Acoustic Scattering by Mildly Rough Unbounded Surfaces in Three Dimensions

Abstract

For a nonlocally perturbed half-space we consider the scattering of time-harmonic acoustic waves. A second kind boundary integral equation formulation is proposed for the sound-soft case, based on a standard ansatz as a combined single- and double-layer potential but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green's function. Due to the unboundedness of the surface, the integral operators are noncompact. In contrast to the two-dimensional case, the integral operators are also strongly singular, due to the slow decay at infinity of the fundamental solution of the three-dimensional Helmholtz equation. In the case when the surface is sufficiently smooth (Lyapunov) we show that the integral operators are nevertheless bounded as operators on $L^2(\Gamma)$ and on $L^2(\Gamma)\cap BC(\Gamma)$ and that the operators depend continuously in norm on the wave number and on $\Gamma$. We further show that for \emph{mild} roughness, i.e., a surface $\Gamma$ which d...