P-SV coupling in inhomogeneous elastic media of Epstein type

Abstract

Summary The coupled wave equations for P– SV propagation in plane strain elastodynamics for inhomogeneous media may be cast in two forms. In the vectorial displacement formulation, the terms containing logarithmic derivatives of inhomogeneity occur explicitly, while in the tensorial stress formulation they are implicit. The displacement formulation is convenient for studying the significance of the logarithmic derivative terms in P–SV coupling and mode conversion. Analytical solutions are obtained for the class of media characterized by the Epstein profiles. The results from the full system of coupled wave equations in the displacement formulation are compared with those from two approximate systems obtained by dropping terms associated with the logarithmic derivative. It is found that the reflection–transmission coefficients from the approximate systems do not satisfy the energy balance equation for subcritical angles of incidence. However, the approximate systems retain the general qualitative features of P–SV propagation given by the full system. The asymmetric waveguide profile displays certain unique features in the vicinity of the velocity minimum for grazing angles of incidence. This is confirmed by comparing the solutions for the full coupled system in the vectorial displacement formulation with the solutions for the coupled system in the tensorial stress formulation.