### Energy distribution of the solutions of elastic wave propagation problems in stratified media $R^{3}$

Senjo Shimizu
1995 Tsukuba Journal of Mathematics
\S 1. Introduction Energy distribution of the solutions of various wave propagation problems has been studied by C. H. Wilcox (,  ,  ,  ). He constructed asymptotic wave functions which approximate the solutions in the sense of $L^{2}$ for large times and calculated asymptotic energy distributions of the solutions in several domain by making use of these asymptotic wave functions. The construction of asymptotic wave functions is based on an eigenfunction expansion theorem which
more » ... s proved by the same author and on the method of stationary phase. J.C.Guillot  studied a Rayleich surface wave propagating along the free boundary of a transversely isotropic elastic half space and showed that the energy of the Rayleich component of every solution with finite energy is asymptotically concentrated along the boundary. In this paper we shall derive energy distributions of the solutions of elastic wave propagation problems in plane-stratffied media $R^{3}$ using methods due to Wilcox. We construct asymptotic wave functions by using spectral integral representations of the solutions and the method of stationary phase. The integral representations are based on an eigenfunction expansion theory which was proved by the author  using methods due to S. Wakabayashi . We calculate asymptotic energy of the solutions for large times of the interface problems for elastic waves and show that the energy of the Stoneley components of the solutions with finite energy is asymptotically concentrated along the interface. We start with the mathematical formulation of the elastic wave propagation problem. Consider the plane stratffied medium $R^{3}=\{x=(x_{1}, x_{2}, x_{3});x_{i}\in R\}$ with the planar interface $x_{3}=0$ , which is defined by \$(\lambda(x_{3}),\mu(x_{3}),\rho(x_{3}))=\{\begin{array}{l}(\lambda_{1},\mu_{1},\rho_{1})(\lambda_{2},\mu_{2},\rho_{2})