Reflective elastic waves at the boundary as a propagation of singularities phenomenon

Abstract

Let us consider an elastic body Ω in R 3 with a smooth boundary ∂Ω. If the medium is isotropic, it is well known that the displacement u(x,t)= t (u 1 , u 2 , u 3 ) of Ω satisfies the following boundary value problem ρ∂ 2 u/∂t 2 =(λ+μ) grad (div u)+μΔu in Ω×R, 3 ∑ i=1 n i (x){λ(div u)δ ij +μ(∂u i /∂x j +∂u j /∂u i )}=0 on ∂Ω×R, where ρ is the density of the medium, λ and μ are the «Lame constants» which are positive, and n(x)=(n 1 (x), n 2 (x), n 3 (x)) is the outer unit normal vector to the boundary ∂Ω. By constructing a special solution in the half space seismology insists the following two phenomena: 1) If a P wave only makes incidence to the boundary, then both a P wave and an S wave reflect from the boundary. 2) There are two kinds of S waves. One of them is called an SV wave, if a reflective phenomenon for its incident wave is similar to one for an incident P wave. The other is called an SH wave, if only an S wave reflects for its incident wave. We shall prove these phenomena from the viewpoint of a propagation of singularities of every solution to the boundary value problem