The Helmholtz equation in a locally perturbed half-space with non-absorbing boundary. http://dx.doi.org/10.1007/s00205-008-0135-3

Abstract

We obtain uniqueness and existence results of an outgoing solution for the Helmholtz equation in a half-space, or in a compact local perturbation of it, with an
impedance boundary condition. It is worth noting that these kinds of domains have unbounded boundaries which lead to a non-classical exterior problem. The established radiation condition is somewhat different from the usual Sommerfeld’s one, due to the appearance of surface waves (in the case of a non-absorbing boundary). A half-space Green’s function framework is used to carry out our computations.
This is an extended and detailed version of the previous article “The Helmholtz equation with impedance in a half-space,” Duran et al. (CR Acad Sci Paris Ser I 341:561–566, 2005).