Variational Approach to Scattering by Unbounded Rough Surfaces with Neumann and Generalized Impedance Boundary Conditions∗

This paper is concerned with problems of scattering of time-harmonic electromagnetic and acoustic waves from an infinite penetrable medium with a finite height modeled by the Helmholtz equation. On the lower boundary of the rough layer, the Neumann or generalized impedance boundary condition is imposed. The scattered field in the unbounded homogeneous medium is required to satisfy the upward angular-spectrum representation. Using the variational approach, we prove uniqueness and existence of solutions in the standard space of finite energy for inhomogeneous source terms, and in appropriate weighted Sobolev spaces for incident point source waves in Rm (m= 2,3) and incident plane waves in R2. To avoid guided waves, we assume that the penetrable medium satisfies certain non-trapping and geometric conditions.