Uniqueness for Elastic Wave Scattering by Rough Surfaces

We consider a two-dimensional elastic wave scattering problem for an unbounded surface represented as the graph of a $C^{1,\alpha}$ function. The total displacement is assumed to vanish on the surface. We present a new radiation condition, the upwards propagating radiation condition, for such problems based on a similar condition recently introduced for acoustic scattering problems. The relation between this radiation condition and more commonly used conditions is discussed. Subsequently we prove uniqueness of solution to the scattering problem under this radiation condition for a general class of incident fields, including plane and cylindrical waves.