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E-raamat: Elastic Waves: High Frequency Theory

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Elastic Waves: High Frequency Theory is concerned with mathematical aspects of the theory of high-frequency elastic waves, which is based on the ray method. The foundations of elastodynamics are presented along with the basic theory of plane and spherical waves. The ray method is then described in considerable detail for bulk waves in isotropic and anisotropic media, and also for the Rayleigh waves on the surface of inhomogeneous anisotropic elastic solids. Much attention is paid to analysis of higher-order terms and to generation of waves in inhomogeneous media. The aim of the book is to present a clear, systematic description of the ray method, and at the same time to emphasize its mathematical beauty. Luckily, this beauty is usually not accompanied by complexity and mathematical ornateness.

Arvustused

"The present book treats high-frequency elastic waves, which occur in many contexts such as in seismic science, acoustics, mathematical physics, variational and tensor calculus, Riemannian and Finsler geometry, functional analysis, etc. [ . . . .] All in all, the book presents the state-of-the-art of the mathematical theory of high-frequency elastic waves."

-Angela Slavova, Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria

Preface xiii
Introduction xv
List of Basic Symbols
xix
1 Basic Notions of Elastodynamics
1(24)
1.1 Displacement, deformation, and stress
1(3)
1.1.1 Displacement vector and strain tensor
1(1)
1.1.2 Stress tensor
2(2)
1.2 Lagrangian approach to mechanical systems
4(2)
1.3 Elastodynamics equations
6(4)
1.3.1 Kinetic and potential energies as quadratic functionals
7(1)
1.3.2 Properties of elastic stiffnesses
7(1)
1.3.3 Derivation of elastodynamics equations
8(2)
1.3.4 Navier and Lame operators
10(1)
1.4 Classical boundary conditions
10(4)
1.4.1 List of boundary conditions
11(2)
1.4.2 Hamilton principle and boundary conditions
13(1)
1.5 Isotropic medium
14(3)
1.5.1 Consequences of the invariance of W with respect to rotations
14(2)
1.5.2 Consequences of the positive definiteness of W
16(1)
1.6 Energy balance
17(1)
1.7 Time-harmonic solutions
18(2)
1.7.1 Basic notions
18(1)
1.7.2 Time-averaging
19(1)
1.8 Reciprocity principle
20(2)
1.8.1 The time-harmonic case
21(1)
1.8.2 Non-time-harmonic case
22(1)
1.9 * Comments to
Chapter 1
22(3)
References to
Chapter 1
23(2)
2 Plane Waves
25(68)
2.1 Plane-wave ansatz
25(2)
2.2 Phase velocity
27(2)
2.2.1 Normal velocity of a moving surface
27(1)
2.2.2 Phase velocity and slowness
28(1)
2.3 Plane waves in unbounded isotropic media
29(10)
2.3.1 Eigenvalue problem
29(1)
2.3.2 Wave P
30(1)
2.3.3 Wave S
31(1)
2.3.4 Time-harmonic waves P and S
31(1)
2.3.5 Polarization of time-harmonic waves P and S
32(3)
2.3.6 Group velocity
35(1)
2.3.7 Energy relations for time-harmonic waves
35(2)
2.3.8 * Potentials
37(2)
2.4 Plane waves in unbounded anisotropic media
39(4)
2.4.1 Eigenvalue problem
39(1)
2.4.2 Phase velocity
40(1)
2.4.3 Group velocity
40(1)
2.4.4 Slowness, slowness surface, velocity surface
41(2)
2.4.5 * Rayleigh principle
43(1)
2.5 * Local velocities and domain of influence
43(7)
2.5.1 Statement of the problem
44(1)
2.5.2 Energy lemma
45(4)
2.5.3 Uniqueness theorem
49(1)
2.6 Reflection of plane waves from a free boundary of an isotropic half-space
50(13)
2.6.1 Upgoing and downgoing waves
50(1)
2.6.2 Waves of polarizations SH and P --- SV
51(1)
2.6.3 The case of polarization SH
51(2)
2.6.4 The case of polarization P --- SV
53(3)
2.6.5 Snell's law
56(1)
2.6.6 Total internal reflection
57(1)
2.6.7 Energy flow in an inhomogeneous wave P
58(1)
2.6.8 * Energy flow under reflection from a boundary and the unitarity of the reflection matrix
58(3)
2.6.9 Reflection of non-time-harmonic waves
61(2)
2.7 Classical plane surface waves in isotropic media
63(9)
2.7.1 Classical Rayleigh wave
64(3)
2.7.2 Classical Love wave
67(3)
2.7.3 * The total internal reflection and constructive interference
70(2)
2.8 Plane surface waves in isotropic layered media
72(2)
2.8.1 Waves P --- SV
73(1)
2.8.2 Waves SH
73(1)
2.9 Plane waves in arbitrarily layered media
74(5)
2.9.1 Eigenvalue problem
74(2)
2.9.2 Virial theorem
76(2)
2.9.3 Group velocity theorem
78(1)
2.10 * Existence of the Rayleigh wave in an anisotropic homogeneous half-space
79(6)
2.10.1 One-dimensional problem and the corresponding energy quadratic form
80(1)
2.10.2 Inhomogeneous plane waves and the continuous spectrum of the operator γ
81(1)
2.10.3 Variational principle
82(1)
2.10.4 Discrete spectrum of γ
83(2)
2.11 Comments to
Chapter 2
85(8)
References to
Chapter 2
87(6)
3 Point Sources and Spherical Waves in Homogeneous Isotropic Media
93(32)
3.1 Delta functions
93(4)
3.2 Scalar point source problems
97(5)
3.2.1 Time-harmonic source
98(1)
3.2.2 Determining a unique solution. Key idea of the limiting absorption principle
99(1)
3.2.3 Nonstationary source
100(2)
3.3 Point sources in a homogeneous, isotropic, elastic medium. Time-harmonic case
102(8)
3.3.1 * Center of expansion and center of rotation as limit problems for spherical emitters
103(3)
3.3.2 Concentrated force
106(4)
3.4 Point sources in a homogeneous, isotropic, elastic medium. Non-stationary case
110(4)
3.5 Conditions at infinity and uniqueness
114(7)
3.5.1 Limiting absorption principle
114(1)
3.5.2 * Radiation conditions
115(5)
3.5.3 Uniqueness theorem in the nonstationary case
120(1)
3.6 * Comments to
Chapter 3
121(4)
References to
Chapter 3
122(3)
4 Ray Method for Volume Waves in Isotropic Media
125(66)
4.1 Ray ansatz and transport equations
125(5)
4.1.1 Ray ansatz and local plane waves
125(3)
4.1.2 Recurrent system
128(1)
4.1.3 Waves P and S
129(1)
4.2 Eikonal equation and rays
130(8)
4.2.1 Fermat functional and rays
130(1)
4.2.2 Solving the eikonal equation with the help of rays
131(1)
4.2.3 Cauchy problem for the eikonal equation
132(3)
4.2.4 Ray coordinates and field of rays
135(1)
4.2.5 * Complex eikonal
136(2)
4.3 Solving transport equations. The wave P
138(11)
4.3.1 Zeroth-order approximation. Consistency condition and the Umov equation
138(2)
4.3.2 Zeroth-order approximation. Formulas for ε and υP0
140(4)
4.3.3 Ray coordinates and the geometrical spreading
144(3)
4.3.4 Anomalous polarization
147(1)
4.3.5 * First longitudinally polarized correction
148(1)
4.3.6 * Higher-order approximations
148(1)
4.4 Solving transport equations. The wave S
149(6)
4.4.1 Zeroth-order approximation. A preliminary consideration. The Rytov law
149(3)
4.4.2 Rytov law. The case of complex υS0
152(1)
4.4.3 Anomalous polarization
153(1)
4.4.4 * First transversely polarized correction
154(1)
4.4.5 * Higher-order approximations
155(1)
4.5 Reflection of the wave defined by a ray expansion
155(5)
4.5.1 Ansatz and the statement of the problem of determining reflected and converted waves
156(2)
4.5.2 Constructing the wavefield in higher orders
158(2)
4.6 * Riemannian geometry in ray theory
160(5)
4.6.1 Riemannian geometry and Fermat principle
160(3)
4.6.2 Parallel translation in Riemannian metric and the Rytov law
163(2)
4.7 Geometrical spreading in a homogeneous medium
165(3)
4.7.1 Lines of curvature and Rodrigues' formula
166(1)
4.7.2 Derivation of a formula for J
167(1)
4.7.3 On the vanishing of the geometrical spreading and caustics
168(1)
4.8 * Geometrical spreading under reflection, transmission, and conversion in the planar case
168(11)
4.8.1 Specific features of the planar case
168(2)
4.8.2 Jacobi equation and geometrical spreading
170(2)
4.8.3 Calculation of the initial data for the Jacobi equation in the case of monotype reflection
172(4)
4.8.4 Calculation of the initial data for the Jacobi equation for the case of reflection with conversion
176(1)
4.8.5 Calculation of the initial data for the case of transmission
176(1)
4.8.6 Case of constant velocities
177(2)
4.8.7 Focusing under reflection
179(1)
4.9 * Nonstationary versions of the ray method
179(2)
4.9.1 High-frequency asymptotics and asymptotics with respect to smoothness
179(2)
4.9.2 Other nonstationary versions
181(1)
4.10 * Comments to
Chapter 4
181(10)
References to
Chapter 4
184(7)
5 Ray Method for Volume Waves in Anisotropic Media
191(16)
5.1 Recurrent system and eikonal equation
191(2)
5.1.1 Recurrent system
191(1)
5.1.2 Eikonal equation
192(1)
5.2 Rays and wavefronts
193(3)
5.2.1 Cauchy problem for a nonlinear equation
193(1)
5.2.2 Characteristic system
194(1)
5.2.3 Special case: eikonal equation
195(1)
5.3 * Fermat principle and Finsler geometry
196(4)
5.3.1 Rays as extremals of a certain functional of the calculus of variations
196(2)
5.3.2 Finsler metric
198(1)
5.3.3 Fermat principle
199(1)
5.3.4 Concluding remarks
199(1)
5.4 Solution of transport equation for u0
200(1)
5.4.1 Consistency condition and the Umov equation
200(1)
5.5 Higher-order terms
201(2)
5.6 * Comments to
Chapter 5
203(4)
References to
Chapter 5
204(3)
6 Point Sources in Inhomogeneous Isotropic Media. Wave's from a Center of Expansion. Wave P from a Center of Rotation
207(24)
6.1 Statement of the problem and elementary consideration
208(3)
6.1.1 Statement of the problem
208(1)
6.1.2 Non-applicability of ray formulas near the source point
209(1)
6.1.3 Elementary locality approach
209(2)
6.2 Structure of the wavefield near the source point in more detail
211(3)
6.2.1 Recurrent system
211(2)
6.2.2 * On solving equations (6.25)--(6.27)
213(1)
6.2.3 Intermediate zone
214(1)
6.3 Preliminary notes on calculating diffraction coefficients Χ1 for a center of expansion and ψ1 for a center of rotation
214(3)
6.3.1 Wavefield in the homogeneous-medium approximation
215(1)
6.3.2 How to find Χ1, or discussion of the matching procedure
215(1)
6.3.3 The case of a center of rotation
216(1)
6.4 * Operator background for constructing solutions of equations (6.25), (6.26), ...
217(1)
6.5 Auxiliary formulas
218(3)
6.5.1 Equations with Helmholtz operators
218(1)
6.5.2 Equations (6.50) and (6.51)
219(1)
6.5.3 Two more identities
220(1)
6.6 Leading nonzero diffraction coefficient for the wave's from a center of rotation
221(3)
6.6.1 Rearrangement of the expression JI1V0
221(1)
6.6.2 Discarding terms unimportant for finding Χ1 in JI1V0
222(1)
6.6.3 Final result
223(1)
6.7 Leading nonzero diffraction coefficient for the wave P from a center of rotation
224(2)
6.7.1 Rearrangement of the expression JI1V0
224(1)
6.7.2 Terms in JI1V0 important for calculation of ψ1
225(1)
6.7.3 Final result
226(1)
6.8 * Comments to
Chapter 6
226(5)
References to
Chapter 6
228(3)
7 The "Nongeometrical" Wave S*
231(16)
7.1 Statement of the problem and qualitative discussion
231(3)
7.1.1 Boundary-value problem
231(1)
7.1.2 Qualitative discussion of arising waves
232(2)
7.2 Derivation of formulas
234(7)
7.2.1 Auxiliary problem and reciprocity principle
235(1)
7.2.2 What is required to find the wave S*?
236(1)
7.2.3 Solution of the auxiliary problem for w
237(1)
7.2.4 Leading term of the asymptotics of the wave S*
238(2)
7.2.5 * On higher-order terms and other refinements
240(1)
7.3 * Comments to
Chapter 7
241(6)
References to
Chapter 7
242(5)
8 Ray Method for Rayleigh Waves
247(26)
8.1 Equations, boundary conditions, and a recurrent system
248(2)
8.2 Boundary value problem for UO
250(10)
8.2.1 Explicit forms of equation and a boundary condition for UO
251(1)
8.2.2 The eikonal equation
251(1)
8.2.3 The Fermat principle, rays, and the group velocity theorem
252(2)
8.2.4 Consistency condition for the boundary value problem for UI
254(2)
8.2.5 Preliminary analysis of the transport equation
256(1)
8.2.6 The Umov equation and an expression for |φ|
256(3)
8.2.7 The Berry phase
259(1)
8.3 On the construction of higher-order terms
260(1)
8.4 Case of an isotropic body
261(5)
8.4.1 Peculiar features of an isotropic case
262(1)
8.4.2 Explicit formulas for the leading-order term
263(2)
8.4.3 Final formula for the leading-order term
265(1)
8.5 * Comments to
Chapter 8
266(7)
References to
Chapter 8
267(6)
Appendix: Elements of Tensor Analysis and Differential Geometry
273(6)
A.1 Definition of tensor
273(1)
A.2 Simple operations with tensors
274(1)
A.3 Metric tensor. Raising and lowering indices
274(2)
A.4 Coordinates (q1, q2, n) associated with a surface in R3. The first and second fundamental forms
276(2)
A.4.1 Coordinates (q1, q2, n)
276(1)
A.4.2 First fundamental form
277(1)
A.4.3 Second fundamental form
277(1)
A.5 Covariant derivative. Divergence
278(1)
References to Appendix 279(2)
Index 281
Vassily M. Babich is a leading Russian expert in mathematical theory of diffraction and wave propagation. He is a co-author of ten monographs, and is the head of the laboratory of Mathematical Methods in Geophysics in the St. Petersburg branch of the Steklov Institute of Mathematics, as well as a part-time Professor at the Mathematical Faculty of St. Petersburg State University.

Aleksei P. Kiselev has authored around 100 papers in diffraction and propagation of waves. He previously worked in seismic exploration, and in mechanical engineering at Leningrad (St. Petersburg). He is now a leading researcher in the Babich Laboratory, a part-time Professor at the Physical Faculty of St. Petersburg State University and a part-time researcher in the Institute of Mechanical Engineering.
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