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

A Nystr\"{o}m method for scattering by a two-layered medium with a rough boundary