Elastic Scattering by Unbounded Rough Surfaces

Abstract

We consider the two-dimensional time-harmonic elastic wave scattering problem for an unbounded rough surface, due to an inhomogeneous source term whose support lies within a finite distance above the surface. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the elastic displacement vanishes. We propose an upward propagating radiation condition (angular spectrum representation) for solutions of the Navier equation in the upper half-space above the rough surface, and establish an equivalent variational formulation. Existence and uniqueness of solutions at arbitrary frequency is proved by applying a priori estimates for the Navier equation and perturbation arguments for semi-Fredholm operators.

Elastic scattering by unbounded rough surfaces: solvability in weighted Sobolev spaces

“…As in our previous paper [22], the boundary := ∂ D of D is supposed to be the graph of a uniformly Lipschitz continuous function f , i.e.…”

“…Assuming that u sc is a linear superposition of outgoing plane waves in D, we shall represent the scattered field in U h in terms of the trace u sc h := u sc | h . Using Fourier transform, it was derived in [22] that…”

“…[32]). Moreover, such a radiation condition can be written in an alternative form that is identical with the Upward Propagating Radiation Condition (UPRC) proposed by Arens [20] (see [22,Remark 1])…”

“…In our previous paper [22], the parametersμ,λ were taken asμ = 0,λ = λ + μ, leading to a minimal loss of coercivity for the corresponding Dirichlet-to-Neumann map on h ; see [22,Remark 4]. Throughout this paper, we setμ = μ,λ = λ so that the operator Tμ ,λ = T μ,λ coincides with the stress operator defined in (2.8).…”

“…These commutator estimates play an essential rule not only in verifying the main Theorems 2.2 and 3.1, but also in establishing equivalent variational formulations in the weighted spaces (see Lemma 2.5). In particular, Lemma 5.4 (i) provides a proof of [22,Lemma 1] in the non-weighted Sobolev spaces.…”