Spectral properies of Christoffel equation for Rayleigh waves

Abstract

Spectral properties of the matrix Christoffel equation are needed for construction of the exponentially decaying surface (Rayleigh) waves. The following analysis covers theorems on root characterization, the structure of eigenvalues, and eigenspaces of the Christoffel equation. The main results are obtained by three-dimensional complex formalism, which goes back to Rayleigh. For the case of anisotropic media this approach was exploited by Farnell in numerical analysis of Rayleigh waves propagating in anisotropic crystals and in analytical study by Stoneley and later by Dieulesaint and Royer. Sextic formalism for Rayleigh wave analysis was proposed by Stroh. This is based on the similarity of solutions for line dislocations in unbounded medium and the propagation of Rayleigh waves. This approach was developed by Barnett and Lothe and later by Chadwick et al., so theorems of uniqueness and existence for Rayleigh waves were proved. Substitution of a complex root in Christoffel’s equation produces an equation for the eigenvector (amplitude vector) determination. Propositions which flow out directly from the analysis of the structure of this equation describe spectral properties and characterization of eigenspaces. One of the most interesting results asserts that algebraic multiplicity of the phase speed equals its geometric multiplicity. [Work was supported by INTAS 96-2003.]