Preface ........................................................ xv
Authors ...................................................... xvii
1 Contents ..................................................... 1
1.1 Introduction ............................................ 1
1.2 Contents Flow Chart ..................................... 2
2 Mathematical Foundations .................................... 13
2.1 Scalar, Vector, and Tensor Fields ...................... 13
2.1.1 Vector of position .............................. 13
2.1.2 Scalar and vector fields ........................ 17
2.1.3 Vector products ................................. 19
2.1.4 Tensor fields ................................... 29
2.2 Vector and Tensor Analysis ............................. 39
2.2.1 Del-operator: Gradient dyadic, gradient,
divergence, and curl ............................ 39
2.2.2 Application of the del-operator to products of
field quantities, chain rules, delta-operator ... 46
2.2.3 Gauss' theorem, Gauss' integral theorems,
Green's formulas ................................ 51
2.2.4 Cylindrical and spherical coordinates ........... 53
2.3 Time and Spatial Spectral Analysis with Fourier
Transforms ............................................. 60
2.3.1 Complex numbers and complex valued functions
of a complex variable ........................... 61
2.3.2 Time domain spectral analysis ................... 66
2.3.3 Fourier transformation rules .................... 70
2.3.4 Analytic signal and Hilbert transform ........... 71
2.3.5 Spatial domain spectral analysis ................ 75
2.4 Delta Function ......................................... 78
2.4.1 Delta function as distribution .................. 78
2.4.2 Delta distribution calculus ..................... 79
2.4.3 Delta function and Fourier transform ............ 81
2.4.4 Three-dimensional delta function ................ 83
2.4.5 Singular function of a surface .................. 84
3 Governing Equations of Elastodynamics ....................... 87
3.1 Newton-Cauchy Equation of Motion and Deformation Rate
Equation in the Time and Frequency Domain .............. 87
3.2 Physical Foundations ................................... 90
3.2.1 Mass conservation ............................... 90
3.2.2 Convective time derivative ...................... 92
3.2.3 Linear momentum conservation: Newton-Cauchy
equation of motion .............................. 94
3.2.4 Angular momentum conservation: Stress tensor
symmetry ........................................ 96
3.2.5 Deformation rate equation ....................... 98
3.2.6 Linear elastodynamics: Newton-Cauchy equation
of motion and deformation rate equation ........ 101
3.3 Transition and Boundary Conditions .................... 102
3.3.1 Discontinuous material properties:
Homogeneous and inhomogeneous transition
conditions ..................................... 102
3.3.2 Infinite discontinuity of material
properties: Boundary conditions ................ 107
3.3.3 Boundary between elastic and fluid materials:
Homogeneous and inhomogeneous transition
conditions ..................................... 108
3.3.4 Boundary between two elastic materials with
fluid coupling: Homogeneous and inhomogeneous
transition conditions .......................... 110
4 Constitutive Equations, Governing Equations,
Elastodynamic Energy Conservation .......................... 113
4.1 Constitutive Equations ................................ 113
4.2 Linear Nondissipative Materials: Cauchy-Hooke Law ..... 114
4.2.1 Anisotropic materials, Voigt notation,
transversely isotropic materials ............... 114
4.2.2 Isotropic materials ............................ 118
4.2.3 Elastodynamic governing equations .............. 119
4.3 Elastodynamic Energy Conservation Theorem for
Nondissipative Materials in the Time and Frequency
Domains ............................................... 119
4.3.1 Elastodynamic Poynting vector in the time
domain ......................................... 119
4.3.2 Complex valued elastodynamic Poynting vector
in the frequency domain ........................ 122
4.4 Linear Dissipative Materials .......................... 126
4-4.1 Maxwell model .................................. 126
4.4.2 Elastodynamic energy conservation law:
Dissipation energy ............................. 129
4.4.3 Rayleigh and Kelvin-Voigt model ................ 131
4.4.4 Relaxation models .............................. 132
4.5 Piezoelectricity and Magnetostriction ................. 133
4.5.1 Piezoelectricity ............................... 133
4.5.2 Magnetostriction ............................... 140
5 Acoustics .................................................. 143
5.1 Governing Equations of Acoustics ...................... 143
5.2 Transition and Boundary Conditions .................... 144
5.3 Wave Equations in the Time and Frequency Domains ...... 146
5.4 Solutions of the Homogeneous Acoustic Wave Equations
in Homogeneous Materials: Plane Longitudinal
Pressure Waves ........................................ 148
5.5 Acoustic Source Fields in Homogeneous Materials:
Point Source Synthesis with Green Functions ........... 150
5.5.1 Green functions for pressure sources ........... 150
5.5.2 Green functions for velocity sources ........... 152
5.5.3 Justification of the distributional term
appearing in the second rank Green tensor of
acoustics ...................................... 155
5.6 Huygens' Principle for Acoustic Scattered Fields
in Homogeneous Materials .............................. 157
5.6.1 Huygens' principle ............................. 157
5.6.2 Acoustic fields scattered by inhomogeneities
with soft and rigid boundaries, Kirchhoff
approximation .................................. 159
5.6.3 Acoustic fields scattered by penetrable
inhomogeneities, Born approximation ............ 163
6 Electromagnetism ........................................... 169
6.1 Maxwell Equations, Poynting Vector, Lorentz Force ..... 169
6.1.1 Maxwell equations .............................. 169
6.1.2 Vacuum Maxwell equations ....................... 170
6.1.3 Poynting's theorem ............................. 171
6.1.4 Lorentz force .................................. 172
6.2 Transition and Boundary Conditions .................... 173
6.3 Constitutive Equations: Permittivity and
Permeability; Dissipation: Susceptibility Kernels
and Conductivity ...................................... 175
6.3.1 Permittivity and permeability .................. 175
6.3.2 Susceptibility kernels ......................... 176
6.3.3 Conductivity ................................... 178
6.4 Wave Equations in the Time and Frequency Domains ...... 179
6.4.1 Wave equations in the time domain .............. 179
6.4.2 Wave equations in the frequency domain ......... 181
6.5 Solutions of Homogeneous Electromagnetic Wave
Equations in Homogeneous Isotropic Materials: Plane
Transverse Electromagnetic Waves ...................... 182
6.5.1 Nondissipative materials ....................... 182
6.5.2 Dissipative materials .......................... 185
6.6 Electromagnetic Source Fields in Homogeneous
Isotropic Materials, Electromagnetic Tensor Green
Functions ............................................. 186
6.6.1 Electric scalar potential and magnetic
vector potential ............................... 186
6.6.2 Electric second rank Green tensor .............. 188
6.6.3 Far-field approximation ........................ 189
6.6.4 Hertzian dipole ................................ 190
6.6.5 Magnetic second-rank Green tensor .............. 191
6.7 Electromagnetic Scattered Fields; Electromagnetic
Formulation of Huygens' Principle ..................... 192
6.7.1 Electromagnetic formulation of Huygens'
principle ...................................... 192
6.7.2 Electromagnetic fields scattered by perfect
electrical conductors: EFIE and MFIE ........... 194
6.7.3 Kirchhoff approximation ........................ 196
6.7.4 Electromagnetic fields scattered by
penetrable inhomogeneities: Lippmann-
Schwinger integral equation .................... 197
6.7.5 Born approximation ............................. 200
6.7.6 Scattering tensor .............................. 201
6.8 Two-Dimensional Electromagnetism: TM- and TE-
Decoupling ............................................ 201
6.8.1 TM-field ....................................... 202
6.8.2 TE-field ....................................... 204
7 Vector Wave Equations ...................................... 205
7.1 Wave Equations for Anisotropic and Isotropic
Nondissipative Materials .............................. 205
7.1.1 Inhomogeneous anisotropic materials ............ 205
7.1.2 Homogeneous anisotropic materials .............. 209
7.1.3 Homogeneous isotropic materials ................ 210
7.1.4 Inhomogeneous isotropic materials .............. 211
7.2 Helmholtz Decomposition for Homogeneous Isotropic
Materials: Pressure and Shear Waves ................... 211
7.3 Decoupling of Scalar SH-Waves for Inhomogeneous
Isotropic Two-Dimensional Materials ................... 214
7.4 Frequency Domain Wave Equations for Nondissipative
and Dissipative Materials ............................. 216
7.4.1 Frequency domain wave equations for
nondissipative materials ....................... 917
7.4.2 Frequency domain wave equations for
dissipative materials .......................... 21
8 Elastic Plane Waves in Homogeneous Materials ............... 219
8.1 Homogeneous Plane Waves in Isotropic Nondissipative
Materials ............................................. 219
8.1.1 One-dimensional plane waves: Primary
longitudinal and secondary transverse waves .... 219
8.1.2 Three-dimensional plane waves: Primary
longitudinal pressure and secondary
transverse shear waves ......................... 232
8.2 Inhomogeneous Plane Waves in Isotropic
Nondissipative Materials .............................. 249
8.3 Plane Waves in Anisotropic Nondissipative Materials ... 257
8.3.1 Plane waves in anisotropic materials ........... 257
8.3.2 Plane waves in transversely isotropic
materials ...................................... 265
8.4 Plane Waves in Isotropic Dissipative Materials ........ 282
8.4.1 Homogeneous plane waves ........................ 283
8.4.2 Inhomogeneous plane waves ...................... 287
9 Reflection, Transmission, and Mode Conversion of Elastic
Plane Waves at Planar Boundaries between Homogeneous
Nondissipative Materials ................................... 291
9.1 Stress-Free Planar Boundary of a Homogeneous
Isotropic Nondissipative Elastic Half-Space ........... 291
9.1.1 Primary longitudinal pressure wave incidence ... 291
9.1.2 Secondary transverse vertical shear wave
incidence ...................................... 304
9.1.3 Secondary transverse horizontal shear wave
incidence ...................................... 323
9.2 Planar Boundary between Homogeneous Isotropic
Nondissipative Elastic Half-Spaces .................... 326
9.2.1 SH-wave incidence .............................. 326
9.2.2 P- and SV-waves incidence ...................... 338
9.3 Planar Boundary between a Homogeneous Isotropic
Nondissipative and a Homogeneous Transversely
Isotropic Nondissipative Half-Space ................... 349
9.3.1 Inhomogeneous elastic plane waves in
isotropic materials ............................ 351
9.3.2 Inhomogeneous plane SH-waves in transversely
isotropic materials ............................ 353
9.3.3 Reflection and transmission of plane
SH-Waves at the planar boundary between
homogeneous isotropic and homogeneous
transversely isotropic nondissipative
materials ...................................... 357
9.3.4 Reflection, transmission, and mode
conversion of plane SV-waves at the planar
boundary between homogeneous isotropic and
homogeneous transversely isotropic
nondissipative materials ....................... 369
10 Rayleigh Surface Waves ..................................... 377
10.1 Planar Surfaces ....................................... 377
10.2 Lightly Curved Surfaces ............................... 381
11 Plane Wave Spatial Spectrum ................................ 385
11.1 Acoustic Plane Wave Spatial Spectrum ................. 385
11.1.1 Plane wave spatial spectrum .................... 385
11.1.2 Propagator as spatial filter ................... 388
11.1.3 Approximate evaluation with the stationary
phase method ................................... 391
11.2 Elastic Plane Wave Spatial Spectrum ................... 394
12 Ultrasonic Beams and Wave Packets .......................... 397
12.1 Gaussian Beams as PaJaxial Approximation of
a Spatial Plane Wave Spectrum ......................... 397
12.2 Pulsed Beams as Exact Solutions of an Approximate
Wave Equation ......................................... 404
12.3 Pulsed Beams as Approximate Solutions of Eikonal
and Transport Equations ............................... 411
12.3.1 Eikonal and transport equations for acoustic
beams .......................................... 411
12.3.2 Eikonal and transport equations for elastic
beams .......................................... 414
13 Point Sources in Homogeneous Isotropic Infinite Space,
Elastodynamic Source Fields ................................ 423
13.1 Homogeneous Infinite Space Scalar Green Function ...... 423
13.1.1 Time harmonic Green function ................... 423
13.1.2 Time domain Green function ..................... 430
13.1.3 Far-field approximation ........................ 435
13.1.4 Point source synthesis of scalar source
fields with the scalar Green function .......... 438
13.2 Homogeneous Isotropic Infinite Space Green Tensors
of Elastodynamics ..................................... 442
13.2.1 Second-rank Green tensor ....................... 442
13.2.2 Particle displacement of a point source force
density, point radiation characteristic ........ 44
13.2.3 Third-rank Green tensor ........................ 457
13.2.4 Particle displacement of a point source
deformation rate, point radiation
characteristic ................................. 461
13.2.5 Fourth-rank Green tensor: Stress tensor of
a point source deformation rate ................ 464
13.3 Two- and Three-Dimensional Elastodynamic Source
Fields ................................................ 471
13.1.1 Elastodynamic point source synthesis ........... 471
13.3.2 Far-field approximations of three-dimensional
elastodynamic source fields .................... 472
13.3.3 Far-field approximations of two-dimensional
elastodynamic source fields .................... 477
13.3.4 -examples for two- and three-dimensional
elastodynamic and acoustic source far-fields:
Planar rectangular, planar circular, and
planar strip-like force density distributions
with constant amplitude ........................ 483
13.4 Elementary Spherical Waves and Plane Waves ............ 496
13.4.1 Spatial plane wave spectrum of the
three-dimensional scalar Green function:
Weyl's integral representation ................. 496
13.4.2 Spatial cylindrical wave spectrum of the
three-dimensional scalar Green function:
Sommerfeld integral ............................ 499
14 Force Density and Dilatation Rate Sources on Surfaces of
Homogeneous Isotropic Half-Spaces, Radiation Fields of
Piezoelectric Transducers .................................. 501
14.1 Acoustic Half-Spaces with Soft or Rigid Surfaces ...... 501
14.1.1 AFIT-wavefronts of the line and strip-like
rigidly baffled aperture radiator .............. 501
14.1.2 Scalar half-space Green functions, Rayleigh-
Sommerfeld integrals, plane wave spectral
decomposition (integral representations of
the Weyl type) ................................. 502
14.1.3 Far-field evaluation of Rayleigh-Sommerfeld
and Weyl integrals ............................. 514
14.2 Strip-Like Normal and Tangential Force Density
Distributions on the Stress-Free Surface of an
Elastic Half-Space: Spectral Plane Wave
Decomposition of the Two-Dimensional Second-Rank
Green Tensor .......................................... 515
14.2.1 EFIT-wavefronts of the linear and strip-like
aperture radiator on the stress-free surface
of an elastic half-space ....................... 515
14.2.2 Strip-like normal and tangential force
density distributions on the stress-free
surface of an elastic half-space ............... 517
14.2.3 Spectral plane wave decomposition of the
two-dimensional second-rank Green tensor ....... 523
14.2.4 Far-field radiation characteristics of normal
and tangential line force densities on the
surface of a stress-free half-space ............ 525
14.3 Circular Normal Force Density Distribution on the
Stress-Free Surface of an Elastic Half-Space: Point
Source Characteristic ................................. 528
14.3.1 Integral representation of the Sommerfeld
type ........................................... 528
14.3.2 Point source characteristics ................... 530
14.4 Radiation Fields of Piezoelectric Transducers ......... 535
15 Scatterers in Homogeneous Isotropic Nondissipative
Infinite Spaces ............................................ 555
15.1 Huygens' Principle .................................... 556
15.1.1 Mathematical foundation of Huygens' principle
of elastodynamics based on physical arguments .. 557
15.1.2 Mathematical derivation of Huygens' principle
for scalar acoustic fields ..................... 564
15.1.3 Mathematical derivation of Huygens' principle
for elastodynamic fields ....................... 569
15.2 Integral Equations for Secondary Surface Deformation
Sources on Scatterers with Stress-Free Surfaces:
Displacement Field Integral Equation and Stress
Field Integral Equation ............................... 572
15.2.1 Integral equations relating secondary sources .. 573
15.2.2 Scatterers with stress-free surfaces: DFIE
and Stress Field Integral Equation (SFIE) ...... 579
15.2.3 Kirchhoff approximation in elastodynamics ..... 586
15.3 Integral Equations for the Equivalent Sources of
Penetrable Scatterers ................................. 592
15.3.1 Lippmann-Schwinger integral equations for
equivalent volume sources of inhomogeneous
anisotropic scatterers ......................... 592
15.3.2 Born approximation for inhomogeneous
anisotropic scatterers ......................... 595
15.3.3 Coupled integral equations for equivalent
surface sources of homogeneous isotropic
scatterers ..................................... 596
15.4 Scattering Tensor; Far-Fields ......................... 598
15.4.1 Scattering tensor .............................. 598
15.4.2 Two-dimensional scalar scattering problems:
Pulsed SH-far-fields of circular cylindrical
voids and strip-like cracks .................... 602
15.4.3 Two-dimensional scattering problems: Pulsed
P-SV-far-fields of circular cylindrical voids
and strip-like cracks .......................... 614
15.4.4 Three-dimensional scattering problems: Pulsed
P-S-far-fields of spherical voids .............. 626
15.5 3D System Model of Pulsed Ultrasonic Scattering
within Kirchhoff's Approximation ...................... 649
16 Inverse Scattering: US-NDT Imaging ......................... 665
16.1 SAFT: Synthetic Aperture Focusing Technique ........... 665
16.1.1 Integration along diffraction curves
(surfaces) and back propagation ................ 665
16.1.2 Pitch-catch and pulse-echo versions of SAFT .... 669
16.1.3 SAFT with Hilbert transformed pulse data ....... 670
16.2 FT-SAFT: Fourier Transform Synthetic Aperture
Focusing Technique .................................... 673
16.2.1 Scalar secondary sources: Contrast sources ..... 674
16.2.2 Contrast source inversion ...................... 677
16.2.3 Generalized holography ......................... 679
16.2.4 FT-SAFT ........................................ 680
16.2.5 Exact derivation of pulse-echo SAFT for
planar measurement surfaces .................... 688
Appendix Collection of Mathematical Definitions and
Identities .................................................... 695
A.l Vector Identities ..................................... 695
A.2 Tensor Identities ..................................... 696
A.2.1 Permutation tensor ............................. 696
A.2.2 Products ....................................... 698
A.2.3 Traces ......................................... 701
A.2.4 Determinants ................................... 702
A.2.5 Adjoints and inverses .......................... 703
A.3 Coordinate Systems .................................... 705
A.3.1 Cartesian coordinates .......................... 705
A.4 Curvilinear Orthogonal Coordinates .................... 708
A.5 Cylindrical Coordinates ............................... 711
A.6 Spherical Coordinates ................................. 715
A.7 Identities for the Del Operator ....................... 720
A.7.1 General scalar, vector, and tensor fields ...... 720
A.8 Special Vector Fields Depending on the Vector of
Position .............................................. 723
References .................................................... 727
Index ......................................................... 737
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