Marine Acoustics: Direct and Inverse Problems

349 Pages·2003·31.32 MB·English

by  James L. Buchanan

MARINE
ACOUSTICS
Direct and Inverse Problems
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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 Mecanique et d'Acoustique
Marseille, France
Yongzhi S. Xu
University of Tennessee at Chattanooga
Chattanooga, Tennessee
siam.
Society for Industrial and Applied Mathematics
Philadelphia
Copyright © 2004 by the Society for Industrial and Applied Mathematics.
109876543 21
All rights reserved. Printed in the United States of America. No part of this book
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.
siam
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 vStrain Tensors and Displacement Vectors 13
1.3.6 Infinitesimal Strains and Rotations 15
1.3.7 Lagrangian and Eulerian Strains in the Framework of In-
finitesimal Deformations 16
1.3.8 Strain Invariants and Principal Directions 17
1.3.9 Area and Volume Changes Due to Infinitesimal Deforma-
tions 18
1.3.10 Kinematics 19
1.3.11 Material Derivatives of Line, Surface, and Volume Inte-
grals over Regions Devoid of Discontinuities 21
1.3.12 Material Derivatives of Integrals over Regions Contain-
ing a Discontinuity Surface 23
1.3.13 Conservation of Mass Law for Uniform Bodies 24
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 Momen-
tum 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
V
vi Contents
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, Ho-
mogeneous 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 Heteroge-
neous, Inviscid Fluids from Navier's Equation 45
1.4.10 Boundary Conditions between Two Linear, Isotropic, Ho-
mogeneous, 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
1.5.3 The Frequency-Domain Radiation Condition at Infinity . 50
1.5.4 Governing Equations for the Frequency-Domain Formu-
lation of Wave Propagation in an Unbounded, Heteroge-
neous, Inviscid Fluid Medium 51
1.5.5 Governing Equations for the Frequency-Domain Formu-
lation of Wave Propagation in Two Contiguous, Semi-
Infinite, Heterogeneous, Inviscid Fluid Media 51
1.5.6 Governing Equations for the Frequency-Domain Formu-
lation of Wave Propagation in an Unbounded, Heteroge-
neous, Isotropic, Elastic Solid 52
1.5.7 Governing Equations for the Frequency-Domain Formu-
lation of Wave Propagation in Two Semi-Infinite, Hetero-
geneous, Isotropic, Elastic Solid Media in Welded Contact 52
1.5.8 Governing Equations for the Frequency-Domain Formu-
lation of Wave Propagation in a Semi-Infinite Domain
Occupied by a Heterogeneous, Inviscid Fluid Contigu-
ous with a Semi-Infinite Domain Occupied by a Hetero-
geneous, 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
Contents VII
2.1.1 Point Source Response in a Constant Depth, Homoge-
neous Ocean 57
2.1.2 Propagating Solutions in an Ocean with Sound-Soft Ob-
stacle 58
2. .3 The Representation of Propagating Solutions 59
2. .4 The Uniqueness Theorem for the Dirichlet Problem ... 61
2. .5 An Existence Theorem for the Dirichlet Problem 66
2. .6 Propagating Far-Field Patterns 69
2. .7 Density Properties of Far-Field Patterns 72
2. .8 Complete Sets in L2( ) 72
2. .9 Dense Sets in L2( ) 74
2. .10 The Projection Theorem in VN 76
2. .11 Injection Theorems for the Far-Field Pattern Operator . . 79
2. . 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 Gener-
alized Sommerfeld Radiation Condition 92
2.2.2 Scattering of Acoustic Waves by an Obstacle in a Strati-
fied 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
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
viii Contents
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 Prob-
lem Arising from Use of the ICBA 158
3.6.5 Method for Reducing the Ambiguity of the Boundary Re-
construction 159
3.7 The ICB A for Shallow Oceans: Objects of Revolution 162
3.7.1 Derivation of the Recurrences for Calculation of the Scat-
tered 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
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
5.4.3 Solution of the Differential Equations for a Poroelastic
Layer 247
5.5 Representation of Acoustic Pressure 252
Contents ix
5.5.1 Differential Equations for Pressure and Vertical Displace-
ment 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 Dis-
placements 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