Variational Approach to Scattering by Inhomogeneous Layers Ab ove Rough Surfaces

In this paper,we study,via variational methods,the problem of scattering of time harmonic acoustic waves by unbounded inhomogeneous layers above a sound soft rough surface.We first propose a variational formulation and exploit it as a theoretical tool to prove the well-posedness of this problem when the media is nonabsorbing for arbitrary wave number and obtain an estimate about the solution,which exhibit explicitly dependence of bound on the wave number and on the geometry of the domain.Then,based on the non-absorbing results,we show that the variational problem remains uniquely solvable when the layer is absorbing by means of a priori estimate of the solution.Finally,we consider the finite element approximation of the problem and give an error estimate.