Analysis of elastic symbols with the Cauchy integral and construction of asymptotic solutions

摘要

This paper deals with the elastic wave equation (D_t~2 -L(x, D_(x′), D_(x_n)))u(t, x′, x_n) = 0 in the half-space x_n > 0. In the constant coefficient case, it is known that the solution is represented by using the Cauchy integral ∫_ce~(ix_nζ)(I - L(ξ′,ζ)~(-1)dζ. In this paper this representation is extended to the variable coefficient case, and an asymptotic solution with the similar Cauchy integral is constructed. In this case, the terms ∂_x~α ∫_ce~(ix_nζ)(I - L(ξ′,ζ)~(-1)dζ appear in the inductive process. These do not become lower terms necessarily, and therefore the principal part of asymptotic solution is a little different from the form in the constant coefficient case.