Acoustic Scattering by Mildly Rough Unbounded Surfaces in Three Dimensions

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 singleand 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 L2(Γ) and on L2(Γ) ∩ BC(Γ) and that the operators depend continuously in norm on the wave number and on Γ. We further show that for mild roughness, i.e., a surface Γ which does not differ too much from a plane, the boundary integral equation is uniquely solvable in the space L2(Γ)∩BC(Γ) and the scattering problem has a unique solution which satisfies a limiting absorption principle in the case of real wave number.