Asymptotic behavior of stationary solutions to elastic wave equations in a perturbed half‐space in ℝ³

Abstract

We study the stationary elastic wave equation with free boundary condition in a locally perturbed half‐space in . Using the stationary phase method, we derive an asymptotic behavior at infinity of the resolvent of the elastic operator uniformly with respect to the direction in . Consequently, the body waves and the Rayleigh surface waves appear simultaneously in the expansion. From the far‐field pattern of the expansion, we obtain the scattering amplitude. We also characterize the space of generalized eigenfunctions in terms of Agmon–Hörmander space and derive their spatial asymptotics.