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.
"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
Chapter 1. Basic notions of elastodynamics
Chapter 2. Plane waves
Chapter 3. Point sources and spherical waves in homogeneous isotropic media
Chapter 4. The ray method for volume waves in isotropic media
Chapter 5. The ray method for volume waves in anisotropic media
Chapter 6. Point sources in inhomogeneous isotropic media. The wave S from a center of expansion. The wave P from a center of rotation
Chapter 7. The "nongeometrical" wave S *
Chapter 8. The ray method for Rayleigh waves
A.1. Definition of tensor
A.2. Simple operations with tensors
A.3. Metric tensor. Raising and lowering indices
A.4. Coordinates (q1, q2, n) associated with a surface in R3. The first and second fundamental forms
A.5. Covariant derivative. Divergence