Author(s): Babich Vassily.M., Kiselev Aleksei.P.
Series: Chapman and Hall/CRC Monographs and Research Notes in Mathematics Ser
Publisher: Chapman and Hall/CRC
Year: 2018
Language: English
Pages: 306
City: Milton
Tags: Elastic waves-Mathematical models.;Elasticity.;Mathematical physics.;Elastic waves -- Mathematical models.
Content: Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Introduction
List of Basic Symbols
1: Basic Notions of Elastodynamics
1.1 Displacement, deformation, and stress
1.1.1 Displacement vector and strain tensor
1.1.2 Stress tensor
1.2 Lagrangian approach to mechanical systems
1.3 Elastodynamics equations
1.3.1 Kinetic and potential energies as quadratic functionals
1.3.2 Properties of elastic stiffnesses
1.3.3 Derivation of elastodynamics equations
1.3.4 Navier and Lamé operators
1.4 Classical boundary conditions
1.4.1 List of boundary conditions 1.4.2 Hamilton principle and boundary conditions1.5 Isotropic medium
1.5.1 Consequences of the invariance of W with respect to rotations
1.5.2 Consequences of the positive definiteness of W
1.6 Energy balance
1.7 Time-harmonic solutions
1.7.1 Basic notions
1.7.2 Time-averaging
1.8 Reciprocity principle
1.8.1 The time-harmonic case
1.8.2 Non-time-harmonic case
1.9 ⋆ Comments to Chapter 1
References to Chapter 1
2: Plane Waves
2.1 Plane-wave ansatz
2.2 Phase velocity
2.2.1 Normal velocity of a moving surface
2.2.2 Phase velocity and slowness 2.3 Plane waves in unbounded isotropic media2.3.1 Eigenvalue problem
2.3.2 Wave P
2.3.3 Wave S
2.3.4 Time-harmonic waves P and S
2.3.5 Polarization of time-harmonic waves P and S
2.3.6 Group velocity
2.3.7 Energy relations for time-harmonic waves
2.3.8 ⋆ Potentials
2.4 Plane waves in unbounded anisotropic media
2.4.1 Eigenvalue problem
2.4.2 Phase velocity
2.4.3 Group velocity
2.4.4 Slowness, slowness surface, velocity surface
2.4.5 ⋆ Rayleigh principle
2.5 ⋆ Local velocities and domain of influence
2.5.1 Statement of the problem
2.5.2 Energy lemma
2.5.3 Uniqueness theorem 2.6 Reflection of plane waves from a free boundary of an isotropic half-space2.6.1 Upgoing and downgoing waves
2.6.2 Waves of polarizations SH and P --
SV
2.6.3 The case of polarization SH
2.6.4 The case of polarization P --
SV
2.6.5 Snell's law
2.6.6 Total internal reflection
2.6.7 Energy flow in an inhomogeneous wave P
2.6.8 ⋆ Energy flow under reflection from a boundary and the unitarity of the reflection matriƀ̌
2.6.9 Reflection of non-time-harmonic waves
2.7 Classical plane surface waves in isotropic media
2.7.1 Classical Rayleigh wave
2.7.2 Classical Love wave
