A Brakhage-Werner-Type Integral Equation Formulation of a Rough Surface Scattering Problem

Abstract. We consider the problem of scattering of time-harmonic acoustic waves by an unbounded sound-soft rough surface. Although integral equation methods are widely used for the numerical solution of such problems, to date there is no formulation which is known to be uniquely solvable in the 3D case. We consider a novel Brakhage-Werner type integral equation formulation of this problem, based on an 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. In the case when the surface Γ is sufficiently smooth (Lyapunov), we sketch how it can be shown that the integral operators are bounded as operators on L(Γ) and, moreover, how it can be shown that the integral equation is uniquely solvable in the space L(Γ). The proof of this latter result uses novel, direct arguments, leading to explicit bounds on the inverse in terms of the wave number, the parameter coupling the singleand 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 if the coupling parameter is chosen proportional to the wave number.