Diffraction of Elastic Waves by Edges

TL;DR

This paper studies how elastic waves diffract at edges in manifolds with edge singularities, revealing that diffracted waves are weaker in Sobolev sense than incident waves, with implications for wave behavior near edges.

Contribution

It provides new mathematical results on the diffraction of elastic waves at edges in manifolds with singularities, including Sobolev regularity estimates for diffracted waves.

Findings

Diffracted p-waves are weaker than incident p-waves in Sobolev sense.

Diffracted s-waves are weaker than incident s-waves in Sobolev sense.

Results extend to more general wave interactions at 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, do not correspond to limits of p-rays 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.