A Nyström method for scattering by a two-layered medium with a rough boundary

This paper presents a study on the integral equation method and the Nyst\"{o}m method for the scattering of time-harmonic acoustic waves by a two-layered medium with an unbounded perturbed boundary. The medium consists of two layers separated by a plane interface, for which we assume transmission boundary conditions. We assume either Dirichlet or impedance boundary conditions for the rough surface boundary. Unlike classical rough surface scattering problems, the presence of a plane interface makes it difficult to establish the well-posedness of the scattering problem and to find a numerical treatment. We introduce the two-layered Green function and prove that this function has similar asymptotic decay properties to the half-space Green function. By using a similar approach to classical rough surface problems, we establish the uniqueness of the scattering problem. We derive the integral equation formulations using the two-layered Green function as the integral kernel and use them to prove the existence of the scattering problem. Furthermore, we propose the Nyst\"{o}m method for discretizing the integral equations and establish its convergence. Finally, we perform numerical experiments to demonstrate the effectiveness of the Nyst\"{o}m method.