Table Of ContentOT84_fmA2.qxd 10/27/2004 3:52 PM Page 3
MARINE
ACOUSTICS
Direct and Inverse Problems
James L. Buchanan
United States Naval Academy
Annapolis, Maryland
Robert P. Gilbert
University of Delaware
Newark, Delaware
Armand Wirgin
Laboratoire de Mécanique et d’Acoustique
Marseille, France
Yongzhi S. Xu
University of Tennessee at Chattanooga
Chattanooga, Tennessee
Society for Industrial and Applied Mathematics
Philadelphia
OT84_fmA2.qxd 10/27/2004 3:52 PM Page 4
Copyright © 2004 by the Society for Industrial and Applied Mathematics.
10 9 8 7 6 5 4 3 2 1
All rights reserved. Printed in the United States of America. No part of thisbook
may be reproduced, stored, or transmitted in any manner without the written
permission of the publisher. For information, write to the Society for Industrial
and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA
19104-2688.
Library of Congress Cataloging-in-Publication Data
Marine acoustics : direct and inverse problems / James L. Buchanan … [et al.].
p. cm.
Includes bibliographical references and index.
ISBN 0-89871-547-4 (pbk.)
1. Underwater acoustics. I. Buchanan, James L.
QC242.2.M37 2004
620.2’5—dc22 2003070359
This research was supported in part by the National Science Foundation through
grants BES-9402539, INT-9726213, BES-9820813, the Office of Naval Research
through grant N00014-001-0853, and the Centre National de la Recherche
Scientifique through grant NSF/CNRS-5932.
is a registered trademark.
Contents
Preface ....................................................................................... xi
Acknowledgments ...................................................................... xii
1. The Mechanics of Continua ............................................... 1
1.1 Introduction ............................................................................. 1
1.2 Survey of Previous Work ........................................................ 5
1.3 Underlying Principles of the Mechanics of Continua .............. 9
1.3.1 Introduction ............................................................. 9
1.3.2 Lagrangian and Eulerian Coordinates, Deformation,
Strain, Displacement, and Rotation ......................... 10
1.3.3 Deformation Gradients and Deformation
Tensors ................................................................... 11
1.3.4 The Cauchy and Green Deformation Tensors ......... 12
1.3.5 Strain Tensors and Displacement Vectors .............. 13
1.3.6 Infinitesimal Strains and Rotations .......................... 15
1.3.7 Lagrangian and Eulerian Strains in the Framework
of Infinitesimal Deformations ................................... 16
1.3.8 Strain Invariants and Principal Directions ................ 17
1.3.9 Area and Volume Changes Due to Infinitesimal
Deformations ........................................................... 18
1.3.10 Kinematics .............................................................. 19
1.3.11 Material Derivatives of Line, Surface, and
Volume Integrals over Regions Devoid of
Discontinuities ......................................................... 21
1.3.12 Material Derivatives of Integrals over Regions
Containing a Discontinuity Surface .......................... 23
1.3.13 Conservation of Mass Law for Uniform Bodies ........ 24
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vi Contents
1.3.14 Conservation of Momentum and Energy Laws ........ 25
1.3.15 External and Internal Loads and Their Incorporation
in the Conservation of Momentum Equation ............ 25
1.3.16 Stress ...................................................................... 26
1.3.17 Global and Local Forms of the Conservation of
Momentum Law in Terms of Stress ......................... 27
1.3.18 Local Form of the Boundary Conditions on
Discontinuity Surfaces ............................................. 28
1.3.19 Thermodynamic Considerations .............................. 29
1.3.20 Constitutive Relations ............................................. 33
1.4 Mechanics of Elastic Media and Elastodynamics ................... 33
1.4.1 Definition of Elastic Media ....................................... 33
1.4.2 Constitutive Equations ............................................ 33
1.4.3 Linear Constitutive Equations (Linear Elasticity) ..... 37
1.4.4 Symmetry Properties of the Elastic Moduli
Tensor ..................................................................... 41
1.4.5 The Wave Equation for Elastodynamics in Linear
Elastic Media ........................................................... 42
1.4.6 Wave Equation for Elastodynamics in
Compressible, Homogeneous Materials .................. 43
1.4.7 Wave Equation for Elastodynamics in
Heterogeneous, Isotropic Solids .............................. 43
1.4.8 Wave Equation for Elastodynamics in
Homogeneous, Isotropic Solids ............................... 43
1.4.9 Obtaining the Wave Equation of Acoustics in
Heterogeneous, Inviscid Fluids from Navier’s
Equation .................................................................. 45
1.4.10 Boundary Conditions between Two Linear,
Isotropic, Homogeneous, Elastic Materials .............. 46
1.5 Forward and Inverse Wavefield Problems .............................. 48
1.5.1 Introduction ............................................................. 48
1.5.2 The Frequency-domain Equation for Propagation
in an Unbounded, Heterogeneous, Inviscid Fluid
Medium ................................................................... 49
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Contents vii
1.5.3 The Frequency-domain Radiation Condition at
Infinity ..................................................................... 50
1.5.4 Governing Equations for the Frequency-domain
Formulation of Wave Propagation in an
Unbounded, Heterogeneous, Inviscid Fluid
Medium ................................................................... 51
1.5.5 Governing Equations for the Frequency-domain
Formulation of Wave Propagation in Two
Contiguous, Semi-infinite, Heterogeneous,
Inviscid Fluid Media ................................................. 51
1.5.6 Governing Equations for the Frequency-domain
Formulation of Wave Propagation in an
Unbounded, Heterogeneous, Isotropic, Elastic
Solid ........................................................................ 52
1.5.7 Governing Equations for the Frequency-domain
Formulation of Wave Propagation in Two Semi-
infinite, Heterogeneous, Isotropic, Elastic Solid
Media in Welded Contact ........................................ 52
1.5.8 Governing Equations for the Frequency-domain
Formulation of Wave Propagation in a Semi-
infinite Domain Occupied by a Heterogeneous,
Inviscid Fluid Contiguous with a Semi-infinite
Domain Occupied by a Heterogeneous,
Isotropic, Elastic Solid ............................................. 54
1.5.9 Eigenmodes of a Linear, Homogeneous,
Isotropic Solid Medium of Infinite Extent ................. 55
2. Direct Scattering Problems in Ocean Environments ....... 57
2.1 The Constant Depth, Homogeneous Ocean .......................... 57
2.1.1 Point Source Response in a Constant Depth,
Homogeneous Ocean ............................................. 57
2.1.2 Propagating Solutions in an Ocean with Sound-
soft Obstacle ........................................................... 58
2.1.3 The Representation of Propagating Solutions ......... 59
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2.1.4 The Uniqueness Theorem for the Dirichlet
Problem .................................................................. 61
2.1.5 An Existence Theorem for the Dirichlet
Problem .................................................................. 66
2.1.6 Propagating Far-field Patterns ................................ 69
2.1.7 Density Properties of Far-field Patterns ................... 72
2.1.8 Complete Sets in L2(δΩ) .......................................... 72
2.1.9 Dense Sets in L2(δΩ) ............................................... 74
2.1.10 The Projection Theorem in VN ................................. 76
2.1.11 Injection Theorems for the Far-field Pattern
Operator .................................................................. 79
2.1.12 An Approximate Boundary Integral Method for
Acoustic Scattering in Shallow Oceans ................... 83
2.2 Scattered Waves in a Stratified Medium ................................ 92
2.2.1 Green’s Function of a Stratified Medium and the
Generalized Sommerfeld Radiation Condition ......... 92
2.2.2 Scattering of Acoustic Waves by an Obstacle in
a Stratified Space .................................................... 96
2.2.3 Reciprocity Relations .............................................. 98
2.2.4 Completeness of the Far-field Patterns ................... 101
3. Inverse Scattering Problems in Ocean
Environments ...................................................................... 107
3.1 Inverse Scattering Problems in Homogeneous Oceans ......... 107
3.1.1 Inverse Problems and Their Approximate
Solutions ................................................................. 108
3.1.2 Inverse Scattering Using Generalized Herglotz
Functions ................................................................ 114
3.2 The Generalized Dual Space Indicator Method ...................... 123
3.2.1 Acoustic Wave in a Wave Guide with an
Obstacle .................................................................. 123
3.3 Determination of an Inhomogeneity in a Two-layered
Wave Guide ............................................................................ 129
3.3.1 Numerical Example ................................................. 133
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Contents ix
3.4 The Seamount Problem .......................................................... 133
3.4.1 Formulation ............................................................. 133
3.4.2 Uniqueness of the Seamount Problem .................... 135
3.4.3 A Linearized Algorithm for the Reconstruction of
a Seamount ............................................................. 139
3.5 Inverse Scattering for an Obstacle in a Stratified Medium ..... 142
3.5.1 Formulation of the Inverse Problem ........................ 142
3.5.2 Uniqueness ............................................................. 144
3.5.3 An Example of Nonuniqueness ............................... 147
3.5.4 The Far-field Approximation Method ....................... 148
3.6 The Intersecting Canonical Body Approximation .................... 154
3.6.1 Forward and Inverse Scattering Problems for a
Body in Free Space ................................................. 154
3.6.2 A Method for the Reconstruction of the Shape of
the Body Using the ICBA as the Estimator .............. 156
3.6.3 Use of the K Discrepancy Functional and a
Perturbation Technique ........................................... 157
3.6.4 More on the Ambiguity of Solutions of the
Inverse Problem Arising from Use of the ICBA ........ 158
3.6.5 Method for Reducing the Ambiguity of the
Boundary Reconstruction ........................................ 159
3.7 The ICBA for Shallow Oceans: Objects of Revolution ............ 162
3.7.1 Derivation of the Recurrences for Calculation of
the Scattered Field .................................................. 163
3.7.2 Numerical Simulation of Object Reconstruction
Using ICBA ............................................................. 166
3.7.3 3D Objects in a Shallow Ocean ............................... 168
4. Oceans over Elastic Basements ........................................ 171
4.1 A Uniform Ocean over an Elastic Seabed .............................. 171
4.1.1 The Boundary Integral Equation Method for the
Direct Problem ........................................................ 174
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4.1.2 Far-field and Near-field Estimates for the Green’s
Function .................................................................. 177
4.1.3 The Far-field Approximation .................................... 180
4.1.4 Near-field Approximations ....................................... 183
4.1.5 Approximating the Propagation Solution ................. 184
4.1.6 Computing the Scattered Solution ........................... 186
4.2 Undetermined Coefficient Problem for the Seabed ................ 189
4.2.1 Numerical Determination of the Seabed
Coefficients ............................................................. 191
4.3 The Nonhomogeneous Water Column, Elastic Basement
System .................................................................................... 193
4.4 An Inner Product for the Ocean–Seabed System .................. 201
4.5 Numerical Verification of the Inner Product ............................ 206
4.6 Asymptotic Approximations of the Seabed ............................. 208
4.6.1 A Thin Plate Approximation for an Elastic
Seabed ................................................................... 208
4.6.2 A Thick Plate Approximation for the Elastic
Seabed ................................................................... 214
5. Shallow Oceans over Poroelastic Seabeds ...................... 217
5.1 Introduction ............................................................................. 217
5.2 Elastic Model of a Seabed ...................................................... 217
5.3 The Poroelastic Model of a Seabed ....................................... 219
5.3.1 Constitutive Equations for an Isotropic Porous
Medium ................................................................... 219
5.3.2 Dynamical Equations for a Porous Medium ............. 220
5.3.3 Calculation of the Coefficients in the Biot Model ..... 222
5.3.4 Experimental Determination of the Biot–Stoll
Inputs ...................................................................... 226
5.4 Solution of the Time-harmonic Biot Equations ....................... 229
5.4.1 Simplification of the Equations ................................ 229
5.4.2 Speeds of Compressional and Shear Waves .......... 232
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Contents xi
5.4.3 Solution of the Differential Equations for a
Poroelastic Layer .................................................... 247
5.5 Representation of Acoustic Pressure ..................................... 252
5.5.1 Differential Equations for Pressure and Vertical
Displacement in the Ocean ..................................... 253
5.5.2 Interface Conditions ................................................ 253
5.5.3 Green’s Function Representation of Acoustic
Pressure ................................................................. 255
5.6 Sound Transmission over a Poroelastic Half-space ............... 257
6. Homogenization of the Seabed and Other Asymptotic
Methods ............................................................................... 267
6.1 Low Shear Asymptotics for Elastic Seabeds .......................... 267
6.1.1 The Wentzel–Kramers–Brillouin Expansion of
the Displacements ................................................... 269
6.1.2 The Regular Perturbation Expansion ....................... 270
6.1.3 A Singular Perturbation Problem for the Love
Function .................................................................. 271
6.2 Homogenization of the Seabed .............................................. 273
6.2.1 Time-variable Solutions in Rigid Porous Media ....... 274
6.3 Time-harmonic Solutions in a Periodic Poroelastic
Medium ................................................................................... 279
6.3.1 Inner Expansion and Homogenized System ............ 281
6.3.2 Interface Matching and Boundary Layers ................ 284
6.4 Rough Surfaces ...................................................................... 290
6.5 A Numerical Example ............................................................. 296
Bibliography ............................................................................. 299
Index .......................................................................................... 333
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