Rayleigh-type Surface Quasimodes in General Linear Elasticity

TL;DR

This paper constructs surface quasimodes for anisotropic elastic waves with traction-free boundary conditions, using a reduction to an eigenvalue problem for a selfadjoint operator, and computes new formulas for the principal and subprincipal symbols.

Contribution

It introduces a novel reduction of the elastic wave problem to a boundary eigenvalue problem and computes new subprincipal symbol formulas, even in isotropic cases.

Findings

Construction of surface quasimodes for anisotropic elasticity.

Reduction to an eigenvalue problem for a selfadjoint operator.

New formulas for subprincipal symbols in the isotropic case.

Abstract

Rayleigh-type surface waves correspond to the characteristic variety, in the elliptic boundary region, of the displacement-to-traction map. In this paper, surface quasimodes are constructed for the reduced elastic wave equation, anisotropic in general, with traction-free boundary. Assuming a global variant of a condition of Barnett and Lothe, the construction is reduced to an eigenvalue problem for a selfadjoint scalar first order pseudo-differential operator on the boundary. The principal and the subprincipal symbol of this operator are computed. The formula for the subprincipal symbol seems to be new even in the isotropic case.