Elastic Waves at High Frequency

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Elastic Waves at High Frequencies Techniques for Radiation and Diﬀraction of Elastic and Surface Waves

Elastic Waves at High Frequencies Techniques for Radiation and Diﬀraction of Elastic and Surface Waves JOHN G. HARRIS Late, University of Illinois at Urbana–Champaign Edited and Prepared for Publication by GARETH I. BLOCK Flemington, NJ RICHARD V. CRASTER University of Alberta ANTHONY M. J. DAVIS University of California, San Diego PAUL A. MARTIN Colorado School of Mines ANDREW N. NORRIS Rutgers University

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521875301 © J. G. Harris 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN-13

978-0-511-78982-3

eBook (NetLibrary)

ISBN-13

978-0-521-87530-1

Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Do not go gentle into that good night, Old age should burn and rave at close of day; Rage, rage against the dying of the light. Though wise men at their end know dark is right, Because their words had forked no lightning they Do not go gentle into that good night. Do not go gentle into that good night By Dylan Thomas, from THE POEMS OF DYLAN c THOMAS, copyright 1952 by Dylan Thomas. Reprinted by permission of New Directions Publishing Corp. and J. M. Dent & Sons Ltd.

Contents

Foreword

page ix

1

Linear elastic waves 1.1 Model equations 1.2 Continuity and boundary conditions 1.3 Flux of energy 1.4 The Fourier and Laplace transforms 1.5 A wave is not a vibration 1.6 Dispersive propagation 1.7 General references

1 1 6 6 9 12 15 20

2

Canonical acoustic-wave problems 2.1 Radiation from a piston in an inﬁnite baﬄe 2.2 Diﬀraction of an acoustic plane wave by an edge 2.3 Summary

23 23 40 50

3

Canonical elastic-wave problems 3.1 The scattering of a spherical wave from a ﬂuid–solid interface 3.2 Rayleigh–Lamb modes and Rayleigh surface waves

52

4

Radiation and impedance 4.1 Reciprocity 4.2 Green’s tensor 4.3 Reciprocity revisited 4.4 Force on a particle from an elastic wave

71 71 72 78 79

5

Integral equations for crack scattering 5.1 Formulation 5.2 A hypersingular integral equation 5.3 Low-frequency scattering

81 81 84 86

vii

52 62

viii

Contents 5.4 5.5 5.6 5.7 5.8

Some strategies Flat cracks as a special case Flat cracks: direct approach Flat cracks: how to compute [u] Curved cracks

88 88 89 91 93

6

Scanned acoustic imaging 6.1 Scanned, reﬂection acoustic microscope 6.2 Fresnel and F number 6.3 Converging spherical wave 6.4 Focused acoustic beam 6.5 Scattered focused beam 6.6 An electromechanical reciprocity relation 6.7 Measurement model 6.8 Acoustic material signature 6.9 Summary

96 97 99 99 101 103 109 113 116 119

7

Acoustic diﬀraction in viscous ﬂuids 7.1 Theory 7.2 Diﬀraction by a half-plane 7.3 Scattering of a spherical wave at a plane interface 7.4 Diﬀraction by an elastic sphere

121 121 123 127 130

8

Near-cut-oﬀ behavior in waveguides 8.1 Shear horizontal and acoustic waveguides 8.2 Elastic waveguides 8.3 Long waves

133 133 136 138

Appendix A

Asymptotic expansions

147

Appendix B

Some special functions

154

References Index

157 163

Foreword

This volume is dedicated to the memory of John G. Harris, whose life ended prematurely on the 6th of May, 2006. John’s friendship and research impacted many people – he was a dedicated and loving husband, an accomplished scientist and applied mathematician, a passionate teacher, and an important mentor to many young scientists. This book was originally intended to be John’s second book on elastic wave theory and diﬀraction. It grew from four lectures that were given at the Department of Mathematics and Mechanics, within IIMAS, at the National Autonomous University of Mexico, in January 2004. After John’s passing, several of his colleagues, inspired by his wife Beatriz, began to convert these unﬁnished notes into a form suitable for publication. We have worked to combine the existing chapters with additional, contributed chapters from experts in the ﬁeld of elastic wave theory. Born and raised in Toronto, John entered McGill University as a mature student and graduated with a Bachelors in Electrical Engineering (Honours). After receiving a Masters of Science in Applied Physics from Stanford University, John traveled to Northwestern University to work toward a doctorate in Applied Mathematics with Jan Achenbach, which he completed in 1979. J. D. Achenbach had a lasting impact on John’s work in elastic wave scattering, which formed the basis of much of John’s research as a professor at the Department of Theoretical and Applied Mechanics at the University of Illinois at Urbana–Champaign between 1979 and 2005. John’s initial research focused on the scattering of ultrasonic elastic waves from cracks, in particular the use of surface waves to interrogate curved shells and surface-breaking cracks in cases of interest to acoustic microscopy. He developed a great interest in WKBJ theory and ix

x

Foreword

asymptotic approximations of waveﬁelds, which allowed him to share his clear and elegant understandings of diﬀraction in an extensive list of publications. John also wrote a graduate-level textbook, Linear Elastic Waves, and a monograph on elastic wave theory during his visit to Mexico. He began this volume in Quaid-i-Azam University in Islamabad, Pakistan, where he spent a semester lecturing in 1994. Two unifying themes are used throughout this book. The ﬁrst is that wave propagation and scattering are among the most fundamental processes that we use to comprehend the world around us. The second is that waves are best understood in an asymptotic approximation, where they are free of the complications of their excitation and are governed primarily by their propagation environment. This book is not intended as a textbook, in the sense that it is not written to accompany a speciﬁc course. However, the chapters do follow a course of increasing complexity, beginning with plane-wave propagation and spectral analyses, which allows for the development of advanced techniques for studying diﬀraction. A short synopsis of the chapters is as follows. John’s writing forms the basis of Chapters 1, 2, and 3. Chapter 1 introduces the topics of elastic wave theory, energy ﬂux, and Fourier and Laplace representations of time-harmonic waveﬁelds. Chapters 2 and 3 solve canonical scattering problems using asymptotic approximations to Fourier integrals. Chapter 4, written by A. Norris, explores the use of reciprocity identities and mechanical impedance to describe radiation and scattering problems. Radiation of elastic waves is examined using Gaussian beam solutions as a model for transducers, while reciprocity is used to derive the force on a particle caused by an incident elastic wave. Chapter 5, written by P. A. Martin, concerns integral formulations of scattering from cracks. In particular, a special case of scattering from a screen is solved in the low-frequency limit by approximating a hypersingular integral equation, and a general strategy for solving more complex problems is proposed to analyze scattering from curved cracks and non-planar geometries. Chapter 6, written by J. G. Harris, develops techniques for scanned acoustic imaging that utilize a converging spherical wave generated by a transducer above a ﬂuid–solid interface. The incident and scattered ﬁelds are written in terms of an angular spectrum of plane waves, and a reciprocity relation is used to express unknown material variability (in a thin ﬁlm attached to the interface, for example) in terms of the measured transducer voltage.

Foreword

xi

Chapter 7, written by A. Davis, explores the eﬀects of viscosity on acoustic diﬀraction. Compressibility and viscosity are seen to be intertwined, as dilatational waves become coupled to vorticity disturbances near scattering interfaces. Scattering solutions are derived for plane waves diﬀracted by a half-plane, and for spherical waves scattered by a plane interface and by an elastic sphere. Chapter 8, written by R. Craster, elucidates the phenomenon of channeling of wave energy along guided structures. The chapter ﬁrst summarizes guided waves in acoustics, and then proceeds to describe elastic waves in straight waveguides (including the Rayleigh–Lamb modes). Asymptotic expressions for wave propagation in waveguides that are inhomogeneous, possibly bent or of varying thickness, are the ultimate focus of this chapter. Appendix A, written by J. G. Harris, discusses asymptotic expansions and asymptotic approximations of integrals, methods that are used routinely throughout the book. Appendix B, also by J. G. Harris, lists, without derivation, properties of the special functions that arise in the book. These nine chapters cover both the necessary introductory material and a broad survey of applications in diﬀraction and scattering theory. John’s dedication, creativity, and clear understanding of these subjects have inspired us to take on the task of completing his ﬁnal work. We thank him dearly for his friendship and collaboration, and hope that future readers will ﬁnd the topics as compelling and captivating as he did. Sincerely, Gareth I. Block Richard V. Craster Anthony M. J. Davis Paul A. Martin Andrew N. Norris

Bibliography of J. G. Harris Achenbach, J. D. and Harris, J. G. (1978). Ray method for elastodynamic radiation from a slip zone of arbitrary shape. J. Geophys. Res., 83:2283–2291. Achenbach, J. D. and Harris, J. G. (1979). Acoustic emission from a brief crack propagation event. J. Appl. Mech., 46:107–112.

xii

Foreword

Harris, J. G. (1980). Diﬀraction by a crack of a cylindrical longitudinal pulse. Zeit. Angew. Math. Physik, 31:367–383, 1980. Correction: 34:253, 1983. Harris, J. G. (1980). Uniform approximations to pulses diﬀracted by a crack. Zeit. Angew. Math. Physik, 31:771–775. Harris, J. G. and Achenbach, J. D. (1981). Near-ﬁeld surface motions excited by radiation from a slip zone of arbitrary shape. J. Geophys. Res., 86:9352–9356. Achenbach, J. D. and Harris, J. G. (1982). Focusing of ground motion due to curved rupture fronts. In J. Boatwright, editor, Proceedings of Workshop XVI: The Dynamic Characteristics of Faulting Inferred from Recordings of Strong Ground Motion, volume 1 (Report 85-591), pages 50–70. U.S. Geological Survey, Menlo Park, CA. Harris, J. G. and Achenbach, J. D. (1983). Love waves excited by discontinuous propagation of a rupture front. Geophys. J. Roy. Astron. Soc., 72:337–351. Correction: 74:647, 1983. Harris, J. G., Achenbach, J. D. and Norris, A. N. (1983). Rayleigh waves excited by the discontinuous advance of a rupture front. J. Geophys. Res., 88:2233–2239. Shield, T. W. and Harris, J. G. (1984). An acoustic lens design using the geometrical theory of diﬀraction. J. Acoust. Soc. Am., 75:1634– 1635. Pott, J. and Harris, J. G. (1984). Scattering of an acoustic Gaussian beam from a ﬂuid–solid interface. J. Acoust. Soc. Am., 76:1829– 1838. Harris, J. G. and Pott, J. (1984). Surface motion excited by acoustic emission from a buried crack. J. Appl. Mech., 51:77–83. Harris, J. G. (1984). Wave-front approximations in a moving coordinate system. J. Appl. Mech., 51:934–935. Achenbach, J. D. and Harris, J. G. (1984). Excitation of near- and far-ﬁeld ground motions by sliding events on a fault plane. In S. K. Datta, editor, Earthquake Source Modeling, Ground Motion, and Structural Response, volume PVP-80, pages 13–27. ASME, New York. Harris, J. G. and Pott, J. (1985). Further studies of the scattering of a Gaussian beam from a ﬂuid–solid interface. J. Acoust. Soc. Am., 78:1072–1080. Harris, J. G. (1987). Edge diﬀraction of a compressional beam. J. Acoust. Soc. Am., 82:635–646.

Foreword

xiii

Achenbach, J. D. and Harris, J. G. (1987). Asymptotic modeling of strong ground motion excited by subsurface sliding events. In B. A. Bolt, editor, Seismic Strong Motion Synthetics, pages 1–54. Academic Press, New York. Harris, J. G. (1988). The waveﬁeld radiated into an elastic halfspace by a transducer of large aperture. J. Appl. Mech., 55:398–404. Choi, H. C. and Harris, J. G. (1989). Scattering of an ultrasonic beam from a curved interface. Wave Motion, 11:383–406. Harris, J. G. (1989). An asymptotic description of an elastodynamic beam. In M. F. McCarthy and M. A. Hayes, editors, Elastic Wave Propagation, pages 505–510. Elsevier, Amsterdam. Choi, H. C. and Harris, J. G. (1990). Focusing of an ultrasonic beam by a curved interface. Wave Motion, 12:497–511. Choi, H. C. and Harris, J. G. (1990). Focusing of an ultrasonic beam by a concave interface. In S. K. Datta, J. D. Achenbach and Y. S. Rajapakse, editors, Elastic Waves and Ultrasonic Nondestructive Evaluation, pages 177–182. Elsevier, Amsterdam. Achenbach, J. D., Ahn, V. S. and Harris, J. G. (1991). Wave analysis of the acoustic material signature for the line focus microscope. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 38:380–387. Ahn, V. S., Harris, J. G. and Achenbach, J. D. (1991). Acoustic material signature for a cracked surface. In Proceedings 1990 IEEE Ultrasonics Symposium, volume 2, pages 921–924. IEEE, New York. Achenbach, J. D., Ahn, V. S. and Harris, J. G. (1991). BEM analysis for the line focus acoustic microscope. In D. O. Thompson and D. E. Chimenti, editors, Review of Progress in Quantitative Nondestructive Evaluation, volume 10A, pages 225–232. Plenum Press, New York. Ahn, V. S., Harris, J. G. and Achenbach, J. D. (1992). Numerical analysis of the acoustic signature of a surface-breaking crack. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 39:112–118. Rebinsky, D. A. and Harris, J. G. (1992). An asymptotic calculation of the acoustic signature of a cracked surface for the line focus scanning acoustic microscope. Proc. R. Soc. A, 436:251–265. Rebinsky, D. A. and Harris, J. G. (1992). The acoustic signature for a surface-breaking crack produced by a point focus microscope. Proc. R. Soc. A, 438:47–65.

xiv

Foreword

Yogeswaren, E. and Harris, J. G. (1994). A model of a conformal ultrasonic inspection system for interfaces. J. Acoust. Soc. Am., 96:3581–3592. Qi, Q., O’Brien, Jr., W. D. and Harris, J. G. (1995). The propagation of ultrasonic waves through a bubbly liquid into tissue: a linear analysis. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 42:28–36. Qi, Q., Johnson, R. E. and Harris, J. G. (1995). Boundary layer attenuation and acoustic streaming accompanying plane-wave propagation in a tube. J. Acoust. Soc. Am., 97:1499–1509. Harris, J. G. and Yogeswaren, E. (1995). A model of a conformal ultrasonic imaging system for solid-solid interfaces. In J. L. Wegener and F. R. Norwood, editors, Nonlinear Waves in Solids, pages 196–200. ASME, New York. Harris, J. G., Rebinsky, D. A. and Wickham, G. (1996). An integrated model of scattering from an imperfect interface. J. Acoust. Soc. Am., 99:1315–1325. Rebinsky, D. A., Harris, J. G. and Wickham, G. (1996). Interrogating a thin layer of heterogeneity with confocal transducers. In Review of Progress in Quantitative Nondestructive Evaluation, volume 15A, pages 1027–1033. Plenum Press, New York. Ti, B. W., O’Brien, Jr., W. D. and Harris, J. G. (1997). Measurements of coupled Rayleigh wave propagation in an elastic plate. J. Acoust. Soc. Am., 102:1528–1531. Harris, J. G. (1997). Modeling scanned acoustic imaging of defects at solid interfaces. In G. Chavent, G. Papanicolaou, P. Sacks, and W. W. Symes, editors, Inverse Problems in Wave Propagation, volume 90 of IMA Volumes in Mathematics and its Applications, pages 237–257. Springer-Verlag, New York. Folguera, A. and Harris, J. G. (1998). Propagation in a slowly varying elastic waveguide. In J. A. DeSanto, editor, Mathematical and Numerical Aspects of Wave Propagation, pages 434–436. SIAM, Philadelphia. Folguera, A. and Harris, J. G. (1999). Coupled Rayleigh surface waves in a slowly varying elastic waveguide. Proc. R. Soc. A, 455: 917–931. Goueygou, M., Harris, J. G. and O’Brien, Jr., W. D. (1999). Timedomain solution of the temperature increase induced by diagnostic ultrasound. In Proceedings 1999 IEEE Ultrasonics Symposium, volume 2, pages 1385–1388. IEEE, New York.

Foreword

xv

Block, G., Harris, J. G. and Hayat, T. (2000). Measurement models for ultrasonic nondestructive evaluation. IEEE Trans. Ultrason. Ferroelectr. Freq. Control, 47:604–611. Asghar, S., Hayat, T. and Harris, J. G. (2001). Diﬀraction by a slit in an inﬁnite porous barrier. Wave Motion, 33:25–40. Harris, J. G. (2001). Linear Elastic Waves. Cambridge University Press, Cambridge, U.K. Harris, J. G. (2002). A biographical note on Jan D. Achenbach with a foreword to the special issue. Wave Motion, 36:307–309. Harris, J. G. (2002). Rayleigh wave propagation in curved waveguides. Wave Motion, 36:425–441. Harris, J. G. and Achenbach, J. D. (2002). Comment on “On the complex conjugate roots of the Rayleigh equation: the leaky surface wave” [J. Acoust. Soc. Am. 110, 2867 (2001)]. J. Acoust. Soc. Am., 112:1747– 1748. Harris, J. G. (2002). Propagation in curved waveguides. In I. D. Abrahams, P. A. Martin, and M. J. Simon, editors, IUTAM Symposium on Diﬀraction and Scattering in Fluid Mechanics and Elasticity, pages 321–328. Kluwer, Dordrecht. Harris, J. G. (2004). Elastic Waves at Microwave Frequencies: Mathematical Models using Asymptotic Methods. Universidad Nacional Aut´ onoma de M´exico. Harris, J. G. and Block, G. (2005). The coupling of elastic, surface-wave modes by a slow, interfacial inclusion. Proc. R. Soc. A, 461:3765– 3783. Block, G. I. and Harris, J. G. (2006). Conductivity dependence of seismoelectric wave phenomena in ﬂuid-saturated sediments. J. Geophys. Res. – Solid Earth, 111:B01304 (12 pages).

1 Linear elastic waves

Chapter 1 provides the background, both the model equations and some of the mathematical transformations, needed to understand linear elastic waves. Only the basic equations are summarized, without derivation. Both Fourier and Laplace transforms and their inverses are introduced and important sign conventions settled. The Poisson summation formula is also introduced and used to distinguish between a propagating wave and vibration of a bounded body. A general survey of books and collections of papers that bear on the contents of the book are discussed at the end of the chapter. A linear wave carries information at a particular velocity, the group velocity, which is characteristic of the propagation structure or environment. It is this transmission of information that gives linear waves their special importance. In order to introduce this aspect of wave propagation, we discuss propagation in one-dimensional periodic structures. Such structures are dispersive and therefore transmit information at a speed diﬀerent from the wavespeed of their individual components.

1.1 Model equations The equations of linear elasticity consist of: (1) the conservation of linear and angular momentum; and (2) a constitutive relation relating force and deformation. In the linear approximation the density ρ is constant. The conservation of mechanical energy follows from (1) and (2). The most important feature of the model is that the force exerted across a surface, oriented by the unit normal nj , by one part of a material on the other is given 1

2

Linear elastic waves

by the traction ti = τji nj , where τji is the stress tensor. The conservation of angular momentum makes the stress tensor symmetric: that is, τij = τji . The conservation of linear momentum, in diﬀerential form, is expressed as ∂k τki + ρfi = ρ∂t2 ui .

(1.1.1)

The term fi is a force per unit mass. In general we use Cartesian tensors such as τij , where the indices i, j = 1, 2, 3, or a bold-face notation τ . The symbol ∂i is used to represent the partial derivative with respect to the ith coordinate; it is the ith component of the gradient operator ∇. Similarly, sometimes dx f is used to represent df /dx. Repeated indices are summed over 1, 2, 3 unless otherwise indicated. For problems engaging only two coordinates, subscripts using Greek letters such as α, β = 1, 2 are used so that a vector component would be written as uβ and a partial derivative as ∂α . When these subscripts are repeated they are summed over 1, 2. At times we use symbols such as cL or cT when there is a need to distinguish between parameters that relate to compressional or shear disturbances, but when that distinction is not important we drop the subscript. Constants such as A are used over and over again and have no special meaning. Deformation is described using a strain tensor ij = (∂i uj + ∂j ui )/2,

(1.1.2)

where ui is ith component of particle displacement. The underlying dependence of the deformation is upon the symmetric part of the displacement gradient ∇u, which ensures that no rigid body rotations are included. For a homogeneous, isotropic, linearly elastic solid, stress and strain are related by τij = λkk δij + 2μij ,

(1.1.3)

where λ and μ are Lam´e’s elastic constants and δij is the Kronecker delta symbol. Substituting (1.1.2) in (1.1.3), followed by substituting the outcome into (1.1.1), gives one form of the equation of motion, namely, (λ + μ)∂i ∂k uk + μ∂j ∂j ui + ρfi = ρ∂t2 ui .

(1.1.4)

Written in vector notation the equation becomes (λ + μ)∇(∇ · u) + μ∇2 u + ρf = ρ∂t2 u .

(1.1.5)

1.1 Model equations

3

Using the identity ∇2 u = ∇(∇ · u) − ∇ ∧ ∇ ∧ u, the equation can also be written in the form (λ + 2μ)∇(∇ · u) − μ∇ ∧ ∇ ∧ u + ρf = ρ∂t2 u.

(1.1.6)

This last equation indicates that elastic waves have both dilatational ∇·u and rotational ∇ ∧ u deformations. If ∂R is the boundary of a region R occupied by a solid then commonly the tractions t or displacements u, or combinations of either, are prescribed on ∂R. When t is given over part of ∂R and u over another part, the boundary conditions are said to be mixed. One very common boundary condition is to ask that t = 0 everywhere on ∂R. This models the case where a solid surface is adjacent to a gas of such small density and compressibility that it is almost a vacuum. When R is inﬁnite in one or more dimensions, special conditions are imposed such that a disturbance decays to zero at inﬁnity or radiates outward toward inﬁnity from any sources contained within R. Another common situation is that in which ∂R12 is the boundary between two regions, 1 and 2, occupied by solids having diﬀerent properties. Contact between solid bodies is quite complicated, but in many cases it is usual to assume that the traction and displacement, t and u, are continuous. This models welded contact. One other simple continuity condition that commonly arises is that between a solid and an ideal ﬂuid. Because the viscosity is ignored, the tangential component of t is set to zero, while the normal component of traction and the normal component of displacement are made continuous. These are only models and are often inadequate. To brieﬂy indicate some of the possible complications, consider two solid bodies pressed together. A (linear) wave incident on such a boundary would experience continuity of traction and displacement when the solids press together, but would experience a traction-free boundary condition when they pull apart (Comninou and Dundurs, 1977). This produces a complex nonlinear interaction. The reader may consult Hudson (1980) for a succinct discussion of linear elasticity or Atkin and Fox (1980) for a somewhat more general view. 1.1.1 One-dimensional models We assume that the various waveﬁeld quantities depend only on the variables x1 and t. For longitudinal strain u1 is ﬁnite, while u2 and u3 are assumed to be zero, so that (1.1.2) combined with (1.1.3) becomes τ11 = (λ + 2μ)∂1 u1 ,

τ22 = τ33 = λ∂1 u1 ,

(1.1.7)

4

Linear elastic waves

and (1.1.1) becomes (λ + 2μ)∂12 u1 + ρf1 = ρ∂t2 u1 .

(1.1.8)

For longitudinal stress all the stress components except τ11 are assumed to be zero. Equation (1.1.3) becomes τ11 = E∂1 u1 ,

E=μ

3λ + 2μ , λ+μ

(1.1.9)

λ . 2(λ + μ)

(1.1.10)

and ∂2 u2 = ∂3 u3 = −ν∂1 u1 ,

ν=

Equation (1.1.1) now becomes E∂12 u1 + ρf1 = ρ∂t2 u1 .

(1.1.11)

Equations (1.1.8) and (1.1.11) are essentially the same, though they have somewhat diﬀerent physical meanings. The longitudinal stress model is useful for rods having a small cross-section and a traction-free surface. Stress components that vanish at the surface are assumed to be negligible in the interior.

1.1.2 Two-dimensional models We assume that the various waveﬁeld quantities are independent of x3 . As a consequence, (1.1.1) breaks into two separate equations, namely, ∂β τβ3 + ρf3 = ρ∂t2 u3

(1.1.12)

∂β τβα + ρfα = ρ∂t2 uα .

(1.1.13)

and

We use Greek subscripts α, β = 1, 2 to indicate that the independent spatial variables are x1 and x2 . The case for which the only non-zero displacement component is u3 (x1 , x2 , t), namely (1.1.12), is called antiplane shear motion, or SH motion for shear horizontal. In this case τ3β = μ∂β u3 ,

(1.1.14)

μ∂β ∂β u3 + ρf3 = ρ∂t2 u3 .

(1.1.15)

giving, from (1.1.12),

Note that this is a two-dimensional scalar equation, similar to (1.1.8) or (1.1.11). The case for which u3 = 0, while the other two displacement components are generally non-zero, (1.1.13), is called inplane motion. The

1.1 Model equations

5

initials P and SV are used to describe the two types of inplane motion, namely, compressional and shear vertical, respectively. For this case (1.1.3) becomes ταβ = λ∂γ uγ δαβ + μ(∂α uβ + ∂β uα ),

(1.1.16)

τ33 = λ∂γ uγ .

(1.1.17)

and The equation of motion remains (1.1.4): that is, (λ + μ)∂α ∂β uβ + μ∂β ∂β uα + ρfα = ρ∂t2 uα .

(1.1.18)

This last equation is a vector equation and contains two wave types, compressional and shear, whose character we explore shortly. It leads to problems of some complexity. These two-dimensional equations are the principal models used. The scalar model (1.1.14) allows us to solve complicated problems in detail without being overwhelmed by the size and length of the calculations needed, while the vector model (1.1.18) allows us enough structure to indicate the complexity found in elastic wave propagation. 1.1.3 Displacement potentials When (1.1.4)–(1.1.6) are used, a boundary condition, such as t = 0, is relatively easy to implement. However, in problems where there are few boundary conditions, it is often easier to cast the equations of motion into a simpler form and allow the boundary condition to become more complicated. One way to do this is to use Helmholtz’ theorem (Phillips, 1933, pp. 182–196) to express the particle displacement u as the sum of a scalar ϕ and a vector potential ψ: that is, u = ∇ϕ + ∇ ∧ ψ,

∇ · ψ = 0.

(1.1.19)

The second condition is needed because u has only three components (the particular condition selected is not the only possibility). Assume f = 0. Substituting these expressions into (1.1.6) gives 1 1 (λ + 2μ)∇ ∇2 ϕ − 2 ∂t2 ϕ + μ∇ ∧ ∇2 ψ − 2 ∂t2 ψ = 0. (1.1.20) cL cT The equation can be satisﬁed if c2L ∇2 ϕ = ∂t2 ϕ,

c2L = (λ + 2μ)/ρ,

(1.1.21)

and c2T ∇2 ψ = ∂t2 ψ,

c2T = μ/ρ.

(1.1.22)

6

Linear elastic waves

The terms cL and cT are the compressional or longitudinal wavespeed, and shear or transverse wavespeed, respectively. The scalar potential describes a wave of compressional motion, which in the plane-wave case is longitudinal, while the vector potential describes a wave of shear motion, which in the plane-wave case is transverse. Knowing ϕ and ψ, do we know u completely? Yes we do. Proofs of completeness, along with related references, are given in Achenbach (2003).

1.2 Continuity and boundary conditions Consider a plane ﬂuid–solid interface oriented by means of a unit normal ˆ pointing into the ﬂuid. The traction acting on the surface of vector n ˆ · τ . The continuity conditions at the interface are thus the solid is ts = n expressed as ˆ = −pf , ts · n

ˆ ∧ ts = 0, n

ˆ = uf · n ˆ. us · n

(1.2.1)

ˆ ∧ uf,s . Because the ﬂuid is ideal, no condition is placed on n The only other boundary condition needed in this work is one at inﬁnity. The waves must in general be outgoing, though when the focused beam is discussed an incoming wave is considered. The principle of limiting absorption (Harris, 2001, pp. 62, 63) is used, in most cases, to determine this. Either by Fourier transforming a signal or by considering a time-harmonic one, in the far ﬁeld, it will have the form ϕ=

A(φ, θ) i(kr−ωt) e , kr

where (r, φ, θ) are spherical coordinates, and k = ω/c is the wavenumber, with c being the wavespeed. The angular frequency is deﬁned such that ω = ω0 + i, ω0 > 0, ≥ 0. The wavenumber then becomes k = (ω0 /c) + i(/c). Therefore |ϕ| ∼ e−r/c et ,

(1.2.2)

as r → ∞ with t ﬁxed; that is, the wave vanishes provided the combination [i(kr − ωt)] appears in some guise. The parameter can be sent to zero at the end of the calculations.

1.3 Flux of energy The remaining conservation law of importance is the conservation of mechanical energy. Again assume f = 0. This law can be derived directly

1.3 Flux of energy

7

from (1.1.1)–(1.1.3) by taking the dot product of ∂t u with (1.1.1). This gives, initially, (∂j τji )∂t ui − ρ(∂t2 ui )∂t ui = 0.

(1.3.1)

Forming the product τkl kl , using (1.1.3) and making use of the decomposition ∂j ui = ji + ωji , where ωji = (∂j ui − ∂i uj )/2, allows us to write (1.3.1) as 1 ∂t {ρ(∂t ui )(∂t ui ) + τki ki } + ∂k (−τki ∂t ui ) = 0. (1.3.2) 2 The ﬁrst two terms on the left-hand side become the time rates of change of 1 1 U = τij ij . (1.3.3) K = ρ(∂t uk )(∂t uk ), 2 2 These are the kinetic and internal energy density, respectively. The remaining term is the divergence of the energy ﬂux, F , where Fj is given by Fj = −τji ∂t ui .

(1.3.4)

Equation (1.3.2) can then be written as ∂E + ∇ · F = 0, (1.3.5) ∂t where E = K + U and is the energy density. This is the diﬀerential statement of the conservation of mechanical energy. To better understand that the energy ﬂux or power density is given by (1.3.4), consider an arbitrary region R, with surface ∂R. Integrating (1.3.5) over R and using Gauss’ theorem gives d ˆ dS. E(x, t) dV = − F ·n (1.3.6) dt R ∂R Therefore, as the mechanical energy decreases within R, it radiates ˆ. outward across the surface ∂R at a rate F · n Because time-harmonic problems are being considered (see the next section), the time average of the ﬂux of energy per unit area, or the intensity, is of interest. The time average of a quantity a(x, t) is deﬁned as 1 t+T a(x, τ )dτ, (1.3.7) a(x, t) := T t where T = 2π/ω, and the time-dependence is e−iωt . Given two terms a(x, t) = Re[a(x)e−iωt ],

b(x, t) = Re[b(x)e−iωt ],

8

Linear elastic waves

the time average of their product is a(x, t) b(x, t) =

1 Re[a(x) b∗ (x)]. 2

(1.3.8)

This is derived by a direct substitution of the product into (1.3.7). The superscript ∗ indicates the complex conjugate. Equation (1.3.8) is especially useful when calculating the time-average ﬂux of energy per unit area, which is also called the intensity. Expressing the stress tensor and particle displacement as τ (x, t) = Re[τ (x)e−iωt ],

u(x, t) = Re[u(x)e−iωt ],

the time average of (1.3.4) is F =

1 Im[ω τ (x) · u∗ (x)]. 2

(1.3.9)

Two additional time-average quantities will be of interest: the timeaverage ﬂux of energy P across a surface ∂S oriented by the unit normal n ˆ , and its complex counterpart P c . These are given by c F · n ˆ dS (1.3.10) P = ReP = ∂S

and P c =

iω 2

(τ · u∗ ) · n ˆ dS.

(1.3.11)

∂S

Lastly, in all the cases treated in this book it can be shown (Auld, 1990a, pp. 201–207; Lighthill, 1965) that F = CE ,

(1.3.12)

where C is the group velocity and the energy density E is given following (1.3.5). Cautionary note. There are waves, such as mode L3 of an elastic plate (see §3.2 and §8.2), whose group velocity C and wave (or phase) velocity c are in diﬀerent directions. In such cases, the principle of limiting absorption, (1.4.2), must be applied with care, and it is more direct to ask that F , or equivalently C, be directed away from the source.

1.4 The Fourier and Laplace transforms

9

1.4 The Fourier and Laplace transforms All waves are transient in time. One useful representation of a transient waveform is its Fourier one. This representation imagines the transient signal decomposed into an inﬁnite number of time-harmonic or frequency components. One important reason for the usefulness of this representation is that the transmitter, receiver, and propagation structure usually respond diﬀerently to the diﬀerent frequency components. The linearity of the problem ensures that we can work out the net propagation outcomes for each frequency component and then combine the outcomes to recreate the received signal. The Fourier transform is deﬁned as ∞ eiωt u(x, t)dt. (1.4.1) u ¯(x, ω) = −∞

The variable ω is complex. Its domain is such as to make the above integral convergent. For t > 0 this domain is Im(ω) > 0. u ¯ is an analytic function within the domain of convergence, though it can be analytically continued beyond it.1 The inverse transform is deﬁned as ∞ 1 e−iωt u ¯(x, ω)dω . (1.4.2) u(x, t) = 2π −∞ Thus we have represented u as a sum of harmonic waves e−iωt u ¯(x, ω). Note that there is a speciﬁc sign convention for the exponential term that we shall adhere to throughout the book. A closely related transform is the Laplace one. It is usually used for initial-value problems so that we imagine that, for t < 0, u(x, t) is zero. This is not essential and its deﬁnition can be extended to include functions that extend through values of negative t. This transform is deﬁned as ∞

u ˜(x, p) =

e−pt u(x, t)dt.

(1.4.3)

0

As with ω, p is a complex variable and its domain is such as to make u ˜(x, p) an analytic function of p. The domain is initially deﬁned as Re(p) > 0, but the function can be analytically continued beyond this. Note that p = −iω so that Im(ω) > 0. The inverse transform is given by +i∞ 1 ept u ˜(x, p)dp, (1.4.4) u(x, t) = 2πi −i∞ 1

Analytic functions deﬁned by contour integrals, including the case in which the contour extends to inﬁnity, are discussed in Titchmarsh (1939, pp. 85–86) in a general way and in more detail by Noble (1988).

10

Linear elastic waves

where ≥ 0. The expressions for the inverse transforms (1.4.2) and (1.4.4) are misleading. In practice we deﬁne the inverse transforms on contours that are designed to capture the physical features of the solution. A large part of this book will deal with just how those contours are selected. But, for the present, we shall work with these deﬁnitions. Consider the case of longitudinal strain. Imagine that at t = −∞ a disturbance started with zero amplitude. Taking the Fourier transform of (1.1.8) gives d2 u ¯1 + k2 u ¯1 = 0, (1.4.5) dx21 where k = ω/cL and cL is the compressional wavespeed deﬁned in (1.1.21). The parameter k is called the wavenumber. Equation (1.4.5) has solutions of the form u ¯1 (x1 , ω) = A(ω)e±ikx1 .

(1.4.6)

If we had sought a time-harmonic solution of the form ¯1 (x1 , ω)e−iωt , u1 (x1 , t) = u

(1.4.7)

we should have gotten the same answer except that e−iωt would be present. In other words, taking the Fourier transform of an equation over time or seeking solutions that are time-harmonic are two slightly diﬀerent ways of doing the same operation. For (1.4.7), it is understood that the real part can always be taken to obtain a real disturbance. Much the same happens in using (1.4.2). In writing (1.4.2) we implicitly assumed that u(x, t) was real. That being the case, u ¯(x, ω) = u ¯∗ (x, −ω), where the superscript ∗ to the right of the symbol indicates the complex conjugate. From this it follows that ∞ 1 u(x, t) = Re e−iωt u ¯(x, ω)dω . (1.4.8) π 0 The advantage of this formulation of the inverse transform is that we may proceed with all our calculations using an implied e−iωt and assuming ω is positive. The importance of this will become apparent in subsequent chapters. Equation (1.4.8) can be regarded as a generalization of taking the real part of a time-harmonic wave (1.4.7). Equation (1.4.6), when the + sign is taken, is a time-harmonic, plane wave propagating in the positive x1 direction. We assume that the wavenumber k is positive, unless otherwise stated. The wave propagates in the positive x1 direction because the term (kx1 −ωt) remains constant, and hence u1 remains constant, only if x1 increases as t increases. The

1.4 The Fourier and Laplace transforms

11

speed with which the wave propagates is cL . The term ω is the angular frequency or 2πf , where f is the frequency. That is, at a ﬁxed position, 1/f is the length of a temporal oscillation. Similarly k, the wavenumber, is 2π/λ where λ, the wavelength, is the length of a spatial oscillation. − Note that if we combine two of these waves, labeled u+ 1 and u1 , each going in opposite directions, namely, i(kx1 −ωt) , u+ 1 = Ae

−i(kx1 +ωt) u− , 1 = Ae

(1.4.9)

we get u1 = 2Ae−iωt cos(kx1 ).

(1.4.10)

Taking the real part gives u1 = 2|A| cos(ωt + α) cos(kx1 ).

(1.4.11)

We have taken A as complex so that α is its argument. This disturbance does not propagate. At a ﬁxed x1 the disturbance simply oscillates in time and at a ﬁxed t it oscillates in x1 . The wave is said to stand or is called a standing wave. Continuing with the case of longitudinal strain, consider the following boundary, initial-value problem. Unlike the previous discussion in which the disturbance began, with zero amplitude, at −∞, here we shall introduce a disturbance that starts up at t = 0+ . Consider an elastic halfspace, occupying x1 ≥ 0, subjected to a non-zero traction at its surface. The problem is one-dimensional, and invariant in the other two, so that (1.1.8), the equation for longitudinal strain, is the equation of motion. At x1 = 0 we impose the boundary condition τ11 = −P0 e−ηt H(t), where H(t) is the Heaviside step function and P0 is a constant. As x1 → ∞ we impose the condition that any wave propagates outward in the positive x1 direction (why?). Moreover, we ask that, for t < 0, u1 (x1 , t) = 0 and ∂t u1 (x1 , t) = 0. Note that, using integration by parts, the Fourier transform, indicated by F, of the second time derivative is ¯1 (x1 , ω) + iωu1 (x1 , 0− ) − ∂t u1 (x1 , 0− ). (1.4.12) F ∂t2 u1 = −ω 2 u In deriving this expression we have integrated from t = 0− to ∞ so as to include any discontinuous behavior at t = 0. Taking the Fourier transform of (1.1.8) gives (1.4.5). Then the inverse transform of the stress component τ11 is given by P0 ∞ ei(kx1 −ωt) dω. (1.4.13) τ11 (x1 , t) = 2πi −∞ ω + iη

12

Linear elastic waves

In the course of making this step we need to choose between the solutions to the transformed equation, (1.4.5). Why is the solution leading to (1.4.13) selected? The integral is readily evaluated by capturing the pole. Note that, if the disturbance is to decay with time, η must be positive. We can also show that τ11 (x1 , t) = −P0 H(t − x1 /cL )e−η(t−x1 /cL ) .

(1.4.14)

The conditions for convergence of the integral, as its contour is closed at inﬁnity, give rise to the Heaviside function. Note how the sign conventions for the transform pair, by aﬀecting where the inverse transform converges, give a solution that is causal. The preceding example indicated the natural association between ikx1 and e−iωt and thereby suggests how we shall deﬁne the Fourier e transform over the spatial variable x. Suppose we have taken the temporal transform obtaining u ¯(x, ω). Then its Fourier transform over x is deﬁned as ∞ ∗ u ¯(k, ω) = e−ikx u ¯(x, ω)dx, (1.4.15) −∞

and its inverse transform is 1 u ¯(x, ω) = 2π

∞

eikx ∗u ¯(k, ω)dk .

(1.4.16)

−∞

Again note the sign conventions for the transform pair. Moreover, note that ∞ ∞ 1 ei(kx−ωt)∗u ¯(x, ω)dω dk . (1.4.17) u(x, t) = 4π 2 −∞ −∞ This shows that quite arbitrary disturbances can be decomposed into a sum of time-harmonic plane waves and thereby indicates that the study of such waves is very central to understanding linear waves.

1.5 A wave is not a vibration A continuous body vibrates when a system of standing waves is established within it. Vibration and wave propagation can be explicitly linked by means of the Poisson summation formula. This formula might better be termed a transform and is quite useful, especially for asymptotically approximating sums.

1.5 A wave is not a vibration

13

Theorem. Consider a function f (t) and its Fourier transform f¯(ω). Restrictions on f (t) are given below. The Poisson summation formula states that ∞ ∞ 1 ¯ f (mλ) = f (2πk/λ), (1.5.1) |λ| m=−∞ k =−∞

where λ is a parameter. This formula relates a series to one comprised of its transformed terms. If we want to approximate the left-hand side of (1.5.1) for λ small, then knowing the Fourier transform of each term enables us to use an approximation based on the fact that λ−1 is large. The left-hand side of (1.5.1) may converge only slowly for a small λ. Proof.2 This proof follows that of de Bruijn (1970). Consider the function ϕ(x) given by ∞

ϕ(x) =

f [(m + x)λ],

(1.5.2)

m =−∞

where the sum converges uniformly on x ∈ [0, 1]. The function ϕ(x) has period 1. We assume that f (t) is such that ϕ(x) has a Fourier series, ∞ ϕ = k=−∞ ck eik2πx . Its kth Fourier coeﬃcient, ck , equals

1

e−ik2πx ϕ(x)dx =

0

0

= =

∞

1

e−ik2πx f [(m + x)λ]dx

m =−∞

∞

(m+1)

e−ik2πx f (xλ)dx

m =−∞ m ∞ 1 −ik2π(x/λ)

|λ|

e

f (x)dx.

(1.5.3)

−∞

(m+t) −ikx Note that the integral m e f (xλ)dx → 0 as m → ± ∞, uniformly in x ∈ [0, 1], as the sum (1.5.2) converges uniformly. This completes the proof. These conditions are more restrictive than needed. Lighthill (1978a, pp. 67–71), among others, indicates that the Poisson summation formula holds for a far more general class of functions than assumed here. 2

A minimum amount of analysis is used, both here and elsewhere, and no attempt at rigorous proofs is made. The conclusions are usually valid under more general conditions than those cited.

14

Linear elastic waves

Consider the impulsive excitation of a rod of ﬁnite length 1. The governing equation is (1.1.11). Assume f1 = 0, set cb and ρ = 1 (c2b = E/ρ), and assume that for t ≤ 0, u1 (x1 , t) = ∂t u1 (x1 , t) = 0. The boundary conditions are τ11 (0, t) = −T δ(t),

τ11 (1, t) = 0.

(1.5.4)

By using a Fourier transform over t and solving the boundary-value problem in x1 , we get3 τ11 = iT H(t)

∞

e−inπt

n =−∞

sin [nπ(1 − x1 )] . cos (nπ)

(1.5.5)

Thus the rod is ﬁlled with standing waves. One usually considers a solution in this form as a vibration. This is a very useful way to express the answer, assuming the pulse has reverberated within the rod for a time long with respect to that needed for one echo from the far end. But the individual interactions with the ends have been obscured. To ﬁnd these interactions we apply the Poisson summation formula to (1.5.5). Break up the sin [nπ(1 − x1 )] term into exponential ones and apply (1.5.1) to each term. The crucial intermediate step is the following, where we have taken one of these terms. ∞

1 −1 e−imπ[t−(1−x1 )] = (π|t + x1 − 2|) cos mπ m =−∞ ∞ ∞

2n dω × exp − iω 1 − |t + x1 − 2| n =−∞ −∞ =2

∞

δ (|t + x1 − 2| − 2n) .

(1.5.6)

n =−∞

The outcome to our calculation is τ11 = T

∞ k=1

δ(t + x1 − 2k) − T

∞

δ(t − x1 − 2k).

(1.5.7)

k=0

This is a wave representation. It is very useful for times on the order of the echo time. For example, if t ∈ (1, 2), then τ11 = T δ[(t − 1) + (x1 − 1)]. We have thus isolated the pulse returning from its ﬁrst reﬂection at the end x1 = 1. However, the representation is not very useful for t large because it becomes tedious to work out exactly at what reﬂection 3

The reader should check that this is the solution.

1.6 Dispersive propagation

15

you are tracking the pulse. Moreover, the representation (1.5.7) would have been awkward to work with if, instead of delta-function pulses, we had had pulses of suﬃcient length that they overlapped one another. Nevertheless, the representation captures quite accurately the physical phenomenon of a pulse bouncing back and forth in a rod struck impulsively at one end. A vibration therefore is deﬁned and conﬁned by its environment. It is the outcome of waves reverberating in a bounded body. A period of time, sometimes a long one, is needed for the environment to settle into a steady oscillatory motion. In contrast, a wave is a disturbance that propagates freely outward, returning to its source perhaps only once, experiencing only a ﬁnite number of interactions. Understanding how an individual wave interacts with its environment and tracking it through each of its interactions constitute the principal problems of wave propagation. Moreover, while one works frequently with time-harmonic propagating waves, one usually assumes that at some stage a Fourier synthesis will be carried out, mapping the unending oscillatory motion into a disturbance conﬁned both temporally and spatially.

1.6 Dispersive propagation 1.6.1 An isolated interaction A basic interaction of a wave with its environment is scattering from a discontinuity. Continue to consider waves in a rod using the longitudinal stress approximation. Consider time-harmonic disturbances of the form ¯1 (x1 )e−iωt , u1 = u

τ11 = ρc2b ∂1 u1 .

(1.6.1)

We shall not indicate the possible dependence on ω of u ¯1 unless this is needed. The equation of motion (1.1.11) becomes d2 u ¯1 + k2 u ¯1 = 0, dx21

k = ω/cb .

(1.6.2)

Assume there is a region of inhomogeneity within x1 ∈ (−L, L). Incident on this inhomogeneity is the waveﬁeld A1 eikx1 , x1 < −L, i (1.6.3) u ¯1 (x1 ) = A2 e−ikx1 , x1 > L, where we have allowed waves to be incident from both directions. Upon striking the inhomogeneity the scattered waveﬁeld

16

Linear elastic waves B1 e−ikx1 , x1 < −L, s u ¯1 (x1 ) = B2 eikx1 , x1 > L,

(1.6.4)

is excited. Note that the scattered waves have been constructed so that they are outgoing from the scatterer. The linearity of the problem suggests that we can write the scattered amplitudes in terms of the incident ones as B1 S11 S12 A1 = , (1.6.5) B2 S21 S22 A2 or, more compactly, as B = SA.

(1.6.6)

The matrix S is called a scattering matrix and characterizes the inhomogeneity. We next consider a speciﬁc example. Imagine that the rod has a crosssectional area 1 and that the inhomogeneity is a point mass M , at x1 = 0. The left-hand ﬁgure in Fig. 1.1 indicates the geometry. The conditions across the discontinuity are u1 (0− , t) = u1 (0+ , t),

M ∂t2 u1 = −τ11 (0− , t) + τ11 (0+ , t);

(1.6.7)

that is, the rod does not break, but the acceleration of the mass causes the traction acting on the cross-section to be discontinuous. Setting tan θ = kM/2ρ, with θ ∈ (0, π/2), the matrix S is calculated to be sin θei(θ+π/2) cos θeiθ . (1.6.8) S= cos θeiθ sin θei(θ+π/2) It is also of interest to relate the wave amplitudes on the right to those on the left. This matrix T, called the transmission matrix, gives 0

1

2

0

L

2L

x1

x1

Fig. 1.1. One or more point masses M are embedded in a rod of cross-sectional area 1. The left-hand ﬁgure shows a single, embedded point mass. The righthand ﬁgure shows an endless periodic arrangement of embedded point masses, each separated by a distance L. The masses are labeled 0, 1, 2, . . . with the 0th mass at x1 = 0.

1.6 Dispersive propagation T

17 T

R = TL, where L = [A1 , B1 ] and R = [B2 , A2 ] . The matrix T is readily calculated from S and is given by 1 + i tan θ i tan θ T= . (1.6.9) −i tan θ 1 − i tan θ Note that the amplitudes A1 and B2 are those of right-propagating waves, while B1 and A2 are those of left-propagating ones.

1.6.2 Periodic structures One of the more interesting aspects of wave-bearing structures is that they often contain several length scales. Propagation in such a structure often can only take place if the angular frequency ω is linked to the wavenumber – a term we must deﬁne a bit more carefully here – in a nonlinear way. To consider this possibility we use the matrix T, (1.6.9), to analyze propagation in a periodic structure. We imagine an inﬁnite rod, of cross-sectional area 1, in which equal point masses, M , are periodically embedded. The right-hand ﬁgure of Fig. 1.1 indicates the geometry and how the masses are labeled. One such mass has the nominal position x1 = 0 and is labeled n = 0. Each mass is separated from its neighbors by a distance L. A cell of length L is thereby formed and is labeled n if the nth mass occupies its left end. There are thus two length scales, the wavelength λ = 2π/k and the cell length L. In this problem we do not concern ourselves with how the waves are excited, but only with the simpler question, what waves does this structure support? Consider the 0th cell, where x1 ∈ (0, L). Within that cell the solution to (1.6.2) is u ¯1 (x1 ) = R0 eikx1 + L0 e−ikx1 .

(1.6.10)

−ikL T

. This can be written as At x1 = L− the waveﬁeld is R0 eikL , L0 e T L1 = PR0 , with R0 = [R0 , L0 ] . The matrix P is called the propagator or the propagation matrix and is given by ikL 0 e . (1.6.11) P= 0 e−ikL At x1 = L+ , within the ﬁrst cell, the waveﬁeld amplitudes R1 = T [R1 , L1 ] are R1 = TPR0 .

(1.6.12)

18

Linear elastic waves T

This relation is readily generalized. If Rn = [Rn , Ln ] then Rn+1 = TPRn .

(1.6.13)

The central feature of the propagation structure is that it has translational symmetry. The central feature of the disturbance we seek is that its phase changes from cell to cell in a way that represents propagation. Speciﬁcally consider propagation to the right. To capture these two features, the waveﬁeld at a point within the (n + 1)th cell can diﬀer from that at a point within the nth cell, where the two points are separated by a distance L, by at most a multiplicative phase factor. This kinematic constraint is expressed by the relation Rn+1 = eiκL Rn ,

(1.6.14)

where κ is unknown: κ is positive, if real, and such as to cause decay, if complex. Combining (1.6.13) and (1.6.14) gives a 2 × 2 system of algebraic equations that has a non-trivial solution if and only if det TP − eiκL I = 0. (1.6.15) Recalling our previous deﬁnition of tan θ = kM/2ρ, this equation can be written compactly as cos κL = cos(kL + θ)/ cos θ.

(1.6.16)

This is a non-linear relationship between the angular frequency ω = cb k and the eﬀective wavenumber κ, though it may not be apparent, as yet, that κ (and not k) is the wavenumber of interest. Note that if κL ∈ [−π, π] is a solution, then κL±2nπ, for n = 1, 2, . . . , is also a solution. Accordingly we need only consider κL ∈ [−π, π]. The term κL is real provided | cos κL| ≤ 1. Therefore the boundaries between real and complex κL are given by cos(kL + θ)/ cos θ = ±1.

(1.6.17)

Taking +1, the solutions are sin(kL/2) = 0 or tan(kL/2) = − tan(θ). That is, kL = 2nπ or kL + 2θ = 2mπ, where n and m are integers. Taking −1, the solutions are cos(kL/2) = 0 or cot(kL/2) = tan(θ). That is, kL = (2n − 1)π or kL + 2θ = (2m − 1)π, where, again, n and m are integers. All these cases are covered by kL = nπ or kL + 2θ = mπ. For kL ∈ [(n − 1)π, (nπ − 2θ)], κL is real. These intervals are called passbands. Elsewhere κL is complex, causing the disturbance to decay as it propagates, and the intervals are called stopbands. At the lower boundary of a passband, L is an integer number of half wavelengths.

1.6 Dispersive propagation

19

If T is real and such as to allow only weak transmission, then all the reﬂected waves add constructively and little or nothing is transmitted. The actual situation is complicated by the complex T, but the constructive interference of the reﬂected waves is the basic physical mechanism giving rise to the stopbands. This phenomenon is referred to as Bragg scattering. Consider the interval x1 ∈ (nL, (n + 1)L). Then u ¯1 (x1 ) = Rn eik(x1 −nL) + Ln e−ik(x1 −nL) = eiκL Rn−1 eik(x1 −nL) + Ln−1 e−ik(x1 −nL) = eiκL u ¯1 (x1 − L).

(1.6.18)

This equation is a restatement of (1.6.14). Further, it indicates that u ¯1 (x1 ) must satisfy the functional equation u ¯1 (x1 + L) = eiκL u ¯1 (x1 ) if the kinematic constraint is to be enforced. Within each cell there are nominally two waves, as indicated in (1.6.18), which we call partial waves. However, we seek a solution for the wave globally propagating to the right along the structure, as distinguished from the right and left propagating partial waves in each cell. With this in mind, the solution to the functional equation is u ¯1 (x1 ) = eiκx1 ϕ(x1 ),

(1.6.19)

where ϕ(x1 + L) = ϕ(x1 ).4 That is, ϕ(x1 ) is a periodic function and can be represented by a Fourier series whose coeﬃcients are cn . Therefore, u ¯1 (x1 ) becomes ∞ u ¯1 (x1 ) = cn eix1 (κ−2πn/L) . (1.6.20) −∞

The time-harmonic waveﬁeld u ¯1 (x1 )e−iωt is thus a consequence of an inﬁnite number of space harmonics. Note that shifting κL by ±2mπ would not change this expression. More importantly, it is clear that it is κ, through the term ei(κx1 −ωt) , that is the wavenumber. Equation (1.6.16) indicates that ω is a function of κ, or κ a function of ω. A relation such as this is called a dispersion relation. Writing κ as ω/c(ω), we see that u ¯1 (x1 , ω) propagates at a diﬀerent speed for each ω. If we excited the structure with a pulse, then the pulse would be comprised of an inﬁnite number of such components, as indicated by (1.4.8). Each component would then propagate at its own speed and the pulse would become dispersed. A pulse is information, 4

This is a partial statement of Floquet’s theorem (Friedman, 1956, pp. 65–67).

20

Linear elastic waves

whereas a sinusoid is not. Hence what we have inferred is that dispersion can cause the distortion of or loss of information from a signal. We shall explore this topic further in Chapter 6. There are many fascinating aspects to propagation in periodic structures. The discussion here has followed parts of Levine (1978, pp. 273–308, 339–345 and elsewhere) and a reader seeking to learn more may wish to read this work further.

1.7 General references Speciﬁc references will be given as the book progresses; there are, however, several general books and reviews that are useful, though not always referred to elsewhere. These are listed here. Propagation of elastic and acoustic waves. There are now several books on elastic waves, including that by Harris (2001). Among those concerned with elastic waves are Achenbach (2003), Auld (1990a; 1990b), Eringen and S ¸ uhubi (1975), Graﬀ (1991), Hudson (1980), Miklowitz (1978), Poruchikov (1993), and Rose (1999). In particular, Achenbach (2003), Miklowitz (1978), Poruchikov (1993) and Rose (1999) discuss many of the advanced techniques used to describe elastic waves, including the scalar Wiener–Hopf method. The two volumes by Auld (1990a; 1990b) are quite original in presenting the subject: elastic waves are approached as an extension of the study of electromagnetic waves. Eringen and S ¸ uhubi (1975), Graﬀ (1991), and Rose (1999) are workmanlike, but do not give imaginative presentations of the mathematics of wave propagation, and are more suited to an applied readership. Lastly, Aki and Richards (2002) is a wide-ranging and very useful textbook, at an advanced level, that describes many aspects of elastic-wave propagation as it applies to seismology. Both Morse and Ingard (1968) and Pierce (1981) describe the equations of linear acoustics and solve many of the standard problems. Pierce (1981) is the more modern in its outlook. Radiation and edge diﬀraction. Bouwkamp (1954) gives a comprehensive overview of formulating and solving diﬀraction problems, with edges, to about 1950, and forms the approximate point in time from which the writer gives references to edge diﬀraction. Geometrical diﬀraction theory (Keller, 1958) has had a deﬁning impact on the study of diﬀraction problems, as is evidenced by the collection of papers in Hansen (1981). Apertures produce well-collimated beams and these beams, which form at a Fresnel length from the aperture, have their own propagation

1.7 General references

21

properties; Felsen and Deschamps (1974) is a collection of papers that describe work on the relation between ray and beam descriptions, up to about 1970. Diﬀraction through an aperture can be described by directing attention to the waveﬁeld ﬁlling the aperture, or by directing attention at scattering from the edges. This latter viewpoint is the one taken in this book; one review of edge diﬀraction, based on physical optics, is that by Uﬁmtsev (1989, pp. 463–474). The study of diﬀraction has a long history, primarily in optics; one helpful place to begin to study this work is the collection of papers in Oughstun (1991). Diﬀraction from edges, and by extension wedges, still remains an important mathematical challenge; Osipov and Norris (1999) review the important work of Malyuzhinets, which echoes Sommerfeld’s original work on edge diﬀraction. Sommerfeld’s original work has recently been translated and combined with various historical and mathematical notes in Sommerfeld (2004); briefer descriptions of his work are given in Baker and Copson (1987, pp. 124–152) and Sommerfeld (1967, pp. 247–272). There are now several books that deal with various aspects of radiation and edge diﬀraction: Achenbach et al. (1982), Babi˘c and Buldyrev (1991), Borovikov and Kinber (1994), James (1980), and Jull (1981). Achenbach et al. (1982) and this book both discuss the edge diﬀraction of an incident compressional wave in three dimensions. Babi˘c and Buldyrev (1991) and Borovikov and Kinber (1994) are comprehensive monographs describing many aspects of the asymptotic solution to linear wave problems: these are not books for the faint-hearted. James (1980) is directed at electromagnetic propagation, as is Jull (1981). Elastic waveguides and surface waves. All the books on elastic-wave propagation discuss guided waves and surface waves to some extent. Miklowitz (1978) in particular discusses elastic waveguides with an excitation at the end of a plate at length; however, his approach and the one taken here are quite diﬀerent. Biryukov et al. (1995) and this book both discuss elastic surface waves on curved surfaces. Malischewsky (1987) deals with guided waves, in somewhat the same spirit as does this book, though he is concerned primarily with anti-plane shear problems. Mathematical methods. Many of the standard techniques of applied mathematics are used in this book; however, some emphasis is given to asymptotic approximations to the Fourier-like integrals that arise in radiation and diﬀraction problems. Borovikov (1994) has as its sole goal to describe such asymptotic approximations. He discusses all the cases of

22

Linear elastic waves

integral approximations that are described in this book. Also of use are chapters 8 and 9 of Bleistein and Handelsman (1975) and most of Copson (1971). The angular spectrum representation of a waveﬁeld plays a significant role in the description of scanning acoustic microscopy. Clemmow (1966) introduces this way of representing a waveﬁeld using examples from electromagnetic propagation, and this technique is given many applications to physical optics by Nieto-Vesperinas (1991). A working knowledge of the Wiener–Hopf technique can be learned from Noble (1988, pp. 11–27) or Weinstein (1969), though the former may be easier to read. Acoustic microscopy. Briggs (1992) is a comprehensive study of many aspects of acoustic microscopy, and its list of references is very complete (up to its publication date). However, the model of the scanning acoustic microscope described in this book diﬀers substantially from that described in Chapter 6, though the imaging equation that is derived, the one describing how imaging works, is identical.

2 Canonical acoustic-wave problems

Four problems form a framework for understanding the excitation and propagation of elastic waves at high frequencies. The purpose of this chapter is to describe two acoustic problems; the next chapter will describe two elastic-wave problems.1 The two acoustic problems are described as follows: (1) The radiation from an oscillating piston, in contact with an ideal ﬂuid, is calculated in three distinct ways, each indicating a diﬀerent interpretation of the source waveﬁeld. The radiation impedance is also calculated to indicate how the source and the radiated waveﬁeld interact. (2) The cylindrical wave diﬀracted from the edge of a rigid barrier when it is struck by an acoustic plane wave is calculated using the Wiener– Hopf method. A uniform asymptotic description of the diﬀracted waveﬁeld is calculated. In each case applied mathematical techniques are introduced, and expanded upon in subsequent chapters.

2.1 Radiation from a piston in an inﬁnite baﬄe The centerpiece of an introductory course on physical acoustics is often a discussion of the piston source. While it is not a very realistic model, its solution informs how one subsequently treats radiation from loudspeakers and ultrasonic transducers. Moreover, the study of this problem allows one to make a number of far-reaching statements about modeling 1

The adjective ‘acoustic’ is used when propagation of a compressional wave in an ideal ﬂuid is of interest; the adjective ‘elastic-wave’ is used when shear-wave propagation must be taken into account.

23

24

Canonical acoustic-wave problems

^e ρ

φ

ρ

x1

r′ r x = (x1, x3, x3)

x2 x3 ^e3

Fig. 2.1. A piston of radius a undergoes small-amplitude, time-harmonic vibrations in a rigid baﬄe. The region R, deﬁned by x3 ≥ 0, is occupied by an ideal ﬂuid. The baﬄe and piston together occupy the boundary region ∂R, deﬁned by x3 = 0, and the piston face occupies the region ∂Sa , deﬁned by ρ ∈ [0, a] and φ ∈ [0, 2π), so that ∂Sa ⊂ ∂R.

sources that extend over several wavelengths. For these reasons radiation from an oscillating piston is taken as a canonical problem. Figure 2.1 shows the geometric arrangement: a piston embedded in a rigid baﬄe radiates sound into an ideal ﬂuid, which occupies the region R. The piston itself need not be rigid, though for some calculations this assumption is made. The problem of radiation from a piston is essentially equivalent to diﬀraction through a circular aperture when the waveﬁeld in the aperture is approximated as being composed of only the incident waveﬁeld. In an exact diﬀraction problem the aperture also contains the edge-diﬀracted waveﬁeld, in addition to the incident one, so that the waveﬁeld in the aperture is not known. However, in the case of radiation by a piston, the motion of the piston is known; it can be considered as an incident waveﬁeld ﬁlling an aperture. Accordingly, much of the very extensive literature on approximate diﬀraction through an aperture is pertinent to the description of radiation from the piston, and names the reader may associate with optics are used here as well.2 2

For example, the approximation to aperture diﬀraction just referred to is a variation of the Kirchhoﬀ boundary conditions. In associating names with equations, approximations, and so on, the writer has tried to be careful, but not necessarily historically accurate. The introduction and notes to Sommerfeld (2004) give some of the history of diﬀraction.

2.1 Radiation from a piston in an inﬁnite baﬄe

25

There are at least three distinct ways to represent mathematically the radiation from a piston. The most common way is to use Huygens’ principle: referring to Fig. 2.1, ∂Sa , the piston face, is considered an initial wavefront, and each point in ∂Sa acts as a source of secondary spherical waves, each of radius r = cΔt. The envelope of these secondary spherical surfaces gives the wavefront of the radiated waveﬁeld after it has propagated at wavespeed c through time interval Δt. This idea was used by Fresnel to explain diﬀraction from an aperture, and by Rayleigh and Helmholtz to describe radiation from a piston. The approach is described in most books on acoustics, among them Morse and Ingard (1968), Pierce (1981), and Skudrzyk (1971, pp. 489–499, 532–537, 603–605). This last book has a useful though discursive discussion of many aspects of the piston radiator, as well as of diﬀraction through an aperture in a screen. Greenspan (1979) is also a useful review article on aspects of radiation from a piston source. A second way to describe the radiated waveﬁeld is to imagine a cylindrical region with base equal to ∂Sa and extending normally into R. Within the cylinder a plane wave radiated by the face of the piston propagates, while outside the cylinder it is zero. Radiating from the edge ρ = a, and compensating for this discontinuity, is a wave with toroidal wavefront. In optical diﬀraction this latter wave is referred to as the boundary diﬀraction wave (Born and Wolf, 1999, pp. 499–503; Skudrzyk, 1971, pp. 519–531; Sommerfeld, 1967, pp. 311–318), though here it is named an edge-diﬀracted wave. Its existence was ﬁrst suggested by Young to explain diﬀraction through an aperture, somewhat before Fresnel proposed using Huygens’ principle, but a mathematical representation was achieved only much later by Rubinowicz (see the citations in Gniadek and Petykiewicz, 1971). This picture could be considered as a beginning for the geometrical theory of diﬀraction (Keller, 1958). A third way is to represent the radiated waveﬁeld as a Hankel transform in planes parallel to ∂R, the plane containing the piston and baﬄe, a calculation ﬁrst undertaken by King (1934) and subsequently extended by Bouwkamp (1946) (see also Oughstun, 1991; pp. 41–67). The Hankel transform can then be re-expressed as an integration over plane waves; this gives a plane-wave or angular spectrum representation of the radiated waveﬁeld (Clemmow, 1966). This particular representation is very useful because knowing how a single plane wave interacts with a boundary or edge allows one to calculate how the beam radiated by a piston interacts with them.

26

Canonical acoustic-wave problems

The problems are formulated throughout this section using ϕ, the particle displacement potential. The end results can readily be expressed in terms of the acoustic pressure p: using p = −ρf ∂t2 ϕ implies the relation p = (ρf c)(ω k)ϕ for time-harmonic waves. Moreover, the amplitude of ϕ is normalized by the constant A. If one imagines that V0 is the magnitude of the particle velocity, then A = V0 /(ω k). ρf is the density of the ﬂuid, c the wavespeed in the ﬂuid, ω the angular frequency, and the wavenumber k = ω/c.

2.1.1 Huygens’ principle and the Fresnel length Figure 2.1 indicates the geometry of the problem being studied. The region R = {x | (x1 , x2 ) ∈ (−∞, ∞), 0 ≤ x3 < ∞}, where x = (x1 , x2 , x3 ) locates the observation point, and is occupied by an ideal ﬂuid. The baﬄe and piston together occupy the region ∂R = {ρ, φ, x3 | ρ ∈ [0, ∞), φ ∈ [0, 2π), x3 = 0}, while the piston face occupies the region ∂Sa = {ρ, φ, x3 | ρ ∈ [0, a), φ ∈ [0, 2π), x3 = 0}, so that ∂Sa ⊂ ∂R. In the time-harmonic approximation, ϕ satisﬁes the reduced wave equation in R, namely, ∇2 ϕ + k 2 ϕ = 0, x ∈ R.

(2.1.1)

This follows from the wave equation ∇2 ϕ = c−2 ∂t2 ϕ. On the boundary ∂R, ϕ satisﬁes ∂3 ϕ = ikA v(ρ/a) H(1 − ρ/a), x ∈ ∂R

(2.1.2)

(here H(ξ) is the Heaviside step function); that is, the function v(ρ/a) is deﬁned in ∂Sa . Equation (2.1.2) is equivalent to the third continuity condition, (1.2.1). The function v(ξ) is normalized so that 1 v(ξ)v ∗ (ξ)ξdξ = 1/2. 0

Lastly, as r = |x| → ∞, then ϕ must satisfy a radiation condition such as that imposed by the principle of limiting absorption, (1.2.2). The surface ∂R also includes a hemisphere of radius r, centered at the origin; however, sending r to inﬁnity removes any contribution from this surface so that it is not discussed further. It is useful to consider an auxiliary problem before solving (2.1.1) and (2.1.2), namely that of the Green’s function g(x, x ) for this problem (Harris, 2001, pp. 56–76). It satisﬁes ∇2 g + k 2 g = −δ(x − x ), x, x ∈ R,

(2.1.3)

2.1 Radiation from a piston in an inﬁnite baﬄe

27

subject to the boundary conditions that ∂3 g = 0, x ∈ ∂R

(2.1.4)

and that a radiation condition as |x| → ∞ be satisﬁed. The solution to this problem is eik|x−x | 1 eik|x−x | + , (2.1.5) g(x, x ) = 4π |x − x | |x − x | where x = (x1 , x2 , x3 ) and x = (x1 , x2 , −x3 ), and x3 > 0. The bilinear form ∇ · (g∇ϕ − ϕ∇g) = ϕ(x)δ(x − x ) is constructed using (2.1.1) and (2.1.3); it is then integrated throughout the half-space R and transformed to a surface integral. Then, using (2.1.2) and (2.1.4), and noting that the Green’s function is symmetric in its arguments, ϕ is represented as ϕ(x) = − g(x, x ) ∂3 ϕ(x ) dS(x ), (2.1.6) ∂Sa

where x ∈ ∂Sa . Substituting (2.1.5) in (2.1.6), with x3 = 0, gives k2 A eikr ϕ(x) = −i v(ρ/a) dS(x ). (2.1.7) 2π ∂Sa kr ˆρ |, and e ˆρ is a unit vector in the ρ direction (Fig. 2.1). Here r = |x − ρ e The element of surface is dS(x ) = ρ dρ dφ. Note that the particular cases solved here are axisymmetric so that the dφ integration can be done immediately. Equation (2.1.7) is a mathematical expression of Huygens’ principle: each point on the initial wavefront ∂Sa is a source of spherical waves, of radius r , weighted at each point by A v(ρ/a). The integral governs the subsequent evolution of the radiated waveﬁeld; in particular, the envelope of the various secondary spherical wavefronts deﬁnes the overall wavefront. As has been indicated by the previous citations, this integral has been extensively studied in numerous ways. There is little to add to the techniques used to evaluate it. Nevertheless, there is one important observation to be made: because the problem is axisymmetric, the observation point can be placed in the x2 = 0 plane, without loss of generality. The distance r can then be approximated as

28

Canonical acoustic-wave problems

r ≈ r 1 −

ρ2 ρ sin θ cos φ + 2 (1 − sin2 θ cos2 φ) , r 2r

where r = |x|. The quadratic term may be dropped, provided ka2 /2r π (that is, less than half a cycle), where max(ρ) = a. This leads to the deﬁnition of the Fresnel length, Fl , as Fl := ka2 /(2π).

(2.1.8)

When r > Fl the waveﬁeld is in the far ﬁeld. In this region, eikr 1 J0 (ka ξ sin θ)v(ξ)ξdξ + O(Fl /r), (2.1.9) ϕ(r, θ) = −i[(ka)2 A] kr 0 where J0 is the Bessel function of order zero. The Hankel transform is deﬁned by ∞ ∗ ϕ(kρ , x3 ) = ρϕ(ρ, x3 )J0 (kρ ρ)dρ, 0 ∞ ϕ(ρ, x3 ) = kρ ∗ϕ(kρ , x3 ))J0 (kρ ρ)dkρ .

(2.1.10) (2.1.11)

0

The radiated waveﬁeld of (2.1.9) appears as a spherical wave radiated by an anisotropic point source; the anisotropy is described by the Hankel transform of the distribution of particle velocity on the piston face. The Hankel transform is useful in deriving solutions to the wave equation, as we will see in §2.1.3. In §2.1.2 the waveﬁeld in the neighborhood of the Fresnel length will be investigated, and there it will be found that it is well collimated and can be described as a beam. When microwave frequency sound is used to probe an environment, it is usual that one wants a beam rather than a spherical wave because when a beam strikes something one can more readily sort out where the echo is coming from. Thus at microwave frequencies the waveﬁeld at the Fresnel length is more important than that in the far ﬁeld, (2.1.9).

2.1.2 The edge-diﬀraction integral To explore the waveﬁeld at the Fresnel length, the edge-diﬀraction interpretation of the radiation from the piston is a more useful starting point. Consider the cylindrical region S = {ρ, φ, x3 | ρ ∈ [0, a], φ ∈ [0, 2π), x3 ∈ [0, ∞)} shown in Fig. 2.2. Its surface ∂S = ∂Sa ∪ ∂Ss ; the surface ∂Ss = {ρ, φ, x3 | ρ = a, φ ∈ [0, 2π), x3 ∈ [0, ∞)}. Within the cylinder a plane wave, the geometrical waveﬁeld ϕg propagates outward,

2.1 Radiation from a piston in an inﬁnite baﬄe

29

x1 ^eρ

α

φ

a

r′ e^3

r x2

ϑ

β

x

x3

Fig. 2.2. A cylindrical region S projects normally from the piston face ∂Sa into R. The surface ∂S = ∂Sa ∪ ∂Ss . Also shown are the angles α, β, and ˆρ |. Note that r does not equal that used in θ, and the length r = |x − a e Fig. 2.1, unless ρ = a.

while outside the cylinder it vanishes. Again there is a surface at the far right in Fig. 2.2 that caps ∂S; however, sending x3 to inﬁnity removes any contribution from this surface, so that it is not discussed further. The geometrical waveﬁeld satisﬁes d2 ϕ g + k 2 ϕg = 0, x ∈ S, dx23

(2.1.12)

dϕg = ikA, x ∈ ∂Sa . dx3

(2.1.13)

subject to

Note that the boundary condition on ∂Sa is equivalent to setting v(ρ/a) = H(1 − ρ/a) in (2.1.2); however, ϕg ≡ 0 outside of S. As x3 → ∞, ϕg represents an outgoing wave, satisfying the principle of limiting absorption, (1.2.2). The solution to this problem is ϕg (x) = A H(1 − ρ/a)eikx3 .

(2.1.14)

Note that ϕg is allowed to jump to zero at ρ = a. ˆ3 be a unit vector in the x3 direction, represent the gradient Letting e operator as ˆ3 ∂3 , ∇ = ∇t + e

30

Canonical acoustic-wave problems

where ∇t indicates the gradient in a plane parallel to ∂R. Using (2.1.3) and (2.1.4) in combination with (2.1.12) and (2.1.13), the bilinear form ∇ · [ g (ˆ e3 ∂3 ϕg ) − ϕg ∇g ] + ∇t g · ∇t ϕg = ϕg (x)δ(x − x ) is constructed, where x, x ∈ R. The term ∇t ϕg contains a delta function because of the discontinuity at ρ = a. Accordingly the second term in this expression can be written as ˆρ ∇t g · ∇t ϕg = − ∇t g · e

A δ(1 − ρ/a) eikx3 . a

Integrating the bilinear form throughout R, taking account of the delta function, and noting that ϕg and g, and their derivatives, go to zero as x3 → ∞ gives g(x, x ) ∂3 ϕg (x) dS(x) ϕg (x ) = − ∂Sa (2.1.15) ˆρ ] ϕg (x) dS(x). [∇1 g(x, x ) · e − ∂Ss

The subscript 1 indicates that the gradient is taken with respect to ˆρ . Because the ﬁrst argument in g(x, x ). The unit normal to ∂Ss is e g c ϕ (x ) ≡ 0 for x ∈ S , the complement of S, there is a corresponding discontinuity located in the surface integral when x lies in ∂Ss . Interchanging the arguments x, x , and noting the symmetry of g(x, x ), allows the previous expression to be written as g(x, x ) ∂3 ϕg (x ) dS(x ) ϕg (x) = − ∂S a (2.1.16) ˆρ ] ϕg (x ) dS(x ). [∇2 g(x, x ) · e − ∂Ss

The subscript 2 indicates that the gradient is taken with respect to the second argument in g(x, x ). From (2.1.6) one notes that the integral over ∂Sa equals ϕ, because ∂3 ϕg = ∂3 ϕ there: compare (2.1.2) with (2.1.13) and note that v(ρ/a) = 1 in ∂Sa (for the problem of this section). Therefore (2.1.16) can be written as ˆρ ] ϕg (x ) dS(x ), ϕ(x) = ϕg (x) + [∇2 g(x, x ) · e (2.1.17) ∂Ss

where x ∈ R.

2.1 Radiation from a piston in an inﬁnite baﬄe

31

2.1.3 The Maggi transformation The next step is to reduce the integral over the surface ∂Ss to a line integral over the boundary of ∂Sa . This transformation is done with varying degrees of sophistication in the previously cited references, and in Baker and Copson (1987, pp. 74–84) and Gniadek and Petykiewicz (1971), though none of these references uses the particular representation given by (2.1.17). The name ‘Maggi transformation’ is used here by analogy with its use in Baker and Copson (1987, pp. 74–84). The points x = (x1 , x2 , x3 ) and x = (x1 , x2 , −x3 ), where x3 > 0, in (2.1.18) now lie in ∂Ss and in its reﬂection in the plane ∂R, respectively. The distances from these points to the observation point x are given by 1/2 . s∓ = (x1 − x1 )2 + (x2 − x2 )2 + (x3 ∓ x3 )2 Referring to (2.1.5), the gradient in (2.1.17) can be expressed as d eiks∓ eiks∓ ˆρ = ·e cos χ∓ , ∇2 (2.1.18) s∓ ds∓ s∓ where ˆρ . cos χ∓ = (∇2 s∓ ) · e The subscript 2 indicates that the derivative is taken with respect to x (the −) or x (the +). Note that ∇2 s∓ = −∇1 s∓ , where the subscript 1 indicates that the derivative is taken with respect to x. ˆρ |, from a point on the boundary of ∂Sa to The distance r = |x − a e the observation point x (see Fig. 2.2), is 1/2 . (2.1.19) r (x, φ) = (x1 − a cos φ)2 + (x2 − a sin φ)2 + (x3 )2 Next, note that the length of a perpendicular dropped from x to a point on ∂Ss is such that s∓ cos χ∓ = r cos α,

ˆρ . cos α = (−∇1 r ) · e

The angle α is shown in Fig. 2.2. We identify a second angle β, also shown in Fig. 2.2, using ˆ3 . cos β = ∇1 r · e By direct computation one ﬁnds that 1 d eik(s∓ +x3 ) eik(s∓ +x3 ) . ik − = (s∓ )2 s∓ dx3 s∓ (s∓ + x3 ∓ r cos β)

(2.1.20)

32

Canonical acoustic-wave problems

Equations (2.1.18) and (2.1.20) constitute the Maggi transformation as used here. Using (2.1.18) and (2.1.20) in (2.1.17), it follows that ˆρ A 2π a ∇1 r · e eikr dφ, (2.1.21) ϕ(x) = ϕg (x) + 2 ˆ3 ) 2π 0 r 1 − (∇1 r · e where x ∈ R. The terms cos α and cos β have been replaced by their deﬁnitions. The distance r is given by (2.1.19). Note that, for an observation point x ∈ ∂Ss , the integrand becomes singular. The integral is then deﬁned so as to maintain the overall continuity of ϕ. Lastly, the geometrical term ϕg is given by (2.1.14). Though it has been complicated to arrive at, (2.1.21) is a remarkable result because it expresses Young’s and Rubinowicz’ ideas very precisely: the radiated waveﬁeld consists of a geometrical term ϕg that vanishes outside S, and an edge-diﬀracted wave, the integral over the boundary of ∂Sa , that eliminates the discontinuity in ϕg across ∂Ss . It is remarkable that two such seemingly diﬀerent representations as (2.1.7) and (2.1.21) are equal when v(ρ/a) = H(1 − ρ/a). Had something more complicated than v(ρ/a) = H(1 − ρ/a) been prescribed, then the result (2.1.21) would not be quite so simple; the reader is invited to consider this case. Alternatively, one could also have begun directly with (2.1.7) and approximated the double integral asymptotically; see Bleistein and Handelsman (1975, pp. 321–366), Borovikov (1994, pp. 148–155), Chako (1965), or Stamnes (1986, pp. 136–162). To understand (2.1.21) more fully, the integral is approximated using the method of stationary phase (Appendix A). The problem is axisymmetric, so that the observation point can again be placed in the plane x2 = 0 without losing any generality. The distance r , given by (2.1.19), can now be expressed as r (r, θ; φ) = (r2 − 2ar sin θ cos φ + a2 )1/2 ,

(2.1.22)

where (r, θ) is the observation point and (a, φ) is a point on the boundary of ∂Sa ; both sets of coordinates are shown in Fig. 2.2. Diﬀerentiating r with respect to φ indicates that the stationary points are φ = 0, π; the ﬁrst indicates the point on the piston edge closest to the observation point, and the second, the point furthest away. Let = r (r, θ; 0), r−

r+ = r (r, θ; π),

and identify . cos α∓ = (a ∓ r sin θ)/r∓

2.1 Radiation from a piston in an inﬁnite baﬄe

33

x1

α– r

r−′

α+ x2

r+′

x = (x1, 0, x3) x3

e^3

Fig. 2.3. The lengths and angles arising from the stationary-phase approximation to the line integral along the piston edge. The lengths r∓ and the angles α∓ are shown in Fig. 2.3. From the edge of the piston a diﬀracted wave with a toroidal wavefront is emitted; the shape of its wavefront will be evident from the stationary-phase approximation. The principal radii of curvature (Kreyszig, 1975, pp. 78–99), ρ1(∓) , ρ2(∓) , of this torus, along the rays emitted from φ = 0, π, are

ρ1(∓) = −

a − r∓ cos α∓ , cos α∓

ρ2(∓) = r∓ .

(2.1.23)

The radii ρ1(∓) carry a sign that is determined by cos α∓ , so that it is necessary to deﬁne the square root: (−ρ1(∓) )1/2 = eiπ/2 (ρ1(∓) )1/2 . It is also helpful to deﬁne the diﬀraction coeﬃcients3 D(α∓ ) as D(α∓ ) :=

ie−iπ/4 1 . (2π)1/2 cos α∓

(2.1.24)

Note that the diﬀraction coeﬃcient has no length scale in its structure, so that it does not change even if a → ∞. With these terms identiﬁed, the stationary-phase approximation to (2.1.21) is given by ϕ(x) ∼ ϕg (x) − A

∓

a/ cos α∓ (kρ2(∓) )(−ρ1(∓) )

1/2

D(α∓ ) eikr∓ , kr∓ → ∞,

3

(2.1.25)

Other names such as ‘excitation coeﬃcient’ are sometimes given to these terms. In this book only the label ‘diﬀraction coeﬃcient’ will be used.

34

Canonical acoustic-wave problems

where ϕg is given by (2.1.14) and the radii of curvature ρ1(∓) and ρ2(∓) by (2.1.23). Looking at Fig. 2.3, it is seen that the (r+ , α+ ) ray, for the observation point shown there, crosses the center axis, making ρ+ change sign; thus, as (2.1.23) indicates, it undergoes a phase shift of π/2. Equation (2.1.25) is the representation of the radiated waveﬁeld that would be obtained using the geometrical theory of diﬀraction: it has a purely geometrical ray, represented by ϕg , followed by edge-diﬀracted ones. There are two such rays, one from the point nearest the observation point and one from the furthest point; this information is contained in the propagation terms exp(ikr∓ ); thus both rays satisfy Fermat’s principle. The amplitudes are determined, in part, by the radii of curvature (2.1.23): the square-root term is a consequence of energy ﬂux being conserved and it thus describes the geometrical spreading of the edge-diﬀracted wave. Lastly the amplitude is determined by a diﬀraction coeﬃcient (2.1.24) that contains no length; it could be calculated by solving a radiation problem formed by the boundary condition ∂3 ϕ = ikA H(1 − x1 /a), x3 = 0.

(2.1.26)

The geometrical ray is expected from geometrical optics, but geometrical optics does not include diﬀracted rays. The notions that diﬀracted rays originating on an edge satisfy Fermat’s principle, and that the diﬀracted amplitudes can be determined by a geometrical spreading factor and a diﬀraction coeﬃcient, are the essence of the geometrical theory of diﬀraction, at least as applied to problems such as the piston radiator. This brief description of the geometrical theory of diﬀraction is complete enough for the purposes of this section, but is by no means thorough. There are now several books on the subject, namely Achenbach et al. (1982), Babi˘c and Buldyrev (1991), Borovikov and Kinber (1994), and James (1980), that provide comprehensive treatments. Moreover, while there are numerous references to the geometrical theory of diﬀraction from apertures, Keller (1957) is arguably the best reference to begin studying. This reference can be found in the collections by Hansen (1981) and Oughstun (1991). However, (2.1.25) is not an accurate representation in many regions of R: the representation fails along the central axis x3 and along the shadow boundaries at α∓ = π/2. The ﬁrst diﬃculty arises because the rays emitted from the piston edge for α± < π/2 focus along the central axis and consequently undergo a second diﬀraction process. This is examined next. The second diﬃculty arises at a shadow boundary where the edgediﬀraction integral undergoes a jump to accommodate the jump in ϕg .

2.1 Radiation from a piston in an inﬁnite baﬄe

35

In §2.2, it will be shown that a region forms around the shadow boundary that steadily grows, eventually choking oﬀ the aperture at a distance equal to the Fresnel length, (2.1.8). For observation points beyond the Fresnel length, (2.1.25) ceases to be a useful representation and (2.1.9) becomes the appropriate one. By the word ‘beam’ is meant a well-collimated waveﬁeld that forms in a neighborhood where the edge-diﬀracted waveﬁeld and the geometrical one begin to merge to form a single waveﬁeld, that is, at the Fresnel length Fl , deﬁned by (2.1.8). If the beam is to remain collimated for a useful propagation length, Fl or, equivalently, ka must be larger than 1. Continuing to leave the observation point in the x2 = 0 plane, the scaled coordinates x ¯1 = x1 /a,

x ¯3 = x3 /Fl

are introduced. The goal here is to approximate the line integral in ¯3 ≈ 1, in inverse powers of (2.1.21) in the neighborhood x ¯1 < 1, x (kFl )1/2 . The resulting waveﬁeld will be a beam in the sense just deﬁned. Note that kx1 x ¯1 = , (2π)1/2 (k Fl )1/2 ¯1 = O[(kFl )−1/2 ]. so that for kx1 ≤ 1 (equivalently ka > 1), x Referring back to Fig. 2.2, with a bit of work, it is readily shown that ˆρ = (−∇1 r ) · e

a(1 − x ¯1 cos φ) ; r

ˆ3 = ∇1 r · e

Fl x ¯3 . r

The distance r , ﬁrst given by (2.1.19), and subsequently by (2.1.22), is rewritten as r (¯ x1 , x ¯3 ; φ) = (a2 x¯1 2 + Fl2 x¯3 2 − 2a2 x ¯1 cos φ + a2 )1/2 π 2 −2 = Fl x ¯3 1 + (¯ x − 2¯ x cos φ + 1) + O[(kF ) ] . 1 l kFl x ¯23 1 Gathering the various pieces, it is readily shown that ˆρ ∇1 r · e a =1+x ¯1 cos φ + O[(kFl )−1 ] ˆ3 )2 r 1 − (∇1 r · e and

2

eikr = eik Fl x¯3 eiπ(¯x1 −2¯x1 cos φ+1)/¯x3 + O[(kFl )−1 ].

36

Canonical acoustic-wave problems

Using the integral representation (B.3) for the Bessel function of zero order, 2π 1 eiz cos μ dμ, (2.1.27) J0 (z) = 2π 0 where arg(z) ∈ (0, π), and substituting the previous approximations in (2.1.21), gives, after removing the scaling, kax1 x1 kax1 g ikx3 ik(x21 +a2 )/(2x3 ) J0 − i J1 e ϕ(x) = ϕ (x) − A e 2x3 a 2x3 + O[(kFl ]−1 ) .

(2.1.28)

Noting that x1 = r sin θ, this expression can be written for an arbitrary observation point. This expression can also be derived by seeking a parabolic approximation to the reduced wave equation (Naze Tjøtta and Tjøtta, 1980; Harris, 1987). Equation (2.1.28) is a mathematical description of a beam: it is a waveﬁeld collimated close to a central axis, propagating as a plane wave on a scale set by the wavenumber k, but having an amplitude – the expression in braces – that evolves on a scale set by the Fresnel length Fl . This expression matches the one given by (2.1.9) for v(ξ) = H(1−ξ), small θ (equivalently kx1 < 1), and x3 Fl . Let the observation point x now be given by the cylindrical coordinates (ρ, φ, x3 ). A solution to (2.1.1) and (2.1.2) is sought by using the Hankel transform pair, (2.1.10) and (2.1.11). One readily ﬁnds that ∗

ϕ = Ceik3 x3 ,

where C is a constant determined from satisfying (2.1.2). The radical k3 is deﬁned as k3 := (k 2 − kρ2 )1/2 ,

Im(k3 ) ≥ 0 ∀ kρ .

(2.1.29)

Figure 2.4(a) shows the Riemann sheet deﬁned by (2.1.29). On this sheet the radiation condition describing outgoing waves is fulﬁlled, and it is referred to as the physical Riemann sheet. Deﬁning the transform 1 ∗ v(kρ a) := ξv(ξ)J0 (kρ aξ)dξ (2.1.30) 0

and using the deﬁnition of the Hankel transform in (2.1.11), ϕ is readily found to be given by ∞ ∗ v(kρ a) kρ J0 (kρ ρ) eik3 x3 dkρ , (2.1.31) ϕ(ρ, x3 ) = A(ka2 ) k3 0

2.1 Radiation from a piston in an inﬁnite baﬄe Im (k3) ≥ 0

kρ plane

37

ξ plane

(b)

∀kρ

Re (kρ) > 0

0

k

π/ 2

–k

Re (kρ) > 0 kρ = k sin ξ

(a)

Fig. 2.4. (a) The kρ plane: the Riemann sheet is selected so that Im(kρ ) ≥ 0, ∀ kρ . (b) The ξ plane: the integration contour runs from 0 to π/2 − i∞. The transformation kρ = k sin ξ has removed the branch cuts present in (a).

where k3 is deﬁned by (2.1.29) and ∗v(kρ a) by (2.1.30), and the integration contour is sketched in Fig. 2.4(a). In §3.1.2 it is shown that, using (2.1.27) and the transformation kρ = k sin ξ, (2.1.31) can be expressed as A(ka)2 ϕ(x) = 2π

0

2π

π/2−i∞

∗

v(ka sin ξ) eikˆp·x sin ξ dν dξ,

(2.1.32)

0

where ˆ2 ) + cos ξ e ˆ3 . ˆ = sin ξ(cos ν e ˆ1 + sin ν e p

(2.1.33)

Figure 2.4(b) sketches the ξ plane: the transformation kρ = k sin ξ, which is sometimes named the Sommerfeld transformation, has eliminated the branch cut present in Fig. 2.4(a). This transformation will be explored in more detail in §3.1.2. For the present, note that the radiation from the piston has been expressed as an integration over plane waves, each ˆ , (2.1.33). Also note that some directions propagating in the direction p of propagation are complex, implying that some of the plane waves are inhomogeneous or evanescent (Harris, 2001, pp. 22, 23). Because (2.1.32) is expressed as an integration over angles, it is named an angular spectrum representation. As previously indicated, this representation is especially useful for exploring the scattering of a beam; it will be the representation used to discuss the scattering of a spherical acoustic wave from a ﬂuid–solid interface in §3.1 and imaging in Chapter 6.

38

Canonical acoustic-wave problems 2.1.4 Radiation impedance

Figure 2.5 shows an idealized model of a rigid piston radiator and its mounting: the piston is treated as point mass m, with particle velocity va (t) = Re(V0 e−iωt ), and its mounting as a spring with constant ks and a dashpot with damping constant cs . Contrary to the suggestion of (2.1.2), it is not the particle velocity of a radiator that is set; rather there is an applied force fa (t) = Re(Fa e−iωt ) imposed by an electromechanical actuator. As the piston face moves, it is loaded by the ﬂuid with a net force fr (t) = Re(Fr e−iωt ). The force Fr is given by

p(x )dS(x ) = Zr (ω)V0 ,

Fr = ∂Sa

where p = (ρf c)(ωk)ϕ, and ϕ is given by any of (2.1.7), (2.1.21), or (2.1.32). This is the force that is excited by the radiated waveﬁeld. The term Zr (ω) is called the radiation impedance; it is a complex function of ω, though the ω dependence will not usually be explicitly indicated. Using (2.1.7), the multiple integral can be explicitly calculated for the rigid piston; this is described in Morse and Ingard (1968, pp. 383– 387), Pierce (1981, pp. 221–225), and Skudrzyk (1971, pp. 663–665).

ds m

fr = Re(Fr e–iωt)

ks fa = Re(Fa e–iωt)

Fig. 2.5. A mechanical model of the rigid piston radiator: m, ks , and ds are the mass, spring constant, and damping constant, respectively, of the piston and its mounting. A force fa = Re(Fa e−iωt ) is applied to the piston by an electromechanical actuator; the ﬂuid loads the piston face, ∂Sa , with a force fr = Re(Fr e−iωt ).

2.1 Radiation from a piston in an inﬁnite baﬄe

39

Approximating p with a plane wave gives Zr = (ρf c)(πa2 ), the speciﬁc acoustic impedance of a plane wave times the surface area of ∂Sa . To understand the importance of Zr , the time-harmonic solution to the problem indicated in Fig. 2.5 is sought; it is expressed as V0 (Zs + Zr ) = Fa , where Zs = ds − i(ωm − ks /ω), Zr = Rr + iXr . The time-average rate at which work is done on the piston P a by the applied force fa is then given by P a =

V0 V0∗ V0 V0∗ ds + P r , where P r = Rr . 2 2

The result (1.3.8) was used here. The ﬁrst term is energy lost to the mounting, while the second, P r , is the energy radiated into the far ﬁeld and therefore lost to the piston radiator. The radiation impedance Zr measures how the waveﬁeld loads the radiator: as the energy balance indicates, its real part Rr describes how much power is lost by the radiator and transmitted into the far ﬁeld, while comparison of Zs with Zr suggests that the imaginary part Xr characterizes the inertial loading by and compressibility of the ﬂuid. For the designer of a transducer, Zr is a very important quantity to estimate because it is needed as part of an equivalent circuit model of the overall radiating system. This brief discussion of transducers is intended only to draw attention to Zr . Actual transducer models are much more complicated; Auld (1990b, pp. 317–349) and Thurston (1974, pp. 257–275) give an idea of how piezoelectric transducers are modeled. To circumvent the need to consider a speciﬁc model of a transducer, following Skudrzyk (1971, pp. 666–669), the complex time-average ﬂux of energy across the surface ∂Sa , given by (1.3.12), is used to deﬁne Zr : Zr :=

2 P cr , V0 V0∗

(2.1.34)

where, in the case of a piston with a general particle velocity distribution across its face, such as indicated by (2.1.2), V0 P cr = p v(ρ/a) dS. 2 ∂Sa Recall that V0 = (ωk)A. The radiator and its electromechanical actuator may exhibit complicated dynamics, so the waveﬁeld speciﬁed on the face of the radiator must usually be estimated. The deﬁnition (2.1.34)

40

Canonical acoustic-wave problems

does not depend on an explicit transducer model and is such that the time-average ﬂux of energy across ∂Sa and transmitted into the far ﬁeld is given, quite generally, by V0 V0∗ Rr , (2.1.35) 2 irrespective of the dynamics of the transducer. Equation (2.1.35) for the piston radiator with arbitrary v(ρ/a) can be evaluated in terms of the Hankel transforms of p and v by using Parseval’s relation, which is given by ∞ ∞ xf (x)g(x)dx = u ∗f (u) ∗g(u)du. (2.1.36) P r =

0

0

After introducing the scaled transform variable ξ, by setting kρ = kξ, the Hankel transform of p can be calculated directly from (2.1.30) as ∗

p(kaξ) = (ρf c)V0

∗

v(kaξ) , (1 − ξ 2 )1/2

where ∗v is given by (2.1.30) and the square root is deﬁned just as was k3 by (2.1.29). Then, from the deﬁnition (2.1.34), it is readily shown that 1 ∗ ∞ ∗ | v(kaξ)|2 | v(kaξ)|2 2 2 ξdξ − i ξdξ . Zr = 2πa (ρf c)(ka) 2 1/2 (ξ 2 − 1)1/2 0 (1 − ξ ) 1 (2.1.37) A very similar use of Parseval’s relation will be made in §6.7.

2.2 Diﬀraction of an acoustic plane wave by an edge A recurrent topic of the book is diﬀraction from edges; see also Chapters 5 and 7. The problem to be discussed here is the simplest one that exhibits the structure of edge-diﬀracted waves; it is thus very much a canonical problem. The presentation uses the Wiener–Hopf technique, and follows that in Harris (2001, pp. 101–103) and Noble (1988, pp. 48–85). Its presentation is intended to introduce the reader to more detailed information about edge diﬀraction, as well as the more complex Wiener–Hopf method that arises in studying edge diﬀraction in general. The original solution to this problem was given by Sommerfeld in 1896 using what he called branched solutions (Baker and Copson, 1987, pp. 124–149; Sommerfeld, 1967, pp. 247–266).4 Several years later, 4

As indicated in §1.7, Sommerfeld’s original paper has been translated, with notes and historical comments, and is available as Sommerfeld (2004).

2.2 Diﬀraction of an acoustic plane wave by an edge

41

Lamb (1906) gave a simpler derivation of Sommerfeld’s solution using parabolic coordinates and a similarity solution. As with radiation from a piston, the edge-diﬀraction problem has been intensively studied, as the general references cited in §1.7 indicate. As in the previous section, §2.1, the problem is formulated using ϕ, the particle displacement potential. The end results can be expressed in terms of the acoustic pressure p = (ρf c) (ωk) ϕ. Moreover, the amplitude of ϕ is again normalized by A = V0 /(ω k).

2.2.1 Problem solution using the Wiener–Hopf technique Figure 2.6 indicates the geometry of the problem being considered. A rigid barrier lies along the x1 ≥ 0 axis. It is struck by a plane acoustic wave described as ϕi = A eiκˆp

i

·x

,

(2.2.1)

where the direction of incidence is given by ˆ i = cos φ0 e ˆ1 + sin φ0 e ˆ2 . p

(2.2.2)

The complex wavenumber κ = k + i, where k = ω/c and > 0, has been introduced, in a manner that is consistent with the principle of limiting absorption, (1.2.2), in order to facilitate the implementation of the Wiener–Hopf technique. The parameter will be sent to 0 at the end of the calculation. Moreover, initially it will be assumed that φ0 ∈ (0, π/2]. The plane x2 = 0 is one of reﬂection symmetry so that the problem can be split into two problems, one antisymmetric and one symmetric ^i p

φ

x1

ρ

x2

Fig. 2.6. A rigid barrier lies along x1 ≥ 0; a plane wave is incident at a ˆ i . (ρ, φ) are polar coordinates identifying a point in the waveﬁeld. direction p φ ∈ {(−π, 0) ∪ (0, π]} and the barrier lies along φ = 0.

42

Canonical acoustic-wave problems

with respect to the reﬂection plane. However, the symmetric part of (2.2.1) excites no scattered waveﬁeld. As a consequence the scattered waveﬁeld must be antisymmetric with respect to the plane of reﬂection. This permits one to consider the solution to the scattered waveﬁeld in the x2 > 0 half-space; the solution in x2 < 0 is just the antisymmetric reﬂection of this one. In terms of the polar coordinates (ρ, φ) shown in Fig. 2.6, φ ∈ (0, π] for x2 > 0, while φ ∈ (−π, 0) for x2 < 0, with the barrier at φ = 0 . The total waveﬁeld ϕt is conveniently expressed as the sum of the incident ﬁeld, ϕi , and the scattered waveﬁeld, ϕ, so ϕt = ϕi + ϕ. The scattered waveﬁeld satisﬁes the reduced wave equation, (2.1.1), in the half-space x2 > 0, with κ2 replacing k 2 . Along x2 = 0 it satisﬁes the mixed boundary conditions ϕ = 0, x1 < 0; ∂2 ϕ = −iκ sin φ0 A eiκ cos φ0 x1 , x1 > 0.

(2.2.3)

In addition ϕ must satisfy the edge condition ∂2 ϕ = O(ρ−1/2 ), ρ → 0, φ = 0.

(2.2.4)

Because reﬂected plane waves are excited along the barrier, in addition to the diﬀracted wave, it may be inferred that ϕ ∼ constant on φ = 0, as x1 → 0+ (that is, as x1 goes to zero through positive values), where the constant can be zero.5 Lastly, ϕ must satisfy the principle of limiting absorption, (1.2.2), as kρ → ∞, which forces the scattered waveﬁeld to be outgoing from the barrier. Introduce two further unknowns ϕ+ (x1 ) and τ − (x1 ) by writing the boundary conditions (2.2.3), at x2 = 0, as ϕ = 0, +

ϕ=ϕ ,

∂2 ϕ = τ − , ∂2 ϕ = τ

+

= −iκ sin φ0 A e

x1 < 0, iκ cos φ0 x1

,

x1 > 0.

(2.2.5)

To seek a relationship between ϕ+ and τ − , the plane-wave spectral representation of ϕ, namely, ∞ 1 ∗ ϕ(β)ei(βx1 +γx2 ) dβ, (2.2.6) ϕ(x1 , x2 ) = 2π −∞ is introduced. Equation (2.2.6) is described as a plane-wave representation, as opposed to an angular one, because the integration is over the Cartesian wavenumber β. The radical γ is deﬁned as 5

The edge condition given by equation (5.89) of Harris (2001) is stronger than necessary; only the condition on ∂2 ϕ is needed. The discussion in Noble (1988, pp. 72–76) of edge conditions is particularly helpful.

2.2 Diﬀraction of an acoustic plane wave by an edge

43

ξ plane

β plane *τ–

κ

π 0

ε cos φ 0

φ0

φ

κ cos φ 0

*ϕ +

(φ) (b)

(a)

φ <ξ1> π + φ

Fig. 2.7. (a) The β plane: both ∗ϕ+ and ∗τ − are analytic in the horizontal strip between the dashed lines. (b) The ξ plane: C is the initial contour; Cs (φ) is the steepest-descents contour passing through the saddle point φ.

γ := (κ2 − β 2 )1/2 ,

Im(γ) ≥ 0 ∀ β;

(2.2.7)

the branch cuts in the β plane are shown in Fig. 2.7(a). This deﬁnition is identical to that for k3 , (2.1.29). Using the inverse Fourier transform, ∞ ∗ ∗ + ϕ(β) = ϕ (β) = ϕ+ (x1 ) e−iβ x1 dx1 . (2.2.8) 0

+

− cos φ0 x1

Note that ϕ (x1 ) = O(e ) as kx1 → ∞, because it is dominated ∗ + by reﬂected waves; therefore, ϕ (β) is analytic for Im(β) < cos φ0 (Noble, 1988, pp. 11–21). In a similar way, ∞ ∗ + τ (β) = τ + (x1 ) e−iβ x1 dx1 0 (2.2.9) sin φ0 κA ; =− (β − κ cos φ0 ) and ∗ −

0

τ (β) = −∞

τ − (x1 ) e−iβ x1 dx1 .

(2.2.10)

Note that τ − (x1 ) = O(e−|x1 | /k|x1 |1/2 ) as kx1 → −∞, because it is composed of a cylindrical, edge-diﬀracted wave; therefore, ∗τ − (β) is analytic for Im(β) > −. Note, in Fig. 2.7(a), the strip of overlapping analyticity and the pole at κ cos φ0 .

44

Canonical acoustic-wave problems

Using (2.2.8)–(2.2.10), the Fourier transform of particle displacement at x2 = 0 is calculated in terms of ∗ϕ+ and ∗τ ± , and the two expressions equated; this gives iγ ∗ϕ+ = −

sin φ0 κA + ∗τ − . (β − κ cos φ0 )

Isolating the pole κ cos φ0 , this functional relation can be written as ∗ − (κ + κ cos φ0 )1/2 − (κ + β)1/2 τ − sin φ κA 0 (κ + β)1/2 (β − κ cos φ0 )(κ + κ cos φ0 )1/2 (κ + β)1/2 = i(κ − β)1/2 ∗ϕ+ +

sin φ0 κA . (β − κ cos φ0 )(κ + κ cos φ0 )1/2

(2.2.11)

The right-hand side is analytic in the half-space Im(β) < cos φ0 , while the left-hand side is analytic in the half-space Im(β) > −; both sides share the common region of analyticity − < Im(β) < cos φ0 . Each side is therefore the analytic continuation of the other and together they represent a function that is entire in the β plane. To learn what this function is, the asymptotic behavior of ∗τ − , as |β| → ∞, in the upper half of the β plane, is needed. Using an Abelian theorem (see Appendix A.2; Abelian and Tauberian theorems as they relate to integral transforms are discussed in Noble (1988)), the condition (2.2.4) implies that ∗ −

τ

= O(β −1/2 ), |β| → ∞, Im(β) > −.

Therefore the left-hand side of (2.2.11) goes to zero. Invoking Liouville’s theorem (Titchmarsh, 1939), the entire function is zero, and it follows then that ∗

ϕ(β) =

i sin φ0 κA . (β − κ cos φ0 )(κ + κ cos φ0 )1/2 (κ − β)1/2

Removing the assumption that x2 > 0, the scattered waveﬁeld is given by i A sin(φ0 /2) κ 1/2 ϕ(x1 , x2 ) = sgn(x2 ) 2π π 1/2 ∞ ei(βx1 +γ|x2 |) × dβ, (2.2.12) 1/2 −∞ (β − κ cos φ0 )(κ − β) where sgn(x2 ) = ±1 as x2 > 0 or < 0, respectively. At this point, → 0, so that κ becomes k; and the restriction φ0 ∈ (0, π/2] can be relaxed and φ0 allowed to lie in (0, π).

2.2 Diﬀraction of an acoustic plane wave by an edge

45

Equation (2.2.12) added to ϕi , (2.2.1), is the total waveﬁeld, ϕt . The integral in (2.2.12) can be expressed in terms of a Fresnel integral, which is deﬁned in Appendix B.2; the steps in this reduction are given by Born and Wolf (1999, pp. 645–648). This reduction is not always possible in more complicated problems, so it will not be undertaken here; rather, asymptotic approximations will be sought because these can almost always be calculated, no matter how complicated the integral. However, before doing so, the problem of diﬀraction of an incident, threedimensional plane wave is considered because the scattered waveﬁeld entails only a slight modiﬁcation of (2.2.12).

2.2.2 Diﬀraction of an incident, three-dimensional plane wave The incident, plane acoustic wave, propagating in three dimensions, is described as ϕi 3D = ϕi eik cos θ0 x3 ,

κ = k sin θ0 + i,

where ϕi is given by (2.2.1) and (2.2.2), provided k is replaced by k sin θ0 , in the expression for κ. The angle θ0 is measured from the positive x3 axis, and will be restricted to be such that θ0 ∈ (0, π/2); the angle φ0 continues to be deﬁned as it is in Fig. 2.6. The boundary conditions for the scattered, three-dimensional waveﬁeld ϕ3D now become, for x2 = 0 and x3 ∈ (−∞, ∞), ϕ3D = 0, x1 < 0; ∂2 ϕ3D = ∂2 ϕ eik cos θ0 x3 , x1 > 0; where ∂2 ϕ is given in (2.2.3), with κ now given by k sin θ0 + i. The boundary conditions do not engage the x3 coordinate, so that solving the three-dimensional problem is equivalent to solving the twodimensional one. In (2.2.12), κ becomes k sin θ0 (after → 0) and the whole expression is multiplied by exp(ik cos θ0 x3 ). Therefore, the scattered waveﬁeld ϕ3D is given by 1/2 i A sin(φ0 /2) k sin θ0 3D ϕ (x1 , x2 , x3 ) = sgn(x2 ) 2π π 1/2 ∞ i(βx1 +γ|x2 |) e × eik cos θ0 x3 dβ. (2.2.13) 1/2 −∞ (β − k sin θ0 cos φ0 )(k sin θ0 − β) In the corresponding elastic-wave problem it becomes evident that the two-dimensional result cannot be trivially extended to three dimensions

46

Canonical acoustic-wave problems

because the boundary conditions engage all three traction components on the barrier. The full solution involves a matrix Wiener–Hopf problem as compared with the scalar Wiener–Hopf equation solved here. The method is conceptually identical in form with the derivation of (2.2.13), although the technical diﬃculty is necessarily greater; Achenbach et al. (1982) provide a thorough discussion of the solution and its application to three-dimensional edge diﬀraction. The related problem of diﬀraction from a semi-inﬁnite screen in a viscous ﬂuid is considered and solved in Chapter 7. 2.2.3 A uniform asymptotic approximation It will be helpful ﬁrst to explain the qualitative features of the total waveﬁeld; the asymptotic analysis will explicitly exhibit each of these features. Figure 2.8 shows a sketch of the various wavefronts incident and scattered by the barrier. In region 1 the scattered waveﬁeld contains both a plane wave and the cylindrical diﬀracted wave; the plane wave cancels the incident plane wave so that the total waveﬁeld is solely composed of the diﬀracted wave. In region 2 the scattered waveﬁeld is composed of the cylindrical diﬀracted wave, but the total waveﬁeld is composed of both this wave and the incident one. In region 3 the scattered waveﬁeld contains both a plane wave, representing the plane wave reﬂected from the barrier, and the cylindrical diﬀracted wave; and the total waveﬁeld has the incident plane wave added to this. The various plane waves do not suddenly turn on or oﬀ: penumbrae form within the parabolic regions

5

e av w

ct

le

f Re

ed

3 Incident wave

Diffracted wave 2

x1

1 φ0 Y

4 X

x2

Incident wave

Fig. 2.8. A sketch of the wavefronts and their directions of propagation. The major disturbances merge into one another across the penumbra, or Fresnel regions, 4 and 5. The local coordinates (X, Y ) describe region 4.

2.2 Diﬀraction of an acoustic plane wave by an edge

47

4 and 5, and are characterized by Fresnel integrals (see Appendix B.2) that combine the plane geometrical waves with the diﬀracted one. In this book the penumbrae are named Fresnel regions. Again, take x2 > 0, allow φ0 ∈ (0, π), and transform (x1 , x2 ) to the polar coordinates (ρ, φ). Moreover, introduce the transformation β = k cos ξ so that γ = k sin ξ; this removes the branch cut. Equation (2.2.12), for φ ∈ (0, π), becomes eiπ/4 A D(ξ, φ0 )eikρ cos(ξ−φ) dξ, (2.2.14) ϕ(ρ, φ) = − (2π)1/2 C where D(ξ, φ0 ), a diﬀraction coeﬃcient, is deﬁned as D(ξ, φ0 ) :=

i e−iπ/4 cos(ξ/2) sin(φ0 /2) . 1/2 sin[(ξ + φ0 )/2] sin[(ξ − φ0 )/2] (2π)

(2.2.15)

Figure 2.7(b) shows the ξ plane and the initial contour C; this contour begins at (0 + i∞) and ends at (π − i∞). Note the similarities between the transformations illustrated by Fig. 2.4(b) and Fig. 2.7(b). Note also the pole at ξ = φ0 . The contour of integration C is next shifted to the contour of steepest descents Cs (φ) (Felsen and Marcuvitz, 1994, pp. 377–391, 399–406; Harris, 2001, pp. 91–94), which is deﬁned by the equation cos(ξ1 − φ) cosh ξ2 = 1, where ξ = ξ1 + iξ2 . The saddle point is ξ = φ. Figure 2.7(b) indicates the pole at φ0 , the saddle point φ, and the contour Cs (φ). To ensure convergence, the contour Cs (φ) must begin in the region φ−π < ξ1 < φ and end in the region φ < ξ1 < π + φ; the ending region is indicated in Fig. 2.7(b). Note that for φ ∈ (0, φ0 ), a pole contribution equal to −ϕi is picked up. Also note that as |φ − φ0 | ranges over some domain, the saddle point and pole can come close to one another, so that (2.2.14) cannot be approximated by the usual saddle-point approximation for all values of this parameter. Stated more precisely, a saddle-point (or steepest-descents) approximation to (2.2.14) is not uniform in the parameter |φ − φ0 |, and a uniform approximation should be sought. To isolate the pole term, (2.2.15) is written as E(ξ, φ0 ) sin[(ξ − φ0 )/2] E(φ0 , φ0 ) E(ξ, φ0 ) − E(φ0 , φ0 ) + . = sin[(ξ − φ0 )/2] sin[(ξ − φ0 )/2]

D(ξ, φ0 ) =

(2.2.16)

48

Canonical acoustic-wave problems

This decomposition splits the integral (2.2.14) so that ϕ = ϕnp + ϕp : one part ϕnp has a removable singularity at ξ = φ0 , while the second part ϕp is deﬁned by this singularity. The integral ϕnp can be approximated by using the standard saddle-point approximation, described in Harris (2001, pp. 91–95), to give ϕnp ∼ −A

E(φ, φ0 ) − E(φ0 , φ0 ) eikρ , kρ → ∞. sin[(φ − φ0 )/2] (kρ)1/2

(2.2.17)

The function E(φ, φ0 ) is deﬁned by (2.2.16). In the limit φ → φ0 , E(φ0 , φ0 ) =

i e−iπ/4 . 23/2 π 1/2

The integral ϕp can be reduced to a Fresnel integral. This integral is deﬁned and its asymptotic properties are derived in Appendix B.2. The reduction of ϕp to a Fresnel integral is also described in Born and Wolf (1999, pp. 645–648); the outcome is ϕp = −A eikρ cos(φ−φ0 ) H(φ0 − φ) + A eikρ cos(φ−φ0 ) sgn(φ0 − φ)

e−iπ/4 F [21/2 sin(|φ − φ0 |/2)(kρ)1/2 ]. π 1/2

The ﬁrst term is the pole contribution, and the second term is the contribution from the contour integral. The Fresnel integral changes smoothly as |φ − φ0 | becomes small. Using (B.7), the two terms can be combined to give ϕp = −A eikρ cos(φ−φ0 )

e−iπ/4 F {21/2 sin[(φ − φ0 )/2](kρ)1/2 }. (2.2.18) π 1/2

In summary, the total scattered waveﬁeld is approximated, uniformly in |φ − φ0 |, by ϕ = ϕnp + ϕp , φ ∈ (0, π],

(2.2.19)

where ϕnp is approximated by (2.2.17), for kρ → ∞. For φ ∈ (−π, 0), φ is replaced by (−φ), and ϕnp and ϕp are multiplied by (−1), in (2.2.17) and (2.1.21) respectively. And to ϕ must be added ϕi to obtain the total waveﬁeld ϕt . Equation (2.2.18) allows one to determine those regions in which the Fresnel function can be replaced by its asymptotic approximation, as given by (B.8), and hence to estimate the sizes of the Fresnel regions 4 and 5 shown in Fig. 2.8. Deﬁne Fr (ρ, φ − φ0 ) := sin[(φ − φ0 )/2] (kρ)1/2 ,

2.2 Diﬀraction of an acoustic plane wave by an edge

49

making 21/2 Fr the argument of the Fresnel function in (2.2.18). For 21/2 |Fr | > π 1/2 the Fresnel integral can be replaced by its asymptotic approximation (B.8); this right-hand side is a reasonable, though not unique, choice. Thus 21/2 |Fr | = π 1/2 deﬁnes the boundaries of the Fresnel regions shown in Fig. 2.8. Using the local coordinates (X, Y ), also shown in Fig. 2.8, it is straightforward to show that the boundary of region 4 is described by the parabola (kY )2 = 4 (π/2)[kX + (π/2)].

(2.2.20)

Outside of the Fresnel regions, (2.2.18) can be asymptotically approximated for kρ large. Combining this approximation with (2.2.17) gives ϕ ∼ −A eikρ cos(φ−φ0 ) H(φ0 − φ) − A D(φ, φ0 )

eikρ , kρ → ∞, (kρ)1/2 (2.2.21)

for φ ∈ (0, π]. The term D(φ, φ0 ), deﬁned by (2.2.15), is the diﬀraction coeﬃcient. For φ ∈ (−π, 0), φ is replaced by (−φ), and (2.2.21) is multiplied by (−1). And to ϕ must be added ϕi to obtain the total waveﬁeld. Were one to solve the radiation problem suggested by (2.1.26) – a less complicated but physically similar problem to the edge-diﬀraction one – one would ﬁnd that a Fresnel region, having the same geometrical shape as described by (2.2.20), projects perpendicularly from the edge x1 = a.

a

α− α+

r−′

x r+′

ka2 2π

Fig. 2.9. The Fresnel region, shown in cross-section, for the piston radiator. Note how the Fresnel region chokes oﬀ the projected aperture at approximately the Fresnel length Fl = ka2 /(2π). This ﬁgure should be considered in conjunction with Fig. 2.3.

50

Canonical acoustic-wave problems

And approximating the radiated waveﬁeld non-uniformly would lead to an expression of the form (2.2.21). Returning to the piston radiator, §2.1.2, and in particular noting the non-uniform asymptotic approximation (2.1.9), one can conclude that a Fresnel region, composed of the parabola (2.2.20) rotated about the x3 axis, is present. As Fig. 2.9 suggests, eventually the Fresnel region grows thick enough that the aperture projected into S is choked oﬀ; this happens for x3 ≈ Fl . Therefore, (2.1.25) is not a particularly useful representation because it is not accurate for much of the region S, as is suggested in Fig. 2.9. One is thus led to consider (2.1.28) as a better description.

2.3 Summary As has been noted previously, the problems were formulated throughout using ϕ, the particle displacement potential. The end results can be readily expressed in terms of the acoustic pressure p = (ρf c) (ωk) ϕ. Moreover, if one imagines that V0 is the magnitude of the particle velocity, then A = V0 /(ω k). As an example, (2.1.7) is expressed in terms of the acoustic pressure as i(ρf c)(k 2 V0 ) eikr p(x) = − v(ρ/a) dS(x ). 2π kr ∂Sa The study of the piston radiator has shown that there are three ways of representing the radiation from an extended source at a boundary: (1) The extended source, or aperture distribution, can be viewed as a distribution of point sources; thus the radiated waveﬁeld (2.1.7) is a consequence of the interference of many spherical waves emitted from a dense set of point sources. This is an adaptation of Huygens’ principle. Moreover, as indicated in its deﬁnition, (2.1.8), the Fresnel length Fl marks the distance at which the radiated waveﬁeld begins to evolve into the spherical wave, represented by (2.1.9). (2) The extended source radiates a geometrical wave and an edgediﬀracted wave; the latter focuses rays along the central axis, as indicated by (2.1.21), and generates a thickening Fresnel region as indicated in Fig. 2.9. It is not hard to show that this region, calculated using (2.2.20), chokes oﬀ the projection of the aperture at approximately the Fresnel length, Fl = ka2 /(2π). This restates the inference originally drawn from (2.1.8). As a consequence, an approximation such as (2.1.25) is not particularly useful. Rather, one can

2.3 Summary

51

take advantage of the representation (2.1.21) to construct a beam waveﬁeld, (2.1.28), in the neighborhood x1 < a, x3 ≈ Fl . It is this region that is of most interest at high frequencies. (3) The extended source can be decomposed into an integration over homogeneous and inhomogeneous plane waves, as indicated by (2.1.7). This representation allows one to derive a general expression for the radiation impedance (2.1.37). This is a quantity of importance in the construction of equivalent circuit models, which in turn are needed to integrate the acoustic source of radiation with its electromechanical actuator. Moreover, as will be shown in subsequent sections, the angular spectrum representation (2.1.32) is an appropriate representation when the scattering of a beam from a boundary or edge is to be calculated. The study of the edge-diﬀraction problem, in addition to introducing the Wiener–Hopf technique for solving mixed boundary-value problems, has suggested the following: (4) The approximation to the diﬀraction integral (2.2.14) has highlighted the distinction between uniform and non-uniform asymptotic approximations. Moreover, it has allowed a determination of the size of the Fresnel region, a geometrical or kinematic property of an edge-diﬀracted waveﬁeld. (5) In the case of diﬀraction from an edge, the Fresnel regions, while important, do not dominate the character of the scattered waveﬁeld. However, when radiation from an extended source such as the piston radiator is studied, then these regions determine much of the character of the radiated waveﬁeld. In both cases these canonical problems show the interplay between physical understanding, intuition, and the mathematical techniques required to distill the formulae into their most insightful form. These techniques are fundamental to the study of wave scattering and diﬀraction phenomena.

3 Canonical elastic-wave problems

This chapter continues the work of the previous one. Its purpose is to describe the following two canonical elastic-wave problems: (1) The scattering of a spherical wave from a ﬂuid–solid interface is calculated using angular spectrum representations of the incident and scattered waveﬁelds. The principal features described are: (a) the integral representations of the incident and scattered waveﬁelds; (b) the inter-relation between the analytic structure of the reﬂection coeﬃcient and the scattered waves, particularly the leaky Rayleigh wave; (c) the importance of the saddle-point method in interpreting the scattered waves. An angular-spectrum representation of the various waveﬁelds is used. (2) As an elastic plate becomes thicker and thicker, the two lowest Rayleigh–Lamb modes combine to form a Rayleigh surface wave. The excitation of a Rayleigh wave is explored using both the eigenmodes of an elastic plate and a Green’s state construction. Again various applied mathematical techniques are introduced.

3.1 The scattering of a spherical wave from a ﬂuid–solid interface The mathematical description of the waves scattered at a ﬂuid–solid interface, when a spherical acoustic wave strikes it, forms the basis for understanding the excitation of a surface wave, and for describing scanned acoustic imaging. By examining this interaction, not only are 52

3.1 The scattering of a spherical wave

53

a number of useful mathematical techniques introduced, but also the underlying physical structure of the scattering at a ﬂuid–solid interface is made manifest. Moreover, it will be shown in Chapter 6 that its solution can be progressively reworked so as to describe the imaging mechanism of a scanned acoustic microscope. This problem is therefore taken as canonical.

3.1.1 Description of the problem The plane x3 = 0 separates an ideal ﬂuid, in x3 < 0, from a homogeneous, isotropic, elastic solid, in x3 > 0, as indicated in Fig. 3.1. A unit ˆ (= −ˆ normal vector n e3 ) points from the solid into the ﬂuid. A spherical acoustic wave is emitted from a very localized source at (0, 0, −b). When it strikes the ﬂuid–solid interface, part of the spherical wave is reﬂected, part is transmitted as compressional and shear waves, and part becomes a leaky wave propagating along the interface. The solution to this problem is not new: a detailed one is described in the book by Brekhovskikh and Godin (1999, pp. 1–40), and another reworking of it is given by Pott and Harris (1984). The equation of motion is δ(r) , (3.1.1) 4πr2 where r = |x + bˆ e3 | and the wavenumber k = ω/c, with ω being the angular frequency and c the ﬂuid’s wavespeed. The particle displacement ∇2 ϕ + k 2 ϕ = −C

(0,0, –b) Ideal fluid

^ n

x1

Elastic solid

x3

Fig. 3.1. The arrangement of the coordinate system. x3 > 0 is occupied by an elastic solid, x3 < 0 by an ideal ﬂuid. The source is placed at (0, 0, −b), where b > 0.

54

Canonical elastic-wave problems

in the ﬂuid is u = ∇ϕ, and the acoustic pressure p = ρf ω 2 ϕ, where ρf is the density of the ﬂuid. The quantity C is a constant parameter giving the strength of the source at (0, 0, −b). The equation of motion describing similar disturbances in the solid is κ2 ∂i (∂k uk ) − eijk ∂j (eklm ∂l um ) + kT2 ui = 0.

(3.1.2)

This is a restatement of (1.1.18) where eijk is the permutation symbol. The compressional and shear wavenumbers are kL = ω/cL and kT = ω/cT , where cL and cT are the respective wavespeeds given by (1.1.21) and (1.1.22). The parameter κ = cL /cT . The continuity conditions to be satisﬁed at the interface are, restating (1.2.1), ˆ = −pf , ts · n

ˆ ∧ ts = 0, n

ˆ = uf · n ˆ, us · n

(3.1.3)

ˆ · τ . The stress tensor τ is related to the where the traction ts = n particle displacement by (1.1.16). The subscripts s and f indicate properties of the solid and ﬂuid, respectively. Equations (3.1.1)–(3.1.3), in combination with the requirement that the scattered waves propagate outward from the interface, constitute the mathematical statement of the problem to be addressed.

3.1.2 Representations for the spherical wave Ignoring for the moment the presence of the elastic solid, the solution to (3.1.1) is ϕ=A

eikr , kr

A=

kC . 4π

The constant A, as in the previous chapter, normalizes the particle displacement potential, but can be related to the magnitude of the particle velocity V0 by A = V0 /ωk. Following Harris (2001, pp. 24–28), the spherical wave can ﬁrst be written as ∞ ∞ iA dk1 dk2 ϕ= ei(k1 x1 +k2 x2 +k3 |x3 +b|) , (3.1.4) 2πk −∞ −∞ k3 where k3 = (k 2 − k12 − k22 )1/2 ,

Re(k3 ) ≥ 0, Im(k3 ) ≥ 0.

(3.1.5)

This choice of the branch of k3 is made so that the spherical wave is outgoing or decays as |x3 + b| → ∞. Second, it can be written as

3.1 The scattering of a spherical wave i A 2π π/2−i∞ ik pˆ i ·x ikb cos ξ i e e sin ξ dν dξ, ϕ = 2π 0 0

55 (3.1.6)

where ˆ1 + sin ν e ˆ2 ) + cos ξ e ˆ3 . ˆ i = sin ξ (cos ν e p

(3.1.7)

Note the similarity between (3.1.6) and (2.1.32), and between (3.1.7) and (2.1.33). The superscript i has been added in (3.1.6) to indicate that this is the incident spherical wave. Equation (3.1.6) assumes that (x3 + b) > 0; it was derived using the transformation k1 = k sin ξ cos ν,

k2 = k sin ξ sin ν,

k3 = k cos ξ,

where the accessible parts of the ξ plane are determined by (3.1.5); Fig. 2.4(b) indicates this. Equation (3.1.6) is an integration of the plane waves eik pˆ

i

·x

not only over all real directions of propagation, but also over a range of complex directions. The contour for ξ (see Fig. 2.4(b)) is taken to be from 0 to π/2, and then to (π/2 − i∞); for ξ ∈ (π/2, π/2 − i∞), ξ = π/2 + iξ2 , with ξ2 ≤ 0. In this case the direction given by (3.1.7) becomes complex: ˆ i = cosh ξ2 (cos ν e ˆ3 . ˆ1 + sin ν e ˆ2 ) − i sinh ξ2 e p ˆ3 direction. The ˆ i produces a decay in the e The imaginary part of p integration over complex angles is essential if the complete curvature of the spherical wave is to be captured by (3.1.6). It will be useful to have two additional representations for the spherical wave. Using cylindrical coordinates (ρ, φ, x3 ), where x1 = ρ cos φ and x2 = ρ sin φ, ϕi can be written as 2π i A π/2−i∞ ik cos ξ (x3 +b) i ikρ sin ξ cos(ν−φ) ϕ = e sin ξ e dν dξ. 2π 0 0 Using the representation (2.1.27) for the Bessel function J0 this integral can be written, ﬁrst, as π/2−i∞ i ϕ = iA eik cos ξ (x3 +b) J0 (kρ sin ξ) sin ξ dξ, (3.1.8) 0

and, second, as i A π/2−i∞ ik cos ξ (x3 +b) (1) i e H0 (kρ sin ξ) sin ξ dξ. ϕ = 2 −π/2+i∞

(3.1.9)

56

Canonical elastic-wave problems

The contour for (3.1.9) passes above the branch cut for the Hankel function; that cut proceeds from 0 to (−π/2 − i∞). Both representations are integrals over cylindrical waves. Equation (3.1.9) is very useful when calculating an asymptotic expansion to ϕi .

3.1.3 Scattered waveﬁelds Consider brieﬂy the expressions for the plane waves reﬂected and transmitted at a ﬂuid–solid interface when a plane wave, incident from the ﬂuid, strikes it. Harris (2001, pp. 37–55) indicates one way to calculate these waves. Reﬂection and transmission, and their respective coeﬃcients, are calculated using the particle displacement u. Each plane wave ˆ and a unit polarization is characterized by a unit propagation vector p ˆ The incident plane wave is written as vector d. ˆ i eik pˆ i ·x , ui = d

ˆi = p ˆ i, d

ˆ i is given by (3.1.7). The reﬂected and transmitted waves must where p all phase-match to the incident wave; thus, k sin ξ = kL sin ξL = kT sin ξT , where the angles ξL and ξT are indicated in Fig. 3.2. These waves are given by ^i p

^r p

ξ

ξ

^

dT

x|| ξL

^L p

ξT ^T p

x3

Fig. 3.2. The geometry of the incident and scattered rays. The incident and scattered propagation vectors, with the deﬁning angles ξ, ξL , and ξT , are ˆ T. shown, as is the polarization vector d

3.1 The scattering of a spherical wave ˆ r eik pˆ ur = R(ξ) d

r

·x

,

ˆ L eik pˆ L ·x , uL = TL (ξ) d ˆ T eik pˆ uT = TT (ξ) d

T

·x

,

ˆr = p ˆ r, d ˆL = p ˆ L, d

57

ˆ r = sin ξ e ˆ − cos ξ e ˆ3 , p ˆ L = sin ξL e ˆ + cos ξL e ˆ3 , p

ˆT = e ˆT ∧d ˆT , p

ˆ + cos ξT e ˆ3 . ˆ T = sin ξT e p (3.1.10)

Two new unit vectors, ˆ1 + sin ν e ˆ2 , ˆ = cos ν e e

ˆT = e ˆ3 ∧ e ˆ , e

(3.1.11)

ˆT , e ˆ3 ) form a right-handed triad. The have been introduced; (ˆ e , e superscripts r, L, and T indicate the reﬂected, transmitted longitudinal, and transmitted transverse waves, respectively. The reﬂection and transmission coeﬃcients, R(ξ) and TL,T (ξ) are given by R(ξ) =

A− (ξ) , A+ (ξ)

TL (ξ) =

TT (ξ) = −

2κ κf (ρf /ρs ) cos ξ cos(2ξT ) , A+ (ξ)

2κf (ρf /ρs ) cos ξ sin(2ξL ) , A+ (ξ)

(3.1.12)

where A± = cos ξ[sin(2ξL ) sin(2ξT ) + κ2 cos2 (2ξT )] ± κ κf (ρf /ρs ) cos ξL . (3.1.13) The parameter κf = c/cT . Recall that κ = cL /cT . Returning to the problem under discussion and using (3.1.6), the incident wave becomes i k A 2π π/2−i∞ ˆ i ui = − d (ξ, ν) eik pˆ ·x eikb cos ξ sin ξ dν dξ. 2π 0 0 Noting that the problem is linear and that the scattered plane waves are given by (3.1.10), the waves scattered from the interface are directly calculated to give the equations r k A 2π π/2−i∞ ˆ r r d (ξ, ν)R(ξ) eik pˆ ·x eikb cos ξ sin ξ dν dξ u =− 2π 0 0 (3.1.14) and u

L,T

L,T k A 2π π/2−i∞ ˆ L,T =− (ξ, ν)TL,T (ξ) eikL,T pˆ ·x eikb cos ξ d 2π 0 0 × sin ξ dν dξ. (3.1.15)

In (3.1.10) are listed the propagation and polarization vectors. It should be clear to the reader that all further outcomes depend on the analytic

58

Canonical elastic-wave problems

structure of the phases in the integrands and the analytic structure of A± (ξ), (3.1.13).

3.1.4 Waveﬁeld in the ﬂuid For the remainder of this chapter only the waveﬁeld in the ﬂuid, (3.1.14), will be considered, because it is the waveﬁeld used to form an image by a scanning acoustic microscope, discussed in Chapter 6. Using the transformation that yielded the representation (3.1.9), ur is written as π/2−i∞ iA (1) r ∇ R(ξ) eik(b−x3 ) cos ξ H0 (kρ sin ξ) sin ξ dξ. u = 2 −π/2+i∞ (3.1.16) Recall that cylindrical coordinates (ρ, φ, x3 ) are introduced to arrive at (1) this representation. Noting the Sommerfeld representation of H0 (see Appendix B.1) it is seen that (3.1.16) is a double integral whose phase is Φ(ξ, μ) = (b − x3 ) cos ξ + ρ sin ξ cos μ. Note: the second, mixed derivative ∂ξ ∂μ Φ = 0 when evaluated at the stationary point (ξ ∗ , μ∗ ), where (b − x3 ) sin ξ ∗ = ρ cos ξ ∗ and μ∗ = 0. As a consequence, (3.1.14) can be treated as an iterated integral, to leading order, when asymptotically approximating it; see Bleistein and Handelsman (1975, pp. 321–366), Chako (1965), and Stamnes (1986, (1) pp. 136–162). Though an asymptotic approximation to H0 is used next, one could work directly with (3.1.14). (1) Using the asymptotic approximation for H0 , and introducing the spherical coordinates (r, θ, φ), where (b − x3 ) = r cos θ and ρ = r sin θ, (3.1.14) can be approximated as ur ∼ −

kA 2

2 πkρ

1/2

e−iπ/4

ˆ r (ξ)R(ξ) eikr cos(ξ−θ) (sin ξ)1/2 dξ, d

C(θ)

kρ → ∞. (3.1.17) The phase of the integrand of (3.1.15) has a saddle point at ξ = θ. The contour C(θ) is the steepest-descents contour; Fig. 3.3(b) is a sketch of a part of this contour as it passes through θ. The method of steepest descents, which is introduced in Appendix A.3, is described more fully in Felsen and Marcuvitz (1994, pp. 370–391), and the saddle-point method (which gives the leading-order term of a steepest-descents approximation) in Harris (2001, pp. 91–94). Lastly, note that the dependence on

3.1 The scattering of a spherical wave ξ plane

ξbL

ξbT

59 ξ plane

ξR ξR

π/ 2 θ

ξS

(a)

(b)

Fig. 3.3. The structure of the complex ξ plane for Re(ξ) ≥ 0. (a) The contour for (3.1.14), the branch cuts beginning at the branch points ξbL and ξbT , the leaky Rayleigh pole ξR , and the Stoneley pole ξS are indicated. (b) The steepest-descents contour C(θ) of (3.1.15) is sketched (the steepestdescents method is described in Appendix A.3). The spatial relation between the steepest-descents contour and the leaky Rayleigh pole ξR is shown.

ˆ r has been dropped, because it is clear, at this point, that the ν in d waveﬁeld is axisymmetric. The subsequent asymptotic approximation of this integral depends on the proximity of θ to the poles and branch points of R(ξ). A+ (ξ), which is given by (3.1.12), has four zeros, ±ξR and ±ξS , and four branch points, ±ξbL and ±ξbT . The branch points are deﬁned by the equations sin ξbL = c/cL ,

sin ξbT = c/cT .

Figure 3.3(a) shows the right half of the complex ξ plane, with these poles and branch points indicated. The poles of R(ξ) correspond to leaky Rayleigh waves (±ξR ) and Stoneley waves (±ξS ), and the branch points to various lateral waves. The wiggly lines in Fig. 3.3 indicate the branch cuts corresponding to these branch points. These cuts, which deﬁne the Riemann sheet on which the problem is solved, are determined by demanding that Im(cos ξI ) ≥ 0, I = L, T , everywhere on the Riemann sheet of interest. Because R(ξ) is a reﬂection coeﬃcient for an incident plane wave, the branch cuts are chosen to ensure that when critical refraction takes place in the solid, the refracted waves decay with depth. In this chapter, only the eﬀect of the leaky Rayleigh-wave pole is considered; a non-uniform expansion is used that does not accurately

60

Canonical elastic-wave problems

account for what happens when the saddle point lies in the neighborhood of a pole, a branch point, or both a pole and a branch point. It was shown in §2.2.3 how a uniform approximation can be calculated if a pole is present. The case here requires a uniform approximation when a pole and branch point are close to a saddle point, which is possible to evaluate; unfortunately, this uniform expansion is itself complicated. For the purposes of this section the non-uniform approximation is adequate. A non-uniform asymptotic approximation to (3.1.17) is given as ˆ r (θ) + χ(ξR , θ) uR d ˆ r (ξR ), ur ∼ ug d where

kr → ∞,

(3.1.18)

χ(ξR , θ) =

1 if C(θ) encloses ξR ,

0 otherwise.

ˆ r (φ) is given by (3.1.10) with ξ replaced by θ; The polarization vector d ˆ r (ξR ) is given by (3.1.10) with ξ similarly, the polarization vector d replaced by ξR . The specularly reﬂected spherical wave ug is given as ug = ikA R(φ)

eikr . kr

(3.1.19)

Note that the condition that kρ → ∞, used in (3.1.17), does not inﬂuence the ﬁnal outcome describing ug ; that is, (3.1.19) remains accurate even as ρ → 0. Describing uR requires somewhat more detail. The leaky Rayleigh pole is ξR = βR + iαR ,

βR , αR > 0.

(3.1.20)

In all cases of interest here, αR /βR 1. Somekh et al. (1985) show that A− (ξR ) ≈ 2iαR . dA+ /dξ(ξR )

(3.1.21)

This approximation will be used frequently. Moreover, except in the phase terms, the approximation ξR ≈ βR is used. Again noting the ˆ R, ˆ R and d various unit vectors introduced in (3.1.10), the unit vectors p given by the expressions ˆ − cos βR e ˆ3 , ˆ R = sin βR e p

ˆR = e ˆR ∧d ˆT , p

ˆ r (ξR ) ≈ p ˆ R (βR ). ˆT is deﬁned by (3.1.11). Note that d are introduced; e Incorporating these approximations and the newly deﬁned unit vectors, uR can be expressed as

3.1 The scattering of a spherical wave

61

Reflected spherical wave (0,0,–b)

βR Leaky Rayleigh wave x||

θ (0,0,b) x3

Fig. 3.4. A drawing indicating the nature of the leaky Rayleigh wave excited by a ray incident at the angle βR . Also shown is the ray describing the reﬂected wave being emitted from the image point (0, 0, b).

R

u = kA

2π krR

1/2

× eikb cos βR

2αR e−iπ/4 eikˆp

cosh αR

R

·x cosh αR

ˆ R ·x sinh αR

ek d

ekb sin βR

sinh αR

,

(3.1.22)

where rR = ρ/ sin βR . Equation (3.1.22) describes an inhomogeneous wave propagating away ˆ R , and decaying in the direction from the interface in the direction p R ˆ R is removed ˆ . The possibility of unlimited growth in the direction d −d from ur , (3.1.16), by the indicator function χ(ξR , φ). Viewing it in the coordinates indicated in Fig. 3.1, (3.1.22) describes a Rayleigh surface wave that steadily radiates, or leaks, into the ﬂuid so that, as it propagates along the interface, it also decays. Further, it only appears outside a right circular cone whose vertex is at (0, 0, b) and which opens upward, cutting the surface at x3 = 0 in a circle of radius b cot βR . Figure 3.4 is a drawing that attempts to indicate these relationships.

3.1.5 Summary of §3.1 (1) Equations (3.1.18), (3.1.19), and (3.1.22) are the principal results of this section. They describe the most important waves scattered from the ﬂuid–solid interface, namely, the specularly reﬂected spherical wave and the leaky Rayleigh wave. (2) By using the angular spectrum representation of the incident wave, (3.1.6), representations of the reﬂected waveﬁeld, (3.1.14), and

62

Canonical elastic-wave problems

the transmitted waveﬁelds, (3.1.15), are readily calculated, and expressed in a form that indicates the underlying role of the scattering of a plane wave. Moreover, these representations indicate that many of the physical outcomes of the problem can be inferred directly from the analytic behavior of the reﬂection and transmission coeﬃcients. The general similarity between the expressions of §2.1.3 and §3.1.2 should be noted. (3) Representations of the spherical wave as an integral over cylindrical waves, (3.1.8) and (3.1.9), are also given. Equation (3.1.8) exhibits the spherical wave as an inverse Hankel transform; (3.1.9) proves quite useful in calculating the asymptotic approximation begun with (3.1.14).

3.2 Rayleigh–Lamb modes and Rayleigh surface waves 3.2.1 Introduction Rayleigh–Lamb modes are described in several books: Achenbach (1973, pp. 220–236), Auld (1990b, pp. 76–94), Brekhovskikh and Goncharov (1985, pp. 75–86), and Miklowitz (1978, pp. 178–209). They are inplane elastic waves that are guided in a uniform, two-dimensional waveguide with traction-free surfaces. The equations describing two-dimensional, inplane, elastic waves are obtained from (1.1.18) by setting u3 and all derivatives ∂3 to zero; that is, the only particle displacement occurs in the (x1 , x2 ) plane and the independent spatial variables are (x1 , x2 ).

3.2.2 Notation The wavenumber kT = ω/cT and is used, initially, to scale lengths; ω is the angular frequency and cT is the shear wavespeed. The Rayleigh wavespeed cR is approximately 0.9cT , so that at high frequencies the shear-wave wavelength approximates the Rayleigh-wave wavelength. Upper-case letters represent the lengths and lower-case letters the scaled lengths, unless otherwise indicated. The coordinates are xα = kT Xα , the particle displacement components are uα = kT Uα , and the nominal thickness of the waveguide is 2h0 = 2kT H0 . The shear coeﬃcient μ is used to scale the stress components: τ1 = τ11 /μ, τ2 = τ12 /μ, and τ3 = τ22 /μ, where the ταβ are the components. The subscripts α, β = 1, 2. Note: this is the only place in this section

3.2 Rayleigh–Lamb modes and Rayleigh surface waves

63

where two subscripts are needed; elsewhere, τp α indicates the pth mode, α component. It is useful to introduce combinations of elastic parameters that will arise frequently. These are: a = λ/(λ + 2μ),

b = μ/(λ + 2μ),

c = 4(a + b).

(3.2.1)

Also of use is the relation τ3 = aτ1 + c ∂2 u2 .

(3.2.2)

3.2.3 Rayleigh–Lamb modes The two lowest Rayleigh–Lamb modes are given by the following expressions, which are taken from Brekhovskikh and Goncharov (1985, pp. 75–86). The lowest antisymmetric mode ua is given as ⎤ ⎡ γL γT sinh(γT x2 ) sinh(γL x2 ) ⎤ ⎡ − ⎢ cosh(γL h0 ) ua 1 (x2 ) pβa cosh(γT h0 ) ⎥ ⎥ ⎢ ⎥ ⎦ = Ca ⎢ ⎣ ⎢ ⎥ , (3.2.3) ⎣ −iγL cosh(γL x2 ) βa cosh(γT x2 ) ⎦ ua 2 (x2 ) − βa cosh(γL h0 ) p cosh(γT h0 ) while the lowest symmetric mode us is given as ⎤ ⎡ cosh(γL x2 ) γL γT cosh(γT x2 ) ⎡ ⎤ − ⎢ sinh(γL h0 ) pβs sinh(γT h0 ) ⎥ us 1 (x2 ) ⎥ ⎢ ⎥ ⎣ ⎦ = Cs ⎢ ⎥ . ⎢ ⎣ x ) x β sinh(γ sinh(γ −iγ L L 2 s T 2 ⎦ us 2 (x2 ) − βs sinh(γL h0 ) p sinh(γT h0 )

(3.2.4)

In (3.2.3), βa is substituted wherever needed; in (3.2.4), βs is substituted wherever needed and Ca , Cs are arbitrary constants. The equations whose solutions give the dispersion relation for each mode are γL γT tanh(γT,L h0 ) = p2 tanh(γL,T h0 ),

(3.2.5)

where the ﬁrst subscript gives the antisymmetric relation and the second the symmetric one. The radicals γL and γT are deﬁned as γL = (κ2 β 2 − 1)1/2 /κ,

γT = (β 2 − 1)1/2 ,

κ = cL /cT = b−1/2 ,

where cL is the compressional wavespeed; it is readily found that γL,T are real and positive. The term p is deﬁned as p = (β 2 − 1/2)/β.

64

Canonical elastic-wave problems

For these two lowest modes it is useful to express βa, s as βa = βR + a ,

βs = βR − s ,

a + s = 2,

(3.2.6)

where βR is the wavenumber of a Rayleigh surface wave, and 2 = (βa − βs ). In general, is not small and must be found from a knowledge ¯ 0 (or more simply h0 ) becomes of βa, s . However, as Fig. 3.5 suggests, as h large, a → s → , and is given by df −1 ¯0 ¯0 R 2 −2γL h −2γT h = 2p e −e . (3.2.7) dβ This approximation is taken from Brekhovskikh and Goncharov (1985, pp. 79–81). βR is used wherever β is needed. The approximation (3.2.7) ¯ 0 > 1 or, more approximately stated, provided is accurate provided γT h ¯ h0 1. The Rayleigh function, given as fR (β) = p2 − γL γT , has roots ±βR . The Rayleigh wavenumber βR is approximated (Auld, 1990b, p. 92) by

cL

cT

h0

cR

ε

β h0 Fig. 3.5. A sketch of the dispersion relation for the two lowest Rayleigh–Lamb modes. Recall that h0 = ωH0 /cT . Solid curve: antisymmetric mode, eigenvalue βa = βR + a . Short-dash, long-dash curve: symmetric mode, eigenvalue βs = βR − s . a + s = 2 measures the horizontal distance between the two curves. At large h0 , a = s = , where 1, as indicated in the ﬁgure. cL , cT , and cR are the compressional, shear, and Rayleigh wavespeeds, respectively; the long-dashed lines indicate the slopes cL /cT , 1, and cR /cT .

3.2 Rayleigh–Lamb modes and Rayleigh surface waves βR = ω/cR ,

65

cR (0.87 + 1.14ν) , = cT 1+ν

where ν = λ/ [2(λ + μ)]. For a uniform waveguide, a and s are constant, x ¯2 = x2 and ¯ 0 = h0 . Therefore, the sum of these two modes gives the net particle h displacement: (3.2.8) u = eiβR x1 C eia x1 ua + e−is x1 us , where Ca = Cs = C. At large h0 , it is very diﬃcult to excite just one mode because a, s ≈ 1. Hence, even for a uniform waveguide, both modes would be excited. Once excited, however, both modes propagate independently. Equation (3.2.8) becomes a wave propagating at the Rayleigh wavespeed, but with a modulated amplitude. Approximating (3.2.3) and (3.2.4) for large h0 would indicate that (3.2.8) has most of its amplitude near the upper surface and decays toward x2 = 0; its amplitude near the lower surface is negligible. It is eﬀectively a Rayleigh wave. When the term ‘Rayleigh surface wave’ was used in §3.1.1, this was the wave being referred to. If the propagation path is long enough, the two terms will move out of phase so that the Rayleigh wave will shift from the upper surface to the lower one; see Auld (1990b, pp. 93, 94) and Brekhovskikh and Goncharov (1985, pp. 79–81). Experimental conﬁrmation of this shifting is described by Ti et al. (1997).

3.2.4 A framework for elastic waveguide problems The equations of motion will be written in a form that is not found in the books cited in §1.7, though it is used by Harris and Block (2005), Kirrmann (1995), and Maupin (1988). This manner of writing the equations of motion was also used for antiplane shear problems by Malischewsky (1987, pp. 54–69). The purpose of ﬁnding an alternative way of writing the equations of motion, (1.1.18), is to eliminate secondorder derivatives of the form ∂1 ∂2 so that, when the pth eigenmode is sought, the eigenvalue βp will appear in an isolated way. The vector U := [u1 , u2 , τ1 , τ2 ]T is introduced, where the superscript T indicates the transpose. The equations of motion are then written as (L − ∂1 ) U = F,

(3.2.9)

66 where

Canonical elastic-wave problems ⎡

0 ⎢−∂2 L := ⎢ ⎣ −1 0

b −a ∂2 0 0 0 0 −c ∂22 − 1 −a ∂2

⎤ 0 1 ⎥ ⎥. −∂2 ⎦ 0

(3.2.10)

F is a vector containing body forces and is described by F = [0, 0, f1 , f2 ]T , where the fα = Fα /(kT μ). The Fα are the body-force components per unit mass. The constants a, b, and c are given by (3.2.1). This system of equations is derived by using both the equation of motion, (1.1.18), and the constitutive relation (1.1.16), and subsequently using (3.2.2) to eliminate τ3 from the equations. An inner product may be deﬁned as h0 ∗ vβ τβ − σα∗ uα dx2 , V, U := −i (3.2.11) −h0

where V = [v1 , v2 , σ1 , σ2 ]T ,

U = [u1 , u2 , τ1 , τ2 ]T .

Using this inner product, the following integration by parts relation is calculated: 0 . (3.2.12) LV, U + V, LU = i [(v1∗ τ2 − σ2∗ u1 ) + (v2∗ τ3 − σ3∗ u2 )] |−h 0

h

τ3 is given by (3.2.2) and σ3 is deﬁned similarly. For the moment, imagine that the waveguide is uniform, with thick¯ 0 . Consider a wave of the form ness 2h U = up (x2 )eiβp x1 . Substituting this into (3.2.9), with F set to zero, gives the following eigenvalue problem: Lup = iβp up , τp 2 = τp 3 = 0, at x2 = −h0 , x2 = h0 ,

(3.2.13)

where the x1 is ﬁxed. up is the eigenmode and βp the eigenvalue. The eigenmodes of (3.2.13) are numbered as follows: they occur in pairs and are labeled so that one member of each pair carries energy or decays exponentially in the +x1 direction, while the other member does so in the −x1 direction. Subscripts p are integers and take plus and

3.2 Rayleigh–Lamb modes and Rayleigh surface waves

67

minus values, but not 0; a plus value is taken for propagation in the +x1 direction and a minus is taken for propagation in the −x1 direction. Thus β−p = −βp . Equations (3.2.3) and (3.2.4) are two eigenmodes that satisfy (3.2.13). However, they have diﬀerent symmetries: β1 = βa satisﬁes the antisymmetric dispersion equation, β2 = βs the symmetric dispersion equation; both dispersion equations are given by (3.2.5). In the development of this section, subscripts 1, 2, . . . will be used to identify the eigenvalues βp , with no indication of the symmetry. Consider next two eigenmodes up and uq . Substituting them into (3.2.12) gives − i(βq∗ − βp )Pqp = 0,

Pqp := uq , up .

(3.2.14)

This is the orthogonality condition. Note that the eigenvalues can be complex, though in practice the two eigenvalues βa,s of interest are always real. A more precise description of the previous expressions is to state that the eigenmodes form a biorthogonal system; that −L, with the same boundary conditions as indicated in (3.2.13), is the adjoint operator; and that (3.2.14) is an expression of biorthogonality. It is assumed throughout this chapter that the up (x2 ) form a complete set, and thus a vector U(x2 ) can be uniquely expressed as cn un (x2 ). U(x2 ) = n

These issues are discussed further by Besserer and Malischewsky (2004) and Kirrmann (1995), and a general discussion of biorthogonal expansions is given by Herrera and Spence (1981). By taking the complex conjugate of (3.2.13), and giving close attention to the mode labeling previously noted, it can be shown that if βq∗ = βp , then u∗q = u−p . Moreover, by asking that the modes of (3.2.13) be symmetric upon reﬂection in x1 = constant, it can be shown that, if βq∗ = βp , then u∗q 1 = −up 1 , u∗q 2 = up 2 , τq∗1 = τp 1 , τq∗2 = −τp 2 , τq∗3 = τp 3 . (3.2.15) The modes are normalized so that the conditions (3.2.15) are satisﬁed. By setting Ca = Cs = i, the eigenmodes ua and us , (3.2.3) and (3.2.4) respectively, are normalized so that these conditions are satisﬁed.

68

Canonical elastic-wave problems 3.2.5 Excitation of a waveguide

In practical terms one is often interested in the excitation of a waveguide by a transducer that applies a known traction on one surface. The discussion of §3.2.3 draws on the connection between the lowest Rayleigh–Lamb modes and the Rayleigh wave; we now turn our attention to exciting such modes within a waveguide and draw upon the treatment given by Folguera and Harris (1998). By using a Rayleigh wave transducer one can strongly excite the lowest two Rayleigh-like modes. To see this, consider a ﬂat, straight waveguide and apply a localized traction on the upper surface, x2 = h0 , of the form τ3 = 0,

c∂2 u2 + aτ1 = AeikT βR X1 [H(X1 + a) − H(X1 − a)], (3.2.16)

where H(X1 ) is the Heaviside function and A is a constant. In the spirit of the eigenvalue problem (3.2.13) and the framework of §3.2.4, we solve (L − ∂1 )U = 0

(3.2.17)

subject to (3.2.16). We use the Green’s states, UG (α) , which for the inﬁnite waveguide satisfy T (L − ∂1 )UG (α) = [0, 0, δ1α , δ2α ] δ(x1 − x1 )δ(x2 − x2 ),

(3.2.18)

where δαβ is the Kronecker delta symbol and α, β = 1, 2. The subscript (α) delineates the two Green’s states that are used. The forcing is now a source located at (x1 , x2 ) and the boundary conditions for the Green’s state are homogeneous, i.e., G = 0, τ3(α)

G c∂2 uG 2(α) + aτ1(α) = 0.

(3.2.19)

The framework (3.2.12) leads to G∗ U, (L − ∂1 )UG∗ (α) + (L − ∂1 )U, U(α) = iu(x1 , x2 ) · δα δ(x1 − x1 ),

(3.2.20) where δα = (δ1α , δ2α ). We are now in a position to use the boundary conditions (3.2.16) and integrate (3.2.20) over x1 . Then, assuming that the waves are damped at inﬁnity, one arrives at kT a (iβR x1 ) uG dx1 . (3.2.21) uα (x1 , x2 ) = A 2(α) (x1 , kT h0 ; x1 , x2 )e −kT a

3.2 Rayleigh–Lamb modes and Rayleigh surface waves

69

As we are primarily exciting the two lowest Rayleigh–Lamb modes, the Green’s state can be expanded to

(−a) iβa |x1 −x1 | (−a) e U + Bα(−s) eiβs |x1 −x1 | U(−s) ]H(x1 − x1 ) UG (α) = [Bα

+ [Bα(a) eiβa |x1 −x1 | U(a) + Bα(s) eiβs |x1 −x1 | U(s) ]H(x1 − x1 ) + . . . , (3.2.22) where here the superscripts a, s denote antisymmetric and symmetric modes, and the associated + or − signs before these letters indicate that they propagate to the right or left, respectively. The waves propagate outwards from the source. These Green’s states involve constants, the B, that are identiﬁed from another application of the inner product framework: (a,s) = [0, 0, δα1 , δα2 ]T δ(x2 − x2 ), U(a,s) δ(x1 − x1 ). (L − ∂1 )UG (α) , U

(3.2.23) Integrating across x1 from x1 − 0 to x1 + 0, and using (3.2.14), gives Bα(±a,±s) = −iu∗(±a,±s) · δα /Pnn ,

(3.2.24)

where n is either a or s, depending on whether the mode is antisymmetric or symmetric. Putting all this together, using (3.2.21), the resulting displacement is u(x1 , x2 ) = −i2kT aA

sin(kT a) (iβR x1 ) e A(x1 , x2 ), kT a

(3.2.25)

with A(x1 , x2 ) =

u∗a2 (h0 ) u∗ (h0 ) ua (x2 )eix1 + s2 us (x2 )e−ix1 . Paa Pss

(3.2.26)

Here ua and us are the lowest antisymmetric and symmetric modes, respectively, as given in §3.2.3. Notably the factor 1/kT a ensures that, although higher-order mode contributions are present, and omitted in the analysis, the contribution from the lowest two modes dominates.

3.2.6 Summary of §3.2 (1) The modes that exist within a straight, uniform waveguide are the Rayleigh–Lamb modes which satisfy a dispersion relation (3.2.5). As the waveguide gets thicker the lowest two modes are perturbations from the Rayleigh surface wave that exists near the surface

70

Canonical elastic-wave problems

of an elastic halfspace. It is diﬃcult to excite these modes independently; the result is that a modulated Rayleigh wave will often be observed. (2) A useful framework for elastic waveguide problems in presented and applied to the excitation of a waveguide by a Rayleigh transducer. The resulting excitation is dominated by contributions from these lowest modes.

4 Radiation and impedance

Radiation from transducers is deﬁned by the near ﬁeld, from which the far ﬁeld can be determined. The near ﬁeld of a general source can be represented as an integral over the source surface in terms of unknowns on the surface. The integral can be derived using the powerful argument that is known as reciprocity. The key ingredient in the integration is the fundamental solution for a point force. The purpose of this chapter is to introduce reciprocity and point-force solution, and to understand them through example. The fundamental solution, known as the Green’s tensor, is used to generate Gaussian beam solutions that are commonly used in practical simulation of transducer ﬁelds. Reciprocity is used to consider the elastic ﬁelds generated by the motion of a ﬁnite-sized particle and, at the same time, to ﬁnd the force on a particle due to plane-wave incidence. A common feature of the point-force solution and the particle radiation is the notion of impedance or its inverse, admittance. This is a direct measure of the energy loss through radiation, and can be interpreted in terms of both near- and far-ﬁeld eﬀects.

4.1 Reciprocity The time-harmonic equations of motion for two independent elastic waveﬁelds, indicated by the superscripts 1 and 2, are 1,2 + ρfi1,2 + ρω 2 u1,2 = 0. ∂j τji i

(4.1.1)

We consider solutions in a bounded region Rx with surface ∂Rx . The 1,2 ˆ to ∂Rx is outward. = τji n ˆ j . The unit normal n tractions on ∂Rx are t1,2 i Take the scalar product of each equation in (4.1.1) with the particle displacement of its reciprocating waveﬁeld, and subtract. This gives 2 1 ρ fi2 u1i − fi1 u2i = −u1i ∂j τji + u2i ∂j τji . (4.1.2) 71

72

Radiation and impedance

Using the relation between stress and strain, (1.1.3), we note that 1 2 2 1 τji i = τji ∂ j ui , so that the right-hand side of (4.1.2) equals ∂j u 1 2 2 1 ui − τij ui . The fundamental statement of elastodynamic reci∂j τij procity follows after integration: 2 1 2 1 f · u − f 1 · u2 ρdV = u · t − u1 · t2 dS. (4.1.3) Rx

∂Rx

One special case of (4.1.3) is of particular interest: point forces at two points and the associated displacements at each point produced by the force at the other. Let us assume that the integral over ∂Rx vanishes and that f 1,2 = a1.2 δ(x − x1,2 ), where the a1,2 are constant vectors. Then the reciprocity statement becomes a2 · u1 (x2 ) = a1 · u2 (x1 ). This is a pointwise version of the general integral identity (4.1.3).

4.2 Green’s tensor We consider the time-harmonic Green’s tensor. The deﬁning equation ˆδ(x), and the associated displacement u follows from (1.1.4), with f = a satisﬁes ˆδ(x) = 0. c2L ∇∇ · u − c2T ∇ ∧ ∇ ∧ u + ω 2 u + a

(4.2.1)

The solution is well known and is obtained, for instance in Harris (2001), using transforms. Here we describe an alternative method based on the Helmholtz decomposition (1.1.19) for u. Using the identity ˆδ(x) = −∇2 a

ˆ a 4πr

= −∇∇ ·

ˆ ˆ a a +∇∧∇∧ , 4πr 4πr

where r = |x|, gives the pair of equations 2 ∇2 ϕ + kL ϕ=

1 1 ˆ·∇ , a 4πc2L r

∇2 ψ + kT2 ψ =

1 1 ˆ∧∇ . a 4πc2T r

These in turn suggest ˆ · ∇ΦL , ϕ=a

ˆ ∧ ∇ΦT , ψ=a

where ΦL and ΦT satisfy similar equations ∇2 ΦI + kI2 ΦI =

1 , 4πc2I r

I = L, T.

4.2 Green’s tensor

73

The solutions must be regular at the origin, be radially symmetric (depend only on r), and contain no incoming waves. Thus, ΦI =

1 1 − G(kI r), ω 2 4πr ω 2

G(kI r) =

1 ikI r e , 4πr

I = L, T.

(4.2.2)

The appearance of the non-radiating ﬁrst term might appear unusual, but recall that the individual potentials are not bound by the same radiation conditions that govern the Green’s tensor. Substituting back into the Helmholtz decomposition (1.1.19) for u, we ﬁnd that the 1/r terms exactly cancel a delta function singularity arising from the second G derivatives of the G functions. The end result is ui = Uik a ˆk , where UG is a second-order tensor, the Green’s tensor (Harris, 2001): UG = ω −2 kT2 G(kT r)I + ∇∇ G(kT r) − G(kL r) . (4.2.3) The identity a2 · u1 (x2 ) = a1 · u2 (x1 ), derived in §4.1 on the basis of the general reciprocity identity (4.1.3), is reﬂected in the symmetry of the Green’s tensor.

4.2.1 Point impedance The Green’s tensor is singular at the point of application of the point force. An expansion of (4.2.3) about r = 0 indicates that the leading term is of order 1/r, and it is identical to the static Green’s tensor UG0 = UG |ω=0 . Proceeding to the next term in the expansion of (4.2.3) yields iω 2 1 + UG (x) = UG0 (x) + I + O(r). 12π c3L c3T The particle velocity near the point of application is also singular. However, the time average of the power dissipated by the force is ﬁnite. Let us assume ρf = δ(x)ReF0 e−iωt ; then the average power radiated per period is equal to the power expended by the force ω P = lim x→0 2π

2π/ω

ReF0 e−iωt · v(x, t)dt,

0

where v(x, t) = Re − iωu(x)e−iωt is the particle velocity. Hence, P =

F0 F0∗ , 2Zr

(4.2.4)

74

Radiation and impedance

where Zr is the radiation impedance, 1 ω2 2 −1 + 3 . Zr = 12πρ c3L cT

(4.2.5)

The inverse, deﬁned by Ar = Zr−1 , is called the drive point admittance. These reﬂect the two-fold interpretation of Zr , or its inverse Ar , as deﬁning the radiation loss at inﬁnity, or the work done by the applied force at the point of application. These measures of power must be equivalent in the absence of material damping.

4.2.2 Complex source: Gaussian beams Let us consider the case of a point force applied at a complex source e3 ), b > 0. In order to understand the Green’s tensor point: f = F0 δ(x−ibˆ for this seemingly unusual forcing, we note that the length s = |x − ibˆ e3 | must be understood as a complex number: 1/2 2 1/2 = r − b2 − 2ibx3 , s = R2 + (x3 − ib)2 where R2 = x21 + x22 is the cylindrical radius about the axis in the x3 direction. We take the branch cut of the square-root function along the negative real axis, and note that the physical coordinates corresponding to this are the disc of radius b: R ≤ b, x3 = 0. We may associate each face of the disc with the √ diﬀerent branches of the square root, thus: s(R ≤ b, x3 = ±0) = ∓i b2 − R2 . The Green’s tensor is therefore discontinuous across the disc. The eﬀect of the complexiﬁcation of the source point is to introduce a ﬁnite region of discontinuity, the size of which grows as the source is moved further into complex space. The far-ﬁeld behavior of the beam follows from the asymptotic form of the complex length s = r −ib cos θ+O(r−1 ), where cos θ = x3 /r. Thus, u=

F0 −2 ikL r kL b cos θ ikT r kT b cos θ ˆ⊗x ˆ) ˆ⊗x ˆ + c−2 · cL e e e (I − x x T e 4πr (4.2.6) + O(r−2 ),

where ⊗ signiﬁes the tensor product. The amplitude in the forward direction θ = 0 is exponentially larger than in the opposite direction. Let us examine the behavior of the beams near the forward symmetry axis. Using cos θ = 1 − θ2 /2 + · · · with θ ≈ sin θ = R/r implies R2 R2 F0 −2 ikT r kT b 1− 2r2 ikL r kL b 1− 2r2 ˆ ˆ ˆ ˆ (I − x ⊗ x ) . · c−2 e e e e u≈ x ⊗ x + c T 4πr L

4.2 Green’s tensor

75

The Gaussian decay away from the central axis makes this type of solution useful in modeling piston-like sources with a suitable summation of beams, e.g. by taking a ﬁnite set of sources with diﬀerent values of F0 and b, say {Fi , bi }, i = 1, 2, . . . , n. The total power then follows from (4.2.6) as P =

n 2 ω2 1 Fi · F∗i 3 I0 (kL bi ) + 3 I0 (kT bi ) , 24πρ i=1 cL cT

where I0 is the modiﬁed Bessel function of order zero. This reduces to the point-force result of (4.2.4) as b → 0, by virtue of I0 (0) = 1. Wen and Breazeale (1988) describe a procedure for choosing a suitable basis set of Gaussian beam parameters for a piston-like transducer in an acoustic ﬂuid. The same basis set was used by Spies for transducers in elastically isotropic (Spies, 1994) and anisotropic (Spies, 1999) materials. The analytic nature of the complex point-source Green’s tensor make it suited for examining Gaussian beam-like interaction with surfaces and interfaces, and scattering from objects such as spheres.

4.2.3 Impedance of a particle The notions developed above, speciﬁcally reciprocity and Green’s tensors, are most powerful when applied to radiation and scattering from ﬁnite regions. We consider in detail the example of a spherical particle with a rigid surface that is free to move. This does not imply that the particle is itself rigid, only that its bounding surface moves as a rigid body. In fact we will speciﬁcally consider later the case of a particle of ﬁnite mass or, more generally, of given impedance. We ﬁrst consider the radiation caused by imposed motion of the particle, and at the same time calculate the forcing required. This replaces the notion of a point force with the realistic model of a forced particle, and its radiation. Let us assume the particle has radius a and moves rigidly so that its center oscillates according to u = u0 with, as usual, the e−iωt term omitted and understood. We are interested in ﬁnding the displacement in the exterior region r > a in an inﬁnitely extended solid. The force F required to maintain the motion must be in the same direction. This is the resultant of the tractions over the surface r = a. The solution was originally found by Oestreicher (1951), and we follow his general approach here. Let us assume the displacement is

76

Radiation and impedance

ui (x) = Uij (x)u0j , where, based on the form of the Green’s tensor (4.2.3), we take (4.2.7) U = −A∇∇G(kL r) + B kT2 I + ∇∇ G(kT r). This only depends on two constants, A and B, which are found by requiring that the displacement equals u0 everywhere on the surface r = a. Note that G(kI r) = (ikI /4π)h0 (kI r), where h0 is the spherical Hankel function of the ﬁrst kind. The ﬁrst three spherical Hankel functions are 2 h0 (z) =

1 iz e , iz

h1 (z) = −

i+z z2

eiz = −h0 (z),

h2 (z) = i

z + 3iz − 3 z3

Using these we may express the displacement as 3 ikL h1 (kL r) 0 1 0 u =A u − h2 (kL r) (u · x)ˆ x 4π kr r 3 0 ikT3 0 0 2h0 (kT r)u − h2 (kT r) u − (u · x)ˆ x . +B 12π r

eiz .

(4.2.8)

Two coupled equations are obtained by requiring u = u0 on r = a, then ˆ equating the u0 component on either side of (4.2.8) and setting the x component to zero. After some algebra, we can express the solutions as (Norris, 2008) Z + ZM Z + ZM 1 1 , B= 2 , A= 2 kL G(kL a) ZL + ZM kT G(kT a) ZT + ZM (4.2.9) where 4 ZM = iω πa3 ρ, 3 ZL = (iω)−1 4πa(λ + 2μ) (1 − ikL a), ZT = (iω)−1 4πaμ(1 − ikT a), and Z satisﬁes 3 1 2 = + . Z + ZM ZL + ZM ZT + ZM

(4.2.10)

This relation between the impedances is similar to the eﬀective impedance of a lumped parameter system. The right member of (4.2.10) is equivalent to (ZL in parallel with ZM ) in series with (ZT in parallel with ZM ) twice. The impedance parameter Z relates the displacement to the force acting on the sphere, according to

4.2 Green’s tensor

77

F = −iωZu0 .

(4.2.11)

This can be veriﬁed by ﬁrst noting that the traction on a spherical surface, ti = τij x ˆj , can in general be written ˆ λ(∇ · u) + (μ/r) ∇(x · u) + r∂r u − u . t=x The resultant of the traction is obviously in the same direction as the imposed oscillation. Therefore, integrating the inner product t · u0 over r = a gives, after considerable simpliﬁcation, −iωZ. The identity (4.2.11) then follows. The non-dimensional tensor or matrix U(x) relates the time-harmonic displacement of the particle of radius a to the displacement in the elastic material. It should reduce to the Green’s tensor if the particle is small or, what is essentially the same limit, when the frequency is small. The proper scaling is in fact kT a 1, meaning that the particle is sub-wavelength and acts eﬀectively like a point. Taking the limit of a vanishingly small particle yields U(x) → iωZρ−1 UG (x),

a → 0.

1

0.8

0.6

A 0.4

0.2

0

0

1

2

3

4

5

kLa

Fig. 4.1. The relative admittance A is the ratio of the real part of −1/Z to Ar of (4.2.5). The quantity Z is the impedance for a rigid spherical inclusion of radius a. The ﬁgure illustrates that the radiation loss from a given timeharmonic force applied to the rigid sphere is a decreasing function of a.

78

Radiation and impedance

In the same limit, the impedance Z simpliﬁes: 2 iω 1 1 1 ω2 iω 3 a 2 2 2 1 = + 2 − + 3 − + +O(a2 ). Z 12πρa c2L cT 12πρ c3L cT 36πρ c2L c2T In this form the impedance acts as three lumped elements in series: a stiﬀness, a damper, and a mass, respectively. Recall that lumped parameter elements in series combine as (Z1−1 + · · · + Zn−1 )−1 . The middle term in (4.2.12), associated with the damper, reproduces the point admittance result of (4.2.4) and (4.2.5); see Fig. 4.1.

4.3 Reciprocity revisited We may apply the general reciprocity formulation to a rigid particle, i.e. one with a bounding surface S that moves as a rigid body. In this particular case the ﬁelds u1 and u2 are constants for the surface integrals in (4.1.3). We again take ﬁeld 1 as the solution for the inclusion undergoing arbitrary rigid-body displacement, with f 1 = 0. Let ﬁeld 2 be the solution for a point force at x in the exterior: ρf 2 (y) = F δ(y − x). The solution u2 is in fact the Green’s function in the presence of the movable inclusion. We can, however, avoid explicit calculation of the Green’s function. The displacement u2S on the inclusion surface must be another rigid-body oscillation, and therefore the reciprocity identity becomes u1S ·

S

τ 2 dS − F · u1 (x) = u2S ·

S

τ 1 dS.

(4.3.1)

The integral involving τ 1 gives the resultant force −iωZu1S . For ﬁeld 2, let Fp denote the resultant caused by the point force at x, τ 2 dS = Fp . S

The displacement at s for ﬁeld 1 follows from the deﬁnition of the tensor U as u1 (x) = U(x)u1S . Using the fact that the rigid-body displacement u1S is arbitrary, we deduce that Fp = U(x)F − iωZu2S .

(4.3.2)

This contains two unknowns: the force and the displacement on the sphere. These are physically related by the nature of the material in the sphere. A wide class of particles is covered by assuming a general impedance ZP of the particle. This could be, at the simplest level, a mass-like impedance ZP = iωm, where m is the particle mass. More

4.4 Force on a particle from an elastic wave

79

generally, the particle impedance could include or model internal structure in the particle. Whatever the precise form, the particle undergoes rigid oscillation, and the equilibrium condition (Newton’s second law) is Fp = iωZP u2S .

(4.3.3)

Eliminating the linear displacement between (4.3.2) and (4.3.3) gives Fp = ZP (Z + ZP )−1 U(x)F.

(4.3.4)

This is the desired result: it provides an expression for the force acting on the particle, caused by the point force in the elastic solid. Note that it involves the fundamental matrix U that was previously interpreted as a generalized Green’s tensor for a force applied to the particle of size a. Equation (4.3.4) is a complementary result, giving the force on the ﬁnite-sized particle, caused by a point force elsewhere.

4.4 Force on a particle from an elastic wave The force on a particle due a remote point load is given directly by the tensor ZP (Z + ZP )−1 U of (4.3.4). Taking the source point to inﬁnity, the eﬀect on the particle is equivalent to a combination of two incident longitudinal and transverse plane waves. The far-ﬁeld form of the tensor follows from (4.2.7) and (4.2.9) as Z + ZM a eikL (r−a) ZP ˆ⊗x ˆ U(x) = ZP x Z + ZP Z + ZP r ZL + ZM eikT (r−a) ˆ⊗x ˆ + O(r−2 ). I−x (4.4.1) + Z T + ZM Consider, for instance, a unit point force in the far ﬁeld at x in the direction n = −ˆ x . This produces a longitudinal plane wave at the origin of the form u = u0 neikL n·x , where u0 = eikL r /(4πμκ2 r). The force on the spherical particle due to an incident longitudinal plane wave u(x) = eikL n·x u0 is therefore Z + ZM 4πaZP e−ikL a u0 . (4.4.2) Fp = (λ + 2μ) Z + ZP ZL + ZM In the same manner, the force on the spherical particle due to an incident transverse plane wave u(x) = eikT n·x u0 , with u0 · n = 0, is Z + ZM 4πaZP e−ikT a u0 . (4.4.3) Fp = μ Z + ZP ZT + ZM

80

Radiation and impedance

A rigid, immovable particle is obtained in the limit as ZP → ∞. The values of the plane-wave-induced forces for the rigid, immovable particle coincide as the frequency tends to zero, the static limit. Despite the relatively simple formulae for the force on the particle from wave incidence, these expressions would be diﬃcult to obtain without the use of arguments based on the reciprocity identity. Related results and other applications of these ideas to particles in elastic solids are given in Norris (2008).

5 Integral equations for crack scattering

This chapter is concerned with cracks. Real cracks in solids are complicated: they are thin cavities, their two faces may touch, and the faces may be rough. We consider ideal cracks. By deﬁnition, such a crack is modeled by a smooth open surface Ω (such as a disc or a spherical cap); the elastic displacement is discontinuous across Ω, and the traction vanishes on both sides of Ω (so that the crack is seen as a cavity of zero volume). We suppose that we have one crack with a smooth edge, ∂Ω, embedded in an inﬁnite, unbounded, three-dimensional solid. Extensions to multiple cracks, to cracks in two dimensions, to cracks in halfspaces or in bounded domains, or to cracks with less smoothness may be made, with varying degrees of diﬃculty. For a variety of applications, see the book by Zhang and Gross (1998). After formulating our scattering problem, we give the governing hypersingular integral equation in §5.2. This equation is solved approximately for long waves (low-frequency scattering) in §5.3. The approach used is elevated to a well-known ‘strategy’ in §5.4 prior to further applications. For ﬂat cracks and screens, we can simplify the governing hypersingular integral equation. This is done in §5.5. Alternatively, we can use a direct approach, using Fourier transforms; see §5.6. Methods for solving the resulting equations are discussed in §5.7. The ﬁnal section is concerned with curved cracks and screens. Results for cracks that are almost ﬂat are described. 5.1 Formulation To keep the analysis relatively simple, we shall focus on analogous scalar problems coming from acoustics. Thus, we suppose that Ω is a thin screen in a compressible ﬂuid. The screen is hard (or rigid), which means that we have a Neumann boundary condition. 81

82

Integral equations for crack scattering

We are interested in scattering time-harmonic waves by the screen. Much is known about how to calculate scattering from objects of nonzero volume (Martin, 2006). Except in a few special cases (such as scattering by a sphere), it is usual to derive and solve (numerically) a boundary integral equation over the boundary of the scatterer. However, special methods are needed for zero-volume obstacles such as cracks and screens. In particular, the Neumann boundary condition means that it is inevitable that we shall encounter hypersingular boundary integral equations over the screen. These equations can be tackled directly (using boundary elements, perhaps), or they may be recast into other equivalent forms. For example, if the screen is ﬂat, various simpliﬁcations can be made. Integral equations can also be used as the basis for various approximation schemes. We consider acoustic scattering by a thin rigid screen Ω surrounded by a compressible ﬂuid; we model the screen as a smooth simply-connected bounded surface with a smooth edge ∂Ω. We write the scattered ﬁeld as Re{usc e−iωt }, where ω is the circular frequency. Then, usc solves the Helmholtz equation in three dimensions, ∇2 usc + k 2 usc = 0 in the ﬂuid,

(5.1.1)

the Sommerfeld radiation condition at inﬁnity, and the boundary condition ∂uin ∂usc + = 0 on Ω. (5.1.2) ∂n ∂n In addition usc is required to be bounded everywhere: we do not permit sources on ∂Ω. Here, k = ω/c, c is the constant speed of sound, uin is the given incident ﬁeld, and ∂/∂n denotes normal diﬀerentiation. The total ﬁeld is u = usc + uin , so that (5.1.2) gives ∂u/∂n = 0 on Ω. Denote the two sides of Ω by Ω+ and Ω− , and deﬁne the unit normal vector on Ω, n, to point from Ω+ into the ﬂuid. Then, deﬁne the discontinuity in u across Ω by [u(q)] = lim+ u(Q) − lim− u(Q), Q→q

Q→q

where q ∈ Ω, q ± ∈ Ω± , and Q is a point in the ﬂuid. Notice that [u] = [usc ] as [uin ] = 0. The scattered ﬁeld has the integral representation 1 ∂ [u(q)] G(P, q) dSq , (5.1.3) usc (P ) = 4π Ω ∂nq

5.1 Formulation

83

where G(P, q) = R−1 exp (ikR)

(5.1.4)

is the free-space Green’s function and R is the distance between P and q ∈ Ω. To be more explicit, we introduce Cartesian coordinates Oxyz and suppose that the surface Ω is given by Ω : z = S(x, y),

(x, y) ∈ D,

where D is a region in the xy-plane with edge ∂D. We deﬁne a normal vector to Ω by N = (−∂S/∂x, −∂S/∂y, 1), and then n = N/|N| is a unit normal vector. Also, if uin represents an incident plane wave, then uin (x, y, z) = eik(xα1 +yα2 +zα3 ) ,

(5.1.5)

where α12 + α22 + α32 = 1. Suppose that P has position vector r = (x0 , y0 , z0 ) and q ∈ Ω has position vector q = (x, y, S(x, y)). Let [u(q)] = w(x, y). Then, we ﬁnd that (5.1.3) becomes, exactly, 1 dA w(x, y) (N(q) · R2 )(1 − ikR2 ) eikR2 3 , usc (x0 , y0 , z0 ) = 4π D R2 where dA = dx dy, R2 = r − q, and R2 = |R2 |. In the far ﬁeld, we have usc (P ) ∼ r−1 eikr f (ˆr)

as r → ∞,

where r = |r|, ˆr = r/r and k [u(q)] {ˆr · n(q)} exp (−ikq · ˆr) dSq f (ˆr) = 4πi Ω k = w(x, y) {ˆr · N(q)} exp (−ikq · ˆr) dA; 4πi D

(5.1.6) (5.1.7)

f is the far-ﬁeld pattern. The formula (5.1.7) is exact. Although the integration is over a ﬂat region, the geometry of the screen enters through w, N, and q. Thus, we can expect that reasonable approximations to w will generate good approximations to f .

84

Integral equations for crack scattering

For example, in some applications, a static approximation to w (or [u]) could be used; this idea, leading to low-frequency approximations, will be developed in §5.3. For another example, we might be able to use high-frequency approximations for [u]. The most popular of these is the Kirchhoﬀ approximation, [u] uK , where uK (q) is the total ﬁeld at q when the incident ﬁeld is reﬂected by an inﬁnite ﬂat plane perpendicular to n(q). This approximation is used widely in models of ultrasonic non-destructive evaluation; see, for example, Schmerr and Song (2007, chapter 10).

5.2 A hypersingular integral equation Application of the boundary condition (5.1.2) to (5.1.3) gives ∂2 ∂uin 1 G(p, q) dSq = − , p ∈ Ω, × [u(q)] 4π Ω ∂np ∂nq ∂np

(5.2.1)

where the integral must be interpreted in the ﬁnite-part sense. Equation (5.2.1) is the governing hypersingular integral equation for [u]; it is to be solved subject to the edge condition [u(q)] = 0 for all q ∈ ∂Ω.

(5.2.2)

Equation (5.2.1) could be solved numerically, using boundary elements combined with regularization techniques. Indeed, this is probably the only option if Ω has a complicated, three-dimensional shape. As a sample, we cite a paper by Tada et al. (2000), in which a formulation for transient elastodynamics and non-planar cracks is developed. The literature on methods for solving hypersingular equations numerically is extensive. ˆ · n(p) when uin is The right-hand side of (5.2.1) reduces to −ikuin α given by (5.1.5); here, ˆ = (α1 , α2 , α3 ) α is a unit vector giving the direction in which the incident plane wave is propagating. What does ‘hypersingular’ mean? This question can be answered precisely, using the notion of pseudodiﬀerential operators acting between function spaces. However, for many purposes, it is enough to gain intuition through simple examples. Suppose we have an expression Lf , where L is a linear operator and f is a function. If L is an integral operator with a weakly singular kernel, Lf will be smoother than f : we

5.2 A hypersingular integral equation

85

usually think of integration as a smoothing process. If L is an integral operator deﬁned by a Cauchy principal-value integral (a singular integral operator), Lf will have the same smoothness as f : in some sense, L is similar to the identity operator, I. Hypersingular operators coarsen. If L is such an operator, Lf will have less smoothness than f : in some sense, L is similar to a diﬀerential operator, and this operator will be of ﬁrst order in our applications. We usually identify hypersingular operators in one of two ways. One is as in (5.2.1): the integral operator is deﬁned in terms of a ﬁnite-part integral (and does not exist as an improper integral or as a Cauchy principal-value integral). The second way involves Fourier transforms. Roughly, we can write (locally) Lf = F −1 {σF{f }}, where σ is called the symbol and F denotes Fourier transformation. Then, if σ is a linear function of the transform variable(s), L is a ﬁrst-order diﬀerential operator; if σ is linear for large values of the transform variable(s), then L will be identiﬁed as one of our hypersingular operators – we will see examples in §5.6. Returning to (5.2.1), let us project onto D (as in §5.1), giving 1 × K(x0 , y0 ; x, y) w(x, y) dA = b(x0 , y0 ), (x0 , y0 ) ∈ D, (5.2.3) 4π D where K = R1−3 (1 − ikR1 ) eikR1 {N(p) · N(q)}

− R1−5 (3 − 3ikR1 − k 2 R12 ) eikR1 (N(p) · R1 )(N(q) · R1 ),

R1 = (x − x0 , y − y0 , S(x, y) − S(x0 , y0 )), R1 = |R1 |, and ˆ exp (ikq · α) ˆ b(x, y) = −∂uin /∂N = −ikN · α

(5.2.4)

when uin is given by (5.1.5). Notice that K(x0 , y0 ; x, y) = K(x, y; x0 , y0 ). Equation (5.2.3) is to be solved subject to the edge condition w(x, y) = 0 for all points (x, y) on ∂D.

(5.2.5)

Let S1 = ∂S/∂x and S2 = ∂S/∂y,

evaluated at (x, y),

(5.2.6)

with S10 and S20 being the corresponding quantities at (x0 , y0 ). Then N(q) = (−S1 , −S2 , 1) and N(p) = (−S10 , −S20 , 1). Let R = {(x − x0 )2 + (y − y0 )2 }1/2

(5.2.7)

86

Integral equations for crack scattering

and Λ = {S(x, y) − S(x0 , y0 )}/R. Also, deﬁne the angle Θ by x − x0 = R cos Θ and

y − y0 = R sin Θ,

whence R1 = R(cos Θ, sin Θ, Λ). Hence eikRX 1 − ikRX K= (1 + S1 S10 + S2 S20 ) R3 X3 Y 2 − 5 (3 − 3ikRX − (kRX) ) , X √ where X = 1 + Λ2 and

(5.2.8)

Y = (S1 cos Θ + S2 sin Θ − Λ)(S10 cos Θ + S20 sin Θ − Λ). This formula for K is exact. If we expand K for small R about p, we ﬁnd that K ∼ R−3 σ(p; Θ),

(5.2.9)

where σ(p; Θ) =

1 + (S10 )2 + (S20 )2 . 1 + (S10 cos Θ + S20 sin Θ)2

In particular, σ ≡ 1 when S is constant. Equation (5.2.9) exhibits the strong singularity in the kernel K, and is typical of hypersingular operators deﬁned over surfaces.

5.3 Low-frequency scattering Before the advent of computers, it was traditional to obtain approximate solutions of scattering problems by assuming that the frequency is low, so that the crack is assumed to be small compared to the wavelength of the incident ﬁeld. These approximations are still useful today. In three dimensions, it is known that the scattered ﬁeld is an analytic function of k: it has a Maclaurin expansion with respect to k. Thus, associated static problems (where k = 0) feature. The basic static problem is as follows. Let φa denote the velocity potential of a given ambient ﬂow. Then, we seek another potential φ, where φ is a bounded solution of ∇2 φ = 0 in the ﬂuid, with ∂φ ∂φa + =0 ∂n ∂n

on Ω

and φ = O(r−1 ) as r → ∞. For a uniform ambient ﬂow, we have φa (x, y, z) = xα1 + yα2 + zα3 .

(5.3.1)

5.3 Low-frequency scattering

87

In general, we can ﬁnd φ by solving a hypersingular integral equation analogous to (5.2.1), namely ∂2 ∂φa 1 × [φ(q)] G0 (p, q) dSq = − , p ∈ Ω, (5.3.2) 4π Ω ∂np ∂nq ∂np with [φ(q)] = 0 for all q ∈ ∂Ω. Here, G0 = R−1 is the static free-space Green’s function; see (5.1.4). The right-hand side of (5.3.2) reduces to ˆ · n(p) when φa is given by (5.3.1). −α Returning to (5.2.1), we seek solutions in powers of k. We can write G = G0 + kG1 + · · ·

and

uin = u0in + ku1in + · · · ,

and then (5.2.1) implies that [u] has a similar expansion, [u] = u0 + ku1 + · · · . Substituting and collecting up powers of k, we obtain a sequence of equations from which uj can be determined. The ﬁrst two are L0 u0 = b0 where

and

L0 u1 = b1 ,

1 ∂2 ∂u0 × u(q) L0 u = G0 (p, q) dSq , b0 (p) = − in , 4π Ω ∂np ∂nq ∂np 1 2 ∂u 1 ∂ × u0 (q) G1 (p, q) dSq , b1 (p) = − in − ∂np 4π Ω ∂np ∂nq

and uj = 0 on ∂Ω for j = 0, 1, 2, . . .. Thus, each uj is obtained by solving a certain static problem. The problem itself could be solved by any convenient method, not necessarily via the equation L0 uj = bj . For an incident plane wave, (5.1.5) gives uin (x, y, z) = 1 + ik(xα1 + yα2 + zα3 ) + · · · . Thus, u0in ≡ 1 and b0 ≡ 0. As the equation L0 uj = bj is uniquely solvable (subject to uj = 0 on ∂Ω), we obtain u0 ≡ 0. Then, the equation for u1 reduces to ˆ · n(p). L0 u1 = −iα Hence, comparison with (5.3.2) gives u1 = i[φ] and [u] ik[φ]

for small k.

This approximation can then be inserted in (5.1.6) to give a lowfrequency approximation for the far-ﬁeld pattern.

88

Integral equations for crack scattering 5.4 Some strategies

In general, we encounter equations, such as (5.2.3), that we can write in operator form as Hw = b,

(5.4.1)

where H is a linear operator, b is known, and w is to be found. We write (5.4.1) as (L0 + L1 )w = b,

with

L1 = H − L0 ,

(5.4.2)

where L0 is another operator. If L0 is invertible, we obtain (I + M)w = g,

−1 with M = L−1 0 L1 and g = L0 b.

(5.4.3)

Several strategies for solving (5.4.1) follow this general pattern. To be eﬀective, L−1 should be available explicitly or it should be easier to 0 compute than H−1 . For example, if L0 is H evaluated at k = 0, we obtain a method with two virtues. First, the operator M will not be hypersingular: the operator H has been regularized. Second, we have access to analytical approximations for low-frequency scattering, as discussed in §5.3. This is because M is small in some norm, so that (I +M)w = g can be solved by iteration. For another example, if L0 is H evaluated for a simpler geometry, we obtain a method for perturbed screens (such as wrinkled discs). We shall see examples of these strategies later.

5.5 Flat cracks as a special case Almost all the literature on scattering by three-dimensional screens (and cracks) assumes that the screen is ﬂat. Thus, we assume that S(x, y) ≡ 0, whence D ≡ Ω. From (5.2.3) and (5.2.8), the governing hypersingular integral equation reduces to eikR 1 × (1 − ikR) 3 [u(x, y)] dA = b(x0 , y0 ), (x0 , y0 ) ∈ D, (5.5.1) 4π D R where R is deﬁned by (5.2.7) and b(x, y) = −∂uin /∂z Noticing that 1 (1 − ikR) eikR = R3

evaluated on z = 0.

∂2 ∂2 + + k2 2 ∂x ∂y 2

eikR = R

∂2 ∂2 + + k2 2 ∂x0 ∂y02

(5.5.2)

eikR , R

5.6 Flat cracks: direct approach we can rewrite (5.5.1) as 2 ∂2 eikR ∂ 2 dA = 4πb(x0 , y0 ), + + k [u(x, y)] 2 2 ∂x0 ∂y0 R D

89

(x0 , y0 ) ∈ D. (5.5.3)

This can be regarded as a kind of regularization, because the ﬁnitepart integral has gone, although it has been replaced by a diﬀerential operator. For an incident plane wave, (5.5.2) gives 2 ∂ 1 ∂2 ik(xα1 +yα2 ) 2 b(x, y) = −ikα3 e = + 2 + k eik(xα1 +yα2 ) ikα3 ∂x2 ∂y (for α3 = 0). Then, (5.5.3) can be integrated to give 4π ik(x0 α1 +y0 α2 ) eikR dA = [u(x, y)] e + Ψ0 (x0 , y0 ), R ikα 3 D where Ψ0 (x, y) solves 2 ∂2 ∂ 2 Ψ0 (x, y) = 0, + + k ∂x2 ∂y 2

(x0 , y0 ) ∈ D, (5.5.4)

(x, y) ∈ D.

Denote the left-hand side of (5.5.4) by S[u]; S is a single-layer operator. The equation Sw = g arises when the analogous scattering problem for a sound-soft screen (Dirichlet condition) is solved. It is known that, in general, the solution of Sw = g is inﬁnite around ∂D, whereas we want w to satisfy the edge condition (5.2.5). This condition can only be satisﬁed by making an appropriate choice for Ψ0 ; it is not clear how to make this choice in practice. It is known that S smooths by one order. Thus, the operator on the left-hand side of (5.5.3) coarsens by one order.

5.6 Flat cracks: direct approach Here, we assume from the outset that the crack or screen is ﬂat and lying in the xy-plane. To proceed, we use two-dimensional Fourier transforms. Thus, deﬁne ∞ ∞ usc (x, y, z) ei(ξx+ηy) dx dy U (ξ, η, z) = F{usc } = −∞

with inverse usc (x, y, z) = F −1 {U } =

1 (2π)2

−∞

∞

−∞

∞

−∞

U (ξ, η, z) e−i(ξx+ηy) dξ dη.

90

Integral equations for crack scattering

Transforming (5.1.1) gives (k 2 − ξ 2 − η 2 + ∂ 2 /∂z 2 )U = 0 with solution U (ξ, η, z) = A± (ξ, η) e±γz

for ±z > 0.

Here, A+ and A− are arbitrary functions, and γ is deﬁned as follows: √ γ=

2 − k2 , s√ s > k, −i k 2 − s2 , 0 ≤ s < k,

with s =

ξ2 + η2 .

This deﬁnition ensures that the radiation condition is satisﬁed. As ∂usc /∂z is continuous across z = 0 for all (x, y), we infer that A+ = − A− . This implies that usc (x, y, z) is an odd function of z, usc (x, y, −z) = −usc (x, y, z), so we can now assume that z ≥ 0 and write ∞ ∞ 1 A(ξ, η) e−γz e−i(ξx+ηy) dξ dη, z > 0. usc (x, y, z) = (2π)2 −∞ −∞ (5.6.1) Let us identify A in terms of [u]. From (5.6.1), we have 2 [u(x, y)] = (2π)2

∞

−∞

∞

−∞

A(ξ, η) e−i(ξx+ηy) dξ dη = 2F −1 {A},

so that 2A = F{[u]}. Explicitly, as [u(x, y)] = 0 for (x, y) ∈ D, we have 2A(ξ, η) = F{[u]} = [u(x, y)] ei(ξx+ηy) dx dy. (5.6.2) D

Then, application of the boundary condition (5.1.2) on z = 0+ yields 1 ∂uin = ∂z (2π)2

∞ −∞

∞

γA(ξ, η) e−i(ξx+ηy) dξ dη,

−∞

(x, y) ∈ D,

or, more concisely, − 12 F −1 {γF{[u]}} = b(x, y),

(x, y) ∈ D,

(5.6.3)

where b is deﬁned by (5.5.2). This is another equation for [u]; it should be compared with (5.5.1). Once solved, usc is given by (5.6.1) with (5.6.2). Equation (5.6.3) can be regarded as a hypersingular equation. This can be seen by noticing that γ ∼ s as s → ∞, so that the right-hand side of (5.6.3) is similar to a ﬁrst derivative of [u]. For an extensive review of the use of (5.6.3) for scattering computations, see Bostr¨om (2003).

5.7 Flat cracks: how to compute [u]

91

Write γ = s + β(s), where β = γ − s. Then, we can write (5.6.3) as (5.4.2), with L0 w = − 12 F −1 {sF{w}},

L1 w = − 12 F −1 {βF{w}},

and w = [u]. The operator L1 is similar to a single-layer operator: its symbol β ∼ − 21 k 2 /s as s → ∞. The operator L0 is hypersingular but it does not depend on k: it is the corresponding static operator. If D has a simple shape (such as a circular disc), L0 has an explicit inverse (subject to the edge condition on ∂D) and this can be used in order to obtain the regularized equation (5.4.3).

5.7 Flat cracks: how to compute [u] For a ﬂat screen D, we found two equations for [u], the discontinuity in u across D, namely (5.5.1) and (5.6.3); write these formally as H[u] = b. A familiar way to solve such equations is to expand [u] with a set of basis functions, writing [u(x, y)] = an un (x, y); n

evidently, the functions un have to be selected and then the coeﬃcients an have to be found. Substitution in H[u] = b gives an (Hun )(x, y) = b(x, y), (x, y) ∈ D, n

and then various methods (such as collocation) immediately suggest themselves for the numerical determination of an . For the expansion functions, un , one option is radial basis functions. Thus, choose N points (xn , yn ) ∈ D, n = 1, 2, . . . , N , put (x − xn )2 + (y − yn )2 , un (x, y) = χ 2

2

and then choose the function χ. Examples are χ(r) = e−r /c (Visscher, 1983) and χ(r) = (1 − r2 /c2 )α H(c − r) (Glushkov and Glushkova, 1996), where c and α are positive constants and H is the Heaviside unit function. Other choices could be made but, to be eﬀective, one should be able to compute Hun eﬃciently if not analytically. One virtue of radial basis functions is that they provide ﬂexibility: the shape of D is relatively unimportant. On the other hand, we know that [u] = 0 around the edge ∂D; in fact, we know that [u] must vanish as the square root of the distance from ∂D. This knowledge could be

92

Integral equations for crack scattering

incorporated by using special approximations near ∂D. However, if D is simple in shape, such as circular, elliptical, or rectangular, it is possible to construct functions un with the correct square-root edge behaviour. As an example, we consider a circular screen D of radius a. For this geometry, we have functions un satisfying L0 un = bn , where L0 is H at k = 0 and bn is known explicitly. Introduce plane polar coordinates (r, θ) so that the crack occupies 0 ≤ r < a. For simplicity, suppose that un (x, y) is an even function of x = r cos θ and put un (x, y) = awn (r/a) H(a − r) cos nθ. Then, a standard calculation (using the Jacobi expansion (Martin, 2006, p. 37) twice) gives ∞ 1 −1 3 F {γF{un }} = cos nθ a γJn (rs) wn (ρ) Jn (asρ) ρ dρ s ds, 0

0

where Jn is a Bessel function. This formula simpliﬁes if we expand wn (ρ) n (ρ), deﬁned by in a series of functions wm n+1/2 n (r) = rn C2m+1 ( 1 − r2 ), wm λ n where Cm is a Gegenbauer polynomial. Each function wm (r) is equal to √ 2 1√− r multiplied by a polynomial in r; in particular, w0n (r) = (2n + 1) rn 1 − r2 . Hence (Krenk, 1979), 1 2 Γ(n + m + 3/2) jn+2m+1 (as) n wm (r) Jn (asr) r dr = , Γ(n + 1/2) m! as 0

where jn is a spherical Bessel function. Then, writing n n [u(x, y)] = aH(a − r) Wm wm (r/a) cos nθ,

(5.7.1)

n=0 m=0

we ﬁnd that (H[u])(r, θ) =

n n Wm tm (r/a) cos nθ,

r < a,

n=0 m=0

where tnm (r/a)

Γ(n + m + 3/2) =− Γ(n + 1/2) m!

0

∞

a2 γJn (rs) jn+2m+1 (as) ds.

The remaining integral must be evaluated numerically. However, in the static limit (replace γ by s), we have (Bell, 1979) ∞ n Γ(m + 3/2) Γ(n + 1/2) wm (r/a) a2 sJn (rs) jn+2m+1 (as) ds = , (n + m)! 1 − r2 /a2 0

5.8 Curved cracks

93

a polynomial in r/a, and this gives the explicit evaluation of L0 [u]. Also, if n n wm (r/a) Bm cos nθ, b(r, θ) = 1 − r2 /a2 n=0 m=0 then the solution of L0 [u] = b is given by (5.7.1) with n =− Wm

n (n + m)! Bm ; Γ(n + m + 3/2) Γ(m + 3/2)

n the coeﬃcients Bm can be found using the orthogonality of the Gegenbauer polynomials, which gives 1 r dr n wln (r) wm (r) √ = hnm δlm , 1 − r2 0 where δij is the Kronecker delta and hnm is a known constant. Thus, in eﬀect, L−1 0 is available, and it can be used as in §5.4. If D is ﬂat but not circular, a possible strategy is the following. Find a conformal mapping that maps the interior of D onto the interior of a disc. Use this mapping in the integral equation (5.5.1); as it is a conformal mapping, it does not change the basic hypersingularity in the kernel, and so the dominant operator is the static operator L0 for the disc. The operator L1 will include some eﬀects due to the mapping and some due to the dynamics. The details have not been worked out, except for static problems (Martin, 1996).

5.8 Curved cracks If the crack or screen is not ﬂat, one may have to resort to solving (numerically) the hypersingular integral equation over the screen Ω, (5.2.1), or the version of this equation obtained by projection onto the ﬂat region D, (5.2.3). Obviously, care must be taken in handling the hypersingularity and with the edge condition, (5.2.2) or (5.2.5). The surface Ω is deﬁned by z = S(x, y) for (x, y) ∈ D. For nonconstant S, the singularity in the kernel of the integral equation (5.2.3) is essentially diﬀerent from that occurring in the integral equation for constant S; this is revealed by the presence of σ in (5.2.9). This diﬀerence means that the equation over D, (5.2.3), cannot be regularized using known results for ﬂat screens. However, we can make analytical progress when Ω is almost ﬂat. Suppose that S(x, y) = εf (x, y),

94

Integral equations for crack scattering

where ε is a small dimensionless parameter and f is independent of ε. Setting Λ = ελ,

with

λ = {f (x, y) − f (x0 , y0 )}/R,

(5.8.1)

we ﬁnd from (5.2.8) that K = R−3 eikR {1 − ikR + ε2 K2 + O(ε4 )}

as ε → 0,

where K2 = (1 − ikR)(f1 f10 + f2 f20 − 32 λ2 ) + 12 λ2 (kR)2

− 3(1 − ikR − 13 (kR)2 )(f1 cos Θ + f2 sin Θ − λ) (f10 cos Θ + f20 sin Θ − λ)

and fj , fj0 are deﬁned similarly to Sj ; see (5.2.6). We expand b similarly. For an incident plane wave, (5.2.4) gives b(x, y) = ik{b0 (x, y) + εb1 (x, y) + · · · }, where, for example, b0 (x, y) = −α3 eik(xα1 +yα2 ) . Then, if we expand w in (5.2.3) as w(x, y) = ik(w0 + εw1 + ε2 w2 + · · · ), we ﬁnd that Hk w0 = b0 ,

Hk w1 = b1 ,

where (Hk w)(x0 , y0 ) =

and

Hk w2 = b2 − K2 w0 ,

1 eikR × (1 − ikR) 3 w(x, y) dA 4π D R

is the basic hypersingular operator for acoustic scattering by a ﬂat soundhard screen D (see (5.5.1)) and 1 eikR × K2 (x, y; x0 , y0 ) 3 w(x, y) dA. (K2 w)(x0 , y0 ) = 4π D R Thus, we have a sequence of hypersingular integral equations, Hk wn = fn , to solve. When k = 0, w0 , w1 , and w2 have been found explicitly for particular geometries, namely inclined elliptical discs and spherical caps (Martin, 1998). The results obtained agree with known exact results. Similar results for perturbed penny-shaped cracks have also been obtained (Martin, 2001). For k > 0, we can see that w0 is simply the solution for a ﬂat screen; however, the far-ﬁeld pattern will be diﬀerent, it being given by (5.1.7)

5.8 Curved cracks

95

with w replaced by w0 . It should be possible to obtain w1 without too much diﬃculty, as f1 = b1 is simple. For higher-order terms, one would have to evaluate K2 w. If it is assumed further that the incident waves are long compared to the diameter of the scatterer, 2a, low-frequency approximations may be made. Then, each wn can be expanded in powers of ka. This approach has been pursued for a shallow crack in the shape of a spheroidal cap (Mikhas’kiv and Butrak, 2006). One diﬃculty with these approximation methods is that there are few results to compare them with. For acoustic scattering by spherical caps, see, for example, the papers by Miles (1971) and Thomas (1963). Numerical results for elastic-wave scattering by cracks in the shape of spherical and spheroidal caps have been given by Bostr¨om and Olsson (1987).

6 Scanned acoustic imaging

An angular spectrum representation of a converging spherical wave is derived, and then combined with a similar representation for a diverging spherical wave, to construct a model of a focused acoustic beam. This beam is directed at a ﬂuid–solid interface and the scattered waveﬁeld calculated, much as was done in §3.1. Representations of the incident and scattered waveﬁelds are also expressed as Hankel transforms. A model of the scanned acoustic microscope is next constructed. Using an electromechanical reciprocity relation, the following imaging equation is derived: βa G2 (ξ) R(ξ, s)ei2kzs cos ξ dξ. δV (s, zs ) = 2iD 0

This expression maps the mechanical reﬂection coeﬃcient R(ξ, s) to the measured change in voltage δV (s, zs ). This change depends on the lateral position of the microscope s, and the relative distance zs between the geometrical focal plane, and the interface that is being imaged. The function G(ξ) and angle βa describe how the ﬁnite aperture of the lens limits the acquisition of information. Varying s produces an image, while varying zs allows the device to be used as an interferometer. The chapter begins with a description of the acoustic microscope, and its deﬁning parameters, such as the F and Fresnel numbers. The model of a converging/diverging focused beam is then presented in detail, followed by the scattered version of the same in §6.5. Before dealing with the acoustic signature, some reciprocity identities are necessary; these are introduced in §6.6 with a sample application. The reciprocity is between pairs of elastic–electromagnetic ﬁelds, which allows determination of the received voltage response, culminating in the acoustic material signature, derived in §6.8. 96

6.1 Scanned, reﬂection acoustic microscope

97

6.1 Scanned, reﬂection acoustic microscope Figure 6.1 shows the acoustic microscope that will be modeled in this chapter. A piezoelectric transducer is placed at one end of a buﬀer rod and an acoustic lens is ground into the opposite end, as shown in Fig. 6.1(a). The transducer, when excited by a tone-burst, converts the incident electromagnetic wave into an elastic compressional wave that propagates down the buﬀer rod to a lens. It is transmitted through the lens into a coupling ﬂuid. If the overall region being scanned is small, the ﬂuid is little more than a drop at the end of the buﬀer rod, as suggested in the ﬁgure, while, if the region being scanned is large, then the whole device is immersed in the ﬂuid. The lens focuses the sound at or near the ﬂuid–solid interface. An echo is excited and subsequently collected by the lens. It propagates back through the buﬀer rod to the transducer, where it is converted into an electromagnetic wave. The buﬀer rod introduces a time delay between the initial excitation and the echo, allowing the two to be separated. The buﬀer rod, with its transducer and lens, is δV Lens aperture

Buffer rod

Reflected focal plane

Lens

b

f

x3 = 2zs

Interface

Fluid

x3 = zs

Focal plane

Solid

Geometric focal point

x1 x3 (a)

x1 βa x3 (b)

Fig. 6.1. (a) A drawing of the transducer, buﬀer rod, and lens that together constitute the acoustic microscope. Note that the origin of the coordinate system is placed at the geometrical focal point. (b) A drawing of the principal geometrical features of the region between the plane of the lens aperture and the geometric focal plane. Note that, in this arrangement, zs < 0.

98

Scanned acoustic imaging

mechanically scanned, in a raster pattern, across the interface. The echo at each point, after being converted to an electrical signal, is displayed as a dot on a screen, having a color or level of gray determined by its strength. The total of all the dots acquired during the scan is the image. As will become apparent, the microscope can also be used as an interferometer. As was shown in §2.3, a leaky Rayleigh surface wave is also excited at the interface. It is collected along with the reﬂected wave, and both waves are added together by the transducer. If the geometrical focal point is placed below the interface, in the solid, at a distance zs , as shown in Fig. 6.1(b), then the signal received from the transducer exhibits interference, between the reﬂected wave and the leaky Rayleigh wave, as zs is varied. If a discontinuity, perpendicular to the surface, is present near the surface, the leaky Rayleigh wave is reﬂected and transmitted by it. The reﬂected wave also radiates into the ﬂuid and combines with all the other signals collected by the lens. For simple geometrical conﬁgurations, ﬁxing zs and varying s in a direction perpendicular to the discontinuity causes the microscope to behave as a standing-wave tube. The work of this chapter is based on two papers by Rebinsky and Harris (1992a; 1992b), and a Ph.D. dissertation by Rebinsky (1991). Many of the ﬁgures shown in this chapter are taken from this dissertation. These references also discuss the role of the leaky Rayleigh wave in enhancing images of surface-breaking cracks, a topic discussed brieﬂy in §6.8. A somewhat broader overview of mathematically modeling acoustic imaging in solids is provided in the review article by Harris (1997). The monograph by Briggs (1992) gives a comprehensive overview of acoustic microscopy, at higher frequencies, along with many interesting photographs. Note that the derivation given in Briggs (1992, pp. 109–123) of the imaging equation diﬀers from that described here. The approach of Rebinsky and the writer was initially taken from Liang et al. (1985). A note on the word ‘microscope’. Throughout this chapter the imaging device is called an acoustic microscope. This is the common name; however, it is not the most descriptive one. The device described here is used in many settings. When used for non-destructive evaluation, the operating frequency is on the order of 10 to 200 MHz. Useful images are made, but their resolution is not so great as to merit calling the device a microscope. It only really behaves as a microscope at frequencies on the order of 500 MHz or higher. However, at these frequencies the material

6.3 Converging spherical wave

99

is either very attenuating, if it is isotropic, or it is anisotropic; isotropy cannot be assumed. Except for the case of a surface-breaking crack, only homogeneous isotropic materials are considered in this chapter; thus, the frequency range of interest is 10 to 200 MHz.

6.2 Fresnel and F number The work of Chapter 3 has shown that, at the Fresnel distance, a plane wave transmitted through an aperture and the waves diﬀracted by its edges begin to merge: it is here that a beam is formed. If b is the aperture radius, and k the wavenumber, the Fresnel distance is kb2 /(2π). Placing a lens in the aperture converts the transmitted plane wave into a spherical wave converging to a point, a distance f away; in this book f is the called the focal length.1 The waves diﬀracted by the edges continue to be present. The resulting waveﬁeld depends critically on whether focusing is made to occur before or after the edge-diﬀracted waves merge with the converging spherical one. The Fresnel number N = kb2 /(2πf ) is a measure of this, and is an important number characterizing an imaging system. A second one is the F number, where F = f /b. Note that kb/(2π) = N F . The scanned acoustic microscope considered in this chapter is such that N 1 and F ≈ 1. This makes the scaled radius of the aperture, kb, large; this is the large parameter that is used for the asymptotic calculations.

6.3 Converging spherical wave A mathematical description of a focused acoustic beam is constructed in this and the next section. To begin, the idealized case of a spherical wave converging to a point and then diverging to inﬁnity is ﬁrst described. Figure 6.2 indicates the geometry: the x3 axis is the centerline. A spherical wave in x3 < 0 converges to the origin and then, in x3 > 0, diverges to inﬁnity. In crossing the plane x3 = 0 it is made to undergo a phase jump of π. It is known that focused beams experience such a phase jump, though the phase variation occurs more gradually within a ﬁnite focal region (Born and Wolf, 1999, pp. 494–499). 1

The focal length can also be deﬁned as the distance from the center of the face of the acoustic lens to the focal point; this is the more common deﬁnition in acoustic microscopy.

100

Scanned acoustic imaging Converging spherical wave

ϕ = A e–iπe–ikρ/kρ

Focal plane x1 ϕ = A eikρ/kρ

Diverging spherical wave

x3

Fig. 6.2. The arrangement of the coordinate system. The form taken by the converging and diverging spherical waves on either side of the focal plane, at x3 = 0, is indicated. Note that there is a phase jump of π across this plane.

Equations (3.1.6) and (3.1.7), with b = 0, describe an outgoing or diverging spherical wave in x3 > 0. Equation (3.1.6) becomes, with b = 0, i A 2π π/2−i∞ ik pˆ i ·x e sin ξ dν dξ, (6.3.1) ϕ= 2π 0 0 ˆ i is given by (3.1.7) but is repeated here: where p ˆ1 + sin ν e ˆ2 ) + cos ξ e ˆ3 . ˆ i = sin ξ (cos ν e p At x3 = 0+ , ϕ reduces to ϕ=A

eikρ , kρ

where x1 = ρ cos θ and x2 = ρ sin θ. Equation (6.3.1) describes a diverging spherical wave in x3 > 0; the accessible regions of the ξ plane, as x3 → ∞, are those for which Re(cos ξ) ≥ 0 and Im(cos ξ) ≥ 0. Recall that k3 = k cos ξ, so that these conditions are equivalent to those in (3.1.5). To construct the incoming or converging spherical wave, in x3 < 0, ϕ is written as ∞ ∞ 1 dk1 dk2 ∗ ϕ= ϕ(k1 , k2 )ei(k1 x1 +k2 x2 +k3 x3 ) , (6.3.2) (2π)3 −∞ −∞ k3 where k3 = (k 2 − k12 − k22 )1/2 ,

Re(k3 ) ≥ 0, Im(k3 ) ≤ 0.

(6.3.3)

Note that the deﬁnition of k3 has changed from that given by (3.1.5). The changed deﬁnition ensures that, as x3 → −∞, the exponential term

6.4 Focused acoustic beam

101

decays or acts as an incoming plane wave. The transform ∗ ϕ is determined by asking that, at x3 = 0− , ϕ be given by the expression (see Fig. 6.2) ϕ = Ae−iπ

e−ikρ . kρ

It is found that ∗

ϕ=

i2πA . kk3

Again introducing the transformation k1 = k sin ξ cos ν,

k2 = k sin ξ sin ν,

k3 = k cos ξ,

the angular spectrum representation of the converging spherical wave is given as i A 2π π/2+i∞ ik pˆ i ·x e sin ξ dν dξ. (6.3.4) ϕ= 2π 0 0 The accessible parts of the ξ plane, as x3 → −∞, are determined by the deﬁnition of k3 , given by (6.3.3), which implies that Re(cos ξ) ≥ 0 and Im(cos ξ) ≤ 0. In fact the only apparent diﬀerence between the diverging spherical wave, (6.3.1), and the converging one, (6.3.4), arises from the deﬁnition of k3 and the contour of integration for ξ.

6.4 Focused acoustic beam A focused wave constructed from simultaneously converging and diverging spherical waves is not physically realizable. First, a spherically converging wave could not be excited because it would demand that all the plane waves in its spectrum, including those propagating at complex angles, be available at some initial aperture. Second, the waveﬁeld in the focal plane should be continuous and not have a jump in phase of π. Such a change is permissible in the sense that, between the source of the converging wave and the far ﬁeld of the diverging wave, a net phase change of π may occur, but the change should not cause a discontinuity at the focal plane. Using the knowledge gained from constructing the representations of the converging and diverging spherical waves, the following is proposed as a model of a focused acoustic beam: i i A 2π π/2 G(ξ)eik pˆ ·x sin ξ dν dξ. (6.4.1) ϕi = 2π 0 0

102

Scanned acoustic imaging

By eliminating the complex legs of the contours, the sudden jump at x3 = 0 has been eliminated, and by introducing a function G(ξ), the focal region at the origin has been modulated to capture more accurately the fact that radiation cannot be focused to a point. The function G(ξ) is deﬁned so that it is symmetric in its argument ξ and is non-zero for ξ ∈ (−π/2, π/2). Lastly, a superscript i has been added because, in subsequent sections, (6.4.1) will become the incident focused beam emitted from the lens aperture. Two additional representations of (6.4.1) are useful. Using the results from Appendix B, the ﬁrst is

π/2

ϕi = iA 0

G(ξ) eikx3 cos ξ J0 (kρ sin ξ) sin ξ dξ,

and the second is iA π/2 (1),(2) i G(ξ) eikx3 cos ξ H0 (kρ sin ξ) sin ξ dξ. ϕ = 2 −π/2

(6.4.2)

(6.4.3)

(1)

The Hankel function H0 is selected when x3 > 0. Its branch cut proceeds from 0 to (−π/2−i∞); the integration contour passes above it. The (2) Hankel function H0 is selected when x3 < 0. Its branch cut proceeds from 0 to (−π/2 + i∞); the integration contour passes below it. The function G(ξ) is speciﬁed by asking that, far from the focal plane, in the halfspace x3 < 0, ϕ match a prescribed waveﬁeld. This waveﬁeld is thought of as that which ﬁlls an aperture, of radius b, placed at a distance |x3 | = f from the focal plane. It is speciﬁed as ϕi = −H(b − ρ) A e(ρ/a)

2n

e−ikr , kr

(6.4.4)

where H(x) is the Heaviside step function and r = (ρ2 + x32 )1/2 . Note that the aperture is ﬁlled with a converging spherical wave. Recall that x1 = ρ cos θ and x2 = ρ sin θ. The coordinate φ has been introduced such that φ ∈ [0, π/2), ρ = r sin φ, and |x3 | = r cos φ. The parameters a and n, where n is a positive integer, allow one to describe how fully the incident waveﬁeld ﬁlls the aperture. Figure 6.3 shows three proﬁles for various values of a/b and n. The larger a/b, the more the waveﬁeld ﬁlls the aperture, and the larger n, the more the function resembles a top-hat. Equation (6.4.3) is asymptotically approximated as kr → ∞, using a procedure identical to that used in §2.4 to approximate the waveﬁeld scattered into the ﬂuid. This approximation gives

6.5 Scattered focused beam

103

2

n = 1, a/b = 0.50

|k2 ϕ i |

n = 2, a/b = 0.75

1

n = 4, a/b = 0.85 0

0

ρ /b

1

Fig. 6.3. The distribution of the waveﬁeld over an imagined aperture at |x3 | = f . A is selected arbitrarily to give the vertical scale shown. The parameters n and a/b measure how fully the converging spherical wave ﬁlls the aperture.

e∓ikr , kr → ∞, x3 ≶ 0. kr Matching (6.4.5) with (6.4.4) gives 2n f tan φ , φ ∈ [0, π/2). G(φ) = H(βa − φ) exp − a ϕi ∼ ∓A G(φ)

(6.4.5)

(6.4.6)

The angle βa is deﬁned by cot βa = F . Equations (6.4.1) and (6.4.6) together deﬁne the model of a focused beam to be used here. Note that the choice of the waveﬁeld in the aperture is such that edge-diﬀracted waves are made small. This choice was based on the observation that the microscope’s acoustic lens did not have sharp edges; accordingly, it was reasoned that only weak edge diﬀraction occurred.

6.5 Scattered focused beam 6.5.1 Representations The reﬂection of the focused beam is needed next. It is calculated in much the same way as was done in §3.1. Noting that the incident focused beam (6.4.1) is an integral over plane waves, an integral representation of the reﬂected focused beam can be written as r iA 2π π/2 G(ξ)R(ξ)eik pˆ ·x ei2kzs cos ξ sin ξ dν dξ. (6.5.1) ϕr = 2π 0 0

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Scanned acoustic imaging

ˆ r is given by (3.1.10), but is repeated here: The propagation vector p ˆ1 + sin ν e ˆ2 ) − cos ξ e ˆ3 . ˆ r = sin ξ(cos ν e p The plane-wave reﬂection coeﬃcient R(ξ) is given by (3.1.12) and (3.1.13). It is important to recall the branch cuts for R(ξ) in the ξ plane; these are discussed in the paragraph following (3.1.15) and partially indicated by Fig. 3.3. These cuts determine the Riemann sheet on which the contour of integration of (6.5.1) is taken; the integration contour passes below the branch cuts whose branch points are ξbL and ξbT . To interpret (6.5.1) it is helpful to refer to Fig. 6.1(b). First recall that the origin of the coordinate system for the focused beam is the geometrical focal point, which in Fig. 6.1(b) is a distance |zs | from the interface. If the incident beam strikes the interface before it focuses, then its reﬂection focuses in a reﬂected focal plane at x3 = 2zs , the mirror image of the focal plane. Combining the phase terms, a distance |x3 − 2zs | emerges, indicating that, for 2zs < x3 < zs , (6.5.1) represents a beam converging to (0, 0, 2zs ) while, for x3 < 2zs , it represents a beam diverging from (0, 0, 2zs ). Again two additional representations of (6.5.1) are useful. Using the results from Appendix A, the ﬁrst is similar to (6.4.2): βa ϕr = iA G(ξ)R(ξ) e−ik(x3 −2zs ) cos ξ J0 (kρ sin ξ) sin ξ dξ. (6.5.2) 0

The second is similar to (6.4.3). However, this representation will be used to asymptotically approximate the reﬂected waveﬁeld; thus, it is important to scale the variables to identify what parameter is large. The scaled variables ρ¯ = ρ/b, x ¯3 = x3 /f , and z¯s = zs /f are introduced. This gives the second representation as iA βa (1),(2) ϕr = G(ξ)R(ξ) e−ikbF (¯x3 −2¯zs ) cos ξ H0 (kb¯ ρ sin ξ) sin ξ dξ. 2 −βa (6.5.3) (1) H0

The Hankel function is selected when x3 < 2zs . In this case the (1) contour passes above the branch cut for H0 , whose cut proceeds from 0 to (−π/2 − i∞). It also passes above the branch cuts of R(ξ), whose branch points are −ξbL and −ξbT , and below those whose branch points are ξbL and ξbT . This is a beam diverging from the reﬂected focal plane. (2) The Hankel function H0 is selected when 2zs < x3 < zs . The contour passes below the branch cut for the Hankel function, which proceeds from 0 to (−π/2 + i∞). However, it passes above the branch cuts of

6.5 Scattered focused beam

105

R(ξ), whose branch points are −ξbL and −ξbT , and below those whose branch points are ξbL and ξbT . This is a beam converging toward the reﬂected focal plane. Recalling the discussion of §6.2, F ≈ 1 and kb 1. Therefore, the integral (6.5.3) can be asymptotically approximated in the same way as was ur in §3.1.4, though in this case kb has been explicitly identiﬁed as the large parameter. However, proceeding in this way will lead to an unexpected diﬃculty for the case of the reﬂected converging beam. It will become apparent that, when the reﬂected focal plane lies above it, leaky Rayleigh waves are excited at the interface that focus at ρ = 0. Therefore, the asymptotic approximation in this latter case must be modiﬁed somewhat from that described in §3.1.4.

6.5.2 Diverging scattered beam, x3 < 2zs (1)

Where x3 < 2zs , the asymptotic approximation for H0 , taken from Appendix A, is introduced into (6.5.3). As well, the spherical coordinates (r, φ, θ) are introduced by setting ρ = r sin φ and |x3 − 2zs | = r cos φ (θ never appears because the waveﬁelds are axisymmetric about the x3 axis). Note that φ ∈ [0, π/2). Equation (6.5.3) is thus approximated as iA ϕ ∼ 2 r

2 πkb¯ ρ

1/2 e

iπ/4

βa

G(ξ)R(ξ) e±ikbF r¯ cos(ξ−φ) sin1/2 ξ dξ,

−βa

kb → ∞, (6.5.4) where r¯ = r/f . Equation (6.5.4) is very similar to (3.1.15). The steepest-descents contour for this case is identical to that shown in Fig. 3.3(b). The contributions from the end points at ±βa are not signiﬁcant, primarily because G(βa ) is small. Carrying out the approximation gives ϕr ∼ ϕg + χd (ξR , φ) ϕR , where

χd (ξR , φ) =

kb → ∞,

1

if C(φ) encloses ξR ,

0

otherwise.

(6.5.5)

This is a non-uniform approximation in the sense indicated in §3.1.4. The specularly reﬂected beam ϕg is given as ϕg = AG(φ)R(φ)

eikr . kr

(6.5.6)

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Scanned acoustic imaging

Using Eqs. (3.1.20) and (3.1.21), and making approximations identical to those used to calculate (3.1.22), the leaky Rayleigh wave ϕR is given as ϕ = −AG(βR ) R

2π krR

1/2

2αR eiπ/4 eikr cos(ξR −φ) ,

(6.5.7)

where rR = ρ/ sin βR . The leaky Rayleigh-wave pole is deﬁned by (3.1.20). The approximation (3.1.21) and the approximation ξR ≈ βR have been used in the amplitude. The phase term can be expanded into real and imaginary parts to exhibit the fact that (6.5.7) is a leaky Rayleigh wave. Equations (6.5.5)–(6.5.7) describe a scattered waveﬁeld that consists of a diverging spherical wave emerging from r = 0 and a leaky Rayleigh wave radiating from the interface. Both these waves are collected by the aperture and transmitted to the transducer. Figure 6.4 shows |k 2 ϕr | as a function of the scaled radius ρ/b, at x3 = −f , the plane of the lens aperture. The spikes in the graph arise from the fact that the asymptotic approximation is not uniform. The oscillations to the right of the large spike indicate the presence of the leaky Rayleigh wave in the aperture. It is important to note that the transducer responds to this entire signal; it does not sense or separate the individual parts.

2

⏐k 2 ϕ r⏐

Leaky Rayleigh waves

1

0

0

ρ /b

1

Fig. 6.4. A plot of the magnitude of |k2 ϕr | against ρ/b. Note the presence of the leaky Rayleigh wave, starting at ρ/b ≈ 0.3, in the waveﬁeld ﬁlling the aperture. Compare this ﬁgure with the corresponding solid curve in Fig. 6.3.

6.5 Scattered focused beam

107

The various parameter values used in Fig. 6.4 are the following: 3

ρs = 2200 kg/m , cL = 5960 m/s, cT = 3760 m/s, 3

ρf = 998 kg/m , c = 1480 m/s, F = 0.75, N = 178, n = 1, a/b = 0.5.

(6.5.8)

The ﬁrst line lists the parameters describing the solid, and the second those describing the coupling ﬂuid; the solid is fused quartz and the ﬂuid water. The third line lists the parameters describing the lens and aperture. The frequency of operation is taken as 225 MHz. The leaky Rayleigh pole ξR is found numerically. The value of zs is chosen such that kzs /2π = −10. The constant A is given the same value as that used in Fig. 6.3.

6.5.3 Converging reﬂected beam, 2z s < x3 < z s To approximate asymptotically the converging beam, where 2zs < x3 < zs , (6.4.3) is again used. The spherical coordinates (r, φ, θ) are introduced, with φ ∈ [0, π/2). The contour in ξ is distorted to the contour (2) Re[cos(ξ − φ)] = 1 with no approximation of H0 being made. This will become the contour of steepest descents. Figure 6.5 shows the complete distortion of the original contour into this contour. The branch cut for (2) the H0 is not shown. The dashed lines indicate that the contour must be distorted onto a lower Riemann sheet in order for the contour to pass through the stationary point ξ = φ. This means that branch cuts are surrounded; however, these contributions are ignored. The pole terms are now extracted, giving ϕR = −i 2πA G(βR ) αR sin βR e−ikr cos φ cos ξR

! (1) (2) × H0 (kρ sin ξR ) − χc (ξR , φ)H0 (kρ sin ξR ) ,

where χc (ξR , φ) =

1 0

(6.5.9)

if C(φ) encloses ξR , otherwise,

and ρ = r cos φ. The leaky Rayleigh-wave pole is given by (3.1.20). The approximation (3.1.21) and the approximation ξR ≈ βR have been used in the amplitude. Note that the indicator χc (ξR , φ) sets the second term to zero when φ is greater than βR , whereas χd (ξR , φ) sets the corresponding term in (6.5.5) to zero when φ is less than βR . When φ is less then βR both Hankel functions are present, indicating that the leaky Rayleigh

108

Scanned acoustic imaging ξ plane

– π /2

–βa

π /2 φ

βa

Fig. 6.5. A sketch of the steepest-descents contour for the case where 2zs > (2) x3 > zs . The branch cut for H0 is not shown. The somewhat complicated path is necessary because the integration contour can be closed at inﬁnity only in the indicated quadrants.

waves are focusing on the interface and along the axis ρ = 0. Moreover, the ﬁrst Hankel function is always present because the pole −ξR is always captured. When φ is greater then βR this term represents an outgoing, cylindrical, leaky Rayleigh wave. Writing the pole terms as Hankel functions gives an expression that is accurate when ρ → 0. Once the pole terms have been extracted, the remaining integral can (2) be approximated by introducing the asymptotic approximation to H0 and noting that the contour of integration is that of steepest descents. The geometrical term is approximated as e−ikr . (6.5.10) kr This expression is accurate when 2zs < x3 < zs , provided zs = 0 or, equivalently, r = 0. Note the phase shift of π in passing from (6.5.10) to (6.5.6). ϕg ∼ −AG(φ)R(φ)

The asymptotic approximation to ϕr is therefore given as ϕr ∼ ϕg + ϕR ,

kb → ∞,

(6.5.11)

6.6 An electromechanical reciprocity relation

109

Aperture βa βR Outermost ray Geometrical ray

φ

Leaky Rayleigh rays x3 ⫽2zS

x3 ⫽zS

Fig. 6.6. A sketch of the rays describing the focusing scattered beam. The solid lines indicate the outermost geometrical rays, and the dashed ones the leaky Rayleigh waves.

where ϕR is given by (6.5.9), with no restriction on ρ, and ϕg by (6.5.10), with the restriction that r = 0. Accordingly, the overall expression is only accurate provided the reﬂected focal plane and the interface are separated. Moreover, (6.5.11) is a non-uniform approximation in the sense indicated in §3.1.4. While (6.5.11) is limited in its usefulness, it does provide a good physical picture of what is happening. Figure 6.6 is a sketch that attempts to illustrate this equation. The solid lines indicate the outermost rays of the focused cone of rays: they reﬂect from the interface and are focused in the reﬂected focal plane. The dashed lines indicate the leaky Rayleigh waves: within a right circular cone, which opens downward and ends at the interface, intersecting it to form a circle with radius |zs | tan βR , they focus on the axis ρ = 0 whereas, outside the cone, they form an outgoing cylindrical wave steadily leaking into the coupling ﬂuid.

6.6 An electromechanical reciprocity relation 6.6.1 Piezoelectric coupling and reciprocity Having considered the waveﬁeld in the ﬂuid and in the solid, we now turn to the question of how to combine this knowledge to determine the electric response of the transducer/microscope. This requires a model of the

110

Scanned acoustic imaging

measurement device which is described in the next section. Here the theory is described that enables us to combine the mechanical and electrical eﬀects. The key is reciprocity, and goes beyond the concepts introduced in Chapter 4 to include a fully coupled ﬁeld theory, as compared with the purely elastic reciprocity identities derived previously. Transducers operating at microwave frequencies frequently use piezoelectric coupling to convert electrical to mechanical signals and vice versa. While the equations of piezoelectricity are not explicitly used in this book, an electromechanical reciprocity identity, which assumes the coupling is piezoelectric, is used; and its use is central to the model of scanned acoustic imaging described in this chapter. To sketch its derivation, a summary of the equations of linear piezoelectricity is needed. A description of piezoelectricity can be found in Auld (1990a, pp. 265–298). The electromechanical reciprocity identity itself is also derived in Auld (1990b, pp. 153,154), and is further discussed in Achenbach (2003, pp. 233–246). Maxwell’s equations, eijk ∂j Ek + ∂t Bi = 0,

eijk ∂j Hk − ∂t Di = Ji ,

are coupled with (1.1.1) through the constitutive relations Di = κij Ej + ρijk jk , τij = −ρijk Ek + cijkl kl . In addition Bi = νHi . The terms E, B, H, and D are the electric intensity, the magnetic induction, the magnetic intensity, and electric displacement. The vector J is an imposed current; the piezoelectric materials are assumed to be non-conducting. Piezoelectric materials must be anisotropic; thus, the various coupling parameters are tensors of the order indicated by their subscripts: κij are the components of the dielectric permittivity tensor, ρijk the components of the piezoelectric tensor, and cijkl the components of the elastic tensor. The scalar ν is the magnetic permeability. All the terms are assumed to be constant. Using energy arguments and the symmetry of τ , it can be shown that cijkl = cklij , cijkl = cjikl , ρikl = ρkli , ρijk = ρjik , κij = κji . Other symmetries can be derived from these.

6.6 An electromechanical reciprocity relation

111

Let (E1 , H1 , τ 1 , u1 ) and (E2 , H2 , τ 2 , u2 ) be two electromechanical states that satisfy the previously listed equations. Moreover, in keeping with the previously stated convention, assume that ∂t → −iω. In essence, the equations satisﬁed by state 1 are multiplied by the variables of state 2, and the equations satisﬁed by state 2 are multiplied by those of state 1; the equations are then added or subtracted to remove common terms, and combined in a way that gives a divergence on one side and forcing terms on the other. The only point to note is that it is the particle velocity v = −iωu, rather than the particle displacement, that is needed. The result of these operations is the following reciprocity identity: 2 1 − u2j τij ) + eijk (Ej1 Hk2 − Ej2 Hk1 ) ∂i −iω(u1j τij = −iωρs (u2j fj1 − u1j fj2 ) + (Ej2 Jj1 − Ej1 Jj2 ).

(6.6.1)

We will return to this identity in §6.7 after a brief example showing application to a simpler but similar conﬁguration.

6.6.2 Example: reﬂection from a rigid surface To demonstrate the use of (6.6.1), reﬂection, from a perfectly rigid surface, of a normally incident sound beam radiated by a circular transducer is considered. This example is taken with little change from Block et al. (2000). Figure 6.7 schematically indicates the conﬁguration: a cylindrical beam is emitted by the transducer, propagates through a ﬂuid used to couple the transducer and rigid surface, and is reﬂected, the reﬂection being received by the same transducer. Equation (6.6.1) is converted to an integral over a surface that cuts the coaxial cable of the transducer along ∂Re , goes outward to inﬁnity, where the waveﬁelds are assumed to vanish, and is closed by the rigid surface ∂Rr . The integral then becomes 1 2 eijk Ej1 Hk2 − eijk Ej2 Hk1 n ˆ i dS = −iω ui p − u2i p1 n ˆ i dS. ∂Re

∂Rr

(6.6.2) The only electrically active part of ∂Re is the cross-sectional area of the coaxial cable. The area of the rigid surface ∂Rr insoniﬁed is essentially that directly below the transducer. Diﬀraction spreading from the transducer is taken as negligible. Accordingly, the incident beam is approximated as ui3 = Aeik(x3 +d) H(1 − ρ/a),

ui1 = ui2 = 0,

(6.6.3)

112

Scanned acoustic imaging

∂Re

a

d

∂Rm

x3

Fig. 6.7. A schematic drawing of the measurement of the reﬂection coeﬃcient of a rigid surface ∂Rm . A cross-section is shown. The surface ∂Re cuts the coaxial cable coming from the transducer. d is the distance to the surface from the transducer aperture; a is the aperture radius.

where ρ = (x21 + x22 )1/2 and a is the radius of the transducer aperture. The Heaviside step function H(ξ) introduces a jump that is not physically present, but for this example this aﬀects nothing. In §2.1.2 this is explored in detail. The reﬂected beam is ur3 = −Ae−ik(x3 −d) H(1 − ρ/a),

ur1 = ur2 = 0,

(6.6.4)

where we have used the fact that the reﬂection coeﬃcient for the particle displacement amplitudes at the rigid surface is (−1). The reciprocating waveﬁeld 1 is taken as the incident waveﬁeld (that when no rigid surface is present), and the reciprocating waveﬁeld 2 as the total waveﬁeld, incident and reﬂected. Although the waveﬁeld is timeharmonic, it is assumed that it is but one component of a temporally separated signal; the waveﬁelds are assumed windowed in time such that only the initially reﬂected beam (very often a tone burst in time) is received, and subsequent reverberations are not measured. It is assumed that the coaxial cable is cut suﬃciently far from the transducer that the overwhelmingly dominant contribution to the waveﬁeld in the cable is

6.7 Measurement model

113

that of the lowest TEM mode (Auld, 1979; Collin, 1991, pp. 247–259). At ∂Re we therefore take as electromagnetic waveﬁeld 1 + + 1 1 Ei , Hi1 = 1 − Ree Hi , (6.6.5) Ei1 = 1 + Ree and as electromagnetic ﬁeld 2 + 2 Ei2 = 1 + Ree Ei ,

+ 2 Hi . Hi2 = 1 − Ree

(6.6.6)

1 2 and Ree . The reﬂection coeﬃcients measured in the coaxial line are Ree Equation (6.6.2) is evaluated using (6.6.3) through (6.6.6). This gives

ΔRee = −

ω 2 ρc |A|2 πa2 ei2(kd+α) , 2P

(6.6.7)

2 1 where ΔRee = Ree − Ree and A = |A|eiα . The argument α is assumed to be introduced by the transducer. The change in voltage, symbolized by ΔV , is proportional to ΔRee , namely, ΔV = C(ω)ΔRee , where C(ω) is a function of the transducer only and not the scatterer. The time-average power P incident on the transducer is eijk Ej+ Hk+ n ˆ i dS, −2P = ∂Re

where the minus sign arises because of the orientation of the unit normal. We can arrange that the integrand be real. 1 = 0. If the transducer is perfectly matched to the ﬂuid, then Ree Moreover, if it is lossless (α real), then the time-averaged incident power 2 = −ei2(kd+α) , the is simply P = (1/2) ω 2 ρc|A|2 πa2 , and therefore Ree displacement reﬂection coeﬃcient for a perfectly rigid surface.

6.7 Measurement model The previous section describes the reﬂected focused beam and leaky surface wave, but does not indicate how that scattered waveﬁeld is transformed into a change in voltage. Looking back at Fig. 6.1(a), it is readily appreciated that the basic function of the lens, buﬀer rod, and piezoelectric transducer is to map the total echo collected by the aperture of the lens to a change in voltage δV . To construct a mathematical model of this mapping, the electromechanical reciprocity identity described in §6.6, and given by (6.6.1), is used. An early use of this identity to construct such models is given by Auld (1979); a more recent application that signiﬁcantly inﬂuenced the writer is Liang et al. (1985). The outcomes

114

Scanned acoustic imaging

from using it are often called measurement models. When no sources are present this identity becomes 2 1 ∂i −iω(u1j τij − u2j τij ) + eijk (Ej1 Hk2 − Ej2 Hk1 ) = 0. (6.7.1) To use (6.7.1), several assumptions must be made. The electromechanical reciprocity identity is assumed to hold throughout R, whatever the internal structure of the real device. The waveﬁelds on the surface ∂R are assumed to be such that all electromagnetic waves enter or leave R through the electrical plane ∂Re , and all mechanical waves enter or leave through the mechanical plane ∂Rm ; elsewhere on ∂R the waveﬁelds vanish. The two planes are considered subsets of the overall surface ∂R and are identiﬁed in Fig. 6.8. The transducer is connected to the electrical source and receiver through a coaxial cable. Only a transverse, electromagnetic plane wave propagates in a coaxial cable (Collin, 1991, pp. 247–251); all higher modes are cut oﬀ. By placing ∂Re far enough from the junction of the coaxial cable with the transducer, it can be assumed that any higher modes that might be excited at the junction have become negligible; therefore, at ∂Re only a transverse plane wave is present. Lastly, it is assumed that the multiple scattering that occurs between the lens aperture and the interface is negligible. This is a reasonable assumption because tone-bursts, not time-harmonic signals, are used, and thus any multiply-scattered signals can often be separated, in time, ∂Re

R

∂R

^ n

∂Rm

Fig. 6.8. The volume R in which the reciprocity relation is applied. The surface ∂R includes the electrical plane ∂Re and the mechanical plane ∂Rm .

6.7 Measurement model

115

from the ﬁrst echo. The δV calculated in this chapter, which uses the time-harmonic assumption, is a Fourier transform of the change in voltage detected in time. At the mechanical plane ∂Rm , therefore, it is assumed that the scattered disturbance has been scattered from the interface only once. Equation (6.7.1) is integrated over the volume R, shown in Fig. 6.8, and transformed into a surface integral over ∂R. This gives 2 1 eijk (Ej1 Hk2 − Ej2 Hk1 )ni dS = iω (u1j τij − u2j τij )ni dS. ∂Re

∂Rm

(6.7.2) Waveﬁeld 1 is taken as 1 = −pi δij . Ei1 = (1 + Re )Ei+ , Hi1 = (1 − Re )Hi+ , u1j = uij , τij

(6.7.3) The ﬁrst two terms are assumed to be evaluated at ∂Re , and the last two at ∂Rm . This is the waveﬁeld that is present when there is no interface. The ﬁrst two terms describe the transverse electromagnetic wave in the coaxial cable; there is both an incident and a reﬂected term, the reﬂection arising from the junction between the cable and the transducer. In practice the designer tries to make Re as small as possible. The second two terms describe the focused sound beam that is radiated from the lens aperture. Waveﬁeld 2 is taken as Ei2 = (1 + Re + δRe )Ei+ , u2j = uij + urj ,

Hi2 = (1 − Re − δRe )Hi+ , 2 τij = −(pi + pr )δij .

(6.7.4)

As with (6.7.3), the ﬁrst two terms are assumed to be evaluated at ∂Re , and the last two at ∂Rm . This is the waveﬁeld that is present when sound is scattered back to the lens aperture from the interface. The ﬁrst two terms again describe the transverse electromagnetic wave in the coaxial cable; there is now an increment to the reﬂection, δRe , arising from the sound collected at the aperture. The second two terms describe a focused sound beam that is incident on and scattered from the interface. Substituting (6.7.3) and (6.7.4) into (6.7.2) gives iω (ur pi − uij pr )nj dS, (6.7.5) δRe = 4 P ∂Rm j where 1 P =− 2

∂Re

eijk Ej+ Hk+ ni dS.

116

Scanned acoustic imaging

The normalization P is the incident power injected into the junction of the cable with the transducer, so that it is known. Despite the assumptions leading to it, (6.7.5) is quite remarkable for at least two reasons. First, it maps the mechanical waveﬁeld at ∂Rm to what is measured at the electrical plane ∂Re . Note that it clearly indicates how the transducer responds only to the complete signal, returning a single scalar quantity δRe . Second, the incident focused beam has a limited angular spectrum because the aperture is ﬁnite; thus, when it combines with the scattered waveﬁelds, it explicitly limits the angular spectrum of the scattered waveﬁeld that can be collected.

6.8 Acoustic material signature Because the incident and reﬂected waveﬁelds in the coupling ﬂuid have been expressed in terms of potentials, it is convenient to express (6.7.5) in terms of them as well. From §1.1, pi,r = ρf ω 2 ϕi,r ,

i,r ui,r j = ∂j ϕ .

It is also convenient to renormalize (6.7.5) so that its maximum absolute value is approximately one. The plane ∂Rm is taken to be the interface x3 = zs . The outcome is an expression for a term labeled δV which is proportional to δRe . δV is named the change in voltage, though it is dimensionless, because it is proportional to the change in measured voltage caused by the presence of the interface and any features on it. The δV is given as δV = E (ϕs ∂n ϕi − ϕi ∂n ϕs ) dS, (6.8.1) ∂Rm

where E −1 = 2

(∂n ϕi ϕi ∗ ) dS.

∂Rm

Equations (6.4.2) and (6.5.2) indicate that the incident and reﬂected waveﬁelds are inverse Hankel transforms. Examining (6.8.1), it becomes clear that it is an integral of the product of Hankel transforms over a semi-inﬁnite domain. This permits Parseval’s relation, given in Appendix A (see also (2.1.36)), to be used to collapse what might seem to be a complicated triple integral into a single one. This is a key step in the derivation, but not one that could be foreseen when the work was ﬁrst begun. Carrying out this step gives

6.8 Acoustic material signature βa G2 (ξ) R(ξ, s)ei2kzs cos ξ dξ, δV (s, zs ) = 2iD

117 (6.8.2)

0

where D = k 5 E or D−1 = 2i

βa

G2 (ξ) sin ξdξ.

0

This is the central result of this chapter, and mentioned in the introduction. A dependence on the position of the interface is indicated by s, and a dependence on the position of the geometric focal plane by zs . If zs ≈ 0 then (6.8.2) describes how the reﬂection acoustic microscope makes an image when it is scanned across the interface. In this case no leaky Rayleigh wave is excited. Assuming that R(ξ, s), as a function of s, changes slowly with respect to wavelength – this is equivalent to assuming that no vertical discontinuities are present – then (6.8.2) maps the reﬂection at s to a change in voltage at the same position. The set of all these reﬂections, when converted to electrical signals, forms the image. Note that δV is a weighted average over the lens aperture of the reﬂection coeﬃcient; the weight function is the square of the aperture function G(ξ). In other words, perfect information about the interface is not recovered. When s is ﬁxed and zs varied, the acoustic microscope acts as an interferometer. A non-uniform asymptotic approximation of (6.8.2), for |zs | = 0, gives δV (s, zs ) ∼ ±DG2 (0) R(0, s)

ei2kzs k|zs |

− H(−zs ) DG2 (βR ) 8πi αR sin βR e−2ik|zs | cos ξR , zs ≷ 0, kb → ∞;

(6.8.3)

here, H(x) is the Heaviside step function. Note that no leaky Rayleigh wave is collected by the lens aperture for zs > 0. The approximations (3.1.19) and ξR ≈ βR have been used in the amplitude, in the second term. For zs negative, (6.8.3) indicates that δV records an interference pattern as zs is varied. Figure 6.9 shows a plot of this interference pattern for fused quartz. The values are identical to those used for Fig. 6.4 and are cited in (6.5.8). The solid line indicates the asymptotic result (6.8.3), and the dashed line a numerical evaluation of (6.8.2). The interference pattern is evident. The distance between the peaks, Δ, is given by the equation

118

Scanned acoustic imaging Δ = 2(1 − cos βR )

−1

.

From this, the variation in the Rayleigh wavespeed can be determined. The graph shown in Fig. 6.9 is named the acoustic material signature. A similar analysis for the line focus can be performed; see Rebinsky and Harris (1992b) for details. Figure 6.10 illustrates the relative magnitude of the acoustic material signature for the two-dimensional and threedimensional focusing microscopes. Much more can be done with (6.8.2). If the geometrical focal plane is placed just slightly below the interface, the leaky Rayleigh wave participates in the image formation, causing vertical surface features, such as the opening of a surface-breaking crack, to be detected acoustically, even when they are diﬃcult to detect with an optical microscope. The monograph by Briggs (1992) gives many examples of this feature. Rebinsky and Harris (1992b) show that the leaky surface wave can be used to set up an interference pattern between the incident surface wave and one reﬂected from a vertical discontinuity. Lastly, (6.8.2) is a more general 1.0

Spectral

0.8

Asymptotic

| δV |

0.6

Δ

0.4

0.2

0 –30

–20

–10

0

10

kzs /(2π)

Fig. 6.9. The acoustic material signature of fused quartz. The solid line indicates the result of plotting the asymptotic approximation, and the dashed line indicates a numerical evaluation of (6.8.2). The distance between the peaks is denoted by Δ.

6.9 Summary

119

1.0 Point focus Line focus 0.8

⏐δVo⏐

0.6

0.4

0.2

0.0 –30

–20

–10 Zs /λ w

0

10

Fig. 6.10. Comparison of the acoustic material signature for line vs. point focus, from Rebinsky and Harris (1992b).

imaging expression than its derivation might suggest. Many scanning, acoustic detection schemes used in industrial non-destructive evaluation work more or less in the same manner as does a reﬂection acoustic microscope, so that (6.8.2) or a minor modiﬁcation of it describes a wide variety of such schemes.

6.9 Summary (1) Equations (6.4.1) and (6.4.6) describe a model of a focused beam. It does not exactly describe focusing through an aperture, as a study of focusing calculations that begin at the plane of the aperture indicates (Born and Wolf, 1999, pp. 484–499), but its form allows one to readily calculate the scattering of a focused beam and, just as importantly, to invoke Parseval’s relation to calculate δV . (2) Equation (6.5.1) and the analysis of §6.5 describe the reﬂected focused beam. Of particular interest is the fact that the leaky Rayleigh waves form a second focal point at the interface. (3) The discussion of §6.6, leading to (6.7.5), indicates how one may relate the measured electrical signal from a piezoelectric transducer

120

Scanned acoustic imaging

to the scattered mechanical waveﬁelds, and as such this plays an important role in many elastodynamic measurement models. (4) Equation (6.8.2) is the principal result of this chapter. It is the equation that describes how a reﬂection acoustic microscope makes an image and acts as an interferometer. In one guise or another it describes acoustic imaging in many other situations that arise in the non-destructive evaluation of manufactured parts.

7 Acoustic diﬀraction in viscous ﬂuids

When a time-harmonic acoustic plane wave in a viscous ﬂuid is incident on a rigid wall, the additional constraint of zero tangential velocity excites a transverse wave, damped by viscosity, of the type described by Batchelor (1967), for example, in a section on unidirectional ﬂows. A semi-inﬁnite viscous ﬂuid is bounded by a wall that oscillates in its own plane, and the purely imaginary square of the wavenumber shows that the induced ﬂow is essentially conﬁned near the wall in what is often described as a Stokes layer. In this chapter, the eﬀects of compressibility and viscosity are seen to be intertwined; the dilatation propagates as a damped acoustic wave and the vorticity propagates with the dilatation along the wall, with its damping in the normal direction consequently modiﬁed. The dilatation satisﬁes a wave equation but, in contrast to the rotation in an elastic solid, the vorticity (time derivative of the rotation) is governed by a diﬀusion equation. However, for a disturbance of given frequency, the governing equations have the same form as in the elastic case and thus the worked examples below relate well to two of the problems solved in Chapters 2 and 3. A contrast is provided by the deduction of the displacement ﬁelds directly from solution forms for the dilatation and rotation, without introducing the potentials.

7.1 Theory The constitutive relation for linearized ﬂow in a homogeneous viscous ﬂuid is expressed by 1 τij = λf ∂k uk δij + 2ν∂t ij − ∂k uk δij , 3 121

(7.1.1)

122

Acoustic diﬀraction in viscous ﬂuids

where ν is the kinematic viscosity (Batchelor, 1967, equation (3.3.11)). Thus, with ρ replaced by ρf and the acceleration f absent, the equation of motion (1.1.1) can be written as

1 c2 + ν∂t ∂i ∂k uk + ν∂t ∂j2 ui = ∂t2 ui . 3

(7.1.2)

The acoustic particle velocity v is expressed in terms of the displacement by ∂t u. On taking the divergence (operator ∂i ) of (7.1.2), the dilatation satisﬁes the damped wave equation

4 c2 + ν∂t ∂i2 − ∂t2 ∂k uk = 0. 3

(7.1.3)

On taking the curl (operator elmi ∂m ) of (7.1.2), the vorticity satisﬁes the diﬀusion equation (ν∂j2 − ∂t )elmi ∂m (∂t ui ) = 0,

(7.1.4)

as in unsteady creeping (Stokes) ﬂow. Vorticity is created at√rigid boundaries and, according to (7.1.4), diﬀuses on the length scale νt. As in earlier chapters, the time dependence e−iωt is assumed and suppressed below. Then (7.1.3) and (7.1.4) reduce to the pair of Helmholtz equations

∂j2 +

where ka =

(∂i2 + ka2 )∂k uk = 0,

(7.1.5)

iω elmi ∂m (∂t ui ) = 0, ν

(7.1.6)

−1/2 4i ω 1 − 2 , c 3

√ =

νω 1. c

(7.1.7)

Since the viscous decay length (ν/ω)1/2 is assumed here to be much smaller than the acoustic length c/ω, as in air or water at normal acoustic frequencies, the viscous eﬀects are essentially conﬁned to a thin Stokes layer adjacent to a boundary. Comparison with (1.1.21) and (1.1.22) shows that c2L and c2T are replaced here by c2 (1 − 4i2 /3) and −i2 c2 , respectively. The reﬂection alluded to in the introduction to this chapter has the damped acoustic plane wave ˆ2 sin φ0 )eika (x1 cos φ0 −x2 sin φ0 ) , e1 cos φ0 − e ui = A(ˆ

0 ≤ φ0 < π/2, (7.1.8)

7.2 Diﬀraction by a half-plane

123

incident on a rigid wall at x2 = 0. The no-slip condition, namely, zero total displacement in the oscillatory ﬂow, requires the reﬂected acoustic wave to have amplitude diﬀerent from A so as to include a vorticity ﬁeld ˆ3 exp[ika x1 cos φ0 ] exp[−x2 (ka2 cos2 φ0 − iω/ν)1/2 ], according parallel to e to (7.1.6). The vorticity generates a unique solenoidal displacement and the reﬂected displacement is found to be ur = [−1 + D(φ0 )(ka2 cos2 φ0 − iω/ν)1/2 ] ˆ2 sin φ0 )eika (x1 cos φ0 +x2 sin φ0 ) × A(ˆ e1 cos φ0 + e ˆ2 ika cos φ0 ] − D(φ0 ) cos φ0 A[ˆ e1 (ka2 cos2 φ0 − iω/ν)1/2 + e 2

2

× eika x1 cos φ0 e−x2 (ka cos

φ0 −iω/ν)1/2

,

(7.1.9)

where D(φ0 ) =

2 sin φ0 . (ka2 cos2 φ0 − iω/ν)1/2 sin φ0 − ika cos2 φ0

But (7.1.7) implies that iω/ν dominates ka2 cos2 φ0 , whence the solenoidal ﬁeld is almost parallel to the rigid wall, with exponential decay factor ∼ (−iω/ν)1/2 . Thus it is indeed an acoustic modiﬁcation of the unidirectional transverse wave generated by an oscillating boundary. Note that (7.1.9) represents ur as a ‘soft’ reﬂection of the incident wave plus a wave combination whose tangential component vanishes at the wall.

7.2 Diﬀraction by a half-plane Suppose that the damped acoustic plane wave (7.1.8) is incident on the semi-inﬁnite barrier x2 = 0, x1 > 0. The geometry is therefore that ˆ i lies in the second quadrant, that is, φ0 depicted in Fig. 2.6 except that p is replaced by −φ0 in (2.2.2). Express the total waveﬁeld ut as ui +u. The scattered waveﬁeld satisﬁes the pair of reduced wave equations (7.1.5) and (7.1.6) and, for no-slip at the barrier, must satisfy u(x1 , 0) = −ui (x1 , 0) ˆ2 sin φ0 )eika x1 cos φ0 , = −A(ˆ e1 cos φ0 − e

x1 > 0.

(7.2.1)

At x2 = 0, x1 < 0, u and ∂2 u are continuous. It is assumed that, at inﬁnity, u has only plane and diﬀracted waves. The plane x2 = 0 is one of reﬂection symmetry so that the problem can be split into two problems, one antisymmetric and one symmetric with respect to the reﬂection plane. Thus write u = ua + us , where ua1 and us2 are odd functions of x2 while us1 and ua2 are even functions of x2 .

124

Acoustic diﬀraction in viscous ﬂuids

The solution in the half-space x2 > 0 then suﬃces and is such that, at x2 = 0, −∞ < x1 < ∞,

ua1 = 0 = us2 ,

(7.2.2)

(us1 , ua2 ) = −A(cos φ0 , − sin φ0 )eika x1 cos φ0 ,

x1 > 0,

(7.2.3)

according to (7.2.1), and (∂2 us1 , ∂2 ua2 ) = (τ1+ , τ2+ ),

x1 > 0,

(7.2.4)

∂2 us1 = 0 = ∂2 ua2 ,

x1 < 0,

(7.2.5)

where τ1+ (x1 ), τ2+ (x1 ) are unknown functions. With implementation of the Wiener–Hopf technique in mind, introduce the Fourier transforms ∞ (∗ua , ∗us ) = (ua , us )e−iβx1 dx1 , (7.2.6) −∞

which, according to (7.2.2)–(7.2.5), are such that ∗ a u1 (β, 0)

= 0 = ∗us2 (β, 0), (7.2.7) 0 [us1 (x1 , 0), ua2 (x1 , 0)]e−iβx1 dx1 [∗us1 (β, 0), ∗ua2 (β, 0)] = −∞ ∞ −A [cos φ0 , − sin φ0 ]ei(ka cos φ0 −β)x1 dx1 0

=

[∂2 ∗us1 (β, 0), ∂2 ∗ua2 (β, 0)]

∗ − [∗u− 1 (β), u2 (β)]

= =

−A

[cos φ0 , − sin φ0 ] , i(β − ka cos φ0 ) (7.2.8)

∞

[τ1+ (x1 ), τ2+ (x1 )]e−iβx1 dx1 0 [∗τ1+ (β), ∗τ2+ (β)].

(7.2.9)

Note that the principle of limiting absorption need not be invoked because ka , deﬁned by (7.1.7), has a positive imaginary part. The integral in (7.2.8) converges provided φ0 is acute, as assumed, due to the damping along the barrier. If φ0 is obtuse, then the reﬂection (7.1.9) must be included in the incident waveﬁeld and a convergent forcing integral arises from the need to achieve a smooth total waveﬁeld along the negative x1 axis where damping occurs.

7.2 Diﬀraction by a half-plane

125

The reduced wave equations (7.1.5) and (7.1.6) and the Fourier transforms (7.2.6) imply that 2 ∞ a d ∂ua2 ∂u1 2 2 0= − β + ka + e−iβx1 dx1 dx22 ∂x1 ∂x2 −∞ 2 d d∗ua2 2 2 ∗ a iβ , = − β + k u + a 1 dx22 dx2 ∞ a 2 ∂u2 iω ∂ua1 d 2 e−iβx1 dx1 − β + − 0= dx22 ν ∂x1 ∂x2 −∞ 2 d iω d∗ua1 2 ∗ a iβ . = − β + u − 2 dx22 ν dx2 Seeking a solution that vanishes as x2 → ∞, it is noted that iβ ∗ua1 +

2 2 1/2 d∗ua2 = ka2 e−(β −ka ) x2 dx2

uniquely yields the irrotational dilatation determined by ∗ a

ˆ2 ]e−(β u = [−iβˆ e1 + (β 2 − ka2 )1/2 e

while iβ ∗ua2 −

2

2 1/2 −ka ) x2

,

2 1/2 d∗ua1 iω = e−(β −iω/ν) x2 dx2 ν

uniquely yields the solenoidal rotation determined by 2 1/2 iω 1/2 ∗ a 2 ˆ1 − iβˆ u = −(β − ) e e2 e−(β −iω/ν) x2 . ν A suitable combination that satisﬁes (7.2.7) is therefore 2 2 1/2 ∗ a ˆ2 ]e−(β −ka ) x2 u = B(β) [−iβˆ e1 + (β 2 − ka2 )1/2 e β2 −(β 2 −iω/ν)1/2 x2 ˆ2 e . + iβˆ e1 − 2 iω 1/2 e (β − ν ) Similarly, the symmetric solution that satisﬁes (7.2.7) is 2 2 1/2 β2 ∗ s ˆ1 − iβˆ u = D(β) − 2 e2 e−(β −ka ) x2 e 2 1/2 (β − ka ) iω 2 1/2 −(β 2 −iω/ν)1/2 x2 ˆ1 + iβˆ . + [(β − ) e e2 ]e ν

(7.2.10)

(7.2.11)

Evidently, in a modiﬁcation of the plane wave representation used in §2.2, the square roots must have positive real part for all β.

126

Acoustic diﬀraction in viscous ﬂuids

The functions B(β) and D(β) are determined by the mixed conditions at x2 = 0, expressed in the β plane by (7.2.8) and (7.2.9). Substitution of (7.2.10) and (7.2.11) shows that 2 D(β) −

β iω A cos φ0 + (β 2 − )1/2 = ∗us1 (β, 0) = ∗u− , 1 (β) − ν i(β − ka cos φ0 ) (β 2 − ka2 )1/2 2

B(β) (β −

ka2 )1/2

β2 − 2 iω 1/2 (β − ν )

= ∗ua2 (β, 0) = ∗u− 2 (β) +

A sin φ0 , i(β − ka cos φ0 )

iω , ∗τ2+ (β) = ∂2 ∗ua2 (β, 0) = B(β)ka2 . ν Elimination of B(β) and D(β) now yields the disjoint pair of scalar Wiener–Hopf equations ∗ + τ1 (β)

= ∂2 ∗us1 (β, 0) = D(β)

[cos φ0 , − sin φ0 ] i(β − ka cos φ0 ) 1/2 β 2 − iω ν∗ + 1∗ + ν = K(β) τ (β) , 2 τ2 (β) , iω 1 β 2 − ka2 ka

∗ − [∗u− 1 (β), u2 (β)] − A

where K(β) is the Wiener–Hopf kernel, given by iω −1/2 ) . (7.2.12) ν The ratio of square roots in the ﬁrst equation, necessitated by viscosity, corresponds to interchanging the square roots in K(β). Note that K(β) is deﬁned so that it reduces, except for the i factor, to (2.2.7) in the inviscid limit. The cuts from ka to function K(β) is analytic except for branch eiπ/4 ω/ν to ∞ eiπ/4 and from −ka to −eiπ/4 ω/ν to −∞ eiπ/4 . The product K(β) = (β 2 − ka2 )1/2 − β 2 (β 2 −

K(β)[2(β 2 − ka2 )1/2 − K(β)] has zeros at ±βK , where, by use of (7.1.7), βK = (ω/c)(1 − 7i2 /3)−1/2 . 2 − iω/ν)1/2 But the chosen branch cuts imply that, at β = ±βK , (βK 2 2 1/2 2 (βK − ka ) = −βK , i.e. K(k) has no zeros. Although general integral representations are available (Noble, 1988) for the decomposition, K(β) = K + (β)K − (β), it is simpler and still highly accurate to derive the functions K + (β) and K − (β) from a Pad´e approximation to the kernel function K(β), as fully discussed by Abrahams (2000). Davis and Nagem (2002) used the Pad´e approximant c1 ω 2 , (7.2.13) K(β) ≈ (β 2 − ka2 )1/2 c2 ω 2 − β 2 c2 2

7.3 Scattering of a spherical wave at a plane interface

127

where the coeﬃcients c1 () and c2 () are, after substituting (7.1.7), given by 2 1 3 2 i+ c , c c1 = = − . 2 1 4i 2 2 2 1 − 3 2 1 − 7i 3 Although (7.2.13) is based on a simple Pad´e approximant with respect to the point at inﬁnity, excellent agreement with the kernel function is obtained over the entire range −∞ < β < ∞, because the poles thus introduced are suﬃciently far from the real β axis. Approximations to the factored kernel functions K + (β) and K − (β) can be obtained directly from the immediate factorization of (7.2.13), and the Fourier integrals for the displacement, u = ua + us , can then be evaluated by standard numerical integration techniques. The Helmholtz representation for u is inappropriate because it becomes problematic at x2 = 0. Inverse Fourier transforms for ˆ 3 ψs , e ˆ3 ψa can be readily deduced from (7.2.10) and (7.2.11), ϕa , ϕs , e but the reﬂections in x2 < 0 of the antisymmetric functions yield discontinuities. In the inviscid case discussed in §2.2, only ϕa is needed and its reﬂective continuity is assured by condition (7.2.7). But in the viscous case, (7.2.7) is satisﬁed by a balance of dilatational and rotational waves whose break-up can be rectiﬁed only by adding conjugate harmonic functions. A similarity solution near the barrier edge shows that the velocity is almost Stokesian, with ψ and ϕ ultimately biharmonic, thus verifying the need for extra terms described above. Stokes ﬂows become unique when the edge singularity is minimized, giving uα = O(|x1 |1/2 ), ∂2 uα = O(|x1 |−1/2 ) on x2 = 0. These estimates are needed for completion of the Wiener–Hopf technique. Thus viscosity eliminates the physically unrealistic inviscid displacement/velocity singularity.

7.3 Scattering of a spherical wave at a plane interface Lighthill (1978b) discusses the sound ﬁeld generated by a volume outﬂow m(t) from the origin (r = 0), showing that the displacement potential is −m(t − r/c)/4πr. With m = m0 e−iωt and the time dependence suppressed, this reduces to ϕ = −m0 eikr /4πr. This form is given in §3.1, where the wavenumber k = ω/c in an ideal ﬂuid is introduced. The somewhat obvious modiﬁcation due to viscosity can be readily determined by inserting a source term in the continuity equation and hence the term c2 m(t)∂i [δ(r)/4πr2 ] on the right-hand side of (7.1.2).

128

Acoustic diﬀraction in viscous ﬂuids

The time-periodic form of the resulting equation can be arranged as δ(r) 4i ∂i 1 − 2 ∂k uk − m0 + i2 (∂i ∂k uk − ∂j2 ui ) + k 2 ui = 0, 3 4πr2 with deﬁned by (7.1.7). The middle term is solenoidal and hence 2 m0 δ(r) ∂k u k = k a 2 −ϕ , (7.3.1) k 4πr2 where ∂i2 ϕ + ka2 ϕ =

ka2 m0 δ(r) . k 2 4πr2

(7.3.2)

Referring to (3.1.1), the solution is ϕ=−

ka2 m0 eika r , k 2 4πr

(7.3.3)

and comparison of (7.3.1) and (7.3.2) shows that ui = ∂i ϕ. Now consider the scattering problem depicted in Fig. 3.1 but with a viscous ﬂuid in the halfspace x3 < 0 and, in the ﬁrst instance, a rigid solid in x3 > 0. As in (7.1.9), the reﬂected ﬁeld consists of a ‘soft’ reﬂection of the spherical wave, easily represented as an image sink (out-of-phase source), and a wave combination that is algebraically similar to the antisymmetric displacement whose Fourier transform is given by (7.2.10). This latter calculation requires the cylindrical wave representation (see (3.1.4) and (3.1.8)) ∞ √ 2 2 eika r kρ dkρ = J0 (kρ ρ)e− kρ −ka |x3 | " , (7.3.4) r 0 kρ2 − ka2 " in which kρ2 − ka2 lies in the fourth quadrant. With this decomposition in use for the dilatation, it is evident that the relevant solution of (7.1.6) is 2 1/2 iω ˆφ , ∇ × u = J1 (kρ ρ)e−(kρ −iω/ν) |x3 | e ν whose solenoidal displacement is 2 1/2 iω 1/2 u = [ kρ2 − sgn(x3 )J1 (kρ ρ)ˆ eρ + kρ J0 (kρ ρ)ˆ e3 ]e−(kρ −iω/ν) |x3 | . ν (7.3.5) Thus, when the source deﬁned by the potential (7.3.3) is placed at (0, 0, −b) in a viscous ﬂuid bounded by a rigid wall at x3 = 0, the total

7.3 Scattering of a spherical wave at a plane interface

129

displacement is, by use of (7.3.4) and (7.3.5), √ k 2 m0 ∞ kρ J1 (kρ ρ) 2 2 " ˆρ + sgn(x3 + b)J0 (kρ ρ)ˆ u= a 2 e e3 e− kρ −ka |x3 +b| 4πk 0 2 2 kρ − ka √ " kρ J1 (kρ ρ) 2 2 ˆρ − J0 (kρ ρ)ˆ e + [B(kρ ) kρ2 − ka2 − 1] " e3 e− kρ −ka (b−x3 ) 2 2 kρ − ka kρ2 J0 (kρ ρ) ˆ + B(kρ ) − kρ J1 (kρ ρ)ˆ eρ + 2 e3 (kρ − iω/ν)1/2 √ 2 2 2 1/2 × e−b kρ −ka e(kρ −iω/ν) x3 kρ dkρ , (7.3.6) for x3 < 0, where

" kρ2 − ka2 − B(kρ ) = 2

kρ2 (kρ2 − iω/ν)1/2

−1 .

(7.3.7)

Note that this denominator has the same form as the Wiener–Hopf kernel in (7.2.12). If, instead, there is an elastic solid in x3 > 0, with wavespeeds deﬁned by (1.1.21) and (1.1.22) and corresponding wavenumbers, kL , kT , the displacement ﬁelds can be similarly written, in the ﬂuid as √ kρ J1 (kρ ρ) ka2 m0 ∞ 2 2 " ˆρ + sgn(x3 + b)J0 (kρ ρ)ˆ e3 e− kρ −ka |x3 +b| u= e 4πk 2 0 kρ2 − ka2 √ " kρ J1 (kρ ρ) 2 2 2 2 " ˆ eρ − J0 (kρ ρ)ˆ + [B(kρ ) kρ − ka − 1] e3 e− kρ −ka (b−x3 ) 2 2 kρ − ka + C(kρ )kρ − (kρ2 − iω/ν)1/2 J1 (kρ ρ)ˆ eρ + kρ J0 (kρ ρ)ˆ e3 √ 2 2 2 1/2 (7.3.8) × e−b kρ −ka e(kρ −iω/ν) x3 kρ dkρ , for x3 < 0, and in the elastic solid as ∞ ka2 m0 E u =

4πk2

D(kρ ) kρ J1 (kρ ρ)ˆ eρ +

0

!

−

2 kρ2 − kL J0 (kρ ρ)ˆ e3 e

2 −k2 x kρ L 3

+ E(kρ )kρ (kρ2 − kT2 )1/2 J1 (kρ ρ)ˆ eρ + kρ J0 (kρ ρ)ˆ e3 √ 2 2 1/2 2 2 × e−(kρ −kT ) x3 e−b kρ −ka kρ dkρ , (7.3.9)

for x3 > 0. The functions B, C, D, E are determined by imposing continuity of displacements and stresses at x3 = 0. The matching of (7.3.8)

130

Acoustic diﬀraction in viscous ﬂuids

and (7.3.9) at x3 = 0 yields 2 1/2 2 − B(kρ )(kρ2 − ka2 )1/2 + C(kρ )kρ2 = D(kρ )(kρ2 − kL ) + E(kρ )kρ2

= F (kρ ),

(7.3.10)

B(kρ ) − C(kρ )(kρ2 − iω/ν)1/2 = D(kρ ) + E(kρ )(kρ2 − kT2 )1/2 = G(kρ ), after writing the common interfacial displacement as ∞ 2 (u)x3 =0 =

ka m0 4πk2

[G(kρ )kρ J1 (kρ ρ)ˆ eρ + F (kρ )J0 (kρ ρ)ˆ e3 ] e−b

(7.3.11)

√

2 −k2 kρ a

kρ dkρ .

0

On noting that the parameter deﬁnitions and the assumed periodicity allow the solid and ﬂuid stress tensors, (1.1.3) and (7.1.1), to be expressed similarly by −2 −2 kL kT τij /ρs 2 (ij − ∂k uk δij ) , =ω ∂k uk δij − 2 τij /ρf ka−2 ν/iω the introduction of the functions F , G enables the two equations that ensure stress continuity to be written concisely as 1 ρf ν ρf kρ2 G(kρ ) + B(kρ ), − (7.3.12) D(kρ ) = 2 kT2 ρs iω ρs E(kρ ) = 2

1 ρf ν − kT2 ρs iω

F (kρ ) +

ρf C(kρ ). ρs

(7.3.13)

7.4 Diﬀraction by an elastic sphere Suppose that the damped acoustic plane wave with potential ϕ = Aeika x3 /ika is incident on the elastic sphere, r < R. This alignment of the axes yields axisymmetric displacement ﬁelds. In terms of spherical Bessel functions and Legendre polynomials, the Rayleigh equation, eikr cos θ =

∞

(2n + 1)in jn (kr)Pn (cos θ),

(7.4.1)

n=0

allows the incident wave to be written as ∞ jn (ka r) Pn (cos θ) sin θˆ (2n+1)in+1 jn (ka r)Pn (cos θ)ˆ er − eθ . ui = −A ka r n=0

7.4 Diﬀraction by an elastic sphere

131

(1)

Note that jn , and similarly hn , is such that # n(n + 1) 2 π Jn+1/2 (x), jn = 0. jn + jn + 1− jn (x) = 2x x x2

(7.4.2)

Suitable axisymmetric solutions of (7.1.5) and (7.1.6) are (1) ∞ Bn hn (ka r) Pn (cos θ), ∇ · ur = ka A (2n + 1)in+1 −jn (ka R) + ka R h(1) n (ka R) n=0 (7.4.3) ∞ (1) hn (r iω/ν) A Pn (cos θ) sin θeφ . (2n + 1)in+1 Cn (1) ∇ × ur = R n=1 hn (R iω/ν) (7.4.4) Following Davis and Nagem (2006), two distinct rearrangements of the diﬀerential equation (7.4.2) enable the irrotational and solenoidal reﬂected displacements to be identiﬁed from (7.4.3) and (7.4.4). Thus, in r > R, ur = −A ×

∞

(2n + 1)in+1 − jn (ka R) + Bn ka R

n=0 (1)

hn (ka r) (1)

(1)

Pn (cos θ)ˆ er −

hn (ka r) (1)

Pn (cos θ) sin θˆ eθ

hn (ka R) ka rhn (ka R) ∞ (1) n(n + 1)hn (r iω/ν) Pn (cos θ)ˆ (2n + 1)in+1 Cn er + AR (1) rhn (R iω/ν) n=1 % d $ (1) 1 rhn (r iω/ν) Pn (cos θ) sin θˆ − (1) eθ . rhn (R iω/ν) dr (7.4.5)

The displacement in the elastic solid, r < R, has a similar form, expressed as uE = −AkL R

∞

(2n + 1)in+1 Dn

n=0

jn (kL r) jn (kL r) Pn (cos θ)ˆ P (cos θ) sin θˆ × er − eθ jn (kL R) kL rjn (kL R) n ∞ n(n + 1)jn (kT r) + AR Pn (cos θ)ˆ (2n + 1)in+1 En er rjn (kT R) n=1 1 d {rjn (kT r)} Pn (cos θ) sin θˆ − eθ . (7.4.6) rjn (kT R) dr

132

Acoustic diﬀraction in viscous ﬂuids

The coeﬃcients Bn , Dn (n ≥ 0), Cn , En (n ≥ 1) are determined by imposing continuity of displacements and stresses at r = R. Writing the common interfacial displacement as ˆr + A u|r=R = AiF0 e

∞

(2n + 1)in+1 Fn Pn (cos θ)ˆ er + Gn Pn (cos θ) sin θˆ eθ ,

n=1

the coeﬃcients Fn and Gn follow, from matching of (7.4.5) and (7.4.6) at r = R, as Fn =

i (1)

(ka R)2 hn (ka R)

= −Dn

− Bn

(1)

ka Rhn (ka R) (1)

hn (ka R)

kL Rjn (kL R) + n(n + 1)En , jn (kL R)

#

Gn = Bn − Cn 1 + R

+ n(n + 1)Cn

(7.4.7)

(1) iω hn (R iω/ν) ν h(1) n (R iω/ν)

kT Rjn (kT R) , = Dn − En 1 + jn (kT R)

(7.4.8)

for n ≥ 0, n ≥ 1 respectively. Mimicking the previous section, the introduction of Fn , Gn enables the stress continuity conditions to be written concisely as 1 ρf ν ρf 1 − [n(n + 1)Gn + 2Fn ] + Bn , (7.4.9) Dn = 2 kT2 ρs iω R2 ρs En = 2

1 ρf ν − kT2 ρs iω

ρf 1 (Gn + Fn ) + Cn , R2 ρs

(7.4.10)

for n ≥ 0, n ≥ 1 respectively. B0 , D0 are determined by (7.4.7) and (7.4.9), and the four equations are solved for Bn , Cn , Dn , En ; n ≥ 1. This construction fully exploits the similar governing equations for viscous acoustic and elastic disturbances, despite the diﬀusive nature of vorticity. Both here and in §7.3, which complements §3.1, the solution does not use the Helmholtz potentials. They are shown to be inappropriate to the half-plane problem discussed in §7.2 because, in contrast to §2.2, the introduction of viscosity generates not only vorticity but also a symmetric component of the gradient ﬁeld.

8 Near-cut-oﬀ behavior in waveguides

Thus far we have seen many applications and examples of the diﬀraction of waves by cracks and sharp edges, and the main interest is often in the ﬁeld far from the scatterer. We now turn our attention to another type of wave interaction; here we are concerned with the channeling of wave energy along guiding structures. Waveguides and the guiding of waves, and thereby energy, through and along structures is a fundamental issue in wave propagation, and arises often in applications. We begin with a summary of guided waves in acoustics, as the equations and ideas are simpler in that context, and then discuss guided elastic waves. We revisit some of the topics discussed in Chapter 3, namely Rayleigh–Lamb modes for straight waveguides. In practice many waveguides are not straight, but are either bent or thicken, or are inhomogeneous, and so we turn our attention to an asymptotic method that can be used in these situations.

8.1 Shear horizontal and acoustic waveguides Many of the basic concepts of guided waves can be understood and seen more readily in acoustics, and hence we consider this ﬁrst. We begin with a perfectly straight inﬁnite waveguide of thickness 2h. It is convenient to non-dimensionalize, so x ˆ = hx, yˆ = hy and the hatted variables x ˆ, yˆ are dimensional. Our ﬁeld variable φ = φ(x, y) satisﬁes the Helmholtz equation φxx + φyy + (k0 h)2 φ = 0,

−∞ < x < ∞, |y| < 1,

(8.1.1)

with k0 = ω/c, ω being the frequency and c the wavespeed. The waveguide itself, of course, can have diﬀerent boundary conditions upon the walls; Dirichlet, φ = 0, or Neumann, φy = 0, are the usual choices. If φ is the out-of-plane displacement and we are considering SH elasticity, then 133

134

Near-cut-oﬀ behavior in waveguides

these would correspond to clamped or traction-free conditions, respectively. But it is important to realize that the Helmholtz equation arises in a host of diﬀerent applications, from acoustics and electromagnetism to quantum mechanics, as well as in elasticity. It is natural, given this waveguide, the Helmholtz equation, and the boundary conditions, to ask what solutions exist. We treat both Dirichlet and Neumann boundary conditions, and it is readily shown that modal solutions can be found in the form ∞ sin nπ ikn x 2 (y + 1) (8.1.2) φ= An e × , cos nπ n=0 2 (y + 1) (the upper/lower solutions being Dirichlet/Neumann) for waves propagating to x = +∞, where each of the discrete modes in the summation has a characteristic wavenumber, kn , satisfying a dispersion relation kn = (k0 h)2 − (nπ/2)2 (8.1.3) connecting it to the non-dimensional frequency k0 h = hω/c. The branch cuts for the square root are chosen such that it is positive imaginary for k0 h < nπ/2. For convenience we have written this as a single cosine or sine, but expanding it reveals that the modes fall into two families: a symmetric (in y) set for n even and an anti-symmetric set for n odd. The dispersion curves from this dispersion relation, for the Neumann case, are plotted in Fig. 8.1, from which we immediately notice several things. First, all the modes except the lowest – it is conventional to number the modes with increasing frequency – cut the frequency axis at non-zero values: nπ/2. These frequencies are called the cut-oﬀ 20

kn

15 10 5 0 0

2

4

6

8

10 k0 h

12

14

16

18

20

Fig. 8.1. The dispersion curves relating frequency and wavenumber for the straight waveguide with Neumann boundary conditions.

8.1 Shear horizontal and acoustic waveguides

135

frequencies for each mode; if the frequency is below that value, then kn is complex and the solution decays exponentially in x, no longer propagates, and thus cannot transport energy eﬃciently. As we approach the cut-oﬀ frequencies from above, the wavenumbers kn tend to zero. Since these are related to the wavelength Δ via kn = 2π/Δn the wavelength is tending to inﬁnity and thus these waves are ‘long.’ Next, we note that all the dispersion curves are not straight lines, except for the lowest mode, and this explains why the relation relating frequency to wavenumber is called a dispersion curve: these modes are dispersive, i.e. the group velocity, deﬁned as cg =

∂ω , ∂k

is not equal to the phase velocity c = ω/k. Thus the energy and wave crests move at diﬀerent speeds. The slope of the curves, in this case, is always positive, so cg > 0 and the group velocity is always positive. The lowest mode, for which n = 0, is diﬀerent: it is absent in the Dirichlet case, it simply corresponds to a plane wave in the x direction, it is non-dispersive, and it automatically satisﬁes the Neumann boundary condition. Thus, even in this simple situation, the dispersion curves have told us useful information encapsulated in a readily understood graph; dispersion curves and relations are fundamental to guided waves, as is the concept that the ﬁeld can be broken down into discrete modes. All of these details are important when it comes to asymptotic methods. If a waveguide is not perfectly straight, contains a defect, or widens, we can no longer expect to separate variables and have a perfect modal structure. Nonetheless one would hope to use the knowledge gained from studying modal solutions and dispersion curves to be of value. For instance, if the waves are at very high frequencies and have short wavelengths relative to geometric or material variations within the waveguide, then, heuristically, one would hope that a local modal structure would persist and that its phase would gradually change as one moved along the waveguide. This is the motivation behind WKBJ and ray-like methods. In the opposite regime, if the waves are very long and the local modes are almost cut oﬀ, then one would hope that expanding around this local situation would give valuable information, and an asymptotic theory could be generated for long waves. In this chapter we shall consider the latter situation, which is not usually covered in books.

136

Near-cut-oﬀ behavior in waveguides 8.2 Elastic waveguides

We consider an isotropic homogeneous elastic straight waveguide occupying the region −h ≤ x2 ≤ h and |x1 | < ∞. The edges x2 = ±h of the waveguide are taken to be stress-free. Before applying the stress-velocity equations, dimensionless variables are introduced: t˜ = cT t/h,

x ˜i = xi /h,

u ˜i = ui /h,

τ˜ij = τij /μ,

ω ˜ = hω/cT , (8.2.1) and in subsequent equations the tilde will be omitted for notational convenience. There are two bulk wavespeeds in isotropic elasticity, of shear/transverse √ waves cT and of compressional/longitudinal waves cL (with cL /cT > 2); the parameter μ in (8.2.1) is the shear modulus. We consider time-harmonic motion, so the multiplicative factor e−iωt , where ω is the frequency, is considered understood and suppressed henceforth. The non-dimensional governing equations of isotropic and homogeneous elasticity (in two dimensions) are: ∂ 2 u1 ∂ 2 u2 ∂ 2 u1 −2 + + (γ − 1) + ω 2 u1 = 0, ∂x21 ∂x22 ∂x1 ∂x2 2 ∂ 2 u1 ∂ 2 u2 −2 ∂ u2 −2 + γ + (γ − 1) + ω 2 u2 = 0, ∂x21 ∂x22 ∂x1 ∂x2

γ −2

(8.2.2) (8.2.3)

where u = (u1 , u2 ) are the displacements in the x1 , x2 directions, and the ratio of the wavespeeds, γ, where γ = cT /cL , emerges as a key parameter. Solutions of (8.2.2), (8.2.3) are modal sums: u1 ikn Bn ikn An = cos αn x2 + sin αn x2 u2 αn An −αn Bn n βn Cn −βn Dn cos βn x2 + sin βn x2 eikn x1 . (8.2.4) −ikn Dn −ikn Cn

+

The parameters αn = ω 2 /γ −2 − kn2 , βn = ω 2 − kn2 , An , Bn , Cn , and Dn are found by applying the boundary conditions at x2 = ±1, as are the values of kn : the summations are carried out over all of the modes. The following analysis is greatly simpliﬁed by considering the even (or symmetric) modes, which have u1n ∼ cos ax2 , u2n ∼ sin ax2 , separately from the odd (or antisymmetric) modes, which have u1n ∼ sin ax2 and u2n ∼ cos ax2 , where a is constant and u1n , u2n are the ﬁrst and second components of displacement associated with the nth mode, respectively.

8.2 Elastic waveguides

137

Physically, excitations can be split into even and odd functions of x2 and considered separately, so there is no loss of generality in doing this.

8.2.1 Rayleigh–Lamb modes We have already seen Rayleigh–Lamb modes in §3.2, with an emphasis on the ﬁrst couple of modes which approximate Rayleigh surface waves at high frequencies. Here we are more interested in all modes, close to cut-oﬀ, which is a diﬀerent regime. For the even modes (8.2.4) becomes (ikn Bn cos αn x2 + βn Cn cos βn x2 ) eikn x1 , (8.2.5) u1 = n

u2 =

(−αn Bn sin αn x2 − ikn Cn sin βn x2 ) eikn x1 .

(8.2.6)

n

Expressions for the stresses in terms of the displacements are τ12 =

∂u2 ∂u1 ∂u1 ∂u2 + , τ22 = (γ −2 − 2) + γ −2 . ∂x1 ∂x2 ∂x1 ∂x2

(8.2.7)

The expression for τ11 has been excluded here as it is not required. Applying the boundary conditions, requiring τ12 and τ22 to be zero at x2 = ±1, leads us to one of the Rayleigh–Lamb dispersion relations, as shown in many textbooks on elasticity, for instance in Achenbach (1973): (k 2 − β 2 )2 tan αn = − n2 n , tan βn 4kn αn βn

(8.2.8)

and Bn = (kn2 − βn2 ) sin βn ,

Cn = 2ikn αn sin αn .

(8.2.9)

This dispersion relation for even modes allows values of kn to be calculated for any given value of ω (or vice-versa). It is a non-trivial relationship, but is easily solved numerically using root-ﬁnding, general purpose packages, or eigenvalue/spectral methods; either way, dispersion curves (such as those of Fig. 8.2) readily emerge. The even eigenmodes are calculated by substituting these expressions into (8.2.5)–(8.2.6) and then using (8.2.7) for the stresses. Similarly, for the odd modes the displacements are written as (ikn An sin αn x2 − βn Dn sinn βn x2 ) eikn x1 , (8.2.10) u1 = n

u2 =

n

(αn An cos αn x2 − ikn Dn cos βn x2 ) eikn x1 ,

(8.2.11)

138

Near-cut-oﬀ behavior in waveguides 10 8

k

6 4 2 0 0

1

2

3

4

5

6

7

8

9

10

ω Fig. 8.2. The Rayleigh–Lamb dispersion curves for aluminum (γ = 0.4593). Even (odd) modes are indicated by solid (dashed) lines.

with An = (kn2 − βn2 ) cos βn ,

Dn = −2ikn αn cos αn ,

(8.2.12)

and the dispersion relation is 4k 2 αn βn tan αn = − 2n 2 2 . tan βn (kn − βn )

(8.2.13)

The odd eigenmodes can be calculated by substituting these expressions into (8.2.10)–(8.2.11) and then using (8.2.7). It is worthwhile inspecting the dispersion curves before embarking upon any analysis. The lowest two modes pass through the origin and are asymptotic to a straight line; these are precisely the modes discussed in Chapter 3, the straight line being associated with Rayleigh waves which are non-dispersive. The other modes have distinct cut-oﬀs, and separate naturally into even and odd modes; near cut-oﬀ the modes are dominated by either compressional or shear terms (this is not evident from the curves alone). The second even mode turns back upon itself, recalling that the group velocity is deﬁned as the gradient of frequency with respect to wavenumber, and thus has negative group velocity; the picture is thus more complex than that for SH modes.

8.3 Long waves We consider a two-dimensional waveguide of inﬁnite extent in the x direction which is bent in the vicinity of the point x = 0 and ﬂattens out at inﬁnity. The geometry of the problem in Cartesian coordinates is

8.3 Long waves

139

η = +1 η

y 0

η = −1

x

σ α

Fig. 8.3. The geometry of a bent waveguide, showing the new coordinates σ, η along and normal to the centerline.

shown in Fig. 8.3. The waveguide thickness is constant and taken as 2, in non-dimensional units. We employ a new coordinate system (σ, η) which is more natural for this type of geometry. Here, 0 ≤ σ ≤ ∞ is the arc length along the centerline, and −1 ≤ η ≤ 1 is the signed shortest distance between the centerline and a point within the waveguide; it takes negative values beneath the centerline. The shape of the centerline is characterized by the angle α between the tangent to the centerline and x axis. Therefore, we assume that the curvature function ασ , where the subscript means the derivative with respect to σ, vanishes as σ → ∞, so considering a bent waveguide that ﬂattens out at inﬁnity. The following derivatives are handy when transforming to the new coordinates: σx = κ cos α,

σz = κ sin α,

ηx = − sin α,

ηz = cos α,

where κ = (1 − ασ η)−1 . Since the waveguide is assumed to be weakly bent, we introduce the small parameter and a new variable ξ such that ξ = σ, which means that the angle function α is a slowly varying function of the coordinate σ. In terms of the new coordinate system (ξ, η) the Laplacian is rewritten as Δ = 2 κ2 ∂ξξ + ∂ηη + 3 κ3 αξξ η∂ξ − καξ ∂η ,

κ = (1 − αξ η)−1 .

Thus for a bent geometry within which the Helmholtz equation applies (we treat SH waves ﬁrst) the equation that one wishes to solve is (8.1.1), which in the new coordinates is Δφ + λ2 φ = 0,

(8.3.1)

where λ2 = (ωh/c)2 , i.e. the squared non-dimensional frequency. This Helmholtz equation then requires boundary conditions, as earlier we

140

Near-cut-oﬀ behavior in waveguides

assumed either Dirichlet or Neumann conditions; we take φ = 0 or φη = 0 on η = ±1, respectively. Long-wave theory explores the physical limit just as one approaches the cut-oﬀ frequency from above, and thus it is natural to consider regular perturbation expansions of φ and the squared frequency λ2 . Posing expansions φ = φ0 + φ1 + 2 φ2 + · · · ,

λ2 = λ20 + λ21 + 2 λ22 + · · ·

leads to a hierarchy of equations: φ0ηη + λ20 φ0 = 0, φ1ηη + φ2ηη +

λ20 φ1 λ20 φ2

=

(8.3.2)

−λ21 φ0

+ αξ φ0η ,

(8.3.3)

= −φ0ξξ + αξ [φ1η + αξ ηφ0η ] −

[λ22 φ0

+

λ21 φ1 ],

(8.3.4)

with the conditions on η = ±1 that φi = 0 or φiη = 0 for the Dirichlet or the Neumann case, respectively (i = 1, 2, 3, . . .). The ﬁrst of these, (8.3.2), has the trivial solution sin (λ0 (η + 1)) nπ φ0 (ξ, η) = f0 (ξ) × , λ0 = 2 cos (λ0 (η + 1)) (the upper/lower solutions being Dirichlet/Neumann, respectively). The equations contain an as yet unknown function f0 (ξ), and the aim of the expansion is to generate an equation that will determine this and the frequency. The second equation, (8.3.3), contains the unknown frequency λ21 , which is determined using a solvability condition: multiplying the righthand side of the equation by the homogeneous solution, i.e. φ0 , and integrating from η = −1 to η = +1 with respect to η must yield zero – this gives λ21 = 0. A little more algebra gives sin((η + 1)λ0 ) αξ η αξ 0 φ1 (ξ, η) = f1 (ξ)+

2

f0 (ξ) ×

cos((η + 1)λ0 )

−

2λ0

f0 (ξ)×

sin((η + 1)λ0 ).

The ﬁnal equation we consider, (8.3.4), can likewise be simpliﬁed using a solvability condition in the same manner as earlier, and this gives an equation to determine λ22 : αξ2 1 2 f0ξξ + λ2 f0 + f0 × = 0. (8.3.5) 4 (−3) This ordinary diﬀerential equation (ODE) in f0 (ξ) yields useful information about the solutions near cut-oﬀ frequencies. If one knows the

8.3 Long waves

141

curvature function, αξ , then this ODE can be solved in either ﬁnite or inﬁnite domains to determine the correction, λ22 , to the cut-oﬀ frequency. For instance, if the waveguide is straight, then αξ = 0 and the ODE simply has solutions f0 (ξ) ∼ exp(iκξ); thus λ22 = κ2 and nπ 2 + 2 κ2 . λ2 = (k0 h)2 = 2 With kn = κ one is led back to the straight-waveguide dispersion relation (8.1.3). If the waveguide has the shape of a circular annulus, i.e. with constant curvature, then αξ = 1 and one may use a polar coordinate system (r, θ) with a < r < b. The centerline, R, is given by R = (a+b)/2 and the halfwidth, h, by 2h = (b−a), with r = R−hη and = h/R 1. The circular annulus has, assuming an exp(ikθ) dependence, a dispersion relation Jk (k0 a)Yk (k0 b) − Jk (k0 b)Yk (k0 a) = 0 (Dirichlet), Jk (k0 a)Yk (k0 b) − Jk (k0 b)Yk (k0 a) = 0 (Neumann), which relates the wavenumber k to frequency k0 h. These relations are easily solved numerically, and long-wave theory gives # 2 nπ 2 − k = (k0 h)2 + 4 2 in the Dirichlet case and k =

# (k0 h)2 −

32 nπ 2 − 4 2

for the Neumann case. These asymptotic approximations are compared to the exact dispersion curves in Fig. 8.4. Also shown in Fig. 8.4 is the straight-waveguide dispersion curve; notably, for a curved Dirichlet waveguide the cut-oﬀ frequency is, relative to the straight-waveguide curve, shifted to the left, with k0 h = (nπ/2)2 − 2 /4, and conversely for the Neumann waveguide it is shifted to the right, with k0 h = (nπ/2)2 + 32 /4. This has implications for modes in varying waveguides as there exists a small window of frequencies for a locally curved waveguide where, in the Dirichlet case, a mode is permitted but is promptly cut oﬀ if it enters a straight region, the converse being true for the Neumann case. This brings us to the topic of trapped modes. Before embarking on that, we

142

Near-cut-oﬀ behavior in waveguides

2.5 2 1.5

k

Dirichlet 1 Neumann

0.5 0

1.56

1.58

1.6

1.62

1.64

1.66

k0 h

Fig. 8.4. The n = 1 dispersion curve relating frequency and wavenumber for an annular waveguide with Dirichlet or Neumann boundary conditions. The parameter has been chosen rather large ( = 0.25) in order to allow some diﬀerence to be seen. The dashed line is the curve for a straight waveguide. The solid lines are calculated from the full dispersion relation and the solid plus dots is from the long wave theory.

mention a generalization: the procedure has been illustrated here for a waveguide of constant width with a localized change in curvature, but similar ideas can be used to deal with localized thickness or material variations. The pedagogic SH/acoustic example we have treated in some detail is quite simple, but nonetheless illustrates an important point: there is a useful asymptotic regime that is readily accessed, limited not to low or high frequencies but to near cut-oﬀ frequencies.

8.3.1 Trapped modes The long-wave asymptotics have distilled all of the physics into a simple ODE, and some quite general properties of the ODE can be deduced using Sturm–Liouville theory and the properties of positive operators; a useful reference, albeit from the perspective of integral equations, is Porter and Stirling (1990). Let us consider an inﬁnite waveguide that has a bend localized at ξ = 0 and which straightens out at inﬁnity, so αξ → 0 as ξ → ±∞. We wish to investigate whether there are modes trapped in the locality of the bend, i.e., mathematically, whether there exist non-trivial eigensolutions that decay at inﬁnity, i.e. f0 → 0, there. If we consider the ODE in (8.3.5) for the Neumann case, we have 3 − f0ξξ + αξ2 f0 = λ22 f0 . 4

(8.3.6)

8.3 Long waves

143

This has been rewritten slightly to emphasize the following point: if one multiplies by f0 and then integrates with respect to ξ from −∞ to ∞, then the left-hand side of (8.3.6) is positive (a positive operator) and thus the eigenvalue λ22 is positive. At inﬁnity we are then left with the ODE f0ξξ + λ22 f0 ∼ 0, which has oscillatory solutions at inﬁnity; thus there are no decaying eigenfunctions. Thus, we immediately deduce that the Neumann bent waveguide has no possibility of trapping. Now we consider the Dirichlet ODE: αξ2 f0 = λ22 f0 , (8.3.7) 4 which is very similar to the Neumann one aside from the curvaturedependent term. This has now changed sign (and dropped a numerical factor); this is important as we no longer have a positive operator, and the eigenvalue could now be negative. Although this does not prove that it is, it allows for the possibility that λ22 < 0, i.e. that an eigenvalue could exist beneath the cut-oﬀ frequency for the straight waveguide and thus that a mode (or modes) could be trapped. This quite general but simple procedure is enlightening and immediately discounts the possibility of localized trapping at a bend in a Neumann waveguide. Numerical investigation of the Dirichlet ODE demonstrates that there is indeed a localized eigenfunction at the bend; see Fig. 8.5. − f0ξξ −

8.3.2 Inplane elasticity Long-wave theory can also be applied to a bent waveguide for inplane elasticity. The algebra is heavier, but the basic idea remains unchanged: identify the cut-oﬀ frequencies, expand about them, and distill the essential physics into an ODE that is then easily studied. We will now derive the analogous ODE to (8.3.5) for an elastic bent waveguide. The asymptotic scheme in elasticity becomes slightly more complicated owing to the higher-order boundary conditions, and it is convenient to use displacement potentials φ, ψ such that u = ∇φ + ∇ × (0, 0, ψ). The governing equations are written as follows: Δψ + Λ2 ψ = 0, (8.3.8) Δφ + γ 2 Λ2 φ = 0, where γ = cT /cL = (1 − 2ν)/(2 − 2ν), Λ2 = (ω/cT )2 , and ν is the Poisson ratio. We assume that the boundaries are traction-free and that

144

Near-cut-oﬀ behavior in waveguides Frequency = 1.5702

10

f0

8 6

0.8 0.6 0.4 0.2 −50

4

0 ξ

50

0

5

2 0 −10 0

0.1

−5 0.2

0.3

0.4

0.5

0.6

0.7

10 0.8

0.9

1

Fig. 8.5. The solution in a Dirichlet waveguide for n = 1 and αξ = (π/4)sech2 (ξ) (a locally bent bar) with = 0.25, excited by a line source oscillating at λ = 1.5702, which is slightly less than π/2; the ﬁgure shows the energy ω 2 |φ|2 /2 normalized to have a maximum of unity. The colorbar shows the magnitude, in grayscale, of the function f0 shown in the inset. The asymptotic scheme predicts the same λ0 , and in both computations strong localization at the bend is observed.

τξη (ξ, ±1) = τηη (ξ, ±1) = 0, where the τ are stresses. In terms of the potentials, these become 2κφξη + 22 κ2 αξ φξ − Δψ + 2ψηη = 0, (γ −2 − 2)Δφ + 2φηη − 2κψξη − 22 κ2 αξ ψξ = 0, evaluated on η = ±1. There are four sub-cases to consider, two sheardominated and two compressional-dominated cases, and these further split into cases that to leading order are symmetric and anti-symmetric about the centerline; we present one of the compressional cases and just quote results for the others later. An inspection of the dispersion relations in the limit as k → 0 shows that the cut-oﬀ frequencies occur at Λ=

π(2n − 1) πn π(2n − 1) , , πn, . 2γ γ 2

These correspond to transverse resonances, where the waves do not propagate but simply bounce back and forth across the waveguide. These also follow from the leading-order asymptotics.

8.3 Long waves

145

For the compressional ﬁeld, we assume the following asymptotic expansion: φ(ξ, η) = φ0 (ξ, η) + φ1 (ξ, η) + 2 φ2 (ξ, η) + · · · , ψ(ξ, η) = ψ1 (ξ, η) + 2 ψ2 (ξ, η) + · · · , 2

Λ =

Λ20

+

Λ21

+

2

Λ22

(8.3.9)

+ ··· ,

which leads to the hierarchy of equations for the components of the expansions. Each level of the hierarchy represents an ODE and uses the result obtained in the previous step. The leading-order calculation leads to φ0 (ξ, η) = f0 (ξ) cos[γΛ0 η],

Λ0 =

π(2n − 1) . 2γ

The function f0 (ξ) is unknown for the moment, and the Λ0 are the cut-oﬀ frequencies in a straight waveguide. At the next order the compressional piece has the solution φ1 (ξ, η) =

αξ αξ f0 f0 η cos[γΛ0 η] − 2 sin[γΛ0 η] + f1 cos[γΛ0 η], 2 Λ0

where f1 is unknown and ultimately not required. The shear piece, ψ1ηη + Λ20 ψ1 = 0, is more interesting, with boundary conditions Λ20 ψ1 + 2ψ1ηη + 2φ0ξη = 0,

η = ±1.

Solving this problem, we obtain the equation for ψ1 : ψ1 (ξ, η) = −2

f0ξ γ sin(γΛ0 ) sin(Λ0 η). Λ0 sin Λ0

The equation of order 2 for φ2 is φ2ηη + γ 2 Λ20 φ2 = −φ0ξξ + ηαξ2 φ0η + αξ φ1η − γ 2 Λ22 φ0 ,

(8.3.10)

with boundary condition 2φ2ηη − 2ψ1ξη − (1 − 2γ 2 )Λ20 φ2 = 0,

η = ±1.

Solving the equation for φ2 and applying the boundary conditions leads to 1 (8.3.11) C (L,s) f0ξξ + 4 − 2 αξ2 f0 = Λ22 f0 , 4γ

146

Near-cut-oﬀ behavior in waveguides

where C

(L,s)

1 8 cot Λ0 . =− 2 + γ Λ0

(8.3.12)

The notation (L,s) denotes that it is the coeﬃcient associated with the longitudinal (L, i.e. compression) piece and that it is symmetric (s). This is clearly a similar ODE to (8.3.5), and again general properties of it can be deduced. In particular, if γ > 1/4 (which is true for realistic media) then one has a positive operator if C (L,s) is negative, and thus no trapping. Conversely, if C (L,s) is negative then there is possibly trapping. Examination of the symmetric Rayleigh–Lamb modes shows that the group velocity cg is cg ∼ −

C (L,s) k, Λ0

and thus localized trapping near the bend is only possible for modes with negative group velocity; this returns us to the dispersion curves shown in Fig. 8.2 and the observation that the second even mode has negative group velocity. This theory is as accurate in its description of near-cut-oﬀ phenomena as that shown for the SH wave/acoustic example. Several examples are found in the literature (Kaplunov et al., 2005; Gridin et al., 2005). The presentation here follows Gridin et al. (2005), where the other ODEs for antisymmetry and shear dominance can be found.

Appendix A Asymptotic expansions

Applications of asymptotic methods and asymptotic expansions in this book are mainly to the evaluation of integrals, particularly Fourier-like integrals. We begin with a brief review of the notation required.

A.1 Notation for asymptotics Most, if not all, of the mathematical deﬁnitions and notations are standard. Nevertheless, the reader is reminded of the following, which are taken, with very little change, from Bleistein and Handelsman (1975, pp. 14–21). The deﬁnitions generally apply in an open angular sector in the complex plane deﬁned as follows. The angular sector Sαβ : Consider an open, two-dimensional region ¯ (the overbar indicates R in the complex plane; let z ∈ R and z0 ∈ R closure). Deﬁne the open sectoral region Sαβ as Sαβ := {z | 0 < |z − z0 | < ρ, α < arg(z − z0 ) < β}. It is sometimes suﬃcient to consider simply a neighborhood of z = z0 without needing to specify α or β. The main tools of asymptotics are the order-of-magnitude notation, o and O, with their associated symbols for comparison, ∼, and equality, =. o[f (z)] An order of magnitude notation: if limz→z0 g(z)/f (z) = 0 as z → z0 in Sαβ , then one writes g(z) = o[f (z)]. O[f (z)] An order of magnitude of f (z): one writes g(z) = O[f (z)] if there exists a real non-negative constant C such that, as z → z0 in Sαβ , |g(z)| ≤ C|f (z)| everywhere in the sector. 147

148

Asymptotic expansions

∼ This is used to indicate an asymptotic expansion of a function, in the angular sector Sαβ , when no explicit estimation of the error is given. For example, g(z) ∼ a0 f0 (z),

z → z0 ,

means that f0 (z) is the ﬁrst term of an asymptotic expansion of g(z) for z as z → z0 in Sαβ . More generally, one writes g(z) ∼ an fn (z), z → z0 , n≥0

where the {fn } form an asymptotic sequence (see below). = In the context of using an asymptotic approximation, this is used when an order of magnitude of the error is given. For example g(z) = a0 f0 (z) + O(f1 ),

z → z0 ,

or g(z) = a0 f0 (z) + a1 f1 (z) + o(f1 ),

z → z0 ,

where again the {fn } form an asymptotic sequence in the angular sector Sαβ .

Types of asymptotic expansions It is useful to distinguish several distinct varieties of asymptotic expansions that are encountered in practice. Asymptotic sequence: The sequence {fn (z)}n≥0 is an asymptotic sequence as z → z0 in Sαβ , if each fn (z) is continuous in R and fn+1 (z) = o[fn (z)], n ≥ 0, uniformly as z → z0 . Asymptotic expansion of Poincar´ e type: The formal series a f (z) is an asymptotic expansion of g(z), as z → z0 in Sαβ , n≥0 n n with respect to the asymptotic sequence {fn }, if the conditions g(z) =

N n=0

an fn (z) + o[fN (z)]

A.2 Abelian theorems for Fourier transforms

149

hold uniformly as z → z0 in Sαβ . When the number N is unimportant, one may also write an fn (z). g(z) ∼ n≥0

Asymptotic expansions not of the Poincar´e type are brieﬂy discussed next. Asymptotic expansion based on an auxiliary asymptotic sequence: The formal series n≥0 an Fn (z) is an asymptotic expansion of g(z), with respect to the auxiliary asymptotic sequence {fn (z)}, if the conditions g(z) =

N

an Fn (z) + o[fN (z)]

n=0

hold uniformly as z → z0 in Sαβ . When the number N is unimportant, one may also write an Fn (z), z → z0 , {fn }. g(z) ∼ n≥0

The auxiliary sequence {fn } may not be explicitly indicated. If the sequence {fn } is to be useful, Fn = O(fn ), Fn = o(fn ), z → z0 , z ∈ Sαβ . Uniform asymptotic approximation: A uniform asymptotic approximation to an integral I(x, κ, α) is one that remains accurate for all values of the parameter α in its domain as κ → κ0 in Sαβ (in practice κ is usually real and κ → ∞).

A.2 Abelian theorems for Fourier transforms The Fourier transform pair used in this book is ∞ 1 ∞∗ ∗ f (k) = f (x)e−ikx dx, f (x) = f (k)eikx dk. 2π −∞ −∞

(A.1)

This pair can be interpreted and used in a slightly more general form (Titchmarsh, 1948, pp. 4–6): ∞, 0 1 ∞±ia ∗ ∗ −ikx f± (k) = f (x)e dx, f± (x) = f± (k)eikx dk, (A.2) 2π −∞±ia 0, −∞

150

Asymptotic expansions

where a is a suﬃciently large positive number. Thus ∗

f (k) = ∗f− (k) + ∗f+ (k),

f (x) = f− (x) + f+ (x).

(A.3)

The function f (x) is said to have the image ∗f (k); similarly, the functions f± (x) are said to have images ∗f± (k). Consider the pair {f+ (x),∗ f+ (k)}. Theorems which relate the known asymptotic behavior of f+ (x) to the asymptotic behavior of its image are called Abelian theorems; those that relate the known asymptotic behavior of the image ∗f+ (k) to that of the function are called Tauberian theorems, and are harder to establish. The following Abelian theorem (Titchmarsh, 1948, pp. 173–174) is suﬃcient for most purposes. Abelian theorem for the pair {f+ (x),∗ f+ (k)}: Let f+ (x) = xα g(x), where α ∈ (0, 1) and g(x) is of bounded variation in (0, ∞). Then ∗

f+ (k) ∼ g(0+ ) (−α)! k α−1 e−iαπ/2 e−iπ/2 , k → ∞;

∗

f+ (k) ∼ g(∞+ ) (−α)! k α−1 e−iαπ/2 e−iπ/2 , k → 0+ .

(A.4)

The function (−α)! := Γ(1 − α), where Γ(z) is the gamma function. An extensive study of Abelian and Tauberian theorems as they relate to the pair {f+ (x),∗ f+ (k)} is given in Doetsch (1974, chapters 32–37).

A.3 The method of steepest descents Integrals of the following two forms frequently arise in the study of linear waves. The ﬁrst is f (z)eκq(z) dz,

I(κ) =

(A.5)

C

where f (z) and q(z) are functions of the complex variable z, having such properties that the integral is convergent, and C is a contour beginning and ending at −∞ ei and ∞ eiζ , respectively, where , ζ ∈ (−π, π] with |ζ| < π/2 < ||. The second is β f (x)eiκp(x) dx, (A.6) I(κ) = α

where f (z) and p(z) are functions of the complex variable z, and p(x) is real on the real axis. In both cases κ is real, positive, and large. These integrals can be asymptotically approximated using one of three interrelated methods: the method of steepest descents, the saddle-point method, and the method of stationary phase.

A.3 The method of steepest descents

151

To advance the discussion, the terms ‘isolated, ﬁrst-order saddle point,’ ‘stationary-phase point,’ and ‘steepest-descents contour’ need to be deﬁned. In doing so, the notation dnz q = dn q/dz n is adopted. Isolated, ﬁrst-order saddle point: zs is an isolated, ﬁrst-order saddle point of q(z) if dz q = 0 at z = zs , but dnz q = 0, n ≥ 2 at z = zs . q(z) is assumed to be an analytic function in a region Q containing zs . Isolated, ﬁrst-order stationary-phase point: xs is an isolated, ﬁrstorder stationary phase point of p(x) if dx p = 0 at x = xs , but dnx p = 0, n ≥ 2 at x = xs . The steepest-descents contour: Assume that q(z) has an isolated ﬁrst-order saddle point at z = zs . The contour, passing through zs , along which Im(q) is constant and along which Re(q) decreases most rapidly is the steepest-descents contour Cs . More explicitly stated, Cs is the curve deﬁned by Re[q(z)] ≤ Re[q(zs )],

Im[q(z)] = Im[q(zs )].

(A.7)

The deﬁnition is readily extended to the case of higher-order saddle points. The method of steepest descents is one in which the steepest-descents contour Cs is determined throughout the complex plane, and is usually used if more than one term of the asymptotic expansion is to be determined. Knowledge of the steepest-descents contour also permits one to accurately take account of possible contributions from poles or branch points. The saddle-point method is one in which the contour of integration is distorted to be near Cs in a neighborhood of zs , and is usually used if only the leading-order, asymptotic contribution is to be determined. Possible contributions from poles or branch points are not always taken account of accurately. This is more or less the distinction made by Copson (1971, p. 64). Because knowing Cs is very useful when interpreting the meaning of an integral, even when only the leading-order term of its asymptotic expansion can be readily calculated, the method of steepest descents will always be used. The method of stationary phase, at least in this book, is used to approximate asymptotically integrals of the form (A.6) and may include contributions both from the end points and from the stationary point (Copson, 1971, pp. 27–35).

152

Asymptotic expansions

The method of steepest descents: Consider (A.5). Assume q(z) has an isolated ﬁrst-order saddle point zs in some region R where q(z) is analytic. Moreover, assume f (z) is analytic in R containing zs and f (zs ) = 0. And, let there be real constants K and b such that |f (z)| < Keb|q(z)−q(zs )| , |z| → ∞, in sectors of the complex plane where Cs begins and ends. Then1 I(κ) =

(−2π)1/2 f (zs ) eκq(zs ) + O(κ−3/2 ), κ → ∞. [κ d2z q(zs )]1/2

(A.8)

The argument of the square-root term is identical to that of dz at zs . The special case q = ir with r a real function of the complex variable z simpliﬁes (A.8), giving I(κ) =

(2π)1/2 f (zs )eiκr(zs ) e±iπ/4 + O(κ−3/2 ), κ → ∞. (A.9) [κ |d2z r(zs )|]1/2

The ± are chosen as d2z r(zs ) is positive or negative. Notes on the method of steepest descents. (1) Harris (2001, pp. 91–95), following Felsen and Marcuvitz (1994, pp. 382–386), constructs (A.8) by ﬁrst deforming C to Cs , picking up whatever pole and branch-point contributions may be present, and then using the mapping s2 = q(zs ) − q(z). This has the eﬀect of converting the integral along Cs into one which can be asymptotically approximated by Watson’s lemma (Copson, 1971, pp. 49–50). The condition placed on f (z) as |z| → ∞ near Cs has been chosen to allow a straightforward application of this lemma. (2) The parameter κ has been assumed real. With care, (A.8) and (A.9) can be analytically continued to an angular sector of the complex κ plane. (3) Further, setting q = ir, where r is a real function of z, leads to an integral very similar to (A.6), considered next. 1

Equation (5.60), p. 93 of Harris (2001) should not have an i before the κ.

A.3 The method of steepest descents

153

The method of stationary phase, including end-point contributions: Consider (A.6). f (z) and p(z) are analytic functions of the complex variable z in a region R containing the segment of the real axis x ∈ [α, β]; p(x) is real on the real axis and has an isolated ﬁrst-order stationary point xs ∈ (α, β). Then I(κ) =

(2π)1/2 f (zs )eiκp(zs ) e±iπ/4 [κ |d2z p(xs )|]1/2 f (β) iκp(β) 1 f (α) iκp(α) e e + O(κ−3/2 ), + − iκ dx f (β) dx f (α)

κ → ∞. (A.10)

The ± are chosen as d2z p(xs ) is positive or negative. Notes on the method of stationary phase. (1) Copson (1971, pp. 29–34) gives a proof of this result without reference to the method of steepest descents. (2) The similarity between the ﬁrst term of (A.10) and (A.9) suggests the two methods are closely related. The method of stationary phase is ideal when the interval of integration is ﬁnite and along the real axis. However, as previously indicated, when the contour integration is inﬁnite and may lie in the complex plane, then the method of steepest descents is the more accurate, as one can be conﬁdent that pole and branch point contributions have been accurately taken account of. (3) Note that if no stationary point appears in x ∈ [α, β] then the leading contributions to the integral come from the end points.

Appendix B Some special functions

Special functions of one sort or another arise frequently, often in the course of seeking uniform asymptotic approximations to more complicated functions. The purpose of this appendix is to collect the formulas or representations used in the text in one place. Magnus et al. (1966) is a well-organized handbook giving most properties of these functions. Borovikov (1994) describes many of the special functions arising in the uniform asymptotic approximation of integrals describing radiation and diﬀraction. The internet site NIST (accessed August 2008) is also a very useful source of information.

B.1 Hankel functions The following facts are taken from Magnus et al. (1966); the section numbers refer to this handbook. (1)

(1) From §3.1.2: The principal branch for the Hankel functions H0 (z) (2) and H0 (z) is deﬁned by arg(z) ∈ (−π, π). The connections among the branches are (1)

(2)

H0 (zeiπ ) = −H0 (z),

(2)

(1)

H0 (ze−iπ ) = −H0 (z).

Hankel functions are used in several places; however, their branch cuts are not always explicitly shown in the diagrams of the complex plane. (2) From §3.6.4: Sommerfeld’s integral representations for the Hankel functions are 1 (1) eiz cos μ dμ (B.1) H0 (z) = π C1 154

B.2 Fresnel integral and (2)

H0 (z) =

1 π

e−iz cos μ dμ.

155

(B.2)

C2

The contour C1 starts at −η + i∞ and ends at η − i∞; the contour C2 starts at η − i∞ and ends at η + i∞, where η is any number between 0 and π; and arg(z) ∈ (−η, π−η). These two integral representations can be combined to show that 2π 1 J0 (z) = eiz cos μ dμ, (B.3) 2π 0 where arg(z) ∈ (0, π), a result that can be arrived at in several other ways. (3) From §3.14.1: Asymptotic expansions for the Hankel functions are (1),(2)

H0

(z) ∼ (πz/2)−1/2 e±i(z−π/4) , |z| → ∞,

(B.4)

where, for the (1) function arg z ∈ (−π, 2π), and for the (2) function arg z ∈ (−2π, π). One additional fact is of use. There is a Parseval’s relation between functions and their Hankel transforms (Sneddon, 1951, pp. 59, 60). The Hankel transform pair are given as ∞ ∗ f (u) = yf (y)J0 (uy)dy, 0 ∞ f (x) = u ∗f (u)J0 (xu)du; 0

and the Parseval’s relation is ∞ xf (x)g(x)dx = 0

∞

u ∗f (u) ∗g(u)du.

(B.5)

0

B.2 Fresnel integral In this book the Fresnel integral F (z) is deﬁned as ∞ 2 F (z) := eiξ dξ,

(B.6)

z

where | arg(z)| ∈ (0, π/2) as ξ → ∞. Note that this integral is, apart from its normalization, usually named the complementary error function (Magnus et al., 1966, §9.2.3). This deﬁnition is used in agreement with

156

Some special functions

its use in Born and Wolf (1999, p. 647). Much use is made of the following property of the Fresnel integral: F (z) + F (−z) = π 1/2 eiπ/4 .

(B.7)

Deﬁning the related function 2

G(z) := e−iz F (z), the asymptotic behavior of F (z), as |z| → ∞, can be expressed as i + O(z −3 ), arg(z) ∈ (−π/2, π/2), 2z (B.8) i 1/2 iπ/4 −3 + O(z ), arg(z) ∈ (π/2, 3π/2). ∼π e + 2z The presence of diﬀerent expansions in diﬀerent sectors of the z plane is an example of the Stokes phenomenon. Moreover, as z → 0, G(z) ∼

F (z) ∼

π 1/2 eiπ/4 − z + O(z 3 ). 2

(B.9)

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Index

Abelian theorems, 150 acoustic material signature, 116, 118 acoustic microscope, 98 admittance, 74 angular frequency deﬁned, 11 angular spectrum spherical wave, 55 antiplane shear motion, 4 asymptotic approximation Abelian theorem, 44 double integral, 32n, 58 non-uniform, pole present, 49 stationary phase, 34 steepest descents, 47–48 uniform, pole present, 46–48 asymptotic expansion, not Poincar´e type, 149 asymptotic expansion, Poincar´e type, 148 asymptotic sequence, 148 beam unfocused, acoustic, 35–36 circular annulus, 141 complex source, 74 compressional wave, deﬁned, 6 continuity conditions, ﬂuid–solid interface, 6–54 cracks, 81, 89 curved waveguide, 141 damped wave equation, 122 diﬀraction edge-diﬀracted wave, 25 edge-diﬀraction integral, 28, 32

through a circular aperture, 24 diﬀraction by an elastic sphere, 130 diﬀraction by a half-plane, 123 diﬀraction coeﬃcient, 33, 47 dispersion, see velocity, group periodic structure, 19 dispersion curve, 134 dispersion relation, 134, 137, 138, 141 dispersion relation, Rayleigh–Lamb, 63 drive point admittance, 74 edge diﬀraction acoustic, 40–49 problem deﬁned, 42 acoustic in three dimensions, 45–46 qualitative features, 47 Wiener–Hopf, 41–45 eigenvalue problem, 66 elastic waveguide, 136 electromechanical reciprocity, 110–113 reﬂection coeﬃcient, 113 reﬂection coeﬃcient calculated, 111 electromechanical reciprocity identity, 113 energy ﬂux, 8 complex, 8 radiation resistance, 40 time average, 8 energy relations conservation law, 7 equations of motion, 7 one-dimensional, 3 two-dimensional, 4 antiplane motion, 4 inplane motion, 4 error function, see Fresnel integral

163

164

Index

F number, 99 focal length, 99 focused beam, converging reﬂected, 107 diverging reﬂected, 105 representations, 101 scattered, 103 Fourier transform space, 12 time, 9 frequency deﬁned, 11 Fresnel integral asymptotic behavior, 155 Fresnel length Fl , 28 beam, 35 Fresnel number, 99 Fresnel region, 47–49 Fr , 49 radiation from a piston, 51 Gaussian beam, 75 geometrical (theory of) diﬀraction radiation from a piston, 32–34 Green’s function piston problem, 27 Green’s tensor, 72, 73 group velocity, 8, 135, 146 Hankel function, 154 Hankel transform, 28, 36, 155 Helmholtz decomposition, 72 Helmholtz theorem, 5 Huygens’ principle, 25, 26–28 hypersingular, 84 hypersingular boundary integral equations, 82

longitudinal strain, 3 longitudinal stress, 4 long-wave theory, 140, 143 low-frequency scattering, 86 Maggi transformation, 31, 32 Maxwell’s equations, 110 measurement model, 113, 115 microscope, acoustic reﬂection, imaging equation, 116 microscope, acoustic reﬂection, qualitative description, 97 mixed boundary condition, 3 Parseval’s relation, 40 partial waves, 19 passbands, 18 periodic structure, 17–20 eﬀective wavenumber, 19 piezoelectricity, equations of, 109–110 plane-wave representation, 42 Poisson summation formula, 12 polarization vector, 57 potentials, displacement, 5 scalar potential, 5 vector potential, 5 propagation matrix, 17 propagation vector, 57

Kirchhoﬀ approximation, 84

radial basis function, 91 radiation from a piston, 24–40 problem deﬁned, 26 impedance, 38–40 radiation impedance, 39 Rayleigh function, 64 Rayleigh wave, focusing, 107 Rayleigh–Lamb dispersion relation, 137 Rayleigh–Lamb modes, 62, 63, 137 reciprocity, 71, 78, 109 elastodynamic, 72 reﬂected plane wave, 57 reﬂection coeﬃcient, 57 poles and branch points of, 59 reﬂection, measured at transducer, 115 Riemann sheet, physical, 36, 43, 59, 104

Lam´ e’s elastic constants, 2 Laplace transform time, 9 leaky Rayleigh pole, 59 leaky Rayleigh wave, 59, 98, 109, 118 leaky Rayleigh wave, ﬁlling lens aperture, 106 leaky wave, 53 limiting absorption, principle of, 6

scalar potential, see potentials, displacement scattering Bragg, 19 from a lumped mass, 15 scattering matrix, 16 shear horizontal motion, 4 shear wave, deﬁned, 6 Sommerfeld transformation, 37

imaging equation, 116 impedance, 73, 76 inplane motion, 4 isolated, ﬁrst-order saddle point, 151 isolated, ﬁrst-order stationary-phase point, 151

Index spectrum angular, 25 spectrum, angular radiation from a piston, 36–37 spherical wave angular spectrum, 55 at a plane interface, 127 converging, 100 cylindrical-wave representation, 56 standing wave, deﬁned, 11 steepest-descents contour, 47, 151 Stokes layer, 122 Stoneley pole, 59 Stoneley wave, 59 stopbands, 18 strain tensor, 2 stress tensor, 2 symmetries, waveguide eigenmodes, 67 Tauberian theorems, 150 time average, 7 product, 8 traction, 2 transforms, Fourier and Laplace, see Fourier transform, Laplace transform deﬁned, 9–12

165

transmission coeﬃcient, 57 transmission matrix, 16 transmitted plane waves, 57 trapped modes, 142 uniform asymptotic approximation, 149 vector potential, see potentials, displacement velocity group periodic structure, 20 vibration, 12 viscosity, 121 viscous ﬂuid, 121 vorticity, 122 waveguides, 133, 136 wavelength, deﬁned, 11 wavenumber deﬁned, 10 eﬀective, periodic structure, 19 wavespeed compressional, 6 shear, 6 Wiener–Hopf acoustic edge diﬀraction, 42–45