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Tables of Contents for Elastic Wave Propagation and Generation in Seismology
Chapter/Section Title
Page #
Page Count
Preface
xiii
Acknowledgements
xviii
Introduction to tensors and dyadics
1
39
Introduction
1
1
Summary of vector analysis
2
5
Rotation of Cartesian coordinates. Definition of a vector
7
4
Cartesian tensors
11
19
Tensor operations
14
2
Symmetric and anti-symmetric tensors
16
1
Differentiation of tensors
17
1
The permutation symbol
18
1
Applications and examples
19
4
Diagonalization of a symmetric second-order tensor
23
5
Isotropic tensors
28
1
Vector associated with a second-order anti-symmetric tensor
28
1
Divergence or Gauss' theorem
29
1
Infinitesimal rotations
30
2
Dyads and dyadics
32
8
Dyads
33
1
Dyadics
34
6
Deformation. Strain and rotation tensors
40
19
Introduction
40
1
Description of motion. Lagrangian and Eulerian points of view
41
2
Finite strain tensors
43
2
The infinitesimal strain tensor
45
5
Geometric meaning of εij
46
3
Proof that εij is a tensor
49
1
The rotation tensor
50
1
Dyadic form of the strain and rotation tensors
51
1
Examples of simple strain fields
52
7
The stress tensor
59
25
Introduction
59
1
Additional continuum mechanics concepts
59
5
Example
63
1
The stress vector
64
3
The stress tensor
67
3
The equation of motion. Symmetry of the stress tensor
70
2
Principal directions of stress
72
1
Isotropic and deviatoric components of the stress tensor
72
1
Normal and shearing stress vectors
73
2
Stationary values and directions of the normal and shearing stress vectors
75
4
Mohr's circles for stress
79
5
Linear elasticity -- the elastic wave equation
84
16
Introduction
84
1
The equation of motion under the small-deformation approximation
85
1
Thermodynamical considerations
86
2
Strain energy
88
2
Linear elastic and hyperelastic deformations
90
2
Isotropic elastic solids
92
4
Strain energy density for the isotropic elastic solid
96
1
The elastic wave equation for a homogeneous isotropic medium
97
3
Scalar and elastic waves in unbounded media
100
29
Introduction
100
1
The 1-D scalar wave equation
100
3
Example
103
1
The 3-D scalar wave equation
103
4
Plane harmonic waves. Superposition principle
107
4
Spherical waves
111
1
Vector wave equation. Vector solutions
112
4
Properties of the Hansen vectors
115
1
Harmonic potentials
116
1
Vector Helmholtz equation
116
1
Elastic wave equation without body forces
117
8
Vector P- and S-wave motion
118
1
Hansen vectors for the elastic wave equation in the frequency domain
119
2
Harmonic elastic plane waves
121
2
P-, SV-, and SH-wave displacements
123
2
Flux of energy in harmonic waves
125
4
Plane waves in simple models with plane boundaries
129
59
Introduction
129
2
Displacements
131
3
Boundary conditions
134
1
Stress vector
135
1
Waves incident at a free surface
136
16
Incident SH waves
136
1
Incident P waves
137
7
Incident SV waves
144
8
Waves incident on a solid-solid boundary
152
16
Incident SH waves
152
5
Incident P waves
157
7
Incident SV waves
164
4
Waves incident on a solid-liquid boundary
168
1
Incident P waves
168
1
Incident SV waves
169
1
P waves incident on a liquid-solid boundary
169
1
Solid layer over a solid half-space
170
18
Incident SH waves
172
7
Incident P and SV waves
179
9
Surface waves in simple models -- dispersive waves
188
46
Introduction
188
1
Displacements
189
2
Love waves
191
11
Homogeneous half-space
191
1
Layer over a half-space
191
7
Love waves as the result of constructive interference
198
1
Vertically heterogeneous medium
199
3
Rayleigh waves
202
10
Homogeneous half-space
202
4
Layer over a half-space. Dispersive Rayleigh waves
206
3
Vertically heterogeneous medium
209
3
Stoneley waves
212
1
Propagation of dispersive waves
213
21
Introductory example. The dispersive string
214
1
Narrow-band waves. Phase and group velocity
215
5
Broad-band waves. The method of stationary phase
220
7
The Airy phase
227
7
Ray theory
234
44
Introduction
234
1
Ray theory for the 3-D scalar wave equation
235
2
Ray theory for the elastic wave equation
237
5
P and S waves in isotropic media
240
2
Wave fronts and rays
242
6
Medium with constant velocity
244
2
Medium with a depth-dependent velocity
246
1
Medium with spherical symmetry
247
1
Differential geometry of rays
248
6
Calculus of variations. Fermat's principle
254
4
Ray amplitudes
258
11
Scalar wave equation
258
3
Elastic wave equation
261
7
Effect of discontinuities in the elastic parameters
268
1
Examples
269
9
SH waves in a layer over a half-space at normal incidence
270
4
Ray theory synthetic seismograms
274
4
Seismic point sources in unbounded homogeneous media
278
38
Introduction
278
1
The scalar wave equation with a source term
279
2
Helmholtz decomposition of a vector field
281
1
Lame's solution of the elastic wave equation
282
3
The elastic wave equation with a concentrated force in the xj direction
285
10
Type of motion
288
1
Near and far fields
289
2
Example. The far field of a point force at the origin in the x3 direction
291
4
Green's function for the elastic wave equation
295
1
The elastic wave equation with a concentrated force in an arbitrary direction
296
1
Concentrated couples and dipoles
297
3
Moment tensor sources. The far field
300
5
Radiation patterns. SV and SH waves
303
2
Equivalence of a double couple and a pair of compressional and tensional dipoles
305
1
The tension and compression axes
306
2
Radiation patterns for the single couple M31 and the double couple M13 + M31
308
3
Moment tensor sources. The total field
311
5
Radiation patterns
313
3
The earthquake source in unbounded media
316
41
Introduction
316
2
A representation theorem
318
3
Gauss' theorem in the presence of a surface of discontinuity
321
1
The body force equivalent to slip on a fault
322
3
Slip on a horizontal plane. Point-source approximation. The double couple
325
4
The seismic moment tensor
329
2
Moment tensor for slip on a fault of arbitrary orientation
331
7
Relations between the parameters of the conjugate planes
338
1
Radiation patterns and focal mechanisms
339
8
The total field. Static displacement
347
5
Ray theory for the far field
352
5
Anelastic attenuation
357
34
Introduction
357
3
Harmonic motion. Free and damped oscillations
360
4
Temporal Q
362
2
The string in a viscous medium
364
1
The scalar wave equation with complex velocity
365
2
Spatial Q
366
1
Attenuation of seismic waves in the Earth
367
3
Mathematical aspects of causality and applications
370
7
The Hilbert transform. Dispersion relations
371
1
Minimum-phase-shift functions
372
3
The Paley-Wiener theorem. Applications
375
2
Futterman's relations
377
4
Kalinin and Azimi's relation. The complex wave velocity
381
3
t*
384
1
The spectral ratio method. Window bias
384
2
Finely layered media and scattering attenuation
386
5
Hints
391
16
Appendices
407
24
A Introduction to the theory of distributions
407
12
B The Hilbert transform
419
3
C Green's function for the 3-D scalar wave equation
422
3
D Proof of (9.5.12)
425
3
E Proof of (9.13.1)
428
3
Bibliography
431
8
Index
439