Diffraction of elastic waves by edges

Abstract

We investigate the diffraction of singularities of solutions to the linear elastic equation on manifolds with edge singularities. Such manifolds are modeled on the product of a smooth manifold and a cone over a compact fiber. For the fundamental solution, the initial pole generates a pressure wave (p-wave), and a secondary, slower shear wave (s-wave). If the initial pole is appropriately situated near the edge, we show that when a p-wave strikes the edge, the diffracted p-waves and s-waves (i.e. loosely speaking, their singularities together with the singularities of the incoming p-wave are not limits of rays associated to the pressure wave speed which just miss the edge) are weaker in a Sobolev sense than the incident p-wave. We also show an analogous result for an s-wave that hits the edge, and provide results for more general situations.