Energy distribution of the solutions of elastic wave propagation problems in stratified media \(\mathbb{R}^ 3\) (Q1916085)

The energy distribution of the solutions of elastic wave propagation problems in plane-stratified media \(\mathbb{R}^3\) is derived using the methods due to Wilcox. Asymptotic wave functions are constructed by means of the spectral integral representations of the solutions and the method of stationary phase. The integral representations are based on an eigenfunction expansion theory which was developed by the author of the present paper. The asymptotic energy of the solutions for large times of the interface problem for elastic waves is computed and it is shown that the energy of the solutions with finite energy is asymptotically concentrated along the interface. The organization of the paper is as follows: In Section 1 the propagation problem of elastic waves in the stratified medium is formulated and the energy of the solution is defined and the main theorem is given. Section 2 deals with the spectral integral representations of the solutions. In Section 3 asymptotic wave functions are constructed by means of the method of stationary phase. The asymptotic wave functions of the P, SV, SH components are given in Section 4. In Section 5 the asymptotic energy distributions of the solutions for large time are calculated.