Application of Steepest Descent Path Method to Lamb’s Solutions for Scattering in Thermo-elastic Half-Plane

Abstract

When an incidence impinges an alluvial valley in half-plane, wave interactions of three inhomogeneities are considered on thermoelastic coupling effects, and the stress concentration along continuous interface is demonstrated. Because of the inhomogeneities, the scattering waves can be deduced by three part, the incidence sources in the half-plane, reflection waves simulated by the image sources in the mirror image half-plane, and the refraction wave inside the alluvial valley. For in-plane problem, two coupled longitudinal waves, of which one is predominantly elastic and the other is predominantly thermal, and a transversal wave are adopted to analyze scattering. This work uses a Rayleigh series of Lamb’s formal integral solutions as a simple basis set. The corresponding integrations of the basis set are calculated numerically by applying a modified steepest descent path integral method, which provides strongly convergence in numerical integrations. Moreover, Betti’s third identity and orthogonal conditions are applied to obtain a transition matrix for solving the scattering. The results at the surface of a semicircular alluvial valley embedded in half-plane are demonstrated to show the displacement fields and the temperature gradient fields. They also indicate that softer alluvial valley is associated with a substantially greater amplification at the interface of the alluvial valley.