  This paper considers the problems of scattering of time-harmonic acoustic
waves by a two-layered medium with a non-locally perturbed boundary (called a
rough boundary in this paper) in two dimensions, where a Dirichlet or impedance
boundary condition is imposed on the boundary. The two-layered medium is
composed of two unbounded media with different physical properties and the
interface between the two media is considered to be a planar surface. We
formulate the considered scattering problems as the boundary value problems and
prove that each boundary value problem has a unique solution by utilizing the
integral equation method associated with the two-layered Green function.
Moreover, we develop the Nystr\"{o}m method for numerically solving the
considered boundary value problems, based on the proposed integral equation
formulations. We establish the convergence results of the Nystr\"{o}m method
with the convergence rates depending on the smoothness of the rough boundary.
It is worth noting that in establishing the well-posedness of the boundary
value problems as well as the convergence results of the Nystr\"{o}m method, an
essential role is played by the investigation of the asymptotic properties of
the two-layered Green function for small and large arguments. Finally,
numerical experiments are carried out to show the effectiveness of the
Nystr\"{o}m method.
