Decoupling Elastic Waves and Its Applications

TL;DR

This paper derives geometric conditions for decoupling shear and pressure elastic waves, enabling improved analysis of inverse scattering problems for polyhedral objects with minimal measurements.

Contribution

It introduces new geometric criteria based on boundary curvatures that ensure wave decoupling in elastic scattering, aiding inverse problem analysis.

Findings

Decoupling conditions depend on mean and Gaussian curvatures.

Decoupling enhances uniqueness in inverse scattering.

Results apply to polyhedral scatterers with minimal measurements.

Abstract

In this paper, we consider time-harmonic elastic wave scattering governed by the Lam\'e system. It is known that the elastic wave field can be decomposed into the shear and compressional parts, namely, the pressure and shear waves that are generally coexisting, but propagating at different speeds. We consider the third or fourth kind scatterer and derive two geometric conditions, respectively, related to the mean and Gaussian curvatures of the boundary surface of the scatterer that can ensure the decoupling of the shear and pressure waves. Then we apply the decoupling results to the uniqueness and stability analysis for inverse elastic scattering problems in determining polyhedral scatterers by a minimal number of far-field measurements.