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 (Γ) and on L 2 (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 L 2 (Γ) ∩ BC(Γ) and the scattering problem has a unique solution which satisfies a limiting absorption principle in the case of real wave number.

We consider the problem of scattering of time-harmonic acoustic waves by an unbounded sound-soft rough surface. Recently, a Brakhage–Werner type integral equation formulation of this problem has been proposed, based on an 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. Moreover, it has been shown in the three-dimensional case that this integral equation is uniquely solvable in the space when the scattering surface does not differ too much from a plane. In this paper, we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope. Moreover, we construct explicit bounds on the inverse of the associated boundary integral operator, as a function of the wave number, the parameter coupling the single- and double-layer potentials, and the maximum surface slope. These bounds show that the norm of the inverse operator is bounded uniformly in the wave number, κ , for κ >0, if the coupling parameter η is chosen proportional to the wave number. In the case when is a plane, we show that the choice is nearly optimal in terms of minimizing the condition number.

We introduce a novel multi-section method for the solution of integral equations on unbounded domains. The method is applied to the rough-surface scattering problem in three dimensions, in particular to a Brakhage-Werner type integral equation for acoustic scattering by an unbounded rough surface with Dirichlet boundary condition, where the fundamental solution is replaced by some appropriate half-space Green's function. The basic idea of the multi-section method is to solve an integral equation Aϕ = f by approximately solving the equation P AP τ ϕ = P f for some positive constants , τ. Here P is a projection operator that truncates a function to a ball with radius > 0. For a very general class of operators A, for which the Brakhage Werner equation from acoustic scattering is a particular example, we will show existence of approximate solutions to the multi-section equation and that approximate solutions to the multi-section equation approximate the true solution ϕ 0 of the operator equation Aϕ = f. Finally, we describe a numerical implementation of the multi-section algorithm and provide numerical examples for the case of rough surface scattering in three dimensions.

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