Copyright by Ronald Edward Kumon 1999 -

Copyright

by

Ronald Edward Kumon

1999

NONLINEAR SURFACE ACOUSTIC WAVES

IN CUBIC CRYSTALS

by

RONALD EDWARD KUMON, B.S.

DISSERTATION

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN

December 1999


IN CUBIC CRYSTALS

Approved by

Dissertation Committee:

Supervisor:

This dissertation is dedicated to my family,

Henry, Rosemary, Karen, and Jim Kumon,

for their continual love and support

and

my other “family,”

the members of Laurel House Cooperative,

who have provided good company and lively conversation

over so many years.

Acknowledgments

I know I haven’t mentioned the thing that Father never stopped

telling us was the most important thing of all: one’s work—wanting

to do something, to achieve something so you can say you weren’t

here in vain, that you’ve left behind some special winding or dis-

covered an unknown wave motion in a crystal, or at least an oscil-

lation within yourself, so that even when God has become distant

and turned His face away from you and people have deserted you,

something lasting remains: such as a passion for the truth.

—Ivan Klima, Judge on Trial, translated by A. G. Brain,

(Vintage International, New York, 1994), p. 309.

This dissertation is the product of three years of intense work and seven years

of dedicated study. However, it could not have been accomplished without the

help of many others.

I would like to thank Dr. Mark Hamilton for his guidance and attention

while working on this project. In particular, I would like to thank him for

accepting me as a new student even though I was already far along in my

program. I will always be grateful to him for this second chance. I would

also like to thank Dr. Yura Il’inskii and Dr. Zhenia Zabolotskaya for teaching

me the intricacies of nonlinear surface acoustic waves and for their continuing

willingness to be consulted over the duration of this project. Many thanks

to my committee members Dr. Marc Bedford, Dr. David Gavenda, Dr. Tom

Griffy, and Dr. Michael Marder for their patience over this long journey.

v

Special thanks go to Dr. Alexei Lomonosov and Dr. Slava Mikhalevich of

the General Physics Institute, Russian Academy of Sciences, Moscow, Russia,

and Dr. Peter Hess, of the Institute of Physical Chemistry, University of Heidel-

berg, Heidelberg, Germany, for sharing their experimental work and answering

my many questions.

Kevin Cunningham, Washington de Lima, Dr. B. J. Landsberger,

Dr. Pennia Menounou, Won-suk Ohm, and Steve Younghouse in the Acoustics

Group are thanked for many useful discussions. The assistance of Fred Bacon,

Dr. Eric Smith, and Dr. Doug Meegan at Applied Research Laboratories during

the various times I was working there is also appreciated.

What would happen to the university without the support staff? Many

thanks to Norma Kotz, Elke Roberts, Jan Dunn, and Olga Vorloou in the

Department of Physics, Cindy Raman in the Department of Mechanical Engi-

neering, and Claudia Darling, Elaine Frazer, Lorrie Polvado, Dottie Beaty, and

Beverly Bavaro at Applied Research Laboratories. They tracked down profes-

sors, sent faxes, filed paperwork, issued emergency paychecks, took care of my

appointments, and helped with all the other administrative details of being a

graduate student.

Finally, I would like to thank the U.S. Office of Naval Research for

providing the funding for this work.

R. E. K., November 1999

vi


IN CUBIC CRYSTALS

Publication No.

Ronald Edward Kumon, Ph.D.The University of Texas at Austin, 1999

Supervisor: Mark F. Hamilton

Model equations developed by Hamilton, Il’inskii, and Zabolotskaya [J. Acoust.

Soc. Am. 105, 639–651 (1999)] are used to perform theoretical and numerical

studies of nonlinear surface acoustic waves in a variety of nonpiezoelectric cubic

crystals. The basic theory underlying the model equations is outlined, quasilin-

ear solutions of the equations are derived, and expressions are developed for the

shock formation distance and nonlinearity coefficient. A time-domain equation

corresponding to the frequency-domain model equations is derived and shown

to reduce to a time-domain equation introduced previously for Rayleigh waves

[E. A. Zabolotskaya, J. Acoust. Soc. Am. 91, 2569–2575 (1992)]. Numerical

calculations are performed to predict the evolution of initially monofrequency

surface waves in the (001), (110), and (111) planes of the crystals RbCl, KCl,

NaCl, CaF2, SrF2, BaF2, C (diamond), Si, Ge, Al, Ni, Cu in the m3m point

group, and the crystals Cs-alum, NH4-alum, and K-alum in the m3 point group.

The calculations are based on measured second- and third-order elastic con-

stants taken from the literature. Nonlinearity matrix elements which describe

the coupling strength of harmonic interactions are shown to provide a powerful

tool for characterizing waveform distortion. Simulations in the (001) and (110)

planes show that in certain directions the velocity waveform distortion may

vii

change in sign, generation of one or more harmonics may be suppressed and

shock formation postponed, or energy may be transferred rapidly to the highest

harmonics and shock formation enhanced. Simulations in the (111) plane show

that the nonlinearity matrix elements are generally complex-valued, which may

lead to asymmetric distortion and the appearance of low frequency oscillations

near the peaks and shocks in the velocity waveforms. A simple transformation

based on the phase of the nonlinearity matrix is shown to provide a reasonable

approximation of asymmetric waveform distortion in many cases. Finally, nu-

merical simulations are corroborated by measured pulse data from an external

collaboration with P. Hess, A. Lomonosov, and V. G. Mikhalevich. Pulsed

waveforms in the (001) and (111) planes of crystalline silicon are quantitatively

reproduced, and two distinct regions of nonlinear distortion are confirmed to

exist in the (001) plane.

viii

Table of Contents

Acknowledgments v

Abstract vii

List of Tables xiii

List of Figures xv

List of Symbols xx

Chapter 1 Introduction 1

1.1 Types and Properties . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Nonlinear Rayleigh, Stoneley, and Scholte Waves . . . . . 6

1.3.2 Nonlinear Surface Waves in Crystals . . . . . . . . . . . . 9

1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Chapter 2 Theory 21

2.1 Description of the Model . . . . . . . . . . . . . . . . . . . . . . 21

2.1.1 Linear Theory . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.2 Nonlinear Theory . . . . . . . . . . . . . . . . . . . . . . 28

2.2 Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.1 Quasilinear Solution . . . . . . . . . . . . . . . . . . . . . 37

2.2.2 Estimates of Nonlinearity Parameters . . . . . . . . . . . 38

2.2.3 Tapered Quasilinear Solution . . . . . . . . . . . . . . . . 41

2.2.4 Coupled Two-Mode Solution . . . . . . . . . . . . . . . . 46

2.3 Time-Domain Evolution Equation . . . . . . . . . . . . . . . . . 49

2.4 Comparison with Isotropic Solids . . . . . . . . . . . . . . . . . 51

2.4.1 Linear Solution . . . . . . . . . . . . . . . . . . . . . . . 52

ix

2.4.2 Nonlinear Solution . . . . . . . . . . . . . . . . . . . . . . 53

2.4.3 Estimates of Nonlinearity Parameters . . . . . . . . . . . 58

2.4.4 Time-Domain Evolution Equation . . . . . . . . . . . . . 60

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 3 Properties of Cubic Crystals 65

3.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Elastic Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3 Cuts and Directions . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.4 Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Chapter 4 Monofrequency SAWs in the (001) Plane 79

4.1 Linear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2 Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.2.1 General Study . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2.2 Study of Si . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2.3 Study of KCl . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2.4 Study of Ni . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121


5.1 Linear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124


5.2.1 General Study . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2.2 Study of Si . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.2.3 Study of KCl . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.2.4 Study of Ni . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147


6.1 Linear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148


6.2.1 General Study . . . . . . . . . . . . . . . . . . . . . . . . 151

6.2.2 Interpretation of Complex-Valued Nonlinearity Parameters151

x

6.2.3 Study of Si . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.2.4 Study of KCl . . . . . . . . . . . . . . . . . . . . . . . . . 176

6.2.5 Study of Ni . . . . . . . . . . . . . . . . . . . . . . . . . . 183

6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Chapter 7 Pulsed SAWs and Experimental Results 192

7.1 Experimental Technique . . . . . . . . . . . . . . . . . . . . . . 192

7.2 Comparison of Theory and Experiment . . . . . . . . . . . . . . 194

7.2.1 Si in (001) plane . . . . . . . . . . . . . . . . . . . . . . . 198

7.2.2 Si in (111) plane . . . . . . . . . . . . . . . . . . . . . . . 206

7.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Chapter 8 Summary 213

Appendix A Anisotropic and Aeolotropic Media 217

Appendix B Surface Acoustic Wave Tutorial 219

B.1 Nondispersive Waves . . . . . . . . . . . . . . . . . . . . . . . . 219

B.1.1 Rayleigh Waves . . . . . . . . . . . . . . . . . . . . . . . 220

B.1.2 Stoneley, Scholte, and Leaky Rayleigh Waves . . . . . . . 222

B.1.3 Generalized Rayleigh Waves . . . . . . . . . . . . . . . . 225

B.1.4 Quasi-bulk Surface Waves and Exceptional Bulk Waves . 232

B.1.5 Pseudo-surface Waves . . . . . . . . . . . . . . . . . . . . 235

B.1.6 Piezoelectric Surface Acoustic Waves . . . . . . . . . . . 237

B.1.7 Bleustein–Gulyaev Waves . . . . . . . . . . . . . . . . . . 242

B.1.8 Piezomagnetic Surface Acoustic Waves . . . . . . . . . . 246

B.2 Dispersive Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 247

B.2.1 Plate Waves (Lamb and SH Waves) . . . . . . . . . . . . 248

B.2.2 Layer Waves (Love, Perturbed Rayleigh, Sezawa Waves) . 249

B.2.3 Other Dispersive Surface Waves . . . . . . . . . . . . . . 251

B.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

xi

Appendix C Surface Acoustic Wave Applications Tutorial 254

C.1 Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 254

C.2 Nondestructive Evaluation . . . . . . . . . . . . . . . . . . . . . 257

C.2.1 Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

C.2.2 Plate and Layer Properties . . . . . . . . . . . . . . . . . 259

C.2.3 Applied and Residual Stresses . . . . . . . . . . . . . . . 261

C.2.4 Adhesive Bonding . . . . . . . . . . . . . . . . . . . . . . 261

C.2.5 Other Material Properties . . . . . . . . . . . . . . . . . . 262

C.2.6 Nonlinear Ultrasonic NDE . . . . . . . . . . . . . . . . . 263

C.3 Chemical Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . 264

C.4 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . 266

C.4.1 Seismology . . . . . . . . . . . . . . . . . . . . . . . . . . 266

C.4.2 Acoustic Microscopy . . . . . . . . . . . . . . . . . . . . . 267

C.4.3 Surface-skimming Bulk Waves (SSBW) Devices . . . . . . 267

C.4.4 Acoustoelectric Applications . . . . . . . . . . . . . . . . 267

C.4.5 Acoustooptic Applications . . . . . . . . . . . . . . . . . 268

C.4.6 Ultrasonic Motors . . . . . . . . . . . . . . . . . . . . . . 268

C.4.7 Surface Acceleration . . . . . . . . . . . . . . . . . . . . . 269

C.4.8 Touch Screen Technology . . . . . . . . . . . . . . . . . . 269

C.4.9 Animal Bioacoustics . . . . . . . . . . . . . . . . . . . . . 269

C.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

Appendix D Crystals and Miller Index Notation 271

Appendix E Integral Transform Between SAW VelocityComponents 274

Appendix F Additional Discussion of Complex-ValuedNonlinearity 279

References 285

Vita 321

xii

List of Tables

1.1 Chronology of some of the experimental work on nonlinear SAWs.Additional details are listed in Table 1.2. . . . . . . . . . . . . . 16

1.2 Chronology of some of the experimental work on nonlinear SAWswith some experimental details. The general topics of these pa-pers are listed in Table 1.1. . . . . . . . . . . . . . . . . . . . . . 17

1.3 Chronology of some of the theoretical work on nonlinear SAWs. 18

2.1 Comparison of various approximate solutions of the spectral evo-lution equations for the fundamental and second harmonic nearthe source. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.2 Conversions and analogies between expressions for the linear so-lutions in the isotropic and anisotropic surface acoustic wavetheories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3 Analogies between expressions for nonlinear acoustical parame-ters in the isotropic and anisotropic surface acoustic wave theories. 59

3.1 Point groups of cubic crystals. . . . . . . . . . . . . . . . . . . . 68

3.2 Lattice types, symmetries, and densities of selected nonpiezoe-lectric cubic crystals. . . . . . . . . . . . . . . . . . . . . . . . . 73

3.3 Second-order elastic (SOE) constants for selected nonpiezoelec-tric cubic crystals in the m3m point group. . . . . . . . . . . . . 74

3.4 Third-order elastic (TOE) constants for selected nonpiezoelectriccrystals in the m3m point group. . . . . . . . . . . . . . . . . . 76

3.5 Second-order elastic (SOE) and third-order elastic (TOE) con-stants for selected nonpiezoelectric crystals in the m3 point group. 78

4.1 Listing of selected nonpiezoelectric cubic crystals with exper-imentally determined third-order elastic constants ordered byanisotropy ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.1 Phases of key linear and nonlinear parameters for the selectedpropagation directions in the (111) plane of Si. . . . . . . . . . . 173

6.2 Phases of key linear and nonlinear parameters for the selectedpropagation directions in the (111) plane of KCl. . . . . . . . . 178

xiii

6.3 Phases of key linear and nonlinear parameters for the selectedpropagation directions in the (111) plane of Ni. . . . . . . . . . 185

7.1 Physical, experimental, and numerical parameters for SAW pulsesin the directions 0◦ and 26◦ from 〈100〉 in the (001) plane of crys-talline silicon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

7.2 Physical, experimental, and numerical parameters for SAW pulsesin the directions 0◦ from 〈112〉 in the (111) plane of crystallinesilicon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

B.1 Summary of some of the theoretical and experimental researchon linear elastic wave propagation in nonpiezoelectric anisotropicmedia of various symmetries. . . . . . . . . . . . . . . . . . . . . 231

B.2 Summary of some of the early theoretical and experimental re-search on linear elastic wave propagation in piezoelectric aniso-tropic media of various symmetries. . . . . . . . . . . . . . . . . 243

B.3 Summary of the various types of surface acoustic waves reviewedin the surface acoustic wave tutorial. . . . . . . . . . . . . . . . 252

xiv

List of Figures

1.1 Horizontal waveforms for nonlinear Rayleigh, Stoneley, and Scholtewaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Vertical waveforms for nonlinear Rayleigh, Stoneley, and Scholtewaves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Coordinate system for plane wave propagation. . . . . . . . . . 23

2.2 Displacement depth profile for silicon in (001) plane in 〈100〉direction for initially sinusoidal wave. . . . . . . . . . . . . . . . 26

2.3 Schematic representation of a generalized Rayleigh wave in apure mode direction. . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Typical particle motion for a generalized Rayleigh wave. . . . . 27

2.5 Plots of the parameters ξ, ξt, ξl, η, ζ , and Λ3 for isotropic mate-rials as a function of Poisson’s ratio σ. . . . . . . . . . . . . . . 57

4.1 Dependence of SAW speed on direction of propagation in the(001) plane for selected materials. . . . . . . . . . . . . . . . . . 84

4.2 Dependence of nonlinearity matrix elements on direction of prop-agation in the (001) plane in selected materials (RbCl, KCl,NaCl, CaF2, SrF2, BaF2, Cs-alum, NH4-alum, K-alum, C, Si,Ge, Al, Ni, Cu). . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.3 Dependence of nonlinearity parameters on direction of propaga-tion in the (001) plane of Si. . . . . . . . . . . . . . . . . . . . . 90

4.4 Magnitudes of the SAW components in the (001) plane of Si. . . 94

4.5 Velocity waveforms in selected directions of propagation in the(001) plane of Si. . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.6 Displacement waveforms in selected directions of propagation inthe (001) plane of Si. . . . . . . . . . . . . . . . . . . . . . . . . 102

4.7 Particle trajectories in selected directions of propagation in the(001) plane of Si. . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.8 Frequency spectra and harmonic propagation curves for selecteddirections of propagation in the (001) plane of Si. . . . . . . . . 105

4.9 Comparison of selected nonlinearity matrix elements calculatedfrom third-order elastic constant data of (1) McSkimin and An-dreatch and (2) Drabble and Gluyas for propagation in the (001)plane of Si. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

xv

4.10 Dependence of nonlinearity parameters on direction of propaga-tion in the (001) plane of KCl. . . . . . . . . . . . . . . . . . . . 108

4.11 Velocity waveforms in selected directions of propagation in the(001) plane of KCl. . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.12 Frequency spectra and harmonic propagation curves for selecteddirections of propagation in the (001) plane of KCl. . . . . . . . 111

4.13 Comparison of selected nonlinearity matrix elements calculatedfrom third-order elastic constant data of (1) Drabble and Stra-then and (2) Chang for propagation in the (001) plane of KCl. . 115

4.14 Dependence of nonlinearity parameters on direction of propaga-tion in the (001) plane of Ni. . . . . . . . . . . . . . . . . . . . . 117

4.15 Velocity waveforms in selected directions of propagation in the(001) plane of Ni. . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.16 Frequency spectra and harmonic propagation curves for selecteddirections of propagation in the (001) plane of Ni. . . . . . . . . 119

4.17 Comparison of selected nonlinearity matrix elements calculatedfrom third-order elastic constant data of (1) Salama and Alersand (2) Sarma and Reddy for propagation in the (001) plane ofNi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122


5.2 Dependence of nonlinearity matrix elements on direction of prop-agation in the (110) plane in selected materials (RbCl, KCl,NaCl, CaF2, SrF2, BaF2, C, Si, Ge, Al, Ni, Cu). . . . . . . . . . 127

5.3 Dependence of nonlinearity matrix elements on direction of prop-agation in the (110) plane in selected alums (Cs-alum, NH4-alum, K-alum). . . . . . . . . . . . . . . . . . . . . . . . . . . . 129







xvi


5.11 Frequency spectra and harmonic propagation curves in selecteddirections of propagation in the (110) plane of KCl. . . . . . . . 140





6.2 Dependence of nonlinearity matrix elements on direction of prop-agation in the (111) plane in selected materials (RbCl, KCl,NaCl, CaF2, SrF2, BaF2). . . . . . . . . . . . . . . . . . . . . . 152

6.3 Dependence of nonlinearity matrix elements on direction of prop-agation in the (111) plane in selected materials (C, Si, Ge, Al,Ni, Cu). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.4 Dependence of nonlinearity matrix elements on direction of prop-agation in the (111) plane in selected alums (Cs-alum, NH4-alum, K-alum). . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.5 Transformation of waveforms for various phase angles 0 ≤ θ ≤180◦ of the transformed nonlinearity matrix elements Sθ

lm =

Slmei(n/|n|)θ, where n = l + m. . . . . . . . . . . . . . . . . . . . 160

6.6 Transformation of waveforms for various phase angles −180◦ ≥θ ≥ 0 of the transformed nonlinearity matrix elements Sθ

lm =

Slmei(n/|n|)θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.7 Comparison of simulated nonlinear waveform distortion betweenvθ

1(x, τ)-transformed nonlinear Rayleigh waves in steel and non-linear SAWs in the direction 0◦ from 〈112〉 in the (111) plane ofSi, Ni, and KCl. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.8 Comparison of simulated nonlinear waveform distortion betweenvθ

1(x, τ)-transformed nonlinear Rayleigh waves in steel and non-linear SAWs in the directions 10◦, 20◦, and 28◦ from 〈112〉 in the(111) plane of KCl. . . . . . . . . . . . . . . . . . . . . . . . . . 165



xvii






6.16 Frequency spectra and harmonic propagation curves for selecteddirections of propagation in the (111) plane of KCl. . . . . . . . 180




7.1 Schematic diagram of the experimental apparatus for photoe-lastic nonlinear surface acoustic wave generation with dual laserprobe detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

7.2 Nonlinearity matrix elements S11, S12, S13 for crystalline siliconin the (001) plane as a function of direction. Due to the symme-tries of this cut, the matrix elements are symmetric about 45◦and periodic every 90◦. . . . . . . . . . . . . . . . . . . . . . . . 199

7.3 Comparison of experiment and theory for a surface acoustic wavepulse propagating in the direction 0◦ from 〈100〉 in the (001)plane of crystalline silicon. . . . . . . . . . . . . . . . . . . . . . 202

7.4 Comparison of experiment and theory for a surface acoustic wavepulse propagating in the direction 26◦ from 〈100〉 in the (001)plane of crystalline silicon. . . . . . . . . . . . . . . . . . . . . . 204

7.5 Comparison of experiment and theory for surface acoustics wavespropagating in the 〈112〉 direction in the (111) plane of crys-talline silicon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

7.6 Comparison of experimental and theoretical longitudinal veloc-ity waveforms reconstructed with shading functions of approxi-mately 700 MHz bandwidth and exactly 3000 MHz bandwidth.The waveforms are the result of propagation in the direction 0◦from 〈112〉 in the (111) plane of crystalline silicon. . . . . . . . . 211

B.1 Schematic representations of a Rayleigh wave and Stoneley wave. 223

xviii

B.2 Schematic representations of a Scholte wave and Leaky Rayleighwave. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

B.3 Typical particle motion for a generalized Rayleigh wave. . . . . 226

B.4 Displacement depth profile for silicon in (001) plane in 〈100〉direction for initially sinusoidal wave. . . . . . . . . . . . . . . . 227

B.5 Schematic representation of a generalized Rayleigh wave in apure mode direction. . . . . . . . . . . . . . . . . . . . . . . . . 227

B.6 Examples of polar plots of relative speed and slowness (based onsilicon in (001) plane). . . . . . . . . . . . . . . . . . . . . . . . 228

B.7 Effect of power flow vector P not being parallel to propagationvector k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

B.8 Schematic diagram of an exceptional bulk wave (surface skim-ming bulk wave) in side and front perspectives. . . . . . . . . . 234

B.9 Coupling between pseudo-surface wave mode and bulk modepropagating at angle θ to the surface. . . . . . . . . . . . . . . . 236

B.10 Schematic representation of a Bleustein–Gulyaev wave. . . . . . 244

D.1 Typical planes for crystal cuts in cubic crystals as specified usingMiller index notation. . . . . . . . . . . . . . . . . . . . . . . . . 273

xix

List of Symbols

A amplitude factor (anisotropic media)A′ linear combination of absorption coefficientsA′′ another linear combination of absorption coefficientsAn dimensionless absorption coefficient (n integer)

Bj =∑3

s=1 β(s)j waveform amplitude at surface (j ∈ {1, 2, 3})

BRj waveform amplitude at surface for Rayleigh wave (j ∈

{1, 2, 3})C constant prefactor of time-domain evolution equation

(isotropic media)Cs coefficients in linear solution (s ∈ {1, 2, 3})CS constant prefactor of time-domain evolution equation

(anisotropic media)D dimensionless diffraction parameterE elastic potential energy densityE2 elastic potential energy density terms of quadratic orderE3 elastic potential energy density terms of cubic orderFs1s2s3 parameters in expansion for Slm (s1, s2, s3 ∈ {1, 2, 3})H Hamiltonian functionH Hilbert transformI amplitude factor (isotropic media)L kernel of integral in time-domain evolution equation

(isotropic media)LS kernel of integral operator in time-domain evolution

equation (anisotropic media)M Mach number or peak strainN number of harmonicsPlm factor in kernel LS (l, m integers)Qlm factor in kernel LS (l, m integers)Rlm nonlinearity matrix (isotropic media; l, m integers)Slm nonlinearity matrix (l, m integers)SR

lm nonlinearity matrix of Rayleigh wave (l, m integers)

Slm = −Slm/c44 dimensionless nonlinearity matrix (l, m integers)T kinetic energy per unit area

xx

T period of waveformT1 taper function for quasilinear solutionV potential energy per unit area terms of quadratic orderVn dimensionless velocity spectral component (n integer)

Vn dimensionless velocity spectral component with taperfunction (n integer)

W kinetic energy per unit area terms of cubic orderW component term of WX dimensionless length variable∆X dimensionless step size in numerical integration∆Xinit dimensionless initial step sizeXswitch range for ∆Xinit

Xmax maximum range for numerical integrationan spectral component functions and generalized coordi-

nates (n integer)bn slowly varying amplitude functions (n integer)c phase speedcR phase speed of Rayleigh waves (isotropic media)cl phase speed of longitudinal bulk wavesct phase speed of transverse bulk wavescijkl second-order elastic (SOE) constants (i, j, k, l ∈ {1, 2, 3})d diameter of beamdijklmn third-order elastic (TOE) constants (i, j, k, l, m, n ∈

{1, 2, 3})eij strain tensor (i, j ∈ {1, 2, 3})f0 characteristic frequencyffund fundamental frequencyi imaginary unit (when not as an index)k wave numberki wave vector components (i ∈ {1, 2, 3})li and l

(s)i penetration depth parameter components (i, s ∈

{1, 2, 3})pn generalized momenta (n integer)t timeui particle displacement components (i ∈ {1, 2, 3})uni depth eigenfunctions (i ∈ {1, 2, 3}, n integer)v0 characteristic velocityvi velocity component in xi-direction (i ∈ {1, 2, 3})

xxi

vl characteristic velocity for lossless bulk wavevn velocity spectral component (n integer)vR

n velocity spectral component for Rayleigh wave (n integer)vx longitudinal velocity componentvx0 amplitude of longitudinal velocity time waveformvy transverse velocity componentvz vertical velocity componentvz0 characteristic measured vertical velocity componentwn1n2n3 coefficients in expansion for Wxi position coordinates (i ∈ 1, 2, 3)x0 characteristic lengthx11 shock formation distance for nonlinear surface wave (as

function of spectral component v0 and nonlinearity ma-trix element S11); equal to xx0

11

xl shock formation distance for bulk waves in a fluidxx0

11 shock formation distance for nonlinear surface wave (asfunction of longitudinal velocity vx0 and nonlinearity ma-trix element S11); equal to x11

xmax probe beam separationx ≡ x1 longitudinal position coordinatey ≡ x2 transverse position coordinatez ≡ x3 vertical position coordinatez0 Rayleigh distance of beamΓ Gol’dberg numberΓik = cijklljll matrix associated with linearized wave equation (i, k ∈

{1, 2, 3})Λ conversion factor between Rlm and Slm

Φ electric potential

αi and α(s)i particle displacement amplitude components (i, s ∈

{1, 2, 3})αn absorption coefficient for n harmonic (n integer)β coefficient of nonlinearity for surface wave (relative to

spectral component v0)βl coefficient of nonlinearity for lossless bulk waveβx0 coefficient of nonlinearity for surface wave (relative to

longitudinal velocity vx0)

β(s)j = Csα

(s)j coefficient in nonlinear solution (j, s ∈ {1, 2, 3})

δij Kronecker delta function (i, j ∈ {1, 2, 3})

xxii

ε acoustic Mach number for surface wave (relative to thelongitudinal velocity vx0)

εl acoustic Mach number for bulk wave in fluidεx0 acoustic Mach number for surface wave (relative to the

longitudinal velocity vx0)ζ numerical factor (isotropic media)

ζs ≡ l(s)3 penetration depth parameters (s ∈ 1, 2, 3)

η numerical parameter (isotropic media)θ phase angleθlong transforming phase for longitudinal velocity componentθtran transforming phase for transverse velocity componentθvert transforming phase for vertical velocity componentλ wavelengthµ bulk shear modulusξ ratio of Rayleigh and shear wave speeds (isotropic media)ξl penetration depth parameter (isotropic media)ξt penetration depth parameter (isotropic media)ρ mass densityσ Poisson’s ratioσij stress tensor (i, j ∈ {1, 2, 3})φBj phase of Bj (j ∈ {1, 2, 3})φvj phase of vj (j ∈ {1, 2, 3})φ

(s)j phase of β

(s)j (j, s ∈ {1, 2, 3})

ω and ω0 angular frequency

xxiii

Chapter 1

Introduction

Surface acoustic waves (SAWs) are waves that exist at an interface between

a solid and a vacuum, gas, liquid, or another solid. They typically have most

of their energy contained close (within approximately one wavelength) to the

interface region. In contrast, the more widely studied bulk acoustic waves,

which do not require an interface in order to travel in a solid, typically have

their energy distributed more broadly throughout the medium in which they

travel.

SAWs were first formally identified and described by Lord Rayleigh1 in

1885. Due to the difficulty of solving the equations to describe these waves even

for simple cases, little additional work was done until the 1950s and 1960s, when

the computation power of the digital computer became available. Since then,

many different kinds of SAWs have been described, and many applications of

SAWs have been invented. Early theories focused on deriving and solving the

simpler linear equations for small amplitude waves. Eventually experiments

began generating the finite amplitude SAWs which are of sufficiently high in-

tensity that more complicated, nonlinear equations are needed to explain the

resulting waves.

SAWs have several properties that distinguish them from bulk waves.

First, SAWs exhibit only two-dimensional geometrical spreading, i.e., the en-

ergy of the SAW spreads out primarily in the two-dimensional interface region

instead of spreading throughout the whole three-dimensional medium like a

1

2

bulk wave. For nonlinear SAWs, this confinement of energy near the sur-

face allows them to maintain larger amplitudes over longer distance than bulk

waves and thereby accumulate more nonlinear effects than bulk waves initially

of the same amplitude. Second, in contrast to many bulk waves which exhibit

spatial confinement (e.g., propagation of bulk waves in waveguides), SAWs in

a homogeneous, semi-infinite half-space∗ are nondispersive, i.e., all frequency

components of the SAW travel at the same speed for each particular direc-

tion. Without this property, features that are often characteristic of nonlinear

velocity waveforms, such as shock fronts, tend to be smeared out as different

frequency components of the wave propagate at different speeds. Finally, non-

linear effects in SAWs are nonlocal, i.e., the global form of the SAW affects its

propagation locally. As shock formation occurs, this nonlocality can be shown2

to cause the formation of sharp cusps in the velocity waveforms which do not

occur in nonlinear bulk waves in fluids and solids.

While all nonlinear SAWs in a homogeneous half-space share the char-

acteristics described above, the symmetries of individual materials can have

a large effect upon how the SAWs propagate in the material. For example,

nonlinear SAWs in isotropic media (materials that are identical in every direc-

tion) can have significantly different properties than those in anisotropic† media

(e.g., crystals). In crystals, the symmetries of the crystalline structure, the ori-

entation of the surface cut, and the direction of propagation on the surface

all affect the behavior of the SAW. If existing applications of nonlinear SAWs

are to be enhanced or additional applications are to be developed, then the

ability to model the propagation of nonlinear SAWs under the aforementioned

∗The existence of multiple interfaces (e.g., plates or layered media) or inhomogeneities(e.g., surface corrugation or impurities) cause SAWs to be dispersive. See Section B.2 for abrief overview of these types of SAWs.

†In some texts, a distinction is made between anisotropic and aeolotropic media. SeeAppendix A for discussion of these terms.

3

conditions must obtained. While much work has been done to understand

nonlinear SAWs in isotropic media and several theories have been advanced

for anisotropic media, few detailed calculations have been performed describ-

ing the propagation of nonlinear SAWs in actual crystals. Performing these

calculations and interpreting the results are the focuses of the present work.

However, because of the wide variety of anisotropic media, the scope of

the dissertation is limited to nonpiezoelectric, cubic crystals. Only materials

with cubic symmetry are considered, not because this is a limitation of the

theoretical model, but because they have the highest symmetry of all crystal

types and are simplest in that sense. In addition, cubic crystals have been

widely studied experimentally, and many data are available on their mechan-

ical properties. Only nonpiezoelectric materials are considered because the

coupling between the mechanical and electrical forces which occurs in piezo-

electric crystals introduces significantly more complexity to the problem. Even

with the above restrictions, many materials fall into this subset of anisotropic

media, including a variety of common dielectrics, semiconductors, metals, and

metallic alloys.

The remainder of this chapter (1) briefly defines and reviews the various

types of SAWs to provide a context for subsequent discussions; (2) briefly

discusses the various applications that have been implemented or proposed for

linear and nonlinear SAWs; and (3) reviews the literature of theoretical and

experimental work on the nonlinear SAWs in crystals.

1.1 Types and Properties

Excellent reviews of the types and properties of SAWs have been given by

Farnell,3,4 Farnell and Adler,5 Stegeman and Nizzoli,6 Feldmann and Henaff,7

and Biryukov et al.8 in the linear regime, and by Parker9 and Mayer10 in the

4

nonlinear regime. Only a brief review of the various cases is given here. A

lengthier tutorial on the types of SAWs is provided in Appendix B.

Depending on the type of interface, SAWs in a homogeneous, isotropic

half-space are classified as Rayleigh waves (solid–vacuum), Stoneley waves

(solid–solid), or Scholte waves (solid–fluid).∗ In these cases, the particle dis-

placement is contained within the sagittal plane, and the amplitude of the

displacement decays exponentially away from the interface into the solid. Due

to the isotropy, the SAW speed is constant in all directions and strictly less

than all the bulk wave speeds in the material.

Several distinguishing linear effects appear in anisotropic media. Par-

ticle displacement is generally no longer confined to the sagittal plane, and it

decays away into the solid as an exponentially damped sinusoid. Such waves

are called generalized Rayleigh waves. In addition, other effects are possible, in-

cluding deeply penetrating surface waves called quasi-bulk surface waves, bulk

waves that satisfy the traction-free surface boundary condition called excep-

tional or “surface-skimming” bulk waves, and unstable surface waves that ra-

diate into the solid, called pseudo-surface waves. In addition, the SAW speed

is a function of both the orientation of the crystal with respect to the surface

cut and the direction of propagation, and the acoustic power flow of the SAW

is no longer necessarily coincident with the direction of the wave vector.

Thus, even in the linear regime, many additional phenomena appear in

anisotropic materials as compared to isotropic media. As shown below, this

trend continues in the nonlinear regime. While nonlinear SAWs in anisotropic

media have some features qualitatively similar to those in isotropic media, their

∗Cases where the sound radiates from the solid into the liquid are usually called leakyRayleigh waves. In these cases the amplitude of the displacement grows exponentially intothe fluid.

5

overall behavior is often distinctly different. These similarities and differences

are discussed at length in the following chapters.

1.2 Applications

A large number of applications for SAWs have been devised, and this section

reviews only some of the developments. SAW devices have been designed to

perform signal processing. Filters, oscillators, and pulse compressors and ex-

panders have been developed using linear effects, while convolvers, correlators,

amplifiers, and memory elements have been developed using nonlinear effects.11

Many of these components are used in mobile and wireless communication de-

vices for personal communication services (e.g., pagers, cellular phones), wide

area networks, and wireless local area networks.12 SAWs have also been used to

perform nondestructive evaluation (NDE). Defects, material properties (den-

sity, elastic constants), applied and residual stresses, adhesive bonding, surface

roughness, plate and layer thickness may all been measured using linear SAWs,

and the use of nonlinear SAWs to test materials is a subject of current research.

SAW sensors have been designed to detect the presence of chemical species and

measure changes in temperature, pressure, and voltage. For a reviews of these

topics, the reader is referred to Oliner,13 Feldmann and Henaff,7 and Biryukov

et al8 (SAW devices), Viktorov,14 Curtis,15 Krautkramer and Krautkramer16

(NDE), and Thompson and Stone17 (SAW sensors). In addition, Appendix C

contains a more detailed tutorial about these applications and many more.

6

1.3 Literature Review

1.3.1 Nonlinear Rayleigh, Stoneley, and Scholte Waves

The nonlinear behavior of Rayleigh waves has been described at length by many

authors, and therefore only a brief review of the most recent work is given here.

Shull et al.18 and Knight et al.19 provide literature reviews of the various the-

oretical and experimental works, as well as a plethora of numerical results for

a variety of cases. Using the theory of Zabolotskaya,20 Shull et al. examined

the behavior of plane,18 cylindrical,18 and diffracting21 nonlinear waves, while

Knight et al. provided a rigorous generalization of the theory19 to nonplanar

waves and pulses, derived approximate expressions for the shock formation

distance,22 and described the propagation of transient waveforms.23 Experi-

mentally, very high amplitude SAWs have been generated photoelastically in

fused quartz and crystalline silicon via pulsed laser excitation.24,25 Calculations

by Knight et al.19,22 were verified by comparison with measurements26 of SAW

pulses in fused quartz.

Investigation of nonlinear Stoneley and Scholte waves has been fairly

limited. Meegan et al.27,28 derived a fully nonlinear theory that includes non-

linear terms up to third order in the energy density and an arbitrary number

of harmonics. They demonstrate that the nonlinear evolution equations for

Rayleigh waves and Stoneley, Scholte, and leaky Rayleigh waves possess the

same basic form. Figures 1.1 and 1.2 show the distortion of initially sinusoidal

longitudinal and vertical velocity waveforms at various propagation distances.

These waveform “snapshots” are taken from the reference frame moving at

the linear SAW speed. The propagation distance is scaled such that X = 1

is approximately the shock formation distance, and the velocity is scaled to a

characteristic value v0. As seen in the figure, similar features are exhibited in

7

-0.6

-0.4

-0.2

0

0.2

0.4

Rayleigh wave:steel

0 π 2π

-0.6

-0.4

-0.2

0

0.2

0.4

0.6steel-glass

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

aluminum-water

0 π 2π

0 π 2π

0.6

ω(t-x/c)

vx

0

vx(1)

v0

vx(1)

v0

1.5

1.0

X=00.5

1.5

1.0

X=0

1.5

1.0

X0.5

Scholte wave:

0.5

(1)

v

Stoneley wave:

=0

Figure 1.1: Horizontal waveforms for nonlinear Rayleigh, Stoneley, and Scholtewaves [based upon Figure 3.3 from Meegan,27 reproduced by permission]. Thepropagation distance is scaled such that X = 1 is approximately one shockformation distance, and velocity is scaled to a characteristic velocity v0.

8

-4

-3

-2

-1

0

1

-2.5

-2

-1.5

-1

-0.5

0

0.5

Rayleigh wave:steel

Stoneley wave:steel-glass

0 π 2π

0 π 2π

0 π 2π

-0.6

-0.4

-0.2

0

0.2

Scholte wave:aluminum-water

ω(t-x/c)

vz(1)

v0

vz(1)

v0

vz(1)

v0

1.51.0

X=00.5

1.51.0

X=00.5

1.51.0

X=00.5

Figure 1.2: Vertical waveforms for nonlinear Rayleigh, Stoneley, and Scholtewaves [based upon Figure 3.4 from Meegan,27 reproduced by permission]. Thepropagation distance is scaled such that X = 1 is approximately one shockformation distance, and velocity is scaled to a characteristic velocity v0.

9

each nonlinear waveform. See Section B.1.1 for additional discussion of some

of the features of these waveforms.

A series of theories by Gusev et al. on the propagation of nonlinear

Rayleigh waves29 and Scholte waves30 (also SAWs in anisotropic media31) has

also been proposed. Their evolution equations differ from those derived using

the dynamical approach of Parker32 and the Hamiltonian approach of Zabolot-

skaya.20 Knight et al.19 proved that the theories of Parker and Zabolotskaya

are equivalent for Rayleigh waves, while Meegan et al.28 have identified incon-

sistencies in the approach of Gusev et al. The interested reader is referred to

their discussions for more details.

1.3.2 Nonlinear Surface Waves in Crystals

Because many practical applications of SAWs require the use of crystals (espe-

cially piezoelectric crystals), extensive work has been done to understand the

propagation of nonlinear SAWs in anisotropic media. For reference purposes,

a tabular summary of much of the experimental and theoretical work is pro-

vided at the end of this chapter. Table 1.1 describes some of the experimental

work that has been done to study nonlinear SAWs. Each entry in Table 1.1

corresponds to an entry in Table 1.2, which lists some of the experimental

parameters and techniques for that work. Table 1.3 summarizes some of the

theoretical work on nonlinear SAWs. None of the tables is exhaustive. In addi-

tion, the tables do not list work that involves dispersion (layer or plate waves),

wedge waves, noncollinear SAW beams, or the interaction of SAWs with bulk

waves.

The most extensive reviews of work on nonlinear SAWs have been given

by Parker9 and Mayer.10 As noted above, Shull et al.18 and Knight et al.19 have

written compact literature reviews that focus primarily on nonlinear Rayleigh

10

waves. However, because these reviews covered much of the experimental work

on nonlinear SAWs, they also cover much of the work that had been done

for crystals. More recently, Hamilton et al.33 reviewed the work on nonlinear

SAWs in nonpiezoelectric crystals. The presentation below includes a summary

of theoretical work on nonlinear SAWs in nonpiezoelectric and piezoelectric

crystals and recent experimental work on nonlinear SAWs in crystals.

Theoretical Work on Nonpiezoelectric Crystals

In the mid-1980s, rigorous theories began to be developed to predict the propa-

gation of nonlinear SAWs in nonpiezoelectric crystals (despite the scepticism of

some earlier researchers∗). Planat35 developed a theory for an elastic solid with

general anisotropy based on a multiple scales approach. He gave numerical

results for the evolution of the amplitude and phase of the first five harmonics

in quartz (neglecting piezoelectric effects), although only a total of eight har-

monics were kept in the calculations. Subsequent work on Rayleigh waves18

shows that an insufficient number of harmonics can introduce anomalies into

the numerical results unless only very weak nonlinearity is considered.

Another theory using multiple scales was developed by Lardner36 based

on his work with nonlinear Rayleigh waves.37 The theory was used by Lardner

and Tupholme38 to investigate the properties of cubic crystals for which the

free surface is a plane of symmetry and the direction of propagation is along

one of the crystal axes. Results were given in terms of tables of coefficients

describing the growth and decay rates of the fundamental, second, and third

∗Interestingly, Lean et al.34 wrote in 1970, “Due to the dispersionless property of Rayleighwaves, the harmonic generation of Rayleigh waves unlike the optical second-harmonic gen-eration does not have any limit in the number of harmonics generated. The large numberof harmonics generated plus the complexity of two-dimensional inhomogeneous waves makestheoretical calculation on the surface acoustic wave harmonic generation extremely compli-cated, if not impossible.”

11

harmonics for a wide variety of cubic crystals, as well as the first three harmonic

propagation curves for the specific example of MgO. Lardner and Tupholme

also showed that for Ge the harmonic growth and decay rates are extremely

sensitive to changes in the third-order elastic constants d111, d112, and d155 and

therefore concluded that measurements using harmonic generation may offer

an accurate way to determine third-order elastic constants. Later papers by

Harvey and Tupholme39,40 extended the method to include multiple waves that

travel in the same or opposite directions, also called co-directional waves. They

presented the following results for MgO: (1) harmonic propagation curves for

the fundamental, second, fifth, and sixth harmonics of co-directional waves

traveling in the directions 0◦ and 40◦ from 〈100〉 in the (001) plane,39 and (2)

growth and decay rates for both single and co-directional waves over the range

of directions 0◦ to 45◦ from 〈100〉 in the (001) plane.40

Independently, Parker32 developed a theory for nonpiezoelectric, ani-

sotropic media that avoids some of the complications and limitations of the

multiple scales approach. This was done by introducing a reference frame

moving at the linear wave speed and then deriving evolution equations for the

wave in the frequency domain. Results for this theory were given only in terms

of waveforms for an isotropic solid (however, see the discussion of piezoelectric

crystals below). A class of waveforms was presented that Parker claimed to be

“non-distorting profiles.” However, Hamilton et al.41 later showed that such

profiles only arise if artificial constraints are imposed on the frequency spec-

trum, such as abrupt truncation or gradual amplitude shading of the Fourier

series representation of a periodic waveform. Kalyanasundaram et al.42 later

extended Parker’s theory to include diffraction effects.

During the mid-1990s, Hamilton et al.43 extended the theory of Zabolot-

skaya20 for nonlinear Rayleigh waves to include nonpiezoelectric, anisotropic

12

media. Longitudinal and vertical velocity waveforms were presented for KCl

in the [112] direction in the (111) plane and the [100] direction in the (001)

plane, although the authors made it clear that this is not a limitation of the

approach. A later paper by Hamilton et al.33 also shows the harmonic propaga-

tion curves for the first five harmonics for both of these cases. Like the theory of

Parker,32 the theory of Hamilton et al. also develops evolution equations in the

frequency domain. However, unlike the theory of Parker, the model equations

of Hamilton et al. are derived using a Hamiltonian approach (see Chapter 2

for a description). Hamilton et al.33 have shown that their theory for non-

linear SAWs in anisotropic media reduces to the theory of Zabolotskaya20 for

nonlinear Rayleigh waves in the isotropic limit.

Gusev et al.31 has also developed a theory for nonlinear SAWs in ani-

sotropic media, but did not provide any numerical calculations demonstrating

their results. The evolution equations are given in the time domain and differ

from those given by Parker32 and Hamilton et al.33

Theoretical Work with Piezoelectric Crystals

Several theories have also been developed to model nonlinear SAW propaga-

tion in piezoelectric crystals (see Section B.1.6 for an introductory discussion).

Based on approaches previously used to model Rayleigh waves, Kalyanasun-

daram44 derived a theory for the special case of Bleustein–Gulyaev waves (see

Section B.1.7) under the assumption that only third-order nonlinearity (quar-

tic anharmonicity) affects the propagation. Mayer45 later extended the theory

to include both second-order and third-order nonlinearity. In 1988, Tupholme

and Harvey46 developed the first general theory for nonlinear SAWs in piezo-

electric crystals by using the method of multiple scales employed previously by

Lardner36 and requiring open circuit electrical boundary conditions (see Sec-

13

tion B.1.6 for a description of electrical boundary conditions at the surface).

Numerical results were later presented by Harvey et al.47 in the form of prop-

agation curves for the fundamental, second, and third harmonics for a wave

propagating along the positive X axis in the Y cut of LiNbO3. Simulations

performed with two different data sets showed significant differences. Addi-

tional papers by Harvey and Tupholme39,40 considered co-directional waves.

They presented the following results for LiNbO3: (1) harmonic propagation

curves for the fundamental, second, fifth, and sixth harmonics of co-directional

waves traveling in the directions 0◦ and 40◦ from the X axis in the Y cut,39

and (2) growth and decay rates for both single and co-directional waves over

the range of directions 0◦ to 90◦ from the X axis in the Y cut.40

Around the same time, Parker and David48 developed a theory for non-

linear SAWs in piezoelectric media based on the approach of Parker32 described

above, and presented simulations showing the evolution of waveforms along the

X and Z axes in the Y cut of LiNbO3 for earthed, open circuit, and free space

electrical boundary conditions. In a later paper by David and Parker,49 “non-

distorting waveforms” were presented for propagation along the X and Z axes

in the Y cut of LiNbO3. However, as noted above, Hamilton et al.41 showed

that stationary Rayleigh waves arise only for the artificial condition of a finite

number of harmonics. Since David and Parker used only forty harmonics in

their simulations, this may also be the cause of the non-distorting profiles in

their results. Diffraction effects were added to the model of Parker and David

by Kalyanasundaram et al.42

Hamilton et al.50 have also developed a theory for nonlinear SAWs in

piezoelectric crystals by generalizing their theory for nonpiezoelectric crystals.33

Free space, shorted, and open circuit electrical boundary conditions may be

included. Simulations have been presented51 for monofrequency waveforms

14

propagating in the X and Y axis directions in the Z cut of LiNbO3. Recent

papers by Hamilton et al.51,52 discuss how all three theories for isotropic solids,

nonpiezoelectric crystals, and piezoelectric crystals are constructed under a

single framework.

Experimental Work with Crystals

Until the mid-1990s, much of the experimental work on nonlinear SAWs in

crystals was limited to measurements of the first few harmonics. In 1996,

Lomonosov and Hess24 presented results of pulsed SAWs generated using a

photoelastic technique. In this approach, an infrared laser is focused into a

small strip on the free surface of the solid. When the laser is pulsed, heat-

ing and radiation pressure cause large amplitude SAWs to be generated. The

vertical velocity waveform is then determined at two neighboring locations by

measuring the deflection of visible laser beams from the surface along the path

of the pulse. Unlike previous experiments, this technique generates extremely

high amplitude (peak strains approaching 0.01) pulses with broadband spec-

tra, and allows the same pulse to be measured at multiple locations. Their

original paper showed waveforms in fused quartz and the 〈112〉 direction in the

(111) plane of crystalline Si that clearly exhibit shock formation. Additional

waveforms in fused quartz were subsequently presented26 with comparison to

the theory of Hamilton et al.,33 and excellent quantitative agreement was ob-

tained.

Additional experiments were performed by Lomonosov and Hess in crys-

talline Si. In the 〈112〉 direction of the (111) plane, good agreement53 was

achieved between the measured pulse data and the theory of Hamilton et al.33

In the (001) plane, it was found54 that the pulses distort in opposite ways,

forming rarefaction shocks in the 〈100〉 direction, and compression shocks in

15

the direction 26◦ from 〈100〉. The same effects are predicted by the theory of

Hamilton et al.,33 which reproduces the waveform evolution in both directions.

(Chapter 7 further discusses the experimental method and shows comparisons

with theory for the cases described in this paragraph.)

1.4 Summary

While several theories have been constructed to model nonlinear SAWs in crys-

tals, none has been used to perform systematic, parametric studies of a variety

of materials, cuts, and directions with the purpose of identifying the types

of nonlinear effects that occur over the whole range of harmonics due to the

anisotropy of the medium. Moreover, until recently, comparisons between the-

ory and experiment have mostly been limited to examining a few harmonics,

and this has made full validation of the theories difficult. This dissertation

addresses both of these issues through an investigation based on the theory of

Hamilton et al.33 Attention is focused on nonlinear SAWs in nonpiezoelectric

cubic crystals in the (001), (110), and (111) planes. Comparisons between the-

ory and experiment are presented for the (001) and (111) planes of crystalline

silicon.

16

Year Author(s) Type Topic1965 Rischbieter55,56 Isotropic 2nd harmonic1968 Løpen57 Piezo 2nd harmonic1969 Lean et al.58 Piezo Parametric mixing1970 Lean et al.34,59 Piezo Parametric mixing1970 Slobodnik60 Piezo Harmonic generation, mixing, saturation1971 Bridoux et al.61 Piezo 2nd harmonic1973 Adler et al.62 Piezo Harmonic generation1973 Vella et al.63 Piezo 2nd harmonic (cw counterpropagating)1974 Gibson et al.64 Piezo Harmonic generation1974 Nakagawa et al.65 Piezo DC electric effects1977 Vella et al.66 Piezo 2nd harmonic, various cuts1977 Alippi et al.67 Piezo 2nd harmonic, various directions1983 Balakirev et al.68 Piezo Weak shocks1983 Nayanov et al.69 Piezo Weak periodic shocks, 19 harmonics observed1984 Brysev et al.70 Isotropic Direct observation of wave motion1994 Telenkov et al.71 Piezo Laser-induced piezoexcitation1996 Lomonosov et al.24 Isotropic Laser-excited pulse generation1996 Meegan et al.27 Isotropic Rayleigh and Scholte waves1997 Kolomenskii et al.25 Isotropic Laser-excited pulse generation1998 Hurley72,73 Nonpiezo Harmonic generation with combs1998 Kumon et al.53 Nonpiezo Laser-excited pulses in crystal1999 Lomonosov et al.26 Isotropic Laser-excited pulse generation2000 Kumon et al.54 Nonpiezo Laser-excited pulses in crystal

Table 1.1: Chronology of some of the experimental work on nonlinear SAWs.Additional details are listed in Table 1.2.

17

Year Author(s) Material f [MHz] Generation Detection1965 Rischbieter55,56 Al, Steel 1.5–14 ? Piezo probe1968 Løpen57 α-quartz 9 IDT IDT1969 Lean et al.58 α-quartz 72 IDT IDT1970 Lean et al.34,59 LiNbO3 200 IDT Opt. diffraction1970 Slobodnik60 LiNbO3 905 IDT Opt. diffraction

367 IDT IDT1971 Bridoux et al.61 Bi12GeO20 50 IDT Opt. diffraction1973 Adler et al.62 LiNbO3 50–150 IDT Opt. diffraction

Bi12GeO20 50–150 IDT Opt. diffraction1973 Vella et al.63 LiNbO3 104 IDT Opt. diffraction

w/Fabry–Perotinterferometer

1974 Gibson et al.64 LiNbO3 160–492 IDT Opt. diffraction1974 Nakagawa et al.65 LiNbO3 134 IDT IDT1977 Vella et al.66 LiNbO3 104 IDT Opt. diffraction

w/Fabry–Perotinterferometer

1977 Alippi et al.67 LiNbO3 91–93 IDT Opt. diffractionα-quartz 79 IDT (transmission &

reflection)1983 Balakirev et al.68 LiNbO3 114 IDT Opt. diffraction1983 Nayanov et al.69 LiNbO3 114 IDT Opt. diffraction1984 Brysev et al.70 Glass 5 Edge trans. EM induction1994 Telenkov et al.71 CdS1−xSex Pulsed laser Piezoeffect1996 Lomonosov et al.24 Fused

quartz∼8 Pulsed laser Laser beam de-

flection (LBD)Si (111) ∼35 Pulsed laser LBD

1996 Meegan et al.27 Al, Cu 1–2 Comb trans-ducer

Contact pin-ducer

Berea sand-stone

0.18 Comb trans-ducer

Contact pin-ducer

1997 Kolomenskii et al.25 Fusedquartz

20 Pulsed laser LBD

1998 Hurley72,73 Al 9.85 Comb Michelson inter-ferometer

1998 Kumon et al.53 Si (111) 50 Pulsed laser LBD1999 Lomonosov et al.26 Fused

quartz20 Pulsed laser LBD

2000 Kumon et al.54 Si (001) 25–35 Pulsed laser LBD

Table 1.2: Chronology of some of the experimental work on nonlinear SAWswith some experimental details. The abbreviation IDT stands for interdigitaltransducer. The general topics of these papers are listed in Table 1.1.

18

Table 1.3: Chronology of some of the theoretical work on nonlinear SAWs. The asteriskindicates that model is used for piezoelectric crystals but piezoelectric effects are ignored inthe model. The abbreviations used are Coupled Amplitude Equations (CAE) and Third-Order Elastic (TOE).

Year Author(s) Type Topic1964 Viktorov74 Isotropic 2nd harmonic effects1968 Løpen57 Piezo∗ Quasilinear, uses TOE constants1969 Lean et al.58 Piezo∗ CAEs for 3 harmonics, parameters from ex-

periment1970 Lean et al.59 Piezo∗ CAEs for 5 harmonics, parameters from exp.1972 Ljamov et al.75 Isotropic Nonlinear corr. to SAW velocity, 2nd har-

monic in pre-shock region, uses TOE con-stants

1973 Adler et al.62 Piezo∗ CAEs for 4 harmonics, parameters from exp.1973 Reutov76 Isotropic Averaged variational principle; later used by

Zabolotskaya1974 Vella et al.77 Piezo∗ Harmonic generation and parametric mixing1974 Tiersten et al.78 Piezo 2nd harmonic generation1976 Anand79 Isotropic 2nd harmonic amp., multiple scales, uses

TOE constants1977 Pavlov et al.80 Isotropic 2nd harmonic generation1979 Normandin et al.81 Piezo∗ Parametric mixing and harmonic generation1980 Kalyanasundaram

et al.82Isotropic Multiple scales, monochromatic line source

1981 Kalyanasundaram83 Isotropic Coupled amplitude theory1981 Kalyanasundaram84 Isotropic Nonlinear mode coupling1982 Kalyanasundaram

et al.85Isotropic Coupled amplitude theory

1982 Kalyanasundaramet al.86

Isotropic Periodic waves, strained coordinates

1983 Kalyanasundaram87 Isotropic Counterpropagating waves1983 Lardner37 Isotropic Evolution equations using multiple scales1983 Parker88 Isotropic Waves of permanent form1984 Kalyanasundaram44 Piezo Bleustein–Gulyaev waves1984 Lardner89 Isotropic Harmonic generation, parametric amplifica-

tion1984 Lardner90 Isotropic Waveform distortion and shock formation1984 Palma et al.91 Nonpiezo Diffraction and harmonic generation1985 David92 Isotropic Uniform asymptotic solution1985 Parker et al.93,94 Isotropic Waves of permanent form1985 Planat35 Nonpiezo Multiple scale analysis1985 Tiersten et al.95 Piezo 2nd harmonic generation revisited1986 Lardner36 Nonpiezo Evolution equations using multiple scales1986 Lardner et al.38 Nonpiezo Numerical results for cubic crystals based on

Lardner36

Continued on next page

19

Continued from previous pageYear Author(s) Type Topic1988 Parker32 Nonpiezo Evolution equations using dynamical ap-

proach1988 Tupholme et al.46 Piezo Evolution equations using multiple scales1988 Tupholme et al.96 General Review of multiple scales approach1988 Harvey et al.47 Piezo Simulations based on Tupholme et al.46

1988 Shui et al.97 Isotropic Reflection method Rayleigh and Stoneleywaves

1988 Solodov98 Piezo Reflection method for crystals1989 Mozhaev99 Isotropic Shear horizontal waves1989 Parker et al.48 Piezo Evolution equations using dynamical ap-

proach1990 Kalyanasundaram

et al.42Piezo Diffraction added to Parker’s theories

1990 David et al.49 Piezo Waves of permanent form1990 Zabolotskaya100 Isotropic Propagation of Rayleigh waves1991 Mozhaev101 Isotropic Shear horizontal waves1991 Harvey et al.39 Piezo Propagation of co-directional waves1991 Mayer et al.45 Piezo Bleustein–Gulyaev waves1991 Parker et al.102 Piezo Dissipative effects1992 Harvey et al.40 Piezo Propagation of single and co-directional

waves1992 Parker et al.103 General Projection method1992 Zabolotskaya20 Isotropic Propagation of plane and cylindrical waves1993 Hamilton et al.41 Isotropic Nonexistence of stationary waves1993 Shull et al.18 Isotropic Harmonic generation in plane and cylindrical

Rayleigh waves1994 Parker9 General Review of dynamical approach1995 Mayer10 General Comprehensive review article1995 Hamilton et al.2 Isotropic Local and nonlocal nonlinearity1995 Hamilton et al.104 Isotropic Time-domain evolution equations for

Rayleigh waves1995 Shull et al.21 Isotropic Diffraction in Rayleigh wave beams1996 Hamilton et al.43 Nonpiezo Evolution equations using Hamiltonian ap-

proach1996 Hamilton et al.50 Piezo Evolution equations using Hamiltonian ap-

proach1997 Gusev et al.29 Isotropic Time-domain evolution equations for

Rayleigh waves1997 Knight et al.19 Isotropic General theory for Rayleigh waves, including

time-domain equations and pulse evolution1997 Knight et al.22 Isotropic Analytical approx. for shock formation dis-

tance1997 Kolomenskii et al.25 Isotropic Comparison of pulse data with theory of Gu-

sev29


20

Continued from previous pageYear Author(s) Type Topic1998 Gusev et al.29 Isotropic Time-domain evolution equations for Scholte

waves1998 Gusev et al.29 Nonpiezo Time-domain evolution for anisotropic media1998 Hamilton et al.52 General Overview of model equations1998 Kumon et al.53 Nonpiezo Comparison of pulse data with theory of

Hamilton33

1999 Lomonosov et al.26 Isotropic Comparison of pulse data with theory ofZabolotskaya20

1999 Hamilton et al.52 Nonpiezo General theory, calculations for cubic crystal1999 Meegan et al.28 Isotropic Theory of Stoneley and Scholte waves2000 Kumon et al.54 Nonpiezo Angular variation of nonlinearity, Compari-

son of pulse data with theory of Hamilton33

2000 Hamilton et al.51 General Review of Hamiltonian approach

Chapter 2

Theory

This chapter describes the theory used throughout the rest of the dissertation.

First, the model equations for nonlinear SAWs in a nonpiezoelectric crystal

are reviewed. Second, approximate solutions to these equations are given,

and estimates for the shock formation distance and nonlinearity coefficient

are described. Third, a time-domain equation is derived corresponding to the

frequency-domain model equations presented in the first part of the chapter.

Finally, the theory for anisotropic media is compared in detail with the theory

of Zabolotskaya20 for nonlinear Rayleigh waves in isotropic media.

2.1 Description of the Model

The model employed here was developed by Hamilton, Il’inskii, and Zabolot-

skaya.33 Because their paper contains a step-by-step derivation of the model

equations, a description of the technique for their numerical solution, and a

detailed comparison to the isotropic case, only an overview of the theory is

given here.

2.1.1 Linear Theory

Because the linear theory forms the basis for the nonlinear theory, it is reviewed

first. From linear theory, the wave speed and ratios of particle displacement

components may be determined.

21

22

The coordinate system is selected such that the (x1, x2) plane coincides

with the surface of the crystal and the x3 axis is the outward normal to the

surface. The crystal occupies the half-space x3 ≤ 0 with a vacuum above.

Without loss of generality, the x1 axis is selected to be in the direction of

propagation (see Figure 2.1).

The equation of motion for the particle displacement ui as a function of

the stress tensor σij in a homogeneous elastic solid with density ρ is

ρ∂2ui

∂t2=

∂σij

∂xj, (2.1)

where the xj are the position variables, t is the time variable, and i, j ∈ {1, 2, 3}.The Einstein convention of summation over repeated indices is assumed. The

stress–strain relation to linear order in the strain for an arbitrary anisotropic

solid is

σij = cijklekl , (2.2)

where σij is the stress tensor,

eij =1

2

(∂ui

∂xj

+∂uj

∂xi

)(2.3)

is the linearized strain tensor, and cijkl are the second-order elastic (SOE)

constants. Substitution of Eqs. (2.3) and (2.2) into Eq. (2.1) yields the linear

wave equation

ρ∂2ui

∂t2= cijkl

∂2uk

∂xj∂xl

, (2.4)

written in terms of the displacement components.

To determine surface acoustic wave solutions of Eq. (2.4), consider in-

homogeneous plane wave solutions of the form

ui = αieik(l·x−ct) . (2.5)

23

c

+x

-x

+x

Vacuum

Crystal

-x

+x1

32

3

2

Figure 2.1: Coordinate system for plane wave propagation. Note that thepositive z axis points out of the crystal.

24

Here, the vector l = (l1, l2, l3) is defined such that l21 + l22 = 1. By choosing

the x1 axis to be the direction of propagation, it follows that l = (1, 0, ζ),

where ζ may be a complex number. For a wave with angular frequency ω, the

phase speed of this wave is c = ω/k and k is the corresponding wave number.

Substitution of Eq. (2.5) into Eq. (2.4) yields

ρc2αi = cijklljllαk . (2.6)

Here the SOE constants are defined with respect to the chosen set of coor-

dinates. Because the values of the SOE constants are usually provided with

respect to the coordinate system defined by the crystalline axes, a transforma-

tion must usually be performed. See Auld105 for a detailed discussion of this

procedure. Equation (2.6) has a nontrivial solution if

det[Γik(ζ)− ρc2δik] = 0 , (2.7)

where

Γik(ζ) = cijklljll (2.8)

and δik is the Kronecker delta function.

Equation (2.7) is a sixth-order algebraic equation in terms of ζ , and c is

a parameter of the equation. Because this equation has all real coefficients, its

roots must be real or complex conjugate pairs. The real solutions correspond to

bulk waves, whereas the complex solutions correspond to surface waves. Fur-

thermore, it is required that Im ζ < 0 for surface waves so that the amplitude

of the wave decays to zero as x3 → −∞. Three such roots exist for any value

of c. Hence the full solution of Eq. (2.4) has the form

ui =3∑

s=1

Csα(s)i eik(ls·x−ct) =

3∑s=1

Csα(s)i eiζskx3ei(kx1−ωt) , (2.9)

25

where ls = (l(s)1 , l

(s)2 , l

(s)3 ) = (1, 0, ζs). The coefficients Cs are determined by

substituting Eq. (2.9) into the stress-free boundary condition

σi3 = 0 at x3 = 0 (2.10)

to yield the condition

ci3kl

3∑s=1

Csα(s)k l

(s)l = 0 . (2.11)

Together Eqs. (2.7) and (2.11) allow for the solution of the real-valued wave

speed c, and complex-valued eigenvalues ζs, eigenvectors α(s)i , and coefficients

Cs. For an arbitrary anisotropic solid, these equations must be solved numeri-

cally. See Hamilton, et al.33 for a detailed discussion of the algorithm employed

to compute the results in this dissertation.

Several differences in physical motion occur between the isotropic and

anisotropic cases even in the linear approximation:

1. While the SAWs are still nondispersive in a crystalline half-space, their

wave speed c is no longer constant as a function of direction. See Fig. 4.1

for an example of this variation in selected cubic crystals.

2. The values of ζs generally have real and imaginary parts. As a result,

the amplitudes of the displacement components decay as exponentially

damped sinusoids and the particle displacement alternates between pro-

grade and retrograde motion. See Figs. 2.2 and 2.3.

3. Generally the values of Csα(s)i imply that the particle trajectories have

longitudinal (x1 direction), transverse (x2 direction), and vertical (x3 di-

rection) components. See Fig. 2.4 for an example of the trajectory at the

surface.

26

-2.5

-2

-1.5

-1

-0.5

0-1 -0.5 0 0.5 1

⟨ ⟩

uz

λz−

uy=0

Retrograde

Prograde

Retrograde

Prograde

Retrograde

Prograde

Type of Motion

ux

100Displacement Depth Profile for Si (001)

Figure 2.2: Displacement depth profile for silicon in (001) plane in 〈100〉 direc-tion for initially sinusoidal wave.

k

Retrograde

x

z

Generalized Rayleigh Wave

Retrograde

Prograde

Solid

ez/λ

∼λ

Figure 2.3: Schematic representation of a generalized Rayleigh wave in a puremode direction [calculations based upon silicon in (001) plane in 〈100〉 direc-tion].

27

φ

-y

z

x

xφz

y Top View of Trajectory at Surface

Side View of Trajectory at Surface

Figure 2.4: Typical particle motion for a generalized Rayleigh wave.

28

4. In crystals, certain special directions can exist in which all values of ζs are

real and no surface wave solution occurs. Instead, there exists a trans-

versely polarized (x2 direction) bulk wave mode which satisfies the stress-

free surface boundary condition. Such a wave is called an exceptional or

surface-skimming bulk wave. See Fig. B.8 for a schematic diagram.

5. The group velocity is not generally coincident with the direction of the

phase velocity. Equivalently, the direction of power flow is not generally

coincident with the direction of the wave vector. Physically this implies

that the plane wave fronts are at an angle to the power flow direction.

Where the group and phase velocities are in the same direction, those

special cases are usually called pure modes.

See Farnell3 for a complete review of the linear properties of SAWs.

2.1.2 Nonlinear Theory

The method used to derive the nonlinear equations of motion is a generalization

of the method that was used to model SAWs in isotropic media by Zabolot-

skaya.20 It involves calculating the Hamiltonian at cubic order in the wave

variables, choosing appropriate generalized coordinates, applying the equations

of motion in canonical form, and deriving evolution equations for the slowly

varying amplitudes in a suitable retarded time frame. This method is very

general—it is applicable to a crystal of any symmetry for which the elastic

constants are known and to any cut and direction in such a crystal. Again,

only an outline of the key equations is given here.

To begin, the particle displacements for the plane SAWs were taken to

29

have the form

uj(x, z, t) =

∞∑n=−∞

an(t)unj(z)einkx , (2.12)

where

unj(z) =

3∑s=1

β(s)j einkζsz , (2.13)

unj = u∗(−n)j , β(s)j = Csα

(s)j , and j ∈ {1, 2, 3}. For simplicity, the notations x ≡

x1 and z ≡ x3 have been introduced in these expressions. Writing the solution

in this form assumes that the nonlinear solution is close to the linear solution; in

particular, the depth dependence unj at each frequency is the same as the linear

solution. It can be shown that nonlinear corrections to the depth dependence

are of higher order than the highest order terms kept in this model.28

Next, expressions for energy density of the crystal must be determined

in preparation for the computation of the system Hamiltonian. The elastic

energy per unit volume is

E =1

2cijkleijekl +

1

6dijklmneijeklemn + · · · , (2.14)

which corresponds to the nonlinear stress–strain relation

σij = cijklekl + cijklmneklemn , (2.15)

an extension of Eq. (2.2). In these equations the exact expression for the

Lagrangian strain tensor is

eij =1

2

(∂ui

∂xj+

∂uj

∂xi+

∂uk

∂xi

∂uk

∂xj

), (2.16)

where eij = eji, and the coefficients dijklmn are the third-order elastic (TOE)

constants. Substitution of Eq. (2.16) into Eq. (2.14) allows the energy density

to be written in terms of the quadratic and cubic contributions as

E = E2 + E3 , (2.17)

30

where

E2 =1

2cijkl

∂ui

∂xj

∂uk

∂xl

, (2.18)

E3 =1

6d′ijklmn

∂ui

∂xj

∂uk

∂xl

∂um

∂xn

, (2.19)

and

d′ijklmn = dijklmn + cijlnδkm + cjnklδim + cjlmnδik . (2.20)

The Hamiltonian of the system is constructed in the form

H = T + V + W , (2.21)

where T is the kinetic energy, V is the potential energy at quadratic order in the

particle displacement, and W is the potential energy at cubic order associated

with the nonlinear interactions. In terms of the particle displacement, these

quantities are given by

T =1

λ

∫ λ

0

dx

∫ 0

−∞

1

2ρu2

i dz , (2.22a)

V =1

λ

∫ λ

0

dx

∫ 0

−∞E2 dz , (2.22b)

W =1

λ

∫ λ

0

dx

∫ 0

−∞E3 dz , (2.22c)

where λ = 2π/k is the wavelength at the fundamental frequency in the Fourier

expansion of ui, the summation convention applies to u2i ≡ uiui, and ui =

dui/dt. The integration over z sums all the contributions of the energy from

the surface to infinite depth while the integration over x averages the expres-

sion over a region that is a characteristic wavelength in size. Substitution of

31

Eq. (2.12) into Eqs. (2.22) yields

T =ρ

2k

∑n

anan

|n| , (2.23a)

V =1

2ρc2k

∑n

|n|anan , (2.23b)

W = 3(W + W∗) , (2.23c)

where

W =∑

n1>0, n2>0,n3=−(n1+n2)

wn1n2n3an1an2an3 , (2.24)

wn1n2n3 = −k2

3

3∑s1,s2,s3=1

n1n2n3Fs1s2s3

n1ζs1 + n2ζs2 + n3ζs3

, (2.25)

Fs1s2s3 =1

2d′ijklmnβ

(s1)i β

(s2)k [β(s3)

m ]∗l(s1)j l

(s2)l [l(s3)

n ]∗ . (2.26)

In addition, the derivation of Eq. (2.23a) imposes the normalization condition

∫ 0

−∞

∣∣∣∣∣3∑

s=1

β(s)i eiζsz′

∣∣∣∣∣2

= 1 . (2.27)

Here again the values of the SOE and TOE constants in d′ijklmn must be trans-

formed into the coordinate system illustrated in Fig. 2.1.

To derive the equations of motion, generalized coordinates are selected

in order that Hamilton’s equations may be applied. The functions an are chosen

to be the generalized coordinates, and hence the generalized momenta are

pn =∂T

∂an=

ρ

k

a−n

|n| . (2.28)

Substituting Eqs. (2.21) and (2.23) into Hamilton’s canonical equations,

an =∂H

∂pn

, pn = −∂H

∂an

, (2.29)

32

then yields the amplitude equations for the system. The calculation may be

simplified without loss of accuracy by considering only progressive wave prop-

agation via the transformation

an =

{bne−inωt for n > 0b∗−ne−inωt for n < 0

, (2.30)

where the functions bn are slowly varying amplitude functions. However, the

desired form of the evolution equations are in terms of the velocity instead of

particle displacement. This can be achieved by defining

vn = an = −inωbn . (2.31)

The combination of the transformations defined by Eqs. (2.30) and (2.31) in

the evolution equations given by Eq. (2.29) yields

vn =n2ω

2ρc3

(n−1∑m=1

Sm,n−mvmvn−m − 2

∞∑m=n+1

S∗n,m−nvmv∗m−n

), (2.32)

for n > 0 and vn = v∗−n for n < 0. The nonlinearity matrix in this expression

is defined as

Slm =

3∑s1,s2,s3=1

Fs1s2s3

lζs1 + mζs2 − (l + m)ζ∗s3

, (2.33)

where Fs1s2s3 is defined by Eq. (2.26). Physically, the nonlinearity coefficients

Slm represent the strength of the coupling between the lth and mth to generate

the (l + m)th harmonic.

Equation (2.33) is useful for solving initial value problems. However,

often conditions are such that a boundary condition is specified instead of an

initial value condition, e.g., a known waveform is radiated from a source at

a given location. For these problems, the transformation from a temporal to

a spatial evolution equation has been shown19 to be equivalent to replacing

33

the temporal derivative vn with the spatial derivative c(dvn/dx). Under this

transformation, Eq. (2.32) becomes

dvn

dx=

n2ω

2ρc4

(n−1∑m=1

Sm,n−mvmvn−m − 2∞∑

m=n+1


), (2.34)

for n > 0 and vn = v∗−n for n < 0. Physically, the first summation on the

right side represents sum frequency generation, while the second summation

represents difference frequency generation. The velocity waveforms are recon-

structed using

vj(x, z, t) =

∞∑n=1

vn(x)unj(z)ein(kx−ωt) + c.c. , (2.35)

where “c.c.” means “the complex conjugate of the previous term” and the

functions unj are given by Eq. (2.13). The expansion in Eq. (2.35) has no

n = 0 term because the bulk of the solid is assumed to be at rest. Because

Eq. (2.34) has no dissipative terms, the total energy of the wave33,41

E =∞∑

n=1

|vn|2n

(2.36)

must be conserved.

However, SAWs in real physical systems are attenuated by thermovis-

cous absorption. It has been shown that SAWs attenuate exponentially with

distance, and the decay coefficient is proportional to the frequency squared.106

This effect can be included in the model in an ad hoc fashion by adding the

term αnvn on the left side of Eq. (2.34)

dvn

dx+ αnvn =

n2ω

2ρc4

(n−1∑m=1

Sm,n−mvmvn−m − 2∞∑

m=n+1


). (2.37)

where αn = n2α1 is the absorption coefficient for the nth harmonic. This

approach has been used previously for SAWs in isotropic media.18 Typically,

34

the absorption is chosen to be sufficiently weak that its primary effect is to

stabilize the portions of the waveform in the neighborhood of a shock without

significantly affecting the rest of the wave. In other words, the absorption acts

as high pass filter for the frequency components that give rise to the shock

front, but minimally attenuate the lowest frequency components that compose

the other parts of the wave.

The system of first-order differential equations of Eq. (2.37) must be

solved numerically. To accomplish this, it is convenient to introduce the di-

mensionless variables

Vn = vn/v0 , X = x/x0 , An = αnx0 , (2.38)

where v0 > 0 is a characteristic velocity of the SAW and

x0 =ρc4

4|S11|ωv0

(2.39)

is a length scale that characterizes the nonlinear distortion for a SAW radiated

at angular frequency ω with amplitude v0. Using these relations, it is possible

to rewrite the evolution equations in the form

dVn

dX+ AnVn =

n2

8|S11|

(n−1∑m=1

Sm,n−mVmVn−m − 2

N∑m=n+1

S∗n,m−nVmV ∗

m−n

).

(2.40)

For numerical purposes, only the first N harmonics are considered, so that only

N differential equations are integrated and the the series expansions are trun-

cated after N terms. The absorption coefficients An were added for purposes

of numerical stability as discussed above.

To integrate the system in Eq. (2.40) the boundary conditions must be

specified. The theoretical development listed above is general in the sense that

it applies for any set of initial spectral amplitudes. For the sake of simplicity,

35

now consider a specific example. It is common that the source condition be

given in terms of the time waveform at a fixed location. Suppose the source

is placed on the surface z = 0 at the position x = 0. Then by Eq. (2.35), the

velocity waveforms at the source may be expressed as

vj(0, 0, t) =

∞∑n=1

vn(0)Bje−inωt + c.c. , (2.41)

where

Bj =3∑

s=1

β(s)j . (2.42)

In much of the analysis and simulations that follows, a single frequency source

of angular frequency ω0 is used. In spectral form, this can be represented as

vn(0) =

v1 for n = 1v∗1 for n = −10 for n 6= ±1

, (2.43)

where v1 is generally complex and n = 1 corresponds to the fundamental an-

gular frequency ω. Applying Eq. (2.43) to Eq. (2.41) gives

vj(0, 0, t) = v∗1B∗j e

iωt + v1Bje−iωt = 2Re

[v1Bje

−iωt]

. (2.44)

If v1 = |v1| exp(iφv1) and Bj = |Bj| exp(iφBj), then Eq. (2.44) may be rewritten

as

vj(0, 0, t) = 2|v1||Bj| cos(ωt− φv1 − φBj) . (2.45)

The amplitudes |Bj| are determined by the normalization given in Eq. (2.27).

Only the relative phases |φBj−φBk| between the values of Bj and Bk are deter-

mined by solution of the linear problem; the absolute phase may be selected as

is convenient. As a result of Eq. (2.45), the amplitudes of the various velocity

components are given by the relations

|vx||vz| =

|B1||B3| ,

|vy||vz| =

|B2||B3| . (2.46)

36

Again, these equations only apply for a monofrequency source at the surface.

As an example, consider the case of an isotropic solid.107 For this class

of materials, monofrequency SAWs have only longitudinal and vertical compo-

nents, and the former leads the latter in phase by π/2. Choose the phase such

that B1 lies along the negative imaginary axis and, therefore, φB1 = −π/2 and

φB3 = 0 (φB2 is undefined because B2 = 0).∗ Furthermore, suppose v1 = |v0|so that φv1 = 0. Then Eq. (2.45) becomes†

vx(0, 0, t) = 2v0|B1| cos(ωt + π/2) = −2v0|B1| sinωt , (2.47a)

vz(0, 0, t) = 2v0|B3| cos ωt . (2.47b)

Similarly, suppose v1 = −|v0| < 0 so that φv1 = π. Then Eq. (2.45) becomes

vx(0, 0, t) = 2|v0||B1| cos(ωt− π + π/2) = 2|v0||B1| sinωt , (2.48a)

vz(0, 0, t) = 2|v0||B3| cos(ωt− π) = −2|v0||B3| cos ωt . (2.48b)

Thus, if vx(0, 0, t) = vx0 sin ωt, then in both cases the amplitude of the time

waveform may be related to the spectral amplitude by

v1 = − vx0

2|B1| . (2.49)

Note again that the results of this paragraph apply only on the surface and at

the source in an isotropic solid.

∗In contrast, Landau and Lifshitz107 select the phase such that the equivalent of B1 liesalong the negative real axis.

†Except for the early papers by Zabolotskaya20,100 on the theory for Rayleigh waves,subsequent papers employing the same approach2,18,19,21,22,41,104 use a series expansion forvj that contains a prefactor of 1/2, and hence the expression corresponding to this equationdoes not contain the factor of 2 [see Eq. (14) in Shull et al.18].

37

2.2 Approximate Solutions

Approximate analytical solutions of Eq. (2.40) may be obtained by considering

only second harmonic generation due to the fundamental frequency term. Con-

sider the monofrequency source of Eq. (2.43). Let the characteristic velocity be

v0 = |v1| so that in nondimensional form the source condition takes the form

Vn(0) =

{e±iφv for n = ±10 for n 6= ±1

, (2.50)

where eiφv = v1/|v1|.

2.2.1 Quasilinear Solution

First consider Eq. (2.40) for N = 1 in the linear approximation:

dV1

dX+ A1V1 = 0 . (2.51)

Solving this equation subject to Eq. (2.50) yields

V1 = eiφve−A1X . (2.52)

Next consider Eq. (2.40) for the second harmonic but only retain the terms on

the right side corresponding to the fundamental:

dV2

dX+ A2V2 =

S11

2|S11|V21 . (2.53)

Solving this equation using Eq. (2.50) and Eq. (2.52) yields

V2 =S11e

2iφv

2|S11|e−2A1X − e−A2X

A2 − 2A1

. (2.54)

Recall that S11 is complex-valued, and so most generally S11/|S11| = eiφS11 for

some phase factor∗ 0 ≤ φS11 ≤ 2π. Close to the source (X � 1), Eq. (2.52)

∗As shown in Chapter 6, the phase of the nonlinearity matrix elements plays an importantrole in determining how the waveform distorts.

38

and Eq. (2.54) can be approximated by the Taylor series expansions

V1 = eiφv

[1− A1X +

1

2A2

1X2 + O(X3)

], (2.55a)

V2 =S11e

2iφv

2|S11|[X − 1

2A′X2 +

1

6A′′X3 + O(X4)

], (2.55b)

where A′ = 2A1 + A2 and A′′ = 4A21 + 2A1A2 + A2

2. In the limit that the

absorption coefficients A1, A2 → 0, Eqs. (2.55) reduce to

V1 = eiφv , (2.56a)

V2 =S11e

2iφv

2|S11| X . (2.56b)

In dimensional form, the magnitudes of the harmonics are then

|v1|v0

= 1 , (2.57a)

|v2|v0

=x

2x0. (2.57b)

Hence within this approximation the second harmonic increases linearly with

range, just as both Rayleigh waves and longitudinal bulk waves in lossless media

do in the nonlinear regime. A weakness of this solution is that the fundamental

does not decrease, clearly violating energy conservation. An improved approx-

imation is given in Section 2.2.3.

2.2.2 Estimates of Nonlinearity Parameters

This similarity of Eq. (2.57b) with bulk waves in lossless fluids makes it possible

to derive an estimate of the shock formation distance. Let xl be the shock

formation distance for a finite amplitude sound wave in a lossless medium

radiated at angular frequency ω and velocity amplitude v0. The quasilinear

solution of the amplitude of the second harmonic component in the fluid then

39

may be expressed as108

|v2|v0

=x

2xl

. (2.58)

Comparison of Eq. (2.58) and the amplitude of Eq. (2.57b) shows that an

estimate of the shock formation distance x11 in the SAW is

x11 = x0 =ρc4

4|S11|ωv0

. (2.59)

As would be expected, x11 depends on the TOE constants via the nonlinearity

matrix element S11. In addition, the shock formation depends inversely on

the strength of the nonlinearity, wave amplitude, and frequency. However, the

amplitude of the time waveform more typically is known from experiment, not

the spectral amplitude. Hence Eq. (2.59) may be rewritten using Eq. (2.49) to

yield

xx011 =

|B1|ρc4

2|S11|ω|vx0| . (2.60)

Equation (2.59) and Eq. (2.60) are equivalent; the latter is given here primarily

for purposes of convenience.

Equation (2.59) has several limitations. First, it is limited in accuracy.

It only contains one term (|S11|) that describes the strength of the nonlinear-

ity, and this factor characterizes only the energy transfer from the fundamental

to the second harmonic. Generation of higher harmonics is necessary for the

formation of a shock, but these harmonics are not accounted for Eq. (2.59).

Second, there exist cases where harmonic generation occurs but shock forma-

tion does not, e.g., near the 〈100〉 direction in the (001) cut of KCl33 (see also

Section 4.2.3). Thus the estimate given by Eq. (2.59) is no longer meaningful.

This limitation is related to the first one because (at least for the KCl case) it

is the low coupling strength of the fundamental to the harmonics higher than

40

second that prevents the shock from forming. Lastly, Eq. (2.59) was derived

assuming a source with a clearly defined angular frequency ω and amplitude

v0. However, it is more common experimentally to have pulses with a range

of spectral amplitudes. While it is possible to apply Eq. (2.59) to this situa-

tion by selecting a characteristic frequency and amplitude for the pulse, this is

essentially an approximation within an approximation. Despite these limita-

tions, Eq. (2.59) provides at least a first-order estimate of the shock formation

distance for many cases of interest and is still useful in this respect. (See Sec-

tion 2.4.3 for comparison of this estimate to the estimates for isotropic solids.)

An expression for the effective coefficient of nonlinearity also may be

derived by drawing an analogy to bulk waves in lossless media. For a finite-

amplitude wave of angular frequency ω and amplitude vl propagating in such

a medium, the shock formation distance xl is related to the coefficient of non-

linearity βl by108

xl =1

|βl|εlk, (2.61)

where εl = vl/c is the acoustic Mach number and k = ω/c is the wave number.

Eq. (2.59) may be rewritten as

x11 =1∣∣∣4S11

ρc2

∣∣∣ v0

cωc

=1

|β|εk . (2.62)

Hence the coefficient of nonlinearity is identified as

β = −4S11

ρc2(2.63)

and depends on the TOE constants, as would be expected, through S11. In

general, β is complex-valued, although cases do exist where it is strictly real-

valued. The negative sign is introduced in Eq. (2.63) so that the sign of the

nonlinearity coefficient is consistent with the theory for isotropic materials of

41

Zabolotskaya20 (see Sections 2.4.2 and 2.4.3 for further discussion). Because

Eq. (2.62) is derived from Eq. (2.59), its limitations are the same as those

discussed in the previous paragraph. (See Section 2.4.3 for comparison of this

estimate to the estimates for isotropic solids.)

Because the definition of the coefficient of nonlinearity is given relative

to a characteristic velocity, different characteristic velocities can give rise to

different expressions for the coefficient of nonlinearity for the same estimate of

the shock formation distance. For example, Eq. (2.60) can be rewritten as

xx011 =

1∣∣∣ 2S11

ρc2|B1|

∣∣∣ |vx0|c

ωc

=1

|βx0||εx0|k , (2.64)

where εx0 = vx0/c is the longitudinal acoustic Mach number. In this formula-

tion, the coefficient of nonlinearity

βx0 = − 2S11

ρc2|B1| (2.65)

is defined relative to the characteristic velocity vx0. Again, the negative sign

is introduced for purposes of consistency. Note that while x11 = xx011 , βx0 6= β

and vx0 6= v0. As described previously, Eq. (2.65) may be more convenient for

practical purposes.

2.2.3 Tapered Quasilinear Solution

One of the disadvantages of the quasilinear solution given by Eq. (2.52) and

Eq. (2.54) is that it does not account for decrease in the amplitude of the

fundamental by transfer of energy to the second harmonic. The approach of

Merklinger109 for finite amplitude waves in fluids, as extended by Shull et al.18

for Rayleigh waves in isotropic solids, may be used to approximate this deple-

tion. Suppose the first harmonic component may be written in nondimensional

42

form as

V1 = T1(X)V1 , (2.66)

where T1 is the taper function that accounts for the decrease in V1 due to

harmonic generation. By the boundary condition Eq. (2.50), V1(0) = eiφv , and

by Eq. (2.52), V1(0) = eiφv . Hence T1(0) = 1 is the appropriate boundary

condition for the taper function. The solution for the second harmonic given

by Eq. (2.54) derives from the forcing function proportional to V 21 on the right

side of Eq. (2.53). Hence the second harmonic term can be approximated by

taking

V2 = T 21 (X)V2 (2.67)

in this modified solution. Consider Eq. (2.40) with N = 2 and n = 1:

dV1

dX+ A1V1 = −1

4

S∗11

|S11| V∗1 V2 . (2.68)

Substitution of Eqs. (2.66) and (2.67) into Eq. (2.68) and simplification via

Eqs. (2.51) and (2.52) yield

dT1

dX= −e−2iφv

4

S∗11

|S11|T∗1 T 2

1 V2 . (2.69)

Substitution of Eq. (2.54) into Eq. (2.69) gives

dT1

dX= −1

8T ∗

1 T 21

e−2A1X − e−A2X

A2 − 2A1

. (2.70)

Let T1(X) = |T1(X)|eiφT (X) where |T1(X)| and φT (X) are real-valued functions.

Then Eq. (2.70) becomes

d|T1|dX

+ i|T1|dφT

dX= −1

8|T1|3 1

A2 − 2A1

(e−2A1X − e−A2X

). (2.71)

Because the right side of this equation is real-valued, it follows that dφT/dX =

0, and hence φT (X) = φT0 = const. The full taper function then has the form

43

T1(X) = |T1(X)|eiφT0 . But by the boundary condition T (0) = 1, it then follows

that φT0 = 0 and T1(X) = |T1(X)|. By separation of variables, the remaining

real part of Eq. (2.71) may be shown to have the solution

T1 = |T1| =[1 +

1

8A1A2

(1− A2e

−2A1X − 2A1e−A2X

A2 − 2A1

)]−1/2

. (2.72)

This equation indicates that if A1A2 � 1, then the nonlinear effects on the

fundamental are negligible because T1 ≈ 1 in that regime. The complete solu-

tion in terms of the velocity amplitude is then given by substituting Eq. (2.72)

into Eqs. (2.66) and (2.67):

V1 = T1(X)eiφve−A1X , (2.73a)

V2 = T 21 (X)

S11e2iφv

2|S11|e−2A1X − e−A2X

A2 − 2A1. (2.73b)

Close to the source (X � 1), these solutions can be approximated by the

Taylor series expansions

V1 = eiφv

[1− A1X +

(1

2A2

1 −1

16

)X2 + O(X3)

], (2.74a)

V2 =S11e

2iφv

2|S11|[X − 1

2A′X2 +

(1

6A′′ − 1

8

)X3 + O(X4)

], (2.74b)

where A′ = 2A1 + A2 and A′′ = 4A21 + 2A1A2 + A2

2 as before. Equations (2.74)

are analogous to Eqs. (2.55) in the non-tapered case. In the limit that the

absorption coefficients A1, A2 → 0, Eqs. (2.74) reduce to

V1 = eiφv

[1− 1

16X2

], (2.75a)

V2 =S11e

2iφv

2|S11|[X − 1

8X3 +

1

256X5

]. (2.75b)

Equations (2.75) are analogous to Eqs. (2.56) in the non-tapered case. Note

that, even in the absence of absorption, the fundamental exhibits a decrease

in amplitude away from the source. Correspondingly, the linear increase in

44

the second harmonic is decreased by the cubic term relative to the non-taper

compensated case. Again, these Taylor series results only apply near to the

source and neglect interactions with all other higher harmonics.

The nonlinear processes in other regimes can be investigated by taking

other limits. Consider the situation where the wave has propagated more than

its shock formation distance (X > 1) but less than the small-signal absorption

length (A1X < 1). Equations (2.73) can then be rewritten as

V1 =

[1 +

1

8X2 − 2A1 + A2

24X3 + O(X4)

]−1/2

V1 , (2.76a)

V2 =

[1 +

1

8X2 − 2A1 + A2

24X3 + O(X4)

]−1

V2 , (2.76b)

where the taper function has been expanded into a Taylor series about X = 0.

Next, suppose the small-signal absorption of the fundamental is sufficiently

small as compared to the nonlinear depletion of the fundamental by energy

transfer to higher harmonics (with the resulting higher absorption at those

higher frequencies) so that the absorption can be neglected. In this approxi-

mation, Eqs. (2.76) combined with Equations (2.56) become

V1 =eiφv√

1 + X2/8, (2.77a)

V2 =S11e

2iφv

2|S11|X

1 + X2/8. (2.77b)

Suppose also that X2 � 1 so that Eqs. (2.77) can be approximated as

V1 ∼ eiφv2√

2

X, (2.78a)

V2 ∼ S11e2iφv

2|S11|8

X. (2.78b)

Eqs. (2.78) are then valid in the range 1 � X2 � A−21 . Converting back to

45

dimensional variables via Eqs. (2.38), Eqs. (2.78) become

v1 ∼ eiφv

√2

ρc3

|S11|kx, (2.79a)

v2 ∼ S11e2iφv

|S11|ρc3

|S11|kx(2.79b)

Hence the velocity amplitudes in this region are independent of the source

amplitude v0. This phenomenon is known as acoustical saturation and is well-

known in nonlinear acoustics.110 Physically, the effect can be explained in the

following way. In this parameter regime, any increase in the source ampli-

tude does not cause an increase in the received amplitude at a fixed location.

Instead, the extra acoustic energy causes shocks to form closer to the source.

The additional dissipation that results offsets the increase in the amplitude that

would be predicted at that location by linear theory. This effect has been de-

scribed theoretically by Shull et al.18 for Rayleigh waves and has been observed

experimentally in LiNbO3.34,60,69

Finally, consider the “old age” region that is many absorption lengths

(A1X � 1) from the source. In this regime, the velocity amplitude of the

fundamental given in Eq. (2.73) becomes

V1 ∼[1 +

1

8A1A2

]−1/2

eiφve−A1X . (2.80)

Assuming that the source amplitude is sufficiently large that shocks form

(A1A2 � 1)∗, this may be further approximated as

V1 ∼√

8A1A2eiφve−A1X . (2.81)

∗Note that in the case of quadratic absorption, the condition for shock formation becomesA1 � 1/2 or 1/A1 � 2. Because 1/A1 = 1/α1x0 = 4|S11|ωv0/ρc4α1 is a parameter whichcharacterizes the ratio of nonlinear to thermoviscous effects, this indicates that nonlineareffects are dominant, as expected. Here then 1/A1 is the analogue of the Gol’dberg numberΓ in fluids.108

46

In dimensional form, this equation has the form

v1 ∼ ρc3

√2|S11|k

eiφve−α1x , α1α2 � 1

x20

=

(4|S11|ωv0

ρc4

)2

. (2.82)

In this region, the amplitude of the wave is proportional to e−α1x as predicted

by linear theory, but is also much further reduced in amplitude due to the

additional dissipation at the shock. The amplitude is again independent of the

source velocity and the small-signal absorption dominates the nonlinear effects.

2.2.4 Coupled Two-Mode Solution

The analysis with the taper function assumes that the propagation of the finite-

amplitude SAW is “typical,” i.e., it exhibits harmonic generation and shock

formation. However, as mentioned briefly above, there exist special cases where

this is not true. Consider the case of propagation near the 〈100〉 direction in

the (001) cut of KCl (see Section 4.2.3). As has been shown,33 shock formation

does not occur in this case because the magnitude of S12 is more than an order

of magnitude smaller than other “neighboring” elements like S11 and S13. As

a result, the generation of the third harmonic is sufficiently inefficient that

shocks do not form. This “trapping” of energy in the fundamental and second

harmonic allows a simpler set of model equations to describe the nonlinear

interactions, at least close to the source.33 Truncating Eqs. (2.40) at N = 2

in the absence of absorption yields the nonlinear, coupled system of equations

that take the form

dV1

dX= − S∗

11

4|S11|V∗1 V2 , (2.83a)

dV2

dX=

S∗11

2|S11|V21 . (2.83b)

47

For a monofrequency source with an initial amplitude of unity, this system was

shown33 to have the analytical solution

V atyp1 =

|S11|S11

sech

(X

2√

2

), (2.84a)

V atyp2 =

|S11|S11

√2 tanh

(X

2√

2

). (2.84b)

For purposes of comparison with the “typical” quasilinear solutions, Eqs. (2.84)

can be rewritten in the limit that X � 1 to yield

V atyp1 =

|S11|S11

[1− 1

16X2 +

5

1524X4 + O(X6)

], (2.85a)

V atyp2 =

|S11|2S11

[X − 1

24X3 + O(X5)

]. (2.85b)

At the source, the V1(0) = |S11|/S11 and V2(0) = 0. Comparison with Eq. (2.50)

implies that for this solution eiφv = |S11|/S11. Hence the Taylor series expansion

of the “typical,” tapered solution with no absorption is given from Eqs. (2.75)

as

V typ1 =

|S11|S11

[1− 1

16X2

], (2.86a)

V typ2 =

|S11|2S11

[X − 1

8X3 +

1

256X5

]. (2.86b)

Thus the solution for the fundamental differs at quartic order in X, and the

solution for the second harmonic differs at cubic order. Hamilton et al.33 plotted

both Eqs. (2.84) and the numerical solution of the full system of Eq. (2.40) and

showed that they matched closely in the range X ≤ 5 for the fundamental and

X ≤ 2 for second harmonic.

The various approximate solutions for the fundamental and second har-

monic components are compared in Table 2.1 for the region near the source.

To simplify the comparison, only the magnitudes of the harmonic components

are given.

48

With absorptionType |V1| |V2|QL 1− A1X + 1

2A2

1X2 + O(X3) X − 1

2A′X2 + 1

6A′′X3 + O(X4)

TQL 1− A1X + (12A2

1 − 116

)X2 X − 12A′X2 +

(16A′′ − 1

8

)X3

+O(X3) +O(X4)

Without absorptionType |V1| |V2|QL 1 X

TQL 1− 116

X2 X − 18X3 + 1

256X5

C2M 1− 116

X2 + 51524

X4 + O(X6) X − 124

X3 + O(X5)

Table 2.1: Comparison of various approximate solutions of the spectral evo-lution equations for the fundamental and second harmonic near the source.A monofrequency source function of angular frequency ω is assumed andA′ = 2A1 +A2 and A′′ = 4A2

1 +2A1A2 +A22. The solution types are quasilinear

(QL), tapered quasilinear (TQL), and coupled two-mode (C2M).

49

2.3 Time-Domain Evolution Equation

A time-domain equation for the evolution of nonlinear SAWs on the surface of

crystals (x3 = 0) may be derived from the frequency-domain evolution equation

in Eq. (2.34). For convenience, Eq. (2.34) may be rewritten with the summation

on the right side performed over all indices, both positive and negative:

dvn

dx=

n2ω0

2ρc4

∑l+m=n

lm

|lm|Slm(−n)vlvm , (2.87)

where ω0 is the angular frequency of the fundamental. From Eq. (2.35), the

velocity component in the xi direction can be similarly rewritten as

vi(x, z, τ) =∞∑

n=−∞vn(x)uni(z)e−inω0τ , (2.88)

where the depth functions uni are given by Eq. (2.13), τ = t − x/c, and ω0 =

2π/T is the angular frequency associated with the waveform of period T . (Note

that the spectral component with n = 0 is zero because the bulk of the solid is

assumed to be at rest.) At the surface, Eq. (2.88) reduces to

vi(x, τ) =

∞∑n=−∞

vn(x)unie−inω0τ , (2.89)

where

uni =

{Bi =

∑3s=1 β

(s)i for n > 0

B∗i =

∑3s=1(β

(s)i )∗ for n < 0

}= |Bi|ei(n/|n|)φBi . (2.90)

Differentiating Eq. (2.89) with respect to x and substituting Eq. (2.87) into the

resulting equation yields

∂vi

∂x=

ω0

2ρc4

∞∑n=−∞

n2unie−inω0τ

∑l+m=n

lm

|lm|Slm(−n)vlvm . (2.91)

50

Equation (2.89) can be inverted by multiplying both sides by einω0τ and inte-

grating over a period. The result is

vn(x) =ω0

2πuni

∫ T/2

−T/2

vi(x, τ) dτ . (2.92)

Substituting Eq. (2.92) into Eq. (2.91) gives

∂vi

∂x=

ω20

4π2c2

1

2ρc2

∞∑n=−∞

(−inω0)(inuni)e−inω0τ

∑l+m=n

lm

|lm|uliumiSlm(−n)

×∫ T/2

−T/2

∫ T/2

−T/2

vi(x, τ1)vi(x, τ2)eilω0τ1eimω0τ1 dτ1 dτ2 .

(2.93)

Identifying (−inω0) with ∂/∂τ , applying the constraint n = l + m to the first

summation, and letting T = 2π/ω0 yields

∂vi

∂x=

CS

c2T 2

∂

∂τ

∫ T/2

−T/2

∫ T/2

−T/2

LS(φBi, τ − τ1, τ − τ2)vi(x, τ1)vi(x, τ2) dτ1 dτ2 ,

(2.94)

where CS = −1/2ρc2|Bi|. The kernel of the integral

LS(φBi, τ1, τ2) =∞∑

l=−∞

∞∑m=−∞

Plm(φBi)Qlm(φBi)Slme−ilω0τ1e−imω0τ2 , (2.95)

where

Plm(φ) = i(l + m)eiφ(l+m)/|l+m| , (2.96a)

Qlm(φ) = − lm

|lm|e−iφl/|l|e−iφm/|m| . (2.96b)

As mentioned in Section 2.1.2, linear theory only fixes the relative phases of the

Bi. Hence without loss of generality, one component can always be chosen such

that Bi0 = −|Bi0 |i = |Bi0|e−iπ/2, i.e., the absolute phase φBi0 = −π/2. Under

this condition, the kernel for the particular velocity component vi0 reduces to

LS(φBi0 = −π/2, τ1, τ2) =

∞∑l=−∞

∞∑m=−∞

|l + m|Slme−ilω0τ1e−imω0τ2 , (2.97)

51

as shown in Section 2.4.4.

Other velocity components vj(x, τ) (j 6= i) may be obtained from inte-

gral transforms of the vi(x, τ) component. The discrete Fourier transform of

vj(x, τ) is

vj(x, τ) =

∞∑n=−∞

vjn(x)e−inω0τ . (2.98)

From Eqs. (2.89) and (2.90), it follows that the nth spectral amplitudes of the

jth and ith velocity components are related by

vjn(x) =

{(Bj/Bi)vin(x) for n > 0(B∗

j /B∗i )vin(x) for n < 0

. (2.99)

Equation (2.99) implies that the velocity waveform components in the xj and

xi directions are related by31

vj(x, τ) = Re(Bj/Bi)vi(x, τ)− Im(Bj/Bi)H[vi(x, τ)] , (2.100)

where

H(f(τ)) =1

πPr

∫ ∞

−∞

f(τ ′)τ ′ − τ

dτ ′ (2.101)

defines the Hilbert transform ,111 Re(z) is the real part of a complex number z,

Im(z) is the imaginary part of a complex number z, and Pr means the Cauchy

principal value of the integral.112 A detailed proof of this relation is given in

Appendix E.

2.4 Comparison with Isotropic Solids

The purpose of this section is to provide a comparison of the theory of Zabolot-

skaya20 for isotropic media (also Hamilton et al.,2,41,104 Shull et al.,18,21 and

52

Knight et al.19,22) with the theory of Hamilton et al.33,43 for anisotropic me-

dia. The anisotropic theory has been shown to reduce to the isotropic theory

in the limit that the material constants correspond to isotropic symmetry,33

but various other differences in notion and notation exist in the various papers

describing the theories. The analysis in this section explicitly examines these

differences.

2.4.1 Linear Solution

The linear solution for the velocity components of a Rayleigh wave of angular

frequency ω and wave number k is given by [Shull et al.18 based on Eqs. (1)

and (2)]

vx(x, z, t) = iI(ξteξtkz + ηeξlkz)ei(kx−ωt) + c.c. , (2.102a)

vz(x, z, t) = I(eξtkz + ξlηeξlkz)ei(kx−ωt) + c.c. , (2.102b)

where

ξt = (1− ξ2)1/2 , (2.103a)

ξl = (1− ξ2c2t/c

2l )

1/2 , (2.103b)

η = −2(1− ξ2)1/2/(2− ξ2) , (2.103c)

ξ = cR/ct , (2.103d)

ξ6 − 8ξ4 + 8ξ2

(3− 2

c2t

c2l

)− 16

(1− c2

t

c2l

)= 0 , (2.103e)

cl is the phase speed of longitudinal bulk waves, ct is the phase speed of trans-

verse bulk waves, cR = ω/k is the Rayleigh wave speed, and I is the appropriate

amplitude to match the boundary conditions at the source. Note that ξ, ξt, ξl,

and η are solely functions of ct/cl. The linear solution for the velocity wave-

forms of a generalized Rayleigh wave under the same conditions is [Hamilton et

53

al.,33 Eqs. (74) and (75); Eq. (2.35) with Eq. (2.13) here]

vx(x, z, t) = A

(3∑

s=1

β(s)1 eikζsz

)ei(kx−ωt) + c.c. , (2.104a)

vz(x, z, t) = A

(3∑

s=1

β(s)3 eikζsz

)ei(kx−ωt) + c.c. , (2.104b)

where A is the appropriate amplitude to match the boundary conditions at

the source. Because of the isotropic symmetry, the wave number representing

the two transverse decay modes is degenerate. Suppose then that ζ1 = ζ2 are

the penetration depth parameters corresponding to ξt and ζ3 is the penetration

depth parameter corresponding to ξl. With these assumptions Eq. (2.104) can

be rewritten

vx(x, z, t) = A[(β

(1)1 + β

(2)1 )eikζ1z + β

(3)1 eikζ3z

]ei(kx−ωt) + c.c. ,(2.105a)

vz(x, z, t) = A[(β

(1)3 + β

(2)3 )eikζ1z + β

(3)3 eikζ3z

]ei(kx−ωt) + c.c. .(2.105b)

Because Eqs. (2.102) and Eqs. (2.105) must give the same result, the various

coefficients must be equivalent. A comparison of these coefficients is given in

Table 2.2.

2.4.2 Nonlinear Solution

Because an explicit comparison of the nonlinear theories for SAWs in isotropic

and anisotropic solids is given in Appendix B of Hamilton et al.,33 the com-

parison given here only introduces the notation and results needed for subse-

quent sections. The velocity components of the series expansions describing

the Rayleigh waves have the form [Shull et al.,18 Eqs. (1)–(4)]

vx(x, z, t) =1

2

∞∑n=−∞

vn(x)uxn(z)ein(kx−ωt) , (2.106a)

vz(x, z, t) =1

2

∞∑n=−∞

vn(x)uzn(z)ein(kx−ωt) , (2.106b)

54

Expression Isotropic Anisotropic

VelocitiesSolution hj(z)ei(kx−ωt) hj(z)ei(kx−ωt)

h1(z) (iI)(ξte

k0ξtz + ηek0ξlz)

A[(β

(1)1 + β

(2)1 )eik0ζ1z + β

(3)1 eik0ζ3z

]h2(z) 0 0

h3(z) I(ek0ξtz + ηξle

k0ξlz)

A[(β

(1)3 + β

(2)3 )eik0ζ1z + β

(3)3 eik0ζ3z

]Decay ξt iζ1 = iζ2

coefficients ξl iζ3

Prefactors ξt

(β

(1)1 + β

(2)1

)AiI

η β(3)1

AiI

1(β

(1)1 + β

(2)1

)AI

ηξl β(3)3

AI

ξl iβ(3)3 /β

(3)1

Surface

h1(0) (ξt + η) (iI)(∑3

s=1 β(s)1

)A

h3(0) (1 + ηξl) I(∑3

s=1 β(s)3

)A

Table 2.2: Conversions and analogies between expressions for the linear solu-tions in the isotropic and anisotropic surface acoustic wave theories. A sinu-soidal source function of angular frequency ω is assumed.

55

where

uxn(z) = i(sgn n)[ξte|n|ξtkz + ηe|n|ξlkz] , (2.107a)

uzn(z) = e|n|ξtkz + ξlηe|n|ξlkz , (2.107b)

and the definitions of ξt, ξl, η, ω, k, and cR are the same as in Section 2.4.1.

Equations (2.106) and (2.107) are analogous to Eqs. (74) and (75) in Hamil-

ton et al.33 and Eqs. (2.12) and (2.13) here. Note the prefactor of 1/2 in

Eqs. (2.106) is not included in Eqs. (2.12). The corresponding nonlinear spec-

tral evolution equations for plane waves are [Shull et al.,18 Eq. (5)]

dvn

dx+ αvn =

n2µω

2ρc4Rζ

(2

∞∑m=n+1

Rm,n−mvmv∗m−n −n−1∑m=1

Rm,n−mvmvn−m

),

(2.108)

where

ζ = ξt + ξ−1t + η2(ξl + ξ−1

l ) + 4η , (2.109)

and µ is the bulk shear modulus. Note that the value of ζ defined here should

not be confused with the values defined in Section 2.1.1. The nonlinearity

matrix Rlm for Rayleigh waves was shown by Hamilton et al.33 to be related to

the nonlinearity matrix Slm for anisotropic media by

Rlm = −(Λ3/µ)Slm , (2.110)

where

Λ = (ζ/2)1/2 (2.111)

and the values of Slm are defined by Eq. (2.33) here. Note that the matrix

elements Rlm are real-valued for all isotropic solids,20 in contrast to the elements

Slm which are generally complex-valued in crystals.

56

The factor of −(Λ3/µ) in Eq. (2.110) arises from several sources. First,

the negative sign results from the choice of sign of the nonlinear terms, as can

be seen from a comparison of the right sides of Eq. (2.37) and Eq. (2.108).

Second, the factor of the shear modulus µ occurs because the nonlinearity

matrix Slm has the units of the elastic constants, whereas Rlm is dimension-

less. The difference in magnitude Λ3 between the nonlinearity matrices arises

out of the fact that the normalizations of the solutions of the linear equa-

tions differ between the two theories [see Eq. (46) in Hamilton et al.33 for

the normalization condition in the anisotropic theory]. From its definition,

Λ = Λ(ζ) = Λ(ξt, ξl, η) = Λ(ct/cl). However, ct/cl can be shown107 to be only

a function of the Poisson’s ratio σ of the material according to the relation

ct

cl

=

√1− 2σ

2(1− σ). (2.112)

All physically realizable isotropic materials107 have a Poisson’s ratio in the range

0 ≤ σ ≤ 0.5. From this constraint, it then follows that 0.579 < Λ < 1.024, and

hence 0.194 < Λ3 < 1.073 for all isotropic materials. The parameters ξ, ξt, ξl,

η, ζ , and Λ3 are plotted as a function of σ in Figure 2.5.

Equations (2.108) were then made dimensionless using the expressions

[Shull et al.,18 Eqs. (8)]

Vn = vn/v0 , X = x/x0 , An = αnx0 , (2.113)

where v0 is a characteristic velocity and x0 = 2ρc3Rζ/µkv0, and αn is the ab-

sorption coefficient of the nth harmonic. The resulting equations take the form

[Shull et al.,18 Eq. (9)]

dVn

dX+ AnVn = n2

(2

N∑m=n+1

Rm,n−mVmV ∗n−m −

n−1∑m=1

Rm,n−mVmVn−m

).

(2.114)

57

0.86

0.88

0.90

0.92

0.94

0.96

0 0.1 0.2 0.3 0.4 0.5

ξ

σ

0.25

0.30

0.35

0.40

0.45

0.50

0 0.1 0.2 0.3 0.4 0.5

ξ t

σ

0.75

0.80

0.85

0.90

0.95

1.00

0 0.1 0.2 0.3 0.4 0.5

ξ l

σ

-0.80

-0.75

-0.70

-0.65

-0.60

-0.55

-0.50

0 0.1 0.2 0.3 0.4 0.5

η

σ

0.40

0.80

1.20

1.60

2.00

2.40

0 0.1 0.2 0.3 0.4 0.5

ζ

σ

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 0.1 0.2 0.3 0.4 0.5

Λ3

σ

Figure 2.5: Plots of the parameters ξ, ξt, ξl, η (Eqs. (2.103)), ζ (Eq. (2.109)),and Λ3 (Eq. (2.111)) for isotropic materials as a function of Poisson’s ratio σ.

58

These expressions are analogous to Eqs. (82) in Hamilton et al.33 and Eq. (2.40)

here.

2.4.3 Estimates of Nonlinearity Parameters

Shull et al.18 define an estimate for the shock formation distance in an isotropic

solid by

x′ =ρc3

Rζ

4R11µv0k, (2.115)

where ζ , R11, and µ are defined in Section 2.4.2. This expression is obtained

by performing a quasilinear analysis in the same manner as described in Sec-

tion 2.2, and hence the analogue is given by x11 in Eq. (2.59) here. Knight et

al.22 rewrite Eq. (2.115) in the form

x11 = x′ =ρc3

Rζ |ξt + η|4R11µvx0k

, (2.116)

where the characteristic velocity v0 = vx0/|ξt+η| is based upon the longitudinal

velocity amplitude vx0 defined such that vx(0, 0, t) = vx0 sinωt. In this disser-

tation, the corresponding expression is denoted by xx011 and given by Eq. (2.60).

Knight et al. go on to derive a more sophisticated estimate of the shock for-

mation distance [denoted by x+ in Eq. (33) of their paper] based upon an

analysis of the kernel of the time-domain evolution equations for the Rayleigh

wave system. This estimate was also used to model the propagation of SAW

pulses in fused quartz.26 The various expressions discussed in this section are

summarized in Table 2.3.

Knight et al.22 propose a coefficient of nonlinearity for an isotropic solid

of the form [Eq. (29)]

β11 =4µR11

ρc2Rζ |ξt + η| . (2.117)

59

Expression Isotropic Anisotropic

Velocity(v0 > 0)

vx(t) = v0(ξt + η) sinωt vx(t) = 2v0|B1| sinωt

ShockFormation

x′ =ρc3

Rζ

4R11µv0k=

1

|β ′|εk x11 =ρc3

4|S11|v0k=

1

|β|εk

NonlinearityCoefficient

β ′ =4

ρc2R

R11µ

ζβ = − 4

ρc2R

S11

Velocity(vx0 > 0)

vx(t) = vx0 sinωt vx(t) = vx0 sin ωt

ShockFormation

x11 =ρc3

Rζ |ξt + η|4R11µvx0k

=1

|β11|εx0kxx0

11 =2|B1|ρc4

4|S11|ω|vx0| =1

|βx0|εx0k

NonlinearityCoefficient

β11 =4

ρc2R

R11µ

ζ

1

|ξt + η| βx0 = − 4

ρc2R

S111

2|B1|

Table 2.3: Analogies between expressions for nonlinear acoustical parametersin the isotropic and anisotropic surface acoustic wave theories. A sinusoidalsource function of angular frequency ω is assumed. Note that in general thequantities between the cases are not equal, only analogous.

60

This coefficient is derived by applying the procedure described in Section 2.2 to

the estimate of the shock formation distance x11 given in Eq. (2.116). Because

x11 of the isotropic theory is analogous to xx0 of the anisotropic theory, β11 is

analogous to the expression for βx0 in Eq. (2.65). Knight et al. go on to derive

a more sophisticated expression for the coefficient of nonlinearity [denoted by

β+ in Eq. (32) of their paper] based on the more sophisticated estimate of

the shock formation distance described in the previous section. The various

expressions discussed in this section are summarized in Table 2.3.

2.4.4 Time-Domain Evolution Equation

In Hamilton et al.,2 it is shown that for nonlinear Rayleigh waves the frequency-

domain evolution equation20

dvn

dx= −n2µω0

2ζρc4R

∑l+m=n

lm

|lm|Rlm (2.118)

has the corresponding time-domain evolution equation for waveforms at the

surface z = 0 given by

∂vx

∂x=

C

c2RT 2

∂

∂τ

∫ T/2

−T/2

∫ T/2

−T/2

L(τ − τ1, τ − τ2)vx(x, τ1)vx(x, τ2) dτ1 dτ2 , (2.119)

where C = −µ/ζρc2R(ξt +η) and ω0 = 2π/T is the angular frequency associated

with a waveform of period T . The kernel of the integral in Eq. (2.119) is given

by

L(τ1, τ2) =∞∑

l=−∞

∞∑m=−∞

|l + m|Rlme−ilω0τ1e−imω0τ2 . (2.120)

The properties of the kernel L were shown to explain the main features of

the harmonic generation and waveform distortion in nonlinear Rayleigh waves.

Equations (2.118), (2.119), and (2.120) are analogous to Eqs. (2.87), (2.94),

and (2.95) in Section 2.3.

61

The relationship between the kernels in the isotropic and anisotropic

cases is determined by comparing Eqs. (2.95) and (2.120). According to Eq.

(2.95), the kernel in the anisotropic case is given by

LS(φBi, τ1, τ2) =

∞∑l=−∞

∞∑m=−∞

Plm(φBi)Qlm(φBi)Slme−ilω0τ1e−imω0τ2 , (2.121)

where

Plm(φ) = i(l + m)eiφ(l+m)/|l+m| , (2.122a)

Qlm(φ) = − lm

|lm|e−iφl/|l|e−iφm/|m| . (2.122b)

For Rayleigh waves, B1 = (ξt + η)i = −|ξt + η|i because ξt + η < 0 for all

isotropic materials (see Figure 2.5 and associated discussion in Section 2.4.2).

Hence |B1| = |ξt + η| and φB1 = −π/2. First, consider Eq. (2.122a) with

n = l + m,

Plm(φ) = ineiφn/|n| =

{i|n|eiφ|n|/|n| for n > 0

−i|n|e−iφ|n|/|n| for n < 0

=

{i|n|eiφ for n > 0

−i|n|e−iφ for n < 0.

(2.123)

With φ = −π/2,

Plm(−π/2) =

{i|n|e−iπ/2 for n < 0

−i|n|eiπ/2 for n > 0

}= |n| . (2.124)

Next, rewrite Eq. (2.122b)

Qlm(φ) = − lm

|lm|e−iφl/|l|e−iφm/|m| . = −

(l

|l|e−iφl/|l|

)(m

|m|e−iφm/|m|

). (2.125)

Observe that

l

|l|e−iφl/|l| =

{ |l||l|e

−iφ|l|/|l| for l > 0−|l||l| eiφ|l|/|l| for l < 0

}=

{e−iφ for l > 0−eiφ for l < 0

. (2.126)

62

Substituting Eq. (2.126) into Eq. (2.125) yields

Qlm(φ) =

−(e−iφ)(e−iφ) for l > 0, m > 0−(e−iφ)(−eiφ) for l > 0, m < 0−(−eiφ)(e−iφ) for l < 0, m > 0−(−eiφ)(−eiφ) for l < 0, m < 0

=

−e−2iφ for l > 0, m > 01 for l > 0, m < 01 for l < 0, m > 0−e2iφ for l < 0, m < 0

.

(2.127)

With φ = −π/2,

Qlm(−π/2) =

−eiπ for l > 0, m > 01 for l > 0, m < 01 for l < 0, m > 0−e−iπ for l < 0, m < 0

= 1 . (2.128)

Substitution of Eq. (2.124) and Eq. (2.128) into Eq. (2.121) yields

LS(−π/2, τ1, τ2) =∞∑

l=−∞

∞∑m=−∞

|l + m|Slme−ilω0τ1e−imω0τ2 . (2.129)

Substitution of Eq. (2.110) into Eq. (2.129) gives

LS(−π/2, τ1, τ2) = − µ

Λ3

∞∑l=−∞

∞∑m=−∞

|l + m|Rlme−ilω0τ1e−imω0τ2 . (2.130)

Equation (2.120) may be substituted into Eq. (2.130) to yield the relation

L(τ1, τ2) = −Λ3

µLS(−π/2, τ1, τ2) . (2.131)

The relationship between the kernels is then the same as that between the

nonlinearity matrices. Note that L(τ1, τ2) implicitly depends on the phase of

the vx velocity component (which was always taken to be φx = −π/2 in previous

work2,18–22,41,100,104).

The relationship between the time-domain evolution equations in the

isotropic and anisotropic cases is determined by comparing Eqs (2.94) and

63

(2.118). According to Eq. (2.94) for the x1 = x direction,

∂v1

∂x=

CS

c2T 2

∂

∂τ

∫ T/2

−T/2

∫ T/2

−T/2

LS(π/2, τ − τ1, τ − τ2)v1(x, τ1)v1(x, τ2) dτ1 dτ2 .

(2.132)

Substituting Eq. (2.131), CS = −1/2ρc2|B1|, c = cR, |B1| = −(ξt + η), and

Λ = (ζ/2)1/2 into Eq. (2.132) yields

∂v1

∂x=

1

Λ

C

c2RT 2

∂

∂τ

∫ T/2

−T/2

∫ T/2

−T/2

L(τ − τ1, τ − τ2)vx(x, τ1)vx(x, τ2) dτ1 dτ2 .

(2.133)

Hence (∂vx

∂x

)iso

= Λ

(∂v1

∂x

)aniso

. (2.134)

Again, this difference arises out of the difference in normalization of the linear

solutions between the theories for isotropic20 and anisotropic33 media.

Finally, it can be shown that the integral transform between velocity

components on the surface for an anisotropic solid reduces to the transform

for an isotropic solid. Hamilton et al.2 state that the vertical and horizontal

velocity components of a Rayleigh wave on the surface of a solid are related by

the expression

vz(x, τ) =1 + ηξl

η + ξt

H[vx(x, τ)] , (2.135)

where

H[f(τ)] =1

πPr

∫ ∞

−∞

f(τ ′)τ ′ − τ

dτ ′ (2.136)

defines the Hilbert transform111 and Pr means the Cauchy principal value112

of the integral. According to Eq. (2.100), the velocity waveform components

64

in the x3 and x1 directions (z and x directions) on the surface of a solid are

related by

v3(x, τ) = Re(B3/B1)v1(x, τ)− Im(B3/B1)H[v1(x, τ)] . (2.137)

For an isotropic solid, Eqs. (2.102) imply that

Re

(B3

B1

)= 0 , Im

(B3

B1

)= Im

(1 + ξlη

(ξl + η)i

)= −1 + ξlη

ξl + η, (2.138)

where ξl, η, and ξl are all real-valued (see Figure 2.5). Substituting Eqs. (2.138)

into Eq. (2.137) immediately gives Eq. (2.135) above.

2.5 Summary

This chapter outlined the theory of Hamilton, Il’inskii, and Zabolotskaya33

for the propagation of nonlinear SAWs in anisotropic media. The frequency-

domain model equations which result from this theory are solved in several

quasilinear approximations, and estimates for the shock formation distance and

nonlinearity coefficient are given. A time-domain formulation of the frequency-

domain model equations is derived and shown in the isotropic limit to reduce to

the same time-domain equation for nonlinear Rayleigh waves given by Zabolot-

skaya.20 A detailed comparison is made between the theory for anisotropic

media presented here and the various papers based on the theory of Zabolot-

skaya.20 Many of the quantities defined in Chapter 2 are used extensively

throughout the rest of the dissertation.

Chapter 3

Properties of Cubic Crystals

In order to model the propagation of nonlinear SAWs in a cubic crystal, the

material properties of the crystal must be known. However, the anisotropy of

a crystal adds a significant level of complexity to the problem as compared to

isotropic media. Not only must more information be determined about the bulk

properties of the material, but surface properties such as the orientation of the

surface with respect to the crystalline axes and the direction of propagation

within that surface must be specified as well. This chapter gives a brief overview

of the types of structures, symmetries, and elastic properties found in cubic

crystals. In addition, tables are provided at the end of the chapter listing the

material constants for all the substances investigated in subsequent chapters.

Cubic crystals are chosen for study for several reasons. First, cubic

crystals exhibit the simplest type of fully three-dimensional anisotropy in the

sense that they are the most symmetric. These symmetries are reflected in the

acoustical properties of the crystal and significantly constrain the parameters

that describe these properties.105 Second, investigations of linear bulk acoustic

waves113 and SAWs3 in cubic crystals have shown that the features exhibited by

waves in these materials are similar in many cases to waves in more complicated

crystal types. Finally, many experimental data exist for the elastic constants

of cubic crystals, possibly more than any other type.114,115 It should be noted

that cubic crystals were not chosen because of any limitation imposed by the

theory of Hamilton et al.33 presented in Chapter 2.

65

66

3.1 Crystal Structure

Crystals are generally composed of a periodic array of points (lattice) with one

or more atoms in a specific composition, arrangement, and orientation placed at

each point (basis). The lattice points are typically grouped into parallelepipeds

(unit cells), often with geometrical properties that reflect the symmetries of the

crystal. As the name would suggest, cubic crystals are conventionally described

in terms of unit cells that are cubes, i.e., the edges of the cells are mutually

perpendicular and have equal length. All cubic crystals can be described in

terms of three different lattices (Bravais lattices):

• simple cubic:

Points are placed on each corner of the cube.

• body-centered cubic:

Points are placed on each corner of the cube and in the center of the cube.

• face-centered cubic:

Points are place on each corner of the cube and in the center of each face

of the cube.

Other lattices for crystals with cubic symmetry exist, but all can be constructed

from these lattices. For example, in the diamond cubic structure each atom has

exactly four nearest neighbors. However, this arrangement can be shown to be

two face-centered cubic lattices displaced from one another by a translation of

(1/4, 1/4, 1/4). Reference is made to these various lattice types in the examples

that follow. The directional properties of the crystal are typically given relative

to a standard set of crystalline axes. For cubic crystals, the x, y, and z axes are

conventionally chosen to lie along the three mutually perpendicular directions

of the cubic unit cell.

67

In terms of symmetry, cubic crystals are characterized by having four

three-fold rotation axes in the directions between opposite corners of the cube

(i.e., tilted approximately 54◦ 44′ from each of the crystalline axes). The other

symmetries of the crystals, like rotation, inversion, and reflection, are described

by point groups, or the groups of symmetry operations that keep at least one

point fixed. Table 3.2 lists the five point groups for cubic crystals. Note that the

m3 and m3m point groups have an inversion center, and hence piezoelectric

effects are not possible in materials with these symmetries.116 In addition,

piezoelectric effects cannot occur in materials with the symmetries of the 432

point group. Even though this point group does not have an inversion center, its

large number of other symmetries precludes piezoelectric effects.116 Because the

theory described in Chapter 2 does not include piezoelectric effects (although

an extension does50), cubic crystals from at most these three point groups are

examined in the present work.∗

While a crystal may be described completely by specifying its lattice

and basis, it may also be described by specifying its space group and the posi-

tions of its atoms.118 Space groups differ from point groups in that they add

translation to the set of possible symmetry operations. Furthermore, space

groups distinguish the type of crystalline lattice and are more specific about

describing all the other types of the crystal’s symmetries. Table 3.2 lists the

space groups for several cubic crystals. For example, as described above, the

diamond crystal structure can be specified by the translation between the two

face-centered cubic lattices. This is explicitly listed in the space group for di-

amond crystal structure by the symbols F in the first position and d in the

second position (see the entry for Si in Table 3.2).

∗In some piezoelectric crystals, there exist cases where SAWs traveling in certain directionsof high symmetry exhibit no piezoelectric effects.117 However, these special cases are notconsidered in this work.

68

International Crystal class name Piezo-symbol electric?

short full

m3m4

m3

2

mCubic hexakis-octahedral No(hexoctahedral)

432 432 Cubic pentagonal icositetra- Nohedral (gyroidal)

43m 43m Cubic hexakis tetrahedral Yes(hextetrahedral)

m32

m3 Cubic dyakis-dodecahedral No

(diploidal)23 23 Cubic tetrahedral-pentagonal- Yes

dodecahedral (tetartoidal)

Table 3.1: Point groups of cubic crystals (from Thurston116). Note that onlythe m3m, m3, and 432 point groups are nonpiezoelectric.

3.2 Elastic Constants

The symmetries outlined above constrain the number of independent constants

that describe the system. In general, the stress–strain relation for an arbitrary

crystalline solid can be written as

σij = cijklekl + dijklmneklemn + · · · , (3.1)

where σij is the stress tensor,

eij =1

2

(∂ui

∂xj+

∂uj

∂xi+

∂uk

∂xi

∂uk

∂xj

)(3.2)

is the Lagrangian (material) strain tensor, cijkl are the second-order elastic

(SOE) constants, and dijklmn are the third-order elastic (TOE) constants. The

elastic constants derive their name from the fact that the corresponding elastic

energy per unit volume of the crystal is

E = 12cijkleijekl + 1

6dijklmneijeklemn + · · · . (3.3)

69

From the form of Eq. (3.2), it is clear that strain tensor is symmetric, i.e., each

element is constant under the operation i ↔ j. Moreover, it can be shown that

the stress tensor must also be symmetric for conservation of angular momentum

to hold.119 From these properties the elastic constant tensors cijkl and dijklmn

must be invariant under the permutations i ↔ j, k ↔ l, and m ↔ n, and

symmetric with respect to the exchange of indices in pairs. As a result of these

symmetries, it can be shown that there are a maximum of 21 independent SOE

constants and a maximum of 56 independent TOE constants (e.g., triclinic

crystal).120 In addition, the symmetries allow Voigt’s notation to be introduced,

in which pairs of indices are mapped to single indices according to ij → I

where (11) → 1, (22) → 2, (33) → 3, (23, 32) → 4, (13, 31) → 5, (12, 21) → 6.

This notation is commonly employed throughout the literature and is used

throughout the rest of this work. See Auld105 for additional discussion and

examples of its usefulness.

It can be shown that additional spatial symmetry conditions reduce the

number of independent constants necessary for specifying the crystal’s dynam-

ics completely. In the case of cubic symmetry, the number of independent SOE

constants is reduced to 3, and the number of TOE constants is reduced to 6 or

8, with the latter value depending on the crystal’s point group. Most commonly

the elastic constants are given relative to a coordinate system associated with

the crystalline axes. For all cubic crystals, the 3 independent SOE constants

are114

c11 = c22 = c33 ,c12 = c13 = c23 ,c44 = c55 = c66 .

(3.4)

For cubic crystals in the 432, 43m, and m3m point groups, the 6 independent

70

TOE constants are115

d111 = d222 = d333 ,d112 = d113 = d122 = d133 = d223 = d233 ,d123 ,d144 = d255 = d366 ,d155 = d166 = d244 = d266 = d344 = d355 ,d456 .

(3.5)

For cubic crystals in the 23 and m3 point groups, the 8 independent TOE

constants are115

d111 = d222 = d333 , d144 = d255 = d366 ,d112 = d133 = d223 , d155 = d266 = d344 ,d113 = d122 = d233 , d166 = d244 = d355 ,d123 , d456 .

(3.6)

In contrast, isotropic materials have only 2 independent SOE constants114:

c11 = c22 = c33 ,c12 = c13 = c23 ,

(3.7)

with the additional dependent relations c44 = c55 = c66 = (c11 − c12)/2, and 3

independent TOE constants115:

d111 = d222 = d333 ,d112 = d113 = d122 = d133 = d223 = d233 ,d123 ,

(3.8)

with the additional dependent relations

d144 = d255 = d366 = (d112 − d123)/2 ,d155 = d166 = d244 = d266 = d344 = d355 = (d111 − d112)/4 ,d456 = (d111 − 3d112 + 2d123)/8 .

(3.9)

More commonly, the SOE constants for isotropic materials are written in terms

of the Lame constants λ and µ as121

c11 = λ + 2µ , c12 = λ , c44 = µ , (3.10)

71

while the TOE constants are written in terms of the constants A, B, and C of

Landau and Lifshitz107 as121

d111 = 2(A + 3B + C) , d144 = B ,d112 = 2(B + C) , d155 = 1

2A + B ,

d123 = 2C , d456 = 14A .

(3.11)

Norris121 lists additional expressions for the TOE constants of isotropic media

in terms of other common conventions.

3.3 Cuts and Directions

Because crystals are anisotropic, the orientation of the planar surface of the

crystal with respect to the crystalline axes and the orientation of the propaga-

tion direction within that plane affect the evolution of the wave. Typically, the

crystal cut is given in terms of Miller indices and the directions in terms of a

vector direction. See Appendix D for details and illustrations.

3.4 Experimental Data

The theory presented in Chapter 2 has no adjustable parameters. Once the

material constants (density, SOE and TOE constants), surface cut, and direc-

tion of propagation are specified, the theory is completely determined. Density

and SOE constant data are available for a wide variety of crystals, and the data

from different sources typically match closely114 (see Tables 3.2, 3.3, and 3.5).

TOE constant data are less common, and in some cases the values can vary

significantly not only in magnitude but even in sign115 (see Tables 3.4, 3.5).

The effect of variations of the data on the calculated results is discussed in

Chapter 4.

72

3.5 Summary

This chapter has discussed some of the basic properties of cubic crystals in-

cluding structure, symmetries, and elastic constants. Of the five point groups

that fall into the cubic class, only three (m3m, 432, and m3) have symmetries

which exclude piezoelectric effects. All three groups have three independent

SOE constants. In contrast, the m3m and 432 point groups have six indepen-

dent TOE constants, while the m3 point group has eight independent TOE

constants. For reference purposes, measured densities, SOE constants, and

TOE elastic constants from the literature are listed in Tables 3.2, 3.3, 3.4, and

3.5 for crystals in the m3m and m3 point groups. Chapters 4, 5, and 6 use the

information contained in this chapter to investigate nonlinear SAWs propagat-

ing in crystals of the m3m and m3 point groups over a variety surface cuts and

directions.

73

Table 3.2: Lattice types, symmetries, and densities of selected nonpiezoelectriccubic crystals. Both the point and space groups are given in short form. Thereferences to the original papers that determined the crystallographic structurecan be found in the general references given in the table.

Cubic CrystalsMaterial Point Space Density Source

group group (kg/m3)RbCl m3m Fm3m 2803 Pies et al.122

KCl m3m Fm3m 1989.1 Pies et al.122

NaCl m3m Fm3m 2167.8 Pies et al.122

CaF2 m3m Fm3m 3180 Pies et al.122

SrF2 m3m Fm3m 4180 Pies et al.122

BaF2 m3m Fm3m 4893 Pies et al.122

C m3m Fd3m 3520 Eckerlin et al.123,a

Si m3m Fd3m 2328 Eckerlin et al.123

Ge m3m Fd3m 5326.74 Eckerlin et al.123

Al m3m Fm3m 2698.01 Eckerlin et al.123

Ni m3m Fm3m 8912 Eckerlin et al.123

Cu m3m Fm3m 8960 Eckerlin et al.123

Cs-alum m3 Pa3 1999.2 Pies et al.124,c

NH4-alum m3 Pa3 1614.8 Pies et al.124,c

K-alum m3 Pa3 1753 Pies et al.124,c

aCrystal in diamond form.cThe hydrous X-alums listed here have the form XAl(SO4)2 · 12 H2O.

74

Table 3.3: Second-order elastic (SOE) constants for selected nonpiezoelectric cubiccrystals in the m3m point group (see Hearmon114,125 and Every and McCurdy126 foradditional data). The constants are written in Voigt’s notation with units of GPaand are given in the reference frame defined by the crystalline axes. The constantsfrom Hearmon are an average over several data sets (not just those listed here), andthe percentage error listed is the standard deviation divided by the average. Paperswhich contain both SOE and TOE constant data have the SOE data listed below(see Table 3.4 for the corresponding TOE constant data). The anisotropy ratio isη = 2c44/(c11 − c12).

Cubic (m3m): SOE ConstantsMaterial η c11 c12 c44 SourceRbCl 0.312 36.4 6.3 4.7 Hearmon114

±0.6% ±7% ±1%KCl 0.373 40.5 6.9 6.27 Hearmon114

±0.9% ±5% ±1%40.76 7.05 6.32 Chang127

40.90 7.04 6.27 Drabble et al.128

±0.06 ±0.04 ±0.01NaCl 0.705 49.1 12.8 12.8 Hearmon114

±1% ±1% ±1%49.34 12.93 12.78 Chang127

49.42 12.69 12.81 Drabble et al.128

±0.06 ±0.03 ±0.0149.8 13.0 12.8 Gluyas129

±0.5 ±0.2 ±0.01CaF2 0.373 184 67 21.8 Hearmon114

±1% ±7% ±1%SrF2 0.803 124 45 31.7 Hearmon114

±0.6% ±0.9% ±0.9%124.61 44.63 31.874 Alterovitz et al.130

±0.05 ±0.11 ±0.01BaF2 1.02 90.7 41.0 25.3 Hearmon114

±0.9% ±4% ±2%89.48 38.54 24.95 Gerlich131


75

Continued from previous pageMaterial η c11 c12 c44 SourceC 1.26 1040 170 550 Hearmon114,b

±6% ±43% ±5%Si 1.57 165 64 79.2 Hearmon114

±0.6% ±0.7% ±0.6%165.64 63.94 79.51 Hall132

±0.02% ±0.02% ±0.02%165.773 63.924 79.619 McSkimin et al.133

Ge 1.66 129 48 67.1 Hearmon114

±2% ±7% ±0.8%128.35 48.23 66.66 Bogardus134

128.528 48.260 66.799 McSkimin et al.133

Al 1.23 108 62 28.3 Hearmon114

±2% ±3% ±0.7%106.75 60.41 28.34 Thomas135

±0.05 ±0.08 ±0.04Ni 2.60 247 153 122 Hearmon114

±2% ±3% ±2%251.6 154.4 122.0 Salama et al.136

250.3 151.1 122.4 Sarma et al.137,c

Cu 3.20 169 122 75.3 Hearmon114

±0.9% ±1.5% ±0.8%166.1 119.9 75.6 Hiki et al.138

168.4 121.4 75.4 Salama et al.136

bCrystal in diamond form.cSample was magnetically saturated.

76

Table 3.4: Third-order elastic (TOE) constants for selected nonpiezoelectric crys-tals in the m3m point group (see Hearmon115,139 and Every and McCurdy140 foradditional data). The constants are given in Voigt’s notation with units of GPa andare given in the reference frame defined by the crystalline axes.

Cubic (m3m): TOE ConstantsMaterial d111 d112 d123 d144 d155 d456 SourceRbCl −617 −67 +87 +25 −26 −38 Prasad

et al.141

KCl (−701) (−22.4) (+13.3) +12.7 −24.5 +11.8 Chang127,a

±0.5 ±0.2 ±0.4−726 −24 +11 +23 −26 +16 Drabble±39 ±4 ±4 ±4 ±2 ±1 et al.128

NaCl (−880) (−57.1) (+28.4) +25.7 −61.1 +27.1 Chang127,a

±1.6 ±0.7 ±1.4−843 −50 +46 +29 −60 +26 Drabble±33 ±7 ±9 ±5 ±4 ±1 et al.128

−823 +2 +53 +23 −61 +20 Gluyas129

±2 ±5 ±7 ±3 ±3 ±1CaF2 −1246 −400 −254 −124 −214 −74.8 Alterovitz

±91 ±30 ±29 ±15 ±9 ±3.8 et al.142

SrF2 −821 −309 −181 −95.1 −175 −42.1 Alterovitz±11 ±5 ±12 ±6.6 ±3 ±2.8 et al.130

BaF2 −584 −299 −206 −121 −88.9 −27.1 Gerlich131

±15 ±14 ±11 ±3 ±1.9 ±0.1C −6260 −2260 +112 −674 −2860 −823 Grimsditch

et al.143,b

Si −744 −418 +2 +29 −315 −70 Drabbleet al.144

−795 −445 −75 +15 −310 −86 Hall132

±10 ±10 ±5 ±5 ±5 ±5−825 −451 −64 +12 −310 −64 McSkimin±10 ±5 ±10 ±25 ±10 ±20 et al.133


aParenthetical data are based upon the Cauchy relations d123 = d456 = d144 and d112 =d166.

bCrystal in diamond form.

77

Continued from previous pageMaterial d111 d112 d123 d144 d155 d456 SourceGe −716 −403 −18 −53 −315 −47 Bogardus134

±20 ±10 ±30 ±5 ±5 ±10−696 −340 +25 +18 −296 −42 Drabble±108 ±62 ±43 ±21 ±22 ±6 et al.144

−710 −389 −18 −23 −292 −53 McSkimin±6 ±3 ±6 ±16 ±8 ±7 et al.133

Al −1076 −315 +36 −23 −340 −30 Thomas135

±30 ±10 ±15 ±5 ±10 ±30−1224 −373 +25 −64 −368 −27 Sarma±60 ±38 ±25 ±8 ±13 ±7 et al.145

Ni −2032 −1043 −220 −138 −910 +70 Salama±40 ±25 ±50 ±55 ±75 ±30 et al.136

−2104 −1345 +59 -180 −757 −42 Sarma±124 ±84 ±56 ±53 ±87 ±23 et al.137,c

Cu −1271 −814 −50 −3 −780 −95 Hiki±22 ±9 ±18 ±9 ±5 ±87 et al.138

−1390 −778 −181 −140 −648 −16 Salama±20 ±10 ±20 ±14 ±10 ±10 et al.136

cSample was magnetically saturated.

78

Table 3.5: Second-order elastic (SOE) and third-order elastic (TOE) constants forselected nonpiezoelectric crystals in the m3 point group (see Hearmon114,115,139 andEvery and McCurdy126,140 for additional data). The constants are given in Voigt’snotation with units of GPa and are given in the reference frame defined by thecrystalline axes. The hydrous X-alums listed here have the form XAl(SO4)2 · 12H2O.

Cubic (3m): SOE ConstantsMaterial η c11 c12 c44 SourceCs-alum 1.06 31.15 15.39 8.39 Hearmon114

NH4-alum 1.12 25.1 10.7 8.06 Hearmon114

±0.5% ±1% ±0.7%K-alum 1.17 24.9 10.4 8.49 Hearmon114

±3% ±3% ±1%

Cubic (3m): TOE ConstantsCs-alum d111 d112 d113 d123 Haussuhl

−212 −111 −126 −90 et al.146

d144 d155 d166 d456

−27 −59 −54 −16NH4-alum d111 d112 d113 d123 Haussuhl

−75 −11 −20 −19 et al.146

d144 d155 d166 d456

−29 −56 −49 −6K-alum d111 d112 d113 d123 Haussuhl

−222 −71 −86 −134 et al.146

d144 d155 d166 d456

−23 −80 −74 −20

Chapter 4

Monofrequency SAWs in the (001) Plane

Consider the case where the crystal surface is the (001) plane, or equivalently,

the xy plane of crystalline axes. (See Figure D.1 for a diagram of this cut.) This

chapter examines the properties of surface acoustic waves in this plane for a

variety of crystals. To begin, some of the basic linear properties of surface waves

in the (001) cut are briefly reviewed. Attention is focused next on the nonlinear

properties, first on the general properties of a variety of materials, and then

on a few materials which show characteristic behaviors. To show the basic

features of the nonlinear processes most clearly, only monofrequency source

conditions are considered. Numerical simulations based on the theory presented

in Chapter 2 are used to compare and contrast the harmonic generation and

waveform distortion between the various cases considered.

4.1 Linear Effects

As mentioned in Chapter 2, crystalline anisotropy manifests itself in SAWs

even in the linear regime. The primary consequences are that (1) the wave

speed is a function of propagation direction and (2) the direction of power flow

is no longer necessarily coincident with the direction of propagation. The latter

effect is also a function of propagation direction.

The properties of linear SAWs are expounded at length in the review

by Farnell.3 Hence the discussion here is brief and discusses only the materials

that are examined later for nonlinear effects.

79

80

Farnell showed that the qualitative and quantitative variation of the

linear wave speed as a function of angle may be characterized by the anisotropy

ratio

η =2c66

c11 − c12. (4.1)

This definition is constructed such that η = 1 for an isotropic material (it is

also sometimes written with c44 instead of c66 for cubic crystals because these

constants are equivalent under cubic symmetry). Farnell also showed that

the SAW speeds for many materials conveniently group by anisotropy ratio.

Qualitatively, a higher anisotropy ratio implies a slower SAW speed relative

to the fast transverse bulk wave speed. Because of this grouping, he was able

to focus on selected materials as characteristic of many others. He chose to

highlight Ni as a characteristic material with η > 1 and KCl as a characteristic

material with η < 1. Table 4.1 lists the anisotropy ratio of many materials,

all of which also have experimentally determined TOE constants. This latter

property makes them candidates for study in the nonlinear regime.

Royer and Dieulesaint117 looked for SAW solutions in orthorhombic,

tetragonal, hexagonal, and cubic crystals under the condition that the dis-

placement of the SAW is confined to the sagittal plane. In these cases, an

analytical solution is possible (similar to the way such a solution is possible

for the propagation of Rayleigh waves in isotropic media). For cubic crystals,

Royer and Dieulesaint show that solutions with sagittal plane polarization exist

in the cases:

1. 〈100〉 direction of the (001) plane and the other cuts and directions that

are equivalent by symmetry [e.g., the 〈010〉 direction of (001) plane, the

〈010〉 direction of the (100) plane],

81

Material ηCu 3.20Ag 2.92Au 2.92Ni 2.60LiF 1.92Ge 1.66Si 1.57MgO 1.54C (diamond) 1.26Al 1.22KAl(SO4)2 · 12 H2O 1.17SrTiO3 1.15NH4Al(SO4)2 · 12 H2O 1.12CsAl(SO4)2 · 12 H2O 1.06BaF2 1.02Y3Fe5O12 0.963RbMnF3 0.851SrF2 0.803NaF 0.772NaCl 0.705Nb 0.502RbF 0.449KCl 0.373CaF2 0.373RbCl 0.312RbBr 0.286RbI 0.254

Table 4.1: Listing of selected nonpiezoelectric cubic crystals with experimen-tally determined third-order elastic constants ordered by anisotropy ratio. Theanisotropy ratio η = 2c44/(c11 − c12) was computed by using the second-orderelastic constants from Hearmon.114

82


are equivalent by symmetry [e.g., the 〈110〉 and 〈110〉 directions of the

(001) plane],


are equivalent by symmetry [e.g., the 〈110〉 direction of the (110) plane],


are equivalent by symmetry [e.g., the 〈001〉 direction of the (110) plane],

While these solutions are only a small subset of all the generalized Rayleigh

wave solutions for a given crystal, their relative simplicity allowed them to be

the first solutions studied for SAWs in crystals.∗

Royer and Dieulesaint also discussed the influence of the anisotropy

ratio η on the solutions with sagittal plane polarization in cubic and tetragonal

systems. They showed that for propagation directions where the magnitudes

of the displacement components decay as an exponentially damped sinusoid,

ui(x, z, t) = f(x, t)e−hkz cos(gkz + φ) , (4.2)

then the SAW speed c relative to the transverse bulk wave speed ct =√

c66/ρ

can be approximated by

c

ct

≈[2

η

(1− c66

c11

1

η

)]1/2

, η � 1 , (4.3)

and the decay coefficient can be written as

h ≈ (1 + c66/c11)/2η , η � 1 . (4.4)

∗Stoneley published studies of SAW propagation in the basal plane of hexagonal crys-tals147 (the plane normal to the sixfold symmetry axis) in 1949 and the 〈100〉 and 〈110〉directions in the (001) plane of cubic crystals148 in 1955.

83

Hence, for large anisotropy ratios, the surface wave travels relatively more

slowly, and the oscillations decay less rapidly. While Eqs. (4.3) and (4.4) strictly

hold only for large anisotropy ratios, Royer and Dieulesant show by a series

of illustrations that the trend of more oscillations with higher anisotropy ra-

tios also holds for values in the range 1 < η < 4. Again these results are

only necessarily true for propagation in the directions listed in the previous

paragraph.

Figure 4.1 shows the SAW speed of selected materials as a function

of the propagation direction. The direction of propagation is measured in

degrees from the 〈100〉 direction, and the speed for each material is scaled by

the characteristic speed cref = (c44/ρ)1/2 for that material. For this surface

cut, cref is also the speed of the fast transverse bulk wave and is constant

for all directions in this plane. Because the normal to the (001) plane is a

fourfold symmetry axis, the SAW speed is periodic every 90◦. In addition, the

〈110〉 direction is a twofold symmetry axis, and therefore the SAW speed is

symmetric about that direction (45◦ in the figure). In most cases (for all the

cases shown here), the speeds group by anisotropy ratio with materials, with

lower anisotropy ratios having higher relative SAW speeds. Materials with

η ≈ 1 are nearly isotropic, and hence have nearly constant SAW speed for all

directions. The dashed lines indicate the crystals in the m3 point group.

For all materials, the directions 0◦ and 45◦ from 〈100〉 are pure modes,

and the particle trajectories are confined to the sagittal plane, as described

above. Some materials have additional pure modes in other directions, but in

these cases the SAW often has a transverse component (e.g., Al, Ni, Cu, Si,

Ge). In some materials, the SAW speed approaches the transverse bulk wave

speed as the direction approaches 45◦ from 〈100〉. This is the source of the

dip in the SAW speeds of Al, Ni, Cu, Si, and Ge observed in the figure [see

84

0.6

0.7

0.8

0.9

1.0

0 10 20 30 40 50 60 70 80 90

Rel

ativ

e V

eloc

ity

Angle from ⟨100⟩ [degrees]

Relative Velocity vs. Angle in (001) planeη=0.312RbCl

CaF2 η=0.373η=0.373KCl

NaCl η=0.705SrF 2 η=0.803

η=1.02η=1.06Cs-alum

-alum4 η=1.12NHK-alum η=1.17

Al η=1.22

C η=1.26

Si η=1.57

Ge η=1.66

Ni η=2.60

Cu η=3.20

BaF2

Figure 4.1: Dependence of SAW speed on direction of propagation in the (001)plane for selected materials. The SAW speed of each material is measuredrelative to cref = (c44/ρ)1/2 and is periodic every 90◦. The dashed lines indicatecrystals in the m3 point group.

85

also Figures 4.3(d) and 4.14(d)]. The SAW then becomes an exceptional bulk

wave in the 45◦ direction. This degeneracy then gives rise to a pseudosurface

wave mode in that same direction (see Appendix B for more discussion of these

phenomena).

All of these are linear effects; they are mentioned here to set a context

for nonlinear effects to be discussed next.

4.2 Nonlinear Effects

This section shows that:

• Plots of the first few nonlinearity matrix elements as a function of direc-

tion can provide a map of the kind of waveform distortion that occurs

in various directions in the surface. This observation is advantageous

because it allows the waveform evolution to be characterized without re-

quiring the numerical integration of any nonlinear differential equations.

• The nonlinearity matrix elements sometimes change in magnitude and

sign as a function of direction. These changes are often proportionally

larger than variations of linear quantities over the same range. In exper-

imental situations where it is difficult to excite waves of large amplitude,

the nonlinear effects may be weak. In these cases large changes in non-

linearity parameters may be easier to measure than small ones.

• Directions may exist where no shock formation occurs even for finite

amplitude waves. This phenomenon arises from weak coupling between

certain harmonics, which prevents the transfer of energy to higher har-

monics necessary for shock formation.

86

• Directions exist where the energy transfer to higher harmonics occurs

more rapidly and shock formation is enhanced.

To some extent, the topics examined here parallel the review by Farnell3 and

provide the nonlinear extension to some of the cases considered there. As is

shown below, cubic crystals exhibit wide diversity in their nonlinear properties.

Fifteen crystals were chosen for study based on their crystalline symmetries

and structure. However, for purposes of simplicity, detailed studies have been

performed only on a smaller set of representative materials. In particular, Si is

studied most intensively due to its practical importance. Attention is also given

to Ni and KCl because these materials were chosen as characteristic examples

by Farnell. It proves useful to define the dimensionless nonlinearity matrix

Slm = −Slm

c44. (4.5)

The negative sign is introduced to be consistent in sign with the nonlinearity

matrix elements Rlm used for nonlinear Rayleigh waves.18 In all cases, the

figures in the remaining chapters use matrix elements defined by Eq. (4.5).

4.2.1 General Study

Figure 4.2 shows plots of the nonlinearity matrix elements for a variety of

crystals. Each graph plots S11 (solid), S12 (long dashed), and S13 (short dashed)

over the range of directions 0◦ to 45◦ from 〈100〉, where the elastic constant

c44 is the corresponding constant for each material type. The vertical scale on

each plot is adjusted to show the curves most clearly for each material. While

plotting only three elements of the nonlinearity matrix certainly does not give

a full description of the nonlinear properties of the SAW, it can give a good

idea of the evolution in many cases, as is shown below.

87

0

0.01

RbCl η=0.312

S11S12S13

-0.02

-0.01

0

0.01

KCl η=0.373

-0.02

0

0.02

0.04

0.06

0.08

NaCl η=0.705

0

0.01

CaF2 η=0.373

-0.01

0

0.01

SrF2 η=0.803

-0.06

-0.04

-0.02

0

BaF2 η=1.02

0

0.05

0.1

0.15

0 10 20 30 40

Al η=1.22

Angle from ⟨100⟩ [deg]

-0.05

0

0.05

0.1

0 10 20 30 40

Ni η=2.60


-0.05

0

0.05

0.1

0 10 20 30 40

Cu η=3.20


0.05

0.1

0.15

C η=1.26

-0.02

0

0.02

0.04

0.06

0.08

Si η=1.57

-0.02

0

0.02

0.04

0.06

0.08

Ge η=1.66

-0.01

0

0.01

0.02

Cs-alum η=1.06

-0.01

0

0.01

0.02

NH4-alum η=1.12

0.05

0.1

0.15

K-alum η=1.17

−Slm

/ c 4

4

Normalized Nonlinearity Matrix Elements in (001) Plane for Selected Materials

Figure 4.2: Dependence of nonlinearity matrix elements on direction of propa-gation in the (001) plane in selected materials.

88

Five groups of crystals are considered here, with three different materials

of similar structure in each group. The materials are ordered by increasing

anisotropy ratio from left to right within each row (see Table 4.1). In addition,

the average anisotropy ratio of each row increases from top to bottom. While

the anisotropy ratio is a linear-based quantity for measuring deviation from

isotropy, it also provides a rough guide for grouping. This may be due to

the relatively strong influence of the linear mechanical properties on nonlinear

properties in some cases. For these plots, the measured third-order elastic

constant data were taken from Prasad and Suryanarayana141 (RbCl), Drabble

and Strathen128 (KCl, NaCl), Alterovitz and Gerlich142 (CaF2), Alterovitz and

Gerlich130 (SrF2), Gerlich131 (BaF2), Grimsditch et al.143 (C), McSkimin and

Andreatch133 (Si, Ge), Thomas135 (Al), Salama and Alers136 (Ni, Cu), Haussuhl

and Preu146 (Cs-, NH4-, and K-alum). Some comparisons with other data sets

are given in Sections 4.2.2, 4.2.3, and 4.2.4.

The first two rows of Figure 4.2 show the nonlinearity matrix elements

for materials with η < 1 or η ≈ 1. The first row shows the nonlinearity matrix

elements for the chlorides RbCl, KCl, and NaCl (η = 0.312, 0.373, 0.705). These

crystals have a face-centered cubic structure, with each alkali atom surrounded

by six equidistant Cl atoms.149 The KCl and NaCl graphs show the same

decreasing trend from 0◦ to 45◦, but the RbCl graph shows the opposite. KCl

is one of the materials chosen for investigation in detail in Section 4.2.3. The

second row shows the matrix elements for the fluorites CaF2, SrF2, and BaF2

(η = 0.373, 0.803, 1.02). In these crystals the non-fluorine atom is at the center

of eight fluorine atoms positioned at the corners of the surrounding cubic cell,

and each fluorine atom has around it a tetrahedron of the non-fluorine atoms.149

All three materials show the same increasing trend from 0◦ to 45◦. Note that

while KCl and CaF2 have the same anisotropy ratio, their nonlinearity matrix

elements display a distinct difference.

89

The last three rows show the nonlinearity matrix elements of materi-

als with η > 1. The third row shows hydrous Cs-alum, NH4-alum, and K-

alum (η = 1.06, 1.12, 1.17) where a hydrous X-alum has the chemical formula

XAl(SO4)2 · 12 H2O. These materials are selected for study because they are

some of the few crystals in the m3 point group of cubics which have exper-

imental TOE constant data available. Their basic structure is described by

a metal atom or small molecule surrounded by six octahedrally coordinated

water molecules.150 Their nonlinearity matrix elements follow the same up-

ward trend as the fluorites although Cs-alum and NH4-alum look more like

SrF2, and K-alum looks more like BaF2. The fourth row shows C, Si, Ge

(η = 1.26, 1.57, 1.66). These crystals have the diamond cubic structure, in

which each atom is surrounded by its four nearest neighbors at the corners of

a regular tetrahedron.149 While all three materials show an increasing trend

from 0◦, the magnitude of the nonlinearity matrix elements in Si and Ge is

greatly decreased and changed in sign near 45◦ due to the convergence of the

SAW and bulk transverse modes in that angular region. This effect is discussed

further in Section 4.2.2, where Si is investigated in detail. Finally, the fifth row

shows Al, Ni, and Cu (η = 1.22, 2.60, 3.20). These are all face-centered cubic

crystals.149 All show an increase in the nonlinearity from 0◦ and all show a sub-

stantial decrease and change in sign near 45◦, much like Si and Ge. Section 4.2.4

investigates the properties of Ni in detail.

4.2.2 Study of Si

Figure 4.3 shows a composite plot of many of the linear and nonlinear param-

eters of SAWs in the (001) plane of Si. Because the format of this figure is

repeated throughout the remaining chapters, a general explanation is provided

here:

90

-0.03-0.02-0.01

0 0.01 0.02 0.03 0.04 0.05 0.06

0 10 20 30 40

−Slm

/c44

(a)

-180

-90

0

90

180

0 10 20 30 40

arg(

−Slm

/c44

) [d

eg]

(b)

-0.15 -0.1

-0.05 0

0.05 0.1

0.15 0.2

0.25 0.3

0 10 20 30 40

β

(c)


0.7 0.8 0.9 1

1.1 1.2 1.3 1.4 1.5 1.6

0 10 20 30 40

c/c r

ef

(d)

0

10

20

30

40

50

0 10 20 30 40

v g d

irect

ion

[deg

] (e)

0

50

100

150

200

0 10 20 30 40

x sho

ck [m

m]

(f)


0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

|Bj/B

0|

(g)

-180

-90

0

90

180

0 10 20 30 40ar

g(B

j/B0)

[deg

] (h)


Si in (001) plane

c44 = 79.2 GPacref = 5829 m/sρ = 2331 kg/m3

For xshock:v0= 40 m/sf0= 40 MHz

Figure 4.3: Dependence of nonlinearity parameters on direction of propagationin the (001) plane of Si. The direction of propagation is measured in degreesfrom the 〈100〉 direction (to the 〈110〉 direction). The parameters are periodicevery 90◦ and symmetric about the 45◦ direction. (See text for keys to thevarious graphs.)

91

• Figure 4.3(a) shows the nonlinearity matrix elements [recall Eq. (4.5)]

S11 (solid), S12 (long dashed), and S13 (short dashed) as a function of

direction. In the (001) plane, the nonlinearity matrix elements are purely

real-valued, symmetric about the direction 45◦ from 〈100〉, and periodic

every 90◦. As in Figure 4.2, the matrix elements are scaled by the SOE

constant c44 (given in the lower right corner of the figure).

• Figure 4.3(b) shows the phase φ, defined by eiφ = Slm/|Slm|, for S11

(solid), S12 (long dashed), and S13 (short dashed). The phase of φ = 0◦

indicates a positive real value, while the phase of φ = 180◦ indicates a

negative real value. (This plot is more useful for materials with complex-

valued nonlinearity matrix elements, as shown in Chapter 6.)

• Figure 4.3(c) shows the nonlinearity coefficient β = −4S11/ρc2 [Eq. (2.63)].

As explained in Section 2.2.2, the nonlinearity coefficient gives an esti-

mate of the strength of second harmonic growth in the SAW.

• Figure 4.3(d) shows the surface acoustic wave speed (solid), the shear

or quasishear bulk modes (long and short dashed), and longitudinal or

quasilongitudinal bulk mode (dotted). The wave speeds are scaled by a

reference speed cref = (c44/ρ)1/2, which is listed in the bottom right corner

(for Si, cref = 5829 m/s). The longitudinal or quasilongitudinal bulk mode

is always the fastest mode, and the SAW mode is always the slowest.

In some cases, the SAW mode becomes degenerate with a shear bulk

mode that is transversely polarized (as opposed to vertically polarized).

When they occur, these exceptional bulk waves can also be seen from

Figure 4.3(g) where the surface wave amplitudes in the longitudinal and

vertical directions go to zero.

92

• Figure 4.3(e) shows the direction of the group velocity versus the direction

of phase velocity (or, equivalently, the wave vector). The direction is given

in terms of the angle from a reference direction specified for each cut. The

dotted line indicates where the direction of the group velocity is the same

as the direction of the phase velocity and, hence, where the pure modes

occur.

• Figure 4.3(f) shows the estimate of the shock formation distance x11 cal-

culated using Eq. (2.59). The characteristic velocity and frequency are

taken to be v0 = 40 m/s and f0 = 40 MHz, which are values typical for

the experiments of Lomonosov and Hess24 using photoelastic SAW gen-

eration. However, the estimated shock formation distance for other char-

acteristic parameters can be calculated by setting up proportions with

Eq. (2.59) and the distance indicated on the graph (e.g., for 20 MHz, x11

is twice as much; for 20 m/s, x11 is half as much, etc.) Where S11 goes

through zero, the estimate of the shock formation becomes large, if not

infinite. In most cases, the other nonlinearity matrix elements do not

also go to zero in the exactly same direction. Near these directions, the

estimate of the shock formation distance given in the figure is no longer

valid.

• Figure 4.3(g) shows the amplitudes of the SAW components at the sur-

face (z = 0) of the solid in the vertical B3 (solid), longitudinal B1 (long

dashed), and transverse B2 (short dashed) directions [Eq. (2.42)]. Note

the components at each angle are normalized such that B21 +B2

2 +B23 = 1,

i.e., B0(θ) = [B1(θ)2 + B2(θ)

2 + B3(θ)2]1/2 where θ is the angle from the

reference direction. The advantage of plotting the components this way is

that it is straightforward to see how the relative component magnitudes

vary in each direction and allows for easier comparison between materi-

93

als. The disadvantage is that the absolute magnitude of the components

between directions is not seen. In particular, this difference becomes im-

portant in cases where the SAW mode converges with a transverse or

quasitransverse bulk mode, and a larger proportion of the surface energy

of the wave moves deeper into the bulk of the solid (e.g., see Figure 4.4).

• Figure 4.3(h) shows the phase of the vertical (solid), longitudinal (long

dashed), and transverse (short dashed) SAW components at the surface

(z = 0) of the solid. By convention, the phase of the longitudinal com-

ponent is set to be −90◦ for most cases (purely negative imaginary). In

a few cases (Ni and Cu), the longitudinal component changes sign. To

prevent a discontinuity in the plots containing the nonlinearity matrix el-

ements by insisting that the phase always be −90◦, the phase was allowed

to switch to +90◦ with a corresponding shift in the other components.

Figure 4.3 is somewhat redundant to the extent in that the nonlinearity coeffi-

cient and shock formation distance can be derived from the nonlinearity matrix

elements. However, they are computed here because they are quantities that

should be measurable by suitably designed experiments. In addition, the infor-

mation about the bulk and surface wave speeds and the pure mode directions

is often used to perform SAW experiments in crystals.

Now consider the properties of Si in particular.∗ Figure 4.3(a) shows

that the nonlinearity matrix elements divide the angular range into three re-

gions in terms of the angle θ between the direction of propagation and the 〈100〉direction. In the first region (0◦ ≤ θ < 20.8◦), the nonlinearity is negative. As

is shown below, this means that positive segments of the longitudinal particle

∗Some of the key results of this section have been previously reported by the author54

although in significantly less detail.

94

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30 35 40 45

|Bj|


Absolute Comparison of Surface Amplitude Components for Si (001)

|B1||B2||B3||B|

Figure 4.4: Magnitudes of the SAW components in the (001) plane of Si. Inthis figure, the components are normalized according to Eq. 2.27. This figuremay be compared with Figure 4.3(g), where each direction is normalized toitself. Here |B| = (B2

1 + B22 + B2

3)1/2 is a measure of the total amplitude of the

wave.

95

velocity waveform steepen backward in space, and negative segments steepen

forward (i.e., opposite to what a sound wave does in a fluid). In the second re-

gion (20.8◦ < θ < 32.3◦), the nonlinearity is positive, with waveform distortion

the reverse of the first region. In the third region (32.3◦ < θ ≤ 45◦), the non-

linearity is again negative although relatively weak. The standard deviation

divided by the mean for |S11|, |S12|, |S13| ranges between 85%–95% over the

angular range shown. In contrast, the standard deviation divided by the mean

for the linear wave speed is around 2% [see Figure 4.3(d)]. Hence not only do

the nonlinear matrix elements change sign (with distinctly different waveform

evolution as a result) but they also vary more widely in magnitude.

Observe that the weakening in the third region is coincident with the

gradual convergence of the SAW mode and transverse bulk mode into an ex-

ceptional bulk wave as seen from the wave speed plot in Figure 4.3(d). This

can also be seen in Figure 4.3(g) where the transverse component grows relative

to the other two. However, as described above, Figure 4.3(g) normalizes each

direction independently. Figure 4.4 shows the absolute magnitudes of the sur-

face components [only normalized by Eq. 2.27] and indicates that an increasing

amount of energy moves away from the surface and into the bulk of the solid

as the 45◦ direction is approached. As a result, it becomes increasingly difficult

to create the surface amplitudes necessary to observe nonlinear effects in this

real SAW mode.∗

Finally, Figure 4.3(a) indicates that S11, S12, S13 pass through zero near

θ ' 20.8◦ and θ ' 32.3◦. Additional calculations show that while all matrix

elements do not go through zero at the same direction, all elements are close

to zero at these angles. Hence propagation is expected to be nearly linear

∗There does, however, exist a pseudosurface wave mode at a higher wave speed.3 In thisdirection, the pseudosurface wave is the mode that is usually excited experimentally (seeAppendix B for more information on pseudosurface waves.).

96

in both these directions even for finite amplitude waves. Because harmonic

generation is suppressed, shocks do not form, or form only over large distances.

This harmonic suppression is reflected in Figure 4.3(f), which indicates vertical

asymptotes for the estimated shock formation distance. For reference purposes,

these kinds of special directions where one or more harmonics are suppressed

and shock formation does not occur are designated quasilinear directions.

To see the effects of the sign and magnitude of the nonlinearity on the

velocity waveforms, simulations were performed by integrating the system given

by Eqs. 2.40 with the monofrequency source condition

Vn(0) =

{1 for n = ±10 for n 6= ±1

. (4.6)

The equations were integrated numerically using a fixed step size, fourth-order

Runge–Kutta routine from X = 0 to X = 10 using N = 200 harmonics. Unless

otherwise indicated, the nondimensional distance X = x/x0 = 1 corresponds

to approximately one shock formation distance as computed using the scaling

factor x0 = x11 = ρc4/4|S11|ωv0 [Eq. (2.59)]. Step sizes were taken to be

sufficiently small to maintain stability, typically between ∆X = 0.002 and

∆X = 0.0005. Setting the step size smaller over the interval X = 0 to X = 0.1

tended to help keep the integration stable. The absorption value A1 = 0.025

was used in all cases. Once the spectra had been generated, the waveforms

were reconstructed using the following nondimensional form of Eq. (2.41)

Vj(0, 0, t) =

∞∑n=1

Vn(0)Bje−inωt + c.c. , (4.7)

where, as before,

Bj =3∑

s=1

β(s)j . (4.8)

97

However, only the first 150 harmonics were used to reconstruct the waveform

to minimize numerical effects due to the truncation of the spectrum. Note that

approach has no adjustable parameters, i.e., once the basic physical parameters

are known (density, SOE constants, TOE constants), then the propagation is

uniquely determined from the starting conditions.

The results for propagation in selected directions are shown in Fig-

ure 4.5. Each row represents the results of propagation in the direction speci-

fied by the angle listed in that row. The first three columns from left to right

give the longitudinal, transverse, and vertical components of the velocity. In

each direction, the waveforms are normalized such that V 2x +V 2

y +V 2z = 1 (and

hence the absolute magnitudes between directions should not be compared).

Each graph of the velocity components contains waveforms at X = 0 (short

dashed), X = 1 (long dashed), and X = 2 (solid) in the retarded time frame,

i.e., a frame moving at the linear SAW speed. The fourth column reproduces

the nonlinearity matrix elements plotted in Figure 4.3(a), with small circles

placed on each curve to indicate the value of the matrix elements in the direc-

tion specified for each row. This same format is followed for all the figures of

velocity components in Chapters 4 and 5.

Each direction has its own interesting features (from top to bottom):

1. 0◦ direction: This direction is in the first region, where most of the non-

linearity elements Slm are negative. As mentioned above, the longitudi-

nal velocity waveform exhibits distortion with the peak receding and the

trough advancing, opposite to that of a sound wave in a fluid. In addi-

tion, the longitudinal waveform exhibits the cusping near the shock front

that is characteristic of SAWs (see B.1.1). The vertical velocity waveform

also exhibits the cusped peak seen in Rayleigh waves. In fact, SAWs in

this direction are considered to be “Rayleigh-type” waves, as defined by

98

-1

-0.5

0

0.5

1

-π 0 π

0°

Vx

-1

-0.5

0

0.5

1

-π 0 π

0°

Vy

-1

0

1

2

3

4

-π 0 π

0°

Vz

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

Slm

0°

-1

-0.5

0

0.5

1

0 π 2π

20.785°

-1

-0.5

0

0.5

1

0 π 2π

20.785°

-1

-0.5

0

0.5

1

0 π 2π

20.785°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

20.785°

-1

-0.5

0

0.5

1

0 π 2π

26°

-0.5

-0.25

0

0.25

0.5

0 π 2π

26°

-4

-3

-2

-1

0

1

0 π 2π

26°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

26°

-1

-0.5

0

0.5

1

0 π 2π

32.315°

-1

-0.5

0

0.5

1

0 π 2π

32.315°

-1

-0.5

0

0.5

1

0 π 2π

32.315°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

32.315°

-0.5

-0.25

0

0.25

0.5

-π 0 π

35°

ωτ

-2

-1

0

1

2

-π 0 π

35°

ωτ

-0.5

0

0.5

1

1.5

-π 0 π

35°

ωτ

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

35°


Nor

mal

ized

Vel

ocity

Wav

efor

ms

in (

001)

Pla

ne fo

r S

i

Figure 4.5: Velocity waveforms in selected directions of propagation in the(001) plane of Si. The velocity components are normalized such that the initialamplitude in each direction is unity. (See text for keys to the various graphs.)

99

Farnell,3 for the following three reasons. First, due to the symmetries in

this direction, B3 = 0 and the motion is confined to the sagittal plane [see

Figure 4.3(g)]. Second, this is a pure mode direction∗ [see Figure 4.3(e)].

Third, the principal axis of the surface particle trajectory is perpendic-

ular to the free surface due to the 90◦ phase difference between B1 and

B3 [see Figure 4.3(h)]. Thus, except for the fact that the amplitudes of

the particle velocities do not decay purely exponentially into the solid,

propagation in this direction is quite similar to the propagation of non-

linear Rayleigh waves in isotropic materials with negative nonlinearity

coefficients (e.g., fused quartz).

2. 20.785◦ direction: This is the first example of a quasilinear direction.

Here S11 ' 2.3 × 10−6, or approximately four orders of magnitude less

than the 0◦ direction. Moreover, the ratio of |S11|/|Slm| for many of the

higher-order matrix elements is between 10−2 and 10−3 (e.g., |S11|/|S12| '0.0084). As a result, the coupling between the fundamental and higher

harmonics is very low, and harmonic generation is suppressed. Because

the characteristic length scale x0 = ρc4/4|S11|ωv0 diverges as S11 → 0,

it is no longer suitable for scaling. Instead, a characteristic length scale

was constructed by selecting another element Slm which was larger in

magnitude than the others in the range 1 < l ≤ 30, 1 < m ≤ 30. In

this particular case, S12 was chosen, and the length scale x0 = x12 =

ρc4/4|S12|ωv0 was constructed in analogy with Eq. 2.59. Even with this

adjustment, the waveforms show almost no distortion. Note that here

the characteristic length scale with v0 = 40 m/s and f0 = 40 MHz is

∗Experimentally, pure mode directions are often preferred over other directions becauseit is typically easier to make measurements when the power flow is in the same direction asthe wave vector.

100

x0 = x12 ' 1600 mm (as compared to the estimate x11 ' 190000 mm!).

Hence the propagation is essentially linear in this direction.

3. 26◦ direction: This direction is in the second region. The nonlinearity

matrix elements Slm are mostly positive, and the longitudinal velocity

waveform exhibits distortion with the peak of the wave advancing and

the trough receding. Accordingly, the vertical velocity forms a cusped

peak in the positive direction. Note that the horizontal scale has been

shifted over by π radians in these waveforms, as compared to the 0◦ case,

so that the shock would not be displayed on the boundaries of the plot.

Here is the first case where the wave has a sizable transverse component

that moves the motion out of the sagittal plane. This is also a pure mode

direction, and hence favorable for propagation measurements.

4. 32.315◦ direction: This is the second example of a quasilinear direc-

tion. Here S11 ' 1.5 × 10−5, or approximately three orders of mag-

nitude less than the 0◦ direction. Again, the ratio of |S11|/|Slm| for

many of the higher-order matrix elements is between 10−2 and 10−3 (e.g.,

|S11|/|S1,24| ' 0.0016), and harmonic generation is suppressed. In this

particular case, the element S1,24 was chosen to construct the charac-

teristic length scale x0 = x1,24 = ρc4/4|S1,24|ωv0. With v0 = 40 m/s

and f0 = 40 MHz, x0 = x1,24 ' 455 mm (as compared to the estimate

x11 ' 273000 mm). Hence the propagation is again essentially linear over

most practical length scales. Once difference between this case and the

20.785◦ direction is that here the SAW has a transverse component.

5. 35◦ direction: In the third region, the nonlinearity elements Slm are neg-

ative again, and the waveform distorts as in the 0◦ case. (Note that the

horizontal scale has been shifted over by −π radians, as compared to the

101

26◦ case, so as to be the same as in the 0◦ case.) However, the magnitude

of the nonlinearity is weaker here, with S11(35◦)/S1(0◦) ' 0.14. Note

that a shift of only a few degrees from the 32.315◦ direction leads to a

significantly different type of waveform distortion.

Thus, the simulations for Si presented above demonstrate both that the SAW

nonlinearity varies significantly in magnitude and direction throughout the

(001) cut and that the nonlinearity matrix elements provide a “map” which

can characterize the nature of the waveform distortion.

Figures 4.6 and 4.7 provide the displacement waveforms and particle

trajectories corresponding to the velocity waveforms in Figure 4.5 and follow

the same format, with plots at X = 0 (short dashed), X = 1 (long dashed),

and X = 2 (solid). The displacement components in Figure 4.6 are computed

by integrating the velocity components over one cycle and, hence, are scaled by

u0 = v0/ω0. As such, they follow the same patterns as the velocity waveforms,

and are included primarily for completeness. Note, however, that shocks and

sharp peaks in the velocity waveforms become sharp peaks and shocks, respec-

tively, in the displacement waveforms. The particle trajectories in Figure 4.6

are constructed from the displacement waveforms. While the particle trajecto-

ries are always confined to a plane, that plane is generally rotated out of the

sagittal plane (e.g., Figure 2.4 shows a plane rotated about the z axis). The

motion in Figure 4.7 is shown in terms of projections into the xy plane (top

view, looking from z > 0), xz plane (side view, looking from x > 0), and yz

plane (front view, looking from y > 0). The motion is initially elliptical in

shape but distorts into an egg-like shape as it propagates, similar to those of

Rayleigh waves.23 The particle trajectories also show clearly the convergence

of the SAW and transverse bulk wave modes. In particular, the top and front

views of the 35◦ direction indicate that the transverse component has started to

102

-0.6

-0.4

-0.2

0

-π 0 π

0°

Ux

-0.4

-0.2

0

0.2

0.4

-π 0 π

0°

Uy

-0.4

-0.2

0

0.2

0.4

-π 0 π

0°

Uz

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

Slm

0°

-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0

0.1

0 π 2π

20.785°

-0.4

-0.2

0

0.2

0.4

0 π 2π

20.785°

-0.4

-0.2

0

0.2

0.4

0 π 2π

20.785°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

20.785°

-0.6

-0.4

-0.2

0

0 π 2π

26°

-0.4

-0.2

0

0.2

0.4

0 π 2π

26°

-0.4

-0.2

0

0.2

0.4

0 π 2π

26°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

26°

-0.4

-0.2

0

0.2

0.4

0 π 2π

32.315°

-1

-0.5

0

0 π 2π

32.315°

-0.4

-0.2

0

0.2

0.4

0 π 2π

32.315°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

32.315°

-0.4

-0.2

0

0.2

0.4

-π 0 π

35°

ωτ

-1

-0.5

0

-π 0 π

35°

ωτ

-0.4

-0.2

0

0.2

0.4

-π 0 π

35°

ωτ

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

35°


Nor

mal

ized

Dis

plac

emen

t Wav

efor

ms

in (

001)

Pla

ne fo

r S

i

Figure 4.6: Displacement waveforms in selected directions of propagation in the(001) plane of Si. The displacement components are computed by integratingthe velocity waveforms of Figure 4.5 over one cycle. (See text for keys to thevarious graphs.)

103

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

0°

Top View

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

0°

Side View

-0.5

-0.25

0

0.25

0.5

-0.5 -0.25 0 0.25 0.5

0°

Front View

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

Slm

0°

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

20.785°

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

20.785°

-0.5

-0.25

0

0.25

0.5

-0.5 -0.25 0 0.25 0.5

20.785°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

20.785°

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

26°

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

26°

-0.5

-0.25

0

0.25

0.5

-0.5 -0.25 0 0.25 0.5

26°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

26°

-1

-0.75

-0.5

-0.25

0

-0.5-0.25 0 0.25 0.5

32.315°

-0.5

-0.25

0

0.25

0.5

-0.5-0.25 0 0.25 0.5

32.315°

-0.5

-0.25

0

0.25

0.5

-1 -0.75-0.5-0.25 0

32.315°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

32.315°

-1

-0.75

-0.5

-0.25

0

-0.5-0.25 0 0.25 0.5

35°

-0.5

-0.25

0

0.25

0.5

-0.5-0.25 0 0.25 0.5

35°

-0.5

-0.25

0

0.25

0.5

-1 -0.75-0.5-0.25 0

35°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

35°


Nor

mal

ized

Par

ticle

Tra

ject

orie

s in

(00

1) P

lane

for

Si

Figure 4.7: Particle trajectories in selected directions of propagation in the(001) plane of Si. The particle trajectories are constructed from the displace-ment waveforms in Figure 4.6. In all cases, the direction of motion is counter-clockwise in side view. (See text for keys to the various graphs.)

104

exceed the other components. In both the displacement waveforms and particle

trajectories, the change in magnitude is primarily due to absorption.

Figure 4.8 shows the frequency spectra and harmonic propagation curves

for this cut. The spectra correspond to the amplitudes |Vn|/|V1(X = 0)| in dB

of each spectral component as a function of the harmonic number n. Spectra

are plotted at X = 0 (short dashed), X = 1 (long dashed), and X = 2

(solid) and, therefore, correspond to the waveforms and particle trajectories in

Figures 4.5, 4.6, and 4.7. The harmonic propagation curves plot the spectral

amplitudes |V1| (solid), |V2| (long dashed), |V3| (short dashed), |V4| (dotted),

|V5| (dot-dashed) as a function of the dimensionless propagation distance from

X = 0 to X = 10. The spectra and propagation curves for the 0◦, 26◦,

and 35◦ cases are similar to those produced by nonlinear Rayleigh waves18

and nonlinear SAWs in the (111) plane of KCl.33 Initially the amplitude of

the fundamental drops and the amplitudes of the higher harmonics rise as

harmonic generation occurs. Subsequently the amplitudes all decrease due

to the effects of absorption. In contrast, the quasilinear directions 20.785◦

and 32.315◦ exhibit clearly the severe harmonic suppression. The higher order

amplitudes are suppressed by four orders of magnitude or more in the spectra

(which has its vertical scale expanded down to −200 dB). Only the fundamental

mode is large enough to appear in the harmonic propagation curves. Note also

that the fundamental decreases more slowly than in the nonquasilinear cases,

with nearly all of the decrease attributable to absorption.

To examine the effects of using the same SOE constants but different

TOE constants, the nonlinearity matrix elements S11, S12, and S13 correspond-

ing to the data of Drabble and Gluyas144 (D&G) are plotted together with

those of McSkimin and Andreatch133 (M&A) in Figure 4.9. While the ampli-

tudes vary at each angle, the overall trend of three regions of nonlinearity still

105

-80

-40

0

0 50 100 150

0°

Spectrum [dB]

0

0.5

1

0 2 4 6 8 10

0°

Harmonic Propagation

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

Slm

0°

-200

-160

-120

-80

-40

0

0 50 100 150

20.785°

0

0.5

1

0 2 4 6 8 10

20.785°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

20.785°

-80

-40

0

0 50 100 150

26°

0

0.5

1

0 2 4 6 8 10

26°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

26°

-200

-160

-120

-80

-40

0

0 50 100 150

32.315°

0

0.5

1

0 2 4 6 8 10

32.315°

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

32.315°

-80

-40

0

0 50 100 150

35°

n

0

0.5

1

0 2 4 6 8 10

35°

x/x0

-0.02

0

0.02

0.04

0.06

0.08

0 10 20 30 40

35°


Spe

ctra

and

Har

mon

ic P

ropa

gatio

n C

urve

s in

(00

1) P

lane

for

Si

Figure 4.8: Frequency spectra and harmonic propagation curves for selecteddirections of propagation in the (001) plane of Si. The initial amplitude of thefundamental is used as the reference amplitude in the spectra. (See text forkeys to the various graphs.)

106

-0.02

0

0.02

0.04

0.06

0 5 10 15 20 25 30 35 40 45

−Slm

/c44


Comparison of Selected Nonlinearity Matrix Elements

S11

S12

S13

McSkimin and AndreatchDrabble and Strathen

Figure 4.9: Comparison of selected nonlinearity matrix elements calculatedfrom TOE constant data of McSkimin and Andreatch133 and Drabble andGluyas144 for propagation in the (001) plane of Si. SOE constant data aretaken from Hearmon114 in both cases.

107

holds. The largest disagreement appears at the extrema. Note also that the

position of the quasilinear directions has shifted a few degrees to the right.

Both sets show directions where the matrix elements cross, although in the

D&G case the crossings do not occur as close to zero as in the M&A case.

In the D&G case, the crossing of the matrix elements occurs at more widely

separated locations. Examination of the nonlinearity matrix elements for other

crystals in Figure 4.2 shows that this separation is probably more common than

the simultaneous crossings of the M&A case. This separated crossing changes

the character of the harmonic generation and waveform distortion as is demon-

strated in the KCl and Ni sections below. Thus, Figure 4.9 indicates that the

detailed waveform predictions for the 20.785◦ and 32.315◦ directions may not

precisely match experimental results. In other words, while the nonlinearity

may be weak in these directions, the resulting waveforms may not be as close to

linear as Figure 4.5 would indicate. Nevertheless, these simulations serve well

the purpose of demonstrating the phenomenon of almost complete suppression

of harmonic generation due to nonlinear effects.

4.2.3 Study of KCl

Additional phenomena can be observed in KCl. Because the linear properties

of KCl are discussed in detail in the review by Farnell,3 they only are discussed

here in so far as they affect the nonlinear properties. Figure 4.10 shows the

nonlinear and linear parameters for KCl in the same format as Figure 4.3 for

Si. As compared to Figure 4.3(a), Figure 4.10(a) for KCl is quantitatively and

qualitatively different. At 0◦, S11 and S12 are positive while S13 and most of

the other elements Slm are negative. As the angle from 〈100〉 increases, S12

goes through zero around 3.205◦ and then finally S11 goes through zero around

5.24◦. At larger angles, most of the matrix elements Slm are negative, although,

108

-0.02

-0.015

-0.01

-0.005

0

0.005

0 10 20 30 40

−Slm

/c44

(a)

-180

-90

0

90

180

0 10 20 30 40

arg(

−Slm

/c44

) [d

eg]

(b)

-0.05-0.04-0.03-0.02-0.01

0 0.01 0.02 0.03

0 10 20 30 40

β

(c)


0.8 1

1.2 1.4 1.6 1.8 2

2.2 2.4 2.6

0 10 20 30 40

c/c r

ef

(d)

0

10

20

30

40

50

0 10 20 30 40

v g d

irect

ion

[deg

] (e)

0

20

40

60

80

100

0 10 20 30 40

x sho

ck [m

m]

(f)


0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

|Bj/B

0|

(g)

-180

-90

0

90

180

0 10 20 30 40ar

g(B

j/B0)

[deg

] (h)


KCl in (001) plane

c44 = 6.3 GPacref = 1775 m/sρ = 1989 kg/m3


Figure 4.10: Dependence of nonlinearity parameters on direction of propagationin the (001) plane of KCl. The direction of propagation is measured in degreesfrom the 〈100〉 direction (to the 〈110〉 direction). The elements are periodicevery 90◦ and symmetric about the 45◦ direction. As discussed in Section 4.2.3,the estimates of the shock formation distance in Figure 4.10(f) are not accuratefor most directions. (See text for keys to the various graphs.)

109

unlike Si, here the magnitude of S11 is less than the magnitudes of S12, S13,

and other matrix elements. Unlike Si, the SAW modes do not converge with

the transverse bulk mode at 45◦. Finally, the standard deviation divided by

the mean for |S11|, |S12|, |S13| ranges between 38%–45% over the angular range

shown. In contrast, the standard deviation divided by the mean for the linear

wave speed is around 0.2% [see Figure 4.3(d)].

Figures 4.11 and 4.12 show the velocity waveforms, frequency spectra,

and harmonic propagation curves for KCl. Several directions are selected for

consideration in detail:

1. 0◦ direction: In this direction, S11 and S12 are positive, and S13 '7.0 × 10−4 is negative but close to zero and smaller than S11 and S12

in magnitude (|S13|/|S11| ' 0.14, |S13|/|S12| ' 0.42). The matrix ele-

ment S13 describes the coupling strength between the fundamental and

third harmonic to generate the fourth harmonic. This suppression of

fourth harmonic generation can be seen in the harmonic propagation

curves in Figure 4.12. Energy that is transferred to the third harmonic

from lower harmonics is not as easily transferred to the fourth harmonic.

The third harmonic (short dashed) curve exceeds the second harmonic

(long dashed) in amplitude around X = 3, while the fourth harmonic

grows more slowly initially (as compared to the propagation curves for

the nonquasilinear directions in Si). The trapping of energy in the lowest

harmonics is also reflected in the multiple peaks in the frequency spec-

trum (note expanded vertical scale to −200 dB). Another consequence of

a small value of S13 is that shock formation does not occur. For example,

while the longitudinal waveform shown in Figure 4.12 is distorting in a

“positive” way, with peaks advancing and troughs receding, it has not

yet formed a shock. Additional calculations indicate that at distances

110

-1

-0.5

0

0.5

1

0 π 2π

0°

Vx

-0.5

-0.25

0

0.25

0.5

0 π 2π

0°

Vy

-2

-1

0

1

0 π 2π

0°

Vz

-0.02

-0.01

0

0.01

0 10 20 30 40

Slm

0°

-1

-0.5

0

0.5

1

0 π 2π

3.205°

-0.5

-0.25

0

0.25

0.5

0 π 2π

3.205°

-2

-1

0

1

0 π 2π

3.205°

-0.02

-0.01

0

0.01

0 10 20 30 40

3.205°

-1

-0.5

0

0.5

1

0 π 2π

5.24°

-0.5

-0.25

0

0.25

0.5

0 π 2π

5.24°

-1

-0.5

0

0.5

1

0 π 2π

5.24°

-0.02

-0.01

0

0.01

0 10 20 30 40

5.24°

-1

-0.5

0

0.5

1

-π 0 π

10°

-0.5

-0.25

0

0.25

0.5

-π 0 π

10°

-1

0

1

2

3

4

5

-π 0 π

10°

-0.02

-0.01

0

0.01

0 10 20 30 40

10°

-1-0.75 -0.5

-0.25 0

0.25 0.5

0.75 1

-π 0 π

45°

ωτ

-0.5

-0.25

0

0.25

0.5

-π 0 π

45°

ωτ

-1

0

1

2

3

4

5

-π 0 π

45°

ωτ

-0.02

-0.01

0

0.01

0 10 20 30 40

45°


Nor

mal

ized

Vel

ocity

Wav

efor

ms

in (

001)

Pla

ne fo

r K

Cl

Figure 4.11: Velocity waveforms in selected directions of propagation in the(001) plane of KCl. The velocity components are normalized such that initialamplitude in each direction is unity. (See text for keys to the various graphs.)

111

-200

-160

-120

-80

-40

0

0 50 100 150

0°

Spectrum [dB]

0

0.5

1

0 2 4 6 8 10

0°


-0.02

0

0 10 20 30 40

Slm

0°

-200

-160

-120

-80

-40

0

0 50 100 150

3.205°

0

0.5

1

0 2 4 6 8 10

3.205°

-0.02

-0.01

0

0.01

0 10 20 30 40

3.205°

-200

-160

-120

-80

-40

0

0 50 100 150

5.24°

0

0.5

1

0 2 4 6 8 10

5.24°

-0.02

-0.01

0

0.01

0 10 20 30 40

5.24°

-80

-40

0

0 50 100 150

10°

0

0.5

1

0 2 4 6 8 10

10°

-0.02

-0.01

0

0.01

0 10 20 30 40

10°

-80

-40

0

0 50 100 150

45°

n

0

0.5

1

0 2 4 6 8 10

45°

x/x0

-0.02

-0.01

0

0.01

0 10 20 30 40

45°


Spe

ctra

and

Har

mon

ic P

ropa

gatio

n C

urve

s in

(00

1) P

lane

for

KC

l

Figure 4.12: Frequency spectra and harmonic propagation curves for selecteddirections of propagation in the (001) plane of KCl. The initial amplitude ofthe fundamental is used as the reference amplitude in the spectra. (See textfor keys to the various graphs.)

112

X > 2 the longitudinal velocity waveform steepens more but does not

ever form a shock. Hence this is another quasilinear direction, although

with a somewhat different character than those described for Si because

the harmonic suppression is less severe and occurs in a harmonic higher

than the second.

2. 3.205◦ direction: Here S11 is positive, S12 ' −5.9×10−7 is close zero, and

S13 and most other matrix elements Slm are negative. In addition S12 is

smaller than S11 in magnitude (|S13|/|S11| ' 2.1× 10−4). The matrix el-

ement S12 describes the coupling strength between the fundamental and

second harmonic to generate the third harmonic. The suppression of third

harmonic generation can be seen in the harmonic propagation curves in

Figure 4.12. The second harmonic (long dashed) grows to be larger in

amplitude than the fundamental (solid) around X = 2, and the third and

higher harmonics grow more slowly initially. The trapping of energy in

the lowest harmonics leads to other unusual phenomena, including the

amplitude of the third harmonic (short dashed) exceeding the fundamen-

tal past X = 5, and the suppression of fourth harmonic (dotted) around

the same location. The complicated interaction of the lowest harmonics

is also reflected in the spectra which have many peaks and valleys at

both X = 1 (long dashed) and X = 2 (short dashed). The waveforms

of Figure 4.11 do not exhibit shock formation but do show some higher

frequency oscillations due to the atypical energy trapping and unusual

spectral structure. (The shown oscillations are not due to spectrum trun-

cation or Gibbs oscillations from the numerical calculations.) Additional

calculations show that at distances X > 2 the waveform does not form

a shock, and the higher frequency oscillations grow in extent and magni-

tude. As mentioned in Section 2.2.4, the amplitudes of the fundamental

113

and second harmonic for this direction can be modelled approximately

for distances X ≤ 2 by a coupled two-mode system.33 In summary, the

simulations identify the 3.205◦ direction as another quasilinear direction,

although with a still different character than either of those described for

Si or for the 0◦ case in KCl above.

3. 5.24◦ direction: This is another example of a quasilinear direction like

the cases of Si. The matrix element S11 ' −1.249× 10−6 is close to zero

and small compared to neighboring elements (e.g., |S11|/|S12| ' 0.0006).

Hence second harmonic generation is suppressed and propagation is es-

sentially linear. In this particular case, the element S1,8 was chosen to

construct the characteristic length scale x0 = x1,8 = ρc4/4|S1,8|ωv0. With

v0 = 40 m/s and f0 = 40 MHz, x0 = x1,8 ' 12 mm (as compared to the

estimate x11 ' 59000 mm).

4. 10◦ direction: Here most of the nonlinearity matrix elements Slm are

negative, but, unlike the 0◦ and 35◦ directions in Si, here the magni-

tude |S11| is less than neighboring elements (e.g., |S11|/|S12| ' 0.84 and

|S11|/|S12| ' 0.68). As can be seen from the spectrum, this inversion in

the magnitudes of the matrix elements causes energy to be transferred

to the higher harmonics at a faster rate (compared the X = 1 curves in

Si to this case). In turn, this increased rate of energy transfer results in

significantly sharper cusping in the waveforms.

5. 45◦ direction: This case is very similar to the 10◦ direction. However,

propagation in this direction occurs in a pure mode (see Figure 4.10(e)).

In summary, the simulations for KCl in the (001) plane indicate that (1) several

different varieties of quasilinear directions exist and (2) care should be taken

with the simple estimate of the shock formation distance given by Eq. (2.59)

114

for all directions in this cut. Note that in both quasilinear cases (e.g., 0◦ and

3.205◦ directions) and matrix element inversion cases (e.g., 10◦ and 45◦) the

simplistic estimate of the shock formation distance x11 = ρc4/4|S11|ωv0 proba-

bly underestimates or overestimates, respectively, the characteristic nonlinear

length scale. Therefore, the values given in Figure 4.10(f) are not accurate for

most directions.

Finally, the effect of different experimental TOE constants is examined.

Figure 4.13 shows a comparison between the nonlinearity elements computed

using the data of Drabble and Strathen128 (used in the simulations above)

and Chang.127 Both sets show the same trend decreasing from positive to

negative, but the location of the zero crossings shifts to the left. In particular,

the 0◦ direction with the Chang data has the same character as the 3.205◦

case with the Drabble and Strathen data. Because Hamilton et al.33 used

the Chang data to perform their simulations for KCl, their paper shows these

more unusual waveforms and spectra in the 0◦ direction. An examination of

Chang’s paper shows that only three of the six TOE constants were determined

experimentally (d144, d155, d456) while the other elastic constants (d111, d122,

d123) were computed by assuming the Cauchy relations d123 = d456 = d144 and

d112 = d166. In contrast, Drabble and Strathen measured all six constants and

showed that not all the Cauchy relations hold (see Table 3.4). In any case,

the lesson here is that the nonlinearity matrix elements and, therefore, the

waveform distortion are fairly sensitive to changes in the TOE constants. Thus

care should be exercised in making detailed predictions about specific directions

without good TOE constant data.

115

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0 5 10 15 20 25 30 35 40 45

−/c

44


Comparison of SelectedNonlinearity Matrix Elements

Chang

Slm

S11

S12

S13

Drabble and Strathen

Figure 4.13: Comparison of selected nonlinearity matrix elements calculatedfrom TOE constant data of Drabble and Strathen128 and Chang127 for propa-gation in the (001) plane of KCl. SOE constant data are taken from Hearmon114

in both cases.

116

4.2.4 Study of Ni

Lastly, consider Ni. As before, because the linear properties of Ni are discussed

in detail in the review by Farnell,3 they only are discussed briefly here. Fig-

ure 4.14 shows the linear and nonlinear properties of SAWs in Ni. Qualitatively,

the properties of the nonlinearity matrix elements plotted in Figure 4.3(a) is

more like Si than KCl. Like Si, the SAW mode approaches the transverse bulk

mode as the propagation direction approaches 45◦ from 〈100〉. Unlike Si, Ni

only has one region where the matrix elements Slm are predominantly nega-

tive. Finally, the standard deviation divided by the mean for |S11|, |S12|, |S13|ranges between 55%–57% over the angular range shown. In contrast, the stan-

dard deviation divided by the mean for the linear wave speed is around 7% [see

Figure 4.14(d)].

Figures 4.15 and 4.16 show velocity waveforms, frequency spectra, and

harmonic propagation curves for Ni. Several directions are selected for consid-

eration in detail:

1. 0◦ direction: Here the nonlinearity elements Slm are predominantly pos-

itive. In the longitudinal velocity waveform, the peaks advance and the

troughs recede, while in the vertical velocity waveform a sharp negative

peak forms. This is “positive” distortion of the type seen in the 26◦

direction in Si, although the propagation here is of the “Rayleigh-type.”

2. 21◦ direction: This direction is very similar to the previous one. While the

wave is no longer of the “Rayleigh-type,” it is still a pure mode direction.

3. 25.15◦ direction: In this direction, S11 and S12 are positive but S13 '2.8× 10−5 is close to zero. This causes the generation of the fourth har-

monic to be suppressed, much like the 0◦ quasilinear direction in KCl.

117

-0.04-0.02

0 0.02 0.04 0.06 0.08

0 10 20 30 40

−Slm

/c44

(a)

-180

-90

0

90

180

0 10 20 30 40

arg(

−Slm

/c44

) [d

eg]

(b)

-0.4 -0.3 -0.2 -0.1 0

0.1 0.2 0.3 0.4 0.5 0.6

0 10 20 30 40

β

(c)


0.6 0.8 1

1.2 1.4 1.6 1.8

0 10 20 30 40

c/c r

ef

(d)

-10 0

10 20 30 40 50

0 10 20 30 40

v g d

irect

ion

[deg

] (e)

0

5

10

15

20

0 10 20 30 40

x sho

ck [m

m]

(f)


0

0.2

0.4

0.6

0.8

1

0 10 20 30 40

|Bj/B

0|

(g)

-180

-90

0

90

180

0 10 20 30 40ar

g(B

j/B0)

[deg

] (h)


Ni in (001) plane

c44 = 122.0 GPacref = 3700 m/sρ = 8912 kg/m3


Figure 4.14: Dependence of nonlinearity parameters on direction of propagationin the (001) plane of Ni. The direction of propagation is measured in degreesfrom the 〈100〉 direction (to the 〈110〉 direction). The parameters are periodicevery 90◦ and symmetric about the 45◦ direction. (See text for keys to thevarious graphs.)

118

-1.5

-1

-0.5

0

0.5

1

1.5

0 π 2π

0°

Vx

-1

-0.5

0

0.5

1

0 π 2π

0°

Vy

-4

-3

-2

-1

0

1

0 π 2π

0°

Vz

-0.05

0

0.05

0.1

0 10 20 30 40

Slm

0°

-1.5

-1

-0.5

0

0.5

1

1.5

0 π 2π

21°

-0.5

-0.25

0

0.25

0.5

0 π 2π

21°

-4

-3

-2

-1

0

1

0 π 2π

21°

-0.05

0

0.05

0.1

0 10 20 30 40

21°

-1

-0.5

0

0.5

1

0 π 2π

25.15°

-1

-0.5

0

0.5

1

0 π 2π

25.15°

-2

-1

0

1

0 π 2π

25.15°

-0.05

0

0.05

0.1

0 10 20 30 40

25.15°

-1

-0.5

0

0.5

1

0 π 2π

26.3°

-1

-0.5

0

0.5

1

0 π 2π

26.3°

-1

-0.5

0

0.5

1

0 π 2π

26.3°

-0.05

0

0.05

0.1

0 10 20 30 40

26.3°

-1

-0.5

0

0.5

1

-π 0 π

29°

ωτ

-2

-1

0

1

2

-π 0 π

29°

ωτ

-1

0

1

2

3

-π 0 π

29°

ωτ

-0.05

0

0.05

0.1

0 10 20 30 40

29°


Nor

mal

ized

Vel

ocity

Wav

efor

ms

in (

001)

Pla

ne fo

r N

i

Figure 4.15: Velocity waveforms in selected directions of propagation in the(001) plane of Ni. The velocity components are normalized such that initialamplitude in each direction is unity. (See text for keys to the various graphs.)

119

-80

-40

0

0 50 100 150

0°

Spectrum [dB]

0

0.5

1

0 2 4 6 8 10

0°


-0.05

0

0.05

0.1

0 10 20 30 40

Slm

0°

-80

-40

0

0 50 100 150

21°

0

0.5

1

0 2 4 6 8 10

21°

-0.05

0

0.05

0.1

0 10 20 30 40

21°

-200

-160

-120

-80

-40

0

0 50 100 150

25.15°

0

0.5

1

0 2 4 6 8 10

25.15°

-0.05

0

0.05

0.1

0 10 20 30 40

25.15°

-200

-160

-120

-80

-40

0

0 50 100 150

26.3°

0

0.5

1

0 2 4 6 8 10

26.3°

-0.05

0

0.05

0.1

0 10 20 30 40

26.3°

-80

-40

0

0 50 100 150

29°

n

0

0.5

1

0 2 4 6 8 10

29°

x/x0

-0.05

0

0.05

0.1

0 10 20 30 40

29°


Spe

ctra

and

Har

mon

ic P

ropa

gatio

n C

urve

s in

(00

1) P

lane

for

Ni

Figure 4.16: Frequency spectra and harmonic propagation curves for selecteddirections of propagation in the (001) plane of Ni. The initial amplitude of thefundamental is used as the reference amplitude in the spectra. (See text forkeys to the various graphs.)

120

Additional calculations indicate that at distances X > 2 the longitudinal

velocity waveform steepens more but does not form a shock. The har-

monic propagation curves for lowest order harmonics show an increase in

magnitude in the region 0 < X < 3 due to trapping that results from the

suppression, although eventually the third and fourth harmonics reach

the same magnitude.

4. 26.3◦ direction: This is another example of the quasilinear directions like

those seen in the 20.785◦ and 32.315◦ directions in Si. Here the ma-

trix element S11 = 3.3 × 10−5, and it is small compared to neighboring

elements (e.g., |S11|/|S12| ' 0.0047). As a result, second harmonic gener-

ation is suppressed and the propagation is nearly linear. In this particular

case, the element S1,4 was chosen to construct the characteristic length

scale x0 = x1,8 = ρc4/4|S1,8|ωv0 for the simulation. With v0 = 40 m/s

and f0 = 40 MHz, x0 = x1,8 ' 8.6 mm (as compared to the estimate

x11 ' 3100 mm).

5. 29◦ direction: Here the matrix elements Slm are predominantly negative

and the motion is mostly in the transverse and vertical directions, similar

to the 35◦ direction in Si. (Note that the horizontal scale has been shifted

over by −π radians as compared to the other cases.)

To summarize, this case shows many of the same nonlinear features as seen in

the Si and KCl cases. These similarities occur despite the fact that (1) Si has a

diamond crystalline structure while Ni does not, and (2) KCl has an anisotropy

ratio η < 1 while Ni has anisotropy ratio η < 1. As before, the estimates of

the shock formation distance given in Figure 4.14(f) may be inaccurate in the

vicinity of quasilinear directions like the 25.15◦ direction. However, in contrast

to KCl, the estimates for Ni should be valid for most directions in the plane.

121

Finally, the effect of different experimental TOE constants is examined.

Figure 4.17 shows a comparison between the nonlinearity elements computed

using the data of Salama and Alers136 (used in the simulations above) and

Sarma and Reddy.137 While the curves show the same overall trend, the devi-

ation is larger than in the previous cases. The deviation may be due at least

in part to the fact that data of Sarma and Reddy were taken while the sample

was magnetically saturated.

4.3 Summary

This chapter has investigated the properties of nonlinear SAWs in the (001)

plane of selected crystals. Si, KCl, and Ni were chosen for detailed study. It

is found that the nature of the nonlinearity is often very sensitive to changes

in direction, and the magnitude of the changes can be larger than linear quan-

tities like the wave speed. For most cases, plotting the first few nonlinearity

matrix elements as a function of direction can provide a guide to the nature

of the nonlinear effects. While particular directions of high symmetry exhibit

harmonic generation and waveform distortion similar to Rayleigh waves, sev-

eral other new effects have been identified including (1) the existence of regions

of “positive” and “negative” nonlinearity within the same cut, (2) several va-

rieties of quasilinear directions where the generation of particular harmonics is

suppressed and shock formation does not occur, (3) directions in which shock

formation is enhanced by rapid transfer of energy to the higher harmonics. As

a result of the latter two effects, the simple estimate of the shock formation

given by Eq. (2.59) may not be valid or accurate in some regions. The choice of

different experimental TOE constants as input to the simulations is shown to

affect the detailed predictions of the nonlinearity matrix elements in any given

direction, but not the trends over the whole angular range. Many of the ideas

122

0

0.05

0.1

0.15

0.2

0 5 10 15 20 25 30 35 40 45

−Slm

/c44

Angle from ⟨100

S

S

S

⟩

11

12

13

[degrees]

Comparison of Selected Nonlinearity Matrix Elements

Salama and AlersSarma and Reddy

Figure 4.17: Comparison of selected nonlinearity matrix elements calculatedfrom TOE constant data of Salama and Alers136 and Sarma and Reddy137

for propagation in the (001) plane of Ni. SOE constant data is taken fromHearmon114 in both cases. Note that the sample in the data from Sarma andReddy was magnetically saturated.

123

presented in this chapter are further investigated in the next chapter with a

different cut.

Chapter 5


This chapter describes the propagation of finite amplitude SAWs in the (110)

surface cut. (See Figure D.1 for a diagram of this cut.) The chapter applies the

same type of analysis used in Chapter 4 to describe and explain the linear and

nonlinear properties of surface waves in this plane. Unless otherwise indicated,

the figures in the chapter have the same format as in Chapter 4, and the reader

is referred to that chapter for detailed explanations. The structure of Chapter 4

is also paralleled: (1) a brief review of linear properties, (2) a general study of

a variety of crystals, and (3) a detailed study of Si, KCl, and Ni.

5.1 Linear Effects

Figure 5.1 shows the SAW speed of selected materials as a function of the

propagation direction. The direction of propagation is measured in degrees

from the 〈001〉 direction, and the speed for each material is scaled by the

characteristic speed cref = (c44/ρ)1/2 for that material. Because the normal to

the (110) plane is a twofold symmetry axis, the SAW speed is periodic every

180◦. In addition, the the 〈110〉 direction is a twofold symmetry axis so that

the SAW speed is symmetric about that direction (90◦ in the figure). In most

cases (and for all the cases shown here), the speeds group by anisotropy ratio,

with materials with lower anisotropy ratios having higher relative SAW speeds.

Materials with η ≈ 1 are nearly isotropic, and hence have nearly constant

SAW speed for all directions. The dashed lines indicate the crystals in the

124

125

0.6

0.7

0.8

0.9

1.0

1.1

1.2

0 20 40 60 80 100 120 140 160 180

Rel

ativ

e V

eloc

ity


Relative Velocity vs. Angle in (110) plane

η=0.312RbClCaF2 η=0.373

η=0.373KCl

NaCl η=0.705SrF 2 η=0.803

η=1.02BaF2η=1.06Cs-alum


Al η=1.22

C η=1.26

Si η=1.57Ge η=1.66

Ni η=2.60

Cu η=3.20

Figure 5.1: Dependence of SAW speed on direction of propagation in the (110)plane for selected materials. The SAW speed of each material is measuredrelative to cref = (c44/ρ)1/2, and the speed is periodic every 180◦. The dashedlines indicate crystals in the m3 point group.

126

m3 point group. According to Farnell,3 SAWs propagating in the 0◦ direction

are “Rayleigh-type” waves (like the direction 0◦ from 〈100〉 in Si; see also

Section B.1.3), whereas SAWs in the 90◦ direction are actually simple Rayleigh

waves (“Rayleigh-type” plus non-oscillatory decay into the solid). As a result,

both the 0◦ and 90◦ directions are pure modes. Pure modes may also exist in

other directions depending on the individual crystal.


This section shows that:

• The nonlinearity changes in magnitude and sign as a function of direction.

These changes are often proportionally larger than variations of linear

quantities over the same range.

• Quasilinear directions exist where no shock formation occurs even for

finite amplitude waves.

• Directions exist where shock formation is enhanced.

These properties are similar in many ways to those in the (001) plane.

5.2.1 General Study

The same materials are considered here as in Chapter 4. Figure 5.2 displays

the nonlinearity matrix elements for all the materials except the alums, which

are shown in Figure 5.3.

As before, Figure 5.2 shows S11 (solid), S12 (long dashed), and S13 (short

dashed) over the range of directions 0◦ to 45◦ from 〈100〉. The vertical scale on

each plot is adjusted to show the curves most clearly for each material. The

127

-0.3

-0.2

-0.1

0

0.1

RbCl η=0.312

0

0.1

0.2

0.3

0.4

KCl η=0.373S11S12S13

0

0.1

0.2

NaCl η=0.705

-0.05

0

0.05

CaF2 η=0.373

0

0.01

SrF2 η=0.803

-0.06

-0.04

-0.02

BaF2 η=1.02

0

0.05

0.1

0.15

0 20 40 60 80

Al η=1.22


0

0.05

0.1

0.15

0 20 40 60 80

Ni η=2.60


0

0.05

0.1

0.15

0.2

0.25

0 20 40 60 80

Cu η=3.20


0

0.05

0.1

0.15

C η=1.26

0

0.05

0.1

Si η=1.57

0

0.05

0.1

Ge η=1.66

−Slm

/ c 4

4Normalized Nonlinearity Matrix Elements in (110) Plane for Selected Materials

Figure 5.2: Dependence of nonlinearity matrix elements on direction of propa-gation in the (110) plane in selected materials.

128

average anisotropy ratio of each row increases from top to bottom. The first

two rows show the nonlinearity matrix elements of materials with η < 1 or

η ≈ 1, while the bottom two rows show the materials with η > 1. In contrast

to the (001) cut, here the materials with the lowest anisotropy ratios exhibit

convergence between the SAW and transverse bulk modes as the direction of

propagation approaches 90◦ (e.g., see Figure 5.9). This is probably the cause of

the large variations of the nonlinearity matrix elements in the region between

70◦ and 90◦ for RbCl, KCl, NCl, and CaF2. Note that SrF2 has η < 1, but

its SAW mode does not converge with a transverse bulk mode. Hence the

properties of its nonlinearity matrix elements in the 70◦ to 90◦ region may

be indicative of what would occur in RbCl, KCl, NCl, and CaF2 without the

convergence effect. The materials with η > 1 show a general trend in which the

maximum values of the nonlinear matrix elements appear in the middle of the

range, and the minimum values nonlinear matrix elements appear at 90◦. In

the cases of Si and Ge, the matrix elements Slm shift from being predominantly

positive to predominantly negative. In contrast, C, Al, Ni, and Cu have Slm

predominantly positive over the entire range.

The nonlinearity matrix elements for the alums for this particular cut

are generally complex-valued. As a result, Figure 5.3 shows the the matrix

elements’ amplitude |Slm| in the top row and phase arg(Slm) = Slm/|Slm| in

degrees in the bottom row. Complex-valued matrix elements occurs because

the alums are in the m3 point group, which has lower symmetry than the m3m

point group of the other crystals. A discussion of the meaning of complex-

valued nonlinearity matrix elements is deferred to Chapter 6, where it is shown

that the matrix elements of all the crystals studied have this property. The

TOE constant data used to generate Figure 5.3 is the same as in Chapter 4.

129

0

0.02

0.04

0.06

Cs-alum

0

0.02

0.04

0.06

NH4-alum

0

0.05

0.1

K-alum

-180

-90

0

0 20 40 60 80


-180

-90

0

0 20 40 60 80


-180

-90

0

0 20 40 60 80


arg(

−Slm

/ c 4

4) [d

eg]

|Slm

| / c

44


Figure 5.3: Dependence of nonlinearity matrix elements on direction of propa-gation in the (110) plane in selected materials. S11 (solid), S12 (long dashed),

and S13 (short dashed) are plotted in each graph.

130

5.2.2 Study of Si

Figure 5.4 for the (110) plane of Si is similar to the figures of linear and non-

linear parameters in Chapter 4. Figures 5.5, 5.6, 5.7, and 5.8 show the velocity

waveforms, displacement waveforms, particle trajectories, and frequency spec-

tra plus the harmonic propagation curves, respectively. Figure 5.4(a) shows

that S11, S12, and S13 start out around zero at 0◦, grow to a maximum around

37◦, and then decrease to a minimum at 90◦. Around 60◦, the matrix elements

go through zero, but not all simultaneously. The standard deviation divided by

the mean for |S11|, |S12|, |S13| ranges between 60%–70% over the angular range

shown. In contrast, the standard deviation divided by the mean for the linear

wave speed is around 4% [see Figure 5.4(d)]. Finally, as described in Chap-

ter 4, the estimates for the shock formation distance shown in Figure 5.4(f) are

probably not accurate in the vicinity of quasilinear directions.

Several directions are selected for consideration in detail:

1. 0◦ direction: In this direction, S11 is positive, S12 ' −0.00050 is small

compared to neighboring elements (|S12|/|S11| = 0.11, |S12|/|S13| = 0.15),

and S13 is negative. The suppression of third harmonic generation results

in the low frequency oscillations seen in the velocity and displacement

waveforms, and in the extra small loop in the particle trajectory. As seen

from the harmonic propagation curves, the weak coupling to the third

harmonic causes more energy to be trapped in the second harmonic, which

eventually grows to exceed the fundamental in amplitude. In addition,

the fourth harmonic is almost completely suppressed around X = 2. The

spectrum displays a correspondingly complicated structure, with many

peaks and valleys at both X = 1 and X = 2. Additional calculations

show that the oscillations increase in amplitude, and that a shock does not

131

-0.03-0.02-0.01

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0 20 40 60 80

−Slm

/c44

(a)

-180

-90

0

90

180

0 20 40 60 80

arg(

−Slm

/c44

) [d

eg]

(b)

-0.2 -0.1 0

0.1 0.2 0.3 0.4

0 20 40 60 80

β

(c)


0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

0 20 40 60 80

c/c r

ef

(d)

0

20

40

60

80

100

0 20 40 60 80

v g d

irect

ion

[deg

] (e)

0

20

40

60

80

100

0 20 40 60 80

x sho

ck [m

m]

(f)


0

0.2

0.4

0.6

0.8

1

0 20 40 60 80

|Bj/B

0|

(g)

-180

-90

0

90

180

0 20 40 60 80

arg(

Bj/B

0) [d

eg] (h)


Si in (110) plane

c44 = 79.2 GPacref = 5829 m/sρ = 2331 kg/m3


Figure 5.4: Dependence of nonlinearity parameters on direction of propagationin the (110) plane of Si. The direction of propagation is measured in degreesfrom the 〈001〉 direction (to the 〈110〉 direction). The parameters are periodicevery 180◦ and symmetric about the 90◦ direction.

132

-1

-0.5

0

0.5

1

0 π 2π

0°

Vx

-0.5

-0.25

0

0.25

0.5

0 π 2π

0°

Vy

-1.5

-1

-0.5

0

0.5

1

0 π 2π

0°

Vz

0

0.05

0.1

0 20 40 60 80

Slm

0°

-1

-0.5

0

0.5

1

0 π 2π

9.66°

-0.5

-0.25

0

0.25

0.5

0 π 2π

9.66°

-2

-1

0

1

0 π 2π

9.66°

0

0.05

0.1

0 20 40 60 80

9.66°

-1

-0.5

0

0.5

1

0 π 2π

37°

-0.5

-0.25

0

0.25

0.5

0 π 2π

37°

-4

-3

-2

-1

0

1

0 π 2π

37°

0

0.05

0.1

0 20 40 60 80

37°

-1

-0.5

0

0.5

1

0 π 2π

59.355°

-0.5

-0.25

0

0.25

0.5

0 π 2π

59.355°

-1

-0.5

0

0.5

1

0 π 2π

59.355°

0

0.05

0.1

0 20 40 60 80

59.355°

-1

-0.5

0

0.5

1

-π 0 π

90°

ωτ

-0.5

-0.25

0

0.25

0.5

-π 0 π

90°

ωτ

-1

0

1

2

3

4

5

-π 0 π

90°

ωτ

0

0.05

0.1

0 20 40 60 80

90°


Nor

mal

ized

Vel

ocity

Wav

efor

ms

in (

110)

Pla

ne fo

r S

i

Figure 5.5: Velocity waveforms in selected directions of propagation in the(110) plane of Si. The velocity components are normalized such that initialamplitude in each direction is unity. (See text for keys to the various graphs.)

133

-0.6

-0.4

-0.2

0

0.2

0 π 2π

0°

Ux

-0.2

0

0.2

0.4

0 π 2π

0°

Uy

-0.4

-0.2

0

0.2

0.4

0 π 2π

0°

Uz

0

0.05

0.1

0 20 40 60 80

Slm

0°

-0.6

-0.4

-0.2

0

0.2

0 π 2π

9.66°

-0.2

0

0.2

0.4

0 π 2π

9.66°

-0.4

-0.2

0

0.2

0.4

0 π 2π

9.66°

0

0.05

0.1

0 20 40 60 80

9.66°

-0.6

-0.4

-0.2

0

0.2

0 π 2π

37°

-0.2

0

0.2

0.4

0 π 2π

37°

-0.4

-0.2

0

0.2

0.4

0 π 2π

37°

0

0.05

0.1

0 20 40 60 80

37°

-0.6

-0.4

-0.2

0

0.2

0 π 2π

59.355°

0

0 π 2π

59.355°

-0.4

-0.2

0

0.2

0.4

0 π 2π

59.355°

0

0.05

0.1

0 20 40 60 80

59.355°

-0.6

-0.4

-0.2

0

0.2

-π 0 π

90°

ωτ

-0.2

0

0.2

0.4

-π 0 π

90°

ωτ

-0.4

-0.2

0

0.2

0.4

-π 0 π

90°

ωτ

0

0.05

0.1

0 20 40 60 80

90°


Nor

mal

ized

Dis

plac

emen

t Wav

efor

ms

in (

110)

Pla

ne fo

r S

i

Figure 5.6: Displacement waveforms in selected directions of propagation in the(110) plane of Si. The displacement components are computed by integratingthe velocity waveforms of Figure 5.5 over one cycle. (See text for keys to thevarious graphs.)

134

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

0°

Top View

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

0°

Side View

-0.5

-0.25

0

0.25

0.5

-0.5 -0.25 0 0.25 0.5

0°

Front View

0

0.05

0.1

0 20 40 60 80

Slm

0°

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

9.66°

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

9.66°

-0.5

-0.25

0

0.25

0.5

-0.5 -0.25 0 0.25 0.5

9.66°

0

0.05

0.1

0 20 40 60 80

9.66°

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

37°

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

37°

-0.5

-0.25

0

0.25

0.5

-0.5 -0.25 0 0.25 0.5

37°

0

0.05

0.1

0 20 40 60 80

37°

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

59.355°

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

59.355°

-0.5

-0.25

0

0.25

0.5

-0.5 -0.25 0 0.25 0.5

59.355°

0

0.05

0.1

0 20 40 60 80

59.355°

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

90°

ωτ

-0.5

-0.25

0

0.25

0.5

-0.75 -0.5 -0.25 0

90°

ωτ

-0.5

-0.25

0

0.25

0.5

-0.5 -0.25 0 0.25 0.5

90°

ωτ

0

0.05

0.1

0 20 40 60 80

90°


Nor

mal

ized

Par

ticle

Tra

ject

orie

s in

(11

0) P

lane

for

Si

Figure 5.7: Particle trajectories in selected directions of propagation in the(110) plane of Si. The particle trajectories are constructed from the displace-ment waveforms in Figure 5.6. The direction of motion is retrograde (counter-clockwise in side view) in all cases. (See text for keys to the various graphs.)

135

-200

-160

-120

-80

-40

0

0 50 100 150

0°

Spectrum [dB]

0

0.5

1

0 2 4 6 8 10

0°


0

0.05

0.1

0 20 40 60 80

Slm

0°

-200

-160

-120

-80

-40

0

0 50 100 150

9.66°

0

0.5

1

0 2 4 6 8 10

9.66°

0

0.05

0.1

0 20 40 60 80

9.66°

-80

-40

0

0 50 100 150

37°

0

0.5

1

0 2 4 6 8 10

37°

0

0.05

0.1

0 20 40 60 80

37°

-200

-160

-120

-80

-40

0

0 50 100 150

59.355°

0

0.5

1

0 2 4 6 8 10

59.355°

0

0.05

0.1

0 20 40 60 80

59.355°

-80

-40

0

0 50 100 150

90°

n

0

0.5

1

0 2 4 6 8 10

90°

x/x0

0

0.05

0.1

0 20 40 60 80

90°


Spe

ctra

and

Har

mon

ic P

ropa

gatio

n C

urve

s in

(11

0) P

lane

for

Si


136

form in the longitudinal velocity waveform. Qualitatively, propagation in

this direction is similar to the 3.205◦ quasilinear direction in the (001)

plane of KCl (see Section 4.2.3).

2. 9.66◦ direction: Here S11 and S12 are positive and while S13 ' 1.5× 10−7

is small compared to neighboring elements (e.g., |S13|/|S11| ' 1.4× 10−5,

|S13|/|S12| ' 3.9 × 10−5). This makes fourth harmonic generation from

the first and third harmonics less efficient and slows energy transfer to

the higher harmonics. The velocity waveforms show distortion, but not

shocks or sharp peaks. Additional calculations indicate that the wave-

forms steepen further but no shocks form. Note also that the transverse

velocity waveform is distorting in the opposite way from the longitudinal

velocity waveform. This occurs because the transverse amplitude B2 is

180◦ out of phase with B1, as can be seen from Figure 5.4(h). Hence

while the distortion is a nonlinear effect, the phase difference is a linear

effect. Qualitatively, propagation in this direction is similar to the 0◦

quasilinear direction in the (001) plane of KCl (see Section 4.2.3) and the

25.15◦ quasilinear direction in the (001) plane of Ni (see Section 4.2.4).

3. 37◦ direction: Here the nonlinearity matrix elements Slm are predomi-

nantly positive. The waveforms distort in the “positive” manner as de-

scribed previously for the 26◦ direction in the (001) plane of Si. Note

that here again the transverse velocity waveform distorts in the opposite

way from the longitudinal velocity waveform.

4. 59.355◦ direction: In this direction, the matrix element S11 = 2.7× 10−6

is close to zero and small compared to other elements (e.g., |S11|/|S16| =0.0010). As a result, most of the harmonic generation is strongly sup-

pressed and the propagation is essentially linear. Hence this direction is

137

similar to the 20.785◦ and 32.315◦ quasilinear directions in the (001) plane

of Si. In this particular case, the element S16 was chosen to construct the

characteristic length scale x0 = x16 = ρc4/4|S16|ωv0. With v0 = 40 m/s

and f0 = 40 MHz, x0 = x16 ' 135 mm (as compared to the estimate

x11 ' 130000 mm).

5. 90◦ direction: Here the nonlinearity matrix elements Slm are predomi-

nantly negative, and waveforms distort in the “negative” manner as de-

scribed previously for the 0◦ and 35◦ directions in the (001) plane of Si.

Note that the horizontal scale on the waveform graphs has been shifted

by −π radians for this direction relative to the other directions.

In summary, the types of nonlinear waveform distortion seen in Si for this cut

are similar in many ways to the types of distortion seen in the (001) cut.

5.2.3 Study of KCl

Figure 5.9 shows the linear and nonlinear parameters of KCl in the (110) plane.

Figures 5.10 and 5.11 show the velocity waveforms and the frequency spectra

plus harmonic propagation curves, respectively. Figure 5.9(a) shows that S11,

S12, and S13 start out negative at 0◦, pass through zero around 10◦, and grow

to a local maximum around 45◦. They decrease to a local minimum around

71◦ and then abruptly rise to a global maximum around 74.5◦ before going to

zero at 90◦. The rapid changes in the matrix elements in the 70◦ to 90◦ range

are coincident with the convergence of the SAW mode with a transverse bulk

mode. This degeneration into an exceptional bulk wave at 90◦ can be seen

in both Figures 5.9(d) and (g). The standard deviation divided by the mean

for |S11|, |S12|, |S13| ranges between 70%–95% over the angular range shown.

In contrast, the standard deviation divided by the mean for the linear wave

138

-0.05 0

0.05 0.1

0.15 0.2

0.25 0.3

0.35 0.4

0.45

0 20 40 60 80

−Slm

/c44

(a)

-180

-90

0

90

180

0 20 40 60 80

arg(

−Slm

/c44

) [d

eg]

(b)

-0.2 0

0.2 0.4 0.6 0.8 1

1.2 1.4

0 20 40 60 80

β

(c)


0.8 1

1.2 1.4 1.6 1.8 2

2.2 2.4 2.6

0 20 40 60 80

c/c r

ef

(d)

0

20

40

60

80

100

0 20 40 60 80

v g d

irect

ion

[deg

] (e)

0

2

4

6

8

10

0 20 40 60 80

x sho

ck [m

m]

(f)


0

0.2

0.4

0.6

0.8

1

0 20 40 60 80

|Bj/B

0|

(g)

-180

-90

0

90

180

0 20 40 60 80

arg(

Bj/B

0) [d

eg] (h)


KCl in (110) plane

c44 = 6.3 GPacref = 1775 m/sρ = 1989 kg/m3


Figure 5.9: Dependence of nonlinearity parameters on direction of propagationin the (110) plane of KCl. The direction of propagation is measured in degreesfrom the 〈001〉 direction (to the 〈110〉 direction). The parameters are periodicevery 180◦ and symmetric about the 90◦ direction.

139

-1

-0.5

0

0.5

1

-π 0 π

0°

Vx

-0.5

-0.25

0

0.25

0.5

-π 0 π

0°

Vy

-1

0

1

2

3

4

5

-π 0 π

0°

Vz

0

0.1

0.2

0.3

0.4

0 20 40 60 80

Slm

0°

-1

-0.5

0

0.5

1

0 π 2π

9.95°

-0.5

-0.25

0

0.25

0.5

0 π 2π

9.95°

-1

0

1

0 π 2π

9.95°

0

0.1

0.2

0.3

0.4

0 20 40 60 80

9.95°

-1

-0.5

0

0.5

1

0 π 2π

45°

-0.5

-0.25

0

0.25

0.5

0 π 2π

45°

-4

-3

-2

-1

0

1

0 π 2π

45°

0

0.1

0.2

0.3

0.4

0 20 40 60 80

45°

-1

-0.5

0

0.5

1

0 π 2π

71.2°

-0.5

-0.25

0

0.25

0.5

0 π 2π

71.2°

-4

-3

-2

-1

0

1

0 π 2π

71.2°

0

0.1

0.2

0.3

0.4

0 20 40 60 80

71.2°

-1

-0.5

0

0.5

1

0 π 2π

74.5°

ωτ

-2

-1

0

1

2

0 π 2π

74.5°

ωτ

-2

-1

0

1

0 π 2π

74.5°

ωτ

0

0.1

0.2

0.3

0.4

0 20 40 60 80

74.5°


Nor

mal

ized

Vel

ocity

Wav

efor

ms

in (

110)

Pla

ne fo

r K

Cl


140

-80

-40

0

0 50 100 150

0°

Spectrum [dB]

0

0.5

1

0 2 4 6 8 10

0°


0

0.1

0.2

0.3

0.4

0 20 40 60 80

Slm

0°

-200

-160

-120

-80

-40

0

0 50 100 150

9.95°

0

0.5

1

0 2 4 6 8 10

9.95°

0

0.1

0.2

0.3

0.4

0 20 40 60 80

9.95°

-80

-40

0

0 50 100 150

45°

0

0.5

1

0 2 4 6 8 10

45°

0

0.1

0.2

0.3

0.4

0 20 40 60 80

45°

-80

-40

0

0 50 100 150

71.2°

0

0.5

1

0 2 4 6 8 10

71.2°

0

0.1

0.2

0.3

0.4

0 20 40 60 80

71.2°

-80

-40

0

0 50 100 150

74.5°

n

0

0.5

1

0 2 4 6 8 10

74.5°

x/x0

0

0.1

0.2

0.3

0.4

0 20 40 60 80

74.5°


Spe

ctra

and

Har

mon

ic P

ropa

gatio

n C

urve

s in

(11

0) P

lane

for

KC

l

Figure 5.11: Frequency spectra and harmonic propagation curves in selecteddirections of propagation in the (110) plane of KCl. The initial amplitude ofthe fundamental is used as the reference amplitude in the spectra. (See textfor keys to the various graphs.)

141

speed is around 6% (see Figure 5.9(d)). As above, the estimates of the shock

formation distances given in Figure 5.9(f) are probably inaccurate near quasi-

linear directions or directions where the amplitude of |S11| is small compared

to neighboring elements.

Several directions are selected for consideration in detail:

1. 0◦ direction: In this direction, the nonlinearity matrix elements Slm are

predominantly negative, and the waveform distorts in the characteristi-

cally “negative” manner. Note that the horizontal scale on the waveform

graphs has been shifted by −π radians for this direction relative to the

other directions in the figure.

2. 9.95◦ direction: Here the matrix element S11 = −4.8 × 10−6 is close to

zero and small compared to other elements (e.g., |S11|/|S15| = 0.0011).

Hence second harmonic generation is suppressed and the propagation is

essentially linear. In this particular case, the element S15 was chosen to

construct the characteristic length scale x0 = x15 = ρc4/4|S15|ωv0. With

v0 = 40 m/s and f0 = 40 MHz, x0 = x15 ' 16 mm (as compared to the

estimate x11 ' 15000 mm).

3. 45◦ direction: In this direction, the matrix elements Slm are predomi-

nantly positive, and the waveform distorts in the characteristically “pos-

itive” manner.

4. 71.2◦ direction: Here the matrix element S11 ' 9.5× 10−3 is small com-

pared to some of the other directions, but still nonzero. However, this

direction is a case where |S11| is less than neighboring elements (e.g.,

|S11|/|S12| ' 0.48, |S11|/|S13| ' 0.38), similar to the 10◦ and 45◦ direc-

tions in the (001) cut of KCl. The larger relative magnitudes of S12 and

S13 result in the transfer of energy to higher harmonics at a rapid rate,

142

and leads to the sharper cusps seen in the waveforms for this direction.

This direction is very close to a pure mode and hence may be favorable

for experimental investigation.

5. 74.5◦ direction: Here the matrix elements are predominantly positive,

and the waveform distorts in the characteristically “positive” manner.

In conclusion, the discussion above shows that the types of waveform distortion

seen in this cut are similar to those in the (110) cut of Si and the materials in

the (001) case.

5.2.4 Study of Ni

Figure 5.12 shows the linear and nonlinear parameters of Ni in the (110) plane.

For several selected directions, Figure 5.13 shows the velocity waveforms, and

Figure 5.14 displays the frequency spectra and harmonic propagation curves.

Figure 5.12(a) indicates that S11, S12, and S13 are positive over almost the

whole range, with a global maximum at 0◦ and a local maximum around 55.5◦.

Near 90◦, |S11| becomes very small and the other elements become negative but

still relatively small in magnitude. The standard deviation divided by the mean

for |S11|, |S12|, |S13| ranges between 55%–65% over the angular range shown.

In contrast, the standard deviation divided by the mean for the linear wave

speed is around 8% [see Figure 5.12(d)]. This proportionally larger variation is

consistent with other materials in the (110) plane.

In all these directions except 90◦, the matrix elements Slm are predom-

inantly positive, and the waveforms distort in the characteristically “positive”

manner. A few additional comments are provided for the directions shown in

the figures:

143

-0.02 0

0.02 0.04 0.06 0.08 0.1

0.12 0.14

0 30 60 90

−Slm

/c44

(a)

-180

-90

0

90

180

0 30 60 90

arg(

−Slm

/c44

) [d

eg]

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0 30 60 90

β

(c)


0.6 0.8 1

1.2 1.4 1.6 1.8

0 30 60 90

c/c r

ef

(d)

-30

0

30

60

90

0 30 60 90

v g d

irect

ion

[deg

] (e)

0

5

10

15

20

0 30 60 90

x sho

ck [m

m]

(f)


0

0.2

0.4

0.6

0.8

1

0 30 60 90

|Bj/B

0|

(g)

-180

-90

0

90

180

0 30 60 90ar

g(B

j/B0)

[deg

] (h)


Ni in (110) plane

c44 = 122.0 GPacref = 3700 m/sρ = 8912 kg/m3


Figure 5.12: Dependence of nonlinearity parameters on direction of propagationin the (110) plane of Ni. The direction of propagation is measured in degreesfrom the 〈001〉 direction (to the 〈110〉 direction). The elements are periodicevery 180◦ and symmetric about the 90◦ direction. (See text for keys to thevarious graphs.)

144

-1.5

-1

-0.5

0

0.5

1

1.5

0 π 2π

0°

Vx

-0.5

-0.25

0

0.25

0.5

0 π 2π

0°

Vy

-4

-3

-2

-1

0

1

0 π 2π

0°

Vz

0

0.05

0.1

0.15

0 20 40 60 80

Slm

0°

-1

-0.5

0

0.5

1

0 π 2π

40°

-0.5

-0.25

0

0.25

0.5

0 π 2π

40°

-5

-4

-3

-2

-1

0

1

0 π 2π

40°

0

0.05

0.1

0.15

0 20 40 60 80

40°

-1

-0.5

0

0.5

1

0 π 2π

55.5°

-1

-0.5

0

0.5

1

0 π 2π

55.5°

-5

-4

-3

-2

-1

0

1

0 π 2π

55.5°

0

0.05

0.1

0.15

0 20 40 60 80

55.5°

-1

-0.5

0

0.5

1

0 π 2π

75°

-0.5

-0.25

0

0.25

0.5

0 π 2π

75°

-5

-4

-3

-2

-1

0

1

0 π 2π

75°

0

0.05

0.1

0.15

0 20 40 60 80

75°

-0.5

0

0.5

0 π 2π

90°

ωτ

-2

-1

0

1

2

0 π 2π

90°

ωτ

-1

0

1

0 π 2π

90°

ωτ

0

0.05

0.1

0.15

0 20 40 60 80

90°


Nor

mal

ized

Vel

ocity

Wav

efor

ms

in (

110)

Pla

ne fo

r N

i


145

-80

-40

0

0 50 100 150

0°

Spectrum [dB]

0

0.5

1

0 2 4 6 8 10

0°


0

0.05

0.1

0.15

0 20 40 60 80

Slm

0°

-80

-40

0

0 50 100 150

40°

0

0.5

1

0 2 4 6 8 10

40°

0

0.05

0.1

0.15

0 20 40 60 80

40°

-80

-40

0

0 50 100 150

55.5°

0

0.5

1

0 2 4 6 8 10

55.5°

0

0.05

0.1

0.15

0 20 40 60 80

55.5°

-80

-40

0

0 50 100 150

75°

0

0.5

1

0 2 4 6 8 10

75°

0

0.05

0.1

0.15

0 20 40 60 80

75°

-200

-160

-120

-80

-40

0

0 50 100 150

90°

n

0

0.5

1

0 2 4 6 8 10

90°

x/x0

0

0.05

0.1

0.15

0 20 40 60 80

90°


Spe

ctra

and

Har

mon

ic P

ropa

gatio

n C

urve

s in

(11

0) P

lane

for

Ni


146

1. 0◦ direction: The wave is of the “Rayleigh-type” in this direction. Its

distortion is then similar to the 0◦ direction in KCl but occurs in the

opposite manner.

2. 40◦ direction: This direction was selected because it is a pure mode, and

hence conducive to experimental investigation. The transverse compo-

nent B2 is nonzero and in phase with B1.

3. 55.5◦ direction: Like the 40◦ direction, the 55.5◦ direction is also a pure

mode direction, and the transverse component B2 is nonzero. However,

the distortion in the transverse velocity waveform is in the opposite di-

rection of the longitudinal velocity. This occurs because B1 and B2 are

180◦ out of phase in this angular region [See Figure 5.12(g)].

4. 75◦ direction: In this region, the nonlinearity is around an order of magni-

tude less than the 0◦ direction. While the waveforms form shocks, they do

so over considerably large distances. The spectrum in Figure 5.13. shows

more energy in the higher harmonics at X = 1 than in previous directions,

correspondingly, the waveforms appear to show a slightly sharper cusps in

Figure 5.14. Additional calculations show that this is caused by increased

relative magnitudes of |S12|/|S11| = 0.80 and |S13|/|S11| = 0.75 in this di-

rection. In contrast, the more typical 55.5◦ case has |S12|/|S11| = 0.62

and |S13|/|S11| = 0.44. This case is a similar but weaker version of the

type of propagation in 71.2◦ direction in the (110) plane of KCl.

5. 90◦ direction: Here S11 = 4.7 × 10−4 is positive but close to zero and

somewhat smaller than neighboring elements (e.g., |S11|/|S1,10| = 0.31).

Higher order elements are predominantly negative. The resulting wave-

form distortion is close to linear but with a slight “negative” trend. For

147

this particular case, the element S1,10 was chosen to construct the char-

acteristic length scale x0 = x1,10 = ρc4/4|S1,10|ωv0 = 32 mm used in the

figures (as compared with the estimate x11 ' 100 mm).

Thus, nonlinear SAWs in the (110) plane of Ni display many of the same

properties as other nonlinear SAWs in the (110) and (001) planes.

5.3 Summary

This chapter has investigated the properties of nonlinear SAWs in the (110)

plane of a variety of cubic crystals. The types of propagation seen in materials

in the m3m point group in this cut are very similar to those seen in the (001)

plane. The largest difference occurs in materials in the m3 point group (alums),

for which the matrix elements become complex-valued. Investigation of the

meaning of complex-valued nonlinearity is the main theme of Chapter 6.

Chapter 6


This chapter discusses nonlinear propagation of monofrequency SAWs in the

(111) plane of selected cubic crystals. (See Figure D.1 for a diagram of this

cut.) Because the (111) plane is not a plane of high symmetry, the nonlinearity

matrix elements are generally complex-valued and the waveform distortion is

significantly more complicated. Nevertheless, the plots of the magnitude and

phase of the nonlinearity matrix elements as a function of direction can still

be used to indicate waveform distortion properties. The chapter begins with a

brief overview of linear properties and a general review of nonlinear effects in

a variety of crystals. Next, an interpretation of the phase of the nonlinearity

matrix elements is proposed based on a simple mathematical transformation.

Last, detailed studies of Si, KCl, and Ni are presented and discussed.

6.1 Linear Effects

The SAW speed of selected materials as a function of the propagation direction

is shown in Figure 6.1. The direction of propagation is measured in degrees

from the 〈112〉 direction,∗ and the speed for each material is scaled by the

characteristic speed cref = (c44/ρ)1/2 for that material. Because the normal to

the (111) plane is a sixfold symmetry axis, the SAW speed is periodic every

∗Note that Farnell3 chooses the reference direction to be 〈110〉. Hence his plots of thewave speed are reversed as compared to Figure 6.1 (i.e., directions cited as θ degrees hereare given there as 30◦ − θ).

148

149

0.6

0.7

0.8

0.9

1.0

1.1

0 10 20 30 40 50 60

Rel

ativ

e V

eloc

ity


Relative Velocity vs. Angle in (111) plane

η=0.312RbClCaF2 η=0.373

η=0.373KCl

NaCl η=0.705

SrF 2 η=0.803

η=1.02BaF2η=1.06Cs-alum


Al η=1.22

C η=1.26

Si η=1.57

Ge η=1.66

Ni η=2.60

Cu η=3.20

Figure 6.1: Dependence of SAW speed on direction of propagation in the (111)plane for selected materials. speed is periodic every 60◦. The dashed linesindicate crystals in the m3 point group.

150

60◦. Note that only the range from 0◦ to 30◦ is shown because the 30◦ direction

is normal to a plane of mirror symmetry. In most cases (for all the cases

shown here), the speeds group by anisotropy ratio, with materials with lower

anisotropy ratios having higher relative SAW speeds. Materials with η ≈ 1

indicate nearly isotropic media, with a correspondingly constant SAW speed

for all directions. Crystals in the m3 point group are plotted using dashed

lines.


This section demonstrates that SAWs in (111) plane have the following prop-

erties:

• Nonlinear SAWs for most directions in the (111) plane are described

by nonlinearity matrix elements which are complex-valued. Both the

magnitude and phase of the matrix elements can be related to the type

of waveform distortion.

• The waveforms generally exhibit asymmetric distortion. In some cases,

this causes the shock formation to be less clearly defined.

• A simple mathematical transformation is proposed to interpret the phase

information in the nonlinearity matrix elements in cases where the phase

of the lowest order matrix elements are the same.

• In cases where the phases of the nonlinearity matrix elements are not the

same, the waveforms exhibit a nonlinear dispersion-like effect in which

oscillations form near the shocks and peaks of the velocity waveforms.

As shown below, nonlinear SAWs in the (111) plane are often significantly

different from those in the (001) and (110) planes.

151

6.2.1 General Study

Figures 6.2, 6.3, and 6.4 display the nonlinearity matrix elements for all the ma-

terials considered previously in Chapters 4 and 5. Because the matrix elements

are complex-valued, two plots are given for each material. The top plot shows

the magnitudes |S11| (solid), |S12| (long dashed), and |S13| (short dashed), while

the bottom plot shows the phases arg(S11) (solid), arg(S12) (long dashed), and

arg(S13) (short dashed). Figure 6.2 contains plots for materials with η < 1

or η ≈ 1. Figure 6.3 covers the materials with η > 1, except for the alums,

which are given in Figure 6.4. In contrast to the (001) and (110) cuts, none of

the crystals considered here exhibit convergence between the SAW mode and

one of the transverse or quasitransverse bulk modes. However, the direction

30◦ from 〈112〉 is normal to a plane of mirror symmetry and, as a result, the

nonlinearity matrix elements are always real-valued in this direction. All of the

crystals in the m3m point group display a generally decreasing trend in the

magnitude |Slm| over the angular range from 0◦ to 30◦. However, the phases

display no such similar pattern. The alums in the m3 point group do not follow

the same trend as the other crystals (e.g., NH4 has a maximum at 30◦, not a

minimum). The phase plots also appear to have little in common with the

other crystals, or with each other. The widely varying phase profiles for each

material is interesting because, as is shown below, the phase of the nonlinearity

matrix elements has a strong effect on the shape of the distorted waveforms.

6.2.2 Interpretation of Complex-Valued Nonlinearity Parameters

Previously only the concepts of “positive” and “negative” nonlinearity have

been described. Examples of waves with a positive coefficient of nonlinearity

β include acoustic waves in fluids, longitudinal bulk waves in many isotropic

solids, SAWs in steel18 (see Figures 1.1 and 1.2), and the direction 26◦ from

152

0

0.1

0.2

0.3

RbClη=0.312

S11S12S13

0

0.05

0.1

0.15

0.2

KClη=0.373

0

0.05

0.1

NaClη=0.705

-180

-90

0

90

180

-180

-90

0

90

180

-180

-90

0

90

180

0

0.05

0.1

0.15

CaF2η=0.373

0

0.01

0.02

0.03

SrF2η=0.803

0

0.02

0.04

0.06

BaF2

η=1.02

-180

-90

0

90

180

0 10 20 30


-180

-90

0

90

180

0 10 20 30


-180

-90

0

90

180

0 10 20 30


arg(

−Slm

/ c 4

4) [d

eg]

|Slm

| / c

44ar

g(−S

lm /

c 44)

[deg

] |S

lm| /

c44


Figure 6.2: Dependence of nonlinearity matrix elements on direction of propa-gation in the (111) plane in selected materials. (See text for keys to graphs.)

153

0

0.05

0.1

Cη=1.26

0

0.05

0.1

Siη=1.57

0

0.05

0.1

Geη=1.66

-180

-90

0

90

180

-180

-90

0

90

180

-180

-90

0

90

180

0

0.1

0.2

Alη=1.22

0

0.05

0.1

0.15

Niη=2.60

0

0.1

0.2

0.3

Cuη=3.20

-180

-90

0

90

180

0 10 20 30


-180

-90

0

90

180

0 10 20 30


-180

-90

0

90

180

0 10 20 30


arg(

−Slm

/ c 4

4) [d

eg]

|Slm

| / c

44ar

g(−S

lm /

c 44)

[deg

] |S

lm| /

c44



154

0

0.02

0.04

0.06

Cs-alum η=1.06

0

0.02

0.04

NH4-alumη=1.12

0

0.05

0.1

K-alumη=1.17

-180

-90

0

90

180

0 10 20 30


S11S12S13

-180

-90

0

90

180

0 10 20 30


-180

-90

0

90

180

0 10 20 30


arg(

−Slm

/ c 4

4) [d

eg]

|Slm

| / c

44



〈100〉 in Si. In these waves, the peaks of the waves advance in a retarded time

frame moving at the linear wave speed, while the troughs recede. Examples of

waves with β < 0 include SAWs in fused quartz26 and SAWs propagating in

the directions 0◦ and 35◦ from 〈100〉 in the (001) plane of Si (see Figure 4.5). In

these waves, the peaks of the wave recede in a retarded time frame moving at

the linear wave speed, while the the troughs advance. However, the situation is

more complicated for the most general case of a SAW in an anisotropic medium.

As shown in Section 2.2.2, the coefficient of nonlinearity for SAWs in crystal is

[Eq. (2.63) with Eq. (4.5)]

β =4c44S11

ρc2, (6.1)

where the nonlinearity matrix element S11 = |S11|eiφS can be complex-valued.

The interpretation of Eq. (6.1) in terms of its effect on waveforms for situations

155

other than φS = 0 (β real and positive) and φS = ±π (β real and negative)

is not immediately obvious. The purpose of this section is to suggest a way of

thinking about this issue.

Ideally, one would like to be able to characterize the type of waveform

distortion by computing just a few parameters, thereby avoiding the process of

numerically integrating a system of nonlinear differential equations for every

material, cut, and direction. As shown in Chapters 4 and 5, the nonlinearity

matrix elements serve as such parameters, allowing a reasonable estimate of

the type of waveform distortion (or lack thereof) to be determined from plots

of the first few elements. The ability to make the same type of estimate is

desired here. The specific objective is to investigate in a simple manner how

the phase of the nonlinearity matrix affects the SAW solutions.

Towards this end, the quantity

Sθlm = Slmeiθ (6.2)

is introduced to represent a nonlinearity matrix constructed by applying a

uniform phase increment θ to a given matrix Slm. Given a solution for a

material with matrix Slm, it is desired to relate that solution to the one obtained

for a material with nonlinearity matrix Sθlm. It is convenient, although not

necessary, to consider the matrix Slm to be real, as is typical for isotropic

solids. The main simplifying assumption here is that there exist materials for

which the phase of the nonlinearity matrix is nonzero yet independent of the

indices l and m. That Eq. (6.2) is a reasonable model of the phase dependence,

for at least the lowest several matrix elements, is supported by Figures 6.2–6.4.

However, in order for Eq. (6.2) to be valid, Sθlm must retain all the

symmetry properties required of the nonlinearity matrix. In particular, the

nonlinearity matrix elements have the symmetry property [Hamilton et al.,33

156

Eq. (80)]

Slm(−n) = S∗(−l)(−m)n , (6.3)

where n = l + m. By straightforward substitution, it is seen that Eq. (6.2)

holds only for n > 0. In contrast, if the definition

Sθlm = Slmei(n/|n|)θ (6.4)

is adopted in place of Eq. (6.2), then Eq. (6.4) holds for all n. Henceforth the

quantity Sθlm refers to the definition in Eq. (6.4).

Next, recall that the evolution equation for a material with nonlinearity

matrix Sθlm [Eq. (2.87)] is given by

dvθn

dx=

n2ω0

2ρc4

∑l+m=n

lm

|lm|Sθlm(−n)v

θl v

θm , (6.5)

where the notation vθn designates that these spectral components are the solu-

tions associated with the matrix Sθlm. Now substitute Eq. (6.4) into Eq. (6.5)

and multiply both sides by ei(n/|n|)θ, let

vn = vθne

i(n/|n|)θ , (6.6)

and rewrite Eq. (6.5) as

dvn

dx=

n2ω0

2ρc4

∑l+m=n

lm

|lm|Slm(−n)vlvm . (6.7)

The spectral components vn in Eq. (6.7) are recognized as the solutions for a

material with nonlinearity matrix Slm. Therefore, the solutions vθn for a material

with nonlinearity matrix Sθlm are related to the solutions vn for a material with

nonlinearity matrix Slm via Eq. (6.6):

vθn = vne

−i(n/|n|)θ . (6.8)

157

The velocity components vθj in the xj direction for a material with nonlinearity

matrix Sθlm are reconstructed from the spectral components vθ

n via Eq. (2.35):

vθj (x, z, τ) =

∞∑n=−∞

vθn(x)unj(z)e−inω0τ , (6.9)

where the depth functions uni are given by Eq. (2.13), τ = t − x/c, and ω0 is

the characteristic angular frequency. At the surface (z = 0), Eq. (6.9) reduces

to

vθj (x, τ) =

∞∑n=−∞

vθn(x)|Bj |ei(n/|n|)φBje−inω0τ , (6.10)

where the linear amplitude factors Bj are defined in Eq. (2.42). For ease of

notation in the remainder of the chapter, define

Sθlm = −Sθ

lm

c44, (6.11)

following the definition of Slm in Eq. (4.5).

As an example of this procedure, take the well-known waveform dis-

tortion for a “positively” distorting nonlinear Rayleigh wave as a reference

case, with reference spectra vRn and corresponding real-valued nonlinearity

matrix elements SRlm under the convention∗ that the linear amplitude factor

BR1 = |BR

1 |e−iπ/2 = −i|BR1 |. For simplicity, only consider the longitudinal ve-

locity waveforms (the other velocity components are considered near the end

of the section). The reference longitudinal velocity waveform for the nonlinear

Rayleigh wave is given by

vR1 (x, τ) =

∞∑n=−∞

vRn (x)

(−i

n

|n| |BR1 |)

e−inω0τ . (6.12)

∗This convention is chosen to be consistent with the nonlinear Rayleigh wave theory ofZabolotskaya.20 Here B1 is analogous to i(ξt + η), where ξt + η < 0 for all isotropic media(see Table 2.3 and Figure 2.5).

158

Suppose there exists a hypothetical crystal with nonlinearity matrix elements

Sθlm = SR

lmei(n/|n|)θ and linear amplitude factors B1 = |B1|e−iπ/2 = −i|B1|. By

Eq. (6.10), the longitudinal velocity waveform at the surface of the crystal is

written in terms of the spectral components of the reference Rayleigh wave as

vθ1(x, τ) =

∞∑n=−∞

vθn(x)

(−i

n

|n| |B1|)

e−inω0τ , (6.13)

or, using Eq. (6.8),

vθ1(x, τ) =

|B1||BR

1 |∞∑

n=−∞vR

n (x)e−i(n/|n|)θ(−i

n

|n| |BR1 |)

e−inω0τ . (6.14)

The summation is the longitudinal velocity waveform of the Rayleigh wave,

except for the factor of e−i(n/|n|)θ. The prefactor |B1|/|BR1 | adjusts for possible

amplitude differences between the linear solutions of the Rayleigh wave and

the SAW in the crystal. Thus, given the crystal’s linear amplitude factor B1

and the phase θ of the crystal’s nonlinearity matrix elements, the waveforms

at the surface in the idealized crystal may be computed by changing the phase

of the spectral components of the nonlinear Rayleigh wave and reconstructing

according to Eq. (6.14).

However, as shown below, the waveforms of real crystals computed from

the Rayleigh waveforms by Eq. (6.14) are not generally the same as the wave-

forms computed directly by integrating Eq. (6.7). Hence in most cases the

expression in Eq. (6.14) is only an approximation to the actual waveform. The

discrepancies occur because crystals rarely have all their nonlinearity matrix el-

ements with identical phase and, even if the phases of the elements are similar,

the magnitudes of the elements may differ. Nevertheless, the overall result can

be qualitatively similar, especially in cases where the dominant matrix elements

have close to the same phase.

159

To gain some intuition about the transformation vθn = vne−i(n/|n|)θ given

in Eq. (6.14), the dimensionless waveforms

V θ1 (x, τ) =

vθ1(x, τ)

|vθ1(0, 0)| =

∞∑n=−∞

vRn (x)

(−i

n

|n|)

e−i(n/|n|)θe−inω0τ (6.15)

are plotted in Figure 6.5 for 0◦ ≤ θ ≤ 180◦ and Figure 6.6 for −180◦ ≤ θ ≤ 0◦.

The reference waveforms (θ = 0◦) are nonlinear Rayleigh waves in steel. The

TOE elastic constants for the simulations are taken from measurements of

“Steel 60 C2H2A” listed in the review by Zarembo and Krasil’nikov151 and

correspond to the same as those used in simulations by Zabolotskaya20 and

Shull et al.18 The simulations were performed under conditions identical to

the crystal simulations described in Section 4.2.2. Each plot contains the di-

mensionless longitudinal velocity waveforms V θ1 (x) at locations X = x/x0 = 0

(short dashed), X = 1 (long dashed), and X = 2 (solid), where the distance x0

is the estimated shock formation distance.

Several observations can be made about these tables of graphs. As

required, the nonlinear distortion is “positive” at θ = 0◦ and “negative” at θ =

180◦. When θ = −90◦, the waveform looks like the vertical velocity component

of a “positively” distorting Rayleigh wave (see Figure 1.2). This similarity

occurs because Eq. (6.8) implies that the phase of the spectral components

vθ=−π/2n = vnei(n/|n|)π/2 = i(n/|n|)vn = H(vn) , (6.16)

where H is the Hilbert transform as expressed in the frequency domain.111

The appearance of H in Eq. (6.16) is not unexpected because, as stated in

Section 2.3, the longitudinal and vertical components of a Rayleigh wave are

related by a Hilbert transform. As shown in Appendix E, the time-domain

transformation corresponding to vθ1(x, τ) is

vθ1(x, τ) = (cos θ)vR

1 (x, τ)− (sin θ)H[vR1 (x, τ)] . (6.17)

160

-2

0

2

4

6

0 π 2π

Steel with θ=0°

-2

0

2

4

6

0 π 2π

Steel with θ=22.5°

-2

0

2

4

6

0 π 2π

Steel with θ=45°

ωτ

-2

0

2

4

6

0 π 2π


-2

0

2

4

6

0 π 2π

Steel with θ=90°

-2

0

2

4

6

0 π 2π


ωτ

-2

0

2

4

6

0 π 2π

Steel with θ=135°

-2

0

2

4

6

0 π 2π


-2

0

2

4

6

0 π 2π

Steel with θ=180°

ωτ

V1θ

V1θ

V1θ

Figure 6.5: Transformation of waveforms corresponding to the various phaseangles 0 ≤ θ ≤ 180◦ of the transformed nonlinearity matrix elements Sθ

lm =

Slmei(n/|n|)θ, where n = l + m. The waveforms are constructed by applying thetransformation vθ

n = vRn e−i(n/|n|)θ to the spectral components of the known wave-

forms of nonlinear Rayleigh waves in steel. Each graph plots the dimensionlesslongitudinal velocity waveforms V θ

1 (x, τ) = vθ1(x, τ)/|vθ

1(0, 0)| for waveforms atX = x/x0 = 0 (short dashed), X = 1 (long dashed), X = 2 (solid), wherex0 is the estimated shock formation distance. Note that θ = 0 corresponds to“positive” nonlinearity and θ = 180 corresponds to “negative” nonlinearity.

161

-6

-4

-2

0

2

0 π 2π

Steel with θ=0°

-6

-4

-2

0

2

0 π 2π

Steel with θ=-22.5°

-6

-4

-2

0

2

0 π 2π

Steel with θ=-45°

ωτ

-6

-4

-2

0

2

0 π 2π


-6

-4

-2

0

2

0 π 2π

Steel with θ=-90°

-6

-4

-2

0

2

0 π 2π


ωτ

-6

-4

-2

0

2

0 π 2π

Steel with θ=-135°

-6

-4

-2

0

2

0 π 2π


-6

-4

-2

0

2

0 π 2π

Steel with θ=-180°

ωτ

V1θ

V1θ

V1θ

Figure 6.6: Transformation of waveforms corresponding to the various phaseangles −180◦ ≥ θ ≥ 0 of the transformed nonlinearity matrix elementsSθ

lm = Slmei(n/|n|)θ. The waveforms are constructed by applying the transfor-mation vθ

n = vRn e−i(n/|n|)θ to the spectral components of the known waveforms

of nonlinear Rayleigh waves in steel. Each graph plots the dimensionless longi-tudinal velocity V θ

1 (x, τ) = vθ1(x, τ)/|vθ

1(0, 0)| for waveforms at X = x/x0 = 0(short dashed), X = 1 (long dashed), X = 2 (solid), where x0 is the esti-mated shock formation distance. Note that θ = 0 corresponds to “positive”nonlinearity and θ = −180 corresponds to “negative” nonlinearity.

162

When θ = −π/2, then Eq. (6.17) reduces to

vθ=−π/21 (x, τ) = H[vR

1 (x, τ)] , (6.18)

as required. The remaining plots show the expected waveform shapes for inter-

mediate values of θ and, therefore, other cases of complex-valued nonlinearity

matrix elements. (See Appendix F for additional discussion about complex-

valued nonlinearity parameters.)

As demonstrated above for the Rayleigh wave case, the tables are not

limited to longitudinal velocity waveforms. If the phase differences between

B1, B2, and B3 are known, then the corresponding waveform distortion in

the vertical and transverse directions can also be read from Figures 6.5. Let

θlong = arg(Slm) be the phase of the matrix elements (in the case that the

phase are not all the same, choose a representative element, typically S11). As

shown in Appendix F, the appropriate values of θtran and θvert to characterize

the transverse and vertical velocity waveforms are given by

θtran = arg(B1)− arg(B2) + θlong , (6.19a)

θvert = arg(B1)− arg(B3) + θlong . (6.19b)

In other words, these are the phases that are appropriate to use with Eq. (6.10)

and Figures 6.5 and 6.6 to determine the nature of the waveform distortion.

The true test of the interpretation is to apply it to real crystals. Fig-

ure 6.7 shows comparisons of the longitudinal waveforms in several real crystals

and the appropriately transformed and scaled waveforms of nonlinear Rayleigh

waves in steel. The figures of the transformed Rayleigh waves were constructed

via Eq. (6.10) in three steps:

1. The linear amplitude factor B1 and nonlinearity matrix element S11 were

computed for each crystal.

163

-1

0

1

2

3

0 π 2π

V1θ


-1

0

1

2

3

0 π 2π

V1θ


-2

-1

0

1

2

0 π 2π

V1θ


ωτ

-1

0

1

2

3

0 π 2π

Vx

Si (111) 0°

-1

0

1

2

3

0 π 2π

Vx

Ni (111) 0°

-2

-1

0

1

2

0 π 2π

Vx

KCl (111) 0°

ωτ

Figure 6.7: Comparison of simulated nonlinear waveform distortion betweenvθ1(x, τ)-transformed nonlinear Rayleigh waves in steel (left column) and non-

linear SAWs in the direction 0◦ from 〈112〉 in the (111) plane of Si, Ni, andKCl (right column).

164

2. The transformation vθn [Eq. (6.8)] was applied to the spectral components

vRn of the nonlinear Rayleigh waves using θ = arg(S11), and the waveforms

were translated so that all the sine waves begin in the same place.

3. The resulting waveforms were scaled using |B1| such that the amplitude

of the undistorted waveforms at X = 0 was equal in magnitude to the

waveform in the corresponding crystal.

The left column gives the transformed waveforms of the nonlinear Rayleigh

waves, while the right column gives the simulations using the full theory. The

rows present comparisons for waveforms propagating in the direction 0◦ from

〈112〉 in the (111) plane of Si, Ni, and KCl. The direction 0◦ from 〈112〉 was

chosen because it most closely satisfies the basic assumption that the nonlin-

earity elements are similar in phase,∗ although in none of the cases do all the

matrix elements have exactly the same value. The transformed Rayleigh wave-

forms reproduce the general shape of the distortion although not all the details.

For example, in the Si waveform the shock is steeper than in the transformed

waveform, and in the KCl waveform the cusping before the shock does not ap-

pear in the transformed waveform. Nevertheless, the similarities between the

waveforms are striking.

In contrast, Figure 6.8 shows several cases where the phases of the non-

linearity matrix elements are less similar. The format of the figure is the same

as Figure 6.8. The primary difference is that the waveforms are shown for the

directions 10◦, 20◦, and 28◦ from 〈112〉 in the (111) plane of KCl. Here the

transformed waveforms do not reproduce at all the extra oscillations that result

∗In addition, the waves propagating in this direction are nearly “Rayleigh-type” waves,except that the principal axis of their initial surface displacement ellipse is not perpendicularto the free surface (see the tilting ellipse in side view of the top row of Figure 6.12).

165

-1

0

1

2

3

0 π 2π

V1θ


-1

0

1

2

3

0 π 2π

V1θ


-1

0

1

2

3

0 π 2π

V1θ


ωτ

-1

0

1

2

3

0 π 2π

Vx

KCl (111) 10°

-1

0

1

2

3

0 π 2π

Vx

KCl (111) 20°

-1

0

1

2

3

0 π 2π

Vx

KCl (111) 28°

ωτ

Figure 6.8: Comparison of simulated nonlinear waveform distortion betweenvθ1(x, τ)-transformed nonlinear Rayleigh waves in steel (left column) and non-

linear SAWs in the directions 10◦, 20◦, and 28◦ from 〈112〉 in the (111) planeof KCl (right column).

166

from the dispersion-like effect of the phase differences introduced between har-

monics during the harmonic generation process. This effect is discussed further

in the studies of individual crystals in Sections 6.2.3, 6.2.4, and 6.2.5.

The rest of Chapter 6 presents detailed studies of Si, KCl, and Ni. The

studies demonstrate the effectiveness of the interpretation of complex-valued

nonlinearity proposed in this section.

6.2.3 Study of Si

As shown above, the waveforms that result from propagation in the (111) plane

can be significantly different than those in the (001) and (110) planes. This

section continues the investigation of nonlinear SAWs in Si started in Sec-

tions 4.2.2 and 5.2.2. Figure 6.9 is similar to the figures of linear and nonlinear

parameters in those sections, with a few differences:

• In the (111) plane, the nonlinearity matrix elements are generally complex-

valued. As a result, Figure 6.9(a) shows the magnitudes of the nonlinear-

ity matrix elements S11 (solid), S12 (long dashed), and S13 (short dashed)

as a function of direction. The matrix elements are symmetric about the

direction 30◦ from 〈112〉, and periodic every 60◦.

• Figure 6.9(b) shows the phase φ = arg(Slm). The phase φ = 90◦ indicates

a positive, purely imaginary value of Slm, while the phase φ = −90◦

indicates a negative, purely imaginary value of Slm. In cases where the

nonlinearity matrix elements are similar in phase, the phase values in the

plot may be used in conjunction with the tables of graphs in Figures 6.5

and 6.6 to characterize the nature of the waveform distortion.

• Figure 6.9(c) shows the magnitude of the nonlinearity coefficient β =

−4S11/ρc2 = 4c44S11/ρc2. The phase of β is plotted in Figure 6.9(b).

167

0 0.02 0.04 0.06 0.08 0.1

0.12

0 10 20 30

|Slm

|/c44

(a)

-180

-90

0

90

180

0 10 20 30

arg(

−Slm

/c44

) [d

eg]

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 10 20 30

|β|

(c)


0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

0 10 20 30

c/c r

ef

(d)

0

10

20

30

0 10 20 30

v g d

irect

ion

[deg

] (e)

0 5 10 15 20 25 30 35 40

0 10 20 30

x sho

ck [m

m]

(f)


0

0.2

0.4

0.6

0.8

1

0 10 20 30

|Bj/B

0|

(g)

-180

-90

0

90

180

0 10 20 30

arg(

Bj/B

0) [d

eg] (h)


Si in (111) plane

c44 = 79.2 GPacref = 5829 m/sρ = 2331 kg/m3


Figure 6.9: Dependence of nonlinearity parameters on direction of propagationin the (111) plane of Si. The direction of propagation is measured in degreesfrom the 〈112〉 direction (to the 〈101〉 direction). The parameters are periodicevery 60◦ and symmetric about the 30◦ direction.

168

• Figure 6.9(f) shows the estimate of the shock formation distance x11 calcu-

lated using Eq. (2.59). However, as shown below, the waveform distortion

is asymmetric in most cases, and it is less clear what is the appropriate

characteristic length scale for nonlinear waveform evolution.

These comments also apply to similar figures in Sections 6.2.4 and 6.2.5.

Figure 6.9(a) indicates that the nonlinearity is strongest in the 0◦ direc-

tion and decreases by an order of magnitude as the direction approaches 30◦.

Figure 6.9(b) shows that the phases of the lowest order nonlinearity matrix el-

ements increase from around 100◦ to 180◦ over the same range. In addition, all

the matrix elements are real-valued in the direction 30◦ from 〈112〉, as a result

of this direction being normal to a plane of mirror symmetry. Figure 6.9(e)

shows that both 0◦ and 30◦ directions are pure mode directions. However, nei-

ther are “Rayleigh-type” modes. In the 0◦ case, Figures 6.9(g) and (h) indicate

that, while the displacement is confined to the sagittal plane (B2 = 0), the

phase difference between B1 and B3 is more than 90◦, and hence the major

axis of initial surface displacement ellipse is not perpendicular to the surface.

In the 30◦ case, Figure 6.9(g) shows that the displacement is not confined to the

sagittal plane. Finally, Figure 6.9(f) shows the shock formation distance esti-

mated from Eq. (2.59). However, given the complicated phase interactions that

result from the complex-valued nonlinearity matrix elements in this case, this

estimate may be less accurate as compared to cases where all the nonlinearity

matrix elements are real-valued.

Figures 6.10, 6.11, 6.12, and 6.13 display the velocity waveforms, dis-

placement waveforms, particle trajectories, and frequency spectra plus har-

monic propagation curves, respectively, for the directions 0◦, 10◦, 20◦, 28◦, and

30◦ from 〈112〉. These plots have two main differences in format as compared to

similar plots in previous chapters: (1) the phases arg(Slm) of the nonlinearity

169

-1

0

1

2

-π 0 π

0°

Vx

-1

0

1

2

-π 0 π

0°

Vy

-1

0

1

2

3

-π 0 π

0°

Vz

-180

-90

0

90

180

0 10 20 30

arg(−Slm)

0°

-1

0

1

2

-π 0 π

10°

-1

0

1

2

-π 0 π

10°

-1

0

1

2

3

4

-π 0 π

10°

-180

-90

0

90

180

0 10 20 30

10°

-1

0

1

2

-π 0 π

20°

-1

0

1

2

-π 0 π

20°

-1

0

1

2

3

4

5

-π 0 π

20°

-180

-90

0

90

180

0 10 20 30

20°

-1

0

1

2

-π 0 π

28°

-1

0

1

2

-π 0 π

28°

-1

0

1

2

3

4

5

-π 0 π

28°

-180

-90

0

90

180

0 10 20 30

28°

-1

0

1

2

-π 0 π

30°

ωτ

-1

0

1

2

-π 0 π

30°

ωτ

-1

0

1

2

3

4

5

-π 0 π

30°

ωτ

-180

-90

0

90

180

0 10 20 30

30°


Nor

mal

ized

Vel

ocity

Wav

efor

ms

in (

111)

Pla

ne fo

r S

i

Figure 6.10: Velocity waveforms in selected directions for propagation in the(111) plane of Si. The velocity components are normalized such that initialamplitude in each direction is unity. (See text for keys to the various graphs.)

170

-0.6

-0.4

-0.2

0

0.2

-π 0 π

0°

Ux

-0.4

-0.2

0

0.2

0.4

-π 0 π

0°

Uy

-0.6

-0.4

-0.2

0

0.2

0.4

-π 0 π

0°

Uz

-180

-90

0

90

180

0 10 20 30

arg(−Slm)

0°

-0.6

-0.4

-0.2

0

0.2

-π 0 π

10°

-0.4

-0.2

0

0.2

0.4

-π 0 π

10°

-0.6

-0.4

-0.2

0

0.2

0.4

-π 0 π

10°

-180

-90

0

90

180

0 10 20 30

10°

-0.6

-0.4

-0.2

0

0.2

-π 0 π

20°

-0.4

-0.2

0

0.2

0.4

-π 0 π

20°

-0.6

-0.4

-0.2

0

0.2

0.4

-π 0 π

20°

-180

-90

0

90

180

0 10 20 30

20°

-0.6

-0.4

-0.2

0

0.2

-π 0 π

28°

-0.4

-0.2

0

0.2

0.4

-π 0 π

28°

-0.6

-0.4

-0.2

0

0.2

0.4

-π 0 π

28°

-180

-90

0

90

180

0 10 20 30

28°

-0.6

-0.4

-0.2

0

0.2

-π 0 π

30°

ωτ

-0.4

-0.2

0

0.2

0.4

-π 0 π

30°

ωτ

-0.6

-0.4

-0.2

0

0.2

0.4

-π 0 π

30°

ωτ

-180

-90

0

90

180

0 10 20 30

30°


Nor

mal

ized

Dis

plac

emen

t Wav

efor

ms

in (

111)

Pla

ne fo

r S

i

Figure 6.11: Displacement waveforms in selected directions of propagation inthe (111) plane of Si. The displacement components are computed by integrat-ing the velocity waveforms of Figure 6.10 over one cycle. (See text for keys tothe various graphs.)

171

-0.6

-0.3

0

0.3

0.6

-0.9 -0.6 -0.3 0 0.3

0°

Top View

-0.6

-0.3

0

0.3

0.6

-0.9 -0.6 -0.3 0 0.3

0°

Side View

-0.6

-0.3

0

0.3

0.6

-0.6 -0.3 0 0.3 0.6

0°

Front View

-180

-90

0

90

180

0 10 20 30

arg(−Slm)

0°

-0.6

-0.3

0

0.3

0.6

-0.9 -0.6 -0.3 0 0.3

10°

-0.6

-0.3

0

0.3

0.6

-0.9 -0.6 -0.3 0 0.3

10°

-0.6

-0.3

0

0.3

0.6

-0.6 -0.3 0 0.3 0.6

10°

-180

-90

0

90

180

0 10 20 30

10°

-0.6

-0.3

0

0.3

0.6

-0.9 -0.6 -0.3 0 0.3

20°

-0.6

-0.3

0

0.3

0.6

-0.9 -0.6 -0.3 0 0.3

20°

-0.6

-0.3

0

0.3

0.6

-0.6 -0.3 0 0.3 0.6

20°

-180

-90

0

90

180

0 10 20 30

20°

-0.6

-0.3

0

0.3

0.6

-0.9 -0.6 -0.3 0 0.3

28°

-0.6

-0.3

0

0.3

0.6

-0.9 -0.6 -0.3 0 0.3

28°

-0.6

-0.3

0

0.3

0.6

-0.6 -0.3 0 0.3 0.6

28°

-180

-90

0

90

180

0 10 20 30

28°

-0.6

-0.3

0

0.3

0.6

-0.9 -0.6 -0.3 0 0.3

30°

ωτ

-0.6

-0.3

0

0.3

0.6

-0.9 -0.6 -0.3 0 0.3

30°

ωτ

-0.6

-0.3

0

0.3

0.6

-0.6 -0.3 0 0.3 0.6

30°

ωτ

-180

-90

0

90

180

0 10 20 30

30°


Nor

mal

ized

Par

ticle

Tra

ject

orie

s in

(11

1) P

lane

for

Si

Figure 6.12: Particle trajectories in selected directions of propagation in the(111) plane of Si. The particle trajectories are constructed from the displace-ment waveforms in Figure 6.11. The direction of motion is retrograde (coun-terclockwise in side view) in all cases. (See text for keys to the various graphs.)

172

-80

-40

0

0 50 100 150

0°

Spectrum [dB]

0

0.5

1

0 2 4 6 8 10

0°

Harmonic Magnitudes

-180

-90

0

90

180

0 2 4 6 8 10

0°

Harmonic Phases

-180

-90

0

90

180

0 10 20 30

arg(−Slm)

0°

-80

-40

0

0 50 100 150

10°

0

0.5

1

0 2 4 6 8 10

10°

-180

-90

0

90

180

0 2 4 6 8 10

10°

-180

-90

0

90

180

0 10 20 30

10°

-80

-40

0

0 50 100 150

20°

0

0.5

1

0 2 4 6 8 10

20°

-180

-90

0

90

180

0 2 4 6 8 10

20°

-180

-90

0

90

180

0 10 20 30

20°

-80

-40

0

0 50 100 150

28°

0

0.5

1

0 2 4 6 8 10

28°

-180

-90

0

90

180

0 2 4 6 8 10

28°

-180

-90

0

90

180

0 10 20 30

28°

-80

-40

0

0 50 100 150

30°

n

0

0.5

1

0 2 4 6 8 10

30°

x/x0

-180

-90

0

90

180

0 2 4 6 8 10

30°

x/x0

-180

-90

0

90

180

0 10 20 30

30°


Spe

ctra

and

Har

mon

ic P

ropa

gatio

n C

urve

s in

(11

1) P

lane

for

Si


173

Direction arg(S11) B1 B2 B3 θlong θtran θvert

0 106 −90 0 9 106 None 710 106 −90 20 8 106 −4 820 107 −90 11 5 107 16 2128 164 −90 2 1 164 72 7330 180 −90 0 0 180 90 90

Table 6.1: Phases of key linear and nonlinear parameters for the selected prop-agation directions in the (111) plane of Si. Plots of various quantities in thesedirections are featured in Figures 6.10, 6.11, 6.12, and 6.13. The parametersθlong, θtran, and θvert may be used to compare longitudinal, transverse, andvertical velocity waveforms to Figures 6.5 and 6.6.

matrix elements are plotted for each direction instead of the magnitudes |Slm|,and (2) harmonic propagation curves in Figure 6.13 are given for both the mag-

nitude and phase of the first five spectral components. As before, the velocity

waveforms, displacement waveforms, particle trajectories, and frequency spec-

tra show results at locations X = 0 (short dashed), X = 1 (long dashed), and

X = 2 (solid). The harmonic propagation curves show the fundamental (solid)

as well as the second (long dashed), third (short dashed), fourth (dotted), and

fifth (dot-dashed) harmonics. The nonlinearity matrix element curves show the

phases of S11 (solid), S12 (long dashed), and S13 (short dashed). In addition to

these figures, Table 6.1 lists the phases of S11 and Bj , as well as the angles θlong,

θtran, and θvert, which can be used to compare the various velocity waveforms

to Figures 6.5 and 6.6. Finally, note that the vertical axis on all the waveforms

is shifted such that −π ≤ ωτ ≤ π (as opposed to the range 0 ≤ ωτ ≤ 2π shown

in waveforms previously in this section).

A description is given for each direction (from top to bottom):

1. 0◦ direction: In this direction, the nonlinearity matrix element |S11| '0.11 and θlong = arg(S11) ' 106◦. In Figure 6.7, the longitudinal velocity

174

waveform in this direction was compared to the corresponding trans-

formed waveform of nonlinear Rayleigh waves in steel and shown to be

in reasonably good agreement. Comparison of the vertical velocity wave-

form in Figure 6.10 with the graphs of Figure 6.5 using θvert ' 7◦ from

Table 6.1 shows that the shape of the waveform indeed fits between the

0◦ and 22.5◦ cases. Figure 6.10 shows that while the particle trajectory

in this case is similar to a Rayleigh wave in that it is confined to the

sagittal plane, it is also different because the major axis of its surface

displacement ellipse is tilted away from the normal to the plane. The

spectra and harmonic magnitude curves in Figure 6.13 look fairly typical

of SAWs. The harmonic phase curves show little variation over the entire

range, as would be expected for nonlinearity matrix elements that have

a high phase similarity.

2. 10◦ direction: In this direction, the nonlinearity matrix element |S11| '0.066 and θlong = arg(S11) ' 106◦. Here the nonlinearity matrix elements

differ in phase more than in the 0◦ direction. As a result, the longitudinal

velocity waveform develops a small oscillation to the right of the the

shock. Here the wave has both vertical and transverse components. While

θvert ' 8◦ appears to be qualitatively appropriate, θtran ' −4◦ appears

to be a little low. This underestimation may be due to the fact that

arg(S11) is lower than arg(S12) and arg(S13), and therefore the overall

effective phase shift is higher. While many of the waveforms displayed

previously had components in all three directions, this is the first example

where the plane containing the particle trajectory is rotated out of the

sagittal plane about more than one axis (e.g., the particle trajectory does

not occur in a plane of the form drawn in Figure 2.4). This is seen in

Figure 6.12 by the fact that the top, side, and front views of the particle

175

trajectory are all elliptical. Figure 6.13 shows that the harmonic phase

curves develop a larger variation as compared to the previous direction.

3. 20◦ direction: In this direction, the nonlinearity matrix element |S11| '0.020 and θlong = arg(S11) ' 117◦. Here the phases of the nonlinearity

matrix elements have further separated. Figure 6.10 shows that this re-

sults in even larger oscillations to the right of the shocks and peaks in all

the velocity waveform components. As Figure 6.8 suggests, these wave-

forms are not reproduced well in detail by the simple vθn transformation

method described above and hence the values of θlong, θtran, and θvert are

less useful. Interestingly, Figure 6.12 shows that the oscillations cause

the particle trajectory to lose its elliptical form and develop into a shape

with a discontinuous tangent vector. Figure 6.13 indicates that spectra

develop small peaks in the lowest harmonics. The harmonic magnitude

curves for the third, fourth, and fifth harmonics move closer together than

in the previous directions, and the harmonic magnitude curves show more

variation and less regular spacing. Note that sharp vertical lines in the

fourth and fifth harmonics are actually discontinuities introduced by the

fact that −180◦ and +180◦ represent the same phase.

4. 28◦ direction: In this direction, the nonlinearity matrix element |S11| '0.0085 and θlong = arg(S11) ' 164◦. In this direction, the velocity wave-

forms begin to look more like the class of waveforms typified by the di-

rection 10◦ from 〈100〉 in the (001) plane of KCl. This occurs for several

reasons. First and foremost, the magnitude |S11| falls below |S12| and

|S13|, just like the previously mentioned case. Second, the phase of the

matrix elements have moved closer together compared to the case at 20◦,

as seen in Figure 6.13. This phase similarity causes the transformed

waveforms corresponding to the phases θtran and θvert to better match

176

the velocity waveforms in Figure 6.10, although they still appear a bit

underestimated. As expected, the frequency spectra indicate that energy

is transferred to the higher harmonics faster than is typical.

5. 30◦ direction: In this direction, the nonlinearity matrix element |S11| '0.0086 and θlong = arg(S11) = 180◦. Due to the symmetries of this direc-

tion, the nonlinearity matrix elements Slm are real-valued and negative

(phase of 180◦). The waveform distortion is “negative” but of the variety

described for the 28◦ direction. Note that the values of θlong, θtran, and

θvert given in Table 6.1 match the waveforms well, as would be expected

from the phase similarity of the nonlinearity matrix elements.

These simulations demonstrate that, in directions in which the phases of the

dominant nonlinearity matrix elements are similar, the velocity waveforms can

be characterized well by the transformations described in Section 6.2.2. Direc-

tions in which the nonlinearity matrix are less similar exhibit oscillations that

appear to right of the shocks and peaks in the velocity waveforms. Finally,

directions where the nonlinearity matrix elements are real-valued or nearly

real-valued exhibit distortion like that seen previously in the (001) and (110)

planes.

6.2.4 Study of KCl

Next, consider nonlinear SAWs in the (111) plane of KCl. Figure 6.14 displays

the linear and nonlinear parameters describing propagation in this plane. Fig-

ure 6.14(a) shows that the magnitudes of the nonlinearity matrix elements are

highest in the 0◦ direction, they dip down to a local minimum around 3◦, rise

to a local maximum around 10◦, and then fall off to a global minimum at 30◦

which is approximately an order of magnitude smaller than at 0◦. The direc-

tion where the local minimum occurs coincides with the direction where the

177

0 0.02 0.04 0.06 0.08 0.1

0.12 0.14 0.16 0.18 0.2

0 10 20 30

|Slm

|/c44

(a)

-180

-90

0

90

180

0 10 20 30

arg(

−Slm

/c44

) [d

eg]

(b)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 10 20 30

|β|

(c)


0.8 1

1.2 1.4 1.6 1.8 2

2.2 2.4

0 10 20 30

c/c r

ef

(d)

-20

-10

0

10

20

30

0 10 20 30

v g d

irect

ion

[deg

] (e)

0

2

4

6

8

10

0 10 20 30

x sho

ck [m

m]

(f)


0

0.2

0.4

0.6

0.8

1

0 10 20 30

|Bj/B

0|

(g)

-180

-90

0

90

180

0 10 20 30

arg(

Bj/B

0) [d

eg] (h)


KCl in (111) plane

c44 = 6.3 GPacref = 1775 m/sρ = 1989 kg/m3


Figure 6.14: Dependence of nonlinearity parameters on direction of propagationin the (111) plane of KCl. The direction of propagation is measured in degreesfrom the 〈112〉 direction (to the 〈111〉 direction). The elements are periodicevery 30◦ and symmetric about the 45◦ direction.

178


0 −150 −90 0 −24 −150 None 14410 60 −90 150 −22 60 −180 −820 71 −90 162 −13 71 179 −628 85 −90 176 −3 85 179 −230 0 −90 −180 0 0 90 −90

Table 6.2: Phases of key linear and nonlinear parameters for the selected prop-agation directions in the (111) plane of KCl. Plots of various quantities inthese directions are featured in Figures 6.15 and 6.16. The parameters θlong,θtran, and θvert may be used to compare longitudinal, transverse, and verticalvelocity waveforms to Figures 6.5 and 6.6.

phase goes from the vicinity of −180◦ to the vicinity of +180◦. In actuality, the

vertical jump shown in Figure 6.14(b) represents a discontinuity which arises

because −180◦ = 180◦ in the plot. Figure 6.9(e) shows that here again both

the 0◦ and 30◦ directions are pure mode directions, but neither is a “Rayleigh-

type” mode for the same reasons as in Si. One marked difference with Si that

occurs at linear order is that the transverse linear amplitude factors B2 have

phases that are closer to 180◦ than to 0◦, in contrast to the case of Si. This

difference alone results in transverse velocity waveforms that look significantly

different. Finally, Figure 6.9(f) shows the shock formation distance estimated

from Eq. (2.59). As before, these estimates should probably be used with care

given the complicated interactions exhibited by these waveforms.

Figures 6.15 and 6.16 display the velocity waveforms and frequency spec-

tra plus harmonic propagation curves, respectively, for the directions 0◦, 10◦,

20◦, 28◦, and 30◦ from 〈112〉. In addition to these figures, Table 6.2 lists the

phases of S11 and Bj as well as the angles θlong, θtran, and θvert, which can be

used to compare the various velocity waveforms to Figures 6.5 and 6.6. Fi-

nally, note that the vertical axis on all the waveforms is shifted back such that

0 ≤ ωτ ≤ 2π.

179

-2

-1

0

1

0 π 2π

0°

Vx

-2

-1

0

1

0 π 2π

0°

Vy

-1

0

1

2

3

4

0 π 2π

0°

Vz

-180

-90

0

90

180

0 10 20 30

arg(−Slm)

0°

-1

0

1

2

3

0 π 2π

10°

-2

-1

0

1

0 π 2π

10°

-1

0

1

2

3

0 π 2π

10°

-180

-90

0

90

180

0 10 20 30

10°

-1

0

1

2

0 π 2π

20°

-4

-3

-2

-1

0

1

0 π 2π

20°

-1

0

1

2

3

4

0 π 2π

20°

-180

-90

0

90

180

0 10 20 30

20°

-1

-0.5

0

0.5

1

1.5

0 π 2π

28°

-3

-2

-1

0

1

0 π 2π

28°

-1

0

1

2

3

0 π 2π

28°

-180

-90

0

90

180

0 10 20 30

28°

-1

-0.5

0

0.5

1

0 π 2π

30°

ωτ

-1

-0.5

0

0.5

1

0 π 2π

30°

ωτ

-1

-0.5

0

0.5

1

0 π 2π

30°

ωτ

-180

-90

0

90

180

0 10 20 30

30°


Nor

mal

ized

Vel

ocity

Wav

efor

ms

in (

111)

Pla

ne fo

r K

Cl


180

-80

-40

0

0 50 100 150

0°

Spectrum [dB]

0

0.5

1

0 2 4 6 8 10

0°

Harmonic Magnitudes

-180

-90

0

90

180

0 2 4 6 8 10

0°

Harmonic Phases

-180

-90

0

90

180

0 10 20 30

arg(−Slm)

0°

-80

-40

0

0 50 100 150

10°

0

0.5

1

0 2 4 6 8 10

10°

-180

-90

0

90

180

0 2 4 6 8 10

10°

-180

-90

0

90

180

0 10 20 30

10°

-80

-40

0

0 50 100 150

20°

0

0.5

1

0 2 4 6 8 10

20°

-180

-90

0

90

180

0 2 4 6 8 10

20°

-180

-90

0

90

180

0 10 20 30

20°

-80

-40

0

0 50 100 150

28°

0

0.5

1

0 2 4 6 8 10

28°

-180

-90

0

90

180

0 2 4 6 8 10

28°

-180

-90

0

90

180

0 10 20 30

28°

-200

-160

-120

-80

-40

0

0 50 100 150

30°

n

0

0.5

1

0 2 4 6 8 10

30°

x/x0

-180

-90

0

90

180

0 2 4 6 8 10

30°

x/x0

-180

-90

0

90

180

0 10 20 30

30°


Spe

ctra

and

Har

mon

ic P

ropa

gatio

n C

urve

s in

(11

1) P

lane

for

KC

l

Figure 6.16: Frequency spectra and harmonic propagation curves for selecteddirections of propagation in the (111) plane of KCl. The initial amplitude ofthe fundamental is used as the reference amplitude in the spectra. (See textfor keys to the various graphs.)

181


1. 0◦ direction: In this direction, the nonlinearity matrix element |S11| '0.18 and θlong = arg(S11) ' −150◦. In Figure 6.7, the longitudinal veloc-

ity waveform in this direction was compared to the corresponding trans-


in reasonably good agreement. As in the Si case, the vertical velocity

waveform compares favorably with the nonlinear Rayleigh wave in steel

transformed by θvert ' 144◦, as listed in Table 6.2. The frequency spec-

tra and harmonic magnitude curves in Figure 6.16 look fairly typical of

SAWs. The harmonic phase curves show relatively little of the variation

during propagation that would be expected from the high phase similar-

ity of the nonlinearity matrix elements in this direction. One interesting

difference here is that the phases of the first five harmonics are not spread

over as wide a range as in the 0◦ direction in Si.

2. 10◦ direction: In this direction, the nonlinearity matrix element |S11| '0.084 and θlong = arg(S11) ' 60◦. The nonlinearity matrix elements

shown in Figure 6.15 are farther apart in phase than before. As shown

in Figure 6.8, this phase dissimilarity leads to a poorer match between

the transformed waveforms of the nonlinear Rayleigh waves than in the

previous direction. As in the Si case, the angles θlong, θtran, and θvert

given in Table 6.2 appear to be too low. The harmonic phase curves

shown in Figure 6.16 have larger variations in the curves, especially near

the source, and are spread over a larger range.

3. 20◦ direction: In this direction, the nonlinearity matrix element |S11| '0.049 and θlong = arg(S11) ' 71◦. The phases of the nonlinearity matrix

elements have moved even farther apart than in the 10◦ direction. This

182

results in oscillations to the right of the peaks and shocks in the velocity

waveforms shown in Figure 6.15. As in the 20◦ direction in Si, Figure 6.16

indicates that the spectra develop small peaks in the lowest harmonics.

The magnitudes of the third, fourth, and fifth harmonics move closer to-

gether, while the phases show larger variation except at the fundamental

frequency.

4. 28◦ direction: In this direction, the nonlinearity matrix element |S11| '0.0098 and θlong = arg(S11) ' 85◦. Like this same direction in Si, the

magnitude of S11 being less than S12 and S13 causes energy to be trans-

ferred to the higher harmonics at a higher rate. However, unlike Si, here

the phases of the nonlinearity elements are much more widely spaced. The

combination of these two factors leads to the sharp, high frequency os-

cillations which occur in the waveform. The harmonic magnitude curves

in Figure 6.16 show that higher frequency oscillations become increas-

ingly strong as the magnitude of the fundamental progressively becomes

less than the second, third, fourth, and fifth harmonics. As would be

expected, there is a strong disagreement between the transformed wave-

forms of the nonlinear Rayleigh waves and the directly calculated wave-

forms in the right column of Figure 6.8.

5. 30◦ direction: In this direction, the nonlinearity matrix element |S11| '0.00017 and θlong = arg(S11) ' 0◦. Here the ratio of |S11|/|Slm| for

many of the higher-order matrix elements is between 10−1 and 10−2 (e.g.,

|S11|/|S17| ' 0.0068). As a result, the coupling between the fundamental

and higher harmonics is low, and harmonic generation is suppressed. This

is clearly seen in the velocity waveforms shown in Figure 6.15. Because

the characteristic length scale x0 = ρc4/4|S11|ωv0 diverges as S11 → 0,

it is no longer suitable for scaling. In this particular case, the element

183

S17 was chosen to construct the characteristic length scale x0 = x17 =

ρc4/4|S17|ωv0 (based on the method described in the section discussing

propagation in the direction 20.785◦ from 〈100〉 in the (001) plane of Si).

With v0 = 40 m/s and f0 = 40 MHz, x0 = x17 ' 3.0 mm (as compared

to the estimate x11 ' 435 mm). This direction of harmonic suppression

is similar to directions of harmonic suppression in the (001) and (110)

planes, except that here some of the nonlinearity matrix elements are

positive and some are negative. However, they are all real-valued. The

frequency spectra and harmonic propagation curves shown in Figure 6.16

are consistent with this quasilinear behavior.

In summary, the simulations in KCl are similar to those in Si from 0◦ to 27◦.

Beyond that region, the waveforms display harmonic suppression, exhibiting

sharp oscillations at 28◦ and quasilinear properties at 30◦. The latter direction

has propagation similar to that seen previously in the (001) and (110) planes.

6.2.5 Study of Ni

Last, consider nonlinear SAWs in the (111) plane of Ni. Figure 6.17 displays the

linear and nonlinear parameters describing propagation in this plane. Like Si

and KCl, Figure 6.17(a) shows that the magnitudes of the nonlinearity matrix

elements are highest at 0◦. As the direction increases, they drop by nearly

an order of magnitude to a small but nonzero value at 30◦. Unlike for Si

and KCl, the amplitude of the fundamental does not drop below the other

harmonics in any direction shown. Despite the similarity in the magnitudes,

the phases of the nonlinearity matrix elements are quite different, starting

at 101◦ in the 0◦ direction, crossing 0◦ around the 18◦ direction, falling to

a minimum around −38◦ near the 25◦ direction, and then returning to 0◦ in

the 30◦ direction. Figure 6.17(e) shows that here again both the 0◦ and 30◦

184

0 0.02 0.04 0.06 0.08 0.1

0.12 0.14 0.16

0 10 20 30

|Slm

|/c44

(a)

-180

-90

0

90

180

0 10 20 30

arg(

−Slm

/c44

) [d

eg]

(b)

0 0.2 0.4 0.6 0.8 1

1.2

0 10 20 30

|β|

(c)


0.6 0.8 1

1.2 1.4 1.6 1.8

0 10 20 30

c/c r

ef

(d)

-20

-10

0

10

20

30

0 10 20 30

v g d

irect

ion

[deg

] (e)

0 2 4 6 8

10 12 14 16

0 10 20 30

x sho

ck [m

m]

(f)


0

0.2

0.4

0.6

0.8

1

0 10 20 30

|Bj/B

0|

(g)

-180

-90

0

90

180

0 10 20 30

arg(

Bj/B

0) [d

eg] (h)


Ni in (111) plane

c44 = 122.0 GPacref = 3700 m/sρ = 8912 kg/m3


Figure 6.17: Dependence of nonlinearity parameters on direction of propagationin the (111) plane of Ni. The direction of propagation is measured in degreesfrom the 〈112〉 direction (to the 〈101〉 direction). The elements are periodicevery 60◦ and symmetric about the 30◦ direction.

185


0 101 −90 0 17 101 None −610 87 −90 43 17 87 −46 −2018 10 −90 28 13 10 −108 −9325 −38 −90 12 6 −38 −140 −13430 0 −90 0 0 0 −90 −90

Table 6.3: Phases of key linear and nonlinear parameters for the selected prop-agation directions in the (111) plane of Ni. Plots of various quantities in thesedirections are featured in Figures 6.18 and 6.19. The parameters θlong, θtran,and θvert may be used to compare longitudinal, transverse, and vertical velocitywaveforms to Figures 6.5 and 6.6.

directions are pure mode directions, but neither are “Rayleigh-type” modes

for the same reasons as in Si. Figure 6.17(g) indicates that in the region near

the 30◦ direction, the transverse linear amplitude factor B2 exceeds that of the

longitudinal component like in KCl, but that its phase approaches 0◦, as for

Si. Finally, Figure 6.17(f) shows the shock formation distance estimated from

Eq. (2.59). As before, these estimates should probably be used with care given

the complicated interactions exhibited by these waveforms.

Figures 6.18 and 6.19 display the velocity waveforms and the frequency

spectra plus harmonic propagation curves, respectively, for the directions 0◦,

10◦, 18◦, 25◦, and 30◦ from 〈112〉. In addition to these figures, Table 6.3 lists

the phases of S11 and Bj , as well as the angles θlong, θtran, and θvert, which can

be used to compare the various velocity waveforms to Figures 6.5 and 6.6.


1. 0◦ direction: In this direction, the nonlinearity matrix element |S11| '0.14 and θlong = arg(S11) ' 101◦. In Figure 6.7, the longitudinal velocity

waveform in this direction was compared to the corresponding trans-


186

-1

0

1

2

0 π 2π

0°

Vx

-2

-1

0

1

2

0 π 2π

0°

Vy

-2

-1

0

1

2

0 π 2π

0°

Vz

-180

-90

0

90

180

0 10 20 30

arg(−Slm)

0°

-1

0

1

2

0 π 2π

10°

-3

-2

-1

0

1

0 π 2π

10°

-3

-2

-1

0

1

0 π 2π

10°

-180

-90

0

90

180

0 10 20 30

10°

-1

0

1

0 π 2π

18°

-3

-2

-1

0

1

0 π 2π

18°

-3

-2

-1

0

1

0 π 2π

18°

-180

-90

0

90

180

0 10 20 30

18°

-1

0

1

0 π 2π

25°

-2

-1

0

1

2

0 π 2π

25°

-2

-1

0

1

2

0 π 2π

25°

-180

-90

0

90

180

0 10 20 30

25°

-1

-0.5

0

0.5

1

0 π 2π

30°

ωτ

-3

-2

-1

0

1

2

0 π 2π

30°

ωτ

-3

-2

-1

0

1

2

0 π 2π

30°

ωτ

-180

-90

0

90

180

0 10 20 30

30°


Nor

mal

ized

Vel

ocity

Wav

efor

ms

in (

111)

Pla

ne fo

r N

i


187

-80

-40

0

0 50 100 150

0°

Spectrum [dB]

0

0.5

1

0 2 4 6 8 10

0°

Harmonic Magnitudes

-180

-90

0

90

180

0 2 4 6 8 10

0°

Harmonic Phases

-180

-90

0

90

180

0 10 20 30

arg(−Slm)

0°

-80

-40

0

0 50 100 150

10°

0

0.5

1

0 2 4 6 8 10

10°

-180

-90

0

90

180

0 2 4 6 8 10

10°

-180

-90

0

90

180

0 10 20 30

10°

-80

-40

0

0 50 100 150

18°

0

0.5

1

0 2 4 6 8 10

18°

-180

-90

0

90

180

0 2 4 6 8 10

18°

-180

-90

0

90

180

0 10 20 30

18°

-80

-40

0

0 50 100 150

25°

0

0.5

1

0 2 4 6 8 10

25°

-180

-90

0

90

180

0 2 4 6 8 10

25°

-180

-90

0

90

180

0 10 20 30

25°

-80

-40

0

0 50 100 150

30°

n

0

0.5

1

0 2 4 6 8 10

30°

x/x0

-180

-90

0

90

180

0 2 4 6 8 10

30°

x/x0

-180

-90

0

90

180

0 10 20 30

30°


Spe

ctra

and

Har

mon

ic P

ropa

gatio

n C

urve

s in

(11

1) P

lane

for

Ni


188

in reasonably good agreement. As in the Si case, the vertical velocity

waveform compares favorably with the nonlinear Rayleigh wave in steel

transformed by θvert ' −6◦, as listed in Table 6.3. The spectra and har-

monic magnitude curves in Figure 6.16 look fairly typical of SAWs. The

harmonic phase curves show relatively little variation during propagation

as would be expected from the high phase similarity of the nonlinearity

matrix elements in this direction.

2. 10◦ direction: In this direction, the nonlinearity matrix element |S11| '0.045 and θlong = arg(S11) ' 87◦. Unlike this direction in the other

crystals considered, the nonlinearity matrix elements are still quite similar

in phase in this direction. As a result, use of the angles θlong, θtran, and

θvert from Table 6.3 to compare the waveforms of Figure 6.18 with the

the transformed waveforms of Figures 6.5 and 6.6 yields favorable results.

Figure 6.19 shows that the harmonic phase curves show little variation

although, except for the fundamental, they are shifted upward slightly as

compared to the 0◦ direction.

3. 18◦ direction: In this direction, the nonlinearity matrix element |S11| '0.00783 and θlong = arg(S11) ' 10◦. Here the nonlinearity matrix ele-

ments start to separate more in phase. This dissimilarity gives rise to the

same result as before, namely, that the waveforms develop oscillations

near the shocks and peaks. One difference between this case and similar

cases in Si and KCl is that here the oscillations form to the left of the

shocks and peaks. This may be due to the fact that the phases of the

matrix elements S12 and S13 are less than S11 instead of more. Note that

the phases of all the nonlinearity matrix elements are near zero, which

indicates that the waveforms should look roughly like “positively” distort-

ing Rayleigh waves. If the oscillations are factored out of the waveforms

189

in Figure 6.18, this “positively” distorting trend can be seen (with some

imagination), especially in the vertical velocity component (θvert ' 93◦).

Figure 6.18 shows peaks in the frequency spectra and more variation in

the harmonic phase curves, especially near the source. The phases of

the odd harmonics are grouped around 0◦ and the even harmonics are

grouped around −180◦.

4. 25◦ direction: In this direction, the nonlinearity matrix element |S11| '0.0035 and θlong = arg(S11) ' −38◦. The nonlinearity matrix elements

are farther apart in phase here than in the 18◦ direction and, as shown

in Figure 6.18, this difference results in even larger oscillations near the

shocks and peaks. Figure 6.19 shows peaks in the frequency spectra and

larger variations in the harmonic phase curves. Note, however, that the

lower magnitudes of the nonlinearity matrix elements result in energy

being transferred to the higher harmonics at a decreased rate, as seen in

the frequency spectra in comparison to the 0◦ and 10◦ cases.

5. 30◦ direction: In this direction, the nonlinearity matrix element |S11| '0.0035 and θlong = arg(S11) ' 0◦. All the nonlinearity matrix elements

are real-valued in this direction. Most of the matrix elements Slm which

describe coupling between the lowest harmonics are positive, although

although some of the higher elements are negative. As seen in Figure 6.18,

this results in waveforms that distort in the “positive” manner seen in

the (001) and (110) planes. Note that the values of θlong, θtran, and θvert

given in Table 6.1 match the waveforms well, as would be expected from

the phase similarity of the nonlinearity matrix elements. Similar to the

25◦ direction, the frequency spectra in Figure 6.18 show that less energy

is transferred to the higher harmonics due to the reduced magnitudes of

the nonlinearity matrix elements. Examination of the harmonic phases

190

shows that the odd harmonics are positive and the even harmonics are

negative. This information is collapsed into the two lines at 0◦ and 180◦

in the harmonic phase curves.

In summary, the simulations in Ni show many features in common with Si

and KCl, including the appearance of oscillations in the velocity waveforms

where the nonlinearity matrix elements are less similar in phase, and distortion

that is similar to (001) and (110) planes near the 30◦ direction. The most

significant difference for Ni is that in the directions where oscillations form in

the waveforms, they form to the right of the peaks and shocks, instead of to

the left.

6.3 Summary

This chapter has studied the propagation of nonlinear SAWs in the (111) plane

for a variety of cubic crystals. The SAWs in this plane differ from those in the

(001) and (110) planes in that the nonlinearity matrix elements are generally

complex-valued. A simple mathematical transformation was shown to provide

an interpretation of the phase information contained in the nonlinearity ma-

trix elements and linear amplitude factors. Comparisons were made between

waveforms approximated by this method and those generated with the full sim-

ulation. The match was shown to be best when most of the nonlinearity matrix

elements of the crystals have the same or approximately the same phase. As

in previous chapters, it was demonstrated that plots of the nonlinearity matrix

elements as a function of direction can be used as an indication of the harmonic

generation and waveform distortion. In most directions, the waveforms distort

asymmetrically and, in some cases, the dissimilar phases of the nonlinearity

matrix elements cause a nonlinear dispersion-like effect whereby oscillations

191

form in the vicinity of the shocks and peaks of the velocity waveforms. De-

tailed analysis was provided for Si, KCl, and Ni, with other crystals expected

to exhibit similar effects.

Chapter 7

Pulsed SAWs and Experimental Results

All of the simulations presented in Chapters 4, 5, and 6 are based upon monofre-

quency source conditions. This was done to show most clearly the basic features

of the harmonic generation and waveform distortion in the crystals. How-

ever, experiments have shown that nonlinear effects also can be seen in SAW

pulses. This chapter discusses the results of a collaboration with experimen-

talists P. Hess of the Institute of Physical Chemistry, University of Heidelberg,

Heidelberg, Germany, and A. Lomonosov and V. G. Mikhalevich of the Gen-

eral Physics Institute, Russian Academy of Sciences, Moscow, Russia. It begins

with a description of their experimental technique and then presents a com-

parison of numerical simulations and their measured data.∗ The comparisons

show that (1) the waveforms predicted by the numerical simulations quanti-

tatively reproduce the features of the measured waveforms in the (001) and

(111) planes of crystalline silicon, and (2) the regions of “positive” and “nega-

tive” nonlinearity predicted to exist in the (001) plane of crystalline silicon are

experimentally corroborated.

7.1 Experimental Technique

The photoelastic technique for measuring nonlinear SAW pulses with distinct

shocks was first demonstrated by Lomonosov and Hess24 in 1996. A schematic

∗The major results of this chapter have been presented previously in several papers,53,54

although in significantly less detail.

192

193

++ --

xy

z

crystal

splitphotodiodes

beamsprobe

absorbinglayer

cw

excitationpulsed laser

Figure 7.1: Schematic diagram of the experimental apparatus for photoelasticnonlinear surface acoustic wave generation with dual laser probe detection.

diagram of the experiment is shown in Figure 7.1. In their method, excitation

is accomplished using an Nd:YAG laser of wavelength 1064 µm, pulse duration

7 ns, and energy up to 50 mJ that is focused into a strip of length 6 to 8 mm and

width 50 µm on the surface of the solid. This geometry creates a SAW beam

which propagates outward from the the excitation region. A strongly absorbing

carbon layer in the form of an aqueous suspension is placed in the strip area to

facilitate energy transfer to the surface and intensify the excitation. To mea-

sure the transient SAW waveforms, a laser probe beam deflection setup using

stabilized cw Nd:YAG laser probes of wavelength 532 nm and power 120 mW

is employed. Two probe beams are focused into spots approximately 4 µm in

diameter located 14 to 16 mm apart, with the closest probe about 5 mm from

the excitation region. As the SAW pulse passes through the area covered by

the probe beams, the deflection of the beams is detected by two photodiodes.

Because the deflection of the beam is proportional to the slope ∂uz/∂x of the

194

surface and because ∂uz/∂x = −1/c∂uz/∂t = −vz/c for a progressive wave, it

follows that the photodiode output is proportional to the vertical velocity com-

ponent vz at the the surface. The bandwidth of the whole detection system is

limited to about 500 MHz. The resulting SAW pulses on this surface are typi-

cally 20–40 ns duration with peak strains between 0.005 and 0.01. The method

has been used successfully to generate surface waves in both isotropic media

(e.g., fused quartz24–26) and anisotropic media (e.g., crystalline silicon24,53,54).

Calibration and alignment of the probe beams is a critical part of the

experiment. To calibrate the probe setup to allow for absolute measurements,

the differential output of each pair of photodiodes is measured for a known

angle shift of the probe beams. In crystalline media, the orientation of the

in-plane crystalline axes must also be determined. This is done by finding

the propagation directions which made the SAW phase speed an extremum for

linear waves. However, reliable data sets could only be taken for pure mode

directions because alignment of the probe beams in the SAW path is more

difficult when the wave vector is not coincident with the power flow vector (see

Figure B.7).

7.2 Comparison of Theory and Experiment

Numerical simulations were performed to compare theory and experiment for

three different data sets taken in crystalline silicon:

• 0◦ from 〈100〉 in the (001) plane



195

To prepare the data for the simulations, first the linear problem was solved

for the particular cut and direction under consideration to obtain the linear

SAW speed c and other parameters of the linear solution, including the lin-

ear amplitude factors B1, B2, and B3. These values were used to compute

the nonlinearity matrix, particularly S11 and the nonlinearity coefficient β,

both of which characterize the strength of the nonlinearity (especially harmonic

self-interaction, such as the fundamental interacting with itself to produce a

second harmonic). All these calculations were performed using the density

ρ = 2331 kg/m3, SOE constants from Hearmon114 (see Table 3.3), and TOE

constants from McSkimin and Andreatch133 (see Table 3.3). Finally, the fre-

quency spectrum of the measured waveform at the probe beam location closest

to the excitation region was computed from the time waveform, appropriately

scaled, and used as the source condition for the dimensionless evolution equa-

tions given by Eq. (2.40).

The scaling parameters were computed in the following way. The char-

acteristic velocity of the vertical velocity waveform was chosen to be

vz0 =max(vz)−min(vz)

2, (7.1)

or half the peak-to-peak vertical velocity. The peak strain or, equivalently,

the Mach number, is then given by M = vz0/c. The characteristic spectral

amplitude was then computed using the vertical velocity analogue to Eq. (2.49):

v0 =vz0

2|B3| , (7.2)

where B3 is the linear amplitude factor for the vertical velocity component

defined in Eq. (2.42), and the factor of two is explained in the footnote near

Eqs. (2.47). The characteristic frequency f0 = ω/2π was chosen to be the

frequency corresponding to the spectral component of highest amplitude. The

196

characteristic length scale x0 was taken to be the estimated shock formation

distance

x0 = x11 =ρc4

4|S11|ωv0, (7.3)

as given by Eq. (2.59). In all cases the absorption was taken to have a quadratic

dependence on frequency152 so that An = n2A1 (see Section 2.1.2 for discussion

of this assumption). The absorption coefficient A1 was chosen differently in

each case, as described below.

The theory presented in Chapter 2 is based on the assumption that the

signal can be expanded in a series of plane waves as defined in Eq. (2.35). This

assumption has two implications. First, the signal must be periodic. In order

to model pulses, they were assumed to repeat with a period Tfund, where this

value was chosen differently based on the individual data set. The fundamental

frequency of the Fourier series expansion is then ffund = 1/Tfund. Second, the

signal must have planar wavefronts, i.e., it cannot exhibit a dependence on the

transverse coordinate. Hence the theory is applicable only if diffraction effects

are negligible, i.e., the total propagation distance z is less than the characteristic

Rayleigh distance∗ z0 of the beam. The ratio of these two lengths is given by

the dimensionless diffraction parameter

D =z

z0=

4zc

πf0d2, (7.4)

where d is the beam width at the source. In all cases discussed, D � 1,

and therefore diffraction effects were negligible. The value of the diffraction

parameter D is provided below for each case.

∗The Rayleigh distance is the approximate distance past which the wavefronts in the beamare no longer essentially planar. It marks the transition between the near-field and far-fieldregions of the beam. For more information on nonlinear sound beams, see Hamilton.153

197

With the assumptions of the theory satisfied, the evolution equations

were then integrated numerically using a fixed step size, fourth-order Runge–

Kutta routine. Several parameters may be varied in the integration routines

including the number of harmonics, step size, and absorption. The frequency

spectra were truncated between N = 400 and N = 1200 harmonics. Because

the pulses are initially broadband, a larger number of harmonics must be used

than with monofrequency waves (N = 200) to minimize errors associated with

the truncation of the spectrum. A smaller number of harmonics is desirable

because computation time is proportional to N2; however, too few harmonics

can introduce spurious oscillations in the waveforms or cause numerical insta-

bility. (Note that the number of harmonics listed in the tables below is not

necessarily the minimum number possible to gain accurate results.) The di-

mensionless step size of the integration was taken to be between ∆X = 0.0001

and ∆X = 0.0005. By trial and error, this range was found to maintain nu-

merical stability while minimizing computation time. Typically, stability was

also enhanced by using a smaller step size ∆Xinit in the region 0 ≤ X ≤ Xswitch

near the source, and then switching to a larger values of ∆X for the rest of the

integration. A suitable value of Xswitch was found to be approximately where

the Nth harmonic achieved its maximum value after the initial spread of energy

over the entire spectral range. Finally, the absorption coefficient A1 was usually

chosen to be as small as possible to maintain numerical stability and provide

reasonable computation time. Because the quadratic dependence of absorp-

tion on frequency implies that most of the energy is dissipated at the highest

harmonics, integrating with lower absorption requires using a larger number of

harmonics which, in turn, quadratically increases computation time. All the

numerical parameters used for the integration are provided below for each case.

198

The equations were integrated out to the dimensionless location of the

remote probe beam

Xmax =xmax

x0

, (7.5)

where xmax is the distance between the probe beams. [An exception to this

occurred for the direction in the (111) plane; see Section 7.2.2 for additional

discussion.] The resulting frequency spectra were then reconstructed into time

waveforms using Eq. (2.35) and compared with the measured waveform at

the remote probe beam location. The longitudinal waveforms are computed

from the vertical velocity waveforms using the linear transformation given by

Eq. (2.100). To further reduce the effects of numerical errors associated with

spectrum truncation, only the first 300 harmonics were used to reconstruct the

waveforms. Even so, this still means that the bandwidth of the simulation is

several times larger than the 500 MHz bandwidth of the experiment. As is

shown below, further truncation of the simulated spectra via shading functions

helped to better match the amplitude of the waveforms, especially near peaks

and shocks.

7.2.1 Si in (001) plane

As described in Section 4.2.2, the nonlinearity in the (001) plane of crystalline

silicon divides into three distinct regions based upon the angle θ between the

propagation direction and the 〈100〉 axis:

• Region I: 0◦ ≤ θ < 21◦, “negative” nonlinearity, vx steepens “backward;”

• Region II: 21◦ < θ < 32◦, “positive” nonlinearity, vx steepens “forward;”

• Region III: 32◦ < θ ≤ 45◦, “negative” nonlinearity, vx steepens “back-

ward.”

199

-0.02

0

0.02

0.04

0.06

0 5 10 15 20 25 30 35 40 45

S

S

S-S

/

c

13

12

1144

lm

Angle θ from [degrees]⟨100 ⟩

Nonlinearity Matrix Elements for Si in (001) Plane

I II III

Figure 7.2: Nonlinearity matrix elements S11, S12, S13 for crystalline silicon inthe (001) plane as a function of direction. Due to the symmetries of this cut,the matrix elements are symmetric about 45◦ and periodic every 90◦.

These regions are shown in Figure 7.2, which is essentially an an enlarged

version of Figure 4.3(a). As seen in Figure 4.3(e), pure modes occur only

in the directions θ = 0◦, 26◦, and 45◦. However, neither of the two surface

modes which occur at 45◦ can be modelled by the theory. The first mode

is an exceptional bulk wave (see Section B.1.4), which does not satisfy the

requirement of the theory that the amplitude decay to zero at infinity. In

addition, because this mode has only a transverse velocity component, it cannot

be measured by the probe beam deflection method. The second mode is a

pseudosurface wave mode (see Section B.1.5). While this mode can be excited

and measured by the probe beam deflection, modelling this mode would require

modifications to the theory to allow coupling with bulk modes. Hence only

propagation in the 0◦ and 26◦ directions is considered here. Table 7.1 presents

200

Table 7.1: Physical, experimental, and numerical parameters for SAW pulsesin the directions 0◦ and 26◦ from 〈100〉 in the (001) plane of crystalline silicon.

Parameter Variable 0◦ 26◦

SAW speed c 4902 m/s 4967 m/sLinear amplitude factors |B1| 0.481 0.398

|B2| 0.000 0.0910|B3| 0.589 0.543

Nonlinearity matrix element −S11/c44 −0.022 0.036Nonlinearity parameter β −0.11 0.18Vert. velocity amplitude vz0 24.2 m/s 22.8 m/sPeak strain M 0.0049 0.0046Char. spectral amplitude v0 20.5 m/s 21.0 m/sCharacteristic frequency f0 30 MHz 40 MHzCharacteristic length x0 50 mm 23 mmFundamental frequency ffund 5 MHz 5 MHzBeam width d 6 mm 6 mmTotal propagation dist. z 20 mm 20 mmDiffraction parameter D 0.115 0.087Probe beam separation xmax 14.6 mm 14.6 mmNumber of harmonics N 1200 1200Step size ∆X 0.0005 0.0005

Initial step size ∆Xinit 0.0001 0.0001Range for ∆Xinit Xswitch 0.05 0.05

Maximum range Xmax 0.29 0.63Absorption coefficient A1 1/3600 1/6400

201

the physical, experimental, and numerical parameters associated with these

directions.

Figure 7.3 shows a comparison of experiment and theory in the direc-

tion 0◦ from 〈100〉 for a SAW pulse in the (001) plane of crystalline silicon.

Figures 7.3(a)–(c) show the experimental data for the location close to the

source. Figure 7.3(a) shows the directly measured vertical velocity waveform

vz. Figure 7.3(b) shows the longitudinal velocity waveform vx, calculated from

the measured vz waveform via Eq. (2.100). Due to the symmetry of this di-

rection, there is no transverse velocity component (B2 = 0) and the particle

displacement is contained in the sagittal plane. Figure 7.3(c) shows the fre-

quency spectrum corresponding to the waveforms, normalized such that the

peak of the spectrum occurs at 0 dB. This was the spectrum taken as the

starting condition for the integration of Eq. (2.40). The harmonic of peak am-

plitude occurred at 30 MHz, corresponding to the harmonic number n = 6.

The absorption coefficient α1 was selected by setting 1/α6 = 100x0, where

x0 = 50 mm is the characteristic shock formation distance. By choosing the

absorption length to be much longer than the shock formation distance and the

total propagation distance, absorption was taken to be weak with respect to the

nonlinearity. Although the computations were performed with 1200 harmonics,

approximately only the first 120 harmonics (600 MHz bandwidth) were used

to reconstruct the waveforms to make a fairer comparison with the experiment

(500 MHz bandwidth). This was done by shading the spectral components ac-

cording to V shaden = Vn exp[−(n/120)16]. The effect of such a shading function

is discussed in more detail near the end of Section 7.2.2.

Figures 7.3(d)–(f) show the experimental data (solid lines) along with

the theoretical spectrum and waveforms (dashed lines) that result from inte-

grating the evolution equations out to Xmax. In this region, the coefficient

202

-40-30-20-10

010203040

0 20 40 60 80

v z [m

/s]

t [ns]

(a)

-40-30-20-10

010203040

0 20 40 60 80

v z [m

/s]

t [ns]

(d)

-40-30-20-10

010203040

0 20 40 60 80

v x [m

/s]

t [ns]

(b)

-40-30-20-10

010203040

0 20 40 60 80

v x [m

/s]

t [ns]

(e)

-30

-25

-20

-15

-10

-5

0

0 50 100 150 200

Spe

ctru

m le

vel [

dB]

f [MHz]

(c)

-30

-25

-20

-15

-10

-5

0

0 50 100 150 200

Spe

ctru

m le

vel [

dB]

f [MHz]

(f)

Figure 7.3: Comparison of experiment (solid lines) and theory (dashed lines) fora surface acoustic wave pulse propagating in the direction 0◦ from 〈100〉 in the(001) plane of crystalline silicon, from the location close to the excitation region(upper row) to the remote location 14.6 mm away (lower row). The referencefor the spectrum levels in (c) and (f) is the peak value of the spectrum at theclose location. The “measured” longitudinal velocity waveforms in (b) and (e)are calculated from the measured vertical velocity waveforms in (a) and (d).[The waveforms are reproduced from Kumon et al.54 by permission.]

203

of nonlinearity is negative. Hence in the longitudinal velocity waveform vx

the peaks should travel slower than the SAW speed and troughs should travel

faster. This behavior can be most easily seen in Figures 7.3(b) and 7.3(e),

where the trough becomes shallower as it advances and the pulse evolves into

an inverted N-shape. The difference in local propagation speeds due to the

nonlinearity results in pulse lengthening of the waveforms as the trough and

peak move apart in time. The lengthening is also reflected in the spectra of

Figures 7.3(c) and 7.3(f) by the shifting of the spectral peaks to lower frequen-

cies. The differences between experiment and theory show up most clearly in

the spectra of Figure 7.3(f), where the measured spectrum has less energy in

the highest harmonics than the predicted spectrum. It is unclear if this is due

to the assumption of the “frequency squared” absorption relation, or due to

the nature of the nonlinearity, or both. One result of this effect is that the

sharp peaks and shocks of the waveforms are reduced in amplitude from what

they might otherwise be if there were more energy in the high frequency com-

ponents. This appears to be the largest difference between the predicted and

measured waveforms in Figures 7.3(d) and (e).

Figure 7.4 shows a comparison of the experiment with theory in the

direction 26◦ from 〈100〉 for a SAW pulse in the (001) plane of crystalline sil-

icon. Figures 7.4(a)–(c) show the experimental data for the location close to

the source for the same quantities as in Figure 7.3. Unlike in the 0◦ case, the

particle motion is tilted out of the sagittal plane by about 13◦. The resulting

transverse velocity component vy is small, however, and is omitted. The spec-

trum of Figure 7.4(c) was taken as the starting condition for the integration

of Eq. (2.40), and the integration and reconstruction parameters were taken to

be the same as in the 0◦ case. The absorption coefficient was also chosen in a

similar way except that the harmonic of peak amplitude occurred at 40 MHz,

corresponding to the harmonic number n = 8. The attenuation coefficient α1

204

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/s]

t [ns]

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/s]

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0 50 100 150 200

Spe

ctru

m le

vel [

dB]

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(f)

Figure 7.4: Comparison of experiment (solid lines) and theory (dashed lines) fora surface acoustic wave pulse propagating in the direction 26◦ from 〈100〉 in the(001) plane of crystalline silicon, from the location close to the excitation region(upper row) to the remote location 14.6 mm away (lower row). The referencefor the spectrum levels in (c) and (f) is the peak value of the spectrum at theclose location. The “measured” longitudinal velocity waveforms in (b) and (e)are calculated from the measured vertical velocity waveforms in (a) and (d).[The waveforms are reproduced from Kumon et al.,54 by permission.]

205

was selected by setting 1/α8 = 100x0, where x0 = 23 mm is the characteristic

shock formation distance.

The predicted frequency spectra and waveforms (dashed lines) are com-

pared with the measurements (solid lines) in Figure 7.4(d)–(f). In this region,

the coefficient of nonlinearity is positive. Hence in the longitudinal velocity

vx the peaks should travel faster than the SAW speed and troughs should

travel slower. This behavior can probably be most easily seen in Figures 7.4(b)

and 7.4(e), where the blunter positive peaks evolve into the sharper positive

peaks. The nonlinearity causes the waveforms to develop into a sawtooth shape

like in a fluid, except for the cusping near the shock front. In addition, pos-

itive nonlinearity causes negative peaks of the vertical velocity vz to become

more negative, as seen in Figures 7.4(a) and 7.4(d). The differences between

experiment and theory show up most clearly in the spectra in Figure 7.4(f).

The decrease in the peak amplitude from its value in Figure 7.4(f) is due to

a combination of energy transfer to higher harmonics and a small amount of

absorption. However, the measured spectrum has less energy in the highest

harmonics than the predicted spectrum, and it also does not exhibit the same

distinct peaks. As in the 0◦ case, it is unclear if this is due to the assump-

tion of the “frequency squared” absorption relation or due to the nature of the

nonlinearity or both.

In summary, the results demonstrate that the theory is corroborated

by the experiment in the 0◦ direction of Region I and the 26◦ direction of

Region II in terms of both the qualitative behavior and quantitative agreement

of the waveforms. It is striking that while the waveforms measured at the close

locations in the 0◦ and 26◦ directions are initially similar in form, they evolve

to waveforms that are significantly different. This is consistent with the regions

of “positive” and “negative” nonlinearity delineated in Figure 7.2.

206

7.2.2 Si in (111) plane

Next consider propagation of SAW pulses in the (111) plane of crystalline

silicon. Figure 6.9(e) shows that only the 0◦ and 30◦ directions are pure modes.

Of these two directions, Figure 6.9(a) shows that the effect of nonlinearity is

approximately an order of magnitude higher in the 0◦ direction than in the 30◦

direction. Note also that the magnitudes of the nonlinearity matrix element

|S11|/c44 and nonlinearity coefficient |β| are several times larger as compared

to the 0◦ and 26◦ directions of the (001) plane. Figure 6.9(b) shows that

the phases of the first few nonlinearity matrix elements are in the vicinity of

106◦ and are relatively close together. As demonstrated for monofrequency

source conditions, this complex-valued nonlinearity results in the asymmetric

distortion of the velocity waveforms, as seen in Figure 6.10. The combination of

heightened nonlinearity and asymmetric distortion should result in significantly

different waveforms than observed in the (001) plane.

Figure 7.5 provides a comparison of the experiment with theory for a

SAW pulse in the (111) plane of crystalline silicon in the direction 0◦ from

〈112〉. As seen in Figure 6.10(g), the wave has no transverse velocity compo-

nent (B2 = 0), and hence only the vertical and longitudinal components are

shown in Figure 7.5. Figures 7.5(a)–(c) display the measured waveforms and

spectrum at distance x = 5 mm from the excitation region. As before, the lon-

gitudinal velocity waveforms are computed from the vertical velocity waveforms

using Eq. (2.100). Figure 7.5(c) shows the corresponding frequency spectrum

that was used as the starting condition for the integration of Eq. (2.40). The

spectrum is normalized so that its peak at 50 MHz occurs has an amplitude

of 0 dB. The absorption coefficient was chosen so that 1/α1 = 40x0 where

x0 = 2.9 mm is the characteristic shock formation distance. The computa-

tions were performed with 400 harmonics, although only the first 70 harmonics

207

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/s]

t [ns]

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/s]

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ctru

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(c)

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ctru

m le

vel [

dB]

f [MHz]

(f)

Figure 7.5: Comparison of experiment (solid lines) and theory (dashed lines) forsurface waves propagating in the 〈112〉 direction in the (111) plane of crystallinesilicon, from x = 5 mm (upper row) to x = 21 mm (lower row). The referencefor the spectrum levels in (c) and (f) is the peak value of the spectrum at theclose location. The “measured” longitudinal velocity waveforms in (b) and (e)are calculated from the measured vertical velocity waveforms in (a) and (d).[The figure is reproduced from Kumon et al.53 by permission.]

208

(700 MHz bandwidth) were used to reconstruct the waveforms. As before, this

was done to make a fairer comparison with the experiment (500 MHz band-

width). The shading function V shaden = Vn exp[−(n/70)16] was applied to the

spectral components prior to reconstruction. All of the physical, experimental,

and numerical parameters associated with this direction are given in Table 7.2.

Table 7.2: Physical, experimental, and numerical parameters for SAW pulsesin the directions 0◦ from 〈112〉 in the (111) plane of crystalline silicon.

Parameter Variable 0◦

SAW speed c 4720 m/sLinear amplitude factors |B1| 0.328

|B2| 0.000|B3| 0.510

Nonlinearity matrix element |S11|/c44 0.11Nonlinearity parameter |β| 0.65Vert. velocity amplitude vz0 37.1 m/sPeak strain M 0.0079Char. spectral amplitude v0 35.9 m/sCharacteristic frequency f0 50 MHzCharacteristic length x0 2.9 mmFundamental frequency ffund 10 MHzBeam width d 6 mmTotal propagation dist. z 20 mmDiffraction parameter D 0.066Probe beam separation xmax 16 mmNumber of harmonics N 400Step size ∆X 0.0001

Initial step size ∆Xinit 0.0001Range for ∆Xinit Xswitch 0.2

Maximum range∗ Xmax 0.8Absorption coefficient A1 1/40

∗This was the range that matched the waveforms best by inspection. The value calculatedfrom Eq. (7.5) is Xmax = 5.44.

209

Figures 7.5(d)–(f) show the measurements (solid lines) compared with

the calculations (dashed lines) for the frequency spectrum and waveforms that

result from integrating the evolution equations out to Xmax = 0.8. This value

was chosen by inspecting the theoretical waveforms after various propagation

distances to determine the one that best fits the experimental data. Unfortu-

nately, the value Xmax = 5.44 predicted by Eq. (7.5) appears to significantly

overestimate the characteristic nonlinear length scale. The cause of this dis-

crepancy is not clear, although it may be related to the more complicated

nature of the waveform distortion that results from complex-valued nonlinear-

ity matrix elements. Note, however, that no other parameters were varied to

achieve these results, other than the ending location of the integration. Despite

this scaling problem, the measured waveforms and spectra are reproduced well

at the remote location, as shown in Figure 7.5(d)–(f). The nonlinear waveform

distortion is predicted accurately by the theory, including the increase in pulse

duration between the close and remote locations. The lengthening is mani-

fested in the shift of the spectral peak from 50 MHz down to about 30 MHz.

In addition, there are notable differences in the distortion of the waveform as

compared to measurements of nonlinear Rayleigh waves in fused quartz26 and

the nonlinear SAWs in the (001) plane shown in Section 7.2.1. In the case

shown in Figure 7.5, the vertical component vz has an N-shaped waveform and

the horizontal component vx has a U-shaped waveform, whereas the reverse

was observed in isotropic solids and in the 0◦ direction in the (001) plane of

silicon. The case shown in Figure 7.5 also differs from the 26◦ direction in the

(001) plane of silicon, where the vz component develops a sharp negative peak

and the vx component develops into a cusped sawtooth wave. Nevertheless, the

evolution of the waveforms is consistent with the distortion seen in the initially

monofrequency waveforms of Figure 6.10, which show a shock forming in the

vz waveform and a positive peak and shallow trough forming the vx direction.

210

Hence Figure 6.10 shows that the change of the measured waveforms in the

(111) plane as compared to previous cases in the (001) plane is a direct re-

sult of the approximately 100◦ phase shift induced by the first few nonlinearity

matrix elements.

For the reasons stated above, the predicted frequency spectra shown in

Figures 7.3, 7.4, and 7.5 incorporate shading to better match the bandwidth

of the experiment. To demonstrate the effect of the shading, the longitudinal

velocity waveforms for the 0◦ direction in the (111) plane were reconstructed

(1) with shading to simulate approximately 700 MHz of bandwidth, and (2)

without shading to simulate 3000 MHz of bandwidth. The results are shown in

Figure 7.6. In both cases the simulations were run with a maximum bandwidth

of 4000 MHz (400 harmonics × 10 MHz fundamental frequency). The largest

change in the waveforms is that the peak heightens and narrows, increasing

from around 35 m/s to around 50 m/s. This is not unexpected because adding

additional high frequency information allows steeper shocks and sharper peaks

to be resolved. The exact form of the exponential shading profile shown in Fig-

ure 7.6 was selected to provide a rapidly decaying function that was not discon-

tinuous. Trials with other functions that retained roughly the same amount of

frequency content (including a step function) were found to cause relatively lit-

tle change in the shape of the reconstructed waveforms. However, reducing the

effective bandwidth further (even as high as 500 MHz) was found to introduce

excessively blunt peaks and overemphasize low frequency oscillations. One pos-

sible conclusion from this result is that an experiment with higher bandwidth

capabilities may measure even larger peaks and steeper shocks than seen in the

experimental waveforms in this chapter.

211

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(c) (d)

(b)(a)

theory

experiment

exp(-[ f / 700 ]16 )

experimenttheory

experiment

theory

experimenttheory

Figure 7.6: Comparison of longitudinal velocity waveforms from experiment(solid) and theory (dashed) reconstructed with shading functions of approxi-mately 700 MHz bandwidth [(a),(b)] and exactly 3000 MHz bandwidth [(c),(d)].The waveforms are the result of propagation in the direction 0◦ from 〈112〉 inthe (111) plane of crystalline silicon.

212

7.3 Summary

This chapter has discussed nonlinear SAW pulses, specifically focusing on prop-

agation in crystalline silicon. Through collaboration with an experimental

group,53,54 measurements were obtained for the directions 0◦ and 26◦ from

〈100〉 in the (001) plane and the direction 0◦ from 〈112〉 in the (111) plane.

The photoelastic technique developed by Lomonosov, Mikhalevich, and Hess

for the generation and detection of nonlinear SAWs is described. In addition,

the procedure used to perform the numerical analysis of their measurements is

reviewed in detail. In all cases, favorable agreement is achieved between exper-

iment and theory. In the (001) plane, the pulses corroborate the notion that

there exist regions of directions with “positive” and “negative” nonlinearity. In

the (111) plane, the waveform distortion is consistent with the phase changes

associated with the predicted complex-valued nonlinearity matrix elements.

Hopefully, further experimental studies will be performed to investigate the

broad range of phenomena that have been proposed to exist in the propagation

of nonlinear SAWs in cubic crystals.

Chapter 8

Summary

This dissertation has undertaken a comprehensive study of nonlinear surface

acoustic waves (SAWs) in nonpiezoelectric cubic crystals based on the theory

developed by Hamilton, Il’inskii, and Zabolotskaya33 for anisotropic media. A

review of the literature reveals that very few simulations of the fully nonlinear

evolution of SAWs have been reported. The reported cases describe only a few

selected materials and particular directions of high symmetry. Moreover, no

work had presented a comparison of experiment and theory for the evolution

of nonlinear SAW waveforms in crystals. Hence the goal of this work was

to use the aforementioned theory to characterize and explain the nature of

the harmonic generation and waveform distortion of nonlinear SAWs over a

wide variety of directions, cuts, and materials, and to validate these results by

comparison with experiment.

Chapter 2 began by outlining the theory of Hamilton, Il’inskii, and

Zabolotskaya.33 By using a Hamiltonian mechanics approach, they derived a

set of frequency-domain model equations to describe the evolution of SAWs in

arbitrary directions and cuts of crystalline media of any symmetry. Quasilin-

ear solutions were derived for these evolution equations under several different

approximations. Via analogy to nonlinear bulk wave propagation in fluids and

solids, expressions were derived for estimates of the shock formation distance

and nonlinearity coefficient. A time-domain evolution equation was developed

from the frequency-domain equations and was shown to reduce to the time-

213

214

domain evolution equation for nonlinear Rayleigh waves20 in the isotropic limit.

Finally, a detailed comparison was made between the various papers derived

from the theory of Zabolotskaya for nonlinear Rayleigh waves and the afore-

mentioned theory for anisotropic media.

Chapter 3 reviewed briefly the basic properties of cubic crystals. Cubic

crystals were chosen for study because they have the highest symmetry of all

crystalline classes and, therefore, the simplest type of fully three-dimensional

anisotropy. In addition, previous investigations have shown that effects exhib-

ited in cubic crystals are often similar to those seen in other crystal types. A set

of tables was given that compiled the measured densities, second-order elastic

constants, and third-order elastic constants for all the materials modelled in

the dissertation.

Chapters 4 and 5 studied the properties of initially monofrequency non-

linear SAWs in the (001) and (110) planes, respectively. Fifteen crystals were

selected for study, including RbCl, KCl, NaCl, CaF2, SrF2, BaF2, C (diamond),

Si, Ge, Al, Ni, Cu in the m3m point group, and Cs-alum, NH4-alum, and K-

alum in the m3 point group [the hydrous X-alums have the chemical formulas

XAl(SO4)2 · 12 H2O]. Si, KCl, and Ni were chosen for study in detail. The mag-

nitudes of the nonlinearity matrix elements were shown to provide a powerful

tool for characterizing the nature of the waveform evolution in these planes.

In most crystals, these elements were demonstrated to be a strong function of

propagation direction. Several interesting effects were found in common, in-

cluding (1) propagation directions in which the nature of the velocity waveform

distortion changes in sign, (2) directions in which the generation of one or more

harmonics are suppressed, thereby causing little or no shock formation, and (3)

directions in which energy is rapidly transferred to the highest harmonics and

shock formation is enhanced. In all cases except the (110) plane of crystals in

215

the m3 point group, the nonlinearity matrix elements were shown to be real-

valued. The choice of different experimental TOE constants as input to the

simulations is shown to affect the detailed predictions of the nonlinearity ma-

trix elements in any given direction, but not the trends over the whole angular

range.

Chapter 6 studied the properties of initially monofrequency nonlinear

SAWs in the (111) plane. For all fifteen crystals studied, the nonlinearity ma-

trix elements are complex-valued. Both the magnitude and phase of the ma-

trix elements were considered to explain the waveform evolution. The relative

phases of the dominant nonlinearity matrix elements were shown to be partic-

ularly important. The phasing produces waveform distortion that is generally

asymmetric, and it can lead to a nonlinear dispersion-like effect whereby low

frequency oscillations form near the peaks and shocks in the velocity waveforms.

A novel interpretation of complex-valued nonlinearity matrices was proposed

and shown to provide a simple method for characterizing the type of waveform

distortion, especially when the dominant nonlinearity matrix elements have

similar phases.

Chapter 7 described the modelling of SAW pulses in crystalline silicon

and compared numerical simulations to experiment. Through a collaboration

with A. Lomonosov, V. G. Mikhalevich, and P. Hess, measurements were ob-

tained with nonlinear SAW pulses in the directions 0◦ and 26◦ from 〈100〉 in the

(001) plane and 0◦ from 〈112〉 in the (111) plane of crystalline Si. A brief de-

scription of the photoelastic technique for generating and measuring the SAWs

was given, followed by a discussion of the numerical analysis of the data. In the

(001) plane, it was found that the measurements validated the predictions of

the theory that there exist regions of “positive” and “negative” nonlinearity as

a function of propagation direction. In the (111) plane, the type of waveform

216

distortion was found to be reversed between the vertical and longitudinal direc-

tions. This result is consistent with the phases of the predicted complex-valued

nonlinearity matrix elements for this case.

Finally, several appendices were provided on auxiliary topics, including

a discussion of the terms “anisotropy” and “aeolotropy,” a tutorial on the basic

properties of linear and nonlinear SAWs in crystals, a review of many appli-

cations of linear and nonlinear SAWs, a description of Miller index notation,

a detailed derivation of the time-domain relation between the velocity compo-

nents of a SAW in a crystal, and additional discussion about the interpretation

of complex-valued nonlinearity parameters.

Future theoretical work in this area will probably include modelling

nonlinear SAW propagation in non-cubic crystals, using an extension of the

theory to study propagation in a variety of piezoelectric crystals, and extending

both theories to include dispersion (e.g., introduced by a thin layer on the

surface of the material) and diffraction. It is hoped that future experimental

work will further validate many of the properties of SAWs presented in this

work.

Appendix A

Anisotropic and Aeolotropic Media

In some texts, a distinction is made between anisotropic and aeolotropic media.

According to Smith and Stephens,154

These two words are often used synonymously in the literature, but

more specifically the term aeolotropic is used in connection with

materials possessing no internal boundaries such as single crystals,

while the term anisotropic is concerned with the bulk properties.

A polycrystalline aggregate in which there is partial alignment of

the individual ‘aeolotropic’ single crystal grains would exhibit ‘ani-

sotropic’ bulk properties, but a randomly oriented aggregate would

show isotropic physical characteristics.

Musgrave113 gives another example:

For example, a cylindrical bar, plastically formed from an aggregate

of crystallites with cubic symmetry may well exhibit preferential

orientation such that directions lying in the circular cross-section

are equivalent but differ from the axial direction. Such a bar is said

to be transversely isotropic and possesses on a macroscopic scale,

the same symmetry as a hexagonal crystal. Thus we see that the

aeolotropy of physical properties implied by the basic structure of

a crystal may be camouflaged or wholly smeared out by preferred

or random orientation.

The theory of Hamilton, Il’inskii, and Zabolotskaya33 used in this dissertation

characterizes completely the investigated media by the specification of the den-

sity and second- and third-order elastic constants. Thus if the elastic constants

217

218

of the medium under consideration reflect the symmetry properties of that

sample, then the nonlinear surface waves generated in the sample are correctly

described by the theory (provided all other assumptions of the theory are met).

For example, consider Musgrave’s example of a bar cut in such a way

that it has a flat surface. It would be incorrect to apply the aforementioned

SAW theory to this sample with elastic constant data for the cubic aeolotropy

of individual crystallites, but it would be appropriate to apply the theory with

elastic constants of hexagonal anisotropy measured from the bulk polycrys-

talline aggregate. Note that the theory could not be applied if the anisotropy

were inhomogeneous (e.g., if the aeolotropy or distribution of the individual

crystallites varied with location in the bar). In most cases considered in this

work, the media are single crystals. In these cases the distinction between ae-

olotropy and anisotropy is not important because the aeolotropy is the same

on both the microscopic and macroscopic scales. Thus the term “anisotropy”

is used exclusively throughout the text as this is consistent with most of the

literature on surface acoustic waves in these types of media.

As might be expected from their similar meaning, the words have a

similar etymology. The word “aeolotropy” comes from the Greek roots αιoλoς+

τρoπια meaning “changeful turning,” while the word “anisotropy” comes from

the Greek roots ανισoς + τρoπια meaning “unequal turning” [Oxford English

Dictionary (Oxford University Press, London, 1933)].

Appendix B

Surface Acoustic Wave Tutorial

Surface acoustic waves can be classified into nondispersive and dispersive waves.

For additional information about the various kinds of SAWs discussed below,

the interested reader may refer to the excellent reviews by Farnell,3,4 Farnell and

Adler,5 Auld,105 Stegeman and Nizzoli,6 Feldmann and Henaff,7 Biryukov et

al.8 in the linear regime and by Parker9 and Mayer10 in the nonlinear regime.

Henceforth, all media considered are assumed to be elastic, i.e., at all times

the internal forces of the medium or stress depend only on the deformation

of the material or strain as measured relative to the undisturbed state. In

addition, the wavelengths of the acoustic waves are assumed to be sufficiently

large compared to the size of the molecules in the material that a continuum

model of the solid is valid. The tutorial focuses primarily on nondispersive

waves, but a short description of dispersive waves is also provided.

B.1 Nondispersive Waves

Nondispersive waves have a wave speed which is independent of their frequency.

Because the nonlinear effects usually occur on a slow time scale in comparison

to the time scale of the wave propagation, significant nonlinear interaction

between the different frequency components or harmonics becomes possible

only when these components propagate together at the same speed. It is this

slow accumulation of effects that gives rise to waveform distortion and shock

formation. The properties of Rayleigh, Stoneley, Scholte, and leaky Rayleigh,

219

220

generalized Rayleigh, quasi-bulk surface and exceptional bulk, pseudo-surface,

piezoelectric surface, Bleustein–Gulyaev, and piezomagnetic surface waves are

summarized here.∗

B.1.1 Rayleigh Waves

Rayleigh waves are SAWs that occur on the interface between half-spaces of an

isotropic solid and a vacuum (see Figure B.1(a)). This type of wave was first

investigated by Lord Rayleigh and tends to be the type of wave to which all

other SAWs are compared. By solving the linearized wave equation subject to

the boundary conditions that (1) the surface is traction-free, i.e., free of forces

acting perpendicular to the undisturbed surface, and (2) the amplitude of the

wave must decay to zero at infinite depth into the solid, Rayleigh was able to

derive expressions for initially sinusoidal surface waves. The particle displace-

ment has only longitudinal (parallel to direction of propagation) and vertical

(perpendicular to surface) components; no transverse (perpendicular to propa-

gation direction and in the surface) component exists. Hence the displacement

is completely contained in the sagittal plane, i.e., the plane defined by the

longitudinal and vertical directions. The particle displacement is retrograde

elliptical near the surface, but it becomes prograde elliptical approximately

one-sixth of a wavelength into the solid. However, the amplitude of the dis-

placement decays away purely exponentially from the surface with most of the

energy contained within approximately one wavelength. The Rayleigh wave

speed is less than both the shear and longitudinal bulk wave speeds. Hence

Rayleigh waves cannot interact with bulk wave modes except in certain cases

∗Wedge waves, i.e., waves that travel along the edge formed by the intersection of twohalf-planes, are also sometimes considered to be in the class of surface waves. While wedgewaves are nondispersive, they are beyond the scope of this review. The interested readershould refer to Mayer10 and Krylov and Parker.155

221

when there is a discontinuity in the surface boundary condition, e.g., reflection

from a corner.4 Because the dispersion relation has solutions for the mate-

rial parameters of any isotropic solid, Rayleigh waves can exist on the surface

of any such solid. General analytical solutions for the linear Rayleigh wave

problem have been explicitly derived by several authors3,105,107,119,156,157 using

a variety of different approaches. See the book by Viktorov14 for a review of

the early theoretical and experimental work on Rayleigh waves, including many

references to the Russian literature.

In the linear regime where the particle displacements are considered

to be infinitesimally small, the initial waveform propagates across the surface

unchanged. However, when finite amplitude waves propagate, energy is trans-

ferred from lower to higher harmonics and the waveform changes in shape. For

simplicity, consider the case of a nonlinear Rayleigh wave with planar wave-

fronts that has an initially sinusoidal time waveform and propagates through a

weakly absorptive medium like steel.20 For such a wave, the horizontal velocity

waveform at the surface distorts into a sawtooth-like wave but with cusps at

the “corners” of the sawtooth which are larger than the initial amplitude of the

sinusoid. Simultaneously, the vertical velocity waveform at the surface evolves

into a sharply cusped impulse. The cusping is attributed to the fact that the

nonlinearly-generated, higher harmonics penetrate less deeply into the solid.

Hence energy from beneath the surface is transferred upwards to the interface

region thereby making the peak amplitude larger.20 The polarity of the saw-

tooth and impulse are dependent upon material parameters. In the reference

frame that moves along at the Rayleigh wave speed, the peaks of the waveform

advance and the troughs recede (like waveform distortion in a fluid108) for some

materials, while for others the opposite occurs. Eventually, thermal absorption

at the shock causes the amplitude of the wave to diminish and the wave re-

turns to the linear regime. The basic features of nonlinear Rayleigh waves

222

described above turn out to be common to many different kinds of SAWs; see

Section 1.3.1 for a discussion of recent work on nonlinear Rayleigh waves.

B.1.2 Stoneley, Scholte, and Leaky Rayleigh Waves

Stoneley waves158 occur on the interface between half-spaces of two different

solids (see Figure B.1(b)). A dispersion relation may be derived by solving the

linearized wave equation in both materials subject to the boundary conditions

that (1) the two solids are bonded such that the particle displacements and

tractions are continuous across the interface and (2) the amplitudes of the waves

in the solids decay to zero at infinite depth into each half-space. However, this

dispersion relation does not necessarily have a solution for any combination

of material parameters; in fact, existence is usually the exception and not the

rule.159–161 This result has been shown in both the cases of isotropic solids27 and

anisotropic solids.162,163 However, whenever there is a solution, that solution

is unique.164 The Stoneley wave speed is always less than the lowest bulk

wave speed in either solid27 but greater than or equal to the slower of two

Rayleigh wave speeds of either solid alone.163 In the case that both materials

are isotropic, the particle displacement in one solid is typically elliptical at the

interface but changes its direction of rotation inside, while in the the other

solid the displacement is elliptical all the way into the half-space. In addition,

the displacement is strictly contained to the sagittal plane. See Meegan27 for

further discussion of Stoneley waves in the linear regime.

Scholte waves165 and leaky Rayleigh waves occur on the interface be-

tween half-spaces of a solid and a fluid (see Figure B.2). They can be consid-

ered to be a limiting case of Stoneley waves when the shear modulus of one of

the media goes to zero. A dispersion relation may be derived by solving the

linearized wave equation in both materials subject to the boundary conditions

223

k

x

k

x

z

Solid

∼λ

z

z/λe

Solid

SolidR

(a) Rayleigh Wave (b) Stoneley Wave

∼λ

z/λe

S

Figure B.1: Schematic representations of a (a) Rayleigh wave and (b) Stoneleywave. [originally Figs. 2.11 and 2.4 from Meegan,27 reproduced by permission].

k

x

z

(a) Scholte Wave

∼ λ

z/λe

-z/λe

z/λeSc k

x

z

SolidSolid

FluidFluid

(b) Leaky Rayleigh Wave

∼ λ

z/λe

lR

Figure B.2: Schematic representations of a (a) Scholte wave and (b) LeakyRayleigh wave. [originally Fig. 2.7 from Meegan,27 reproduced by permission].

224

that (1) the vertical components of the particle displacements are zero at the

interface, (2) the shear stress in the solid is zero at the interface, (3) the per-

pendicular stresses are continuous across the interface, and (4) the amplitudes

of the waves in both media decay to zero at infinite depth into each half-space.

In the case that a root of the dispersion relation is real, then the solution is

called a Scholte wave. For Scholte waves, the particle displacement is retro-

grade elliptical at the surface but becomes prograde elliptical into the solid,

while in the fluid the displacement is retrograde elliptical all the way into the

half-space. In both media, the amplitude of the displacement decays away ex-

ponentially away from the interface. In the case that the root is complex, then

the solution is called a leaky Rayleigh wave. For this case, the amplitude of the

wave in the solid decays exponentially not only into the solid but also in the

direction of propagation. The wave is attenuated in the direction of propaga-

tion because the energy is radiated into the fluid. Accordingly, the amplitude

of the wave in the liquid grows exponentially away from the interface. The

shape of the particle displacement is similar to Scholte waves except that the

elliptical displacement is tilted towards the direction of power flow out of the

solid. A review of some of the earliest work along with a coordinated experi-

mental and theoretical investigation of linear Scholte and leaky Rayleigh waves

(called “Stoneley-like” and “pseudo-Rayleigh waves” in the paper) was per-

formed by Roever, Vining, and Strick.166 More recently Meegan27 performed

an extensive theoretical study of the behavior and existence of linear Scholte

and leaky Rayleigh waves in isotropic solids. In contrast to the Stoneley wave

case, there always exists a solution to the dispersion relation for Scholte waves.

However, leaky Rayleigh waves only exist for certain combinations of material

parameters. In the same work, Meegan also reported that under circumstances

where the interface region is insonified, a “doubly leaky” wave may propagate

along the interface region with energy radiating into both the liquid and solid.

225

Nonlinear Stoneley, Scholte, and leaky Rayleigh waves have been inves-

tigated; see Section 1.3.1 for a discussion of recent work.

B.1.3 Generalized Rayleigh Waves

Generalized Rayleigh waves occur on the interface between half-spaces of an

anisotropic solid and a vacuum. While the interface boundary condition is

still that the surface be traction-free, the anisotropy of the medium causes

generalized Rayleigh waves to differ in several ways from Rayleigh waves. First,

the particle displacement generally has components in the longitudinal, vertical,

and transverse directions, although the particle trajectory does remain confined

to a plane (Figure B.3). Secondly, the amplitude of the wave typically decays

as an exponentially damped sinusoid, thereby giving rise to alternating regions

of quasi-retrograde and quasi-prograde elliptical displacement (see Figures B.4

and B.5 for the simplified case of a pure mode direction). In addition, the

orientation of the plane of the displacement ellipse generally also varies with

depth. The SAW speed is still less than the bulk wave speed of the medium

and so usually no coupling can occur between the modes to linear order∗ except

under special conditions (see Section B.1.5). However, the SAW speed is a

function of both the direction of propagation and the orientation of the crystal

with respect to the surface or crystal cut (see Appendix D). This anisotropy is

often expressed graphically via a polar plot of inverse wave speed or slowness

as a function of angle from a given reference direction105 (see Figure B.6).

One of the reasons that slowness is usually plotted instead of speed is that

the direction of power flow for any given propagation direction can be found

∗It is possible in three-wave nonlinear interactions for generalized Rayleigh waves to in-teract with bulk waves.10 For example, a surface wave and bulk wave can generate a bulkwave,167 two surface waves can generate a bulk wave,168–170 and two bulk waves can generatea surface wave.171

226

φ

-y

z

x

xφz

y Top View of Trajectory at Surface

Side View of Trajectory at Surface

Figure B.3: Typical particle motion for a generalized Rayleigh wave.

227

-2.5

-2

-1.5

-1

-0.5

0-1 -0.5 0 0.5 1

⟨ ⟩

uz

λz−

uy=0

Retrograde

Prograde

Retrograde

Prograde

Retrograde

Prograde

Type of Motion

ux

100Displacement Depth Profile for Si (001)

Figure B.4: Displacement depth profile for silicon in (001) plane in 〈100〉 di-rection for initially sinusoidal wave.

k

Retrograde

x

z

Generalized Rayleigh Wave

Retrograde

Prograde

Solid

ez/λ

∼λ

Figure B.5: Schematic representation of a generalized Rayleigh wave in a puremode direction (calculations based upon silicon in (001) plane in 〈100〉 direc-tion).

228

2

1

0

1

2

2 1 0 1 2

cSAW

cFT Slowness

Speed

k

P

<100>

<010>

cFT

cSAW

Si on (001) plane

Figure B.6: Examples of polar plots relative speed and slowness (based onsilicon in (001) plane). The speed and slowness are scaled by the fast transversebulk mode speed cFT which, in this case, is constant for all directions. Noticethat the normal to the slowness curve P is not parallel to the wave vector k inthe direction shown (32◦ from 〈100〉).

transducer

P

k

transmitting

transducerreceiving

wave packet

wavefronts

Figure B.7: Effect of power flow vector P not being parallel to propagationvector k. If a pulse is generated at the transmitting transducer, the receivingtransducer must be placed along the line parallel to the power flow vector todetect the resulting wave packet. Note that while k is perpendicular to thewavefronts, P is not.

229

by taking the normal to the slowness curve at that given direction. Thus for

most generalized Rayleigh waves, the power flow is not parallel to the direction

of propagation. Because it can be shown that the power flow vector is always

parallel to the group velocity (velocity of any given wave packet), it then follows

that the group velocity is usually not parallel to the phase velocity (velocity of

the wave fronts). Physically, this indicates that the wave fronts are at an angle

with respect to the power flow direction (see Figure B.7). This is actually a

general property of waves in anisotropic media and has been observed in both

bulk105 and surface172 acoustic waves. Directions in which the phase and group

velocity are parallel are called pure mode† directions.3 For particular pure

mode directions on particular surface cuts where the displacement is confined

to the sagittal plane and the principal axis of the surface displacement ellipse in

perpendicular to the free surface, the waves are additionally called “Rayleigh-

type” waves.105 These Rayleigh-type modes typically occur in special directions

of high symmetry.117

The work on the propagation of SAWs in anisotropic media grew out

of studies of the properties of bulk acoustic waves in anisotropic media. Much

of this research on bulk waves occurred in the 1950s,∗ precipitated by the in-

creased use of acoustic delay lines in electronic systems during World War II

and the emerging powers of the digital computer as a research tool. Much of

the work in this area prior to 1970 is discussed in the book by Musgrave,113 so

only a brief summary is given here. Theoretical studies on bulk waves were per-

†Note that Auld105 defines pure modes in bulk waves to be those in which the velocity isstrictly parallel or perpendicular to the direction of motion. He does not use the term in thecontext of surface acoustic waves.

∗See the historical sketch provided by Musgrave113 for review of wave propagation in non-piezoelectric, anisotropic, elastic media prior to the 1950s. More generally, see Howard173

for a discussion of Rayleigh’s work on SAWs and Auld174 for a discussion of early work onSAWs in various geometries by Cauchy, Poisson, Rayleigh, Lamb, and others.

230

formed by Carrier,175,176 Mapleton,177 Musgrave,178–182 Synge,183 Buchwald,184

Waterman,185 Lighthill,186 Duff,187 Stroh,188 and Federov189 in which the basic

properties of the velocity surface, wave surfaces, displacement vectors, and the

effects of a point source disturbance were investigated. Results were derived

for media with hexagonal, cubic, tetragonal, orthorhombic, trigonal, and mono-

clinic symmetries (See Table B.1). Experimental work generally lagged behind

theoretical developments during this period; see the reviews by Musgrave182,113

for descriptions of some of the experiments and their results. In the 1970s and

1980s, the study of the propagation of ballistic phonons (very high frequency

lattice vibrations with macroscopic mean free paths) renewed interest in the

study of wave propagation through anisotropic media and eventually led to

further study of linear ultrasonic bulk and surface acoustic waves in crystals.

It is beyond the scope of this tutorial to provide a complete review

of the extensive literature on linear SAWs in anisotropic media. However, a

brief discussion of the the earliest and most fundamental work is summarized

here (for more information see the review by Farnell3). Stoneley was the first

to consider the problem of SAWs in anisotropic media, first in a transversely

isotropic (isotropic in planes parallel to the surface but anisotropic with depth;

also called orthotropic) medium147 and later in a cubic crystal.148 In the latter

case he limited his investigation to selected symmetry planes and directions

of the crystal and only considered solutions that decayed purely exponentially,

thereby missing what later became known as the “generalized” Rayleigh wave

solutions. Shortly thereafter, Gold196 showed that more general solutions could

be derived by transforming the problem into a coordinate system in which one

of the axes was in the direction of propagation; however, he was still only

able to provide explicit solutions for particular cuts and directions of cubic

crystals. Later authors183,197,193,190,198,199 included the possibility of exponen-

tially damped sinusoidal amplitudes, but some concluded that certain cuts and

231

Type Bulk waves Surface WavesGeneral Musgrave178,113 (1954,1970) Synge183 (1957)

Synge183 (1957) Buchwald190 (1961)Buchwald184 (1959) Lim et al.191 (1968)Waterman185 (1959) Ingebrigtsen et al.192 (1969)Lighthill186 (1960) Farnell3 (1970)Duff187 (1960)Stroh188 (1962)Fedorov189 (1968)

Hexagonal Carrier175,176 (1944,1946) Stoneley147 (1949)Musgrave179,113 (1954,1970) Synge183 (1957)Waterman185 (1959) Buchwald193 (1961)Duff187 (1960) Ingebrigtsen et al.194(1966)Fedorov189 (1968) Farnell3 (1970)

Royer et al.117 (1984)Cubic Mapleton177 (1952) Stoneley148 (1955)

de Klerk et al.195 (1955) Gold196 (1956)Miller et al.180 (1956) Gazis et al.197 (1960)Musgrave181,113 (1957,1970) Buchwald et al.198 (1963)Waterman185 (1959) Tursunov199 (1967)Duff187 (1960) Farnell3 (1970)Fedorov189 (1968) Royer et al.117 (1984)

Tetragonal Musgrave181,113 (1957,1970) Farnell3 (1970)Waterman185 (1959) Royer et al.117 (1984)Fedorov189 (1968)

Trigonal Waterman185 (1959) Farnell3 (1970)Farnell200 (1961)Fedorov189 (1968)Musgrave113 (1970)

Orthorhombic Musgrave181,113 (1957,1970) Farnell3 (1970)Royer et al.117 (1984)

Monoclinic Deresiewicz et al.201 (1957)

Table B.1: Summary of some of the theoretical and experimental researchon elastic wave propagation in nonpiezoelectric anisotropic media of varioussymmetries. Note that in many papers about specific crystal classes solutionsare only presented for selected directions.

232

directions were “forbidden” from having SAW solutions. Lim and Farnell191

later showed numerically that for a wide variety of crystals that these “forbid-

den” regions were really regions where the surface wave solution degenerated

into a bulk wave solution, now often called an exceptional bulk wave (see Sec-

tion B.1.4). See Table B.1 for a summary of the work done on linear SAWs in

nonpiezoelectric anisotropic media of various symmetries.

In subsequent years, it was proved rigorously that the traction-free sur-

face boundary condition can be satisfied for arbitrary surface cuts and direc-

tions on any anisotropic elastic solid,202,203 although in some cases the only

solution is the exceptional bulk wave. The proofs involved formulating the lin-

earized surface wave problem in formal linear algebraic terms and then deriving

general statements about the possible states of such systems.204 It is interesting

to note that many key developments in the existence and uniqueness problem

came from recognizing the analogies between the the propagation of surface

waves and the propagation of a line dislocation (discontinuity in the elastic

medium) in uniform motion through anisotropic media.188

The nonlinear behavior of SAW in anisotropic media is the subject of

this dissertation. See Section 1.3 for a review of the experimental and theoret-

ical work on this topic.

B.1.4 Quasi-bulk Surface Waves and Exceptional Bulk Waves

In anisotropic media with a free surface, there typically exist three different

bulk waves: one longitudinal or quasi-longitudinal and two shear or quasi-shear

modes.105 In some media, the speed of the the SAW approaches the speed of one

of the shear modes as the propagation direction approaches certain directions

on certain cuts. Typically these are configurations that have a high degree

of symmetry,204 e.g., the [110] direction of the (001) plane of nickel.3 As the

233

particular direction is approached, the penetration depth of the SAW gradually

increases and becomes much greater than one wavelength. Because the energy

of such a wave is distributed throughout the solid more like a bulk wave, it is

called a quasi-bulk surface wave. At the particular direction, the only mode

that satisfies the traction-free boundary condition is a shear horizontal (SH)

bulk wave, i.e., a shear bulk wave that is polarized transversely to its direction

of propagation. Such a wave is called an exceptional bulk wave204 or “surface

skimming” bulk wave8 (see Figure B.8). Interestingly, it can be shown that the

change in the magnitudes of penetration depth and the difference between the

speed of propagation of the quasi-bulk surface wave and exceptional bulk wave

modes are independent of the symmetry of the crystal as the special direction

is approached.205 The existence of exceptional bulk waves has been studied for

non-piezoelectric202,206 and piezoelectric crystals.207 See Lewis208 and Biryukov

et al.8 for additional discussion of properties and applications of exceptional

bulk waves.

The nonlinear behavior of exceptional bulk waves has also been studied.

Mozhaev99 showed that exceptional bulk waves in an isotropic media with a

small but finite amplitude have their depth dependence modified so that the

amplitude of the displacement of the wave decays into the solid. This local-

ization of the wave to the surface region is caused by the nonlinearity of the

medium. He later generalized this result to crystals with orthorhombic sym-

metry.101 His results assume that the amplitude of the wave is sufficiently

weak that terms higher than cubic nonlinearity may be neglected. See Mayer10

for a review of other work on nonlinear, shear horizontal, bulk waves and a

derivation of some basic results.

234

no decaywith depth

Exceptional Bulk Wave(Surface Skimming Bulk Wave)

z

x y

kk

z

of DisplacementSide View

of DisplacementFront View

Figure B.8: Schematic diagram of an exceptional bulk wave (surface skimmingbulk wave) in side and front perspectives.

235

B.1.5 Pseudo-surface Waves

Pseudo-surface waves arise in conjunction with the exceptional bulk waves

described above. In the situation that the normal surface wave becomes de-

generate with the normal bulk wave, it is then possible for a new surface wave

solution to exist at a speed higher than the lowest shear wave mode. Away from

the direction associated with the exceptional bulk wave, a pure surface wave

mode is no longer possible, and the pseudo-surface wave mode arises. Close

to the surface, this mode appears like a surface wave in that it decays rapidly

in amplitude away from the surface. However, it differs in that (1) hundreds

or thousands of wavelengths into the solid the wave grows in amplitude and

(2) near the surface its components decay in amplitude as it propagates. This

behavior is consistent with a wave that has its energy flowing or “leaking” at

an angle to the surface into a quasi-shear bulk wave mode. For this reason,

pseudo-surface waves are also called leaky surface waves∗ in some papers.209

This coupling between the surface and bulk modes becomes possible because

the pseudo-surface wave speed is the same as the projection of the speed of

the lowest shear bulk wave traveling at the power flow angle (see Figure B.9).†

Thus the growth of the wave amplitude deep into the solid can be visualized by

observing that energy that reaches a point deep into the solid has come from

areas on the surface where the energy was larger (i.e., larger than the energy

at the surface immediately above that point).3

Pseudo-surface waves have been experimentally observed in several ma-

terials209–213 and predicted to exist in many others.214,3 The experimental ob-

∗Note that pseudo-surface waves differ from leaky Rayleigh waves. In the latter case, theenergy is being radiated into bulk modes of fluid whereas in the former case the energy isbeing radiated into the solid itself.

†In an isotropic medium, bulk wave speeds are the same in all directions. Hence this kindof speed matching cannot occur, and pseudo-surface waves are not possible.

236

pseudosurfacev

vbulk

=cos θ

vbulk

wavefronts

x

solidθ

z

Figure B.9: Coupling between pseudo-surface wave mode and bulk mode prop-agating at angle θ to the surface.

237

servation is possible because there exist cases where the attenuation is so low

(e.g., 0.025 dB/wavelength in the Y Z plane of quartz209) that the wave can be

essentially detected like an ordinary surface wave. As the direction of propa-

gation is moved further and further away from the direction of the exceptional

bulk wave, the pseudosurface wave speed slows to the speed of the lowest shear

bulk wave until it is completely degenerate with that mode. This latter tran-

sition has also been observed experimentally.210 However, Stegeman215 has

shown theoretically that in some crystals regular surface wave solutions can

exist on branches of the pseudo-surface wave solutions. As a result, there may

be directions in which two surface wave solutions may exist each with a different

speed; however, the stability of such modes was not explored.

B.1.6 Piezoelectric Surface Acoustic Waves

In some materials, the mechanics of wave propagation cannot be fully described

by the normal stress–strain relationship alone. Piezoelectric crystals exhibit

several new significant properties including the direct piezoelectric, converse

piezoelectric, and electrostrictive effects.105 The direct piezoelectric effect oc-

curs when mechanical strain causes the molecules in the crystal to become

polarized, thereby by giving rise to an electric field within the crystal. As

might be expected, the converse piezoelectric effect occurs when an electric

field causes mechanical strain in the crystal. Both the direct and converse

piezoelectric effects vary linearly with the electric field and hence can be quite

significant in materials that have strong piezoelectric coupling like quartz and

lithium niobate. The electrostrictive effect occurs when an applied electric

field causes a mechanical strain. However, it differs from the converse piezo-

electric effect in that (1) it occurs in all materials, not only those that exhibit

piezoelectricity and (2) it varies quadratically with the electric field strength

238

and is therefore weaker than the converse piezoelectric effect for the same field

strength. While not a linear effect, electrostrictive effects must be included

for higher-order nonlinear theories of piezoelectric SAWs.∗ Because all of these

effects allow for coupling between electric and mechanical forces, they can be

employed in devices for a wide variety of uses.

The SAW problem can be formulated in a way similar to a nonpiezoelec-

tric crystal but with extra terms describing the electromechanical interactions

listed above. While the full set of equations can be very complicated, they can

be simplified without losing significant accuracy by assuming that the electric

field has relatively slow variation in time. This is called the quasistatic approx-

imation. The approximation is valid because the ratio of the speed of sound to

the speed of light is so low (on the order of 10−5 for most solids) that effects

due to the time variation of the electric field can be considered negligible. In

this approximation it also follows that magnetic effects are negligible and that

electric effects can be described by the electric potential function. This poten-

tial coupled with the particle displacements is sufficient to provide an accurate

description of SAW effects in most cases.

As a simple example, consider the case of a half-space of a piezoelectric

crystal bounded by a vacuum. The mechanical boundary condition for the

piezoelectric case is the same as for the anisotropic case, i.e., that the surface

must be traction-free. The electrical boundary conditions at the surface of the

solid are that: (1) the electrical potential is continuous across the interface and

(2) the component of the electric displacement field normal (perpendicular) to

the surface is continuous across the interface. The potential on the free space

side of the surface is determined by solving Laplace’s equation (∇2Φ = 0, where

∗Electrostrictive effects on SAWs have been measured in nonpiezoelectric crystals (e.g.,SrTiO3) by several authors,216–218 and a SAW transducer has been built in a nonpiezoelectriccrystal based on this effect.219

239

Φ is the electrical potential) subject to the additional condition that the poten-

tial decays to zero at infinite distance from the plane. This type of boundary

condition is usually called the electrically free4 or free space9 condition. How-

ever, in some circumstances idealized boundary conditions which are simpler

can suffice. If the potential at the surface can be taken to be identically zero at

the boundary, e.g., in the case of a thin metallic coating,9 then the boundary

condition is called shorted4 or earthed .9 If the normal component of the elec-

trical displacement is taken to be identically zero at the boundary, e.g., in the

case that the polarization field inside the crystal is parallel to the surface,220

the boundary condition is called open circuited .9 The shorted and open cir-

cuited conditions are actually just limits of the electrically free condition as the

permittivity of free space goes to zero or infinity, respectively.

The mechanical displacements for piezoelectric SAW are similar to those

for generalized Rayleigh waves and are affected by the electrical boundary con-

ditions.4 The electrical potential also varies according to the type of boundary

condition. In the case of the free space boundary condition, the potential in

the solid generally has a maximum within a wavelength of the surface and then

decays exponentially into the solid while the potential in the free space decays

away purely exponentially.221,222 The SAW speeds are again a function of prop-

agation direction and can be substantially affected by the electrical boundary

conditions. In fact, the relative change in the SAW speed between the free

space and shorted boundary conditions is commonly used as measure of the

electromechanical coupling strength of a material.4,222 This approximation is

particularly useful because exact calculations and measurements of the electric

fields involved are often difficult. Another approximation which has proven

useful in some cases is to employ stiffened elastic constants. For bulk acoustic

waves it is possible to derive a simple analytical expression which shows that the

addition of piezoelectricity increases or “stiffens” the effective elastic constants

240

of the material.4 Computation of the SAW speeds by this method can be done

within a few percent while computation of relative amplitudes of displacement

components can be found within around fifteen percent even for directions of

low symmetry.3 Pseudo-bulk, exceptional, and pseudo-surface waves can also

occur in piezoelectric crystals, e.g., quartz.3,209 In fact, piezoelectric materials

may have two independent pseudo-surface wave modes.7

The work on piezoelectric SAWs grew out of more general studies of

piezoelectric bulk vibrations. Piezoelectricity was first formally studied by

the brothers Pierre and Jacques Curie in 1880.223 It remained essentially a

scientific curiosity until World War I when Langevin realized that electrically-

excited quartz plates could be used as emitters, and later as receivers, of high-

frequency sound underwater, thereby giving birth to the field of ultrasonics.220

After the war, investigation of piezoelectric effects, especially in regards to the

piezoelectric resonators, gave rise to components that have been critical parts

of modern electronic technology including oscillators, stabilizers, filters, and

transducers. The books by Cady,220 Mason,224 and Hunt225 review much of the

work in this field. Some of the first studies of linear bulk wave propagation in

piezoelectric crystals were done by Kyame226,227 and Koga et al.228 This work

was later extended to piezoelectric semiconductors by Hutson et al.229 They

also showed that it was possible for acoustic waves in such semiconductors to

be amplified or attenuated by the application of a DC electric field parallel to

the propagation direction if at the same time a piezoelectric field is generated

parallel to the propagation direction.230–232 This effect was later employed to

make a SAW device for signal amplification.233

It is beyond the scope of this tutorial to provide a complete review of

the extensive literature on linear piezoelectric SAWs. However, a brief discus-

sion of the the earliest and most fundamental theoretical work is summarized

241

here. In 1963, Tiersten solved the closely related problem of linear wave prop-

agation in piezoelectric plates.234,235 Later Coquin and Tiersten analyzed the

problem of linear SAW excitation and generation in quartz by means of surface

electrodes,236 thereby establishing a simple model of an interdigital transducer.

Several authors examined cases of piezoelectric materials but only considered

elastic effects. Such work included Deresiewicz and Mindlin201 who analyzed

the case of SAWs on the AT cut of quartz,∗ Vervekina et al.238 who investigated

the Y Z plane of quartz, and Ingebrigtsen and Tonning194 who analyzed various

cuts and directions in quartz and CdS. Subsequently Tseng and White239 calcu-

lated the properties of SAW in the basal plane of the hexagonal crystals CdSe,

CdS, ZnO for the fully piezoelectric case with free space boundary conditions,

and Tseng221 extended the work to free space and metallized surfaces of CdS,

ZnO, and the strongly piezoelectric lead zirconate titanate (PZT-4). Campbell

and Jones222 computed the amplitude profiles of the mechanical displacements

and electrical potential for several directions of LiNbO3 under the shorted and

free space electrical boundary conditions. They also showed that the elec-

tromechanical coupling strength of a material was directly proportional to the

relative change in the SAW speed between the shorted and free space configu-

rations. An alternative approach was developed by Ingebrigtsen192,240 to deal

with more general electrical boundary conditions via a “surface impedance”

which relates the electrical potential to the normal component of the electrical

displacement. This approach later proved useful both as a method of demon-

strating the existence and uniqueness of SAWs163,203 and of describing SAWs in

device design7 and inhomogeneous and layered media.8 Finally, while most of

the work described above only allowed for plane wave propagation, Day et al.241

considered theoretically and experimentally the problem of cylindrical SAWs

∗Commonly used cuts of quartz have been named by letters (A-Type or AT, B-Type orBT, etc.). A diagram of many of these cuts is given in Fig. 20 of Mason.237

242

in PZT-5. See Table B.2 for a summary of the work done on linear SAWs in

piezoelectric anisotropic media of various symmetries.

The existence of piezoelectric SAWs has also been studied. The prob-

lem becomes more complicated than the nonpiezoelectric case not only by the

addition of the electrical aspect but also because the various mechanical and

electrical boundary conditions have to be taken into account. Lothe and Bar-

nett249 showed that for the case of a mechanically free surface (1) at most one

solution is possible for the electrically open circuited condition and (2) at most

two solutions are possible for the electrically short circuited condition. If two

solutions exist, one must be a Bleustein–Gulyaev wave (see Section B.1.7). No

solutions exist for mechanically clamped surfaces. Alshits et al.250 extended

these results to include piezomagnetic (see Section B.1.8) and piezoelectric–

piezomagnetic materials and provide a table describing the solutions under

all the possible combinations of mechanical, electrical, and magnetic surface

boundary conditions.

B.1.7 Bleustein–Gulyaev Waves

A different kind of SAW occurs in piezoelectric crystals when the sagittal plane

is perpendicular to an axis of twofold rotation. Due to the symmetry of this

situation, the longitudinal and vertical displacements decouple from the trans-

verse displacement and electric potential. The longitudinal and vertical com-

ponents remain intercoupled but propagate as if the piezoelectric constants of

the crystal were set to zero. The transverse displacement and electric potential

also remain intercoupled and give rise to a transversely polarized surface wave

called a Bleustein–Gulyaev wave (BGW)242,243 (See Figure B.10). This type

of wave cannot occur on the surface of a non-piezoelectric solid under normal

243

Type Bulk waves Surface WavesGeneral Kyame226,227 (1949,1954) Ingebrigsten240 (1969)

Koga et al.228 (1958) Farnell3,4 (1970,1978)Hutson et al.229 (1962)

Hexagonal Hutson et al.230 (1961) Tseng239,221 (1967)(CdS, Hutson et al.229 (1962) Bleustein242 (1968) [BGW]CdSe White231 (1962) Gulyaev243 (1969) [BGW]ZnO) Berlincourt232 (1964) Koerber et al.244,245 (1972) [BGW]

Morozov et al.246 (1970) [BGW]Soluch et al.247 (1977) [BGW]

Cubic Kyame226 (1949) Tseng248 (1970) [BGW](ZnS, Hutson et al.229 (1962) Koerber et al.244,245 (1972) [BGW]GaAs)Tetragonal Kyame227 (1954) Koerber et al.244,245 (1972) [BGW]

Soluch et al.247 (1977) [BGW]Orthorhombic Tseng248 (1970) [BGW](Ba2NaNb5O15)Trigonal Koga et al.228 (1958) Coquin et al.236 (1967)(α-quartz, Campbell et al.222 (1968)LiNbO3)

Table B.2: Summary of some of the early theoretical and experimental researchon linear elastic wave propagation in piezoelectric anisotropic media of varioussymmetries. Note that most of the papers in each crystal class, solutions areonly presented for selected directions. The notation [BGW] indicates that thereference is discussed in Section B.1.7 on Bleustein–Gulyaev and related waves.

244

decay with depthslow exponential

Φ/Φ0 y

k

∼10λ

z z

shorted surface (Φ(0)=0)

Electrical Potentialwith

Front View

Displacementof

Bleustein-Gulyaev Wave

Figure B.10: Schematic representation of a Bleustein–Gulyaev wave.

245

conditions.∗ As in the case of the piezoelectric SAW considered in the previous

section, the exact form of the BGW depends upon the electrical boundary con-

dition at the surface. However, in both the ‘free space” and shorted boundary

conditions, the amplitude of transverse component decays away purely expo-

nentially from the surface, although the depth of penetration is much greater in

the former than in the latter.4 The potential grows in amplitude until it reaches

an extremum, but then decays away purely exponentially. The depth of pene-

tration in both cases is inversely proportional to the strength of the electrome-

chanical coupling.4,8 Because this coupling is relatively weak even for many

piezoelectric materials, the penetration depth of the waves is large (sometimes

several hundred or thousand wavelengths) as compared to Rayleigh or general-

ized Rayleigh waves. This property, combined with the fact that BGWs have a

larger energy flux as compared to Rayleigh waves, makes BGWs more sensitive

to imperfections in surfaces.8 With both boundary conditions, the velocities of

the BGWs are very close to the velocities of the transversely-polarized shear

bulk waves of the material. Hence, in some ways, BGWs can be considered

to be transversely-polarized shear bulk waves† perturbed into surface waves by

the piezoelectric effects of the medium.4,8

The existence of BGWs has been investigated theoretically,207 and it

has been shown that they exist in materials with several different symmetry

classes.244,245,248 Generalizations of BGWs have been found to propagate along

the interface between two piezoelectric solids252 and two dielectric solids placed

in an external electric field.253 Under circumstances where two piezoelectric

solids are separated by a vacuum gap, BGW can couple from one solid to the

∗Gulyaev and Plessky251 have shown that BGWs may be electrostrictively induced inisotropic dielectrics under the influence of a externally applied electric field.

†Note that BGWs are not the quasi-bulk surface waves described in Section B.1.4 becausethose waves have non-transverse components.

246

other solid via the potential in the gap. Such “gap” waves have been studied

both theoretically251,254,255 and experimentally.8 Acoustoelectric effects and the

amplification of BGWs have also been studied.246,247,256 For more information

on the properties and applications of linear BGWs, see Biryukov et al.8

B.1.8 Piezomagnetic Surface Acoustic Waves

Piezomagnetism, the magnetic analogue to piezoelectricity, occurs when the

mechanical forces couple to the magnetic forces in the material and vice-versa.

The analogue to the electrostrictive effect is the magnetostrictive effect which

occurs when an applied magnetization causes a mechanical strain. It also oc-

curs in all materials and varies quadratically with the magnetization. Gulyaev

et al.257 have shown that in a magnetically ordered crystal with a compen-

sated magnetic moment (i.e., an antiferromagnet) purely shear piezomagnetic

SAWs can exist in certain directions.∗ Examples259 of such piezomagnetic ma-

terials include MnF2 and CoF2. By assuming that (1) the magnetic field is

quasistatic, (2) the surface is mechanically free, and (3) the magnetic potential

is continuous across the surface, they derive expressions for the transverse dis-

placement and magnetic potential inside and outside of the crystal. They show

that unlike Bleustein–Gulyaev waves, which exhibit a superposition of rapidly

and slowly decaying terms in the expression for the electrical potential, the

magnetic potential only has a single term which decays slowly. This property

of deep penetration may make them useful for practical applications.

A limited amount of other work has been performed on piezomagnetic

waves. Parekh260,261 has shown that magnetostrictively induced, purely trans-

∗These piezomagnetic waves should not be confused with the magnetic spin waves ormagnons that can occur on the surface of ferromagnetic materials that are purely magnetic.For more information about magnons, see the review by Mills.258

247

verse shear waves can exist, thereby providing the magnetic analogue to elec-

trostrictively induced Bleustein–Gulyaev waves. Alshits et al.250 has inves-

tigated the conditions for the existence of piezomagnetic SAWs as well as

piezoelectric–piezomagnetic materials and provides a table describing the solu-

tions under all the possible combinations of mechanical, electrical, and magnetic

surface boundary conditions. A few additional references from the Russian lit-

erature on work done on magnetoacoustic waves are also given in this same

paper.

To the knowledge of this author, no work has been performed in the

area of nonlinear piezomagnetic SAWs. However, some work has been done on

linear and nonlinear magnetostrictive SAWs by Abd-Alla and Maugin.262

B.2 Dispersive Waves

Dispersive waves have the property that their wave speed depends upon fre-

quency. Hence a wave packet with many different frequency components may

have each component traveling at a different speed. In the case of strong dis-

persion, wave packets do not form shocks because the harmonic components

of the waves separate relatively quickly compared to the time scale on which

the nonlinear effects occur. Under weaker dispersion, it is possible for effects of

nonlinear distortion to balance the effects of dispersive distortion, thereby giv-

ing rise to waveforms that can propagate without any change in shape. These

waves of permanent form are also called solitons. In contrast with nondispersive

waves, systems that give rise to dispersive waves must define a characteristic

length scale over which the dispersion occurs. For SAWs, this scale is typically

given by a layer depth in layered materials or ridge spacing for corrugated sur-

faces. Because this dissertation focuses exclusively on nondispersive waves, the

brief discussion given below is included only to describe the basic types and

248

properties of dispersive SAWs. Lamb, shear horizontal (SH), Love, perturbed

Rayleigh, Sezawa, and other waves are described below.

Introductions to SAWs with dispersion are given in works by Auld,105

Farnell,4 Lewis,208 and Biryukov et al.8 The books by Ewing, Jardetzky, and

Press263 and Brekhovskikh156 cover some of the earliest work on layered media.

An excellent review on linear surface wave propagation in thin layers is given

by Farnell and Adler.5 Parker9 and Mayer10 review work on nonlinear SAWs

with dispersion. However, the topic of nonlinear SAWs with dispersion is still

very much a topic of current research. For example, Neubrand and Hess264 have

experimentally extracted information about surface layers using laser-excited

SAWs. Even more recent work by Gusev et al.265 and Eckl et al.266 has indicated

that nonlinear SAWs of permanent form may exist in layered half-spaces under

certain conditions.

B.2.1 Plate Waves (Lamb and SH Waves)

Subsequent to his famous paper on surface waves in a semi-infinite half-space,

Rayleigh267 solved the problem of linear waves propagating in a freely vibrating,

infinite, isotropic elastic plate of arbitrary thickness.∗ Around the same time,

Lamb268 also published a paper on the same topic. Because the thickness of the

plate introduces a characteristic length scale, plate waves are dispersive. For

any given wave number, plate waves may only propagate at a set of discrete

frequencies. Such discrete modes can be further classified into even and odd

modes that are symmetric (extension without bending of the middle plane) and

antisymmetric (bending without extension of the middle plane), respectively,

about the centerline of the plate.

∗Auld174 notes that Rayleigh was aware that Cauchy and Poisson had already addressedthe wave propagation problem for thin plates. Moreover, Auld notes that Cauchy had evenconsidered anisotropic plates.

249

Plate waves in isotropic media divide into two independent classes, now

commonly called Lamb waves and shear horizontal (SH) plate waves. Lamb

waves are similar to Rayleigh waves in that they have displacement components

in the longitudinal and vertical directions only (where the vertical direction is

perpendicular to the plate and the longitudinal direction is the direction of

wave propagation). As might be expected from their name, SH waves only

have their displacement in the plate transverse to the direction of propagation.

Physically, the propagation of SH waves can be understood by drawing an

analogy to the propagation of acoustic or electromagnetic waves traveling in

a waveguide, i.e., as plane waves bouncing back and forth off the sides of the

plate at such an angle that the boundary conditions are satisfied. The frequency

cut-off and dispersion exhibited in those systems also occurs with the SH wave

modes. A similar interpretation may be made for Lamb waves but is much

more complicated because mode conversion occurs between the vertical and

longitudinal components at each bounce.

Much additional work has been done on plate waves to extend descrip-

tions to other situations, e.g., curved plates14 and nonpiezoelectric183 and piezo-

electric235 anisotropic plates. See the general references on dispersive SAWs

listed above for additional information and references.

B.2.2 Layer Waves (Love, Perturbed Rayleigh, Sezawa Waves)

A configuration which occurs in many practical situations is a layer of one

solid on the substrate of a different solid with an “empty” half-space above the

layer. Examples include layers that exist in the earth’s crust and layers used to

construct microelectronic components. The thickness of the layer introduces a

characteristic length scale and makes the wave propagation dispersive. In some

ways, the half-space problem and the plate problem can be considered limits

250

of the layered half-space problem as the layer thickness goes to zero and the

substrate’s density or elastic constants go to zero, respectively. As a result,

the waves that can occur in various circumstances are similar in many ways to

those already described. Love waves, perturbed Rayleigh waves, and Sezawa

waves in isotropic materials are briefly described below.

Love waves269 are shear horizontal (transverse displacement only) waves

that occur in the layer and substrate. The displacement typically oscillates

with depth into the layer and then decays into the substrate with the number

of oscillations determining the mode number. They may occur only if the shear

velocity of the layer is less than the shear velocity of the substrate. Suppose

the thickness of the layer is L and the wavelength is λ. When λ/L � 1, the

layer is effectively only a small perturbation to the wave, and the Love wave

speed approaches the substrate shear wave speed. When λ/L � 1, most of the

wave displacement occurs in the layer, and the Love wave speed approaches

the layer shear wave speed. The Love wave speed changes continuously from

one extreme to the other and is, therefore, dispersive.

In contrast, perturbed Rayleigh waves and Sezawa waves are similar to

Rayleigh waves in a half-space in that they have displacement components in

the longitudinal and vertical directions only. They may occur regardless of

whether the layer wave speeds are faster or slower than those of the substrate.

If λ/L � 1, the layer is effectively only a small perturbation to the wave,

and the mode speed approaches the substrate Rayleigh wave speed. When

λ/L � 1, most of the wave displacement occurs in the layer, and the mode

speed approaches the layer Rayleigh wave speed. However, if the Rayleigh wave

speed of the layer is less than the Rayleigh wave speed of the substrate, higher

order modes can occur in which the displacement of the wave oscillates with

depth in the layer and then decays into the substrate. Such higher order modes

251

divide into two classes which correspond to symmetric and antisymmetric Lamb

wave modes in the limit that the density or elastic constants of the substrate

go to zero. The lowest order symmetric mode is called a perturbed Rayleigh

wave, while the lowest order antisymmetric mode is called a Sezawa wave.

Under certain circumstances (typically when the shear wave speeds of the layer

and substrate are similar), the behavior of the Sezawa wave can approach the

behavior of a Stoneley wave near the layer–substrate interface.

Much additional work has been done on surface waves in layered media

to extend descriptions to other situations, e.g., curved surfaces263 and nonpi-

ezoelectric5 and piezoelectric5 anisotropic media. For anisotropic media, the

solutions generally are much more complex because they cannot be decoupled

into Love waves and perturbed Rayleigh and Sezawa waves.270,4 See the gen-

eral references on dispersive SAWs listed above for additional information and

references.

B.2.3 Other Dispersive Surface Waves

Dispersion can arise in other situations than those described in the previous

sections. Composite materials with periodically layered media can have very

complicated mode structures.271,272 Surfaces with periodic corrugations or ran-

dom roughness can introduce a length scale which can cause dispersion; such

corrugations are important for the modeling of SAW devices. See the reviews

of the work in this area by Maradudin,273 Gulyaev and Plessky,274 Biryukov et

al.,8 and Mayer10 for additional information and references.

B.3 Summary

Many different kinds of surface acoustic waves can arise in elastic media due

to interfaces of various kinds. As shown in Table B.3, the waves divide into

252

Substrate Particle motion Particle motionconfiguration mainly in mainly

sagittal plane shear horizontalIsotropic Rayleigh wave SH plane wavehalf-space (SSBW)Isotropic Stoneley wave SH plane wavessolid–solid in solids (SSBW)half-spacesIsotropic Scholte wave SH plane wavesolid–fluid [also leaky Rayleigh] in solid (SSBW)half-spacesAnisotropic Generalized Rayleigh wave Exceptional bulk wavehalf-space [also pseudo-SAW] [also pseudo-SAW]Piezoelectric Generalized Rayleigh wave Bluestein–Gulyaev wavehalf-space with quasistatic electric wave

[also pseudo-SAW] [also pseudo-SAW]Piezoelectric Generalized Stoneley wave Maerfeld–Tournois wavesolid–solid with quasistatic electric wavehalf-spaces [also pseudo-SAW] [also pseudo-SAW]Piezomagnetic Generalized Rayleigh wave Gulyaev–Kuzavko–half-space with quasistatic magnetic wave Oleınik–Shavrov waveIsotropic Lamb wave SH plate waveplate Symmetric AntisymmetricIsotropic Perturbed Rayleigh Sezawa wave Love wavehalf-space wave plus higher plus higherwith layer order modes order modes

Table B.3: Summary of the various types of surface acoustic waves reviewed inthe surface acoustic wave tutorial.

253

two main categories: those that have particle motion primarily in the sagittal

plane and those that have particle motion that is primarily shear horizontal∗

This tutorial has attempted to clearly describe the basic types and properties

of surface waves which the reader may find mentioned throughout the vast

literature on the subject of surface waves.

∗Maugin275 has specifically reviewed the topic of shear horizontal SAWs and discusses ina unified way many of the waves listed in the table.

Appendix C

Surface Acoustic Wave Applications Tutorial

This tutorial provides an overview of some the basic concepts used to develop

applications for surface acoustic waves (SAWs). It is by no means a complete

review of the field. For more technical and detailed information, see references

listed at the end of each section.

C.1 Signal Processing

Probably the most well-known application of SAWs is a class of solid-state

electronic components generally called SAW devices. These devices process

electrical signals by transforming an electrical input to an electrical output

via the transmission of a SAW across the surface of a piezoelectric crystal.

Electromechanical conversion at both the input and output is usually done via

the use of the interdigital transducer (IDT).276 As Gerard277 describes,

[IDTs] generally consist of interleaved combs of metal electrodes,

with each set extending from a common contact pad. The ID

transducers are photodeposited on the highly polished surface of

a precisely oriented piezoelectric crystal. Upon application of a

voltage to the contact pads, an electric field distribution having

the same spatial period of the electrodes is established between

the electrodes. By means of piezoelectric coupling, these surface-

concentrated fields produce a corresponding elastic strain distribu-

tion.

254

255

At the output, converse piezoelectric coupling causes the elastic strain of the

crystal to induce an electric signal in the IDT. By designing and combining

these IDTs in particular ways, it is possible to construct a wide range of useful

devices including filters, oscillators, pulse compressors and expanders (FM chirp

filters) using linear effects and convolvers, correlators, amplifiers, and memory

elements using nonlinear effects.11 These components have found uses in many

applications, especially mobile and wireless communication devices for personal

communication services (e.g., pagers, cellular phones), wide area networks, and

wireless local area networks.12

A few simple examples are given here to provide a basic understanding

of how such signal processing is accomplished.

In the linear regime, much of the signal processing capability comes from

creative uses of the acoustic transmission across the surface as a delay mech-

anism. Because the speed of sound is so much slower than the speed of light,

the electrical time variation of an input signal can be converted into an acous-

tical spatial variation over a relatively small length. Different points of this

spatial variation can then be sampled and manipulated by IDTs at particular

positions and with particular electrode lengths to process the signal in time.

Consider a SAW device for which an input signal f(t) gives the output signal

f(t) + f(t + τ), where τ is the desired delay time.11 This task is accomplished

by using an IDT to convert the signal f(t) to a SAW of the form f(x−ct) where

c is the SAW speed in the particular direction of propagation. By sampling

the wave with two other IDTs at two points separated by the distance cτ , the

signals f(t) and f(t + τ) are recorded. If these two IDTs are then connected

together the resulting signal is the sum f(t)+ f(t+ τ), as desired. Varying the

lengths of the electrodes varies the amount of electromechanical coupling and

hence the strength of the signal read at any given IDT sampling point.

256

In the nonlinear regime, signal processing is accomplished by exploit-

ing the nonlinearity of the crystalline medium. Consider the case of a SAW

convolver. The two input signals to be convolved are converted to counterprop-

agating SAWs via IDTs on each end of a piezoelectric crystal. A third IDT,

placed between the input IDTs, records the electrical output. If the two input

signals have sufficiently high power levels, then nonlinear harmonic generation

occurs. It can be shown that in such a process the power density of the non-

linearly produced signal is proportional to the product of the power densities

of the generating signals.11 Sampling of the wave by the middle IDT at the

appropriate intervals can then add up the product of the signals (the addition

occuring as described in the previous paragraph) and result in a convolved

signal output.11

Of course, the same effects can be achieved by standard digital electron-

ics. However, the SAW device approach has several advantages over other more

traditional methods. First, the digital electronics equivalent of a SAW device

can often be significantly more complicated.11 Second, traditional approaches

may not work well or at all at very high frequencies, whereas SAW devices

can easily function over a bandwidth from 100 MHz to 2 GHz.11,7 Moreover,

high-performance SAW devices can be cheaply and reliably produced in large

quantities.277,7 Finally, SAW devices can be integrated with other acoustic

and nonacoustic devices. Multiple functions can be implemented on the same

crystal surface and the wave traveling on the surface can be made to inter-

act with other types of external waves (e.g., light, electric, magnetic waves).7

All of these features make SAW devices very attractive for use in solid state,

microelectronic devices. As Ash7 notes,

Every modern television receiver contains at least one SAW filter;

many telecommunications systems—in particular satellite systems—

are dependent on extremely high performance SAW filters; large,

257

and to an increasing extent smaller radar systems use pulse com-

pression SAW filters; one finds high frequency SAW oscillators pen-

etrating electronic systems where previously one was forced to start

with a low frequency quartz crystal source followed by a frequency

multiplier chain. The nonlinear properties of SAW materials form

the basis of signal multiplication, which has opened up a much

wider range of signal processing capabilities. Surface acoustic wave

devices are here to stay.

For more detailed information on SAW devices, see the review by Stegeman

and Nizzoli6 and the books of Oliner13 (editor), Feldmann,7 and Campbell.278,12

C.2 Nondestructive Evaluation

Ultrasonic testing has proven to be a useful tool in the nondestructive evalu-

ation (NDE) of physical systems. By relating acoustic properties to material

or physical properties, it is possible to develop techniques to convert the (of-

ten difficult) task of nondestructively determining physical properties to the

(sometimes easier) task of making acoustic measurements. For example, mea-

surement of the density, transverse sound speed, and longitudinal sound speed

of an isotropic solid allows the Poisson’s ratio, shear modulus, and Young’s

modulus all to be computed.16 Sound velocity measurements have also been

used to determine other quantities including the mixing ratio in composites,

porosity in porcelain, moisture content in plastics, and structure and tensile

strength of gray cast iron.16 As discussed below, measurements of nonlinear

acoustic parameters may provide additional useful information about physical

properties of systems.

While both bulk acoustic waves (BAWs) and SAWs have been used to

perform NDE in wide variety of situations,16,279 the use of SAWs can be more

advantageous than BAWs in certain circumstances. Because SAWs have their

258

energy localized near the surface or interface (within approximately one to

two wavelengths), they are particularly well suited to probe the properties of

surfaces or interfaces. In fact, for thin walled or layered structures, SAWs may

be the only way to probe the sample because BAW testing requires that all

dimensions of the sample to be much larger than a wavelength.14 In addition,

because SAWs only spread two dimensionally and have a relatively smaller

cross-section than BAWs, the amplitudes produced for a given power input

may often be larger for the BAW. This can be important in NDE because often

the detected signals are very small, and the size of the initial amplitude may

decide whether a measurement may be possible at all.

SAWs have been used to characterize a variety of physical properties.

The following sections describe primarily work in the linear regime, but studies

using nonlinear SAWs have been performed as well. The brief descriptions

provided below are by no means a complete review of the extensive work that

has been done in this area; see the references cited in each section for more

information.

C.2.1 Defects

SAWs have been used to detect cracks, pits, voids, segregates, and impuri-

ties.14,280,15 A variety of methods have been developed, but two of the most

common are the pulse–echo and shadow methods. In the pulse–echo method,

defects are detected by measuring the echo reflected from them, and typically

the SAW source also acts as the detector. In the shadow method, separate SAW

source and detectors are used, and the acoustical shadow of the source signal is

measured by the detector. SAWs can exhibit strong reflection and absorption

from surface defects because their surface localization means that nearly the

entire wave is affected. They can also measure small defects quite accurately

259

over a large distance, e.g., a 0.025 mm defect has been detected at a distance

of 4 m using a pulse-echo technique.14 In sheets and tubes, Lamb waves (see

Section B.2.1) may be used with similar results, e.g., inclusions of 1 mm2 or

more in a 4 mm steel plate were reliably detected at radius of 1 m from the

source.14 In addition, Lamb waves can be used to detect segregation of surface

layers of relatively constant thickness because excitation of the surface causes

Lamb modes to occur where they would otherwise not.14 However, SAW meth-

ods have the disadvantages that (1) the presence of foreign materials in the

defects may mask their existence by allowing the SAWs to propagate through

with little change, (2) a rough surface may cause enough scattering that de-

tection of defects may be impossible, (3) visual inspection or liquid penetrant

methods may often be used more easily for many defects.16 In addition, the

interaction of even linear SAWs with very small defects (the size of the mean

free path of air molecules or about 0.1 µm) can be complicated by the nonlin-

ear relation between the compressibility and density of the air in these confined

regions.281 For more detailed information on defect detection with SAW, see

descriptions by Viktorov14 of some of the earliest work and Curtis,15 Doyle and

Scala,282 Stegeman and Nizzoli,6 Kino,280 Krautkramer and Krautkramer,16

and Schmerr283 for more recent efforts.

C.2.2 Plate and Layer Properties

Plates and layers arise in a wide variety of industrial applications. Plates

are common in applications that require metallic sheets or tubes including

ship hulls, airplane bodies, turbine blades, automotive chassis, combustor cas-

ings, and nuclear reactor pressure vessels. Layered systems are important to

many modern applications including very large scale integrated (VLSI) circuits,

micro-electro-mechanical systems (MEMS), reflective coatings on optical com-

ponents, chrome plating for corrosion protection, enamel glazing, low-friction

260

coatings, magnetic thin films for data storage, and hardening surface treatments

in steels. However, the mechanical characteristics of plates and layers, espe-

cially thin films, are not necessarily the same as those measured in the bulk

because the magnitude and directional distribution of mechanical, electrical,

magnetic, and chemical forces change as surfaces or interfaces are approached.

For example, thin films formed by vapor deposition have different properties

than layers formed by other means. Because of their localization in space near

surfaces and interfaces, SAWs have proven useful for determining the proper-

ties of plates and layers. It is well known that SAWs in plates and layers are

dispersive because the plate or layer thickness provides a characteristic length

scale for the system.5 The dispersion causes the SAW speed to vary with fre-

quency and certain modes of propagation to exist only in certain frequency

regimes. Because the dispersion relation and mode structure are functions of

the physical parameters of the system, measurements of the former can give

information about the latter.

Two examples are given here. First, plate thickness can be deduced

by exciting the sample with narrow-band Lamb waves (see Section B.2.1) suc-

cessively over a range of frequencies and looking for mode cut-on. With this

approach, plate thickness measurements from 0.015 mm to several millimeters

with precision of 0.0003 mm have been achieved.14 Secondly, measurements

of dispersion and attenuation in thin films have been performed by examining

the spectra of broad-band, laser-generated SAW pulses at various propagation

distances.264 From this data, the thickness, density, and transverse and longi-

tudinal sound speeds of the layer were extracted for a variety of layer–substrate

combinations. For additional examples see Viktorov,14 Curtis,15 Krautkramer

and Krautkramer,16 and Mayer.10

261

C.2.3 Applied and Residual Stresses

The study of stress in materials is important for many applications. Stresses

caused by the application of an external force are called applied stresses while

stresses that remain even with no force applied are designated residual stresses.

The latter are sometimes induced artificially, e.g., glass has its strength in-

creased by several orders of magnitude by the introduction of surface com-

pressional stresses due to thermal quenching or ion exchange.15 Due to the

acoustoelastic effect, the acoustic velocity of a material changes when a mate-

rial is under stress. For example, when an isotropic material is stressed, the

degeneracy of its transverse bulk wave speed velocities is broken so that differ-

ent polarizations have different speeds.16 Surface waves also exhibit this effect

and have been used to measure applied and residual stresses in solids. One ad-

vantage of SAWs for stress measurements is that they penetrate more deeply

than traditional methods like x-ray diffraction (millimeters vs. microns) and

are less strongly affected by surface roughness.280 However, the acoustic ap-

proach requires careful measurements because the changes in sound speed are

often a fraction of a percent even for large loads.15 An additional difficulty in

such velocity measurements is that stress may also cause changes in the grain

structure from the bulk material and these grain structure changes, in turn,

affect the velocity measurements.15 Residual stress measurements are possible

but tend to be more difficult for a variety of reasons.284,280 For more informa-

tion on ultrasonic stress measurements see the reviews by Curtis,15 Crecraft,284

Kino,280 and Mayer.10

C.2.4 Adhesive Bonding

Quantitative measurement and description of the strength of adhesion (bond-

ing between layers) and cohesion (bonding within a layer) is generally a com-

262

plicated and difficult problem. While adhesion and cohesion can vary with

interface chemistry, contamination, applied stresses, and roughness, it is pos-

sible to minimize and control many of these factors so as to make reasonable

studies possible.285 In layers that are relatively defect-free, bonding strength

is often proportional to elastic modulus.286 With microscopic imperfections,

the strength may be more dependent upon the type and quantity of the de-

fects.286 In both cases, acoustic waves may be useful because their velocity and

attenuation may be related to elastic modulus and defect-induced scattering.

Traditional techniques for measuring adhesion like the scratch and peel test,

while relatively simple, are destructive and often hard to reproduce.285

Some efforts have been made to use linear SAWs to investigate adhesion

between solids. Detection of Stoneley waves via an optical method287 was used

to correlate the attenuation of the waves with surface roughness and hence bond

strength.288 Shear modulus variations in the vicinity of adhesive–adherend

interfaces were determined by measuring the velocity of Rayleigh waves as

a function of adhesive thickness.289,290 Despite work in this area, problems

remain due to the complex nature of the problem and the difficulty of extracting

bond information from acoustic measurements. For example, one problem with

acoustic-based measurements of adhesion is that some organic adhesives are

highly absorptive and may mask poor bonding.16 See Curtis,286 Stegeman

and Nizzoli,6 and Krautkramer and Krautkramer16 for more information on

ultrasonic testing of adhesively bonded structures.

C.2.5 Other Material Properties

Many studies attempt to correlate velocity measurements with material prop-

erties, but attenuation measurements may also be useful. SAW attenuation

is affected by many factors including surface roughness, interaction with ther-

263

mal phonons, interactions with electrons in metals and semiconductors, ion

implantation, molecular in-diffusion, dislocations, and impurities. Often the

sensitivity can be quite high, to the extent that “just breathing on the sur-

face directly can completely attenuate a wave in a few tens of wavelengths.6”

Changes in attenuation have also been used to study phase transitions, e.g.,

superconductive thin layers.6

C.2.6 Nonlinear Ultrasonic NDE

Application of nonlinear BAWs and SAWs to NDE is a topic of current re-

search. Nonlinear effects and BAWs have been used to successfully study crack

size,6 the adhesion of composites,291 precipitate hardening in aluminum,292 and

the carbon content of quenched martensitic steels.72 Less work has been done

with SAWs, probably due to the experimental difficulties of exciting waves

that are large enough to exhibit nonlinear effects but do not destroy the sam-

ple. Several examples are given here. First, SAW excitation of a surface with

microcracks has been shown to generate a significant second harmonic signal,

often as early as 10–20% of the total fatigue life of the sample.293 Secondly,

in systems with “weak” dispersion (i.e. the mode coupling between the fun-

damental and higher harmonics is only slightly perturbed), the combination of

dispersion and nonlinearity typically gives rise to oscillations in the intensity

of the higher harmonics294–296 and has been used to investigate surface rough-

ness due to mechanical polishing.294 Lastly, measurements of the nonlinear

distortion in fused quartz have been performed by examining the spectra of

broad-band, laser-generated SAW pulses at various propagation distances.25

By curve-fitting data to theory, the various third-order elastic constants could

be deduced.

264

C.3 Chemical Sensing

SAW sensors have been developed to detect and identify chemicals. Concep-

tually, these SAW chemical sensors consist of two parts: (1) a material that

contains sites for binding the desired atomic, ionic, or molecular species and

(2) a transducer that has its response changed when the chemicals are present

in the binding material. The binding material is typically a thin film that is

deposited on a piezoelectric substrate. The SAWs are excited and detected

by interdigital transducers (see Section C.1) The adsorption and absorption

of the chemical molecules onto and into the film causes the properties of the

film to change (mass, conductivity, viscoelasticity, etc.). These changes cause

the amplitude and phase of the SAWs to be modified between the input and

output signal. Alternatively, only one IDT may be used with reflector(s) to

set up a SAW resonant cavity, typically with a resonant frequency in the range

20–300 MHz.17 In this method, the presence of the chemical species causes a

shift in the resonance frequency of the device.

Consider, for the moment, SAW devices that measure changes in reso-

nance frequency. Because an IDT transducer produces SAWs at a fixed wave-

length λ (determined by its finger spacing), measurement of the resonance

frequency f0 is effectively a measurement of the SAW velocity v0 = λf0. As

Thompson and Stone write,

The key to SAW chemical sensors, then, is understanding which

factors can affect the acoustic propagation velocity and, therefore,

result in changes of phase and resonance frequency. Following the

notation of Ricco,297 et al., these can be expressed as

∆v

v0=

1

v0

[∂v

∂m∆m +

∂v

∂c∆c +

∂v

∂ε∆ε +

∂v

∂σ∆σ +

∂v

∂T∆T +

∂v

∂P∆P + · · ·

]where m is mass, c is stiffness, ε is dielectric constant, σ is surface

conductivity, T is temperature, and P is pressure. It is possible to

265

obtain expressions for ∆v/v0 for a variety of different cases using

a perturbational approach and the appropriate boundary condi-

tions.105

Many of the earliest SAW sensors used the effects of mass loading alone, but

later sensors have taken more of these factors into account. In fact, the depen-

dence of SAW velocity on these properties has been exploited outside of their ef-

fect on chemical sensing to develop sensors specifically designed to measure tem-

perature,298,299 pressure,300 voltage,301,302 acceleration,303 and magnetic field.304

The small size and resulting portability of SAW sensors allows the

“putting of the instrument into the chemistry, not the chemistry into the instru-

ment.17” Applications have included monitoring workplaces for occupational

hazards, detecting chemical warfare agents, analyzing gas fuels, and determin-

ing ion levels in drinking water.17

The basic design of the SAW chemical sensor was introduced by Wohtjen

and Dessy305,306 in 1979. Subsequently sensors that employ other varieties of

SAWs have also been developed including Lamb waves,307 shear horizontal (SH)

waves,308 and Love waves.309 Each type has advantages and disadvantages. For

example, Lamb wave sensors have the advantages over traditional SAW devices

that they use lower frequencies (2–7 MHz as compared to 30–500 MHz), can

be constructed using standard microfabrication techniques, and can be used

for applications in both the gas and liquid phases (the latter occurs because

the speed of the plate waves is typically less than the compressional wave

speed of gases and liquids and hence no “leaky” waves are produced).17 See

the references at the end of this section for further discussion of the different

devices.

This section is by no means a complete review of this very wide-ranging

field. For more information on SAW chemical sensors, see the book by Thomp-

266

son and Stone.17

C.4 Other Applications

Many other applications have been developed using SAWs. While it is not

possible to provide a complete listing, some brief descriptions and references

are provided here. Note that inclusion in this section does not necessarily imply

that these applications are more important. The division simply expresses the

interests of the author.

C.4.1 Seismology

Seismology is the study of the acoustic disturbances generated by the move-

ments of the earth’s crust. The measurement and understanding of the propa-

gation of SAWs in the earth’s crust often play important roles in seismological

signal analysis. Because SAWs spread two dimensionally and because the near-

earth surface often has little effect on their propagation, they can often prop-

agate long distances and still be detected. Studies have shown that the most

powerful vibrations from earthquakes are in the frequency range of 0.5–5 Hz,

and, close to their source, the strongest ground motions are produced by shear

bulk waves and Love waves.310 In particular, the large horizontal motion of Love

waves is often particularly damaging to the foundations of structures.311 By

combining measurements of BAWs and SAWs at multiple locations and knowl-

edge of the wave speeds of the various signals, seismologists may determine an

earthquake’s location. By similar techniques, the location and magnitude of

underground explosions of nuclear weapons may also be monitored.312,313 How-

ever, recent work has shown that certain kinds of rock are highly nonlinear;314

this nonlinearity has been shown to cause a significant transfer of energy to

higher frequencies in seismic signals.

267

C.4.2 Acoustic Microscopy

Focusing ultrasonic beams on surfaces can excite SAWs. These waves interact

with surface features and, at high frequencies, can be used to produce detailed

acoustical images. These techniques have been used to produce images of many

different media from microelectronic circuits to biological tissues. Acoustic

microscopy has also been used to measure the elastic constants of thin films

and elastic properties of stressed materials.315 For more information, see review

articles by Bertoni,316 Chubachi,317 and Kino280 and the book by Briggs315

(editor).

C.4.3 Surface-skimming Bulk Waves (SSBW) Devices

Surface-skimming bulk waves (SSBW) (see Section B.1.4) combine many of the

advantages of SAWs and BAWs. For example, because the SAW velocity is less

than the BAW velocities, the thermal stability of the delay time for SAWs is

worse than that for BAWs in certain cuts of quartz.8 SSBW devices are used

as high frequency delay lines, filters, and resonators. For more information, see

Biryukov8 (Chapter 6) and Mayer.10

C.4.4 Acoustoelectric Applications

SAWs that propagate along the surface of a piezoelectric crystal have an accom-

panying electric field (see Section B.1.6). This makes it possible for acoustic

signals to interact with the electrons in semiconductors. The effects can often

be highly nonlinear due to the nonlinear relation between the current and field

in semiconductors. Applications include amplifiers, correlators, and optical im-

agers. For more information, see the review articles by Kino318 and Stern319

and Biryukov8 (Chapter 2).

268

C.4.5 Acoustooptic Applications

SAWs that propagate across the surface of certain materials modulate the index

of refraction of the material. This makes it possible for acoustic signals to

interact with light. Applications include optical diffraction gratings, deflectors,

modulators, and filters. As an example, modulators are used for switching light

beams in laser printers and laser communications.320 For more information, see

review articles by Chang,321 Schmidt,322 and Stegeman.323

C.4.6 Ultrasonic Motors

An ultrasonic motor is a type of actuator that uses mechanical vibrations in

the ultrasonic range as its drive source.324 For example, suppose two rings are

placed on top of one another so that they touch. Then the excitation of a

traveling Lamb wave in one ring (stator), due to frictional forces, causes the

other ring (rotor) to move in the direction of the traveling wave. Such traveling

wave motors have been developed commercially for use in auto-focusing camera

lenses. Alternatively, the frictional interaction between a small object on a

surface and the movement of a SAW can cause the object to move. As a

result, micromechanical motors can be produced that can rapidly make position

changes of an object with the accuracy of nanometers, e.g., see Kurosawa et

al..325 Ultrasonic motors also have the advantages that they are low-speed

but high-torque, have small size, and operate quietly. In addition, they are

unaffected by the presence of high magnetic fields (unlike traditional electric

motors) and, therefore, may be used in applications like magnetic resonance

imaging machines and magnetic levitating trains. See the books by Sashida and

Kenjo324 and Ueha et al.326 for comprehensive overviews of ultrasonic motor

research.

269

C.4.7 Surface Acceleration

Laser-generated, nonlinear SAWs have been used to produce local surface ac-

celerations that are high enough to throw off weakly adhering microparticles.327

Applications of this technique include cleaning microparticles from the surface

of semiconductor wafers in a vacuum, measurement of the adhesive forces of

microparticles, and visualization of the distribution of surface acoustic wave

pulses. The latter application was realized to image SAW focusing in crys-

tals.328

C.4.8 Touch Screen Technology

SAWs may be used to create screens that are sensitive to a user’s touch. A set

of SAW transducers and reflectors is arranged around the perimeter of a glass

panel. When a finger touches the screen, the SAWs are absorbed at that point

causing a touch event to be detected at that location. This type of screen has

the highest resolution and durability of any type of touch screen technology,

although it can be sensitive to large amounts of dirt, dust, and water. See the

paper by Adler329 for a brief review and discussion of this topic.

C.4.9 Animal Bioacoustics

A variety of animals are known to employ SAWs for short range communication.

Certain species of insects, frogs, kangaroo rats, mole-rats and chameleons are

able to generate and detect low frequency SAWS. For more information, see

the article by Narins.330

270

C.5 Summary

A remarkable variety of applications have been developed for SAWs in both the

linear and nonlinear regimes. With the ongoing research in the generation and

modeling of nonlinear SAWs, it is not unreasonable to expect that additional

applications will be found in future.

Appendix D

Crystals and Miller Index Notation

The properties of surface acoustic waves in anisotropic crystals depend critically

upon how the surface is oriented with respect to the crystal axes and the direc-

tion that the wave is propagating. The surfaces of cut crystals have tradition-

ally been described using a crystallographic convention called Miller indices.

Miller indices are defined by finding three noncollinear atoms on the surface

that intersect the crystal axes and then applying the following method331:

1. Find the intercepts of the three basis axes in terms of the

lattice constants.

2. Take the reciprocals of these numbers and reduce to the small-

est three integers having the same ratio. The result is enclosed

in parentheses (hkl).

Note that if the Miller indices are interpreted as vector components, the re-

sulting vector is normal to the surface of the cut. Directions are specified in a

different way331:

The indices of a direction in a crystal are expressed as the set of the

smallest integers which have the same ratios as the components of

a vector in the desired direction referred to the axis vectors. The

integers are written in square brackets, [uvw]. The x axis is the

[100] direction; the −y axis is the [010] direction. A full set of

equivalent directions is denoted this way: 〈uvw〉.

271

272

Figure D.1 shows some typical crystal cuts and how they are specified

using the Miller index notation. For example, the (001) plane intersects the

crystal axes at, x = ∞ (1/∞ = 0), y = ∞, and z = 1. Similarly, the (111)

plane intersects the crystal axes at x = 1, y = 1, and z = 1.

273

Typical Planes

for Crystal Cuts

in Cubic Crystals

(Miller index notation)y

x

z

θ

<112>

(111) plane

θ

y

x

z

<100>

(001) plane

y

x

z

<001>

(110) plane

θ

Figure D.1: Typical planes for crystal cuts in cubic crystals as specified usingMiller index notation. The reference directions used for specifying directionsof propagation are indicated by the vector direction in angle brackets.

Appendix E

Derivation of the Integral Transform Between

Velocity Components of a SAW in a Crystal

As stated in Section 2.3, the velocity components of SAW waveforms in a crystal

can be related by the integral transforms given by Eq. (2.100). In this section,

a derivation of this result is provided.

According to Eq. (2.89), the velocity component on the surface x3 = 0

in the xj direction is given by

vj(x, τ) =

∞∑n=−∞

vn(x)unje−inω0τ =

∞∑n=−∞

vjn(x)e−inω0τ , (E1)

where

unj =

{Bi =

∑3s=1 β

(s)i for n > 0

B∗i =

∑3s=1(β

(s)i )∗ for n < 0

}= |Bi|ei(n/|n|)φBi , (E2)

and the constants β(s)i from linear theory are defined in Section 2.1. Equa-

tion (E1) can be rewritten as

vj(x, τ) =

∞∑n=−∞

[unj

uni

][vn(x)uni]e

−inω0τ . (E3)

Substitution of Eq. (E2) into Eq. (E3) immediately yields

vjn(x) =

{(Bj/Bi)vin(x) for n > 0(B∗

j /B∗i )vin(x) for n < 0

, (E4)

which is the result relating the spectral amplitudes given in Eq. (2.99).

274

275

Calculations relating the time waveforms can be most easily carried out

by considering the problem in terms of Fourier transforms instead of Fourier

series. The continuous analogue to the series of Eq. (E1) is

vj(x, τ) =1

2π

∫ ∞

−∞v(x, ω)uj(ω)e−iωτ dω =

1

2π

∫ ∞

−∞vj(x, ω)e−iωτ dω , (E5)

where

uj(ω) =

{Bj for ω > 0B∗

j for ω < 0

}= |Bi|ei(ω/|ω|)φBi . (E6)

Correspondingly, the continuous analogue of Eq. (E3) is given by

vj(x, τ) =1

2π

∫ ∞

−∞

[uj(ω)

ui(ω)

][v(x, ω)ui(ω)]e−iωt dω . (E7)

Let F (x, ω) = v(x, ω)ui(ω) and

G(ω) =uj(ω)

ui(ω)=

{Bji = Bj/Bi for ω > 0B∗

ji = B∗j /B

∗i for ω < 0

, (E8)

so that

vj(x, τ) =1

2π

∫ ∞

−∞G(ω)F (x, ω)e−iωt dω . (E9)

Here G(ω) and F (x, ω) can be interpreted as Fourier transforms of undeter-

mined functions g(τ) and f(x, τ). Under this interpretation, the Convolution

Theorem111 can be applied to Eq. (E9) to yield

vj(x, τ) = g ∗ f =

∫ ∞

−∞f(x, τ ′)g(τ − τ ′) dτ ′ . (E10)

Hence, the computation of f(x, τ) and g(τ) fully determines the integral trans-

form.

The function f(x, τ) is found by taking the inverse Fourier transform of

F (x, ω) = vi(x, ω). This can be computed immediately from Eq. (E5)

f(x, τ) =1

2π

∫ ∞

−∞F (x, ω)e−iωτ dω =

1

2π

∫ ∞

−∞vi(x, ω)e−iωτ dω = vi(x, τ) .

(E11)

276

The function g(τ) is found by taking the inverse Fourier transform of G(ω):

g(τ) =1

2π

∫ ∞

−∞G(ω)e−iωτ dω . (E12)

However, substitution of Eq. (E8) followed straightforward evaluation of this

integral fails because the integral is not absolutely convergent. Instead, com-

pute the inverse transform

gr(τ) =1

2π

∫ ∞

−∞G(ω)e−r|ω|e−iωτ dω , (E13)

and take the limit

limr→0

gr(τ) = g(τ) . (E14)

Evaluation of the integral of Eq. (E13) is done as follows:

gr(τ) =1

2π

∫ ∞

−∞G(ω)e−r|ω|e−iωτ dω

=1

2π

[∫ 0

−∞B∗

jie(r−iτ)ω dω +

∫ ∞

0

Bjie(−r−iτ)ω dω

]=

1

2π

[B∗

ji

e(r−iτ)ω

r − iτ

∣∣∣∣0−∞

+ Bji

e(−r−iτ)ω

−r − iτ

∣∣∣∣∞0

]

=1

2π

[B∗

ji

r − iτ+

Bji

r + iτ

].

(E15)

In order to properly evaluate the limit as r → 0, Eq. (E15) may be further

rewritten as

gr(τ) =1

2π

[B∗

ji

r − iτ+

Bji

r + iτ

]=

1

2π

[B∗

ji(r + iτ) + Bji(r − iτ)

r2 + τ 2

]=

1

2π

[(Bji + B∗

ji)r − (Bji − B∗ji)iτ

r2 + τ 2

]=

(Bji + B∗

ji

2

)1

π

r

r2 + τ 2+

(Bji − B∗

ji

2i

)1

π

τ

r2 + τ 2

= Re(Bji)1

π

r

r2 + τ 2+ Im(Bji)

1

π

τ

r2 + τ 2.

(E16)

277

It then follows that

g(τ) = limr→0

gr(τ) = Re(Bji) limr→0

1

π

r

r2 + τ 2+ Im(Bji) lim

r→0

1

π

τ

r2 + τ 2. (E17)

Evaluating the first limit on the right side of Eq. (E17) yields

limr→0

1

π

r

r2 + τ 2= lim

r→0

1

π

1/r

1 + (1/r)2τ 2= lim

n→∞1

π

n

1 + n2τ 2= lim

n→∞δn(τ) = δ(τ) ,

(E18)

where δ(τ) is the Dirac delta function defined such that112∫ ∞

−∞f(τ)δ(τ) dτ = f(0) . (E19)

The last equality in Eq. (E18) is a well-known result from distribution theory.112

Evaluating the second limit on the right side of Eq. (E17) yields

limr→0

1

π

1

r2 + τ 2=

1

πτ(E20)

Substituting Eqs. (E18) and (E20) into Eq. (E17) gives

g(τ) = Re(Bji)δ(τ) + Im(Bji)1

πτ. (E21)

Substitution of Eqs. (E11) and (E21) into the convolution of Eq. (E10) then

gives the result

vj(x, τ) =

∫ ∞

−∞f(x, τ ′)g(τ − τ ′) dτ ′

=

∫ ∞

−∞vi(x, τ ′)

[Re(Bji)δ(τ − τ ′) + Im(Bji)

1

π(τ − τ ′)

]dτ ′

= Re(Bji)vi(x, τ) + Im(Bji)1

π

∫ ∞

−∞

vi(x, τ ′)τ − τ ′

dτ ′

= Re(Bji)vi(x, τ)− Im(Bji)1

π

∫ ∞

−∞

vi(x, τ ′)τ ′ − τ

dτ ′ .

(E22)

By definition,111 the Hilbert transform is

H[h(τ)] =1

πPr

∫ ∞

−∞

h(τ ′)τ ′ − τ

dτ ′ . (E23)

278

Substituting Eq. (E23) and Bji = Bj/Bi into Eq. (E22) yields

vj(x, τ) = Re(Bj/Bi)vi(x, τ)− Im(Bj/Bi)H[vi(x, τ)] . (E24)

This is the result listed in Eq. (2.100).

Appendix F

Additional Discussion of Complex-Valued

Nonlinearity Parameters

The basic results of Section 6.2.2 can be derived in an alternate way. This

alternative method provides additional insight into relationship between the

various linear and nonlinear quantities described by the theory presented in

Chapter 2.

Instead of determining the effect of phase changes on the spectral am-

plitudes, here the focus is on phase changes in the linear amplitude factors

β(s)j . As stated in Section 2.1.2, the absolute phase of the β

(s)j terms is not

determined by the theory; only relative phases between the β(s)j are fixed. To

see this, recall from Eq. (2.9) that the particle displacement for a linear SAW

in a crystal can be written in the form

uj =

3∑s=1

β(s)j eik(ls·x−ct) =

3∑s=1

β(s)j eiζskx3ei(kx1−ω0t) , (F1)

where ls = (l(s)1 , l

(s)2 , l

(s)3 ) = (1, 0, ζs), and the β

(s)j = Csα

(s)j are defined by

Eq. (2.11):

ci3kl

3∑s=1

β(s)j l

(s)l = 0 , (F2)

as a result of the stress-free boundary condition in Eq. (2.10). To examine the

effects of changing the absolute phase, let

β(s)θj = β

(s)j eiθ (F3)

279

280

for some phase angle 0 ≤ θ ≤ 2π. Multiplying both sides of Eq. (F2) by eiθ

and substituting Eq. (F3) into the result yields

ci3kl

3∑s=1

β(s)θj l

(s)l = 0 , (F4)

which shows that the β(s)θj are also valid solutions for the linear problem. The

only other condition on the β(s)θj terms is the normalization condition of the

nonlinear theory given in Eq. (2.27)

∫ 0

−∞

∣∣∣∣∣3∑

s=1

β(s)j eiζsz′

∣∣∣∣∣2

= 1 . (F5)

However, because |β(s)θj | = |β(s)

j |, it follows immediately that

∫ 0

−∞

∣∣∣∣∣3∑

s=1

β(s)θj eiζsz′

∣∣∣∣∣2

= 1 . (F6)

Thus, it has been shown if the linear amplitude factors β(s)θj are solutions of the

linear problem and satisfy the normalization condition of the nonlinear theory,

then the amplitude factors β(s)θj defined by Eq. (F3) are also valid solutions.

However, changing the phase of linear amplitude factors, also changes

other quantities in the theory which depend upon them. In particular, the rep-

resentation of the nonlinearity matrix elements, depth dependence functions,

and waveforms are all changed.

First, consider changes in the nonlinearity matrix elements under the

transformation caused by introducing β(s)θj in place of β

(s)j . Eq. (2.26) then

becomes

F θs1s2s3

= Fs1s2s3eiθ . (F7)

281

According to Eq. (2.33) for n = l + m > 0, it then follows that

Sθlm(−n) =

3∑s1,s2,s3=1

F θs1s2s3

lζs1 + mζs2 − nζ∗s3

= Slm(−n)eiθ . (F8)

For n = l + m < 0, the symmetry property of Eq. (6.3) [also Hamilton et al.,33

Eq. (80)] may be used to show that

Sθlm(−n) = (Sθ

(−l)(−m)n)∗ = (S(−l)(−m)neiθ)∗ = S∗(−l)(−m)ne−iθ = Slm(−n)e

−iθ .

(F9)

Combining Eq. (F8) and Eq. (F9) implies that for all n,

Sθlm(−n) = Slm(−n)e

i(n/|n|)θ . (F10)

By multiplying both sides of Eq. (F10) by −1/c44 and using Eq. (4.5), it im-

mediately follows that

Sθlm(−n) = Slm(−n)e

i(n/|n|)θ . (F11)

Equation (F11) matches the notation introduced previously in Eq. (6.4).

Next, consider the effect of transforming the depth functions under the

substitution β(s)θj = |β(s)θ

j | exp(iφ(s)θj ) for β

(s)j = |β(s)

j | exp(iφ(s)j ). By Eq. (F3),

it follows that

|β(s)θj | = |β(s)

j | , (F12a)

φ(s)θj = φ

(s)j + θ . (F12b)

Using Eqs. (F12), the depth functions given in Eq. (2.13) transform as

uθnj(z) =

3∑s=1

|β(s)θj | exp[i(n/|n|)φ(s)θ

j ]

=3∑

s=1

|β(s)j | exp[i(n/|n|)(φ(s)

j + θ)]

= unj(z)ei(n/|n|)θ .

(F13)

282

Note that uθnj = [uθ

(−n)j ]∗, as required by Eq. (2.13).

Finally, if the transformed velocity waveforms are defined as

vθj (x, z, τ) =

∞∑n=−∞

vn(x)uθnj(z)e−inω0τ , (F14)

then Eq. (F14) is the analogue of Eq. (6.9). The only difference between them

is that here Eq. (F14) introduces the phase factor ei(n/|n|)θ, whereas Eq. (6.9)

introduces the phase factor of e−i(n/|n|)θ from the transformed spectral ampli-

tudes of Eq. (6.8):

vn(x) = vn(x)e−i(n/|n|)θ . (F15)

In other words, Figures 6.5 and 6.6 can just as well be derived from Eq. (F14)

by replacing θ with −θ.

The close relationship between Eq. (F14) and Eq. (6.9) is not coinciden-

tal. Because the transformation of the nonlinearity matrix given in Eq. (F11)

above also implies Eq. (F15) (by the derivation in Chapter 6), the waveforms

should, strictly speaking, be reconstructed using both the transformations of

the spectral amplitudes vn(x) and the depth functions unj(z). Define the wave-

form function

vθj (x, z, τ) =

∞∑n=−∞

vθn(x)uθ

nj(z)e−inω0τ . (F16)

Substituting Eqs. (F13) and (F15) into Eq. (F16) immediately implies that for

all θ,

vθj (x, z, τ) = vj(x, z, τ) . (F17)

In other words, the velocity waveforms are invariant under the change in the ab-

solute phase of the β(s)j . This result is expected because the internal mathemat-

ical representation of the absolute phase cannot affect the physical waveforms.

283

The partial waveform transformations [partial in the sense that only vθn(x) or

only uθnj(z) are used] show how breaking the invariance of Eq. (F17) is similar

to introducing anisotropy into the physical system and thereby changing the

phase relationship between the nonlinearity matrix elements, linear amplitude

factors, and spectral components from the isotropic case.

To close, this section provides a detailed derivation of Eqs. (6.19). These

equations give the values of the transformation phase θ that are appropriate use

in conjunction with Figures 6.5 and 6.6 to characterize the various waveforms

of the velocity components. Assume that the phases differences between the

surface linear amplitude components B1, B2, and B3 are known. For example,

consider the vertical velocity component at the surface, and the idealized cases

where all the nonlinearity matrix elements have the phase θ = θlong = arg(Slm).

(If the matrix elements are not all the same phase, use a matrix element rep-

resentative of the average phase of the elements.) Then Eq. (6.10) can be

rewritten as

v3(x, τ) =

∞∑n=−∞

vθn(x)|B3|ei(n/|n|)φB3e−inω0τ

=|B3||B1|

∞∑n=−∞

vθn(x)|B1|ei(n/|n|)φB1ei(n/|n|)(φB3−φB1)e−inω0τ

=|B3||B1|

∞∑n=−∞

ei(n/|n|)(φB3−φB1)vRn (x)ei(n/|n|)θ|B1|ei(n/|n|)φB1e−inω0τ ,

(F18)

where the last step uses Eq. (6.8) with the reference Rayleigh wave spectrum

vRn (x). If θvert = φB1 − φB3 + θ, then Eq. (F18) can rewritten as

v3(x, τ) =|B3||B1|

∞∑n=−∞

vRn ei(n/|n|)θvert |B1|ei(n/|n|)φB1e−inω0τ =

|B3||B1|v

θvert1 (x, τ) .

(F19)

284

A similar derivation can be done for the transverse component. Therefore, the

transformation phases corresponding to the transverse and vertical velocity

components on the surface are given by

θtran = arg(B1)− arg(B2) + θlong , (F20a)

θvert = arg(B1)− arg(B3) + θlong , (F20b)

which is the result listed in Eqs. (6.19). As discussed at length in Chapter 6,

Eqs. (6.19) are only expected to give reasonable results when the dominant

matrix elements are similar in phase.

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Vita

Ronald Edward Kumon was born in Grosse Pointe, Michigan, on April 11, 1970,

the son of Henry Victor Kumon and Rosemary Cynthia Kumon. He grew up in

the rustic town of Hartland, Michigan. After graduating as valedictorian from

Hartland High School, Hartland, Michigan, in 1988, he entered Michigan State

University in East Lansing, Michigan. He received the degree of Bachelor of

Science with Honors in Physics and Mathematics from Michigan State Univer-

sity in June 1992. In August 1992, he matriculated into the graduate program

in the Department of Physics at The University of Texas at Austin. During his

graduate studies, he worked for three years at Applied Research Laboratories

on the thermoacoustics of the Rijke tube and for three years in the Department

of Mechanical Engineering on nonlinear surface acoustic waves in crystals. He

graduated from The University of Texas of Austin with a Ph.D. in Physics in

1999.

Permanent address: 12521 Erika, Hartland, MI 48353, U.S.A.

This dissertation was typed by the author.

321

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