# Elastic scattering from rough surfaces in three dimensions

## TL;DR

This paper develops a three-dimensional angular spectrum representation for elastic wave scattering by rough surfaces, proving uniqueness and existence results using variational methods and Green tensor analysis.

## Contribution

It introduces a 3D angular spectrum representation for elastic scattering and establishes uniqueness and existence results for rough surface problems.

## Key findings

- Derived the 3D angular spectrum representation for elastic waves.
- Proved uniqueness using a Rellich-type identity.
- Established existence results for perturbed flat surfaces.

## Abstract

Consider the elastic scattering of a plane or point incident wave by an unbounded and rigid rough surface. The angular spectrum representation (ASR) for the time-harmonic Navier equation is derived in three dimensions. The ASR is utilized as a radiation condition to the elastic rough surface scattering problem. The uniqueness is proved through a Rellich-type identity for surfaces given by uniformly Lipschitz functions. In the case of flat surfaces with a local perturbation, we deduce an equivalent variational formulation in a truncated bounded domain and show the existence results for general incoming waves. The main ingredient of the proof is the radiating behavior of the Green tensor to the first boundary value problem of the Navier equation in a half space.