Lamb modes were considered in Section С but these modes may, of course, also propagate along any direction in the plate. For waves propagating in directions other than z, the SH mode dispersion relation (10.15) and the Lamb wave dispersion relations (10.18) (10.21) are modified by making the simple substitution Fxact solutions to the free rectangular strip waveguide problem in Fig. 10.37 can then be obtained (for special frequencies and special ratios a/b) by treating the SH and Lamb modes as partial waves and reflecting them back and forth between the waveguide sides r = 0 and a. Boundary conditions may be satisfied by suitably combining (1) Four SH modes, defined by kt.i = ~T~ . kx = ± — , fc2 = 6' b a and kts = ~r~ > kx -j- — , kz = 8 b a (2) Two Lamb modes, defined by . Ш77 1177 PTT i о i Ktl — . > kx — ± » Яг — P • b b a
210 ACOUSTIC WAVEGUIDES Verify that this is possible when m, n, p, q are positive integers q - ff The calculation is carried out most conveniently by transforming the previously given SH and Lamb mode field distributions. 13. Rigid boundary conditions (v = 0) are applied at the surfaces of the rotated hexagonal plate in Fig. 10.48. Show that the SH mode solutions in this case arc sin b n = 2, 4, 6, птту ., - cos , n = \, 5,3, . h with Wcor у + sin>)g ^ + (Cg6 cos, v + ^ sin, V)M2 _ ^ = o. (ceG cos2 y> + cu sin у) \ b / 14. Derive (10.93) and (10.94). 15. Noting that ce6 = K<n - <1г)- show that the solutions to [9.100(b)] and [9.100(c)] area pure longitudinal wave, with a dispersion relation and a pure shear wave, with a dispersion relation Verify that this result allows (10.34) to be used for Rayleigh waves propagating along the F-axis of an X-oriented hexagonal (6mm) substrate. 16. Find plane wave solutions for propagation in the XY plane of hexagonal crystals belonging to classes 6, 622, 6 and Ът2. Since all hexagonal crystals have uniaxial symmetry about the Z axis (sec Example 2 in Chapter 4 and Example 7 in Chapter 8 of Volume I), the analysis may be simplified by assuming propagation along X. The solutions obtained are then converted to an arbitrary propagation dir tion by transforming t v nd Ф fi d with a coordinate rotation about the Z axis. Using these solutions as partial waves, show that F-propagating surface waves on an A'-oriented substrate (with an electrical short circuit boundary condition) have the following properties PROBLEMS 211 Crystal Class Bleustein-Gulyaev Wave Rayleigh Wa*c 6 Yes 622 No 6 No 6m 2 No Inactive piezoelectrically Inactive piezoelectrically Active piezoelectrically Active piezoelectrically The incident wave amplitudes may be set equal to zero directly in the boundary condition equations, instead of calculating the scattering coefficients and proceeding as in Example 3. The latter approach is advantageous only when the scattering coefficients are already available. 17. There are two general methods for analyzing surface wave problems: (a) by finding singularities of the scattering coefficients (as in Example 3), or (b) by matching boundary conditions with a set of outgoing partial waves (as in Problem 16). In some special cases the method of singularities does not seem to provide all possible solutions, but a closer examination of the equations shows that this is not so. Consider, for example, the problem of an [010]-propagating Bleustein-Gulyaev wave on a [100]-oricnted cubic substrate with short circuit electrical boundary conditions. The relevant boundary condition equations and scattering coefficients were given at the end of Example 4 in Chapter 9. In particular, (9.89) stated that the reflection coefficient for a horizontally polarized shear wave is equal to unity. From method (a), one would then conclude that there is no Bleustein-Gulyaev solution. Using method (b), show that the solution is a horizontally polarized uniform plane shear wave propagating parallel to the boundary, and that this corresponds to the scattering equation (9.89) with 0S = тг/2. In this case the Bleustein-Gulyaev wave has degenerated into a nonpiezoelectric solution with infinite penetration depth in the substrate. 18. Equations (9.100)(a) and (d) in Example (5) of Chapter 9 take the form lor fields with arbitrary functional dependence on y, z (Problem 11 Chapter 9). Show that the substitution Ф = у + ч 1
212 ACOUSTIC WAVEGUIDES reduces these equations to where _ Compare these reduced differential equations (Reference 73 at the end of the chapter) with those given in Problem 11, Chapter 9. Use one or the other set of equations to solve for K-propagating Bleustein-Gulyaev waves on an A'-oriented Hexagonal (6mm) substrate with short circuit electrical boundary conditions. 19. Derive (10.103), (10.104), and (10.105). 20. Suppose that the boundary conditions in Fig. 10.68 are T • у = Ъл • v Ъл symmetric and pure imaginary at у — 0, b and Ф = ZK(iwD - y) ZE pure imaginary at у — —Л, b + h. Show that the orthogonality relation (10.120) still applies. 21. Show that the real reciprocity relation (10.108) is valid, but the complex relation (10.111) is not, when acoustic losses (cu -*ctJ + itorjij) and dielectric losses (e,7 -*- ei3 + a Jim) are present. Verify that the complex relation applies to fields varying as e'wl in lossless ferrite and semiconductor media with a dc magnetic field (fitj = //*, etj = «*), but the real relation docs not. 22. Starting from the real reciprocity relation (10.109), derive an orthogonality relation that applies to lossy waveguides with lossy boundary conditions (that is, Z , and ZE in Problem 20 are pure real). 23. Derive expressions for the Rayleigh wave amplitudes excited by the following time harmonic distributions of surface forces, (/■', N</ (a) T - у = (b) T$ = (c) Т-у=(/ О, |г| > * F sin Kz, \z\ < I 0, \z\ > I Z F, \z\ < I 10, |z| > / PROBLEMS 213 24. Two thin-wire electrodes, arranged in the configuration shown, are used to excite Rayleigh waves on a piezoelectric substrate. Assuming that the diameter of the electrodes is much smaller than the Rayleigh wavelength, derive expressions for the Rayleigh wave amplitudes excited by a charge Qe™* per unit length on one wire and —Qe110* on the other. j ^Piezoelectric \ Thin wire electrodes 1/ 25. Prove that the orthogonality relation (10.120) is generalized to guides of arbitrary cross sectional shape by performing the integral Pmfl over the entire cross section. 26. Extend the formalism of Section M,4 to piezoelectric waveguides with reflection symmetry. Include Ф in the "voltage" field and ia>Dz in the "current" field. 27. Show that the SFI mode dispersion relation (10.15) becomes (/?-ia)2+ L- mrY pax2 when viscous damping is included. 28. The wth mode of a nonpiezoelectric lossy waveguide with stress-free side boundary conditions is vm(a\ y)e ^'e'iPnZ T,„ • z(.r, y)e-"mze iPnZ. Apply the complex Poynting Theorem to an elemental length dz of the guide mid, after substituting the wth mode fields, show that where S* dS 2 (P,/)av = w js ~ is the average power loss per unit length and -v*.T,n.zdS is 2 JS is (he average power flow. For low-loss waveguides, this formula may be used Hi lind am by substituting lossless modal solutions into (P^av and i\v- Apply this method to the lossy SH modes in Problem 27. Show that the same
214 ACOUSTIC \\ AVFGUIDES approach can be used for piezoelectric waveguides with the general side boundary conditions given in Problem 20. REFERENCES Methods of Analysis 1. T. R. Meeker and A. H. Meitzler, "Guided Wave Propagation in Elongated Cylinders and Plates/' pp. 112 119, 130-134, Physical Acoustics 1A, W. P. Mason, Ed., Academic Press, New York, 1964. 2. M. Redwood, Mechanical Waveguides, Pergamon, New York, 1960. 3. 1. Tolstoy and E. Usdin, "Dispersive Properties of Stratified Elastic and Liquid Media: A Ray Theory," Geophysics 18, 844-870 (1953). 4. A. A. Oliner, "Microwave Network Methods for Guided Elastic Waves," IEEE Trans. МТГ-17, 812 826 (1969). Free Isotropic Plate 5. W. M. Ewing, W. S. Jardetzky, and F. Press, Elastic Waves in Layered Media, pp. 281 288, McGraw-Hill, New York, 1957. 6. Reference 1, pp. 119-129. 7. A. H. Meitzler, "Backward-Wave Transmission of Stress Pulses in Elastic Cylinders and Plates," /. Acous. Soc. Amer. 38, 835-842 (1965). 8. R. D. Mindlin, "Waves and Vibrations in Isotropic, Elastic Plates," pp. 199-232, Structural Mechanics, Pergamon, New York, 1960. 9. Reference 3, pp. 845 853. 10. I. A. Viktorov, Rayleigh and Lamb Waves, Ch. II, Plenum, New York, 1967. Isotropic Rayleigh Waves 11. Reference 5, Ch. 2. 12. L. D. Landau and E. M. Lifshilz, Theory of Elasticity, pp. 105 109, Pergamon Press, New York, 1959. 13. G. Nadeau, Introduction to Elasticity, pp. 233 236, Holt, Rinehart, andj Winston, New York, 1964. 14. Reference 4, pp. 822-823. 15. Reference 10, Ch. 1. Isotropic Plate on an Isotropic Half Space 16. J. D. Achenbach and S. P. Keshava, "Free Waves in a Plate Supported by a Semi-infinite Continuum," /. Appl. Mech. 34, 397^104 (1967). REFERENCES 215 17. J. D. Achenbach and II. 1. Epstein, "Dynamic Interaction of a Layer and a Half-space," J. of the Engineering Mechanics Division, Proc. ASCE 93, pp. 27-42 (1967).' 18. L. M. Brekhovskikh, Waves in Layered Media, Academic Press, New York, I960. 19. Reference 5, Ch. 4. 20. G. W. Farncll and E. L. Adler, "Elastic Wave Propagation in Thin Layers," Chapter 2 in Physical Acoustics 9, W. P. Mason and R. N. Thurston ed. Academic Press, New York, 1972. 21. C. Lardat, C. Maerfcld. and P. Tournois, "Theory and Performance of Acoustical Dispersive Surface Wave Delay Lines," Pioc. IEEE 59, 355 368 (1971). 22. Reference 13, pp. 237-241. 23. Reference 4, p. 824. 24. K. Sezawa, "Dispersion of Elastic Waves Propagated on the Surface of Stratified Bodies and on Curved Surfaces," Bull. Earthquake Res. Inst. 3, 1-18 (1927). 25. K. Sezawa and K. Kanai, "Discontinuity in the Dispersion Curves of Rayleigh Waves," Tokyo Bull. Earthquake Res. Inst. 13, 237 244 (1935). 26. H. F. Tiersten, "Elastic Surface Waves Guided by Thin Films," J. Appl. Phys. 40, 770-789 (1969). 27 Reference 3, pp. 857-861. Isotropic Stoneley Waves 28. Reference 5, pp. 107-113. Ю Reference 20. 30. Т. E. Owen, "Surface Wa\e Phenomena in Ultrasonics," Prog. In Appl. Mat. Res. 6, 69 87 (1964). 31 J. G. Scholte, "The Range of Existence of Rayleigh and Stoneley Waves," Roy. Atsron. Soc. London, Monthly Notices Geophys. Suppl. 5,120-126 (1947). 32 K. Sezawa and K. Kanai, "The Range of Possible Existence of Stoneley Waves and Some Related Problems," Tokyo Bull. Earthquake Res. Inst. 17, 1-8 (1939). 33- R. Stoneley, "Elastic Waves at the Surface of Separation of Two Solids," Roy. Soc. Proc. London, Series A 106, 416-428 (1924). Free Isotropic Cylinder fl. Reference 1, pp. 130 141. 15 Reference 2. J6. R. A. Waldron, "Some Problems in the Theory of Guided Microsound Waves,' IEEE Trans. MTT-I7, 893 904 (1969).
216 ACOUSTIC WAVEGUIDES Rayleigh Waves on Curved Surfaces 37. L. M. Brckhovskikh, "Surface Waves Confined to the Curvature of the Boundary in Solids," Soc. Phys.- -Acoust. 13, 462-472 (1968). 38. B. Uulf, "Rayleigh Waves on Curved Surfaces," J. Aeons. Soc. Amer. 45, 493 499 (1969). 39. Reference 10, pp. 29 41. Isotropic Rectangular Strip 40. M. A. Medick, "One Dimensional Theories of Wave Propagation and Vibration in Elastic Bars of Rectangular Cross section," J. Appl. Mech. Series E 33, 489 495 (1966). 41. R. D. Mindlin and M. A. Medick, "bxtcnsional Vibrations of Elastic Plates," J. Appl. Mech. Scries E 26, 561-569 (1959). 42. Reference I, pp. 142-145. 43. N. .1. Nigro, "Steady-State Wave Propagation in Infinite Bars of Noncircular Cross Section,"/. Acous. Soc. Amer. 40, 1501 1508 (1966). 44. С. C. Til and G. W. Farnell, "Thickness Effects in Overlay Elastic Waveguides," IEEE Trans: SU-19, p. 394 (1972). 45. R. A. Waldron, "Mode Spectrum of a Microsound Waveguide Consisting of an Isotropic Rectangular Overlay on a Perfectly Rigid Substrate," IEEE Trans. SU-I8, 8 16 (1971). Microsound and Related Waveguide Structures 46. L. R. Adkins and A. J. Hughes, "Elastic Surface Waves Guided by Thin Films: Gold on Fused Quartz," IEEE Trans. MTT-17, 904 911 (1969). 47. A. .1- Hughes, "Flastic Surface Wave Guidance by ДЕ/К Effect Guidance Structures," /. Appl. Phys. 43, 2569-2586 (1972). 48. E. A. Ash, R. M. De La Rue, and R. F. Humphryes, "Microsound Surface Waveguides," IEEE Trans. MTT-17, 882 892 (1969). 49. H. Engan, "Experiments with Elastic Surface Waves in Piezoelectric Ceramics," FLAB Report T-I28, The Royal Norwegian Institute of Technology, Trondheim, 1969. 50. F. G. Marshall and E. G. S. Paige, "Novel Acoustic Surface Wave Directional Coupler with Diverse Applications, * Electron Lett 7,460-462 (1971). 51. I. M. Mason, R. D. DeLa Rue, R .V. Schmidt, E.A. Ash, and P E. Lagasse, "Ridge Cinides for Acoustic Surface Waves," Electron Lett. 7, 395 397 (1971). 52. J. D. Ross, S. J. Kapuscienski, and К.. B. Daniels, "Variable Delay Line Using Ultrasonic Surface Waves," IRE Nat. Cone. Rec. pi. 2, 118 120 (1958). 53. Reference 26, pp. 776 785. REFERENCES 217 54. Reference 44. 55. R. A. Waldron, "Some Problems in the Theory of Guided Microsome Waves" IEEE Trans M1T-I7, 893 904 (1969). 56. R. M. White, "Surface Elastic Waves,*' Proc. IEEE 58, 1238 1276 (1970). Anisotropic Waveguides 57. N. G. Einspruch and R. Truel, "Propagation of Traveling Waves in a Circular Cylinder Having Hexagonal Elastic Symmetry," J. Acous. Soc. Amer. 31, 691-693 (1959). 58. G. W. Farnell, "Properties of Elastic Surface Waves," Ch. 3 in Physical Acoustics 6, W. P. Mason and R. N. Thurston, ed., Academic Press, New York, 1970. 59. Reference 20. hO. D. C. Gazis and R. F. Wallis, "Exlensional Waves in Cubic Crystal Plates," Proc. 4th U.S. National Congress of Applied Mechanics, vol. 1, pp. 161-168 (1962). 61. W. W. Johnson, "The Propagation of Stoneley and Rayleigh Waves in Anisotropic Elastic Media," Bull. Seis, Soc. Amer. 60, 1105 1122 (1970). 62. Reference 21. 63. Т. C. Lim and G. W. Farnell, "A Search for Forbidden Directions of Elastic Surface Wave Propagation in Anisotropic Crystals," J. Appl. Phys. 39, 4319^325 (1968). 64. Т. C. Lim and G. W. Farnell, "Character of Pseudo Surface Waves on Anisotropic Crystals." J. Acous. Soc. Amer. 45, 845-851 (1969). 65. T С Lim and M. J. P. Musgiave, "Stoneley Waves in Anisotropic Media," Mature 225, 372 (1970). 66. D. S. Loftus, "Elastic Surface Waves on Layered Anisotropic Crystals," Appl. Phys. Letters 13, 323-326 (1968). 67. M. J. P. Musgrave, Crystal Acoustics, Ch. 12, Holden-Day, San Francisco, 1970. 68. N. J. Nigro, "Wave Propagation in Anisotropic Bars of Rectangular Cross Section Part I. Longitudinal Wave Propagation," J. Acous. Soc. Amer. 43, 958 965 (1968). 64. N. J. Nigro and R. Andrews, "Wave Propagation in Anisotropic Bars of Rectangular Cross Section II. Flexural Wave Propagation," J. Acous. Soc. Amer. 46, 639 642 (1969). 70 F R Rollins, Jr., Т. C. Lim, and G. W. Farnell, "Ultrasonic Reflectivity and Surface Wave Phenomena on Surfaces of Copper Single Crystals." Appl. Phys. Letters 12, 236 238 (1968). 71 A. Ronnckleiv. "Propagation of Elastic Waves Along Cylindrical Rods and Holes in Hexagonal and Isotropic Media," ELAB Report TE-I4I, The Norwegian Institute of Technology, Trondheim, 1970.
218 ACOUSTIC WAVEGUIDES 72. A. J. Slobodnik, Jr. and E. D. Conway, "Microwave Acoustics Handbook, vol. 1 Surface Wave Velocities," Physical Sciences Research Paper No. 414, Air Force Cambridge Research Laboratories, Bedford, Mass (1970). Piezoelectric Waveguides 73. J. L. BIcustein, "A New Surface Wave in Piezoelectric Materials," Appl. Phys. Letters 13, 412-413 (1968). 74. .1. L. Bleustcin, "Some Simple Modes of Wave Propagation in an Infinite Piezoelectric Plate," /. Aeons. Soc. Amer. 45, 614-620 (1969). 75. J. J. Campbell and W. R. Jones, "A Method for Estimating Optimal Crystal Cuts and Propagation Directions for Excitation of Piezoelectric Surface Waves," IEEE Trans. SU-15, 209-217 (1968). 76. H. Deresiewicz and R. D. Mindlin, "Waves on the Surface of a Crystal," J. Appl. Phys. 28, 669-671 (1957). 77. H. Engan, K. A. Ingebrigtsen, and A. Tonning, "Elastic Surface Waves in a-Quartz: Observation of Leaky Surface Waves," Appl. Phys. Letters lb, 311-313 (1967). 78. Reference 57. 79. C. A. A. J. Greebe, P. A. van Dalen, T. J. B. S. Swanenburg, J. Wolter, "Electric Coupling Properties of Acoustic and Electric Surface Waves," Phys. Lett. 1С, 235-268 (1971). 80. Yu. V. Gulyaev, "Elcctroacoustic Surface Waves in Solids," JETP Letters 9, 37 38 (1969). 81. K. A. Ingebrigtsen and A. Tonning, "Numerical Data for Acoustic Surface Waves in a- Quartz and Cadmium Sulfide," Appl. Phys. Letters 9, 16 18 (1966). 82. K. A. Ingebrigtsen, "Surface Waves in Piezoelectrics," J. Appl. Phys. 40, 2681-2686 (1969). 83. G. Kocrber, "Uncoupled Piezoelectric Surface-Wave Modes," IEEE Trans. SU-18, 73 78 (1971). 84. Reference 21. 85. C. Maerfeld and P. Tournois, "Pure Shear Elastic Surface Wave Guided by the Interface of Two Semi-Infinite Media," Appl. Phys. Lett. 19, 117-118 (1971). 86. R- C. Rosenfeld, "Bleustein Waves on Cylindrical Surface," IEEE Trans. SU-18, p. 48 (1971). 87. R. V. Schmidt and F. W. Voltmer, "Piezoelectric Elastic Surface Waves in Anisotropic Г ayered Media." Trans. IEEEMTT-П, pp. 920 926 (1969). 88. Reference 72. 89. H. F. Tiersten, Linear Piezoelectric Plate Vibrations, Ch. 10, Plenum, New York, 1969. REFERENCES 219 90. С. C. Tseng and R. M. White. "Propagation of Piezoelectric and Elastic Surface Waves on the Basal Plane of Hexagonal Piezoelectric Crystals," J. Appl. Phys. 38, 4274-4280 (1967). 91. С. C. Tseng, "Elastic Surface Waves on Free Surface and Metallized Surface of CdS, ZnO, and PZT-4," /. Appl. Phys. 38, 4281-4284 (1967). 92. L. V. Verevkina, L. G. Merkulov, and D. A. Ttirsunov, "Surface Waves in a Quartz Crystal," Son. Phys. Acoustics 12, 254-258 (1967). 93. F. W. Voltmer, E. P. Ippcn, R. M. White, Т. C. Lim, and G. W. Famell, "Measured and Calculated Surface Wave Velocities," Proc. IEEE 56, 1634-1635 (1968). 94. R. M. White, "Surface Elastic-wave Propagation and Amplification," IEEE Trans. ED-14, 181 189 (1967). Transducers 95. B. A. Auld and G. S. Kino, "Normal Mode Theory for Acoustic Waves and its Application to the Interdigital Transducer," IEEE Trans. ED-18, 898 -908 (1971). 96. A. J. Bahr, R. E. Lee, A. F. Podell, "The Grating Array; A New Surface Acoustic Wave Transducer," Proc IEEE 60, 443 -444 (1972). 97. H. L. Bertoni and T. Tamir, "High Efficiency Wedge Transducers," IEEE Trans. SU-19, 413 (1972). 98. G. A. Coquin and R. E. Dean, "Low Loss Excitation of Surface Waves with Wedge-Type Mode Converters," IEEE Trans. SU-18, 52 (1971). 99. H. Engan, "Excitation of Elastic Surface Waves by Spatial Harmonics of Interdigital Transducers," IEEE Trans. ED-16, 1014-1017 (1969). 100. C. S. Hartmann, W. S. Jones, and H. Vollers, "Wideband Unidirectional Interdigital Surface Wave Transducers," IEEE Trans SU-19, 378 381 (1972). 101. F. S. Htckernell, "Piezoelectric Film Surface Wave Transducers," IEEE Trans. SU-19, 413 (1972). 102. G. S. Kino and R. Wagers, "interdigital Transducers on Nonpiezoelectric Substrates," IEEE Tians. SU-19, p. 413 (1972). 103. W. R. Smith, H. M. Gerard, J. H. Collins, Т. M. Rccdcr, and H. J. Shaw, "Analysis of Interdigital Surface Wave Transducers by Use of an Equivalent Circuit Model," IEEE Trans. MTT-17, 856-864 (1969). 104. F. G. Marshall, E. G. S. Paige, and A. S. Young, "New Unidirectional Transducer and Broadband Reflector of Acoustic Surface Waves," Electron Lett. 7, 638-640 (1971). 105. R. A. Waldron, "Principles of Wideband Unidirectional Piezoelectric Transducers," MIT Lincoln Laboratory, Technical Note 1969-54, 1969. 106. R. M. White, "Surface Elastic Waves," Proc. IEEE 58, 1238-1276 (1970).
220 ACOUSTIC WAVEGUIDES Chapter 11 ACOUSTIC RESONATORS A. FREE MODFS OF OSCILLATION 221 B. METHODS OF ANALYSIS 222 С ORTHOGONALITY RELATIONS FOR RESONATOR MODES 250 D. ELECTRICAL EXCITATION OF PIEZOELECTRIC RESONATORS 253 E. STORED ENERGY AND POWER LOSS 259 PROBLEMS 262 REFERENCES 268 A. FREE MODES OF OSCILLATION In Chapter 10 free and forced wave propagation in acoustic waveguides was discussed. Waveguide structures are of infinite extent in one dimension (the direction of propagation) and finite in at least one of the transverse dimensions, and the source-free modes of propagation are traveling wave solutions. I'hey are characterized by a relationship between frequency < > and propagation factor ft (the dispersion relation). This chapter is concerned with free and forced acoustic field solutions in structures which are finite in all direc- tionst —that is, acoustic resonators. In this case the free modes are acoustic oscillations of the structure, and each mode is characterized by a natural frequency of oscillation. These are the resonant frequencies of the forced oscillation problem. For a lossless resonator the free oscillations are undamped and the natural frequencies are all pure real. These frequencies are found by solving the field equations, subject to the appropriate mechanical and t In some cases the structure may be idealized by allowing it to be inbnite in one or two dimensions; but only resonant, rather than propagating, solutions are considered 221 107. С. В. Willingham, M. G. Holland, M. B. Schub-, and J. H. Matsinger, "ZnO Overlay Film Surface Acoustic Wave Transducers," IEEE Trans. SU-I9,413 (1972). 108. J. C. Worley and H. Matthews, "Broadband Unidirectional Surface Wave Transducer," IEEE Trans. SU-18, 52 (1971). 109. К. H. Yen and R. С. M. Li, "Broadband Efficient Excitation of Thin Ribbon Waveguide for Surface Acoustic Waves, "Appl. Phys. Lett. 20, 284 286 (1972). Transmission Line Model 110. R С. M. Li, A. A. Oliner, К. H. Yen, and H. L. Bcrtoni, "Properties and Applications of the Acoustic Wave Junction between Plated and Unplated Substrates," IEEE G-MTT Intl. Micro. Symp. Digest, 54-55 (May, 1971). 111. T. Makimoto and S. Sato, "Generalized Treatment of Piezoelectric Waveguide," Proc. IEEE 60, 733-734 (1972). 112. A. A. Oliner, H. L. Bertoni, and R. С. M. Li, "A Microwave Network Formalism for Acoustic Waves in Isotropic Media," Proc IEEE 60, 1503 1512 (1972). 113. A. A. Oliner, R. С. M. Li, and H. L. Bertoni, "Catalogue of Acoustic Equivalent Networks for Planar Interfaces," Proc. IEEE 60, 1513-1518 (1972). 114. C. G. Montgomery, R. H. Dicke, and E. M. Purcell, Principles of Microwave Circuits, pp. 166-167, Dover, New York (1965).
222 ACOUSTIC RESONATORS electrical boundary conditions at the resonator boundaries. The quasistatic field equations (10.12) and (10.13) can usually be used. That is, V • (cK: V» = -p (,/y - iroV • e - У7Ф «rV . (€'s • УФ) = -iwV ■ e : Vsv. The free mode problem is more difficult to solve in resonators than it is in waveguides. As a result there are even fewer exact solutions available. Although some of the analytical techniques used in solving the waveguide problem (Section 10.B) are applicable to some of the simpler problems and will be discussed in this chapter, it is generally necessary to use sophisticated approximation techniques. Perturbation and variational methods of attacking these problems will be treated in Chapters 12 and 13. B. METHODS O* ANALYSIS B.l Unbounded Nonpiezoelectric Plate The simplest and most direct method of resonator analysis is to set up standing waves which match the required boundary conditions at the resonator surface. This technique can be carried through to an analytic solution in closed form only for a very few cases, and the unbounded isotropic plate resonator (Fig. 11.1a) is the only problem in which simple analytic solutions can be obtained for the complete mode spectrum. In this case the analysis is just a specialization of the transverse resonance method for isotropic plate waveguides (part 3 of Section 10.B). у у = + 6/2 — H **-T'y = 0 ^ v = - 6/2 (a) У ^ > z 6 < *Vr- y = 0 (b) FIGURE 11.1. Isotropic and aniso- tropic unbounded free plate resonators. B. METHODS OF ANALYSIS 223 EXAMPLE 1. Thickness Resonances in an Unbounded Isotropic Plate. These resonances are found by setting up standing longitudinal and shear waves across the plate. The stress fields may be written as tyy = Ateiki" + b*rikt» (11.2) for the longitudinal standing wave, and тух = /V**** + bje-*** tvz = As.eilc-v + Ве.е~ш'у (' ''^ for the two shear standing waves. For a free plate resonator, the stress-free boundary conditions tvy = ту, = tVZ=0 (11.4) at у = ±/j/2 can be satisfied independently for each of these standing waves. The symmetry argument of part 3 Section 10.B is first used to show that В = ±A (11.5) in each case. It is then sufficient to consider boundary conditions at only one surface of the plate. Taking the longitudinal case first, the boundary condition at у = b/2 is At(cik'm ± e <kffZ) = 0 from (11.2) and (11.4). This requires that e4.lb = ±i and where v = 1, 2, 3,..., is a mode index labeling the resonant frequencies. The stress field patterns are then tm ~ sin — 0/ + b/2) (11.7) h from (11.2), (11.5), and (11.6). Similarly, the properties of the shear modes are (Os„ vtt and vtt sin— (?/+-) (11.9) ('4 't(» + 2)- or v-n i h\ (11.10) Each of the three families of modes separates into an even symmetry class and an odd symmetry class (Fig. 11.2).
224 ACOLSTIC RESONATORS v = 3 • = 2 „ = 4 Longitudinal Modes z-Polarized Shear Modes v = 4 3 Ш . = 3 v = 2 %-Polanzed Shear Modes ■ ^ Symmetric Anbsymmetric FIGURE 11.2. Particle velocity field distributions cos ■— (v + W2), p„ b cos — (v + fc/2), в. ft c0S Z (.. + /,/2)] for the resonator shown in figure b ' 11.1a. EXAMPLE 2. Thickness Resonances in an Unbounded Anisotropic Plate. The standing wave method can also be applied to anisotropic plate resonators. Here the problem is much more difficult and it is not usually convenient to seek an .analytical solution in closed form. The most common technique is to derive Л' cnaracteristic equation for the resonant frequencies and then to solve this numerically. In the nonpiezoelectric cubic plate resonator of Fig. I l.lb, plane wave solution* for propagation along </ are obtained from Section D of Chapter 7 in Volume I. These solutions consist of one pure shear wave polarized along .», and quiKishoB B. METHODS ОГ ANALYSIS 225 and quasilongitudinal waves polari/ed in the yz plane. Standing waves are constructed by superposing waves traveling in the positive and negative у directions. v.*,, = y.(A^Heik^" + flsll<--,'*sHi') v« = (''„>' + i'sA){A0lfiik°*" + £,se-'A"-») v„i = (v„S + vltmA,lte,k-"« + В,ле-^у). The total velocity field is then taken to be a superposition of these three standing waves, l'J: = Аяце1"*** + B<ne ""sn" iv = ^(/V"'"" + В„е-''*«-«) + vly(AtJleu«» + BQie (11.11) r* = VsM^*"'1 + Ва?-'***) + elz(Avleik'0' Role *»«■*) and the stresses required for the boundary conditions (11.4) arc obtained from the consliiulive relations, using the transformation laws on the front cover papers to find the stiffness constants in the rotated coordinate system. This gives I 3r, T = — 141 c™ dy i fa, v22 dy + C'2i By j ( dv, where the subscripts on the stiffness constants are referred to the coordinate axes •г, y, z in Eig. I l.lb rather than the crystal axes X, E, Z. Since the stress component Tyx involves only the wave amplitudes Лнп and fi8H in (11.11), this part of the problem can be treated separately. It follow* the isotropic analysis given above, and the result is thai the SH modes have resonant frequencies vt7 FSH h ' where K?„ = CgJp. For the remaining two stress components there are four boundary conditions T, = 0, у = ±b\2 (11.12) Tzy = 0, у = ±ft/2 involving the four unknown amplitudes, A,,i.< liJsy in (11.11), and these cannot, in general, be simplified by means of symmetry krgumenls. Because of the wave coupling imposed by the boundary condition equations (11.12) the resonant modes have both quasishear and quasilongitudinal field component*. The characteristic equation for the resonant frequencies is
226 ACOUSTIC RESONATORS obtained by setting the determinant of the four linear equations (11.12) equal to zero and substituting where i/os and Kol are obtained from Example 4 of Chapter 7 in Volume I. In this example the resonant frequencies may be found analytically in closed form. More generally, however, all three types of standing waves are coupled by the boundary conditions, and numerical solution of the characteristic equation is required. B.2 Bounded Nonpiezoelectric Plates The unbounded plate structure is an idealization suitable for analyzing thickness vibrations in plate resonators with dimensions / and w that are very much larger than the plate thickness / (Fig. 11.3). Analysis of the other vibration modes in thin plates and of the general problem for thick plates is very difficult and approximation methods are usually required. Because of the great practical importance of the bounded plate resonator structure a great deal of effort has been devoted to its analysis and there is a large literature on the subject. General references are given at the end of this chapter, and application of the variational method is discussed in some detail in Chapter 13. Only two of the direct analytical methods will be illustrated here. EXAMPLE 3. Resonances in an Isotropic Plate Bounded in One Transverse Direction. In this case / — да in Fig. 11.3 and simple solutions may be obtained for some special cases by considering the structure as a section of plate waveguide of length w, with stress-free boundary conditions at the ends (Fig. 11.4). As in the example above, the problem is analyzed by setting up standing waves, where the waves arc now waveguide modes of the plate structure rather than plane waves. For the nth SH mode (part 1 of Section 10C) the stress field of the FIGURE 11.3. Bounded plate resonator. B. METHODS OF ANALYSIS 227 FIGURE 11.4. Standing SH plate wave. standing wave is where T„ = - ^ c« cos ^ (у + A/2)J (Ae - Be*"*') У w f" S'n \~b (2/ + Ь/2) + Stress-free boundary conditions at z = ±w/2 demand that T„ =0 in (11.13). That is /te-'""" 2 Ве^,г1г = 0 Ae*** 2 - Be-fl>«w/s = 0 which requires The solution is Pn = m-njw and В = Ae m = 0, 1,2, Combining (11.14) and (11.15) then gives the resonance condition (11.13) (11.14) (11.15) (11.16) mid the field distribution is found by substitution (11.15) into (11.13).f I orthe Lamb modes of a plate waveguide (part 2 of Section 10.C) simple standing wuve resonance solutions cannot be set up by using incident and reflected waves of the same mode type. The stress-free boundary condition at the end of a semi- iiifmiic plate waveguide cannot be satisfied by such a simple reflection, because a t 'I hese arc called iliickness-iwisi modes of the plate (Reference 6).
228 ACOUSTIC RESONAIORS particular Lamb mode incident upon a free boundary reflects into an infinite number of other modes. In this case approximation techniques, such as the variational method (Chapter 13) m st be used. EXAMPLE 4. Contour txtensional Modes in Thin Plates. An especially important family of modes for the bounded thin plate resonator (Hg. 11.5a) is the set of contow-extensional modes. In these vibrations the mid-plane of the plate remains fixed and, throughout the plate, the particle motion is primarily in the xz plane (Fig. 11.5b). The stress components are zero at the upper and lower surfaces of the plate and, if the plate is very thin, these stresses are very nearly zero throughout the plate. In the thin plate approximation Tvv, Tvz, and Tyx are assumed to be exactly zero at all points. The elastic properties of the plate can then be described by a "planar" Ffooke's Law, A- 'hi л13 s3 = s13 ■S33 *1S A_ JlS S1S Л. (a) FIGURE 11.5. (a) Bounded thin-plate resonator, (b) Plan view of (a) showing one of the resonant modes of contour-cxtensional type. B. METHODS OF ANALYSIS 229 To obtain a "planar" wave equation for this problem it is most convenient to invert (11.17), giving (11.18) The planar stiffness matrix in (11.18) is the inverse of the truncated compliance matrix in (11.17) and the planar stiffness constants ctJ differ from the bulk stiffness constants Cjj, which arc obtained from the inverse of the complete compliance matrix. In the isotropic case the "planar" stiffness constants are "'"11 C|3 A~ r3 = S3 Jl5 '~55_ A- сзз — ct, С12 cnCia ~ C12 Си (11 19) ru = 0 A "planar" wave equation is derived by substituting the "planar" stiffness relation TT = crjSj into the source-free acoustic field equations^ a 'dtVi with 1, J =x,z LJ = I, 3,5. elimination of the strain from these equations gives v>7 ?u ^Ji'b = <'>2pVi I, J = I, 3,5 (11.20) for fields with ei,at time dependence. At the edges of the plate in Fig. 11.5, stress- free boundary conditions require that T - np =0 where ft" is the normal to the plate edge. (11.21) See I .Л on the front cover papers.
230 ACOUSTIC RESONATORS FIGURE 11.6. Isotropic thin-strip resonator, AH boundaries arc stress free. As an illustration consider the isotropic resonator in Fig. (11.6), where the particle velocity field is entirely along z and is independent of the x coordinate. In this case (11.20) reduces to eh - c?2 Э2 Cii Oz subject to the boundary condition _ 1 & - ff2 dvt - 0, 2 = 0, w * zz -~v ICO Сц 02 from (11.21). A general solution to (11.22) is vz = A sin kz + В cos kz. where ">Vii 2 2 ' I'll Cfj. p =i 1,2,3, (11.22) (11.23) The boundary conditions (11.23) require Л =0 = ,i=l,2,3,... w and the resonant frequencies are given by For plates of more general shape it is not usually possible to obtain a closed form solution such as this, and the variational methods of Chapter 13 must be used. (П.24) B. METHODS OF ANALYSIS 231 B.3 Isotropic Cylinder of Finite Length 1 his problem (Fig. 11.7) is another case where some of the resonances may be analyzed as standing waveguide modes. The simplest waveguide modes for a free circular rod waveguide are the torsional modes T0g (part I of Section 10.E). These have a velocity field Wt^-J-I^V (П.25) a \ a I with ■w = 0t <7= 1,2,3-•■ and There is also a mode (q = 0) with = «~*г"" (П-26) and PToo — у • Since the isotropic stiffness matrix has the same form in cylindrical and rectangular coordinates (Part D оГ Appendix 1 in Volume I) it follows from (10.59) that the stress field corresponding to (11.25) and (11.26) has two components Тгф and Тгф. T- 5 = 0 at z = 0, IX T • f = 0_ at r = о FIGU R E 11.7. Free isotropic cylinder resonator. I Sec Problem 10.9
232 ACOUSTIC RESONATORS Only the first of these enters into the stress-free boundary conditions at z = 0 and L in Fig. 11.7. These conditions can therefore be satisfied with a simple standing wave field (Problem 3), and the resonant frequencies for the torsional vibrations of the cylinder are given by For the other cylindrical waveguide modes, coupling takes place at the free end surfaces of Fig. 11.7 and a simple standing wave solution is not possible. A variational method for analyzing modes of this class is discussed in Section В of Chapter 13. B.4 Isotropic Sphere Surprisingly, this problem (Fig. 11.8) is simpler than the cylindrical resonator, and the complete mode spectrum can be found without using approximate methods, although the characteristic frequency equation has to be evaluated numerically. The separation of variables method is used to solve the acoustic scalar and vector potential problem (part 1 of Section 10.B), v = VO + V x W (11.28) where У2ф + = () (Ц.29) t-f = 0 at r = о x HGURE П.8. Free isotropic sphere resonator. B. METHODS OF ANALYSIS 233 and V'W + -\ 4> = 0. (11.30) Solution of the Potential Equations by Separation of Variables. Spherical coordinate solutions of the scalar potential equation (11.29) have the general formf 0, ф) = zn P,[" (cos 0) е""ф, (11.31) where и = 0, 1, 2, . . . — n < m < n. The functional dependence on 0 is given by associated Legendre polynomials of the first kind, defined by the relation with 1ш1 d m PlTkcos 0) = (1 - cos2 6) l—± - — Pjcos 0), (11.32) 2 d cos1™" r„(cos 0) = — (cos- в - 1)». 2"я1 dcos В" Table 11.1 lists the first few functions of this kind. The radial functions in (11.31) are radial Bessel functions zri(p). Some of the radial solutions of first and second kinds are given in Table 11.2 and the others can be found from the recursion relation d dp n+i(/») = -Pn — (/> %(/»))■ (11.33) TABLE 11.1 Associated Legendre Polynomials of the First Kind Pj(cos 6) = I Pj(cos в) = COS 0 P{(cos в) = sin 0 Pg(cos 0) =4(3 cos 20 + 1) P-](cos в) = i sin 26 Pi(cose) .](! - cos 26) t reference 15 at the end of llie chapter.
234 ACOUSTIC RESONATORS First kind, zn(p) = j„(p) Second kind, zn(p) = nn(p) sin p cos p n = 0 n = 1 n = 2 P p я sin p cos p _ sin p _ cos p "7" ~~ ~~P~ v P2 if— - л sin p - ^-cos p - ^sin p - -(-2- IJ cos p p\p2 / p2 p Л? / In deriving (11.30) in Section 10.B.1, it was found that V-4? may be chosen arbitrarily. Here it is convenient to use the condition Y-Y - 0. Solutions for the vector potential can then be constructed in a simple manner from the scalar functions (11.31), with Vl replaced by Vs\ that is ^mn(r, в, ф) = Z„(y)plm,(cos 0) e"* (11.34) It is easily checked by direct substitution into (11.30) that М=УхгТ_ (11.35) and NM = -'VxM„ (H.36) where4rmn is defined by (11.34), are vector potential solutions with V • 4! = 0 (Problem 4). When V • 4? = 0, (11.30) is equivalent to -VxVx»F+^ = 0; (U-37) У s and it follows from this and (11.36) that Mm„ and Nmn are also related by the equation MM = -'VxNm (11.38) со Pure Compressional (S0I) Modes. For a solid spherical resonator, the field must remain finite at r = 0 and only the spherical Bessel functions of the B. METHODS OF ANALYSIS 235 first kind in Table 11.2 can be used. In this case the scalar potential (11.31) is Ф,т„(г, 0, ф) = jB ^ Pj^cos 0) e""* (11.39) These solutions take a particularly simple form when m = n = 0, namely, (eor\ , д . f(or\ Voo = V^-)=r-7o(_J (11.40) According to Appendix 1 in Volume I the strain field corresponding to (11.40) has only three components с ±?Ei ^rr — . _ II!) ог s ко r с J_£r ьфф — . 1(0 r and the traction force on the spherical surface is therefore! T -r = T„ — cuSrr + c^CSm, + S^). Consequently, the stress-free boundary condition is C,1cV at r = a; and substitution of (cor\ _ sin (cor/F;) Vj- corjV, from Tabic 11.2 gives the characteristic frequency equation /сосЛ (oalV, tan ы = ~^2- (И-41) 4c„\ VJ If the roots of (11.41) are designated as s00l, where the first two subscripts indicate m = n = 0 in (11.39), the natural frequencies of this family of modes are , -Z'c ""(JO! — S00!- a t The isotropic stiffness matrix in spherical coordinates has the same form as in rectangular coordinates. (0)r\ , 2c12 д . /тг\ TABLE 11.2. Spherical Bcssel Functions of Tirst and Second Kinds
236 ACOUSTIC RESONATORS According to common convention these modes are labeled S0l on the mode chart of Fig. 11.9. The particle motion is entirely radial. In the lowest order mode Sm the entire sphere expands and contracts radially (Fig. 11.10). For the higher order modes there are spherical shells of oppositely directed motion. Pure Shear (Tn ,_j) Modes. The spherical shear waves defined by (11.36). ▼m« = V x Nmn, are also fairly simple in nature. From (11.36) and (11.37) vm„ = (-', V x V x Mmn = f Mmn. (11.42) Vol V„ Poisson s Rat о, о — Z(cn - cu FIGURE 11.9. Mode chart for an isotropic spherical resonator of radius a, with stress-free boundary conditions. The first mode subscript n refers to the 0 variation (Figure 11.8), and for each value of л there exist 2л 4- 1 degenerate modes with ф indices |m| < |n|. (After Eraser and I.eCraw). В MFTHODS OF ANALYSIS 237 FIGURE 11.10. Lowest frequency spherical resonator mode of the compressional family (Sol), often called the "breathing" mode. Since Мгая = V x rV (11 35) the particle motion is always normal to the radial coordinate, and sin в дф ЭУ„Я (11.43) V*~~ дв from (11.34). The spherical Bcssel function of the first kind is again chosen in (11.34) because the field must remain finite at r = 0. According to Appendix 1 in Volume I the strain field corresponding to (11.43) has components Soo > Sre< Sr4>> and the traction force on the spherical surface is therefore ito \dr r) ioj \dr r / (11.44) Substitution of the velocity field (11.43) and imposition of the stress-free boundary condition T-f = 0atr = a gives a characteristic equation in the variable v»a\Vs. If the roots of this equation are designated as /,„ the resonant frequencies of this family of modes are a These frequencies are shown as a function of c„ and c44 in Fig. 11.9, where the conventional mode designation Tn , г has been used. Because the frequency «'тт does not depend upon m, there are 2n + 1 degenerate modal solutions (—n ^ m ^ n) for each root /„,.
238 ACOLSTIC RESONATORS FIGURE 1 Ell. Lowest order pure shear (or torsional) mode of an isotropic spherical resonator (7\ x). Figure 11.9 shows that this does not have the lowest resonant frequency of the Tn pure shear family. This family of modes is called torsional because there is no radial motion. The torsional motion is illustrated in an especially simple manner by the case m = 0,n= 1. From (11.35) and (11.42) the particle velocity field ist v24l = 4>-|TM=-d>cosO./1(^) (И-45) and the frequency relation reduces to tan—= WalK ... (П46) V _ l/amY- 3\fJ The motion in this case is a rigid rotation of spherical shells about the 2-axis, with adjacent shells moving in opposite directions (Fig. 11.11). From the spherical symmetry of the problem it is obvious that similar motions about the x and у axes have the same natural frequencies. These correspond to linear combinations of the degenerate mode solutions, with m = ± 1 rather than m — 0. Mixed (5„ м,л?« 0) Modes. It is not possible to match the stress-free boundary conditions at the sphere surface by using just the spherical shear wave solutions obtained from the vector potential M,„„ in (11.35), v -VxM = — N (11-47) To solve this problem it is necessary to combine (11.47) with the congressional solution obtained from (11.39), v = A V0> + SNmm. t When n = 1 the second mode subscript is chosen to be / rather than / — 1. B. METHODS OF ANALYSIS 239 FIGUR E 11.12. Lowest frequency spherical resonator mode of the mixed family (S20). This is often called the oblate-prolate mode. Imposition of the stress-free boundary conditions at r = a then leads to a complicated frequency relation, which must be solved numerically for specific cases. Figure 11.9 gives the resonant frequencies for some of the lower order modes of this family, designated S„ (n Ф 0); and Fig. 11.12 illustrates the field pattern for the lowest frequency mixed mode (S^). B.5 Unbounded Piezoelectric Plate Addition of piezoelectricity makes the resonator problem very much more difficult to solve, but it is still possible to obtain a closed-form frequency equation for the unbounded plate by setting up standing waves. Both ine- han cal and electrical boundary cond t ons must now be satisfied at the surfaces of the plate. As in piezoelectric waveguide problems, this generally requires use of all three of the acoustic wave solutions, as well as an electrostatic wave solution (part 2 of Section 10.B), and for most problems the frequency equation must be solved numerically. EXAMPLE 5. Thickness Resonances of an Unbounded Piezoelectric (Hexagonal 6mm) Plate. Equations appropriate to this problem (Fig. 11.13) have already been discussed at some length in Example 5 of Chapter 9. All three of the acoustic wave solutions are obtained by taking kz 0 in (9.100). From (l>.100b) and (9.100c) a pure longitudinal acoustic wave v, - ye кг = <o(plcnrn
240 ACOUSTIC RESONATORS Perfectly conducting- electrode Stress-free^ boundaries" Perfectly conducting'' electrode X,y\~~ Hexogonal (6mm) plate 7///^v//m///////w////////.v/////////////m -..■////// ■ x Г G U undcd ul h xa na ) p ate resonator. and a pure shear acoustic wave ksv = fotp/сдд)1'2 are obtained. Since these do not interact with the electric potential, they generate nonpieroelectric resonances, as in Examples 1 and 2, and will not be considered further. Elimination оГ Ф from (9.100a) and (9.100d) gives the stiffened acoustic shear wave solution v*n <Psii s (11.48 with aii1 = к , i I -s- ) • \c.w + <?ЫЕл'л/ In Example 5 of Chapter 9 the evanescent (or electrostatic) solution was given by (9.101). This would seem to indicate that k., = 0 when k, = 0; but this is not, in fact, correct. The reason becomes clear if one returns to (9.100) and recalls that this equation was obtained by making tlto substitutions djdy * —ikv djdz ► —ik., B. METHODS OF ANALYSIS 241 — Ф = -Л? , ду* io, ду2' (11.49) when the fields are independent of the r and z coordinates. This shows that the solution corresponding to kz =0 in (9.101) must be u, - 0 — =0 Ф = А,у + В <V cJ e (11.50) ЭФ T^ = eX6^. To satisfy the boundary value problem in Fig. 11.13, the fields within the plate are described by a standing SH wave (7^)SH =kjf(c?x + pjr-)(V*«"» - Bj **•) *sn = (A**** + (M-51) constructed from (11.48), or by taking 0S =0 in (9.103), and the electrostatic solution Фв = Acy + B, (£>„)„ = -e%xA, x* (11.52) (T„)e exs~dy =ехьЛ" hum (11.50). Outside the plate, the electrical potential satisfies Laplace's equation V^I> = 0, and the solutions arc b b - h <y < - - 2 2 Ф «= /Ц. [y + ^ + hj, / b \ (11.53) I lie boundary conditions to be satisfied arc T =0 Ф, continuous Dy, continuous for exponential wave solutions, in the original field equations. For general non- c.\poncntial solutions, the equations are
242 ACOUSTIC RESONATORS \ <xx/ = ("-54) and cj вде^ у 2 - 4л-4« = -VU (П.55) respectively. These equations are most easily solved by considering separately the odd-symmetry modes (Bs = —As, Be = 0) and the even-symmetry modes (B„ = As, Ac = 0). If the SH subscript on k is dropped, the frequency relations can be written as s к b \ e0 b} к b tan^_=-i ?— '— (11.56) 2 *A'5 2 js II for the odd modes and kvb = 2>чг »' = 1, 2, 3, for the even modes. In both cases (П.57) The spectrum of the odd modes is most easily visualized by means of a graphical solution to the transcendental equation (11.56). In Fig. 11.14 the left and right-hand sides of (11.56) are graphed separately as functions of A„A/2, and the intersections give the odd natural frequencies of the resonator. As the piezoelectric coupling constant ел'ь goes to zero, the roots of (11.56) approach the poles of the tangent function. That is b fcv-=0 + .)-, or kvb = О + 1)тг (11.58) »■ =0,1,2,3, ... where kv is calculated from the unstiffened constant c'^. B. METHODS OF ANALYSIS 243 kyb/Z 7Г/2 Зтг/2 5тг/2 FIGURE 11.14. Graphical solution of (11-56) for the odd mode frequencies of the resonator shown in rigure 11.13. tan kyb/2 It this is combined with (11.57) the complete mode spectrum Tor the nonpiezoelectric ease may be given as (11.59) »' = 1, 2, 3, As would be expected, this is the same as the result obtained in Example 1. In (11.56), removing the electrodes to infinity (A ->■ ш) has the same effect as letting еХъ — 0. Гог electrodes at a finite distance from the plate the odd-symmetry mode frequencies arc shifted, the amount of frequency shift being different for each at у = ±Ь/2 (Fig. П. 13). That is,
248 ACOUSTIC RESONATORS у (Ь) FIGURE 11.19. Energy trapping in the SH mode plate resonator of Figure 11.4. (a) Thickened region on a nonpiezoelectric plate, (b) Elcctrodcd region on a piezoelectric plate. (-w'j2 < z < w'12) the stress field is P, from (11.13), and the corresponding particle velocity field is t, = cos[^//F^](^ «««+&*.•>, r---'S'«cM[?(*+^]°*H with Forz < w'\2 vr = cos (11.65) (11.66) (11.67) B. METHODS OF ANALYSIS 249 with and for z > w'/2 ■ (;)' со I?, = cos ^ ^ + ^ Jxte-»»". TIZ = I - C„ CO! CO (11.68) Boundary conditions at the planes z = — w'/2 and w'/2 require continuity of Txz and iv Because of the discontinuity in the upper and lower surfaces of the plate, the fields (11.65) to (11.68) cannot satisfy these boundary conditions exactly and other higher order modes must be added to obtain a rigorous solution. Variational methods are available for solving problems of this kind. However, if b' » b it is permissible in the lowest approximation to assume b' = h for the field expressions, but not for the waveguide dispersion relations. The boundary conditions at z = ±w'j2 then state that Pi Ae-ifi*w'l- - Be*"*™'1* = — — De-'"u,'lt ft Solution of the problem is simplified by considering separately the even symmetry modes (B = A, D = C) and the odd symmetry modes (B = — A, D = — C). It is then sufficient to consider boundary conditions at only one edge of the thickened region, and the frequency relations are P\w y\ tan-i- = -J (11.69) 2 Pi for the even symmetry modes and cot^- = - 1 (11.70) 2 px for the odd symmetry modes, with As anticipated, the mode spacing is determined by w' rather than w. In a piezoelectric resonator, plate thickening can also be used to achieve energy trapping but another, more convenient, technique is also available.
250 ACOUSTIC RESONATORS Electroded ' regions Input Output FIGURE 11.20. Trapped energy monolithic filter on a piezoelectric plate. According to Example 2 in Chapter 10, conducting electrodes on the surfaces of a hexagonal (6mm) plate waveguide lower the cutoff frequencies for the SH modes. The trapped energy region in Fig. 11.19b can therefore be defined by means electrodes on the plate surfaces, and the frequency relation is derived by following the same field-matching procedure used in obtaining (11.69) and (11.70), (Problem 9). Another important feature of the energy trapping principle is its application to the design of monolithic multimodc filters. These are constructed by placing several trapped energy resonators in close proximity to each other on a, single piezoelectric plate (Fig. 11.20). The individual resonators are coupled to each other through their evanescent fields and, in this way, a coupled resonator chain is realized.f C. ORTHOGONALITY RELATIONS FOR RESONATOR MODES Orthogonality relations for resonator modes can be derived from the reei-, procity relations (Section J of Chapter 10) in a manner which parallels very closely the waveguide derivation. The existence of these orthogonality relations allows the forced oscillation problem for a resonator to be solveij by expanding the fields in a series of modes. As in the waveguide problem, this transforms the partial differential field equations into ordinary difTerentinl equations for the mode amplitudes and leads to an equivalent circuit reproJ sentation of the acoustic resonator. The practical usefulness of this approacli has been very convincingly demonstrated in microwave electromagneti theory. Only the complex reciprocity relation (10.113) will be used to deriv resonator orthogonality relations. A different set of relations can be obtaintr from the real reciprocity relation (10.109). Except for notational details, the approach used here will be the same . that developed by Lewis and Lloyd.J The resonator configuration, show t For a survey of multiple resonator filter applications see W. J. Spencer, "Monolilli Crystal Filters," Ch. 4 in Physical Acoustics 9, W. P. Mason and R. N. Thurston, euV. Academic Press, New York, 1972. J References 30 and 31 at the end of the chapter. C. ORTHOGONALITY RELATIONS FOR MODES 251 SurfacesS0 ■■•«д. If are underneath ° 1 the electrodes Ф FIGURE 11.21. Iree piezoelectric resonator. i\Tfh- П'2г' 1S, a" arbitrari,y shaPed> l^sless piezoelectric body with N o7ut on" ?2 ?durg; mTss e,ectrodcs dep°sited °» ^ solutions 1 and 2 m the complex reciprocity relation, v. (-y*. tl _ vl. t*+ф* ^+ф] m\ V " ' dt 1 dt) P о о 0 :s:R :d- .0 • d : - e + v* • F, + vx • F* + ф* ^ + ф,д-& dt 1 dt (1171) are taken to be free oscillation modes. That is, the volume sources are zero, F, - F2 = 0 Pel = Pel = 0, and the fields are assumed to have the form V] = еи°»\н(х> y> Z), etc. v2 = ^'vv(.r, y, z), etc. (.4.72) Since the resonator is lossless, the free oscillations are undamped and the ural requcncies to „,v arc real. After these substitutions have been made '» (П.71), an mtcgrahon ,s perlormed over the volume V of the resonator
252 ACOUSTIC RESONATORS Use of the divergence theorem converts the result into j{-v* - T, - ■ Tv* + ф,*(.7-)„В„) + 0„(/£ovDv)*} • & dS = — i(« — <u J '[v* T* -у"ф*] 0 C2J -V<&j </F (11.73) where the boundary surface 5 is assumed to pass under the electrodes in Fig. 11.21. The boundary 5 in Fig. 11.21 can be divided into N electroded regions 5в and an unelectroded region Sv. Because the electrodes are perfect conductors, the potential function ф must be a constant on each electroded boundary region Sp. The mass of the electrodes is ignored, and mechanical boundary conditions are assumed to be either free or rigid. For a lossless resonator, the electrical boundary conditions on the electrodes must be chosen so that there is no average power flow out of the terminals. This permits connection of an arbitrary reactive circuit to each terminal. However, only the simplest terminations, short-circuit (фк = 0) and open-circuit (/„ = 0), will be considered here. Boundary conditions at the electrodes are summarized in Table 11.3. On the unelectroded boundary region Su any set of lossless boundary conditions, including capacitance loading, may be used; but only the boundary conditions in Table 11.4 will be considered here. These boundary conditions may have discrete or continuous variations with position on Su. Substitution of the boundary conditions from Tables 11.3 and 11.4 into (11.73) gives '[v* T* -ЪФЩр о /(-/>„ — to,,) 0 0 :s L0 - V T, .-^ф„. dV (11.74) With short-circuit or open-circuit terminations on the electrodes, the right- hand side of (11.74) is zero, and i(to„ — cov)4UMy = 0, D. ELECTRICAL EXCITATION OF RESONATORS 253 where Mechanical Free T - ft = 0 Electrical Фв = constant Rigid v = 0 \ 8 D • n dS = - - л = _/ 'К т* — дф*] P о 0 0 : s :B : d ■ .0 -d: Jf \ L0 -d: .€.TJL-V^ An orthogonality relation for resonator modes is therefore \dV. (11.75) ^, = 0, юц9£ыг. (11.76) From the complex Poynting's Theorem for piezoelectric media (Section G of Chapter 8 in Volume 1) Uyy is equal to the stored energy of the *>th mode. ( alculation of this quantity will be discussed in Section E. 1). ELECTRICAL EXCITATION Ob PIEZOELECTRIC RESONATORS The primary device applications of acoustic resonators are as transducers and filters. For simple geometries, such as the thin disk transducer, it is possible to analyze the resonator excitation problem and find an equivalent circuit representation without using resonator mode theory. Modal analysis, mi the other hand, provides a circuit representation without restriction to any specific resonator geometry. TABLE 11.4. Boundary Conditions for the Unelectroded Region Su Mechanical Electrical Free T • n = 0 Short-circuit Ф = 0 Rigid v = 0 Open-circuit D ■ n = 0 TABLE 11.3. Boundary Conditions for the Electroded Regions Sp
254 ACOLSTIC RESONATORS фу ** FIGURE 11.22. Driven piezoelectric resonator. The theory of forced oscillations can be developed from the complex reciprocity relation (11.71). For simplicity the analysis is restricted to the case of purely electrical excitation, and Fig. 11.22 is taken as a typical geometry with N electrical terminal pairs, or circuit ports. The theory pre-i sentcd here is therefore applicable only to electrical filtering, but it may be generalized without difficulty to both transducer and mechanical filter problems D.l Mode Amplitude Equations Viewed from its electrical terminals, the resonator of Fig. 11.22 is simply an electrical circuit with N terminals; and the excitation may be specified by giving either the terminal voltages <v'3', p = 1,2, ... N or the terminal currents Ive™\ p = 1,2,... N, but not both. In the first case the response of the system to the excitation is described by the terminal currents, which are related to the excitation voltage^ by the admittance equations /s = iv, p,q=\,2,...N. (1i.77j For current excitation, the voltage response is described by the impedan- equations Ф, = 1ад. P,q=l,2,--.N. Only the first case will be considered here. (11.79) D. ELECTRICAL EXCITATION OF RESONATORS 255 The analysis begins by expressing the forced oscillation field as a superposition of the modal field distributions discussed in the preceding section; that is, v(x, y, - ^ a^x, y, z)j eUat T(.r, у, «у»' = a^x, y, Ф(х, у, zy«" = (J а„ФД*. У> *)) e'"" Щх, у, z)euat - (2 а^х, у, г)) etc. In electromagnetic resonator theory it is well-knownt that such a normal mode expansion does not always completely describe the forced oscillation Held, and it is often necessary to include a "static" field solution. This requirement also occurs in piezoelectric resonator problems, where the "static" field term is a solution У, 2) = 0 Ts(*,«/, z) 7* О (Ц.80) Ф,(аг, у, z) j£ 0 Ds(x, y, z) Ф 0 to the static (to >-0) boundary value problem with the same boundary conditions as the forced solution. In particular, Ф = Ф„ p=l,2, .../V on the electrodes. The complete forced oscillation fields at frequency a> are therefore given by j(x, y, z)eiai = а^(х, y, z))e'»« T(.r, y, z)eu°l = fax, y, z) + 2 aj^x, у, z))<T' Ф(х, у, z)eito1 = (фг{х, у, fc) + 2 aflJL*, у, zjj eitoi Щх, у, г)е*°>' - у, z) + £ fl„D„(*, у, *)j сГ1. etc., rather than (11.79). (11.81) Reference 25 at the end of the chapter.
256 ACOUSTIC RESONATORS The next step is to substitute (11.81) for solution "1" in the reciprocity relation, with no volume sources, and to let solution "2" be one of the free normal modes v2 = vv(x, //, z)e'"", etc. of the resonator. An integration is then performed over the resonator volume and the divergence theorem is applied, as in deriving (11.73). The forced oscillation field "1" and the free mode field "2" satisfy the same electrical boundary conditions on the unelectroded part of the resonator boundary (Su in Figs. 11.21 and 11.22) and the same mechanical boundary conditions at all parts of the boundary. Consequently the surface integral over S\ reduces to 2 f {ф* (i«iD) + <!>,,(k»vDv)£} • n dS. where Ф,7, is the rth normal mode potential on the pth electrode. The result of performing the integration on (11.71) is therefore - i(o, - ю,)\ли„ + i 4a„uJ = i f {<1>*,(.V„D) + ф„(ia,vDv)*} • ft dS L * j » ij.s\, (11.82) where Usy is given by (11.75) with v„ ■> v„, T„ - Ts, ф,, - ф, from (11.80). In a voltage-excited resonator problem, the first term in the integral on the right-hand side of (11.82) cannot be evaluated directly from the applied voltages фг„ and it must therefore be eliminated. Since the electrical boundary conditions for the normal modes have not yet been specified on the electrodes, this is easily accomplished by imposing short-circuit boundary conditions at the electrodes for all of the modal fields; that is фу„ = 0, /> = 1,2 N. This reduces (11.82) to i(w — cof) 41/* + 2 4e,l/*l = 2фво? qf)i, (11-83) f j » 1 where the superscript S indicates a short-circuit resonator mode and (Qf)B= - f Df-rWS is the rth resonator mode charge on the pth electrode. According to the orthogonality relation (11.76) only the term p = j contributes to the summation on the left-hand side of (11.83). Furthermore, 4£/* = 2ф„(с??)* D. FLECTRICAL EXCITATION OF RESONATORS 257 according to Problem 12. The amplitude of the ?lh resonator mode is therefore given in terms of the applied voltages by i4U%(co — co%) D.2 Admittance Matrix for a General Л' Terminal Pair Resonator The current responses at the electrodes, given by /„ = -но D-fidS, are calculated by substituting the mode amplitudes (11.84) into the expression for electric displacement in (11.81), This gives where Df>, y, z) = Ds(.r, y, 2) + J a^D„(:r, y, z). 1>-1 ц iW^Uo — со") = - j D,-n dS, the "static" contribution to the charge on the qth electrode, is related to the applied voltages фр by the "static" capacitances Cav ея = 2сл (П.86) These "static" capacitances are defined under the same mechanical boundary conditions as those applied to the resonator problem. The frequency со in the numerator of (11.84) has been approximated by «»f, because the vth mode amplitude is significant only near resonance. For each modal solution v — v„(.r, у, 2)е'ш"', etc., another mode solution can always be obtained by taking the complex conjugate. That is, the modes always occur in complex pairs, with frequencies <•!„ and «>_ц = —(Op. Furthermore, the normal mode fields can always be chosen so that all the electrode charges (Q%, (Qf)„ in (11.85) are pure real. The terms in the summation (11.85) can then be combined in pairs, giving /, = 2 '»>Cqv + но X гт-^-т-^ ; (11-87)
258 ACOUSTIC RESON4TORS Elements of the admittance matrix in (11.77) are therefore If L к 21/л (со*) (11.88) D.3 Resonator Equivalent Circuits In computing equivalent circuit elements from (11.88) it is usual to normalize the resonant mode fields so that s\2 2uf. - 1. The admittance matrix elements are then where (6g),(6*) 7(e»*)2-*- (11.89) (ej?)« is the free charge on the <yth electrode for the /ith short-circuit mode. EXAMPLE 8. One Terminal-pair Resonator. In this case (11.88) reduces tol у ■ \r 4- V 1 (11.90) This is equivalent to the circuit shown in Fig. 11.23, where the shunt capacitam C„ contributes the first term in (11.90). Each of the series resonant branches in thejj figure contributes an admittance C„(LUC„) Y» U" (Lucu) which corresponds to the /<th term in the summation of (11.90) if the resonan frequency is chosen according to (L,CJrin = rf (ii.9i; and the equivalent capacitance is defined by (f- (11.92) This is called the motional capacitance of the //th resonator mode. If a mode « has a large motional capacitance C„ the input admittance will be large in the vicinity t(f the resonant frequency «у That is, a strong resonance will be observed. Tin E. STORED ENERGY AND POWER LOSS 259 FIGURE 11.23. Equivalent circuit for a one terminal-pair piezoelectric resonator. motional capacitance of a mode is therefore a measure of its piezoelectric coupling strength. A more useful measure is the normalized motional capacitance с„х = г^г. (11.93) From the circuit diagram in Eig. 11.23 the normalization factor in the denominator of (11.93) is easily seen to be the difference between the zero frequency capacitance ct and the motionally-clampcd capacitance C„. FXAMPLE 9. Two Terminal-pair Resonator. In this case ^=4^+?(^b] (i,-94) and the general circuit representation is very complicated. However, if all resonant terms except one can be ignored, the circuit of Fig. 11.24 is applicable. Because there arc only three independent motional parameters, " one of the circuit elements must be specified arbitrarily. In this case the motional capacitance Сц has been taken to be unity. E. STORED ENERGY AND POWER LOSS In order to calculate the stored energy in the ilh resonator mode, f/w = IJ (pv,. • v* 4- T* : (sB : Tv - d • VФV) - VO* ■ (d : Tv e7' • V4>v)) dV, = I Г (/°vv • v* + 1 * : Sv - \"Ф* • Dv) dV, (11.95)
260 ACOUSTIC RESONATORS Сц — с,, + Cl2 Co2 = C22 + Cl2 jv,, = e„i/4 N„2 = {?„»/«>„ FIGURE 11.24. Equivalent circuit for a two terminal-pair piezoelectric resonator near one of its resonant frequencies (after Holland and Eer Nisse). it is necessary to know the particle velocity, the stress, and the potential fields. This calculation can be greatly simplified by using the complex Poynting's Theorem (8.194), in Section G of Chapter 8 in Volume T. For any combination of the boundary conditions in Tables 11.3 and 11.4 the Poynting vector is zero at all points on S, and the surface integral on the left-hand side of (8.194) is zero. Since Jf = о • E = 0 for a lossless resonator and F = pt = 0 for a modal solution, one has J dv _ J (LIS _ ™v py*) dK (, i.96) where pv = pvv. Using the complex conjugate of (11.96) it is possible to rewrite (11.95) as U„=ljp»r-»r^; (11.97) that is, the stored energy can be calculated from the v field alone. The integrals in (11.96) represent peak stored energies in the velocity, strain, and electrostatic fields. Since the instantaneous modal fields are periodic functions of time with frequency cov, the instantaneous stored energy terms are periodic F. STORED ENERGY AND POWER LOSS 261 functions of time with frequency 2o>v. In a lossless resonator the total stored energy is, however, constant. One must conclude from (11.96), therefore, that the instantaneous kinetic stored energy is 90° out of phase with the sum of the instantaneous strain and electrostatic stored energies. That is to say, the strain and electrostatic stored energies are zero when the kinetic stored energy is at its peak. The stored energy in the resonator can thus be calculated when it is all in kinetic form. This is the meaning of (11.97). Up to this point only lossless resonators have been considered, although it has been noted that all acoustic materials have internal energy losses. In a lossless resonator a free oscillation, once initiated, will persist indefinitely and the resonant modes therefore have purely real natural frequencies. On the other hand, if the resonator is lossy there is continuous dissipation of the energy stored in the oscillating field and it eventually dies away to zero. This behavior can be illustrated by the simple mechanical system in Fig. 11.25, which is governed by the equation of motion K'~ + K = 0. (11.98) or at Assuming an exponential solution х=Аеи°', (11.99) one finds that the characteristic equation for the oscillation frequency is m(u»f + io,K' + К = 0, (11.100) which has solutions '« = - — ± ,(— -—)) . (11.101) 2M \m \2mJJ This displacement of the mass in Fig. 11.25 is found by taking the real part of (11.99). If A is pure real, this gives an exponentially damped cosine function *■> £-(з#Г'- <1U02) м Si K ^ FIGURE 11.25. Damped mechanical resonator.
262 \COUSriC RESONATORS which dies away in the manner described. As the viscosity K' is reduced to zero the oscillation persists for a longer and longer time. In Chapter 12, approximate methods will be given for analyzing acoustic resonators with small losses and it will be found that all resonant modes exhibit the same exponentially damped behavior. Tf the rate of loss of energy by an acoustic resonator mode is small compared with its stored energy, it is permissible to assume that the rate of decrease of stored energy is equal to the power loss averaged over several cycles; that is _4Ш = (Р) (Ц.103) dt Assuming that the fields are damped exponentially as e-m and noting that stored energy is a quadratic function of field strength, one can express the peak stored energy in (I I 103) as (t/vv(0) = «ио»-*"- Therefore, 0P„.)av - 2oc(lUO)) and the damping constant (which is equal to the imaginary part of the natural frequency, as in (11.101)) is given by (Р*,)а\' (11.104) 2(UVV(0)) This is usually expressed in terms of the quality factor or Q of the mode, defined as _ «>0(Stored energy) (11.105jj Average power loss where mB is the real part of the natural frequency Accordingly a = ^. (11.106) 2Q Examples illustrating calculations of these quantities will be given in Chapter 12. PROBLEMS 1. Under certain special conditions, the Lamb wave solutions of Section 10.C reduce to SV waves reflecting back and forth at 45° to the plate surfaces. These are the Lame waves discussed in Problems 3 and II of Chapter M. PROBLEMS 263 Using (9.43), verify that an incident Lame wave at a free boundary reflects into a reflected wave of the same type. If w/b in Fig. 11.4 is equal to p\q, wherep and q are any integers, show that standing Lame wave resonances can be constructed by reflecting Lame waves back and forth between the boundaries z = ±u/2, and prove that the resonance condition is where a is arbitrary 2. Starting from Problem 4, Chapter 10, construct P-type standing wave resonances for Fig. 11.4 when wjb = pjq (p,q any integers). Are these solutions valid for the completely bounded plate resonator in Problem 1 ? 3. Derive the v field distribution for the torsional modes of a finite circular cylinder and verify the resonance condition (11.27). 4. Verify by direct substitution that M = V x гЧ' and N - V V x M aj
264 ACOUSTIC RESONATORS PROBLEMS 265 then use the resulting expressions to show that pv? • v„ dV = 0, T* : s : T„ dV = 0, % ^ «,„ 12. Prove that 4<л? = 1Флад in (11.83). (Use short-circuit electrode conditions for the vth mode in (11.73) and replace the /nth mode with the static solution (11.80), noting that co^ is replaced by со >-0.) 13. Verify that the orthogonality relation |*pv* • v„ dV = 0, со„ Ф V)y derived in Problem 11 applies to the torsional modes of a finite circular cylinder (Problem 3) and to the pure compressional modes of a sphere (11.40). 14. By using open-circuit resonator modes in (11.82), derive impedance matrix equations corresponding to (11.88). 15. Assume that the electrodes in Fig. 11.13 are ungrounded and that h — 0. Use the modal solutions obtained in Problem 8 and the impedance equations in Problem 14 to find the electrical input impedance between the electrodes. Compare with the input impedance obtained by imposing stress-free acoustic boundary conditions in (8.247). The series expansions . a 1 , ^ 20 cot 0 = +2, e »-i в- - (птг)2 1 * (-1Г20 esc 0 — - 4- У e £i в2 - (птг)2 will be found useful in this connection. 16. The unbounded isotropic plate resonator in Fig. 11.1(a) is viscously dampled by elastic losses. That is, c,., c44 + icorju. Show that the frequency equation (11.6) is now are dieergenceless solutions to (11.30), provided that *t+(?)v-0- Show also that M = K V x N. со 5. Show that the spherical Bessel functions of the first kind in Table 11.2 remain finite at p = 0. 6. Starting from (11.40), derive the characteristic frequency equation (11.41) for the purely comprcssional modes of a sphere. Derive the torsional mode frequency equation (11.46) from (11.45). 7. Derive the modal field distributions (11.60) and (11.61) in Example 5. 8. Assume that the electrodes in Fig. 11.13 are ungrounded; that is, the electrical boundary conditions are open-circuit. Find the resonant modes of the structure and their resonant frequencies. Show that the frequency differences between the odd-symmetry short-circidt modes (Example 5) and the odd-symmetry open-circuit modes are measures of the piezoelectric coupling. 9. Assume that the piezoelectric plate in Fig. 11.19(b) is Hexagonal (6mm) with the Z axis parallel to x. Find the trapped mode resonances corresponding to the fundamental SH0 plate waveguide mode (Example 2 in Chapter 10). Boundary conditions at z = ±vv'/2 should be approximated as in Example 7 of this chapter. 10. Verify that the resonator modes in Examples 1 and 5 satisfy the orthogonality relation (11.76). 11. For a nonpiezoelcctric resonator, the orthogonality relation (11.76) reduces to J\pvv* • v„ + T* : s : T„) dV - 0, (o„ * <or Apply the complex Poynting's Theorem (Section F.2 of Chapter 5 in Volum I) to the field distributions (vteto*' + v/'"«') (TveiMv-f 4- 1>MV) and (т/и>' + iv/'V) (TvetoV + ,T(lC"V);
266 ACOUSTIC RESONATORS in all cases where ">res// <5C c. 18. In the neighborhood of a particular acoustic resonance, the input admittance of a one-terminal pair resonator is Y = i to C0 + s.., —; , L u<v" — J according to (11.90) and (11.92) in Example 8. If the resonator is viscously damped, the substitution should be made. Assuming that Q* » I, show that Y has a pure real value at rumnx — aif and a pure real value Gu,in = —5— at "tab. = ("H 1 + 77) Verify that the real part of У is equal to G^/VS at frequencies о>® ± Дт/2, where S provided that S.<» is small compared with <отах — f'Jmin- PR OBI EMS 267 Stress-free boundary Show that the frequency equation for longitudinal resonances is tan Lj-^'b) = i-^s—. After substituting V W 7 ^ —4,+2fJr) and applying the trigonometric formula show that 1ап(Л + В)- tan/t + tang I — tan Л tan В V —I =— n = J, «и/ £> GiJ° = - ;i7r lor 2 tanh-1 " Z„ < (pc,,)1 (n - lb 2 tanh 1 (pen)1 lor ^Xpci,)lrt. and has solutions ^тмт)(1+ш" I Verify that the quality factor Q in (11.106) is ft.--5"- "'resell «rr.s = ^ «>iv> when furmr/u « cu. 17. Repeat Problem 16 for the torsional modes of a lossy isotropic cylindrical resonator and for the compressional and torsional modes of a lossy isotropic spherical resonator. Show that 19. The isotropic plate resonator in Fig. 11.1a is bonded to an infinite lossless medium with characteristic acoustic impedance Zn.
268 ACOUSTIC RESONATORS 2c2,(:" 20™ kt — 1 1 (t-j — t-j,v) 2GJV1) + 2Q™ + 1 Note that there is no reflection at w = «>„ when the radiation Q equals the] unloaded Q. REFERENCES Free Resonators 1. J. S. Arnold and J. G. Manner, "Description of the Resonances of Short Solid Barium Titatiate Cylinders," Jour. Acous. Soc. Amer. 31, 217 226 (1959 2. J. L. Bleustein and H. F. Tiersten, "Forced Thickness Shear Vibrations 1 Discontinuously Plated Piezoelectric Plates," J. Acous. Soc. Amer. 43, 131 I-J 1318 (1968). 3. R. J. Byrne, P. Lloyd, and W. J. Spencer, "Thickness-Shear Vibration ml Rectangular AT-Cut Quartz Plates with Partial Electrodes," J. Acous. Soc. Amer. 43, 232-238 (1968). 4. II. Ekstein, "Free Vibrations of Anisotropic Bodies," Phys. Rev. 66, 108 1 IK (1944). 5. D. B. Fraser and R. C. LeCraw, "Novel Method of Measuring Elastic and Anclastic Properties of Solids," Rev. Sci. Inst. 35, 1113-1115 (1964). 6. R. Holland and E. P. EerNisse, "Design of Resonant Piezoelectric Devices, Chapter 2-5, Research Monograph No. 56. MIT Press, Boston (1969). REFLRENCES 269 7. R. Holland, "Piezoelectric Effects in Ferroelectric Ceramics," IEEE Spectrum 7, 67-77 (April 1970). 8. W. H. Horton and R. C. Smythe, "The Work of Mortley and the Energy- trapping Theory for Thickness-Shear Piezoelectric Vibrators," Proc. IEEE (Letters) 55, p. 222 (1967). 9. P. Lloyd and M. Redwood, "Finite Difference Method for the Investigation of the Vibrations of Solids and the Evaluation of the Equivalent-Circuit Characteristics of Piezoelectric Resonators," Parts I, II, J. Acous. Soc. Amer. 39, pp. 346-361 (1966), Part III, J. Acous. Soc. Amer. 40, 82-85 (1966). 10. G. W. McMahon, "Experimental Study of the Vibrations of Solid, Isotropic, Elastic Cylinders," Jour. Acous. Soc. Amer. 36, 85 92 (1964). 11. R. D. Mindlin, "Waves and Vibrations in Isotropic Elastic Plates," pp. 220-230, Structural Mechanics, Pergamon, New York, 1960. 12. P. M. Morse and H. Fcshbach, Methods of Theoretical Physics, pp.1872-1874, McGraw-Hill, New York, 1953. 13. G. Nadeau, Introduction to Elasticity, Ch. 10, Holt, Rinehart and Winston, New York, 1964. 14. Y. Sato and T. Usami, "Basic Study on the Oscillation of a Homogeneous Elastic Sphere," Ceophys. Mag. 31, 15-62 (1962). 15. J. A. Stratton, Electromagnetic Theory, pp. 399-406and pp. 414-416, McGraw- Hill, New York, 1941. 16. M. Onoe and H. Jumonji, "Analysis of Piezoelectric Resonators Vibrating in Trapped Energy Modes," Electronics and Communications in Japan 48, 84-93 (September 1965). 17. W. Shockley, D. R. Curran, and D. J. Koneval, "Energy Trapping and Related Studies of Multiple Electrode Filter Crystals," Proc. J7th Ann. Symp. Frequency Control, pp. 88 126 (1963). 18. E. A. G. Shaw, "On the Resonant Vibrations of Thick Barium Titanate Disks," Jour. Acous. Soc. Amer. 28, 38 50 (1956). 19. K. Shibayawa and Y. Kikuchi, "Studies on Vibration of Short Columns Part I Theoretical Consideration for Longitudinal Modes and their Electromechanical Constants," Sci. Rep. RITU (Tohoku University), B-(Elect. Comm.) 8, 133 -150(1956). 20 K. Shibayawa and Y. Kikuchi, "Studies on Vibration of Short Columns Part II Experimental Consideration of Longitudinal Modes and their Electro-mechanical Constants," Sci. Rep. RITU (Tohoku University), B-(Elect. Comm.) 9, 113-122 (1957). 21. K. Shibayawa and Y. Kikuchi, "Studies on Vibration of Short Columns Part III—On the Longitudinal Mode with Nodal Circles and the Related Thickness Modes," Sci. Rep. RITU (Tohoku University), B-(Elect. Comm.) 11, 203 217 (I960). The quality factor g"0, which relates to radiation loss from the resonator, is called the radiation Q and should be distinguished from the unloaded Q (or Q\l') which was calculated in Problem 16. 20. Assume that the resonator considered in Problem 19 is viscously damped; that is c~n -* cn + 'e)4n- If « сц (f*.i)"" » zn show that the acoustic input impedance looking into the resonator at у — h is and that the stress reflection coefficient at the same point is 1 1 (t» — o),v) + i— —
270 ACOUSTIC RESONATORS 22. Т. R. Sliker and D. A. Roberts, "A Thin-Film CdS Quartz Composite Resonator," J. Appl. Phys. 38, 2350 2358 (1967). 23. Ft. F. Tiersten, Linear Piezoelectric Plate Vibrations, Chapter 9 16, Plenum, New York, 1969. Forced Resonators 24. D. A. Berlincourt, D. R. Curran, and H. Jaffee, "Piezoelectric and Piezo- niagnetic Materials and their Function in Transducers," Ch. 3, Physical Acoustics I-A, W. P. Mason Ed., Academic Press, New York, 1964. 25. R. N. Ghose, Microwave Circuit Theory and Analysis, Ch. 8, McGraw-Hill, New York, 1963. 26. K. Haruta. P. Lloyd, and J. L. Hokanson, "Monolithic Crystal Filter II— Normal Mode Frequencies and Displacements," IEEE Trans. SL-16, p. 21 (1969). 27. Reference 6, Ch. 2, and pp. 129 145, 167-179, 237-244. 28. С. C. Johnson, Field and Wave Electrodynamics, Ch. 6. McGraw-Hill, New York, 1965. 29. Y. Kikuchi, Ultrasonic Transducers, Tohoku University Electronics Series II, Corona, Tokyo (1969). 30. J. A. Lewis, "The Effect of Driving Electrode Shape on the Electrical Properties of Piezoelectric Crystals," B.V77 40, 1259-1280 (1961). 31. P. Lloyd, "Equations Governing the Electrical Behavior of an Arbitrary Piezoelectric Resonator having N Electrodes," BSTJ 46, 1881-1900 (1967). 32. P. Lloyd and K. Haruta, "Monolithic Crystal Filter (I) -Theoretical Model," Trans. SU-16. p. 21 (1969). 33. W. P. Mason, "Use of Piezoelectric Crystals and Mechanical Resonators in Filters and Oscillators," Ch. 5 in Physical Acoustics I A, W. P. Mason, ed.. Academic Press, New York, 1964. 34. J. W. May, Jr., "Guided Wave Ultrasonic Delay Lines," Ch. 6 in Physical Acoustics I A, W. P. Mason, ed., Academic Press, New York, 1964. 35. M. Опое, II. Jumonji, and N. Kobori, "High Frequency Crystal Filters Employing Multiple Mode Resonators Vibrating in Trapped Energy Modes." Proc. 20th Annual Syrnp. Trequency Control, pp. 266-287, 1966. 36. R. C. Smythe, "Communications Systems Benefit from Monolithic Crystal Filters," Electronics, 48-51 (Jan. 31, 1972). 37. R. A. Sykes and W. D. Beaver, "High Frequency Monolithic Crystal Filters with Po<*ihlc Application to Single Frequency and Single Side Band Use.* Proc. 20th Annual Symp. Frequency Control, pp. 288 308 (1966). 38. R. A. Sykes, W. L. Smith, and W. J. Spencer, "Monolithic Crystal Filters," 1967 IEEE International Conv. Record 15, Pt. II, 78 93. 39. Reference 22, pp. 132-136. chapter 12 PERTURBATION THEORY A. INTRODUCTION 271 B. WAVEGUIDE PROBLEMS 272 C. WAVE SCATTERING PROBLEMS 302 D. RESONATOR PROBLEMS 315 PROBLEMS 324 REFERENCES 331 A. INTRODUCTION Chapters 10 and 11 have shown that the number of waveguide and resonator problems with exact analytical solutions is not large, especially for anisotropic and piezoelectric media. This difficulty can be overcome to a large degree by using electronic computers to obtain numerical solutions of the field equations. However, such computations do have certain disadvantages. They do not show clearly the effects of changing physical parameters in the problem and also they are not easily applicable to problems with complicated boundaries.! For instance, numerical computation is easily applied to the surface wave and layered media waveguides in Chapter 10, but not to the general piezoelectric resonator geometries in Sections С and D of Chapter 11. There is therefore a real need for analytical approximation techniques, to provide physical guidelines for numerical computation and to solve prohlems that are not readily attacked by direct computation. The two most powerful approximation techniques of applied mathematics are perturbation theory and the variational method. In electromagnetism these methods have been applied 10 a wide variety of waveguide and resonator problems, and this work is almost directly transferable to the acoustic problems of interest here. In i Sec, however, the second foolnole at the beginning of Section I0.G. 271
272 PERTURBATION THEORY this chapter the basic principles of perturbation theory are presented and applied to a number of problems. The variational method is treated in similar fashion in Chapter 13. Perturbation theory is concerned with small changes in a solution, caused by small changes in the physical parameters of the problem. In the context of acoustic wave problems the theory is particularly useful for calculating the effects of small parameter changes on a numerically computed solution. It might, for example, be used to find the attenuation of a surface wave solution that has been solved numerically for the lossless case. Another application would be to evaluate the temperature coefficient of the propagation velocity for a numerically computed surface wave solution. Perturbation theory can also serve as a guideline for computation. Examination of various perturbations of a numerical solution shows trends which are useful in selecting other cases to be carried through a full scale computation. B. WAVEGUIDE PROBLEMS The complex reciprocity relation (10.113) is a useful starting point for the derivation of perturbation formulas. Application of these formulas will, of course, be limited to conditions under which the reciprocity relation is valid. This means that the unperturbed material parameters must be lossless, but ferrite and semiconducting media in a dc magnetic field are permitted It will be seen that these restrictions do not apply to the perturbed system. That is, the formulas can be applied to lossy perturbations. An alternative set oil perturbation formulas may be derived from the real reciprocity relation (10.109). These formulas (Problem 1) are subject to a different set of restrictions; the unperturbed material parameters may be lossy but ferrite and semiconducting media in a dc magnetic field are excluded. B.l Boundary Perturbations This first class of problems deals with perturbations at or outside the transverse boundaries of a waveguide; material parameters within the waveguide, interior are assumed to be unchanged. Perturbations of this kind are relevand to the microsound waveguide problem (Section G of Chapter 10) and to coupled piezoelectric waveguides (Section 1.3 of Chapter 10). They are also of central importance in the theory of the surface wave electroacoustic amplifier. In the complex reciprocity relation (10.113), it is assumed that no sources are present and that solutions "1" and "2" both vary with time as eiml\ This gives V • {—v* • Tj — vx • T* + otfrtadx) + <I\(<«'D2)*} =0. (12.1)| From this simple relation, boundary perturbation formulas can be derived1 B. WAVEGUIDE PR OBI EMS 273 FIGURE 12.1. General layered waveguide structure. The waveguide cross section is infinite in the direction and the fields are uniform along с. for waveguides of arbitrary structure. However, as in Sections J, K, and L of Chapter 10 a restriction will be made to layered structures such as Fig. 12.1, which have fields that are independent of the x coordinate. In such cases the divergence operation of (12.1) contains only у and z derivatives, and integration from у = 0 toy = b in Fig. 12.1 gives (12.2) { } = {-▼? ■ t, - v, • t2* + фд/тВ.) + o.owd,)*}. Solution "2" in (12.2) is taken to be some unperturbed waveguide mode of the structure! v., = e-,p"*vrt(y), etc. The exponential time factor assumed in deriving (10.113) has now been dropped.
274 PERTURBATION THEORY Jo 4P„ - 2 ?Ae \\-vl ■ t„ + ф,/>>в,Л - i dy Jo where Pn is the average unperturbed power flow per unit width along x. Unfortunately, the same simple approximation cannot be made in the numerator of (12.3), because this would simply give Д/5,, = 0. A more accurate approximation must therefore be found. Various approximation techniques will be demonstrated in the following examples. Once the perturbed fields have been found by use of these methods, an approximate value of A/?„ can be calculated from the general boundary perturbation formula _ -j{-< ■ t; - yn ■ т; + ф;(шр;,) + ф:,(.о>рд)*} • пи (124) Mechanical Surface Perturbations. Since small perturbation effects are directly additive, it is convenient to treat mechanical and electrical perturbations separately. For mechanical problems, the unperturbed waveguide is assumed to have stress-free boundary conditions at the upper and lower surfaces у — 0, h in Fig 12.1 and the electrical boundary conditions art- assumed to be unperturbed. Under these conditions the second term in (12.41 is zero and the third and fourth terms are omitted. If only the upper surface % Note that this does not violate any of the restrictions on the derivation of (12.2). B. WAVEGUIDE PROBLEMS 275 0 is perturbed, the change in the propagation factor is '■({-v;-t;}.& о Д/5„ = 4p„ The change in sign on the right-hand side arises from the fact that у = 0 is the lower limit in (12.4). To evaluate Д/5и it is necessary to relate the perturbed surface traction force — T'„ • у to the unperturbed fields. By analogy with the acoustic impedance defined in Section К of Chapter 7 in Volume I, an acoustic surface impedance Z'A for the perturbed case is defined by the relation -t;-y = Z^-v;. (12.5) This can be used to evaluate the perturbed surface traction force if the particle velocity field at the surface is assumed to be unchanged by the perturbation; that is, Ю„-о = (v„U- (12.6) The perturbation formula is then where Д/Зи reduces correctly to zero for the (unperturbed) stress-free boundary condition Z'A - ъА - 0. The remaining step in the calculation is to evaluate the mechanical surface impedance for the perturbed problem. In the most common practical problem, a thin film overlay on the surface, this is a relatively simple matter. Several illustrations will be given. LXAMPLE /. Mechanical Surface Perturbation by a Thin, Lossless Isotropic Overlay.^ In this problem (Fig. 12.2) the overlay is characterized by a mass density p', Lame constants '/.' = c'V2 and /(' = c'^, and a thickness A. The boundary conditions at the upper surface of the overlay are, of course, stress-free. Within the overlay the perturbed fields satisfy the acoustic field equations on the rront cover paper, with F = 0. For fields that arc independent of a; and vary with z t If the unperturbed waveguide is piezoelectric, a thin film overlay will perturb both the mechanical and electrical boundary conditions. Because of the additive property of small perturbations, the electrical perturbation can, however, usually be ignored in evaluating the mechanical perturbation and vice versa. An exception is the short circuit electrical boundary condition on a strong coupling piezoelectric, which has a strong influence on the particle velocity field v at the surface and must therefore be taken into account for an accurate evaluation of the mechanical perturbation due to a conducting overlay. This is assumed to be a lossless, propagating wave; that is /Sn is pure real. Solution "1" is the corresponding perturbed wave vT = е-<гу„(у), etc., where /S« will be complex if the perturbation is lossy.! Substitution into (12.2) then gives Д/?п = fi'n - fin -i{-< - t; - v; • т; + ф^шад + ф;,('о>р„)*} • ?к j {-< • т„ - v; • т* + ФГХ^Ю + Щ™ъп)*} ■ z dy о This is an exact expression; but the perturbed fields v.;„ T'n, etc., must be known in order to use it. If an exact calculation of the perturbed problem is available, the perturbation calculation is unnecessary. It is therefore necessary to seek an approximate solution for the perturbed fields. This is always the central problem in applying perturbation theory. Since the perturbation in (12.3) is assumed to be small, the perturbed fields in the denominator may be replaced by the unperturbed fields, giving
276 PERTURBATION THEORY FIGURE 12.2. Perturbation of the upper mechanical surface (y = 0) in Fig. 12.1 by a thin film isotropic overlay. as e~ip'z, the first field equation takes the form э , dy vx - w't'zx = nop vx д , дуТт - iP'T^ t 1 = КО p Vy г,,J- - ip'TL (12.8), Since S\ = {\luo)(dv'xjdx) = 0 because the fields are independent of x, the seco^ field equation can be reduced to . /■sii-^i2„, siic*u — s12) л — = no I ; lvy + ; lzz\ ду \ s,i su ) . (s'vzis'n - s[.z) , (s'd - Sis) , \ — Vz = 1(0 I ; Iyv h 7 lzz] \ Sn Su / — IP Vy = 1(1)544/ . dy dv'x dy (12.' by eliminating the stress component Txx. B. WAVEGUIDE PROBLEMS 277 1 yz - iP'T™ — ;<„„'.> to) - J(Op vx r<i> 1 yy = icljp't'j,0' (12.11) t<1) * zv - Ф'т™ = »v40), and the only equations in (12.9) that involve zero order velocity components arc ■sll -ifl'v™ = ias'uT™. (12.12) This shows that, in the zero order approximation, v is uniform throughout the film. The sets of equations (12.11) and (12.12) are now solved for T(1) • f as a function of v(0). For the nth waveguide mode this gives («•■«)* ( T<n)xy = ifohyp' — = uohp'(v^)y cK)Zy = io)h\ p' — kv1+twjj(^' (1213) where the approximation P'„ = Pn = «>IVr, (12.14) has been used and the compliances s'a are expressed in terms of the Lame constants in Fig. 12.2. These equations, the Tiersten boundary conditions^ for a very thin isotropic film overlay, define the acoustic surface impedance Z'A in (12.5). From (12.7) and (12.13) the perturbation formula for a very thin isotropic overlay is (12.15) This result is often expressed as a perturbation of the phase velocity Vn rather than the propagation factor /?„; from (12.14) да д К Pn Vn t Reference 23 at the end of the chapter. For a thin overlay, an approximate solution to these equations is found by expanding the fields as power scries in the variable (»/ + A), v'(,/) = v«» + >d)(,y + A) + v'2>('/ + A)2 + • • • (12.10) T'fr) = T'°> + Т<»(у + A) + T<2>(?/ + A)2 + • • • Because of the stress-free boundary conditions at у = —A in Fig. 12.2, t(0) . у = 0 in (12.10). The zero order terms in (12.8) are therefore
278 PERTURBATION THEORY Formula (12.15) has been applied most frequently to Rayleigh surface wave problems (n = R), and it can be used for substrate materials with arbitrary aniso- tropy. An important special case is that of an isotropic substrate, where \vllx\ = 0 from (10.36) and (10.37), and ApR Д1/к VHh\ , I, V X + //\ "I In this case the normalized particle velocity components at the surface, (l'l\y\=0 (vhz\ о рш ' pi/2 *r r can be evaluated analytically in terms of the mechanical properties of the substrate. Expressions for these quantities are given in Part A of Appendix 4. Using these results, the perturbation Д/Зи//?к can be evaluated for any combination of arj isotropic substrate and an isotropic overlay. Figure 12.3, for example, compares perturbation and exact velocity-curves for a polycrystalline gold overlay on fusa" quartz. The quantity (10.80) which determines whether an isotropic overlay i. "fast" or "slow" can be derived by substituting the particle velocity components al у = 0, from part A of Appendix 4, and expressing the right-hand side of (12.171 in terms of wave velocities. For Rayleigh waves on anisotropic substrates the normalized particle velocily] components at the substrate surface arc not usually obtainable in analytic form and must be computed numerically. In part С of Appendix 4 these quantities ard] listed for a number of substrate materials and propagation directions. These can bJ used in (12.15) to calculate ЪРа1ри. for a thin isotropic overlay on an anisotropic FIGURE 12.3. Comparison of perturbation theory with an exact calculation of Ravkigh wave propagation. Fused silica substrate with a gold overlay (after Skcie). FIGURE 12.4. The phase velocity as a function of finh for fused silica on YAG, with propagation along Z. Both first and second order perturbation solutions are shown (after Wolkerstorfer) The computer solution is by Solie. substrate. Comparisons of perturbation curves with computer solutions are given in Figs. 12.4, 12.5, and 12.6 for a fused quartz overlay on YAG, YIG, and lithium niobate substrates. In these cases it is seen that the first order perturbation theory, defined by the impedance relation (12.13), fits the computer solution only over a small range of the parameter PHh. To obtain a better approximation, second order terms in h may be retained in the impedance relation (12.13) and this gives a second order term in (12.15) (Problem 4). Calculations based on this theory, shown as dashed curves in Figs. 12.4, 12.5, and 12.6, arc seen to agree much more closely with the computer results. In applications of Rayleigh wave propagation on a substrate covered with a film overlay, it is often necessary to distinguish between the group velocity Vg = d<»ldp and the phase velocity У„ = w/p (Section О of Chapter 10). For a Rayleigh wave on tin unplated surface, pn is linearly proportional to o>; Vg and Vv are therefore both the same and are independent of cj. The phase velocity curves in Figs. 12.3-12.6 show that this is no longer true when a film overlay is deposited on the surface. In this case the phase velocity Vv is given as a function of ph by the perturbation
280 PERTURBATION THEORY 3600 3550 3500 t Second order / / i / Fused silica on YAG (Z propagation) Computer solution 0.0 0.2 0.4 0.6 0.8 1.0 1.2 FIGURE 12.5. The phase velocity as a function of fSKli for fused silica on YIG with propagation along Z. Both first and second order perturbation solutions are shown (after VVoIkerstorfer). The computer solution is by Solic. formulas (12.15) or (12.17), and the group velocity can be calculated from V,, + ph dph (Problem 5). A comparison of perturbation and computed curves for Vv and V„ is given in Fig. 12.7. EXAMPLE 2. Mechanical Surface Perturbation by a Thin, Lossless Anisotropic Overlay. In this problem the perturbed acoustic surface impedance is no longer given by the Tiersten boundary conditions (12 13), but impedance boundary conditions for the anisotropic case can be derived in the same general w-лЩ For fields that arc independent of x and vary with s as e.-ip"'z, the first acouste field equation takes the form a 7" by B д a/2 iPin H»pv'x i°>f*>v (I2.IH B. WAVEGUIDE PROBLEMS 281 Fused silica on YZ lithium n obate 5200 - ■£ 4600 - E - 4400 - FIGURE 12.6. The phase velocity as a function of pKh for fused silica on YZ LiNb03. The upper curves (a) are for no shorting plane at the interface, while the lower curves (b) do have an electrical shorting plane at the interface (after VVoIkerstorfer). The computer solution is by Solie. f>ince S, = (\\w)0v'Jcw) = 0. T1 can always be expressed in terms of the other itress components by using the first component of the elastic constitutive relation. In this way, 7\ is eliminated and the second field equation becomes <4 ду = iioTjS? IPX = "°TjS9 = i"T'js{ IP'n< = i«>Tjsi dy 3650
282 PERTURBATION THEORY where summation over J is, of course, implied and / s'iis'ij ~~ susn s'i = ; • % Fields in the overlay are then expanded as a power series in the variable (y + it) and the zero order set of equations are solved, as in Example 1, for T(1) ■ у as a function of v(fl). This gives the Wolkerstorfer boundary conditions^ for a thin anisotropic film overlay. For the nth waveguide mode these are t Reference 26 at tlie end of the chapter. B. WAVEGUIDE PR OBI EMS 283 5700 1 1 1 1 \ Y7 lithium niobate on \^ ZY sapphire 5500 — \\ VP in meters/second Ul s \ n. Computer so ution 5100 — First order \ _ perturbation \ theory \ 4900 0.0 0.25 0.50 0 75 100 FIGURE 12.8. The phase vclocit) as a function of pafi for an overlay of YZ lithium niobate on a substrate of Z У sapphire. The first order approximate theory is compared with the computer curve (after W olkerstorfer). The computer solution is by Vх agers. Both overlay and substrate arc anisotropic. where d = 44 - 44. These define the acoustic surface impedance Z'A in (12.5), and substitution into (12.7) gives the perturbation formula for a thin anisotropic overlay (12.21) Comparisons of perturbation and computer curves are given in Figs. 12.8 and 12.9. Ii\AMPLE 3. Rayleigh Surface Wave Attenuation due to Surface Loading by a Nonviscous Gas or Liquid. In this example the Rayleigh wave is perturbed by a semi-infinite nonviscous fluid above the substrate surface (Fig. 12.10). Power is radiated from the substrate surface into the fluid and the Rayleigh wave is .iltcnualed by virtue of the power lost in this way.
284 PrRTURBATION THEORY 5700 5500 5300 5100 4900 zx zinc oxide on zy sapphire First order perturbation theory 1.2 HGURE 12.9. The phase velocity as a function of flnh for ZX ZnO on Z Y sapphire. The first order approximate theory is compared with the computer curve (after Wolkcr- storfer). The computer curve is by W agers. Both overlay and substrate are anisotropic. To apply (12.7) in this problem it is necessary to find an approximate value for the perturbed acoustic surface impedance Z'A. As in the previous examples, this requires an approximate solution to the perturbed field problem. A nonviscous fluid supports only waves of compressional (or pressure) type. For this problem, the compressional field in the fluid medium is assumed to be a plane wave with a particle velocity field v' = fcexp(-i(fc>/Fc')fc-r) and a pressure field (12-22) p = p Fc' exp (-/(«>/Ррк ■ r), where p is the mass density and V'e the compressional propagation velocity in the fluid, radiating at an angle 6' =sin-1 V'lV^. B. WAVEGUIDE PROBLEMS 285 Radiated „ ._,,.,,„ compress onal ^=sin Vc/Vjfj-^ waves Fluid Solid substrate Rayleigh wave FIGURE 12.10. Fluid loading of a Rayleigh wave. This angle is determined by the boundary conditionst (12.23) at the substrate surface, which require that the Rayleigh and compressional wave fields are both proportional to <hv*. If the attenuation is small, it is permissible to assume that From (12.22) and (12.23) the surface mechanical impedance is 7' — L i — . — and cos С y/V*-{Vp д0в = -i ЛР« № - (КУ- from (12.7). The perturbation is imaginary (or attenuative), and the attenuation constant is = (20 ,og e) ^ dB/m. (12.24) For air at standard temperature and pressure (20°C and one atmosphere) p = 1.21 kg/m3 V'c = 343 m/sec. Figure 12.11 shows a comparison of perturbation, computer, and experimental attenuation curves for air loading of a Rayleigh wave on K-oriented Z-propagating t Only the normal components of particle velocity and traction force are continuous at a l'riclionless interface.
286 PERTURBATION THEORY T x _ 1000 1500 2000 2500 Frequency (MHz) FIGURE 12.11. A comparison of perturbation, exact computer and experimental attenuation curves for air loading of Ravleigh wave propagation on ) -oriented Z-propagating lithium niobate. lithium niobate.t The variation of attenuation with frequency arises from the frequency dependence of given in part С of Appendix 4. Discrepancy between the theoretical and experimental results may be due to the neglect of collision damping in air. Electrical Surface Perturbations. In these problems the mechanical boundary conditions (T„ • у = 0 at у = 0, b in Fig. 12.1) are unperturbed/)- The first two terms in (12.4) are therefore zero. If the electrical boundary conditions are perturbed only at the upper surface у = 0, the perturbation t More recent information is available in A. J. Slobodnik, Jr., "Attenuation of Microwave Acoustic Waves due to Gas Loading," J. Appl. Phys. 43, 2565-2568 (1972). t If an overlay is actually placed on the surface of a piezoelectric waveguide, neglect of ihc mechanical boundary perturbation is justified by the additive property of small perturbations. The electrical and mechanical perturbations can thus be evaluated independently. B. WAVEGUIDE PROBLEMS 287 v • eT • w = 0 (12.29) formula is then ^Pn — ' — - (12.2Э) As in the mechanical examples, the central problem is to find an approximate relationship between the perturbed and unperturbed fields. For mechanical problems with a stress-free unperturbed boundary, the assumption was made. In the electrical case, the corresponding procedure would be to require that either (Ф„)„_0 or (D„ • y)y _0 be unchanged by the perturbation. The difficulty is that the unperturbed electrical boundary conditions are not usually either short-circuit [(Ф„Х,=0 = 0] or open-circuit [(Dn • y)„__0 = 0], and it is not clear whether the potential or the electrical displacement should be approximated by its unperturbed value. Tt has been found that neither of these choices is, in fact, optimum. Best agreement between perturbation theory and exact numerical calculations is obtained by using the so-called weak coupling approximation in which the stress field T is assumed to be unchanged by the perturbation. From D = eT - E + d : T and the condition v . D = 0, for a region with no free charge density p„, the relation between electrical potential and mechanical stress is V ' eT ■ У7Ф = v • d : T (12.26) for the unperturbed problem and v- ет-\7Ф' = v -d :T (12.27) for the perturbed problem. If the difference between the perturbed and unperturbed potential distributions is «F = ф' _ ф (12.28) and the weak coupling approximation i' = j is made, it follows from (12.26)-(12.28) that
288 PERTURBATION THEORY l -j The perturbed fields in (12.25) are now related to the unperturbed fields by writing Ф' = Ф 4- 1 + (12.31) D' = D - eT • VF and using the arbitrary constants in (12.30) to satisfy the perturbed electrical boundary conditions. This procedure is a slight variant of the derivation given in Reference 14. To illustrate the method, a perturbation formula will be derived for the important practical case of a piezoelectric Rayleigh wave (Fig. 12.12). In doing this it is most convenient to express the electrical boundary conditions in terms of the electrical surface impedance Like Z'A in (12.5), this is an impedance per unit area. It will be assumed, in general, that the unperturbed problem has free electrical boundary conditions at the substrate surface. That is, the region above the substrate (// < 0 in Fig. 12.12) is vacuum and extends to у = —со. For this region (12.26) reduces to Laplace's equation V20 - 0. (12.33) The unperturbed potential function is therefore Ф = Фн(у)е^Ркг = e"*ve *a"', у < 0 and the normal component of e ectrical displacement is Dv = -Pn^e '"KV, у < 0. В. WAVEGUIDE PROBLEMS 289 FIGURE 12.12. Perturbation of a piezoelectric Rayleigh wave by a change in electrical surface impedance. Consequently, the unperturbed surface impedance is z,;(0) = - —j— . (12.34) The perturbed boundary conditions will be specified in terms of the normalized surface impedance For a surface wave problem, the potential must go to zero as у -»■ со (lig. 12.12); therefore В = 0 in (12.30). From (12.31) where (Ф Vo = (Ф)„ „ + A A general solution to (12.29) of the form has W(y) = Ле r* + Bev+* (12.30) where /4, are arbitrary constants and
290 PERTURBATION THEORY <*V. = ~'4(0) г кч +:iml(%o. (12.36) УО---*™^^ (12-37) fa + eft l>„ - i^(0)] From (12.35) the perturbed electrical displacement is then Substitution of the perturbed fields (12.36) and (12.37) into (12.25) and use of the impedance relation (12.34) then gives AAi._AK.__ (Щ £o 1 + ЬЦО) , (12.38) where the unperturbed boundary condition is electrically free, ei = vv;s - («£), and /afr\ \ fk /вс -w(eo + ei>) 4PR is the perturbation due to an electrical short circuit on the boundary. This is the Ingebrigtsen formulae for electrical surface perturbations of piezo- electrical Rayleigh waves. It is also applicable to other types of piezoelectric surface waves when the appropriate normalized surface wave potential is used in (af/f)sc. Since the unperturbed wave satisfies free electrical boundary conditions, a/5R -»■ 0 in (12.38) when zM) = —— —>-1 !Zk(0)| EXAMPLE 4. Perturbation of a Piezoelectric Rayleigh Wave by an Electrical Short Circuit Boundary at the Substrate Surface. The simplest application of (12.38) is acalculation of the short circuit perturbation (Д У-sJVxdsc- It was seen in Section L of Chapter 10 that this useful parameter provides a figure of merit for the coupling of a Rayleigh wave to an interdigital transducer. To evaluate this perturbation it is necessary to know only the normalized electrical potential at the substrate surface (Part D of Appendix 4) and the effective permittivity e^T, which can be calculated from constants given in Part C.2 of f Reference 10 at the end of the chapter. B. WAVEGUIDE PROBLEMS 291 Appendix 2 in Volume I. A comparison of perturbation and exact numerical calculations of (^fr/flt)sc is given in Part E of Appendix 4. For all but a few cases, the agreement is very good. These exceptional cases are characterized by the existence of a Bleustein-Gulyaev surface wave under free electrical boundary conditions. The propagation velocities of Rayleigh and Bleustein-Gulyaev waves differ by only a few percent. Under these circumstances a perturbation of the system introduces appreciable coupling between the Rayleigh wave and the Bleustein-Gulyaev wave, and the perturbed fields cannot be approximated by assuming that T remains constant. This is a difficulty that must always be watched for in performing perturbation calculations (Problem 15). EXAMPLE 5. Perturbation of a Piezoelectric Rayleigh Wave by an Arbitrary Impedance Boundary Close to the Substrate Surface. Figure 12.12 shows a typical illustration of this problem, which is relevant to calculation of the beat wavelength in the piezoelectric surface wave coupling described in Section 1.3 of Chapter 10 and also to the Rayleigh wave amplifier in the following example. To apply (12.38) to this problem, the impedance at у = —h is transformed into an equivalent impedance at the substrate surface= 0. The transformation law, which corresponds to the impedance transformation along an electrical transmission line, is derived by writing general expressions for Ф' and D'v in the region 0 > у > —к. Since Ф' satisfies Laplace's equation, these are ф' = (Ае"к* + Be-ify-'pr'- дФ' л; = -«о--£- = -Ръ^АеРв? - Ве-»к*)е-^ where the perturbed propagation constant P'K has been approximated by pK. From (12.35) the normalized impedance at arbitrary у is therefore 'I \ I It* е-»Н* i A The constant BjA is evaluated by setting у = — h in (12.39), and this gives the mpedancc transformation aw i tanh pRh + z'( h) ^-.-^(%«JThM- (12-40) After substitution of (12.40) into (12.38), one has lАУЛ lАУЛ J I -tanhfaA \ / 1 +iz'E(-h) \ \ Ун /V< u \ "и Лзс (" Vo + 4 tanh pKh) u - i$z'K(-K)J' 11 л ' where T _ «f + <r0 tanh PKh C'' ~ e0 + tanh pKh " and the approximation р'н = pR has been made. The electrical displacements D„ and D'v are expressed in terms of potentials by means of (12.34) and (12.35), and elimination of A gives
292 PERTURBATION THEORY fib Radians FIGURE 12.13. Comparison of perturbation theory with an exact numerical calculation for YZ lithium niobate and z^(-A) = 0. (After Kino and Reeder) Figure 12.13 shows a comparison of perturbation theory and numerical computation for YZ lithium niobate and zjj(—Л) = 0. To calculate the beat wavelength for coupled piezoelectric Rayleigh waves it is necessary only to evaluate (12.41) forz^(^A) = 0 and z^(-A) = со (Fig. 12.14). The even and odd velocities required in (10.102) are then simply F++ = (F„Xa'(-m=oo = V\i + (A Vv)zE-i-b\ со (12 42) ..4 = 0 f h v 2 (Ь) FIGURE 12.14. Perturbed Rayleigh wave configurations used to calculate piezoelectric wave coupling (a) 4(-A) =0,(Ь)4(-Л) = со. B. WAVEGUIDE PROBLEMS 293 EXAMPLE 6. Rayleigh Wave Amplifier. Electronic amplification of a piezoelectric Rayleigh wave can be achieved by placing a semiconducting material carrying a dc current in the vicinity of the substrate surface. A typical arrangement is shown in Fig. 12.15. Analysis of this device has been one of the most important applications of Ingcbngtscn's perturbation formula (12.38). A detailed treatment of current carrier dynamics! in the semiconductor film of Fig. 12.15 gives an expression for the normalized admittance y'v(—h) = ~—-— гК(-Л) at the semiconductor surface. For a film of arbitrary thickness , . 1 Pn(0c€s tanh pRd »it-A) = TT—= fo«rfB JL-L^ (12-43) **<-*> (V{V n , . /<■», tanh M where K„ is the drift velocity, is the diffusion frequency, <°c = "J и is the dielectric relaxation frequency of the semiconductor, and 'a = Уц1\!соссоп is the Debye length. After substitution of (12.43) into (12.38), the gain per unit length ol0 is calculated from aff = (20 log е)(У»мАРъ) dB/m. (12.44) Figures 12.16 and 12.17 compare perturbation calculations with experimental results. ; D electric, td '■ d >■ V0 Semiconductor film, ts, <rs, <£ Air gap. eo Raylegh wave ■ Piezoelectric tp . r FIGURE 12.15. Rayleigh wave amplifier structure. (After Kino and Reeder) [ Reference 12 at the end of the chapter.
294 PERTURBATION THEORY -Я 60 20 1— i 1 1 1 ii 11 ° experiment 1 1 1 i ш1 — theory ^л = 560а о >v _ / о s ° 1 1 1 1 \ 1 1 11 h = 1580 a i i i f i i i i frequency in ghz FIGURE 12.16. Comparison of theoretical and experimental acoustic gain versus frequency for a silicon-lithium niobate Rayleigh wave amplifier. (After Lakin and Shaw) b.2 interior perturbations In this class of problems the perturbations are entirely inside the boundaries у = 0, b in Fig. 12.1 and the boundary conditions are assumed to be lossless. To derive perturbation formulas for this case the complex reciprocity relation (10.113) must be slightly modified. The fields v2, T2 etc., now represent a solution corresponding to lossless material parameters p2, s^, etc., and the fields vb т1э etc., correspond to a different set of material 0 40 80 120 lbu 200 240 280 320 360 400 frequency (mhz) FIGURE 12.17. Comparison of theory and experiment for a Rayleigh wave amplifier using a silicon-on-sapphirc semiconductor and a lithium niobate substrate. (After Kino and Reedcr) B. WAVEGUIDE PROBLEMS 295 v • {-v* • t, - y, • t* + tdfo-todj + ф^ъб*)*} t[vf t* -уф*][др о 0 " 0 :As:E :ad- where _0 -ad: .Де.гл_-?Ф, ДР = pi — Pi, etc. (12.45) This is the starting point for derivations of interior perturbation formulas. In the special case of waves on a layered structure (Fig. 12.1), with fields independent of x, only the у and z derivatives appear in the divergence operation and integration of (12.45) from у = 0 to у = b in Fig. 12.1 gives jo3^ )'idy= ~{ } ' yj - ioj & dy, (12.46) tth and { } = {-v? ■ t, - v, • t* + ©ftiodo + Ф^а.Т),)*} ^ = Apv% ■ v, + T* : (AsE: 1\ - ad - уф,) + vф* • (AeT • уф1 - ad : tj. Solution "2" in (12.46) is taken to be a lossless unperturbed waveguide mode v2 = e-{li"z\n{y), etc., and solution "1" is the perturbed wave v, = е-*""'х(2), etc. since the boundary conditions are unperturbed the boundary terms in parameters Pl, sf\ etc., which may be lossy. There are no sources present and solutions "1" and "2" both vary as e'«". Under these assumptions, a repetition of the steps leading to (10.113) now gives
296 PERTURBATION THEORY (12.46) are zero when these fields are substituted, and one is left withf со P(APv* • v;, + T* : (As" : T'„ + Ad ■ EJ,) Jo A/S„=- + е;-(А6у-е; +Ad:T;))^y f {-v* • t; - v; ■ т* + Ф*п(иоЮ + Фп(1юъ„)*} ■ z dy Jo (12.47) where Ap = p' - p A_s* = se- _ sk etc. Just as in (12.3), the denominator of (12.47) can be approximated by AP„. Methods of approximating the perturbed fields in the numerator will be considered in the examples below; but, before doing this, it will be convenient to express the second and third terms of the numerator in terms of v rather than T. This is easily accomplished in the following way. In (12.47) the second and third terms of the numerator can be rearranged as T: : (sE': T; + d' • e;) - T; : (e* : T„ + d • en)* + e* ■ (£t' • e;, + d' : T;) - e; • (eT • e„ + d : tj* (12 48) if all of the lossless constitutive parameters sK, eT, and d for the unperturbed waveguide arc pure real.t The piezoelectric relations on the front cover papers can then be used to convert (12.48) to т*: s; - t; : s* + e* • d; - e; - d* = s; : T: - S: : T'n + e* • - e; . D* (12.49) Substitution of the piezoelectric constitutive relations for T*, T^, and D*, D'„ in terms of the strain and rearrangement of terms then gives the volume perturbation formula AS. = P(Apv* - v; - S* : Ac* : S'n + E* ■ AeT - e; 4f„ Jo + e:-Ae:S; + S*:Ae.e;)^ (12.50) Ар = p — p, etc., Vsv S = — ico where and t As in Section M of Chapter 10, functions bearing mode subscripts are independent of Incoordinate 2. For this reason, the electric field should be expressed as E„(y), rather than —V0>„(y). The modal potential function Фп(у) gives only а у component of the electric field when the gradient operation is taken. J This excludes ferrite and semiconducting media in a dc magnetic field. B. WAVEGUIDE PROBLEMS 297 Mechanical Volume Perturbations. In these problems the electrical and piezoelectric properties of the waveguide are unperturbed. The third, fourth, and fifth terms under the integral in (12.50) are therefore zero. EXAMPLE 7. Temperature Dependence of the Rayleigh Velocity. In surface wave delay line applications, minimizing the changes of delay with temperature ,9~ is often an important practical consideration. To do this it is necessary to evaluate the temperature coefficient of the surface wave velocity for various substrate materials and orientations. A direct computation of the velocity temperature variation from temperature coefficients for the substrate material parameters requires very high precision, because small differences of large quantities are involved. Perturbation theory provides a useful means for avoiding this difficulty. Although piezoelectric substrates are of primary interest in this problem, the simpler nonpiezoelectric case will be used to illustrate the method. For a Rayleigh wave on a nonpiezoelectric substrate (12.50) reduces to Д£ "ft where hi Vn f™ - = - -ГГ = 7# • v t - «г* *>J:Ac: V^rf,. ('2-51) dp _ dc The upper limit of integration may be taken to be infinity, as the thickness of the substrate in most instances is very much larger than the Rayleigh wave penetration depth. For very small changes Д/> and Дс the approximation vn = VR can be made, and (12.51) then gives bW = -4^J0 [W W 02.52) This is the Rayleigh velocity temperature coefficient for an anisotropic, but nonpiezoelectric, substrate. In the isotropic case (12.52) can be evaluated analytically, using (10.36) and (10.37). For anisotropic substrates the field distribution must be obtained by numerical computation. EXAMPLE 8. Rayleigh Wave Attenuation Owing to Viscous Damping in the Substrate. Another important problem is the calculation of Rayleigh vsmve attenuation due to a lossy substrate material. In this example the substrate is allowed to be piezoelectric. If viscous damping is assumed Дсл = ;Ът)Л,
298 PERTURBATION THEORY where r\E is the viscosity tensor defined in Section E of Chapter 3 in Volume I, and (12.51) then becomes -/ f m when the approximation SR = SK is made. According to Section F.2 of Chapter 5 in Volume I the integral is twice the average power loss (Pd)KV per unit surface area. Therefore aR = (20 log e) (_^IdB/m (12.53) with г со Jo V>ft:4B: in watts/m2 and Pn in watts/m. This is the King-Sheard surface wave attenuation formula.^ It can also be stated in terms of an effective viscosity coefficient; that is, where 2/>P| ' (12.54) is a function of the particle velocity field vR and the viscosity matrix componentsj For an isotropic medium there are two independent viscosity coefficients, the compressional viscosity rju and the shear viscosity »j44- The attenuation constant for a Rayleigh wave on an isotropic surface can thus be evaluated analytically] and is found to be 8.686co2 with 4<£(1 -4)2 " = T 2 я-о -<)[44 - a +«?/] / = 4«(2, + [4^(1 - «2г)/(1 - <)] (1 + </, where ats = a-tjfai and an = *ti!Pn (see Section C.3 of Chapter 10). For most isotropic insulators С « Я, and Rayleigh wave attenuation is therefore dominated by the shear viscosity »;44. In anisotropic materials the number of independent viscosity coefficients is equal to the number of independent stiffness constants. For quartz there are 6 independent t Reference 11 at the end of the chapter. B. WAVEGUIDE PROBLEMS 299 TABLE 12.1. Viscous Damping of Ravleigh Surface Waves on Quartz Substrates (After King and Shcard) Substrate Orientation* Attenuation at 1 GHz (dB/cm) n** ''elf Perturbation Theory Experiment*** YX 0.33 7.0 8.6 ZX 0.33 6.3 — XY 0.36 6.9 — ZY 0.39 5.2 ■— XZ 0.33 4.1 — YZ 0.34 4.1 — * The first letter gives the surface normal, and the second the propagation direction. ** In centipoises (units conversion ratios on back endpaper). *** E. Salzmann, T. Plieninger, and K. Dransfeld, appl. Phys. Lett. 13, 14-15 (T968). coefficients, with values 'in E37 !?la=0.73 4» = 0-97 vis = 0.72 ?jM = 0.36 nu = 0.01 in centipoises. Table 12.1 lists attenuations calculated from these viscosity coefficients and numerically computed Rayleigh wave fields. It should be noted that the I \BLE 12.2. Room Temperature Propagation losses in LiNb03 at 1 GHz (After Slobodnik et al.) Surface Waves, У-Cut, Z-Propagating Volume Waves, Z-Propagation* '1 emperature- dependent attenuation 0.7 dB/microsecond (Shear 0.9 dB/microsecond Temperature- |Compressional 0.3 dB/microsecond independent attenuation (Best sample) 0.2 Air loading 0.2 Total 1.1 dB/microsecond • Л. B. Smith, M. Kcstigian, R. W. Kedzie, and M. I. Grace, j. appl. Phys. 38, 4928^1929 (1467).
300 PERTURBATION THEORY effective viscosity coefficient docs not differ widely from the shear viscosity just as in the isotropic case. (Measurements of Rayleigh wave attenuation on a lithium niobate substrate (Table 12.2) indicate the same behavior.) Table 12.1 also gives a comparison of perturbation theory with experiment for the VX orientation. The discrepancy may be due to the surface defects and to polishing damage to a depth of several Rayleigh wavelengths into the substrate. In Table 12.2 the temperature-independent part of the attenuation is attributed to surface mechanisms of this kind. Electrical Volume Perturbations. Tn this case just the electrical properties of the waveguide are perturbed. All but the third term in (12.50) is equal to zero. EXAMPLE 9. Perturbation of a Piezoelectric Rayleigh Wave by a Perfectly Conducting Layer within the Substrate. In Example 4 the perturbation of a Rayleigh wave by a perfectly conducting layer at the substrate surface was seen to provide a figure of merit for operation of an intcrdigital transducer on a piezoelectric substrate surface. For some applications one would like to use an inter- digital transducer to excite Rayleigh waves on a nonpiezoelectric substrate.^ One way of doing this is to place the interdigital array directly on the substrate and then cover it with a piezoelectric overlay. Tn this case, the relevant figure of merit is given by the perturbation of the Rayleigh wave velocity by a perfectly conducting layer at the interface between substrate and overlay (Fig. 12.18). The problem may be approached by assuming that in the thin perturbing layer of thickness d, and then letting a' — со. From (12.50) ■ eh+d &fi*=7ir\ En-Jn^ 02.56) where is the conduction current density in the perturbing layer. Before taking the limit of infinite conductivity, the integrand in (12.56) must be rearranged. Let У (if, *) = jb&V~*»' <12-57> and Ф(У, г) = Фк(«/)«г г"кг (12.58) t Sec, for example, Reference 24 at the end of the chapter. B. WAVEGUIDE PROBLEMS 301 thin conducting layer FIGURE 12.18. Surface wave perturbation by a thin conductive layer in the interior of a Rarteigh waveguide In the example treated, the electrical boundary condition at the plane у = 0 is assumed to be short circuit. be the current density and the unperturbed potential in the perturbing layer. Because V ■ J' =0 within the layer, V ■ (ф*Л') = r£*V • J' + УФ* . J' = ?ф* . J'. Use of (12.57) and (12.58) converts this relation to V ■ (*,*fc)jr(S/V-<-V*) = -E*M • jR(j/)e-<-V*, (12.59) where -Vfcfjf.*) =Et,(y)e-il>^. In the zero order approximation (P'R = pR), (12.59) becomes and substitution into (12.56) gives = ~ л „ ~ (фКЛ J- (12-60) For a layer of high, but finite, conductivity the boundary conditions at у = A, A + d require that
302 PERTURBATION THEORY C. WAVE SCATTERING PROBLEMS Perturbation theory can also be used for wave scattering problems. Although problems of a very general nature may be treated in this way, only two- dimensional examples will be considered here. Figure 12.19 illustrates • typical situation. The waves arc uniform along the x coordinate but, by contrast with the problems of Section B, the perturbation now varies along the direction of wave propagation. A different approach to the analysis will therefore be required. In Fig. 12.19, the unperturbed waveguide is uniform along the z direction and supports pure traveling wave solutions v = e^"vM(y), etc. C. WAVE SCATTERING PROBLEMS 303 These modal solutions are governed by the quasistatic field equations v • T = icopv v> = ia>sE : T — ioA • v<$ (12.63) V.(-€T-VO + d:T) = 0, with unperturbed boundary conditions at у = 0 [defined in terms of surface impedances (12.5) and (12.32)] assumed to be stress-free, ZA = 0 and electrically free (12.34), '-e . q For the perturbed structure, the field equations arc V • T = io>(p + f, Ao)v' vy = ko(se + e usb): I" - iw(d + в Ad) • vg>' (12.64) V . (-(eT + £ AeT) • УФ' + (d + e Ad) : T') = 0, and the boundary conditions are Z'A = e AZA Z'E — + f.l\ZK. l«/5„e0 Here e is a perturbation parameter, introduced to separate the various orders of approximation. After serving this purpose it is set equal to unity. If the layer is very thin, Фд is the same at both boundaries; (12.60) then becomes /;n K n 4iK and this result remains unchanged when the conductivity is allowed to become infinite. In (12.61) the perturbed electric displacement field D'M cannot be approximated by the unperturbed field, which is continuous through the perturbing layer. To find D[, the weak coupling approximation ((12.28) (12.31)) is again used. For Region 1 in Fig. 12.18 smh (Me i\ )h is the solution to (12.29), subject to short circuit boundary conditions applied to ф; = фг + Tj at у = 0 and h. The solution in Region II is Tu = (Фи), A^«V^. According to (12.31) the perturbed electrical displacements on either side of the perturbation are therefore U>iiJ*-ft = (OuAmi + (OiAw^iWk /?) coth y+A) and and (12.61) gives vvhere
304 PERTURBATION THEORY (12.65) terminal 2 z = I perfectly conducting strip perturbation p', X, j"' terminal 1 z = 0 transmitted rayleigh wave FIGURE 12.20. Rayleigh wave scattering from a thin, isotropic strip overlay. (A' = c'lit P = £44). t The perturbation parameter e has now been set equal to unity. C. WAVE SCATTERING PROBLEMS 305 These define the scattered wave fields, which are driven by distributed volume source" and "surface source" terms calculated from the zero-order or incident wave) fields. As will be illustrated in the following examples, these forced wave equations can be solved by the normal mode method of Section К in Chapter 10. EXAMPLE 10. Scattering of a Piezoelectric Rayleigh Wave at a Very Thin Isotropic Strip Overlay. In this problem there is no volume perturbation and the surface impedances are perturbed just at the upper boundary у = 0 (Fig. 12.20). Only the scattering into reflected and transmitted Rayleigh waves will be calculated, but the method can also be used to find the scattering into Lamb waves. Underneath the perturbing strip the mechanical surface impedance at # = 0 is p'-fi'lV^ 0 0 -icoh 0 '-1%\ттг?)_ from (12.5) and (12.13). If the strip is perfectly conducting, the electrical surface impedance perturbation is д-^е = — C^a')free bound пгт = ~7, - The source terms in the first order boundary conditions of (12.66) are therefore [/- мфк»*, n в '"it (12.67) eopit Solutions to (12.64) are constructed by superposing modal solutions for the unperturbed waveguide. The problem is solved by expanding the total fields in terms of the perturbation parameter e, v' = v'01 + f?v(1) + e2v(2> + - • T' = T"» + eT1' + e4m + • ф' = ф<п> + еФ*11 + *2Ф(2) + D' = D(n> + eD(1> + k2D<2> + These fields are then substituted into (12.64) and equal powers of rj are equated. In the zero-order equations, which are solution is taken to be an incident wave, vIU) = е-*"ти(У)> etc., The equations of first order in e arc thent V • T'1' = uoPv(1> + ito Apv"" V/1' = i<osK : Тш - hod • ТФ'1' + iw AsK : T(0) - iv> Ad - УФ'0) (12.66) V*. (-eT • W + d : T(1)) = -V • (-Ae7' - VФ(0) + Ad : T""), subject to the first order boundary conditions _ т(1) • у = AZ,, • v(w ф'" = (Г,, - bZK)io>Oa> • у + AZKm>D(01 • y.
306 PERTURBATION THEORY r( •Т"'w + «&-<toD<" <*» + Ф<1ч*)(/™Юк+)*^ o • У Лк+0)=< / - a l + a 2 <Z< 2 0, elsewhere (12.68) and (^-#u)^k-M = < ' —( -v* -Т'1»^) + Ф* (i»D«4*» + Фичз)(|0>О,^*j 4"r( 1* 0 / - a i + a 2 <Z< 2 0, elsewhere (12.6ЧЦ where eip^\i^_(i/), etc., are the fields of a negative-traveling Rayleigh wave with unit amplitude. From (12.65) the perturbed electric potential and displacement are Ф' = oR + фш D' = DR + D<" in the first order approximation.* Substitution into (12.68) and (12.69), and use of the boundary conditions 1 (12.70) 1 k0e0/3k Ф' = 0 (kudlt -y) converts the driving terms to _l( v*+-T»>(z) + Ф in (12.68) and *(imD'(z))} у A- ( • т<1>(г> + *r-('»>D'(e))] • У 4/r ( j«-0 (12.71) (12.72) in (12.69), where d' is the pcrluibed electrical displacement in (12.70). In (12.71) and (12.72), T(1) • у is related to the unperturbed field by the boundary condition (12.67), but the perturbed electrical displacement D' • у is as yet unknown t After the perturbation parameter с has been set equal to unity. Г. V< AVE SCATTERING PROBLEMS 307 where the zero-order field has been taken as a positive-traveling Rayleigh wave with unit amplitude, v(0) = <r!/,r*vH+(,'/), etc. According to (10.134), the forward- and backward-scattered Rayleigh wave amplitudes are calculated from the differential equations It seems reasonable at this point to follow the weak-coupling approximation used in deriving Ingebrigtsen's perturbation formula (12.38). That is, the stress field is assumed to be unchanged by the electrical perturbation. The perturbed electrical displacement is then (D' • y),J=Q = -pE(eo + ^W*. (12.73) since z'K(0) = 0 in (12.37). According to (10.164) the potential and the particle velocity for the negative- traveling Rayleigh wave arc given by (ФкЛо = (Ф*ло (12.74) (vK-.)*=o = -(v,^)*_0. After substitution of (12.67), (12.73), and (12.74), the driving terms (12.71) and (12.72) become /e~1pr" = 47н к " ■ v»+ _ + Я |фн+11_0 c"'pr* (,275) for (12.68) and f c ~ 4^ { ~v,l+ " ^ ' vl1^ ~ 'ft>/ir(e° + е?)(фк )2} _0е"!ркг п2л6) for (12.69). Integration of the differential equations (12.68) and (12.69) by the methods used in Section L of Chapter 10 gives the scattered wave amplitudes (Ml+oJ/2 d* = afe-*^ _ u-n)/2 for z > (/ + a)/2, and (12.77) fu+n)/2 sin Я n Jfl-a)f* Pil for z < (/ - a)j2. Solutions (12.77) can now be used to find the scattering coefficients in Fig. 12.20. 'ITie reflection coefficient at г = 0 is sin fi.>a „ , Su = Лк-(0) = p^f_e pit and the transmission coefficient from z = 0 to z = / is S21 = е-1"*1 + All+(l) = (1 + afy »•». Го evaluate the scattering of a negative-traveling incident wave, it is necessary only to substitute R_ for R in (12.67) and (12.73), and to make appropriate xign changes in the exponents. More useful formulas for the scattering coefficients are obtained by expressing the driving amplitudes fv and f in terms of simple physical paiainctcis. From (12.38) the second terms in/+ and/_ arc
308 PERTURBATION THEORY and ... /(Фк+П /ДРк\ respectively. The first term in/^. is I ДРц lc3.3 from (12.15), and the first term in f_ is -ip. Д Kf APi: Stiffness/ where /C^ is defined below. Scattering coefficients at z = 0 and z = / in Fig. 12.20 are therefore 5„ - -/sin /?Ra A e '"r1 522 = -«sin^BflA*r-*«l 5]2 = S21 = (1 -1р>Д> (12.78) Here <4 = л = -л: ^R Д^т R + A", |ДРр + КГ sc ДР, 11 |stiffiicss R SO with **~\ l»nja JU /(У + 2fi){vK+x? + 4(л' + /Q(tW2\ + 2//)|rw+J2 + 4(Л' + ^OIpjbhI1 Pr |Мазя PB* 4Pu p' Ipr+IiI-o Stiftuess and v hi /so is given by (12.38). If there is no crystal symmetry plane normal to the z axis, KM and KE are generally complex. This means that Д is also complex; and 5U, S22 have different phase angles. According to Example 6 of Chapter 10, the phase angle of the third teniij C. WAVE SCATTERING PROBLEMS 309 in Д is 190° for Z propagation on a — У cut lithium niobate substrate. For a thin aluminum strip perturbation the first two terms in Д_ arc negligible compared with the third term; and the reflection coefficients Sxl and 522 for positive- and negative- traveling Rayleigh waves differ, therefore, by a phase angle of 20°. When a symmetry plane does exist, Pr^ and vRJI y are pure imaginary and v Ф11ч_ arc pure real (Part 1 of Section 10.M). In this case KM, KE, and Kc are all real and, as required by the symmetry, Su = S'i2. The surface wave reflection and transmission coefficients at the strip now have the same form as for a bulk acoustic wave scattering at a layer with a slightly different impedance and velocity. (Compare Sxl with the bulk wave reflection coefficient obtained by following the method described by Problems 8-10 of Chapter 6 in Volume 1). This physical interpretation of surface wave scattering at a strip has been used to analyze the effects of scattering in interdigital transducer structures (W. R. Smith, H. M. Gerard, and W. R Jones, "Analysis and Design of Dispersive Interdigital Surface Wave Transducers," IEEE Trans. MTT-20, 458-471 (1972); W. S. Jones, С S. Hartmann, and T. D. Sturdivant, "Second Order Effects in Surface Waves Devices," IEEE Trans. SU-19, 368 377 (1972) Reflection at arrays of strips and grooves has also been used to realize surface acoustic wave filters (R. C. Williamson and H. I. Smith, "Large-Time-Bandwidth-Product Surface-Wave Pulse Compressor Employing Reflective Gratings," Electronics Letters 8, 401-402 (1972). A review article on these reflection filters appears in Reference 22 in the bibliography. EXAMPLE 11. Rayleigh Wave Attenuation Owing to Surface Roughness. Example 10 dealt with scattering of a Rayleigh wave by a localized perturbation on the substrate surface. In this example the substrate is assumed to be semi-infinite and isotropic, and the perturbation is a roughening of the entire surface (Fig. 12.21). A calculation will be made of the energy scattered from a Rayleigh wave into longitudinal and transverse plane waves propagating into the substrate. The problem is approached by first finding the perturbed boundary condition at the substrate surface, and then using (12.66) to evaluate the longitudinal and shear wave radiation fields. Unperturbed surface \ Perturbed FIGURE 12.21. Characterization of a roughened substrate surface by the roughening function fr(z). с is a perturbation parameter.
310 PERTURBATION THEORY dfrW ftO) = yn„(z) + z,h(z) ъ у - if -±—. (12.8 Since the roughened surface is stress-free, the exact boundary condition for the! perturbed field is T'(.'/, *) • fi(z) - Т'(?У, *) - f - e T%, z) ■ i = О. у yT(z), (12.81) from (12.80). As in (12.65), the field is expanded in powers of c, T'O/, z) = Т'°Чу/, z) + <Т(1)0/, z) + ■ ■ • . The boundary condition (12.81) then becomes T' • fi(V) = T'°> ■ у + f ^T(1» - у - T<°> ij + - ■ - = 0 (12. 82 at у = yT(z). Each term in this perturbation series is now expanded as a powe series in y; for example, x*.or.«>-» - or« - л- + as-l.+ ■ The final expression for the boundary condition at у = yT(z) is then (t'-aw = a™ -y%=o + {/,(^r^)yo+ ст(1'-у)^ - §(т<°> .«^1 = 0, (12.8. where only the first order term in и is shown. From the unperturbed boundary condition 7,A =0, у = 0, the zero order term in (12.83) is zero; and the first order boundary condition ' therefore (T<»-a=o = -г. /ат'°> • y\ (12.8- The exact boundary condition on the perturbed field at у = yr(z) has thus hern converted to an approximate boundary condition at the substrate surface, у =■ 0. Setting the perturbation parameter в equal to unity, the radiated field is calculate! by using (12.84) as a boundary source term in (12.66). C. WAVE SCATTERING PROBLEMS 311 Once the radiation field has been obtained, the power loss can be calculated. If this is small and is averaged over many corrugations of the perturbed surface, the average power loss (Pd)AT per unit surface area can be treated as a uniformly distributed viscous loss, and the attenuation is calculated from aR = (20 log e) dB/m, (12.53) where (Ра)±у = average radiation loss per unit surface area and PK = average Rayleigh wave power per unit width normal to the propagation direction. To evaluate the attenuation factor aR, the roughening function is expanded in a bourier series, fT(.z) 1/ги/(2""г/Ч (12.85) m— со where >.T is the fundamental corrugation period of the surface. Attenuation factors for the individual Fourier components are then weighted according to f*m and added, giving CO l»=—co Tor the case of an isotropic substrate the with Fourier component of the roughening function gives an attenuation 4** (s - l)Vs(s - q) {vq - ? IZ-12 + VT - v* I Г|2} aRm = ~, :; "—, dB/m, (12.87) K, 4s2 - 3qs - 3s + Zq - His - 1) V(s - q)(s - i) where * = (Я5/ЛЕ)3 ч = (W к к v = — + m —, /R f.r А,, А,, Лк are the longitudinal, shear, and Rayleigh wavelengths, /„„ГС - 2>f)(2s - 1) f(l+4™-4i) ЪпК \\ l -2 u 4.vttt_+2""1" "1 д ~ " " 41 f _ ^r,fa - r^? + _ m + % - 4.) _ 2^ j-! v^l 2sv's - 1 i 2v'5 *r j j Д = -(2j;2 - If - 4j?2V(9 - if)(l - rf) mid A,, is the fundamental period of the surface roughness. This is Brekhovskikh's attenuation formula^ for a roughened isotropic substrate. In Fig. 12.22 the calculated attenuation for a sinusoidal surface is shown as a function of the corrugation period Ar. The normalized attenuation a}lAs is a function t The expressions for L and T incorporate corrections in the sign of terms containing m (Reference 18). An analytic description of the perturbed surface in the figure is given by the roughening function уЛ*)=&тЮ> fl2-79> where f is the perturbation parameter and it is assumed that: (a) yr is much le than the Rayleigh wavelength AR, and (b) the slope of the perturbed surface is smallJ (dfTjdz) « i. The normal to the perturbed surface is therefore
312 PERTURBATION THEORY 10.0 r- FIGURE 12.22. Theoretical Rayleigh wave attenuation due to sinusoidal roughening of an isotropic substrate. (After Sabine) only of the Poisson ratio of the substrate 1 -2(VjV.f "га-WW (12-88) and is shown for two values of this parameter. The curves exhibit two maxima, which are high and narrow for small values of a but broaden out as с becomes larger. These peaks occur when the scattered longitudinal wave propagates parallel to the surface, in the same direction as the Rayleigh wave for the left-hand peak anil in the opposite direction for the right-hand peak. This behavior is fully discussed in Reference 17. Figure 12.23 shows attenuation calculations for a sawtooth profile on an alunii num substrate, including only the first and third harmonic components in (12.86). Experimental results from Reference 17 are also shown for comparison. Agreement is better for the smaller amplitude sawtooth, as would be expected from the assumptions of the theory. Large discrepancies occur at the above mention- attenuation peaks, where the assumption of weak scattering breaks down, and f points where the fundamental period of the sawtooth is some integral multiple | AR/2. In the latter case there is constructive scattering of the Rayleigh wave from th periodic perturbation (Problem 18). This accounts for the large experiment- attenuation at <IJK) = 1-07 in Fig. 12.23a and at (2s//r) = 2.14 in Fig. 12.23b. /3= 10* <r = 0 33 Theoretical results x x Rischbieter s exeiimental results 0 0.5 10 1.5 К/к (a) 0 05 10 1.5 2.0 2.5 (t>) FIGURE 12.23. Comparison of theoretical and experimental Rayleigh wave attenuation due to a sawtooth roughening function. (After Sabine)
314 PERTURBATION THEORY EXAMPLE 12. Rayleigh Wave Excitation by Plane Wave Scattering at a Substrate Surface with Periodic Mass Loading. In this example a plane wave is normally incident on the periodically loaded surfacct (Fig. 12.24). A perturbation formulation of the problem is established by using (12.63) (12.66), where the zero order equations describe scattering of the plane wave at the unperturbed boundary. Solution of the latter problem gives the unperturbed velocity field at the boundary, (v(0'w Mechanical boundary conditions for the first-order equations (12.66) arc then -<T«>to • y),_o - AZ^ • (v<°vo> (1289> where AZ.( = — iwhp'<z) ir only mass loading is considered. It will be assumed that there is no electrical perturbation. The radiated Rayleigh waves are calculated by applying (12.68) and (12.69), with appropriately modified driving terms. With the purely mechanical perturbation assumed here, AR (*) = - {-<_ • T(,)<s) • *L 0 (12.91) For a single mass-loading strip (Fig. 12.25) i о . p'w = — < 2 < - 2 2 > dl2\ 7/////////У//У/'// Radiated Rayleigh wave Incident plane wave Thin str p array Reflected plane wave FIGURE 12.24. Scattering of a normally incident plane wave into Rayleigh waves by an array of thin isotropic strips on the surface. f Reference 2 at the end of the chapter. Mass density p". Rayleigh wave amplitude AR_ D. RESONATOR PROBLEMS 315 Rayleigh wave amplitude Да+ Incident plane wave Reflected plane wave FIGURE 12.25. Plane wave scattering into Rayleigh waves at a single strip. and, from (12.89) to (12.91), iwhp 'K+V itohp rd/2 . o-(v(0vo^z е'Ы dt. itohp' ^ , sin B,, dl2 = - -йГ КЛо • (v""Vo 7 ' e~^ (12.92) ioihp' „ sin 6„ dl2 AR& = (v*a=„ • (v<°% „ 7 ' (12.93) The wave amplitudes (12.92) and (12.93) can be evaluated from the normalized Rayleigh wave particle velocities in part С of Appendix 4 and the power density Рг of the incident longitudinal wave. If the incident wave is pure longitudinal (2P, V1- ~pTj ' (12-94> and the power radiated into the Rayleigh waves, Р+ = ик_:1гл, P_ = /vi'''ills proportional to Pt. Since the perturbation calculation is based on a weak-scattering assumption, scattering by an array of strips is calculated by combining amplitudes (12.92) and (12.93) for all of the strips. 1). RESONATOR PROBI EMS To obtain perturbation formulas for resonator problems, it is necessary to further modify the complex reciprocity relation (10.113). In deriving (12.45), solutions "I" and "2" were both assumed to vary as eiat. If they are now allowed to have different frequencies ro, and co2, a repetition of the steps d
316 PERTURBATION THEORY = [у* Tf -*Ф*] ш>2[ ]3-ift>i[ Ь т (12.95) where [ ] = О О О : s :й : d _0 • d : • е •: A simple rearrangement of the right-hand side of (12.95) leads to V ■ {-v* ■ Ti — vx • T* + ФЮсоЕЧ) + Ф,(и»1Ч>*} '[у* T* -VOJ] = —i Aw where and 0 0 0 : s :л : d _0 • d : • e T .-toJ (12.96) & = APv„* • V! + T* : (AsE : T, — Ad • TOE) + УФ} • (Дег • TOX - Ad : Т,). with Др = pi - рг, etc., as in (12.47). Solution "2" in (12.96) is now taken to be a lossless unperturbed resonator mode v, — eiw»*vv(a:, y, s), etc., and solution "1" is the perturbed mode v, = е*°»У.(.т, ?/, z), etc. D. RESONATOR PROBLEMS 317 /Ас .к 1 dV -то;_ = ~j{ }-*dS - ic.,'vj-&dV where 0s is defined above and { } = {-▼«• Ъ - v, • T* + *Jf>D,) + ФДкоЦ,)*}. When the perturbed fields under the volume integral on the left are approximated by unperturbed fields, this term becomes /4Г7 Aok where Urr is the stored energy of the unperturbed mode (Section e of Chapter 11). Since the unperturbed mode is lossless, can be expressed entirely in terms of v, following the derivation of (12.50); and the final result is Aft>v = 4lT l'~v*'T: ~v;" T*+ ф*(^в,У + *№^DV)*} ■ ft dS - ~j- JW: ■ v; - sv*: Ac75: s; + e* . дег. e; + E*.Ae:S; + S:-Ae.e;)dE (12.97) where to'v has been approximated by to„ in the second term on the right-hand side, and Др = p — p, etc., S — VjV/iVj). D.l Boundary Perturbations If perturbations occur only on or outside the resonator boundary, (12.97) i educes to A«v = 77Г f {-v* ■ Ti - v; ■ T* + 0*(i«D()' + OtfwD,)*}. ft dS. 4t/vs Js (12.98) which led to (12.45) now gives After integration over the volume of the resonator and conversion of the divergence term into a surface integral, one obtains К т* -УФ2*]ГР о о
318 PERTURBATION THEORY Mechanical Surface Perturbations. As in the waveguide case (Section В. 1) mechanical and electrical surface perturbations may be treated independently. Mechanical perturbations are governed by the first two terms under the integral in (12.98). EXAMPLE 13. Resonant Frequency Perturbation by Mass Loading. If the resonator is nonpiezoelectric or if there is no electrical surface perturbation, the electrical terms do not appear in (12.98). With stress-free boundary conditions on the unperturbed resonator, Tv • n = 0 on S, and The perturbed stress at the boundary is evaluated by assuming that vv = vv on 5 and expressing T, • n in terms of \'v, as in Problems (8) and (9). If the perturbation is a thin overlay of thickness ft and mass density p deposited on the surface of the resonator (Fig. 12.26) and only the mass loading effect is significant, T^ • n = —hoyp'h \[ and FIGURE 12.26. Perturbation of anisotropic spherical resonator by mass loading. D. RESONATOR PROBLEMS 319 urn- 1 ill l^t from (11.41). The stored energy in (12.99) is therefore, according to (11.97), given by and the velocity field under the integral is obtained by setting r = a in (12.100) Electrical Surface Perturbations. In this case (12.98) is Ae»v = - J- Г {<P*(/wvDv)' + Ф;('соД>у)*} • ft dS. (12.102) 4UYVJx As in the waveguide surface perturbation formula (12.25), the central problem here is to approximate the perturbed fields under the integral. For the waveguide analysis good agreement between perturbation theory and numerical calculations was obtained by assuming that the stress field T is unchanged by the perturbation. In the resonator problem, by contrast, it is best to assume that the strain field S is unchanged. The validity of this choice will be demonstrated by an example. EXAMPLE 14. Frequency Perturbation due to a Change in Electrical Boundary Conditions. The fully electroded unbounded plate resonator (Part 5 of Section 11 .B), one of the few piezoelectric resonator problems with an exact solution, will be used as an illustration. For the unperturbed resonator the electrodes are chosen to be in direct contact with the plate (Fig. 12.27a). In this case the unperturbed fields for the piezoelectrically active (odd symmetry) modes of an X- oriented hexagonal (6mm) plate arc given by Example 5, Chapter 11. That is, (ux)v = ho sin kvy (JxV)v = + pf^j (cos kvy - cos kvb/2) m e**(- Z y ■ ТыЛ (l2103> Фу = \sm kvy - — sin kvb!2j 2exs — (Dv\ = —г— sin kvbl2. d For the Sol modes of an isotropic sphere with radius a and stress-free boundary conditions (Part 4 of Section ll.B), the particle velocity field is entirely radial, . д fejr\ „ <a /a>r\ •*"-r ~гуАу) (12л00) where ша am] Vt
320 PERTURBATION THEORY electrodes electrodes Г -Y,z hexagonal (6mm) "piezoelectric plate (a) unperturbed resonator П Х,У v,\v////////////// //////////////// 1 1 _ hexagonal (6mm) piezoelectric plate (b) perturbed resonator FIGURE 12.27. Frequency perturbation of an unbounded piezoelectric plate resonator by a change in electrical boundary conditions. where [ей + exr> 4у and the unperturbed frequency relation is tan kyb 2 П 2.1041 For the perturbed resonator (Fig. 12.27b), the electrodes arc separated from the plate surfaces by a small distance A. The boundury condition perturbation at Л can be defined in terms of the electrical surface impedance (12.32) used for the waveguide problem; namely, = ) • ZF D. RESONATOR PROBLEMS 321 In this case the unperturbed impedance atу = ±bjl is and the perturbed impedance h = ~ 7 (12.105) »»„€„ is obtained from the potential solution for the region between plate and resonator. From D = es E + e:S and V • D =0 the relation between electrical potential and mechanical strain is V.e's'. ?Ф = V-e:S (12.106) for the unperturbed case, and V.e*. V0>' = V-c:S' (12.107) for the perturbed case. The difference between the perturbed and unperturbed potentials is expressed as lF = Ф' - Ф. (12.108) If the strain field is assumed to be unperturbed in the lowest order approximation, S' = S. it follows from (12.106) (12.108) that V-es'.V4=0. (12.109) In the present example, where the fields depend only on the у coordinate, the general solution to (12.109) is г = Ay + B. This function must satisfy the same odd symmetry requirements as the unperturbed potential, and therefore П = 0. Consequently, the perturbed electrical potential and displacement are фу = ^;(sin kv'J ~ Щ sin M>/2^ + Ay (D»)f = 7~ sin kv Ы2 - 4A-/1. b Imposition of the perturbed boundary condition (12.105) at у = ±6/2 gives (2\ 2h exi sin kvbj2 b b
322 PERTURBATION THEORY /sin кф!!^ Am, Abl'o \ m/2 ) 2Л to, pVf I sin /с„/>/2 cos kvb/2\ h, (12.112) m/2 / where an approximation has been nride in the denomina or Tl i agrees w th the pe ttirbation о taincd by differentiating the exact frequency relation (11.56) with respect to Л and assuming h ъ 0. D.2 Interior Perturbations If perturbations occur only within the volume of the resonator, (12.97) reduces to —v = - f (APv? ■ v-;. - s* : Ac*: £ + e* ■ Де*'. E'v + e* • Де : s;, + s* • Де • e^rfF (12.113) Mechanical V olume Perturbations. If only the mechanical properties of the resonator are perturbed the third, fourth, and fifth terms under the integral in (12.113) are zero. EXAMPLE 15. Q-Factor of a Viscously Damped Resonator. In this problem Ac = iwr\. From (12.113) and (11.106), дй, i /tu. f — = 4rT = IiT^ v.*?'-4: rfF, (12.114) wv 2(2v 4Lrvv(yv |r where the perturbed fields have been approximated by the unperturbed fields. D. RESONATOR PROBLEMS 323 As an illustration, (12.114) will be evaluated for the SDl modes used in Example 13. The viscosity matrix for an isotropic medium is where 'hi Viz Vl2 0 0 0 ~~ 4iz Vn 0 0 0 Vl2 Vl2 Vu 0 0 0 0 0 0 'lii 0 0 0 0 0 0 Пи 0 0 0 0 0 0 wr]u = Qi "»Ht = c. COth2 = - 2'i«) and (2,, Gs are the longitudinal and shear wave Q's. These matrix components arc directly applicable in spherical coordinates (Appendix I in Volume I), and substitution of the velocity field (12.100) into the perturbation formula gives /Д"Л Ы Г" I I dv, + 2(>hl + »)12) - \)r*dr (12.1 15) where U?Ql and t> arc given in Example 13. Electrical Volume Perturbations. In this case, the perturbation is Де7, and only the third term in (12.113) is nonzero. EXAMPLE 16. Q-Factor of a Piezoelectric Resonator with Finite Conductivity. The unperturbed resonator is assumed to be lossless and electrically isotropic. From (12.113) and (11.106) the £>-factor is given by and the perturbed potential is therefore From (12.103), (12.105), and (12.110) the integrand in the numerator of (12.102) is evaluated at ?/ = ±h/2 and the stored energy in the denominator is, from (11.97), t/w = iP"'l \jrsin1 *v'J dV. (12.111) The integrals arc taken over a unit surface area of the plate and, since the surface integral of S includes both upper and lower faces, . 2
324 PERTURBATION THEORY This expression will be evaluated for the piezoelectrically active modes of the fully electroded, -V-oriented hexagonal 6mm plate in Example 14. From (12.103) ЭФ, куехл J - sin k~№\ (E„)v = - — = — cos kv!, - ) ду 4A V Kbll } and r/vv is given by (12.111). Performance of the integral in (12.116) over unit surface area of the plate gives / sinM/2 /2 sin *vr3/2\2\ Q, 2"'exx Pvs ( sin fcvi/2 cos At V ~" \h\2 PROBLEMS 1. Starting from the real reciprocity relation for a source-free region [(10.109) with Fj = F2 = 0 and pel = pe2 = 0], derive an expression corresponding to (12.3). Assume that solution "2" is the unperturbed nth mode propagating in the -f-z direction, v, = e iP"z\„ (y), etc., and that solution "1" is the perturbed wth mode propagating in the —z direction, v, _ e'rV_„(y), etc. Derive a perturbation formula corresponding to (12.4) and state the conditions under which it is valid. (Refer to Section J and Problem 22 in Chapter 10.) 2. Derive (10.80) from (12.17). 3. The isotropic plate waveguide considered in Section Ю.С is perturbed by placing a thin isotropic overlay of thickness h on one surface. Derive an expression giving Д/9//3 for the SH modes and arrange it in a form corresponding to (12.17). Solve the same problem exactly by extending the Love wave analysis given in Section 10.D, and compare with the perturbation solution. 4. Extend the derivation of (12.13) to the next order of approximation and show that this adds a second order term PROBLEMS 325 lo and a second order term -№Mv>h)\P' ± _«L_l(ia L S|j + s« V-(sj, - SjJ)J to (TrXy Find, the corresponding second order term in (12.15), expressing the i/j's in terms of Lame constants. 5. Show that dV dp Л 6. Derive (12.20). 7. An acoustic waveguide with arbitrarily shaped cross section is aligned along the z axis. Boundary contour С Cross section S Show that the complex reciprocity relation (12.1) can be expressed as where oz dx dy Rdl and { }T lies in the xy plane. Obtain the relation Bz by iniegrating over the waveguide cross seclion S, and show that this leads lo the perturbation formula f { }-idS=- \ { } J я Jo f {-v: • t; -»; • т; + ф*о«о;) + ф'Лшъх} • n di for the tith waveguide mode. 4p„
326 PERTURBATION THEORY 8. It was seen in Section E of Chapter 10 that the torsional modes of a free isotropic circular cylinder have particle velocity fields of the form v = &ф(г)е Starting from the result obtained in Problem 7, assume that the surface of the cylinder is uniformly perturbed by a surface acoustic impedance Z'A, defined by -Ti -7s'A-\', and derive a perturbation formula corresponding to (12.7). Note that the surface impedance here is defined relative to the outward normal f at the boundary, whereas the impedance (12.5) was defined with respect to the inward normal y. This will lead to a change in sign of the perturbation formula relative to (12.7). 9. The circular cylinder waveguide in Problem 8 is perturbed by a thin isotropic overlay of thickness h. Obtain a first order formula for the perturbed boundary impedance Z'A in this case by using the cylindrical coordinate representation of the acoustic field equations (see Appendix 1 in Volume I) and applying the method used in Example 1. 10. A piezoelectric Rayleigh wave is perturbed by an isotropic dielectric slab, with thickness и and permittivity e, placed at a distance h above the substrate surface. Show that the Rayleigh wave velocity is changed by an amount given by (12.41), with € + tanh /?Rw z,,{-h) = i — € 1 + — tanh /?Rw PROBLEMS 327 11. According to (12.50), the perturbation of /9 produced by a small change in the electrical permittivity within the interior of a waveguide is 4P„ Jo Aft This formula (with the integration extended from у = — со to + со) may be applied to Problem 10. If —u«l, it may be assumed that e;: = er- -У((Фй),_0Л*е *v>, for »/ < 0. Show that f1-- — -- \**&+ e-^\e-^ - 1) Pr Pn 4PX and verify that the formula used in Problem 10 gives the same result when й«1. 12. In Problem 11, the approximation Er — KK was based on the condition ^ « 1. When this condition is not valid, one may still obtain a simple solution when P> « I. Show that In € (Ej,)2 ~ (ErL. in this case, and that ApR _ to (e~ - <;g) А» 4PH e Verify that the calculation performed in Problem 10 gives the same result when Rltw « 1. 13. Derive the attenuation formula (12.55), using the normalized Rayleigh wave fields given in Part A of Appendix 4.
328 PERTURBATION THEORY 14. The free isotropic plate waveguide considered in Section 10.C is perturbed by an internal layer with different mechanical properties. ft cll, c44 For SH modes on this structure, show that the general perturbation formula (12.50) reduces to Д9я = -2- Г+7(р' - Р)(»ХЫ» - - cn)[(s5)*(s5); + (s6)*n<sj„])<iy 4P„ Jit \ with (0« /177 = cos у e b (ы! fttiZ) (S5)« Pr. = — — cos (o —у e b ' (Se)„ п77 . - - — sin h If p - p «) c44 ~ Cli « 1 the perturbed fields can be approximated by the unperturbed fields. Find A/?n under these conditions. 15. For a nonpiezoelectric waveguide, { } = {-y*-T, -v.-tf} and &> = Apvt ■ v, + t* : As : t, in (12.46). Take solution "2" to be any unperturbed waveguide mode v2 = e^y,(y), etc. and solution "1" to be Vl = e "V(y), etc., where v'(«/) is a superposition of unperturbed modal field distributions v'(y) = 2/1mv»«C*)- PROBLEMS 329 Assuming that solutions "1" and "2" satisfy the same boundary conditions at у = 0, b, show that (12.46) becomes m 4Pt where P, = Pn is defined in (10.119), Klm = f• v,„ + t* : As : t J dy, Jo and / is arbitrary. This infinite set of linear equations defines modal solutions for the perturbed waveguide in terms of coupling between the unperturbed modes, and approximate solutions may be obtained by utilizing the properties of coupled wave systems. In particular, it should be noted that weakly coupled waves interact significantly only when they have approximately the same values of со and p" (See Problem 2 in Chapter 10). If, for example, there are no unperturbed modes with propagation factors close to /?„, one may ignore all terms except m = n in the nth coupled wave equation. That is, '(Pn ~ P)An = j~ K„„A„. Show that this is equivalent to the result obtained from (12.50) by taking v» = у„ and t^, — t„. When there is an unperturbed mode with ftp /?„, show that the nth and pth coupled wave equations are i(a - (i)An = ~ (K„nAn + K.npA„) KP» - P)A„ = ~ (KvnAn + K„AJ. Solve for the perturbed propagation factors f>'n, p"v and compare with the previous result. 16. In Problem 14 the perturbed fields were approximated by the unperturbed fields. This is valid only when the conditions P С'ы ~ C" « 1 are applicable. When these conditions arc not satisfied, show that
332 PERTURBATION THEORY Chapter 13 VARIATIONAL TECHNIQUES A. INTRODUCTION 333 B. RESONATOR PROBLEMS 334 C. WAVEGUIDE PROBLEMS 361 D. TRANSDUCER PROBLEMS 364 PROBLEMS 36g REFERENCES 373 A. INTRODUCTION For calculating small changes (5 or 10 percent) in physical quantities such as propagation velocity or resonant frequency, the perturbation theory of Chapter 12 is completely satisfactory. If larger changes are required, or if no exact solution is available as a starting point for the perturbation, a variational analysis is called for.| This is basically a technique for calculating a desired physical quantity from an estimated (or trial) solution to the problem, w here the mechanism of the method provides that a relatively large error in the trial solution will give only a small error in the calculated quantity. A substantial advantage of this technique is that the trial solution does not have to satisfy the exact boundary conditions of the problem. However, a significant amount of numerical computation is usually required; and the variational method does not, therefore, provide physical insights as readily as perturbation theory does. t Oiher approximation methods for problems of this kind are discussed in References 10 niul 11 at ihe end of the chapter. 333 9. К. A. Ingebrigtsen, "Surface Waves in Piezoelectrics," J. Appl. Phys. 40, 2681 2686 (1969). 10. K. A. Ingebrigtsen, "Linear and Nonlinear Attenuation of Acoustic Surface Waves in a Piezoelectric Coated with a Semiconducting Film," J. Appl. Phys. 41, 454-459 (1970). 11. P. J. King and P. W. Sheard, "Viscosity Tensor Approach to the Damping of Rayleigh Waves," /. Appl. Phys. 40, 5189-5190 (1969). 12. G. S. Kino and Т. M. Reeder, "A Normal Mode Theory for the Rayleigh Wave Amplifier," IEEE Trans. ED-18, 909 920 (1971). 13. К. M. Lakin and H. J. Shaw, "Surface Wave Delay Line Amplifiers," IEEE Trans. МТГ-17, 912-920 (1969). 14. К. M. Lakin, "Perturbation Theory for Electromagnetic Coupling to Elastic Surface Waves on Piezoelectric Substrates," J. Appl. Phys. 42, 899-906 (1971). 15. M. F. Lewis, G. Bell, and E. Patterson, "Temperature Dependence of Surface Elastic Wave Delay Lines," /. Appl. Phys. 42, 476-477 (1971). 16. F. Press and J. Hcaly, "Absorption of Rayleigh Waves in Low-Loss Media," J. Appl. Phys. 28, 1323-1325 (1957). 17. F. Rischbicter, "Messungen an Oberflachenwellen in festen Korpern," Acoustica 16, 75-83 (1965). 18. P. V. H. Sabine, "Rayleigh-Wave Propagation on a Periodically Roughened Surface," Electronics Letters 6, 149-151 (1970). 19. M. B. Schulz, B. J. Matsinger, and M. G. Holland, "Temperature Dependence of Surface Acoustic Wave Velocity on a Quartz," J. Appl. Phys. 41, 2755 2765 (1970). 20. H. Skeie, "Electrical and Mechanical Loading of a Piezoelectric Surface Supporting Surface Waves," J. Acous. Soc. Am., 48, 1098-1109 (1970). 21. A. J. Slobodnik, Jr., P. H. Carr, and A. J. Budrcau, "Microwave Frequency Acoustic Surface-Wave Loss Mechanisms on LiNb03," J. Appl. Phys. 41, 4380-4387 (1970). 22. L. P. Solie, "Piezoelectric Effects in Layered Structures," to be published i Appl. Phys., 44 (1973). 23. H. F. Tiersten, "Elastic Surface Waves Guided by Thin Films," Appl Phys. 40, 770-789 (1969). 24. G. S. Kino and R. S. Wagers, "The Theory of Interdigital Couplers oA Nonpiezoelectric Substrates," to be published in J. Applied Phys., 44 (1973).] 25. R. A. Waldron, "Perturbation Formulas for Elastic Resonators and WavcJ guides," IEEE Trans. SU-18, 16 20 (1971). 26. D. C. Wolkerstorfer, "Methods for Measuring the Acoustic and Optica Properties of Organic Crystals," PhD Thesis, Dept. of Applied Physics,, Stanford University, August 1971.
336 VARIATIONAL TECHNIQUES (13.4) pv ■ v d V It will now be shown that this is a variational expression. Suppose that vv is some exact solution to (13.1) with a resonant frequency cjv. An approximate (or trial) solution v — vv + <5v = vv + ef is assumed. Here e is a measure of the error in the trial solution, this error] being described by function f Substitution into (13.4) gives o>2 as a function] Of e, (Ae) - «>v + K«2)(U + tW)(2) + • " • J V,vv : с : Vsvv dV + 2ejvj: с : Vsvv dV + Л ) j pvv • vv dV + 2e f pi • vv dV + e-'( ) (13.5 In order for w2 to be stationary, the first order coefficient (ro2)a> in (13.5 must be zero. When evaluating this coefficient, the r2 terms may be ignoredj since only first order terms are significant; and use of the Binomial Theorem gives ^) = Jr , + , с : V,vv dV P\,-y\.dV pv,-vvdV J у Jr J V,v,: с : Vevv dV f:c:V,vvdF- pf-v.dV, dV From (13.5) and (13.6) it follows that V,vv :c :X\vv dV 2 J г "К - —y~— I pvv • v„ dV + A ) + (13.( B. RESONATORS 337 v ' = vv + sf c:Vsvvn 1 = 0 (Rigid Boundary Conditions) = 0 (Stress-Free Boundary Conditions) at every point on the resonator boundary 5, the surface integral in (13.7) is zero. Therefore and Vv :c : V.vdF = f - (13.8) 1 pv • v is a variational expression for the resonant frequency of any resonator lhat has an arbitrary distribution of rigid and stress-free boundary conditions on its surface. This has been shown to be true only if the trial solution satisfies precisely the same boundary conditions as the exact solution. In a resonator ol complicated shape this is a limitation which makes it difficult to perform exact and approximate solutions satisfy the same conditions. Since c:Vsv = iroT, the surface integral in (13.3) is zero under these conditions; and V,v : с : Vs\ dV .2 Jr and 2 Г (V,f :c:V5v, - a»2Pf.vv) </F / 2.(1) Jr (w ) = p\y-vvdV Jv The expression for (<w2)(1) can be transformed to a more useful form by applying the identity V • (f - [c:V>]) = f • (V • [c:V,v]) + [c:V,v]:VJ to the first term in the numerator, and V • с : Vsvv — — i:o2pvv (from (13.1)) to the second term. Use of the divergence theorem then gives the final result 2 Г f.[c:\>,J-*dS ("v" = • (13-7) pv,, • v, Jr Under the assumptions that
338 VARIATIONAL TECHNIQUES the variational calculation. Fortunately, boundary condition restrictions on the trial solution can be removed by adding a suitably chosen surface integral to the numerator of (13.3). For example, the right-hand side of (13.4) is obtained from the right-hand side of (13.3) by adding an identical surface integral to the numerator. One may then show, by substituting a trial solution, that this is a variational expression for a>2 when the surface integral on the right-hand side of (13.7) is zero. This condition is satisfied for any resonator with completely stress-free boundaries. Notice that there are now no boundary condition restrictions on the trial solution. To apply (13.8) to a resonator with stress-free boundaries one may use trial solutions with arbitrary boundary conditions. In a similar way, variational expressions with unrestricted trial functions may be found for other classes of boundary conditions (Problem 4). Removal of boundary condition restrictions on the trial solution greatly simplifies the task of choosing trial functions. However, this advantage is gained at the cost of slower convergence of the approximation process. To illustrate the use of (13.8) it is best to consider some specific examples. EXAMPLE L Unbounded Isotropic Plates. In mathematical physics texts the standard example of a variational calculation is the vibrating string problem. The analogue in acoustic field theory is an unbounded isotropic plate with rigid boundary conditions (Fig. 13.2). The exact solution is trivial; but the problem serves, nevertheless, as a useful vehicle for illustrating the basic features of the variational method. To reduce the problem to its essentials, only the compressional wave resonances are considered. Exact solutions, obtained by the method used in Section B.l of Chapter 11, are vv = у sin Virjf b* p (13.9 where »> = 1, 2. 3 The exact boundary conditions are rigid, and the same boundary conditions must therefore be applied lo the trial solution if (13.8) is to used. Rigid boundary 77- Rigid boundary- FIGURE 13.2. Unbounded isotropic plate resonator with rigid boundary conditions. B. RFSONATORS 339 As a first approximation, choose the parabolic trial function v = y(b - ?/)»/. The integrand in the numerator of (13.8) is then Vsv:c:V> = cn{b - 2yf. Since the fields are independent of * and z, (13.8) becomes f Ф - 2,/)- , Jo Jo (Ь" ~ 1Л Comparison with (13.9) shows that this differs by less than 2 percent from the exact value for the fundamental resonance г ь ">i = -г— b P V = fi»)v(l) + fiU)v<2, = £ в,1)у{я) (13.10) The first term is taken to be the function already used above, v(1> = y(b - y)y, and the second is v«) = уф - y)hf. Both functions satisfy the required rig d boundary conditions at у =0,/>; and substitution into (13.8) gives j В™В^Р(*,Р) 2 B(x)B(p>H{a,fi) where (13.12) tf(a, ft) = pv"" • V(/" dV Jf arc matrix elements related, respectively, to the strain and kinetic energies of the lesonator.
340 VARIATIONAL TECHNIQUES The stationary condition is imposed on (13.11) by requiring that ^_1_*£.-0 (13-13) This step is most easily carried out by rearranging (13.11) as й(«)Л^)(Р(а. /Г) - ш2Жа, p)) = 0. а-/) 1 Differentiation with respect to B(1> and B(2>, and use of (13.13), then gives* 6 = 1.2. This is a matrix equation [P о;2Я][В] = [0], with the explicit representation -P(l, 1) - o>*H0, D P(L 2) - w4f(\,2) _P(2, 1) «>2tf(2, 1) P(2, 2) - «>2tf(2, 2)J (13.14) = 0. Solution of (13.14) gives the values of Bn) and B('-> required for an extremum of (13.11). For a nontrivial solution the determination of the matrix must vanish; that is' |P — m*ff| = b, (13-15) which gives two solutions. The significance of this fact will be considered below. For the trial functions assumed above, the matrix elements in (13.14) are P(l, 1) = c„*V3; P(1, 2) = P(2, 1) = <_*•/!5; «2, 2) = 2Cll67/l05; Я(1, 1) = pfc5/30; H(\, 2) = Я(2, 1) p/>7/140; tf(2, 2) = /А'/бЗО. The dcterminantal equation (13.15) is therefore cn/>s/3 - <>rplfftQ c,,ft*/IS - ^2р67/140 rufc5/l5 - to2 РЛ7/140 2c„AT/105 - w*pb»l630 0, or (V -тЫ+|С"-0: and this has two roots, and , (56 - 2 v 532) cn _ 9.86974 c„ ^= ^-7"~^7 (56+2V532)cu 102 rn = м 7^^T- f Mote that P(cc, 0) = />(& a) and //(a, jS) = //(ft =0 >n (13.12). B. RESONATORS 341 From (13.9) the frequency of the fundamental resonance (v = 1) is given by 2 = ^ cn _ 9.8696044 c„ tUj ~ 7 Ь* я ' With the two-term trial function, this has been approximated to the order of one part in 100,000 by the root mj. The meaning of the second root m*t next comes into question. A check against (13.9) shows that this is a fair approximation to the frequency of the second overtone resonance (»■ = 3), given by 2 (3«)*<_ 89 cn fO„ = . 3 h* p b2 p At this point it is natural to ask what happened to the first overtone resonance (v = 2). A clue is obtained by examining the symmetry of the solutions in (13.9). This shows that the fundamental and all the even-order overtones (r odd) have even symmetry, while all the odd-order overtones (v even) have odd symmetry (Fig. 13.3). On the other hand, the trial functions used in calculating w\ and to,2, both have even symmetry. It is, therefore, at least plausible that the use of these trial functions will lead only to approximations for mode solutions with the same symmetry. These results also suggest that approximations to more and more of the even-order I ) \ Fundamental (v = 1) \ < I 2nd Overtone (v = 3) 4th Overtone (v = 5) 6th Overtone (v = 7) Even Symmetry Vy = sin -j- у v = 1, 2, 3, 1st Overtone (v = 2) 3rd Overtone {v = 4) 5th Overtone (i» = 6) [ 7th Overtone (v = 8) Odd Symmetry ] FIGURE 13.3. Particle velocity field distributions for the compressional resonant modes of an unbounded isotropic plate with rigid boundary conditions (Figure 13.2).
344 VARIATIONAL TECHNIQUES that was derived in Problem 11.11. The approximate solutions obtained from the Rayleigh-Ritz method can also be normalized with respect to (13.16). Solution of (13.14) gives only the ratios of the weighting coefficients B(a) in (13.10), and a common multiplying factor can always be used to achieve this normalization (Problem 8). EXAMPLE 2. Thick Anisotropic Disks. Example I was an academic exercise chosen to illustrate the principle of the variational method. It is necessary now to consider some practical examples; that is, problems for which there are no analytic solutions. In such problems it is not always easy to pick trial functions, and convergence of the Rayleigh-Ritz method is not generally as rapid as in the above example. However, the availability of electronic computers makes it possible to use a large number of suitably chosen trial functions in the trial solution, thereby assuring accurate results. At the same time, calculations using many trial functions give the resonant frequencies for many modes. In this example the variational method will be applied to the thick anisotropic disk resonator with stress free boundary conditions (Fig. 13.5). The elastic proper- tics are assumed to correspond to a ferroelectric ceramic poledf along the г-axis in the figure, с = 012 <13 0 0 0^ c,. cn 0 0 0 C13 f33 0 0 0 0 0 0 C44 0 0 0 0 0 0 0 0 0 0 0 0 However, the piezoelectric effect will be neglected for the purpose of the example. This calculation is therefore a first approximation to the thick piezoelectric disk treated below in Example 4. It was noted in Example 1 that the choice of trial functions hinges to some extent on the symmetry of the resonator modes that are to be approximated. In this example attention will be directed to modes having the following symmetry properties iV: independent of ф and described by an even function of z 1>ф = 0 r-: independent of Ф and described hy an odd function of z. These are called the angularly symmetric dilatational modes of the disk. t This material has the same properties as a hexagonal (6mm) crystal, with the poled direction corresponding to the Z axis. FIGURE 13.5. Nonpiezoelectric thick- disk resonator. The approximation calculation follows exactly the procedure outlined in Example 1. A trial solution is assumed to have the form of (13.10), with a now summed up to some upper limit,Sf m. The coefficients required to minimize «>2 are then determined by (13.14), where the matrix elements (13.12) have indices a and p running up to <f„■. An initial choice of trial functions which satisfies the symmetry requirements is t v(ra) = wmsin [(2mx - lWi-]] 4«>=o j „..>=0 * = 1,2,...^ (,3.17) i>'a) = J0(kxrla) cos [(2nx - 1)^/t] * =sr1 + 1, &1+г,...уп where ma, n„ = 1,2,3,... m0(a) = (I - Си/cuvica.) JB(ka) = 0. These trial functions were chosen for ease in computing the matrices P(a, p) and H(a. p) in (13.12). They do not closely resemble the exact solutions nor do they satisfy stress-free boundary conditions. From the discussion following (13.8) it is t Only a brief summary of this calculation is given here. Full details are given in Reference 7. where the problem is formulated in terms of the displacement field u rather than the velocity field v. These fields differ only by the scalar factor 1Y0 and are therefore equivalent.
346 VARIATIONAL TECHNIQUES nol necessary that they should do so; but, if they do not, it is necessary to pay close attention to convergence of the approximation. One necessary condition for good convergence is that the set of trial functions be complete. The question is discussed in Reference 7 in full detail. Here it will bd simply stated that the set of trial functions (13.17) is complete in the Fourier sense but is notpointwise complete. The latter condition, which requires that not all of the trial functions be zero at the same point, is essential to convergence of the approximation, ft is difficult to find simple trial functions which satisfy this condition, anil a way out of the difficulty is to add correction terms which make the set of functions over-complete in the Fourier sense but give the desired pointwise completeness. The selection of correction terms cannot in general be made on the basis of a simple recipe. For the present case the following set of over-complete trial function^ was found to work well. p<«' =y,(V/«)sin [(2ma - lW-r]) i-i" = 0 j a = 1,2,...^ •{.«> =Л(Ы")) t<->=0 j a=^ + I, £/\ +2,...У2 (13.181 .•<*»= 0 \ rj" = A,^'W«)cos [(2иа - 1W-]) a = if, + 1.^ + 2,...-^, r<*> = 0 \ 4a) = cos K2nx - l)^/r]j a = ,У3 + \,<?3 + 2,...£Гт where mx, иа, ha, kx are as given in (13.17). In (13.18) the correction terms vlra) j Jr1(A,r/e) and г'"' = cos [(2^ - 1)-яг/т] are functions of only a single variable. This has been found to be a satisfactory arrangement in most cases. Complete numerical calculations have been carried out for Clevitc Ceramic A| (BaTi03) disks. This material has the following material parameters (MKS units) с& = 1.5 x 10" eg = 0.44 x 1011 eft = 0.66 x 10" ей = 0.43 x 10" ей = 0.66 x 1011 p = 5.7 x 103. eg = 1.46 x 1011 A total of 30 trial functions were used. These are, referring to (13.18), У\ = 12, with (Д,, ma) (hlt 1), (A5, 1), V's, О, ("*, 0 (Aj, 2), (Л2, 2), (ft3, 2), (Л4, 2) (A„ 3), (A2, 3), (A3, 3), (A4, 3) .9".. - .S^ = 6, with A« = Alf А», A3, A4, A5, A. (13.1«/) Уз - У» = 8, with (&,,«*) = (fci, I), (*г, 1), (*з. О. (A-4. 1) №l5 2), (fr2, 2), (a3, 2), (A4, 2) -Г,,, - У3 = 4, with ла = 1, 2, 3, 4. В. RESONATORS 347 Figure 13.6 compares the calculated resonant frequencies with experimental results and with a variational calculation including piezoelectricity, to be treated in Example 4. The agreement with experiment is very good, even with piezoelectricity neglected, and is found to be excellent when the piezoelectric coupling is included. Results are expressed in terms of normalized dimensions and arc therefore applicable to any disk of this material. EXAMPLE 3. Rectangular Contour-extensionul Thin-plate Resonators The problem of contour-extensional modes in thin plates is of such great practical importance that a special thin plate formalism has been developed to deal with it. Field equations for nonpiezoelectric cases are given by (11.20) and (11.21). For thin piezoelectric plates that arc completely covered with short-circuited electrodes, the electric field is approximately zero; and (11.20). (11.21) arc modified by simply
348 VARIATIONAL TECHNIQUES appending a subscript E to the stiffness constants. For most problems of this kind the plate equations cannot be solved exactly, and the variational method must be used. Since the plate approximation reduces the resonator problem to two dimensions, the integrals in (13.8) are performed over the surface area of the plate. Except for this, the procedure is exactly the same as in the previous example. The particular problem illustrated here is a rectangular ferroelectric ceramic plate, with the polarization axis normal to the platef (Fig. 13.7). In this case the medium is planarly isotropic, with only two independent "planar" stiffness constants, This problem differs from Example 2 in that the piezoelectric effect is not ignored. After an approximate solution has been obtained by the variational method, the motional capacitances (Section D.3 in Chapter 11) can be calculated directly from (11.62), (11.83), and (11.92). A calculation of this kind will be illustrated for the thick disk resonator structure in Example 4. First, the symmetry of the desired resonant modes must be decided upon. One particular class of modes is characterized by vx: odd with respect to x and even with respect to у i;„: even with respect to x and odd with respect to y. These are dilation-type contour modes. In this case use of an over-complete trial function scries is found to be desirable (even through a pointwise-complete series is easily found), since convergence is greatly improved by the addition of correction FIGURE 13.7. Thin-plate fully electroded piezoelectric resonator (тг <^ тж, т„). In Example 3 resonant frequencies are calculated for the short-circuited case shown here. t Poled ceramics have the same properties as a hexagonal (6mm) crystal, with the polari/a- tion axis corresponding ю z B. RESONATORS 349 Mode 1 Mode 2 FIGURE 13.8. The Ave lowest-order short-circuit contonr- dilatational modes of a square thin-plate resonator (тх = Ty in Figure 13.7) with fully electroded surfaces (a = c{\lc[\ = 0.30). (After Holland and Eer Nisse) terms.f In Example 2, by contrast, the addition of correction terms was essential. Otherwise, convergence would not have been obtained. Calculations were carried out using a total of 40 trial functions. Figure 13.8 shows the contour deformation patterns obtained for the first 5 modes of a square plate, assuming a "planar" Poisson's ratio c12 sVi Figure 13.9 shows the normalized resonant frequencies of the lowest order modes as a function of length-to-width ratio of the plate. B.2 Piezoelectric Case In the quasistatic approximation, the free oscillation modes of a piezoelectric resonator are governed by the quasistatic field equations J V • (cE : V» = -pco2u — V . e • У"Ф V.£s-VO = V.(e:V,u) plus the mechanical and electrical boundary conditions at the surface S of the resonator. The method used in deriving a variational expression from (13.1) for the nonpiezoclcctric case cannot be used here and it is necessary to lollow the more fundamental Lagrangian approach. t Reference 7 at the end of the chapter. * 11 is somewhat more convenient to formulate piezoelectric variational calculations in lerms of ihc displacement field u rather than the velocity field v = ittm used in (I I.]).
350 VARIATIONAL TECHNIQUES Tjc/7jp FIGURE 13.9. Contour-dilatational mode spectrum for the resonator shown in Figure 13.7 (h = efg/cf, = 0.3). (After Holland and Eer Nissc) It has been shownj- that a Lagrangian density function appropriate Equations (13.20) is jsf = —kinetic energy density \p — dielectric potential energy density \D • E + clastic potential energy density : T. The Lagrangian L Гиг a piezoelectric resonator is obtained by integral in J & over the volume V of the resonator and adding surface integrals over S to take care of boundary conditions. This is a stationary quantity, in thd f References 7 and 17 at the end of the chapter. B. RESONATORS 351 Boundary surface S FIGURE 13.10. General piezoelectric resonator with short-circuited electrodes. The outward normal to the boundary surface is given by the unit vector n. sense defined above, for the exact solution defined by (13.20) and the boundary conditions. Thorough discussions of Lagrangian theory for piezoelectric problems, and derivations of Lagrangian functions Li for a variety of boundary conditions are given in References 7 and 17. Here, only the short-circuit resonator modes used in Section D of Chapter 11 will be considered. The boundary conditions for this case are shown in Fig. 13.10. Volume V is defined as the total volume of the resonator, including the electrodes, while V' is the volume of the piezoelectric medium. Tn this case, by contrast with the treatment of Section D in Chapter 11, the enclosing surface Spasses outside the electrodes. This allows for inclusion of mass loading by the electrodes. Stress-free boundary conditions T • n = 0 are applied at all points on the surface 5. As is common in piezoelectric resonator analysis, electric fields outside the resonator arc neglected. That is, open-circuit electrical boundary conditions D • ft = 0 are applied everywhere on S. This is not an unreasonable assumption since many of the piezoelectric materials, notably the ferroelectric ceramics, have very high dielectric constants. For this general configuration the Lagrangian function is L=j i(Y> : c7- : Vsu - Pw2u*) dV + f (2Vcf>.c:Vsu V<f> • es ■ \Ф) dVf <I>D.firfS, (13.21) t This is distinguished from the lagrangian density function by the fact that it involves volume and surface inlegrals. (See (13.21) below).
352 VARIATIONAL TECHNIQUES where the surfaces Sj, between the electrodes and the piezoelectric medium are identical with the surfaces Sp in Fig. 11.21 and D = e • Vsu — cs • W>. (13.22) Derivations of (13.21) and proof of its stationary property are given in the references cited. The procedure for using the variational expression (13.21) is exactly the same as for (13.8). Trial solutions for the particle displacement and electrical potential fields are taken to be linear combinations of trial functions, 7] (13.23) ф =2c(/,w'. As before, the trial functions must be pointwisc complete but do not have to satisfy the exact boundary conditions of the problem. However, it is advantageous if the trial functions do satisfy some of the boundary conditions. For example, if the trial functions ф"" satisfy the correct boundary condition ф = 0 on all of the electrode interfaces in Fig. 13.10, the surface integrals in (13.21) are all zero. Convergence of the approximation is improved by satisfying as many boundary conditions as possible. The stationary property of L requires that dB{x) dL ec"1 = 0 a - 1,2, . . .У„ and, as before, this leads to sets of linear equations which must be satisfied by the weighting coefficients BU) and C(/" in (13.23). In this case these are expressed as the matrix equations [P-w*H][B] = -[K][C] [E][C] = [K][B], (13.24) B. RLSON \TORS 353 (13.27) The potential weighting coefficients Cj" are then obtained from (13.26) and the modal electric potential distributions are (13.28) EXAMPLE 4. Thick Piezoelectric Disks, (a) Resonant frequencies. This is a generalization of the thick disk problem of Example 2 above. The same material (Clevite Ceramic A), with e = 0 0 0 0 0 0 0 em ers 0 0 0 <?,i e~ 0 0 0 where the P (clastic), /7 (kinetic), E (electric), and A' (piezoelectric) interaction matrices are P(«, p) =]ЧУа) : cK : Vsu(/,> dV Я(а,/3) -|р"Ы ' u<p)dV £(a, Я) = j" V^P**' • сл' ■ уф"" dV - £ I (фый • cs • уф«" + ф^'п • es - уфы) dS v=l JSv X(a, /3) = Г уф"" • с : V,uM dV - £ Г ф(/1)й • e : у5иы dS The simplification achieved by requiring фы = ф"" = 0 on all Sp becomes obvious at this point. It is convenient to rearrange the equations (13.24) by eliminating [С]. This gives [P + KE'K - oj2H] [В] = 0 (13.25) and [С] - [E-*K][B]. (13.26) In the absence of a piezoelectric term ([KE^K] = 0) this is simply an alternative derivation of (13.14), which was previously obtained from (13.8). Solution of (13.25) gives the resonant frequencies and the particle displacement field distributions for a number of modes equal to .Ут, the number of trial functions used to approximate u. That is,
354 VARIATIONAL TECHNIQUES FIGURE 13.11. Symmetric piezoelectric thick-disk resonator with three electrodes. (After Holland and Eer Nisse) and 4 XX о 0 ezzJ (13.29) is assumed. Figure 13.11 shows the electrode configuration, where the gap between electrode 1 and electrode 2 is assumed to be negligible. Symmetry of the modal solutions is taken to be the same as in Example 2 (angularly symmetric dilatatwnal modes). This problem has been solved for the short-circuit resonanccst by using the trial functions (13.18) for the particle displacement field and the electric potential trial functions Ф<"> = ЛЛУ/fl) sin Pj,W^I P = 1> 2, 3,.. -У5 Ф"» = sin Раряг/т] p=<fi + \, 2>b + 2, ... Sft with qf = 1,2, 3,... and W = o. These electrical trial functions are complete in both the Fourier and pointwise sense, and no correction terms arc needed. They also satisfy the correct short- circuit boundary conditions. Ф</» = 0 on the electrodes. This eliminates the surface integrals in E(a, P) and K(x, p) in (13.24), and also provides improved convergence. f Reference 7 at the end of the chapter. B. RESONATORS 355 An accuracy of approximately 3 percent for the short-circuit resonant frequencies of the first 10 modes was achieved by using the 30 trial functions of (13.19) for u and the 22 trial functions, referring to (13.29), У& = 16, with (lf, qf) = (Л, 1), (l2,1), (/3,1), (/4, I) (A, 2), (4,2),(.3,2),(/4,2) (4,3), (4, 3),(/3, 3),(/4,3) (/,,4), (4,4), (/3,4),(/4,4) Ус - = 6- with ^ = 1, 2, 3,4, 5, 6 for Ф. In Fig. 13.6 the results are compared with experiment and with the nonpiezo- electric calculation. Piezoelectric and dielectric parameters used are ezl -- -4.35 ez3 = 17.5 = 11-4 4-U15«o 4 = 12«ч in MKS units. (A) Motional Capacitances. In Part 3 of Section D in Chapter 11 it was shown that the equivalent circuit of a tw o-clcctrode piezoelectric resonator is characterized by its resonant frequencies and its motional capacitances. The structure considered here (Fig. 13.11) falls into this two-electrode category if electrodes 1 and 2 are connected together. The resonant frequencies have already been obtained in (a) above and only the dynamic capacitances Cv=-^f- (11.92) remain to be calculated. The free charge on electrodes 1 and 2 connected together is given by 2ir a -IS (Dv-z)z Trdrd$ (13.30) и и where However, convergence difficulties may arise if Qv is calculated by directly substituting the series (13.27) and (13.28) into (13.30). A better procedure is to note, from V ■ D = 0 and the boundary condition D • n = 0 on the sides of the disk, that (13.30) is independent of the г coordinate. Averaging over г therefore gives 6„ = \ J D • idV, О г Electrode 1 Electrode 2
356 VARIATIONAL TECHNIQUES and this extra integration eliminates the convergence difficulties. Figure 13.12 gives the normalized motional capacitances (11.93) for the first seven angularly symmetric dilatational modes of the resonator. In the more difficult case of a three-electrode resonator the equivalent circuit representation is given by Fig. 11.24. The important parameters are now tov and NVJI = , with p = 1, 2. It is possible to evaluate as However, the convergence difficulties mentioned in the preceding paragraph may again occur. This can be avoided by generalizing the previous technique. Consider the closed surface defined by z = -r, -/ and r = a (Fig. 13.13). Since the divergence of D is zero, the flux of D through the upper surface z = т is equal to the inward flux over the remainder of the surface. Because this inward flux is Diameter-to-thickness ratio га/т FIGURE 13.12. Normalized motional capacitances for the first seven angularly symmetric dilatational modes of the resonator shown in Figure 13.11. (After Holland and Eer Nisse) B. RESONATORS 357 Electrode 2, S2 v Electrode 1,\ 1 T 1 r' ! Electrode 0 FIGURE 13.13. Method of averaging used to calculate values of Qvi for the resonator shown in Figure 13.11. therefore independent of т — r it can be averaged over 0 < / < т. The averaged value of ■i.dS is then Gvi = ~t jo -J 2na'tD ■ r)T_a dz + j 2Mli ■ -l)z_T. dr dr. As in the previous example, this additional integration eliminates convergence difficulties. The corresponding value of £>v2 can be obtained by subtracting Qvl from the result for the fully electroded resonator calculated above. That is, Qv2 = (GJ. Fully electroded These quantities can then be used to calculate the normalized transformer ratios of Fig. 11.24, which are given in Fig. 13.14 for the second azimuthally symmetric dilatational mode of a Clevitc Ceramic A disk. EXAMPLE 5. Rectangular Piezoelectric Parallelepipeds. In this problem the thin plate problem of Example 3 is generalized by allowing the thickness to be arbitrary. The upper and lower faces arc fully electroded (Fig. 13.15). Dilatational modes having the following symmetry Kx:odd in x, even in у and z uv:odd in y, even in x and z n2:oddinz, even in x and у are assumed. A detailed evaluation of this problemt has been given for PZT-5A material. t Reference 7 at the end of the chapter. z
а'/а FIGURE 13.14 Normalized transformer ratios for the v = 2 angularly symmetric dilatational mode of the resonator shown in Figure 13.11. See Figure 11.24 for the equivalent circuit diagram. C. - C0 is equal to the denominator of (11.93) for the single electrode case. (After Holland and Ecr Nisse) z,Z FIGU R E 13.15. Fully electroded piezoelectric parallelepiped resonator. As in Figure 13.7 the material is a piezoelectric ceramic poled along the coordinate axis. B. RESONATORS 1.00 0.90 0.80 0 70 0.60 0.50 0 40 0.30 0.20 0.10 0 ft FIGURE 13 1 Dilatational mode spectr m for th resonator shown n Figure 13.15, with material PZT-5A and rx = T„ = w. Even symmetry (subscript e) and odd symmetry (subscript o) arc defined with respect to the diagonal x = y. (After Holland) An accuracy of about 3 percent for the first 30 modes was obtained by using 33 trial functions for each component of the panicle displacement and 33 trial functions for the electrical potential. Resonant frequencies for structures with square cross section in the xy plane are shown in Fig. 13.16. The even modes have the symmetry "svO, </>z) = Uyv(y, x, z) Uzv(x, У, z) = MzvO/, x, z) and the odd modes have uxv(x, y, z) = -UyV(y, x, z) "zvO. ?/, z) = -u„(y, x, z) In Fig. 13.17 are shown level (or contour) curves for the particle displacement component щ in the plane z = 0. The first odd mode and the first four even modes arc shown. For comparison, the results for an isotropic cube arc also given.
Isotropic cube , . PZT-5 cube 1st odd symmetry mode C. WAVEGUIDES 361 Isotropic cube ^ PZT 5 cube 3rd even symmetry mode Isotropic cube , PZT-5 cube (e) 4th even symmetry mode FIGURE 13.17. Contour curves of the particle displacement component и, at plane z 0 in isotropic (cu/Cm = 3.5) and PZT-5 cube shaped resonators, тх = ry = rz in Figure 13.15. (After Holland and Eer Nisse.) C. WAVEGUIDE PROBLEMS By contrast with the acoustic resonator problems just considered, acoustic waveguide problems have not often been treated by the variational method. As shown in Chapter 10, most waveguides of current interest are layered structures. In this kind of geometry, problems which cannot be solved exactly or by perturbation theory are easily attacked by direct numerical computation. This approach has the disadvantage, already noted, of not providing the physical insights available from exact analytical and perturbation solutions. However, variational analysis is not much more helpful in this respect. The examples of resonator variational analysis given above show that a computer is usually required Consequently, direct numerical
362 VARIATIONAL TECHNIQUES This is applied to waveguide problems by assuming a standing wave resonance of the waveguide mode in Fig. 13.18. Because of mode conversion effects (Section N of Chapter 10), such a standing wave pattern cannot be established by simple physical boundary conditions at the ends of the resonant section. However, this point is of no importance here, where it is necessary only to assume that a standing wave pattern is somehow established and has the form Ъ = vh(*, y)e^ + vjx, y)j»', (13.31) for the particle velocity field. ■f Sec, for example, References 9, 10, 14, J 8, and 19 at the end of the chapter. C. WAVEGUIDES 363 FIGURE 13.18. Resonant length of a nonpiezoelectric waveguide with stress-free boundaries. The volume V in (13.3) is defined by a length L along the waveguide, such that flL = 2ir. (13.32) From this condition, the surface integrals on the end-faces in Fig. 13.18 will cancel. Under these conditions, these surface integrals will also cancel for a traveling wave solution. One can therefore simplify the calculation by using only the first term in (13.31). The surface integral over the sides of the waveguide also vanishes because of the assumed stress-free conditions, and (13.3) therefore becomes Vsvp : с : V,vp dV wp = . (13.33 j^pVp - y„ dV The arguments of Section B.l can again be used to show that this is a variational expression, even when the trial function does not satisfy stress-free boundary conditions on the sides of the waveguide. For some simple types of acoustic waveguide modes, such as the SH waves in Section C.l of Chapter 10, the complete dispersion relation is determined by the cutoff frequency, where ft = 0. In such cases, a calculation of the cutoff frequency from (13.33) gives the complete dispersion relation. The transverse resonance approach to waveguide analysis used in Section В. 3 of Chapter 10 shows that an SH mode at cutoff is simply a plane standing wave resonance across the plate, and the waveguide problem at cutoff therefore becomes identical with the infinite plate problem of Example 1 in this chapter. A variational calculation of the cutoff frequencies for the SH modes of a transversely uniform plate then follows exactly the steps of Example 1. This is, of course, a trivial problem, but the same approach can be applied to the nontrivial problem of SH wave propagation on a transversely analysis is usually the most efficient procedure in geometries where it is easily performed. For more complicated geometries (such as the rectangular strip and microsound waveguides in Chapter 10) direct numerical computations are, however, difficult. In such cases variational methods prove to be useful.! In resonator problems, each free oscillation mode is characterized by a resonant frequency and a modal field distribution. For waveguides the situation is more complicated. The propagation characteristics of a waveguide mode cannot, in general, be specified by a single number. It is necessary, instead, to give a functional relationship between the propagation factor /? and the frequency w. Each waveguide mode is therefore characterized by a dispersion relation and a modal field distribution in the plane transverse to the propagation direction. In applying the variational method to a waveguide problem the preferred procedure would be to find a variational expression for the propagation constant fi at a specified frequency w, in terms of the transverse modal field distribution. Waveguide variational expressions of this kind are frequently used in electromagnetism. However, compared with resonator variational expressions, they require more involved derivations and are of more complicated form. An alternative, and simpler, procedure is to regard the propagation characteristic /s as an independent variable and to use a resonator-type variational expression to find the corresponding frequency «> and transverse modal field distribution. This second approach will be followed here. A nonpiezoelectric waveguide with stress-free boundaries (Fig. 13.18) is taken as an example. The starting point for the derivation of nonpiezoelectric variational expressions has been shown to be vjv : с : V„v — v - (c : Vsv) • fl dS Js . (13.3)
364 VARIATIONAL TECHNIQUES nonuniform plate, which is solved in the same way as the nonuniform plate resonator (Problem 11). A variational analysis of dilatational wave propagation in a thin isotropic plate is considered in Problem 12. Piezoelectric waveguide problems may also be analyzed by specifying 8 as an independent variable and solving for w. Tn this way they are converted to resonator problems, which can be treated by the Lagrangian methods of Section B.2. D. TRANSDUCER PROBLEMS Variational methods play a very important part in the theory of electromagnetic antennas and the approach used there is directly applicable to acoustic transducers, which may be quite properly regarded as acoustic antennas. Certain very simple transducer problems can be analyzed exactly; more complicated structures, on the other hand, usually require approximations such as the weak coupling approximation used in analyzing the interdigital transducer (Section L of Chapter 10). In problems of this kind the variational method provides a powerful tool for obtaining solutions having the required degree of accuracy, without using simplifying physical assumptions. Application of the variational method to transducer problems is best illustrated by a specific case, the interdigital transducer. It is first assumed that the transducer and the source driving it are enclosed by a surface S0 (Fig. 13.19a), and the system is taken to be perfectly lossless. From the real form of Poynting's theorem VI = i ExH-udS + — , (13.34) Jso dt where dUjdt is the rate of change of electromagnetic stored energy within S0 If the surface S0 is shrunk closely around (but does not touch) the surface. (a) D. TRANSDUCERS 365 (b) FIGURE 13.19. Derivation of a variational expression for the electrical input admittance of an interdigital transducer. of the transducer lingers and leads in Fig. 13.19b, the stored energy U becomes negligible. The input admittance of the transducer is then given by I E x H - fl dS Y~v = • <13-35> where E, H, V, and / all vary as eUot. This is the same as the standard variational expression for the input admittance of an electromagnetic antenna. Similarly, the input impedance can be written as| i Exti-ndS Z = -^-J, • (13-36) Since the entire transducer structure in Fig. 13.19 is much smaller than an electromagnetic wavelength, it is permissible to use the quasistatic approximation E = -УФ. In this case the integrands in (13.35) and (13.36) may be replaced by, ф(/шБ) • ii according to the quasistatic approximation. If the fringing fields are ignored, the only significant contributions to the integrals come from the part of the surface (ST) under the transducer, and the input admittance and impedance These expressions are often derived by using the reaction concept (Reference 6), which, is also applicable lo acoustic problems
366 V ART A HON AT TECHNIQUES D. TRANSDUCERS 367 since there are no stress forces on the surface. When (13.40) is substituted for the electric displacement fields, (13.42) becomes jj Ф1(г)^(г, г')Ф2(г') dS dS' = JJ Ф2(т)Щг, г'Ж(г') dS dS'. Since this is true for arbitrary Ф, and Ф2 and the integral on the right-hand side is equivalent to jj Фг(г')%', г)Фх(г) dS ds; sT+ss it follows that Щг', г) = &(r, r'). (13.43) It can now be demonstrated that (13.41) is a variational form. The proof follows the method of Section B.l. A trial solution Ф = Ф„ + ef is first assumed. Substitution into (13.41) gives Y = Г° + \Ыi0J jj Чг; г')(Фо(г)/(г') + Ф0(г')/(г)) dS dS' - 2Y0V0Vf)j + e\ ) + -•■ (1345) where «» jj ^(г,г')Ф0(г)Ф0(г'.) dSdS' Y0~ is the exact input admittance. Because of the condition (13.43), the coefficient of e in (13.45) can be rearranged as K\ Я Г')/(г)Фо(Г,) dS dS' - (13.46) From (13.40), this is the same as _2 V. are given by у =— f ФО -ft dS (13.37) V- Jsr Z = — Г ФБ • fi dS. (13.38) /2 Js, Proof that these are variational expressions will be given only for the admittance case. The impedance case follows completely parallel lines. The piezoelectric fringing field at small distances from the substrate is negligibly small, and the electric potential Ф and displacement D can therefore be assumed to be zero on the surface 5^ in Fig. 13.19b. Integration in (13.36) may thus be extended over both 5T and Ss, giving У = — J ФВ-hdS. (13.39) Since no sources are enclosed, the acoustic and electric fields in the volume enclosed are completely specified by the distribution of Ф on ST + Sgi (Problem 13). In particular, the normal component of electric displacement can be expressed formally as n • D(r) - j Щг; г')Ф(г') dS, (13.40) where rS(t, r') is called the Greerts function of the problem. Substitution of (13.40) into (13.39) then gives У = -p jj ФШ(т; г')Ф(г') dS dS'. (13.41) &r+Sg To prove the variational character of (13.41) it is necessary only to show that ^ is a symmetric function, that is SF(r; r') - Щг'; r). By using the real reciprocity relation (10.109), this can be done without requiring an explicit formulation of 3?(r;r'). For a source-free region (10.109) becomes V • (yl • T2 - v2 - T, + Ф1(/шВ4) - Ф2(/соБ0) = 0. Integration over the volume enclosed by S-, + Ss in Fig. 13.19b and use of the divergence theorem gives Г Ф^ - n dS = I Ф2В, • f. dS (13.42)
368 VARIATIONAL TECHNIQUES In deriving (13.39) it was assumed that the fringing field outside the substrate in Fig. 13.19b is completely negligible. According to this assumption, the only contribution to the integral in (13.47) comes from the parts of the surface under the fingers; D-й is zero at all other points. The integral is therefore ho j* ar)D(r) • ft dS = ^ j iwD(r) ■ ft dS - |" iwD(r) • ft dS, Fingers Positive Negative Fingers Fingers since each finger is an equipotential. Each of the integral terms is equal in magnitude to the total current /n flowing into the transducer, and it follows that \o> f (./(r)D(r) • n) dS = ^ (/„) - ^ (-/„) = Y0V0Vf. JsT+ss 2 2 The coefficient of e in (13.45) is thus equal to zero and (13.41) is. consequently, a variational expression. To evaluate the transducer admittance from (13.41) the integration is performed only over S,, since the contribution of Ss is negligible. If the fields are uniform along the fingers the result is Kino\ variational formula (13.48) where 21 is the length of the transducer array. This gives the input admittance for fingers of unit length (as in (10.156) of Chapter 10).| problems 1. Starting by taking the scalar product of (13.1) with v*, derive the expression f y>* :c : VjrdV o> = —— . pv*-vde j г Show that the variational proof (13.5) (13.8) breaks down in this case, unless v is pure real. 2. Expression (13 8) applies to the modal particle velocity field \(т, у, г)еш t Another variational analysis of the interdigital transducer is given in Reference 13 lit the end of the chapter PROBLEMS 369 that was assumed in (13.1). According to parts 1-4 of Section ii.B, v(x, y, z) is pure real. Prove that (13.8) is therefore equivalent to where fer^g fs(0:c:s(0 Jv 2 Jv 2 u(t) = &lc (j- «*"") and the bar denotes a time average. Show that this result is consistent with the statements made in the first paragraph of Section 1 I.E. 3. Substitute into (13.3) a trial function v = vv + ef and show that (13.7) now becomes (о/)'" - j(f • [(c : vsvv) • ft] - vv ■ [(c : Vsf) ■ ft]) dS j^pvv ■ vv dV Prove from this that the trial solution and the exact solution must both satisfy the same spring-loaded boundary conditions t • n = jfvu. 4. Starting from the calculation in Problem 3 find a variational expression for rigid-boundary resonators, allowing arbitrary trial solution boundary conditions. 5. A nonpiezoelectric resonator has either rigid (v = 0) or stress free (t' • ft — 0) boundary conditions at every point on its boundary S Assume a trial function v = 2 "a' where vm are correct modal functions for the problem, and show that с»* -" ;r li Jv The identity V" • (v„ • tv) - v„ - (t • t.) + tv : vjv^, where ico tv = с : vsvv, and the orthogonality relation of Problem 11 Chapter 11 will be found useful in this connection. Note that v,, can be taken as pure real.
170 Л ARIA ТЮХА L TECHNIQUES 6. Assume that the trial solution v in Problem 5 is very close to the Ath modal solution (that is, aja?, = ru « I, fi Ф A). Normalize the integrals in Problem 5, and derive the expression (l)2 _ wi + 2 u% — °A.)rl + ■•■ ■ Prove from this that the stationary value in the neighborhood of the lowest- frequency mode is an absolute minimum. 7. Show that solutions ^TT and L«TT J to the generul two-by-two matrix equation (13.14) are orthogonal in the sense that e<2>] на, о h(i,2) 'в\[г _ g(2) = 0 and prove from this that the approximate modal solutions + b'2V2' satisfy the orthogonality relation Pvi • yn dV = 0. Generalize to the case of an n-term trial solution in (13.10). 8. Find the multiplicative factors required to normalize the solutions in Problem 7 so that Pi?vdV = 1, 9. Assume a trial function v = I, II. у = f (Aa2r - Br3) for the lowest-frequency compressional mode Sm of an isotropic sphere withJ radius a and material constants p, cu, c44 Use (13.14) to find the ratio A\B, and show that the trial solution docs not satisfy stress-free boundary conditions at the resonator surface. Note that с has the same form in spherical coordinates as in rectangular coordinates. Evaluate Vl/2 aa /c44V2 — = tool — I and compare with the curve S0l in Fig. 11.9 for a = 0.1, 0.2, 0.3, 0.4. PROBLEMS 371 10. Assume a trial function v - ф(Ло2г - Br3) for the v = 0 modes of the circular cylinder resonator in Fig. 11.7 and use (13.14) to calculate the two lowest resonant frequencies. Compare with the exact solutions given by (11.27), where tu = 0, 5.136,8.418, etc, from Problem 9 in Chapter 10. Repeat the calculation using a trial function v — &(Aa~r — Brs) cos L for the v = 1 modes. 11. Figure 13.2 is modified by making the lower boundary (y = b) stress- free rather than rigid, and the material constants are assumed to vary with У as Cu(y) - ct0 - У b) c*i(y) - <?.(! У\Ь) p(y) = Po(I - У lb) For compressional vibrations, show that (13.1) reduces to Id2 1 д юлрЛ „ f = h - y. Verify that solutions matching the boundary conditions are with w/j(p0/c,)1/2 equal to one of the zeros of Jn, /„(«) = о a = 2.4048, 5.5201, 8.6537, Use (13.8) and a trial solution у sin —=- 2b solution31' Ше frCqUCnCy °f the lowest modc> and compare with the exact _ 2.4048/сД1 2 CO Repeat the calculation for shear vibrations.
372 VARIATIONAL TECHNIQUES 12. In Fig. 10.12 there exists only one propagating symmetric wave and one propagating antisymmetric wave when >b cob « 1. Show that the symmetric Rayleigh-Lamb frequency relation (10.18) reduces under these conditions, and use the auxiliary conditions (10.20) and (10.21) to show that 2 . c44 (с» — c41) 2 со = 4 — - - P ■ Assuming a trial solution v = (A + Bij) cos fiz in (13 33), solve for the ratio A\B by using (13.14), and give the lowesj frequency solution. Compare the dispersion relation with the exact lovl frequency relation given above. 13. Show that Poynting's Theorem for a source-pee piezoelectric mediul reduces to cE :s s* :е-(-уф) + - (-уф) • cs • (-уф)* - (-vo) • e : s*| dV ) Jr when the lossy constitutive relations t = —e • (-^ф) + (c*: + hovf) : s D = ^cs + -—^ • (-уф) + e : s are used. Assume that \u Tlt s„ Dj and v2, T2, s2, D2 are two solutions to the piezoelectric field equations, and substitute v = yt — v2, etc., into Poynting's Theorem. Verify that solutions "1" and "2" are identica throughout the volume V when yt — v2 (or tx • fi = t2 • ft) and фа = фш REFERENCES 373 (or Dx • ft = D2 - n) on S. This statement specifies the boundary conditions needed to uniquely specify the piezoelectric field in a lossy, source-free region. 14. Derive a variational expression for the input admittance of the transducer. у ' /' / ' / '' . /л Piezoelectric plate Assume that the electrodes are massless and the electroded plate is semi- infinite. 15. Show that the input admittance and impedance in Fig. 13.19 can also be expressed as ico VV* J a* • ф*0 -ft .sv. dS ico С ~ II* j» Prove that these are not variational expressions Z = - — i ФО* ■ ft dS. st REFERENCES 1. B. A. Auld and G. S. Kino, "Normal Mode Theory for Acoustic Waves and Its Application to the Interdigital Transducer," IEEE Trans. ED-18,898-908 (1971). 2. M. Becker, "The Principles and Applications of Variational Methods," Research Monograph No. 27, MIT Press, 1964. 3. M. Ben-Amoz, "Variational Principles in Anisotropic and Nonhomogeneous Elastokinetics," Quart. Appl. Math. 24, 82-86 (1966). 4. A. D. Berk, "Variational Principles for Electromagnetic Resonators and Waveguides," IRE Trans. AP-4, 104-111 (1956). 5. LL. G. Chambers, "An Approximate Method for the Calculation of Propagation Constants for Inhomogeneously Filled Waveguides," Quart. J. of Mech. and Applied Math. 7, 299-316 (1954). 6. R. F. Harrington, Time-Harmonic Electromagnetic Fields, pp. 331-365, McGraw-Hill, New York, 1961. 7. R. Holland and E. P. EerNisse, "Design of Resonant Piezoelectric Devices," Ch. 3 and 4, Research Monograph No. 56, MIT Press, 1969. 8. R. Holland, "Piezoelectric Effects in Ferroelectric Ceramics," IEEE Spectrum 7,67-77 (April, 1970).
374 VARIATIONAL TECHNIQUES 9. G. J. Kynch, "The Fundamental Modes of Vibration of Uniform Beams for Medium Wavelengths," Brit. J. Appl. Phys. 8, pp. 64-73 (1957). 10 P E. Lagasse, "A Higher-Order Finite Element Analysis of Topographic Guides Supporting Elastic Surface Waves," to be published in /. Acous. Soc. Amer. S3 (1973). 11 L Meirovitch, Analytical Methods in Vibrations, Macmillan, New York, 1967. 12. S. G. Mikhlin, Variational Methods in Mathematical Physics, Pergamon, New York, 1964. 13. R. F Milsom and M. Redwood, "The Piezoelectric Generation of Surface Waves by an Interdigital Array: A Variational Method of Analysis," Proc. IEE 118, 831-840 (1971). 14 N. J. Nigro, "Steady State Wave Propagation in Infinite Bars of Noncircular Cross Section," J. Acoust. Soc. Amer. 40, 1501-1508 (1966). 15. С. H. Page, Physical Mathematics, pp. 65 76, van Nostrand, New York, 1955. 16. M. P. Stallybrass, "A Variational Approach to a Class of Mixed Boundary-I value Problems in the Forced Oscillations of an Elastic Medium," Proc. 4th U.S. Nat. tongr. Appl. Mech. 1, pp. 391-400 (1962). 17. H. F. Tiersten, Linear Piezoelectric Plate Vibrations, Ch. 6 and pp. 137 139, Plenum, New York, 1969. 18. G. O. Stone, "Coupling Matrices for High-Order Finite-Element Analysis of Acoustic-Wave Propagation," Electronics Letters 8, 466-468 (1972). 19. R. Burridge and E. J. Sabina, "The Propagation of Elastic Surface Waves Guided by Ridges," Proc. Roy. Soc. London, Л.330, 417^141 (1972). Appendix 4 RAYLEIGH WAVE PROPERTIES A. ISOTROPIC SUBSTRATES 375 B. CONVENTION FOR SPECIFYING ANISOTROPIC SUBSTRATE ORIENTATIONS 376 С PROPAGATION VELOCITY AND NORMALIZED PARTICLE VELOCITY COMPONENTS 379 D. NORMALIZED ELECTRIC POTENTIAL AND DISPLACEMENT 389 E- (AFJFhXsc 393 To calculate the performance of surface wave transducers (Section L of Chapter 10) or the effects of a thin plating on the substrate surface (Sections В and С of Chapter 12), it is necessary to know the wave field components at the substrate surface in terms of the power flow. In this appendix, field components (normalized with respect to power flow) are given for Rayleigh surface waves propagating on a variety of different substrates. A. ISOTROPIC SUBSTRATES In this case analytic expressions for the normalized particle velocity components at the substrate surface can be derived from (10.36) and (10.37); that is, for a wave traveling in the +z direction (£тф_0 = (A;V ™"2 /o° (4-2) where P,( = power flow per unit width along x ЫУ-HlVsf - i щ 375
376 RAYLEIGH WAVE PROPER! IES VJVi FIGURE 4.1. Field parameters fv and fz in (4.1) and (4.2) as a function of the bulk shear wave velocity V and the bulk longitudinal wave velocity V,. The normalized particle velocity components at the surface are functions oil p, Vs, and V, of the substrate. Plots offy and fz as functions of VJV, атЩ given in Fig. 4.1. W'ave propagation is along the z coordinate and the inware normal to the surface is in the у direction. Exact and approximate expression J for the Rayleigh wave velocity fu are given in Fig. 4.2. В CONVENTION FOR SPECIFYING ANISOTROPIC SUBSTRATE ORIENTATIONS The characteristics of Rayleigh wave propagation on anisotropic substrates! depend on both the crystal class of the substrate material and its orientatioJ with respect to the surface normal and the propagation direction. The orientation of a crystal medium can be specified in terms of three rotation anglcsj (either the IRE angles or the Euler angles) described in Section 3.D of Voluniu] I. These conventions can also be used to specify substrate orientation unfl В. ANISOTROPIC SUBSTRATE ORIENTATIONS 377 0.960 0.950 0.940 0.930 0.920 Э ; 0.910 0.900 0.890 0.880 0.870 0.860 ■ Exact, Eq. (10.34) ■ Approximate, Eq. (10.35) Range of physical realizability _L 0.1 0.2 0.3 0.4 VJVi 0.5 0.6 0.7 FIGURE 4.2. Isotropic Rayleigh wave velocity KH as a function of the bulk shear wave velocity Vs and the bulk longitudinal wa\e velocity surface wave propagation direction but, for the cases considered here, it is more convenient to use a simplified convention. Coordinate axes arc chosen such that у is the inward normal to the substrate surface and z is along the propagation direction (Fig. 4.3 and Fig. 4.4). The substrate normal is often parallel to one of the crystal axes. In such cases the substrate orientation is described by giving the crystal axis direction corresponding to the outward normal to the surface. In Fig. 4.3 for example, the X crystal axis lies in the direction of the outward normal and the substrate A' FIGURE 4.3. Illustration of the surface wave orientation X-ctit 0° F-propagation. X, Y, Z are crystal axes; x, y, z are coordinate axes.
378 RAYLEIGH WAVE PROPERTIES FIGURE 4.4. Illustration of the surface wave orientation ф° rot X. X, Y, Z are crystal axes; x, y, z arc coordinate axes. is called ЛГ-cut. If the X crystal axis had pointed in the direction of the inward normal, this would have been called — A"-cut. The remaining parameter to be specified is the propagation direction, which lies along the г coordinate axis. This can be defined by giving the angle в between one of the crystal axes lying in the surface and the z coordinate axis. As in Fig. 4.3, в is always measured in the clockwise direction with respect to the outward normal. For substrates of cubic symmetry, crystal axis directions are defined by the notation [100], [111], [ПО], etc. described in Fig. 9.12a. Crystal axes for other symmetries are defined in Chapter 7 of Volume I. The other substrate configuration considered in the following tables of Rayleigh wave properties is shown in Fig. 4.4. Tn this case the propagation direction z is parallel to the X crystal axis and the У and Z crystal axes occupy a rotated position specified by angle ф in the figure. Initially, the substrate surface is oriented with the X crystal axis lying in the plane of the surface and the Z axis along the outward normal. The surface is then rotated clockwise- through an angle ф about the X axis. И Z О On о и и о > а U Р < Он О ы N -1 < а О Й а z. < Н D о Ы > о Б -< о I си I о _о 4> я Й <2 Е 3 с (/} ■— •о 3 4J -—- N О . ^ 'Л и i 8 о ° I ^1 о. ш (Г ел о 1 О о О 1 * 1 о |о О О ?Г > 3 1 \ Ъ 1 а "з С. О Ч; ос о со гч гч гч' гч' 14 с о о\ > О О >ы з "з i а г- о 1 - ГГ> —< ГО ^ го со ГО гп о г- о 3^3 I Й I с в а, о Б. о О о OS ч. > \ о "з 1-« 3 ,8151 0086 о о у §• у о ^ о. I N «о ГО Р 5 и С _ Он с 5 51 И i х о я О _ о —* (/, О я jo s •о . с та I <£ Is О <L> о "С < В it -с о |*- £ *. С с S и *— О. Л 1) s- U. о I* ^: с -А -с £ „ с с ■— р s 5 ? с - 1Л jj 01 а: С 1) л О ел-о Й д <ч -5: £ Ь » -5 х: -°S ? О зд^ ■а . ii — = # ca. " ■с « £ a « ^ ■IS*- ц и и = " я „ м ° я с -а о с- 5. S -5 ц, с -а ■а Ь R S < га '■ и С и С J= 55 3 Я -г; j- S - СЗ О . ■£ ^™ ■ ■ с < Z -о о В Г - ь- а> - -3 -о " ^ "С о и ^ ^ +- cl .£ ++ о > vc-, J; > о В- 5 о -С £ сд — "5 с ь. с О о — ■£ Й я га л S о о.<« — •5 Й § л J3 5 &£■ о х: с: — с - ° Э Ё 1) " Р
m sec Material Germanium Indium Arsenide Indium arsenide Indium antimonide Orientation Propagation velocity [110]-cut [I10]-prop LTlO]-cut [001]-prop Z-cut 21° AT-prop Z-cut 21° AT-prop [TlT]-cut 30° [110]-prop [TlTj-cut 30 [110] prop [Tl0]-cut [001]-prop [TW]-cut [001]-prop [TH]-cut 30° [110]-prop [TiT]-cut 30°[110>prop [TlORut [001]-prop Normalized surface particle velocity /ш_б (су in radians/sec) ^ (watts/m): Electrical boundary conditions (°ItrV pl/2 Ml 2648.4 0 3.602 cy1/2/90° 2.050 ^'VO0 3013.7 0 3.705 ft.1'2/90" 3.172 (у1 уг/ 2116.8 free 0.7428 O)112/CP 4.685 о)1,3/90с 3.454 иА'УР" 2116.7 short 0.7434 ш1/а/00 4.685 mui/9(f 3.454 щ'''2/0° 1986.7 free 0 4.245 v1'2/104' 2.428 шь'2/0° 1986.6 short 0 4.245 си1'2/104° 2.428 су1'2/£ 2192.0 free 0 4.622 а>"У90° 4.021 «Л'УО0 2192.0 short 0 4.622 a?i2/9(f 4.021 су1,г/0° 1751 2 free 0 4.824 cu1'2/103° 2,766 су"2/0° 1751.1 short 0 4.824 су"2/103° 2.766 су^/О0 1923.8 free 0 5.252 су"8/90° 4.472 й^уг/ Lithium niobate <5 [Tl0]-cut [001]-prop 1923.7 short 0 5.252 at'*/90° 4.471 со'/уо0 AT-cut 114° У-ргор 3397,4 free 0.4502 оЯ'уо" 2.572 су1''у 90° 1.774ю1Й/0с A'-cut 114° У-ргор 3318.8 short 0.2553 ft>1/2/0° 2.996 су''2/90с 1.597 су]'У0° - У-cut A'-prop 3769.0 free 0.7298 to1'2/ -90° 3.150 си1 у 90° 2.219 w1'2/0° - У-cut AT-prop 3739.5 short 0 4420 со1/2/900 3.152 су1'2/90° 2.311 t^'yo0 - У-cut 70° AT-prop 3428.3 free 0.9484 су1/2/-134' 2.614 су1'2/88° 1.831 су1/2/0° - У-cut 70° AT-prop 3367.4 short 0.9072 to1'2/-126е 2.909 ш1 у 83° 1.655 оЛ'*/0с - У-cut Z-prop 3487.7 free 0 2.625 су1 2/90с 1.777 оЛ'*/0й - У-cut Z-prop 3403.7 short 0 3.056 а>'Л/89° 1.638 со^/О0 Z-cut AT-prop 3797.6 free 0.8554 ео'/уэО" 2.866 (о1/а/90° 2.302 су1 у 0° Z-cut AT-prop 3787.8 short 0.6265 са^уэО0 2.982 су1'У 90° 2.261 tuVa/o° Z-cut У-ргор 3902.2 free 0 3.041 су,/2/97.2° 2.480 су1 'у 0° Z-cut У-ргор 41.5° rot AT 3858.5 short 0 3.170 ш1 у 90.8° 2.377 со12/ 0° 3999.5 free 0.2996 су1'2/-90° 3.122 («''УЭО0 2.802 су1/2/0° 41.5° rot AT 3888.6 short 0.2446 тЧ*/90° 3.432 а>У*/90° 2.571 су»'2/0° Vr (m/scc)
material orientation propagation velocity normalized surface particle velocity («> in radians/sec) electrical boundary conditions ity ,|0~6 m-sec \ [ (watts.'m)1'2, pl/2 nl/2 1 li bismuth germanium oxide cadmium sulfide diamond [001]-cut [110]-prop [001]-cut [110]-prop [tlt]-cut [1101-prop [tlt]-cut []10]-prop - у-cut 60° лг-ргор -у-cut 60° a"-prop - у-cut z-prop - у-eul z-prop z-cut .v-prop z-cut a"-prop z-cut a*-prop 1680.7 1669.2 1707.9 1693.9 1702.2 j 699.5 1715.7 1711.2 1728.9 1724.8 10971 free short free short free short free short free short 4.163 со1'2/90c 4.265 ш1/2/90с 2.468 о;1/г/90° 3.680 со1'у90е 2.726 <а*''8/90° 3.578 ю*л/90° 0.4304 wi/y-180° 5.582 т1/2/90с 0.3006 -180е 5.162 со1'2/90° 0 5.083 си1/2/90° 5.095 со1/2/90° 5.324 соу2/90с 5.297 №1/2/90° 1.405 w]/V 90е 2.535 с^'Уо" 2.540 ft^Vjf 2.409 (ц1;г/0° 2.388 w12/0c 3.163 w1 у 0° 3.155 со1/8/0° 2.600 о)] у 0° 2.636 цлг/0° 2.861 ал'2/0° 2.868 м1/г/0° 1.143 ш^/о0 europium iron garnet gallium arscnids germanium Z-cut 45° jt-prop [tlt]-cut [moj-prop [tl0]-cut [110]-prop [tl0]-cut [00lj-prop [tl0]-cut [110]-prop [tl0]-cut [0011-prop z-cut 22.5е л"-ргор z-cut 22.5° *-prop [tlt]-cut 30° [110]-prop [IlTJ-cut 30° [110]-prop [tl0]-cut [001]-prop [tl0]-cut [001]-prop z-cut x-prop [tlt]-cut [110]-prop 11120 10756 10602 11063 3158.7 3207.4 2763.4 2762.8 2605.2 2605.0 2822.0 2821.8 2934.3 2687.2 free short free short 0 1.398 a^y 90° 0.2414 ая/у-90° 1.322 w1/2/90a 0 1.381 со]/а/90° 0 о о 1.363 0^/90" 2.994 cu^yjxf 3.018 щ]/г/90° free 0.4978 ю1/2/0° 3.853 wl<*/90° short 0.5033 ш1'2^ 3.852 сош/90° 3.627 со'/3/102° 3.626 to1 'а/102° 3.849 wlfi/90c 3.849 ш^/эо0 1.080 ш^/оЦ 9.599 ая^/о0 9.816 a>1/2/0° 1.123 ш"у0° 1.928 t»'/yoc 2.039 со^ур0 2.883 со'^/0с 2.881 а>'''уо° 2.201 со1'у 0" 2.202 сцууо0 3.318 wiyo0 3.317 со]'у0о 0 3.925 со1'2/90° 3.285 «л'*/о° 1.323 а^у-90с 3.050 а^2/90° 1.741 a^'/p0 (m/sec)
9~ I О 1 в о тз £ 2 3 -S ■о 3 £ о ° о \| м а. "з as г<1 О С х> — » 3 с Гп S ° о "о вя rt CL. О (Л jO о /ш О > — U о с о £- D. CL, ft О _ о — о — о о. г4 у у 9 0.0^ ^ ~ч — О ^ .j ч1, «л I— — |" " t а ~2 с N D. NORMALIZED ELECTRIC POTENTIAL AND DISPLACEMENT! Material Barium sodium niobate Bismuth germanium oxide Cadmium sulfide el/2 I kilovolts \ / Pi/2 10 12 coulombs/m2' boundary \(wa«s/m^ \ (watts/m)"* Orientation conditions (to in radians/sec) — Г-cut 38" X-prop — K-cut 38° X-prop — F-cut 90° X-prop — K-cut 90° X-prop 45° rot x 45u rot x I001]-cut [110] prop [001]-cut [1IO]-prop [TlT]-cut [110]-prop [TlT]-cut Lll0]-prop — K-cut 60° X-prop — K-cut 60° X-prop — F-cut 90° X-prop — K-cut 90е X-prop Z-cut X-prop Z-cut X-prop free short free short free short free short free short free short free short free short 0.9416 w-1'2/—90° 0 1.368 to 1 у 90° 0 2.501 w-l'2/0° 0 8.698 o; ''2/0° 0 9.335 (a "2/0° 0 8.209 oi 1 y-90° 0 10.51 щ-1 У—90° 0 10.04 о *«/180° 0 2 504 си1'У91° 580.4 ш1,8/90 3.834 ш1/2/-90° 481.3 оЛ'У-90° 6.589 си1'2/—180° 1285 о1-'2/—180° 45.80 шУ/—180° 1897 со1 У—180° 48.37 о1/2/—180° 2089 р)г/У—180° 42.68 со1 у 90° 462.7 «1^/90° 54.21 о1''2/90' 579 5 г)1 У90° 51.41 о'Я/О" 547.4 cu1/2/0° t After A. J. Slobodnik, Jr., and E. D. Conway, "Microwave Acoustics Handbook Vol. 1," Physical Sciences Research Papers No. 414, Air Force Cambridge Research Laboratories, Bedford, Mass, 1970. % Sec Part C. 389
Material Quartz Rutile Sapphire Silicon orientation - V-cut A'-prop - У-cut A'-prop дг-cut У-ргор - У-cut A'-prop - У-cut Z-prop Z-cut A-prop Z-cut 20° A'-prop Z-cut 45° A-prop - У-cut A-prop Z-cut AT-prop Z-cut 30° A'-prop Z-cut A'-prop Propagation velocity Fr ' (m/sec) Electrical — boundary conditions 3159.3 free 3156.4 short 4144.0 4144.0 4193.5 4788.9 4751.3 5066.3 5639.4 5555.2 5706.4 4921.2 Normalized surface particle velocity (со in radians/sec) fV» m/scc \ \ (watts/m)1'2/ (vrt)v o. -У1/2 + (vhy)y 0 1 e 1.436 fu^V-90° 4.337 ы^У90° 1.440 uA'*/-90° 4.339 w1 '8/90° 0 2.362 w1,2/9Q- 0 2.362 co]/2/90" 0 2.698 tu1'У 90° 0 2.321 wU*/90° 0.1624 a)*/*/180" 1.962(^^/90° 0 2.190 cu1'2/90° 0.9714 ю^»/-90° 1.505 to1 ■''/ 90° 0.6472 0)^/90° 1.841 wLy_90° 0 2.045 аЛу84° 0 3.512 ft)1 'У 90" -Й7Г-* 2.897 f^yo- 2.902 w^'yO0 2.265 ta^/P0 2.265 cu'VO0 1.591 au/0° 2.029 cu^/P 1.463 Ц)1'У0° 1.425 co''2/0° 0.9349 w1 У0 1.075 юш/0° 1.265 aW/V 2.862 ц)]/2/0° spinel Yttrium aluminum garnet yttrium gallium garnet [Tl !]-cut [lioj-prop Z-cut A'-prop [TlT]-cut [!10]-prop [TlOJ-cut [110]-prop [TlOJ-cut [001]-prop Z-cut A'-prop [TlT]-cut [110]-prop [Tl0]-cut [110]-prop [Tl0]-cui [001]-prop Z-cut A'-prop [TlT]-cut [110]-prop [TlOJ-cut [110]-prop [Tl0]-cut [OOlj-prop 4546.1 4910.9 4154.9 4120.5 5259.5 4598.6 4580.6 4566.5 4602.8 3699.2 3643.4 3614.2 3710.9 1.155 cu'/y-90° 2.784 0^/90° 1.615 ш'-'уо0 0 2.572 at^/W0 0.9222 (о"У-90° 1.666fti1/a/90° 0 2.169 ю''У90° 0 2 384 to1'У 90° 0 2.565 to1'У 90° 0.1200 <о1/г/-90° 2.541 со1 ''/90° 0 2.548 at>*/90° 0 2.555 ш1/а/90° 0 2,797 а^у90° 0.4089 ^/-90° 2.674 to1'2/90 0 2 729 шт/90° 0 2,763 «Р'У90° 2.292 со1 2/0э 0.7147 tu^yp' 0.9804 м"уо° 2.199 аА'ь/О0 1.775 си^ур0 1.724 щ^уо0 1.719 со1 у 0° 1 769 со1 yjf 1.946 со1 уо° 1.737 су1 2/0° 1.748 гд^уо" 1.923 со1 у 0°
Material Indium arsenide Indium antimonide Lithium b Ml pl/2 Ml Orientation Electrical boundary conditions (kilovolts \ /10 12 coulombs (watts/my12j \^ (walts/m)"2' (cu in radians/sec) Z-cut 22.5° X prop Z-cut 22.5° A'-prop [TlT]-cut 30" [H0]-prop [TlT]-cut 30° [110]-prop [DO] cut [OOtj-prop [Tt0]-cut [001] prop Z-cut 21" A'-prop Z-cut 21 AT prop [ilt]-cul 30° [110] prop lTlT]-cut 30° [U0]-prop [Tl0]-cut [001]-prop I110]-cut [001]-prop [TlT]-cut 30' [П0]-ргор [TlTj-cut 30 1110]-prop [Tl0]-cul [001]-prop [Tl0]-cut [00l]-prop AT-cut 114е K-prop X-cut 114° У-ргор У-cut AT-prop free 2.770 cu I2/180 short 0 free 1.803 to-1'2/—207° short 0 free 1.704 to-1'a/90° short 0 free 0.8200 cu~"2/180° short 0 free 0.5463 огш/—204° short 0 free 0.5347 т-1,й/9й" short 0 free 0.9013 к)-1'2/— 204° short 0 free 0.9087 tt)-1/2/90° short 0 free 13.23 at "2/90° short 0 free 4.511 со 1 2/0° 8.876 to"2/0° 106.8 со"2/0° 6.132 to"2 73.65 со"2/ — 2 5.35 со1'2/ -90 64.28 co"2^ 3.409 со"2/0° 53.06 ^"уо0 2 454<»"2> 37.68 ш"2/ 2.163 co"2^ 33.44 о)1 y_-jj 4.567 col/2/ 77.47 co"2£- 4.187 со"2 71.08 со1'2 34.45 to"2/-. 2037 со"2/ -9j 10.59 to"2/180" Omiw At,), pi 2 Mi Material lithium niobate Ilium tenia late Orientation — K-cul AT-prop — K-cut 70° AT-prop — K-cut 70' X prop — У-cut Z-prop — K-cut Z-prop Z-cut A'-prop Z-cut A'-prop Z-cut K-prop Z-cut У-ргор 41.5° rot AT 41.5° rot AT AT-cut 131.5° r-prop A^cut 131.5 К prop — K-cut AT prop K-cut A'-prop — K-cut 36.5° A'-prop — K-cut 36.5 AT-prop — У-cut 5ff A"-prop — K-cut 56' A'-prop — K-cut Z-prop Electrical boundary conditions / kilovolts \ /10 12 coulombs/m^X ^(watts/m)"^ ^ (watts/m)1'2 J (со in radians/sec) short free short free short free short free short free short free short free short free short free short free 11.87 со "2/75° 0 14.50 to-,/s/95a 0 4.501 со- '-yo" 0 10.33 со-"2/—26° 0 14.88 co-i/yo° 0 6.625 eo-1/g/90° 0 1.663 w-"2/0° 0 4.068w "2/33° 0 2.830 г0-1Л2/56° 0 5.548 cu-'y96° 1829 to'-'8/180° 30.63 cu"2/—105° 1740 со1-2/—Ю5° 36.79 со"2/—85° 1900 to1'2/—83^' 10.49 o>'/yi80° 607.6 со1'3/180° 23.44 со1'2/—206° 1150 to"2/—210° 32.92 to"2/180° 1991 со"2/180" 17.78 со"2/—90° 906.0 со1'2/—90° 4.682 to"2/180° 280.3 со"2/!80° 11.26 со"2/—147° 606.0 со"2/—148° 7.886 п>"у_ 124° 397.9 to1'2/ — 124° 15.21 со"2/—84° 391 Gallium arsenide
(^Ktr)y-ot Material Quartz Zinc oxide Orientation — У-cut Z-prop Z-cut X-prop Z-cut X-prop Z-cut 30° X-prop Z-cut 30° X-prop 22° rot X 22° rot X X-cut 27° У-rot X-cut 27° K-rot — У-cut X-prop — У-cut X-prop 45" rot X 45е rot X short free short free short free short free short free short free short t>l/2 Mt pin 1 к Electrical boundary conditions (kilovolts \ / (watts/m)1'^ \ 10 12 coulombs/ (watts/m)1') (со in radians/sec) 0 3.152 eo-^/O0 0 7.401 с»-'/2/9° 0 4.722 со 1/2/<f 0 7.190 в)-1Д/180° 0 8.726 со 1/2/ —90° 0 9.035 oj-^y 180° 0 748.1 со"2/—84 8.719 arVyi80° 454.4 со1'2/180° 19.67 со1'2/—I7i 970.5 со1'2/—1? 12.66 со1''2/180" 704.9 со1-"2/]80" 20.10 со^ур" 110.8 co^yo" 24.44 со1/2/9(/ 134.7 о№/Щ? зо.зо о"уо" 330.0 m1;2/0° Е. (AFR/Kn)Hf. 393 Е. (AFk/^r)s,- It was seen in Section L of Chapter 10 and Section С of Chapter 12 that the fractional change in Rayleigh wave velocity produced by a short circuit at the substrate surface, (A^'r/E'iOsc' ls a fundamental parameter in the theories of interdigital transducers and of scattering from metallic gratings. A perturbation method for calculating this quantity was given in Examples 4 and 5 of Chapter 12. The table below lists perturbation theory and exact results for a number of commonly used substrate configurations. У* , in percent sc €v Perturbation Exact§ Substrate material Oriental ionf e0 theoryj calculation Barium sodium 45° rot X 187 0.26 0.27 niobate — У-cut Z-prop lit 0.46 0.52 Bismuth [Ul]-cut [Н0]-ргор 43 0.830 0.816 germanium oxide [001]-cut [I10]-prop 43 0.725 0.683 Cadmium sulfide — У-cut Z-prop — E-cut 60° X-prop 98 0.264 0.260 9.7 0.160 0.162 Z-cut X-prop Z-cut 22.5° X-prop 9.8 0.242 0.236 Gallium arsenide 11.0 0.02 0.024 Lithium niobate — T-cut Z-prop — У-cut 70° X-prop 50.1 2.38 2.41 55 1.75 1.77 41.5° rot X 67 3.34 2.77 — T-cut X prop Л'-cut 131.5° T-prop 84 0.38 0.795 Lithium tantalate 49.4 0.489 0.496 Z-cut 30° X-prop 47.9 0.593 0.589 22° rot X 48.4 0.244 0.274 Quartz — У-cut X-prop 4.5 0.091 0.093 t Figures 4.3 and 4.4 X After К. M. Lakin. § Alter A. J. Slobodnik, Jr. and E. D. Conway. Lithiufn tantalate
Bibliography Reference lists at the ends of the chapters contain only material that is closely related to the subject matter discussed. Even in these restricted areas this constitutes only a small fraction of the material available in the literature. In addition, there arc areas such as aoousto-optics, nonlinear effects, and general applications that are not covered in the text at all. This bibliography (twenty- two in number) is directed toward a broader coverage of the subject of acoustics. References arc restricted to recently published review articles and reference works, which themselves contain extensive bibliographies. SUBJECT INDEX BY NUMBER Acoustic Amplification: 2, 3, 5, 6, 7, 11, 16, 21 Acousto-optics: 2, 5, 6, 7, 11, 13, 16, 21 Biomedical: 2, 6, 7, 15, 21 Filters, Signal Processing, and Memories: 2, 3, 5, 6, 7, 8, 10, 16, 21, 22 Geology and Seismology: 6 Industrial Processing: 2, 6, 7, 8, 21 Materials Properties: 1, 2, 4, 6, 7, 9, 11, 12, 21 Nodestructive Testing: 2, 6, 7, 8, 21 Nonlinear Effects: 2, 7, II, 19, 21, 22 Physics: 2, 6, 7, 11, 17, 18, 20, 21 REFERENCES 1. K. S. Aleksandrov and Т. V. Ryzhova, "The Elastic Properties of Crystals," Sov. Phys.-Crystall. 6, 228-252 (1961). 2. Documentation in Ultrasonics, vol. 1 (1968), vol. II (1969), vol. Ill (1971), vol. IV (1972), R. Pohlman Ed., S. Hirzel Vcrlag, Stuttgart. 3. M. Epstein, "Surface Microacoustics," Information Processing and Control Systems Laboratory, Northwestern University, Evanslon, 111. 1970. 4. H. B. Huntington, "The Elastic Constants of Crystals," in Solid State Physics, F. Scitz, and D. Turnbull, eds., vol. 7, pp. 213-351, Academic Press, New York, 1958. 5. IEEE Trans. MTT-17, November 1969, Special Issue on Microwave Acoustics. 6. Proc. IEEE, October 1965, Special Issue on Ultrasonics. 395
396 BIBIIOGRAPEH INDEX Acoustic antenna, 175, 364 Acoustic boundary conditions, 2, 8, 39,43, 44,45,47 Acoustic field equations, matrix form, 152 symbolic notation, 152 Acoustic impedance, 22-23, 38-39, 192, 275, 284, 305, 326, 330 Acoustic polarization, 11-19, 21, 23, 24, 25, 28, 29, 43, 136, 139, 142, 146 Acoustic reflection, anisotropic, 6-21, 38-43 coefficient, 22-24, 27-38, 39, 42, 268 critical angles, 5-7, 22-23, 27, 32, 34, 35 isotropic, 2, 3-4, 5-7 Snell's Law, 3, 6, 22, 26, 27, 28, 29 stress-free boundary, 8-12, 30-38, 46-57 transmission line model, 35-38 Acoustic refraction, anisotropic, 6-21, 38-43 critical angles, 5-7, 22-23, 27 isotropic, 2, 3-4, 5-7 Snell's Law, 3, 6, 22, 26, 27, 38, 39 Acoustic scattering coefficients, 22-24, 27-38, 39,42 Acoustic wave equation, 66, 229, 245 Admittance, interdigital transducer, 175-177, 365-368, 373 matrix, 257-259, 266 resonator equations, 254, 257 resonator input, 266 Amplifier, Rayleigh wave, 291, 293-294 surface wave, 272, 293-294 Anisotropic, Frcsncl equations (nonpiezoelectric), 38-43 Ircsnel equations (piezoelectric), 43-57 lamb waves, 130, 133 Love waves, 133 plate waveguide, free, 128-130 Anisotropic (Continued) plate waveguide, on a half space, 133 Rayleigh waves, 130-131, 133 SH modes, 128-130 Stoneley waves, 133, 135 substrate, 376-378 unbounded plate resonator, 224-226 Anisotropic waveguide, 69-70, 128-151, 155-160,177-189 mode orthogonality, 155-160, 185, 187 transmission line model, 177-189 Antenna, acoustic, 175, 364 end fire, 175 Attenuation, Brekhovskikh's perturbation formula, 311 fluid loading, 283-286 Rayleigh wave, 283-286, 297-299, 309-313 surface roughness, 309-313 viscous, 297-299 Backward wave, 158 Beat wavelength, 94, 127, 145 Bcssel functions, 208, 233-234, 264 Birefringence (double refraction), 2, 4, 6 Bleustcin-dulyaev wave, 139-142, 145-146, 148, 210-212, 291 cylindrical, 139 dispersion relation, 141, 148 velocity, 291 Bound wave, 102 Boundary conditions, acoustic, 2, 8, 39, 43, 44,45,47 Brekhovskikh, 330 electric displacement, 45, 178, 190. 252, 253 electric field, 43 electric potential, 45, 50, 178, 190, 252, 253 397 7. IEEE Ultrasonics Symposium Programs 1967, IEEE Trans. SU-15, pp. 56 79 П 968); 1968, IEEE Trans. SU-16, pp. 20-39 (1969;) 1969, IEEE Trans. SU-17, pp. 53-70 (1970); 1970, IEEE Trans. SU-18, pp. 44 67 (1971); 1971, IEEE Trans. SU-19, pp. 390 418 (1972). 8. IEEE "Invited Proceedings 1970 Ultrasonics Symposium," Publication 70C69SU, 1971. 9. Landolt-Bornstein, "Numerical Data and Functional Relationships in Science and Technology"—Group III, in Crystal and Solid State Physics vol. 1, K-H Hcllwege and A. M. Hcllwege, eds.. Springer, Berlin, 1966. 10. Microwave Journal 13, March 1970, Special Issue on Microwave Acoustics. 11. Physical Acoustics vol. 1-vol. 9, W. P. Mason and R. N. Thurston, eds., Academic Press, New York, 1964-1972. 12. G. Simmons, "Single Crystal Elastic Constants and Crystal Aggregate Properties," /. Crad. Res. Center, Dept. of Geol. and Geophys., 34, 1-152 (1965). 13. State of the Art Reports No. 4, T. Kallard cd., "Acoustic Surface Wave and Acousto-Optic Devices," Optosonic Press, New York, 1971. 14. A. B. Smith and R. W. Damon, "A Bibliography of Microwave Ultrasonics," IEEE Trans. SU-17, 86-111 (1970). 15. P. N. T. Wells, Physical Principles of Ultrasonic Diagnosis, Academic Press, New York, 1969. 16. R. M. White, "Surface Elastic Waves," Proc. IEEE 58, 1238-1276 (1970). 17. R. T. Beyer and S. V. Letcher, Physical Ultrasonics, Academic Press, New York, 1969. 18. A. S. Nowick and B. S. Berry, Anelastic Relaxation in Crystalline Solids, Academic Press, New York, 1972. 19. L. D. Rozcnberg, High-Intensity Ultrasonic Fields, Plenum Press, New York. 1971. 20. R. Trucl, C. Elbaum, and В. B. Chick, Ultrasonic Methods in Solid State Physics, Academic Press, New York, 1969. 21. IEEE "1972 Ultrasonics Symposium Proceedings," Publication 72 CHO 708- 8 SU, 1972. 22. IEEE Trans. MTT-21 SU-20, April 1973, Special Joint Issue on Microwave Acoustic Signal Processing.
398 INDEX Boundary conditions (Continued) electrically free, 47 electromagnetic, 43, 44 lossless, 178, 212, 252 magnetic field, 43 mass-loading, 314 mechanically free (stress free), 8, 30-31,46-47,68, 252, 253 mixed. 80-81, 113, 116-117 open encuit, 142, 143, 157, 252, 253, 2R7 particle velocity, 2, 43, 45,178. 190, 252, 253 quasistatic, 44, 45, 47, 178, 190. 252, 253 rigid, 157, 252, 253, 335, 337-338, 338-343, 369 short circuit, 51, 56, 136,142, 143, 157, 162, 252, 253, 287 spring loaded, 369 stress. 2, 43, 45, 178, 190, 252, 253 stress-free, 30-38, 66, 99, 104, 113, 130, 131, 136, 157, 162, 223, 227, 232, 235, 237, 239, 245, 252, 253, 274, 335, 337-338, 351, 362-363 Tiersten, 277, 280, 330 variational techniques, 335-336, 337 338, 350-353 Wolkerstorfer, 282 Boundary perturbations, resonator, 315-322 scattering, 302-315 waveguide, 272-294, 325, 326 Breathing mode, 237 Brekhovskikh boundary conditions, 330 Brekhovskikh's attenuation formula, 311 Bulk (plane) wave, 132, 176 transducer, 176 Capacitance, dynamic (motional), 355 intcrdigital transducer, 170, 176 motional, 258, 259, 348, 335-357 motionally-clamped, 259 "static", 257 zero frequency, 259 Channel waveguide, I 20 Characteristic equation, compressional sphere modes, 235 dilatational modes, free cylinder waveguide, 110 generalized Lamb waves, 100 Lamb waves, 79 Characteristic equation (Continued) Love waves, 96 SH modes. 137 SH trapped energy modes, 249 stripe waveguide, 123 thickness modes, anisotropic, 226 thickness modes, piezoelectric, 242 torsional sphere modes, 238 torsional waveguide modes, 110, 208 Characteristic impedance, 22-23, 38, 39, 183, 184, 192, 196 Christoffel equation, isotropic, 66 stiffened, 47-48, 53, 54 Closed waveguide, 95 Complete set of functions, 151, 342, 346 Fourier completeness, 346, 354 pointwise completeness, 346, 354 Complex natural frequency, 261-262 Cvomplex Poynting's Theorem, 205-206, 213, 253, 260, 264, 372 Complex propagation constant, 82-83, 158 159, 186 Complex reciprocity relation, 154-155, 204, 212, 250, 254 , 256 , 27 2, 294,315 Compressional sphere modes, characteristic equation, 235 field distributions, 235, 237 frequency curves, 236 frequency equation, 235 perturbed, 318-319, 322-323 Q, 266, 322-323 quality factor, 266, 322-323 variational techniques, 370 Conductance, interdigital transducer, 174-176 radiation, 175 Conical refraction, external, 19-21 internal, 11-20 Constant phase surface (wave front), 9, 10, 12 Constitutive matrix, 52-53 Constitutive relations, piezoelectric, 50, 152 Contour extensional modes, dilatation type, 347-349 field distributions, 230, 349 frequency curves, 350 frequency equation, 230 piezoelectric, 347-349 INDEX 399 Contour extensional modes (Continued) variational techniques, 347-349 Coordinate transformations. 42 Coupled waves, 81-82, 93-94, 113, 125-128, 137-138, 144-145, 150, 207, 272, 291, 328-329 beat wavelength, 94, 127, 145 coupling length, 127 crossover point, 81, 207 dispersion curves, 207 electrostatic and stiffened acoustic, 137-138 perturbation theory, 328-329 Rayleigh waves, 93-94, 144-145, 150, 272, 291 SA and P modes, 113 SVand P modes, 81-82 Coupling length, 127 Critical angles, anisotropic acoustic, 9, 11, 12 electromagnetic, 5 isotropic acoustic, 7, 22-23, 27, 32, 34, 35 piezoelectric, 44 Crossover point, 81, 207 Cubic crystal, conical refraction, 11 -21 nonpiezoelcctric shear wave scattering (fl Ill-axis), 11-21 nonpiezoelcctric wave scattering (YZ-plane), 8-11 piezoelectric shear wave scattering (XY-pIane), 46-52 Current, waveguide, 183 1 84,187, 188,196 Cutoff frequency, 75, 97 Cutoff mode, 75, 81-82, 158-160, 196 Cylindrical Bleustein-Gulyaev wave, 139 Cylindrical coordinates, scalar mechanical potential, 106-107 strain, 108 vector mechanical potential, 106 108 Cylindrical Rayleigh wave, 113-114 Cylindrical waveguide, 104-114 Damping constant, resonator, 262 Decay constant, transverse, 96 Degenerate waves, 11, 14, 87, 88 Dilatational Lamb waves, 79, 80, 82, 83-87, 88, 93, 111, 112 Dilatational modes (free cylinder waveguide), characteristic equation, IJ 0 field distributions, 11 2 Dilatational modes (free cylinder waveguide) (Continued) Pochhammer frequency equation, 110 Dilatational modes (rectangular parallelepiped resonator), field distributions, 357, 359-361 frequency curves, 359 variational techniques, 357-361 Dilatational modes (thick disk resonator), field distributions, 344, 354 frequency curves, 347 motional capacitances, 355-357 variational techniques, 344-347, 353-357 Dilatation type contour modes, 348 Dispersion curves, coupled waves, 207 Lamb waves, 80, 82, 83, 87, 158, 159 Love waves, 99, 101 SH modes, 76, 137-138 Dispersion relation, Bleustein-Gulyaev wave, 141, 148 definition, 221 generalized Lamb waves, 102 implicit, 203 Lamb waves, 77-83 Macrfcld-Tournois waves, 148, 151 Pochhammer frequency equation, 110 Rayleigh wave, 88-92 Rayleigh-Lamb frequency equations, 79-84,372 SH modes, 69, 73, 130 stiffened shear wave (XY-planc, cubic crystal), 49 stiffened shear wave (XY-plane, hexagonal 6mm crystal), 53 Distributed sources, surface, 161-162, 305-307, 310, 314 volume, 161-162, 305 Divergence (Gauss's) Theorem. 205, 252. 256, 317,335, 337, 366 Elastic isotropy conditions, 26 Electric displacement, boundary conditions, 45, 178. 190, 252, 253 perturbation, 287-289, 307, 321-322 Electric field boundary conditions, 43 Electrical perturbations, resonator surface, 319-322 resonator volume, 323-324 wave scattering, 302-309 waveguide surface. 286-294, 325. 326
400 INDEX Electrical perturbations (Continued) waveguide volume, 300-302 Electrical potential, boundary conditions, 45, 50, 178, 190, 252, 253 definition, 45 perturbation, 287-289, 321-322 piezoelectric plane wave, 47-48, 53,136 quasistatic equations, 54, 71 Electrical sources, 161-162, 163 Electrically free boundary, 46-57,288 definition, 47 reflection at, 46-57 Electromagnetic boundary conditions, 43,44 Electromagnetic field equations, matrix form, 152 symbolic notation, 152 Electromagnetic reflection, critical angle, 5 Snell's Law, 3, 6 Electromagnetic refraction, critical angle, 5 Snell's Law, 3, 6 Electrostatic stored energy, 260-261 Electrostatic wave, 70-71, 136, 137-138, 139-141, 147, 239, 240, 241 definition, 70-71 Elliptic integral, 173 End fire antenna, 175 Energy density, kinetic, 350 potential, 350 Energy flow direction (ray vector), 9, 10, 12 Energy velocity, 12, 199-207 group velocity, 205-206 SH modes, 206-207 waveguide, 205-206 Equivalent circuit, free boundary scattering, 33, 35-38 one terminal-pair resonator, 258-259 two terminal pair resonator, 259. 356-358 waveguide junction, 193, 196,198, 200 Evanescent wave, 5, 6, 9, 11, 23, 44, 47, 49, 50, 51, 52, 55, 5G, 57, 59, 70, 136, 147, 240 definition, 5 incident, 55, 57 reflected, 9, 11, 44, 47. 50, 52, 54, 56, 59 transmitted, 5, 6, 23, 44, 47, 49, 50, 52 Even symmetry modes, 242-245, 341-342 Faraday rotation, 155 Filter, general, 253-259 monolithic, 250 multimodc, 250 surface wave reflection, 309 First order perturbation theory, 304. 310, 314 Flcxural Lamb waves, 79, 80, 84-87, 88,93, 102 1 lexural modes (free cylinder waveguide), field distributions, 112-113 Forced oscillation, 221, 250, 254, 255, 256 Forced wave, 161-177, 221 Fourier series expansion. 151 Free mode, 156, 221-222, 251, 334, 349, 362 Free oscillation, 221, 251, 256, 261-262, 334, 349, 362 Free wave, 211 Frequency, natural, 221, 251, 261-262, 334 resonant, 221, 334 Frequency curves, compressional sphere modes, 236 contour extensional modes, 350 dilatational modes, rectangular parallelepiped, 359 dilatational modes, thick disk, 347 mixed sphere modes, 236 torsional sphere modes, 236 Frequency dispersion, 202 Frequency equation, compressional sphere modes, 235 contour extensional modes, 230 SH modes, bounded plate, 227 SH trapped energy modes, 249 thickness modes, anisotropic, 225-226 thickness modes, isotropic, 223, 265, 267,338 thickness modes, piezoelectric, 242-243, 320 thickness-twist modes, 227, 246-247 torsional cylinder resonator, 232 torsional sphere modes, 238 Fresnel equations, anisotropic nonpiezoelectric, 38^13 anisotropic piezoelectric, 43-57 definition, 2 isotropic, 21-38 Fundamental resonance, 173, 174, 341, 343 INDEX 401 Fundamental resonance (Continued) interdigital transducer, 170, 174 Gauss's (Divergence) Theorem, 205, 252, 256, 317, 335, 337, 366 Generalized Lamb waves, characteristic equation, 100 dispersion relation, 102 field distributions, 102-103 Generalized Rayleigh waves, 131 Green's function, 366-367 Group velocity, 8, 12, 14, 158, 199-207, 279, 280 energy velocity, 205-206 SH modes, 202-203 waveguide, 158, 202, 205-206, 279, 280 Hexagonal crystal, nonpiezoelectric shear wave scattering (X-axis), 39-43 nonpiezoelectric shear wave scattering (YZ-plane), 39 piezoelectric shear wave scattering (XY-planc, 6mm crystal), 52-57 Hooke's Law, planar, 229 Hybrid wave, 43 Imaginary propagation constant, 74, 81-82, 158-159, 186 Immittance, transducer, 163 Impedance, acoustic, 22-23, 38-39, 192, 268, 275, 284, 305, 326, 330 characteristic, 22-23, 38, 39, 183, 184, 192, 196 equivalent transmission line, 33, 37-38 interdigital transducer, 365, 366, 373 matrix, 38-39, 57 Rayleigh wave, 123 relation to reflection coefficient, 22, 37, 39, 192-193 relation to transmission coefficient, 22, 39 resonator equations, 254 resonator input, 268 shear wave, 22-23, 76, 192 surface, acoustic, 275, 284, 305, 326. 330 surface, electrical, 288, 289, 291, 305, 320-321 surface, mechanical, 275, 284, 305, 326, 330 transformation, 57 Impedance (Continued) transverse resonance analysis, 73 waveguide, 184, 187, 188, 192, 195 Implicit dispersion relation, 203 Index (subscript), resonator modes, 235-237 waveguide modes, 82-83, 158 lngebrigtscn perturbation formula, 176, 290, 293 Instantaneous stored energy, 260-261 Interaction matrix, 353 Interdigital transducer, 163-177, 290, 309, 364-368, 375, 393 admittance, 175-177, 365-368, 373 capacitance, 173, 176 conductance, 174-176 fundamental resonance, 173, 174 impedance, 365, 366, 373 susccptance, 175, 176 variational techniques, 176-177, 364-368 Interface wave, 102 Interior perturbations, resonator, 315-317, 322-324 scattering, 302-305 waveguide, 294-302 Isotropic, bounded plate resonator, 226-230 Christoffel equation, 66 cylinder resonator, 231-232 cylinder waveguide, free, 104-114 Fresnel equations, 21-38 plate waveguide, free, 66-94 plate waveguide, on a half space, 94-104 sphere resonator, 232-239 strip waveguide, free, 115-118, 209 strip waveguide, on a substrate, 1 I 8, 119, 121 substrate, 375-376 unbounded plate resonator, free, 222-224 unbounded plate resonator, rigid, 338-343 Isotropic waveguide, 66-128 mode orthogonality, 155-160, 185, 187 transmission line model, 180-189 lsotropy, elastic, 26 Junction, waveguide, 190, 193. 196, 198, 200, 201
402 INDEX Kinetic energy density, 350 Kinetic stored energy, 260-261 Kino's variational formula, 176-177, 368 Lagrangian, density function, 350, 351 function, 350, 351 methods, 349-353, 364 Lamb reciprocity relation, 153 Lamb waves, 76-88, 130, 133, 136, 139, 145, 158, 198, 203, 207, 227-228, 262, 305, 372 anisotropic, 130, 133 antisymmetric (flexural), 79, 80, 84-87, 88, 93, 102, 372 characteristic equation, 79 dilatational (symmetric), 79, 80, 82, 83-87, 88, 93, 111, 112 dispersion curves, 80, 82, 83, 87, 158, 159 dispersion relation, 77-83 field distributions, 83-88 flexural (antisymmetric), 79, 80, 84-87, 88, 93, 102 Rayleigh-Lamb frequency equations, 79, 84, 372 scattering, 198, 305 slowness curves, 78 symmetric (dilatational), 79, 80, 82, 83-87, 88, 93, 111, 112, 372 variational techniques, 372 Lame, constants, 275, 277 velocity, 85, 116 wave, 85, 86, 207, 209, 262, 263 Laplace wave, definition, 71 Laplace's equation, 49, 71, 241, 288, 291 Layered waveguide, 156 Leaky wave, 95, 96-97, 98, 102, 114, 119, 123, 131 Legendrc polynomials, 173, 233 Lewis reciprocity relation, 154 Longitudinal wave, 7, 28-32, 223-224 Lorentz reciprocity relation, 153 Lossless boundary conditions, 178, 212, 252 Lossy media, 155, 212, 213, 261-262, 264, 266 resonators, 261-262, 264, 266 waveguides, 212, 213 Love waves, 95-98, 99, 101, 133, 145, 201, 208 anisotropic, 133 Love waves (Continued) characteristic equation, 96 dispersion curves, 99, 101 scattering, 201 slowness curves, 98 Love-type mode, velocity dispersion curves, 119 Maerfeld-Tournois wave, 145-151 conditions for existence, 149, 151 dispersion relation, 148, 151 Magnetic field boundary conditions, 43 Magnctostrictive coupling, biased, 155 Mass loading, 314, 318 boundary conditions, 314 Matrix, acoustic field equations, 152 constitutive, 52-53 electromagnetic field equations, 152 impedance, 38-39, 57 interaction, 353 planar stiffness matrix, 229, 245 resonator admittance, 257-259 scalar product. 152 stress boundary condition, 39 Mechanically free (stress-free) boundary, 8-12, 30-38,46-57,68 definition, 8 reflection at, 8-12, 30-38, 46-57 Mechanical perturbations, resonator surface, 318-319 resonator volume, 322-323 wave scattering, 302-315 waveguide surface, 274-286, 325, 326 waveguide volume, 297-300 Mechanical potential, cylindrical coordinates, 106-108 definition, 67 scalar, 67-68, 107 spherical coordinates, 233-234 vector, 67-69, 108 Mechanical sources, 161-162, 163 Microsound waveguides, 118-128, 272 Mixed boundary conditions, 80-81, 113,116-117 Mixed sphere modes, field distributions, 238, 239 frequency curves, 236 Modal analysis, complete set of functions, 151 orthogonal functions, 151, 250-253 resonator, 250, 253-259 waveguide, 65, 151. 163 INDEX 403 Modal field distributions, compressional sphere modes, 235, 237 contour extensional modes, 230, 349 dilatational modes, free cylinder waveguide, 112 dilatational modes, rectangular parallelepiped, 357, 359-361 dilatational modes, thick disk, 344, 354 flexural modes, free cylinder waveguide, 112-113 generalized Lamb waves, 102-103 Lamb waves, 83-88 Maerfeld-Tournois waves, 147 mixed sphere modes, 238, 239 Rayleigh wave, 88, 92-93, 146, 147, 148, 149, 180, 182, 278, 375-392 SH modes, bounded plate, 227 SH trapped energy modes, 248-249 SH waveguide modes, 76, 137-139 Stoneley waves, 104 thickness modes, anisotropic, 225 thickness modes, isotropic, 223-224, 338, 339, 343 thickness modes, piezoelectric, 244 thickness-twist modes, 227 torsional cylinder resonator, 231-232 torsional sphere modes, 238 torsional waveguide modes, 111 Mode, angularly symmetric dilatational (thick disk), 344-347, 353-357 antisymmetric, 72, 79, 137, 144, 150, 223, 224, 242-245 backward wave, 158 Bleustein-Gulyaev wave, 139-142, 145-146, 148, 210-212, 291 breathing mode, sphere, 237 compressional sphere, 234-236, 266, 318-319,322-323, 370 contour extensional, 228-230, 245-246, 347-349 cutoff, 75, 81-82, 158-160, 196 dilatational, free cylinder waveguide, 110-112 dilatational, free plate waveguide, 79, 80, 82, 83-87, 88, 93 dilatational, rectangular parallelepiped, 357-361 dilatational, thick disk resonator, 344-347, 353-357 even symmetry, 242-245, 341-342 excitation, 161 162 Mode (Continued) flexural, free cylinder waveguide, 112-113 flexural, free plate waveguide, 79, 80, 84-87, 88, 93 free, 156, 221-222, 251, 334, 349, 362 function, 184,187, 188-189 generalized Lamb waves, 97-103 generalized Rayleigh waves. 131 index (subscript), 82-83, 158, 235-237 Lamb waves, 76-88. 130, 133, 136, 139, 145, 158, 198, 203, 207, 227-228, 262, 305, 372 Love waves, 95-98, 99, 101, 133, 145, 201, 208 Love-type, strip waveguide, 118, 119 Maerfeld-Tournois wave, 145-151 mixed modes, sphere, 236, 238-239 nonpropagating, 75, 158-160, 179, 186 normalization, 188, 189 oblate-prolate mode, sphere, 239 odd symmetry, 242-245, 264, 341-342 open circuit, 264, 265 orthogonality, resonator, 250-253, 264-265, 343, 370 orthogonality, waveguide, 155-160, 185, 187 P.77, 80-81, 82,113 perturbed, 274 propagating, 75, 158-160, 179, 186 pseudosurface waves, 131-134, 135, 142 Rayleigh wave, 88-94, 130-131, 139, 141-144. 145, 163-177, 198-199, 200, 201, 210-211, 278-294, 305-315, 326, 375-393 Rayleigh-type, strip waveguide, 118, 119 resonator, 222 SA, 113 SH, bounded plate, 226-227 SH, free plate waveguide, 68-69, 71-73, 136 139 shear modes, sphere, 236-238 short-circuit, 239-246, 256, 258, 264, 347, 351,354, 357-361 Stoneley waves, 103-104, 105, 106, 133, 135, 145 subscript (index), 82-83, 158, 235-237 SV, 77, 80-81,82,113 symmetric, 72, 79, 137, 144, 150, 223, 224. 242-245 thickness, nonpiezoelectric, 222-224, 224-226, 265, 267, 268, 338-343
404 INDEX Mode (Continued) thickness, piezoelectric, 239-245, 319-322, 323-324 thickness-twist, 226-227 , 246 torsional, cylinder resonator, 231-232, 266, 371 torsional, sphere, 236-238, 266 torsional waveguide modes, 110-111 trapped energy, 246-250 unperturbed, 273 waveguide. 65, 222 Mode amplitude, resonator, 250, 254-257 waveguide, 161-162, 183 Mode amplitude equations, resonator, 254-257 waveguide, 162, 170 Mode expansion, resonator, 250, 254-258 waveguide, 65, 161 Monolithic filter, 250 Motional capacitance. 258, 259, 348, 355-357 normalized, 259, 356 Motionally clamped capacitance, 259 Natural frequency, 221, 251, 261-262, 334 complex, 261-262 real, 221, 251 Nonpropagatingmode, 75, 82, 158-160, 179, 186 power flow, 159-160, 186 Normal mode, amplitudes, 183, 255-258 equations, 183, 255-258, 305 Normalized motional capacitance, 259, 356 Normalization of waveguide modes, 188, 189 Oblate-prolate mode, 239 Odd symmetry modes, 242 245. 264, 341-342 Open circuit boundary conditions, 142, 143, 157, 252. 253, 287 Open-circuit mode, resonator, 264, 265 Open waveguide, 95, 118 Orthogonal functions, 151, 250-253 Orthogonality, nonpropagating modes, 159 propagating modes, 158 resonator modes, 250-253, 264-265, 343, 370 waveguide modes, 155-160, 185, 187 Oscillation, angularly symmetric dila- tational (thick disk), 344-347, 353-357 breathing mode, 237 Oscillation (Continued) contour extensional, 228-230, 245-246, 347-349 dilatational, rectangular parallelepiped, 357-361 forced. 221, 250, 254, 255, 256 free, 221, 251, 256, 261-262, 334, 349, 362 mixed, isotropic sphere, 236, 238-239 oblate-prolate mode, 239 pure compressional, isotropic sphere, 234-236, 266, 318-319, 322-323, 370 pure shear, isotropic sphere, 236-238 SH, bounded plate, 226-227 thickness, unbounded anisotropic plate, 224-226 thickness, unbounded free isotropic plate, 222-224. 265, 267, 268 thickness, unbounded piezoelectric plate, 239-245, 319-322, 323-324 thickness, unbounded rigid isotropic plate, 338-343 thickness-twist, 226-227, 246 torsional, isotropic cylinder, 231-232, 266, 371 torsional, isotropic sphere, 236-238, 266 trapped energy, 246-250 Overlay, anisotropic, 280-283 isotropic, 274-280, 326 Overtone resonance, 341-343 Pmodes, 77, 80-81, 82,113 Pwave, 28, 30, 31,32 Partial wave, 69-71, 71-72, 74-75, 77-78, 86, 89-90, 95-96, 99, 101, 122-123, 128-130, 136, 142, 146-148 definition, 70 electrostatic, 70-71, 136, 139-141, 147 evanescent, 70, 136, 147 longitudinal, 77 quasilongltudinal 70 quasishear, 70 Rayleigh, 122-123 shear, 72, 77 superposition, 128-130, 131, 142, 146-148 transverse resonance, 71-72, 77-78. 89-90, 96,99, 101, 136-137, 139-141 INDEX 405 Particle velocity, boundary conditions, 2,43,45, 178, 190, 252, 253 coordinate transformation, 41-42 polarization, 11-19, 21, 23, 24, 25, 28, 29, 43 polarization conversion at a boundary, 27-32, 34-35, 36, 37, 43 reflection coefficient, 22-24, 27-38, 39, 42, 191 transmission coefficient, 22-24, 27-38, 39, 42 Peak stored energy, 260, 262 Perturbation theory, 176, 271-324 acoustic (mechanical) surface impedance perturbation, 275, 277, 283, 284-285, 305, 314, 326, 330 anisotropic lossless overlay, 280-283 boundary perturbation formula, resonator, 317 boundary perturbation formula, waveguide, 274, 275, 287, 325, 326 Brekhovskikh's attenuation formula, 311 conductive piezoelectric plate resonator, 323-324 coupled waves, 328-329 electrical surface impedance perturbation, 290-293, 305,321 electrical surface perturbations, resonator, 319-322 electrical surface perturbations, waveguide, 286-294, 325, 326 electrical volume perturbations, resonator, 323-324 electrical volume perturbations, waveguide, 300-302 first order, 304,310,314 fluid loading, 283-286 Ingebrigtsen formula, 176, 290, 293 interior perturbation formula, resonator, 317, 322 interior perturbation formula, waveguide, 296 internal layer perturbation, 300-302, 326-328 isotropic lossless overlay, 274-280, 326 mass loading, 114,318 mass-loaded isotropic sphere resonator, 318-319 mechanical surface perturbations, resonator, 318-319 Perturbation theory (Continued) mechanical surface perturbations, waveguide, 274-286, 325, 326 mechanical volume perturbations, resonator, 322-323 mechanical volume perturbations, waveguide, 297-300 perturbation parameter, 303, 310 Rayleigh wave attenuation, 283-286, 297-299, 309-313 resonator boundary perturbations, 315-322 resonator interior perturbations, 315-317, 322-324 thickness modes, piezoelectric, 319-322, 323-324 viscously damped isotropic sphere resonator, 322-323 wave scattering, 302-315 waveguide boundary perturbations, 272-294, 325,326 waveguide interior perturbations, 294-302 Phase velocity, waveguide, 158, 202, 204, 277, 299 Piezoelectric constitutive relations, 50, 152, 245 planar, 245 Piezoelectric coupling, 155 Piezoelectric media, conditions for unique field solution, 372-373 Fresnel equations, 43-57 plane waves, 43-57, 136-137 Poynting vector, 57 quasistatic equations, 54, 59, 71, 178, 222, 303, 349 scattering, 43-57 Snell's Law, 45-46, 49, 50, 54 Piezoelectric reflection, critical angles, 44 cubic crystal, 46-52 hexagonal (6mm) crystal, 52-57 Snell's Law, 45, 46, 49, 50, 54 Piezoelectric refraction, critical angles. 44 Snell's Law, 45, 46 Piezoelectric resonator, equivalent circuit, 258-259,356-358 general, 250-253 open-circuit mode, 264, 265 rectangular parellelepiped, 357-361
406 INDEX Piezoelectric resonator (Continued) short-circuit mode, 239-246, 256, 258, 264, 347, 351, 354, 357-361 "static" solution, 255-257, 265 thick disk, 353-357 thin plate, 245-246, 347 349 trapped energy, 249-250 unbounded plate, 239-245, 319-322, 323-324 Piezoelectric scattering coefficients, 51, 55, 56, 57 Piezoelectric waveguide, 70-71, 134-151, 155-160 free plate, 134-139 general, 178-180 mode orthogonality, 155-160, 185 Rayleigh wave, 139, 141-144, 145, 146, 147, 148, 149, 210-211, 272, 288-294, 379-393 SH modes, 136-139 transmission line model, 177-186 Piezoelectrically stiffened velocity, 55 Ficzomagnetic coupling, 155 Planar, constitutive relations, 245 Hooke's Law, 229 Poisson's ratio, 349 stiffness matrix, 229, 245 wave equation, 229, 245 Plane wave, nonpiezoelcctric media, 1-43 piezoelectric media, 43-57, 136-1 37 scattering, 1-57 scattering into Rayleigh waves, 314-315 transducer, 176 See also, Bulk wave Plate waveguide, 66 104 Pochhammer frcquen у equation 110 Poisson's ratio, 92, 236, 312, 349 numerical values, 92 planar, 349 Polarization, acoustic, 11-19, 21, 23, 24, 25, 28, 29, 43, 136, 139, 142, 146 particle velocity, 11-19, 21, 23, 24, 25, 28, 29, 43 particle velocity conversion at a boundary, 27-32, 34-35, 36,37, 43 Potential, cylindrical coordinates. 106-108 electric, 45, 47-48, 50, 53, 67, 71, 136 scalar mechanical, 67-68, 107 scalar wave equation, 67, 106-107 spherical coordinates, 233-234 vector mechanical, 67-69, 108 Potential (Continued) vector wave equation, 67, 106-107 Potential energy density, 350 Potential theory, free isotropic cylinder, 104-110 free isotropic strip, 115-118 isotropic sphere, 232-234 SH modes, free plate, 68-69 Power density vector, 14 Power flow, nonpropagating modes, 159-160, 186 propagating modes, 159, 186 Rayleigh wave, 170 reflection-symmetric waveguide, 185-186 transmission line model, 184-185 waveguide, 184-186, 274 Poynting vector, nonpiezoelectric, 14-18 piezoelectric, 57, 260 Poynting's Theorem, complex, 205-206, 213. 253, 260, 264, 372 real, 364 Propagating mode, 75, 82, 158-160, 179, 186 power flow, 159, 186 Propagation constant, complex, 82-83, 158-159, 186 definition, 64, 158, 159 imaginary, 74, 81-82, 158-159, 186 real, 74, 81 82, 158, 186 Pseudosurface waves, 131-134, 135, 142 definition, 131 Pure mode, 39, 60 Pure shear wave, 8,11-18, 39-43 Q, compressional modes (sphere), 266. 322-323 definition, 262 radiation, 268 thickness modes, piezoelectric, 323-324 torsional modes, cylinder, 266 torsional modes, sphere, 266 unbounded isotropic plate, 265-266 unloaded, 268 Quality factor, compressional modes (splicirl. 266, 322-323 definition, 262 thickness modes, piezoelectric, 323-324 torsional modes, cylinder, 266 torsional modes, sphere, 266 unbounded isotropic plate, 265 266 Quasiacoustic wave, 43-46 Quasielectromagnctic wave, 43-46 Quasilongitudinal wave, 8. 70, 225 Quasishear wave, 8, 70, 224-225 Quasistatic, approximation, 45-46, 47, 59-60, 70, 136, 349, 365 boundary conditions, 44, 45, 47, 178, 190, 252, 253 equations, 54, 59, 71, 178, 222, 303, 349 reciprocity relations, 153-154, 155 Quaitcr-wavc matching transformer, 57 Radial Bessel functions, 233-234 Radiation conductance, 175 Radiation Q, 268 Ray vector (energy flow direction), 9, 10, 12 Rayleigh wave, 88-94, 130-131, 133, 139, 141-144, 145, 163-177, 198-199, 200, 201, 210-211, 278-294, 305-315, 326, 3 75-393 amplifier, 291, 293-294 anisotropic, 130-131, 133 attenuation, 283-286, 297-299, 309-313 coupling, 93-94, 144-145, 150, 272, 291 cylindrical, 113-114 dispersion relation, 88-92 excitation by plane wave scattering, 314-315 field distribution, 88, 92-93, 146, 147, 148, 149, 180, 182, 278, 375-392 impedance, 123 perturbed, 278-286, 288-294, 393 piezoelectric, 139, 141-144, 145, 146, 147, 148, 149, 210-211, 272, 288-294, 379-393 power flow 172 reflection coefficient, 307-309 scattering, 198-199, 200, 201, 305-315,330,393 lowness curves, 1 22-124 Smith's transducer model, 176 temperature dependence, 297 transducer, 163-177, 290, 309, 364-368, 375, 393 transmission coefficient, 307-309 velocity, 91-92, 112, 113, 121, 143, 144, 145, 291, 377, 379-388, 393 Rayleigh-Lamb frequency equations, 79, 84, 372 Raylcigh-Ritz method, 339-344 INDEX 407 Rayleigh-type mode, velocity dispersion curves, 119 Real Poynting's Theorem, 364 Real propagation constant, 74, 81-82, 158, 186 Real reciprocity relation, 152-154, 212, 250, 272, 324, 366 Reciprocity relation, complex, 154-155, 204, 212, 250, 254, 256, 272, 294, 315 Lamb, 153 Lewis, 154 Lorentz, 153 quasistatic, 153-154, 155 real, 152-154, 212, 250, 272, 324, 366 Rectangular parallelepiped resonator, 357-361 Reflection, anisotropic acoustic, 6-21, 38-43 electromagnetic, 2, 3, 4-6 isotropic acoustic, 2, 3-4, 5-7 piezoelectric, 43-57 total internal, 5, 23 Reflection coefficient, equivalent current, 37-38 particle velocity, 22-24, 27-38, 39, 42, 191 plane wave, normal incidence, 42 plane wave, oblique incidence, 22-24, 27-38, 39,46-57 Rayleigh wave, 307-309 relation to impedance, 22, 37, 39, 192-193 stress, 192, 268 transverse resonance analysis, 72-73 waveguide, 191-193 Reflection-symmetric waveguide, 180-189 mode orthogonality, 155-160, 185, 187 power flow, 185-186 Refraction, anisotropic acoustic, 6-21, 38-43 birefringence, 2, 4, 6 electromagnetic, 2, 3,4-6 external conical, 19-21 internal conical, 11 -20 isotropic acoustic, 2, 3-4, 5-7 piezoelectric, 43-46 birefringence, 2,6 Resonance, fundamental, 173, 174, 341, 343 overtone. 341-343 Resonant frequency, 221, 334
408 INDEX Resonator, acoustic input impedance, 268 admittance equations, 254, 257 admittance matrix, 257-259, 266 angularly symmetric dilatational modes, thick disk. 344-347, 353-357 boundary perturbation formula, 317 boundary perturbations, 315-322 bounded isotropic plate, 226-230 breathing mode, sphere, 237 compressional modes, sphere, 234-236, 266, 318-319, 322-323, 370 contour extensional modes, 228-230, 245-246, 347-349 damping constant, 262 dilatational modes, rectangular parallelepiped, 357-361 electrical input admittance, 266 equivalent circuit, 258-259, 356-358 impedance equations, 254 interior perturbation formula, 317, 322 interior perturbations, 315-317, 322-324 isotropic cylinder, 231-232 isotropic sphere, 232-239, 318-319, 322-323, 370 lossy, 261-262, 264, 266 mixed modes, sphere, 236, 238-239 modal analysis, 250, 253-259 mode, 222 mode amplitude, 250, 254-257 mode amplitude equations, 254-257 mode expansion, 250, 254-258 mode orthogonality, 250-253, 264-265, 343, 370 oblate-piolate mode, sphere, 239 open-circuit mode, 264, 265 perturbation theory, 315-324 potential theory, 232-239 Q, 262, 264, 266 quality factor, 262, 264, 266 rectangular parallelepiped, 357-361 rigid boundary, 338-344, 369 SH modes, bounded plate, 226-227 shear modes, sphere, 236-238 short-circuit mode, 239-246, 256, 258, 264, 3-17, 3S1, 354, 357-361 stored energy, 253, 259-262, 319, 322 thick anisotropic disk, 344-347, 353-357 thickness modes, anisotropic, 224-226 thickness modes, isotropic, 222-224, 265, 267, 268, 338-343 Resonator (Continued) thickness modes, piezoelectric, 239-245, 319-322, 323-324 thickness-twist modes, 226-227, 246 thin plate, 228-230, 245, 347-349 torsional modes, cylinder, 23 1-232, 266, 371 torsional modes, sphere, 236-238, 266 trapped energy, 246-250 unbounded anisotropic plate, 224-226 unbounded isotropic plate, free, 222-224, 265, 267, 268 unbounded isotropic plate, rigid, 338-343 unbounded piezoelectric plate, 239-245, 319-322, 323-324 variational techniques, 334-361 Ridge waveguide, 119-120 Rigid boundary conditions, 157, 252, 253, 335, 337-338, 338-343,369 Rotary activity, 155 Sandwich waveguide, 120 Scalar product, matrix, 152 Scattering, acoustic plane waves, 1-43, 49 coefficients, acoustic, 22-24, 27-38, 39, 43 coefficients, piezoelectric, 51, 55, 56, 57 electrical, 43-57, 302-309 equivalent circuit, 33, 35-38, 193, 196, 198, 200 Lamb waves, 198, 305 longitudinal to shear, 28-30, 32, 36, 37 Love waves, 201 mechanical, 1-57, 190-201, 302-315 perturbation theory, 302-315 piezoelectric plane waves, 43-57 plane wave into Rayleigh waves, 314-315 plane waves, 1-57 Rayleigh wave, 198-199, 200, 201, 305-315,330,393 SH (horizontal shear) modes, 190-198 shear to longitudinal, 27-29, 30-31, 34-35 waveguide, 190-201 weak-scattering assumption, 315 Sezawa wave. 102-103 SH (horizontal shear) modes, 68-69. 71-73, 73-76, 128-130, 136-139, 190-198, 202-203 anisotropic, 128-130 antisymmetric, 72, 137-138 INDEX 409 SH (horizontal shear) modes (Continued) bounded plate resonator (thickness-twist modes), 226-227 characteristic equation, 137 dispersion curves, 76, 137-138 dispersion relation, 69, 73, 130 energy velocity, 206-207 field distributions, 76, 137-139 group velocity, 202-203 piezoelectric, 136-139 scattering, 190-198 slowness curves, 74 symmetric, 72, 137-138 trapped energy resonator, 246-250 SH trapped energy modes, characteristic equation, 249 field distributions, 248-249 frequency equation, 249 SH wave, 21, 23, 24 Shear sphere modes, 236-238 See also. Torsional sphere modes Shear wave, 7, 21-29, 30-31, 49-52, 53-57, 223-224 horizontally polarized (SH). 21, 23, 24 vertically polarized (SV), 23, 25, 28, 29 Short circuit boundary conditions, 51, 56, 136, 142, 143, 157, 162, 252, 253, 287 Short-circuit mode, resonator, 239-246, 256, 258, 264, 347, 351, 354, 357-361 Slot waveguide ("fast-on-slow" stripe waveguide), 121-127,199 coupled, 125-128 velocity dispersion curves, 126-127 Slowness curves, I,amb waves, 78 Love waves, 98 purely acoustic, 8-13, 70 purely electromagnetic, 4, 5, 70 quasiacoustic, 45, 46 quasielectromagnetic, 45, 46 Rayleigh waves, 122-124 SH modes, 74 stiffened acoustic, 45, 46, 70 surface waves, 122-124 two-dimensional waveguides, 203-204 Slowness surface, 3, 6, 8 Smith's Rayleigh wave transducer model, 176 Snell's Law, acoustic, 3, 6, 22, 26, 27, 38. 39 Snell's Law (Continued) definition, 2 electromagnetic, 3, 6 piezoelectric, 45, 46, 49, 50, 54 slowness surface, 3, 6, 8, 45, 46 Sources, distributed, 161-162, 305-307. 310,314 electrical, 161-162, 163 mechanical, 161-162, 163 surface, 161 162 volume, 161-162 Spherical Bessel functions, 233-234, 264 Spherical coordinates, scalar mechanical potential, 233 strain, 235 vector mechanical potential, 234 Spring-loaded boundary conditions, 369 Standing wave, 222, 223, 225-226, 227, 239, 241, 362 "Static" capacitance, 257 "Static" resonator solution, 255-257, 265 Stationary property, 334, 335, 340, 350-351 Stiffened acoustic wave, 43-46, 47, 52, 136, 137, 147 coupling, 137-138 Stiffened Christoffel equation, 47-48, 53, 54 Stiffness matrix, planar, 229, 245 Stoneley waves, 103-104. 105, 106, 133, 135, 145 anisotropic, 133, 135 conditions for existence, 106, 135 field distributions, 104 velocity, 103-104, 105 Stored energy, electrostatic, 260-261 instantaneous, 260-261 kinetic, 260-261 peak, 260-262 resonator mode, 253, 259-26 2, 319,322 strain, 260-261 waveguide mode, 205-206 Strain, cylindrical coordinates, 108 spherical coordinates, 235 stored energy, 260-261 Stress, boundary conditions, 2, 43, 45, 178, 190, 252, 253 coordinate transformation, 41-42 reflection coefficient, 192 transmission coefficient, 192
410 INDEX Stress-free boundary, 30-38, 66, 99, 104, 113 130, 131, 136, 157, 162, 223, 227, 232, 235, 237, 239, 245, 252, 253, 274, 335, 337-338 See also. Mechanically free boundary Strip waveguide. 114-118. 128-151, 209 Stripe waveguide, characteristic equation, 123 coupled, 125-128 "fast-on-slow" (slot), 120-127, 199 "slow-on-fast", 120-127, 199 velocity dispersion curves, 125-126 Subscript (index), resonator modes. 235-237 waveguide modes, 82-83, 158 Substrate, anisotropic, 376-378 definition, 94 isotropic, 375-376 orientation convention, 376-378 velocity, 102-103 Superposition of partial waves, Maerfeld- Tournois waves, 146-148 piezoelectric surface waves, 142. 146-148 SH modes, 128-130, 131 Surface impedance, acoustic, 275, 284, 305, 326, 330 electrical, 288, 289, 291, 305, 320-321 mechanical, 275, 284, 305, 326, 330 perturbation, 275-294, 305, 314, 321, 326, 330 transformation, 291 Surface sources, 161-162 Surface waves, 87, 88-94, 94-97, 97-103, 103-104, 118-128, 130-134, 135, 139-144, 145-151, 163-177, 200, 201, 210-211, 272-315, 375-393 Surface wave, amplifier, 272, 293-294 coupled, 93-94, 125-128, 144-145, 150 slowness curves, 122-124 Smith's transducer model, 176 transducer, 163-177, 290, 309, 364-368, 375, 393 Susceptance, interdigital transducer, 175,176 SV modes, 77, 80-81, 82, 113 SV wave, 23, 25, 28, 29 Symbolic notation, acoustic field equations, 152 electromagnetic field equations, 152 Symmetry, arguments, 72, 79, 137, 242-245 Symmetry (Continued) even, 242-245, 341-342, 348, 357, 359-361 odd, 242-245, 264, 341-342, 348, 357, 359-361 Temperature dependence, Rayleigh wave velocity, 297 Thick disk resonator, 344-347, 353-357 Thickness modes (anisotropic), characteristic equation, 226 field distributions, 225 frequency equations, 225-226 Thickness modes (isotropic), field distributions, 223-224, 338, 339, 343 frequency equations, 223, 265, 267, 338 0, 265-266 quality factor, 265-266 variational techniques, 338-343 Thickness modes (piezoelectric), characteristic equation, 242 field distributions, 244 frequency equation, 242-243, 320 perturbed, 319-322, 323-324 Q, 323-324 quality factor, 323-324 Thickness-twist modes, field distributions, 227 frequency equation, 227, 246-247 Thin bounded plate resonator, 228-230, 245-246, 347-349 Thin plate resonator, 228-230, 245, 347-349 Tierstcn boundary conditions, 277, 280 33(1 Topographic waveguide, 120 Torsional cylinder resonator modes, field distributions, 231-232 frequency equation, 232 Q, 266 quality factor, 266 variational techniques, 371 Torsional sphere modes, characteristic equation, 238 field distributions, 238 frequency curves, 236 frequency equation, 238 Q, 266 quality factor, 266 Torsional waveguide modes, Uiaracterislii: equation. 110, 208 Transducer, general, 253-254 immittance, 163 interdigital, 163-177, 290, 309, 364-368, 375, 393 Kino's variational formuM, 176-177, 368 Rayleigh wave, 163-177, 290, 309. 364-368, 375, 393 surface wave, 163-177, 290, 309, 364-368, 375, 393 unidirectional, 177 variational techniques, 364-368 waveguide, 163-177 wedge, 177 Transformation, coordinate, 42 impedance, 57 surface impedance, 291 Transformer, matching, 57 Transmission coefficient, particle velocity, 22-24, 27-38, 39,42, 191 plane wave, normal incidence, 42 plane wave, oblique incidence, 22-24, 27-38, 39, 191 Rayleigh wave, 307-309 relation to impedance, 22, 39 stress, 192, 268 waveguide, 191, 192 Transmission line, characteristic impedance, 183, 184, 192, 196 model for reflection at a free boundary, 35-38 model, general piezoelectric waveguide. 177 180 model, reflection-symmetric piezoelectric waveguide, 180-182 normal mode amplitudes, 183 normal mode equations, 183 power flow, 184-185 voltage-current (VI) equations, 183 Transverse decay constant, 96 Transverse resonance, Bleustein-Gulyaev wave, 139-141 definition, 71 generalized Lamb waves, 102-103 impedance. 73 Lamb waves, 78-79 Love waves, 96 partial waves, 71-72, 77-78, 89-90 96, 99, 101, 136-137, 139-141 Rayleigh wave, 88, 89-90 reflection coefficient, 72-73 Sll modes, 71-73, 136-137 INDEX 411 Trapped energy resonator, 246-250 Trapped wave, 96-97, 98. 102, 121, 123, 131 Traveling wave, 221, 363 Trial function, 339, 344-346, 348, 352, 353, 354, 359, 363 Trial solution, 333, 334, 336-338, 339, 352, 367 boundary conditions, 335-336, 337-338 Trircfringence (triple refraction), 2, 6 Unidirectional transducer, 177 Uniqueness theorem, 372-373 Unloaded Q, 268 Variational expression, 334, 336-338 Variational techniques, 176-1 77, 197, 333-368 boundary conditions, 335-336, 350-353 compressional sphere modes, 370 contour extensional modes, 347-349 convergence, 338, 342, 346, 348-349, 352, 354, 355-357 dilatational modes, rectangular parallelepiped, 357-361 dilatational modes, thick disk. 344-347, 353-357 interdigital transducer, 176-177, 364-368 Kino's variational formula, 176-177, 368 Lagrangian density function, 350, 351 Lagrangian function, 350, 351 Lagrangian methods, 349-353, 364 Lamb waves, 372 lower bound, 342 nonpiezoelectric resonator, 334-349 nonpiezoelectric waveguides, 362-364 piezoelectric resonator, 349-361 piezoelectric waveguide, 364 Rayleigh-Ritz method, 339-344 resonator, 334-361 rigid-boundary resonator, 369 stationary property, 334, 335, 340, 350-351 thickness modes, isotropic, 338-343 torsional cylinder resonator modes, 371 transducer, 364-368 trial function, 339, 344-346, 348, 352, 353,354, 359, 363
412 INDEX Variational techniques (Continued) trial solution, 333. 334. 336-338, 339, 352, 367 upper bound, 342 variational expression, 197, 334, 336-338 waveguide, 361-364 Vector mode function, 184, 187, 188-189 Velocity, Bleustein-Gulyaev, 291 energy, 12,199-207 group, 8, 12, 14, 158, 199-207, 279, 280 Lame, 85, 116 phase, 158, 202, 204, 277, 299 piezoelectricaUy stiffened, 55 Rayleigh wave, 91-92, 112, 113, 121, 143, 144, 145, 291, 377, 379-388, 393 stiffened acoustic, 55 Stoneley, 103-104, 105 Velocity dispersion, 202 Velocity dispersion curves, Love-type mode on strip waveguide, 119 Rayleigh-type mode on strip waveguide, 119 stripe waveguide, 125-126 slot waveguide, 126-127 Voltage, waveguide, 183-184,187, 188,196 Volume sources, 161-162 Wave, backward, 158 Bleustein-Gulyaev, 139-142, 145-146, 148, 210-212, 291 bound, 102 bulk (plane), 132, 176 coupling, 81, 93-94, 113, 125-128, 137-138, 144-145,150, 207, 272, 291, 328-329 degenerate, 11, 14, 87, 88 electrostatic, 70-71, 136, 139-141, 147, 239, 240, 241 evanescent, 5, 6, 9, 11, 23, 44, 47,49, 50.51,52, 55, 56, 57, 59, 70, 136, 147, 240 excitation by distributed sources, 161-162. 305-307, 310, 314 forced, 161-177, 221 free, 221 front, 9, 10, 12 generalized Lamb, 97-103 generalized Rayleigh, 131 Wave (Continued) guided, 64 horizontally polarized shear (SH), 21, 23, 24 hybrid, 43 interface, 102 Lamb, 76-88, 130, 133, 136,139, 145, 158,198, 203, 207, 227-228, 262, 305, 372 Lame, 85, 86, 207, 209, 262, 263 Laplace, 71 leaky, 95, 96-97, 98, 102, 114, 119,123, 131 longitudinal, 7, 28-32, 223-224 Love, 95-98,99, 101, 133, 145, 201, 208 Maerfeld-Tournois, 145-151 P, 28, 30, 31, 32 partial, 69-71, 71-72, 74-75, 77-78, 86, 89-90, 95-96, 99, 101,122-123, 128-130, 136, 142, 146-148 plane (bulk), 132, 176 plane waves in nonpiezoelcctric media, 1-43 plane waves in piezoelectric media, 43-57, 136-137 pseudosurface, 131-134, 135, 142 pure shear, 8, 11-18, 39-43 quasiacoustic, 43-46 quasielectromagnetic, 43-46 quasilongitudinal, 8, 70, 225 quasishear, 8, 70, 224-225 Rayleigh, 88-94,130-131, 139, 141-144, 145, 163-177, 198-199, 200, 201, 210-211, 278-294, 305-315,326,375-393 reflected, 2 refracted, 2 scattering, 1-57, 190-201 scattering perturbation theory, 302-315 Sezawa, 102-103 SH, 21,23, 24 shear, 7, 21-29, 30-31,49-52, 53-57, 223-224 spherical shear, 238 standing, 222, 223, 225-226, 227, 239, 241, 362 stiffened acoustic, 43-46, 47, 52, 136, 137, 147 Stoneley, 103-104,105, 106, 133. 135. 145 INDEX 413 Wave (Continued) surface, 87, 88-94, 94-97, 97-103, 103-104, 118-128, 130-134, 135, 139-144, 145-151, 163-177,200, 201, 210-211, 272-315, 375-393 SV, 23, 25, 28, 29 transmitted, 2 trapped, 96-97, 98, 102, 121, 123,131 traveling, 221, 363 vector, 8, 19-21, 70 vertically polarized shear (SV), 23, 25, 28, 29 Wave equation, acoustic, 66, 229, 245 planar, 229, 245 scalar potential, 67, 106-107 vector potential, 67, 106-107 Wave vector, 3, 5, 8, 19-21, 49, 53, 54. 70 boundary conditions, 3, 20 imaginary component, 5, 49, 53, 54, 75, 78, 88 Snell's Law condition, 3, 49 Waveguide, anisotropic, 69-70, 128-151 anisotropic plate on a half space, 133 backward wave, 158 Bleustein-Gulyaev wave, 139-142, 145-146, 148, 210-212, 291 boundary perturbation formula, 274, 275, 287, 325, 326 boundary perturbations, 272-294, 325, 326 channel, 120 closed, 95 components, 64 current, 183-184, 187, 188, 196 definition, 64 dilatational modes, free cylinder, 110-112 dilatational modes, free plate, 79, 80, 82, 83-87, 88, 93 energy velocity, 205-206 excitation, 161-162 flcxural modes, free cylinder, 112-113 flexural modes, free plate, 79, 80, 84-87, 88, 93 free anisotropic plate, 128-130 free isotropic cylinder, 104-114 free isotropic plate, 66-94 free isotropic strip, 115-118, 209 free piezoelectric plate, 134-139 frequency dispersion, 202 generalized Lamb waves, 97-103 Waveguide (Continued) generalized Rayleigh waves, 131 group velocity, 158, 202, 205-206, 279, 280 impedance, 184, 187, 188, 192, 195 interior perturbation formula, 296 interior perturbations, 294-302 isotropic, 66-128 isotropic plate on a half space, 94-104 isotropic strip on a substrate, 118, 119,121 junctions, 190, 193, 196, 198, 200, 201 Lamb waves, 76-88, 130, 133, 136, 139, 145, 158, 198, 203, 207, 227-228, 262, 305, 372 Lame wave, 85, 86, 207, 209, 262, 263 layered, 156 lossy, 212, 213 Love waves, 95-98, 99, 101, 133, 145, 201, 208 Love-type mode, strip waveguide, 118. 119 Maerfeld-Tournois wave, 145-151 microsound, 118-128, 272 modal analysis, 65, 151, 163 mode, 65 mode amplitude, 162, 183 mode amplitude equations, 162. 170 mode expansion, 65, 161 mode orthogonality, 155-160, 185. 187 open, 95, 118 P modes, 77, 80-81, 82, 113 perturbation theory, 272-302, 325, 326 phase velocity, 158, 202, 204, 277, 279 piezoelectric, 70-71,134-151, 155-160 potential theory, 66-69 power flow, 184-186, 274 pseudosurface waves, 131-134, 135, 142 Rayleigh wave, 88-94, 130-131, 139, 141-144, 145,163-177, 198-199, 200, 201, 210-211, 278-294, 305-315,326,375-393 Rayleigh-typc mode, strip waveguide, 118. 119 reflection coefficient, 191-193 reflection-symmetric, 180-189 ridge, 119-120 SA modes, 113
414 INDEX Waveguide (Continued) sandwich, J 20 scattering, 190-201 SI I modes, 68-69, 71-73, 136-139 slot ("fast-on-slow stripe"), 121-127,190 Stoneley waves, 103-104, 105, 106, 133, 135, 145 stored energy, 205-206 stripe, 120-127,199 superposition of partial waves, 69-71, 128, 134 surface sources, 161-162 SV modes, 77, 80-81, 82, 113 symmetry analysis, 72, 79, 137 three-dimensional guiding, 65 topographic, 120 torsional modes, 110-111 transducer, 163-177 Waveguide (Continued) transmission coefficient, 192 transmission line model, 177-189, 192 transverse resonance, 71-73, 128, 134, 139, 222 two-dimensional guiding, 65, 203-204 variational techniques, 361-364 velocity dispersion. 202 voltage, 183-184, 187, 188, 196 volume sources, 161-162 Waveguide junction, equivalent circuit, 193, 196, 198, 200 Weak coupling approximation, 173. 175, 177, 287,302, 307, 319,364 Wedge transducer, 177 Wolkerstorfer boundary conditions, 282 Zero frequency capacitance, 259