The matrix sign function for solving surface wave problems in homogeneous and laterally periodic elastic half-spaces

TL;DR

This paper introduces the matrix sign function as a straightforward tool to analyze surface wave problems in anisotropic and periodically inhomogeneous elastic half-spaces, simplifying derivations and generalizations.

Contribution

It presents a novel application of the matrix sign function to derive fundamental surface wave results and extend the Barnett-Lothe formalism to complex inhomogeneous media.

Findings

Simplifies derivation of surface wave formulas

Provides explicit solutions to the Riccati equation

Enables analysis of periodically inhomogeneous half-spaces

Abstract

The matrix sign function is shown to provide a simple and direct method to derive some fundamental results in the theory of surface waves in anisotropic materials. It is used to establish a shortcut to the basic formulas of the Barnett-Lothe integral formalism and to obtain an explicit solution of the algebraic matrix Riccati equation for the surface impedance. The matrix sign function allows the Barnett-Lothe formalism to be readily generalized for the problem of finding the surface wave speed in a periodically inhomogeneous half-space with material properties that are independent of depth. No partial wave solutions need to be found; the surface wave dispersion equation is formulated instead in terms of blocks of the matrix sign function of i times the Stroh matrix.