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.

翻译：本文研究了时谐声波在具有无界扰动边界的双层介质中散射的积分方程方法与Nyström方法。该介质由被平面界面分隔的两层构成，我们对此施加传输边界条件。粗糙表面边界则分别考虑Dirichlet或阻抗边界条件。与经典粗糙表面散射问题不同，平面界面的存在使得散射问题的适定性建立与数值处理方法变得困难。我们引入双层格林函数，并证明该函数具有与半空间格林函数相似的渐近衰减性质。通过采用类似经典粗糙表面问题的分析方法，我们建立了散射问题的唯一性。利用双层格林函数作为积分核导出积分方程形式，并据此证明了散射问题的存在性。进一步地，我们提出采用Nyström方法离散积分方程并证明其收敛性。最后通过数值实验验证了Nyström方法的有效性。