Acoustic scattering by mildly rough unbounded surfaces in three dimensions

Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283, Heinemeyer, E. and Potthast, R. ORCID: https://orcid.org/0000-0001-6794-2500 (2006) Acoustic scattering by mildly rough unbounded surfaces in three dimensions. Siam Journal on Applied Mathematics, 66 (3). pp. 1002-1026. ISSN 0036-1399

Abstract/Summary

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 G) boolean AND BC(Gamma) and that the operators depend continuously in norm on the wave number and on G. We further show that for mild roughness, i.e., a surface G which does not differ too much from a plane, the boundary integral equation is uniquely solvable in the space L-2(Gamma) boolean AND BC(Gamma) and the scattering problem has a unique solution which satisfies a limiting absorption principle in the case of real wave number.