PROPAGATOR MATRICES IN ELASTIC WAVE AND VIBRATION PROBLEMS

Abstract

The boundary value problems most frequently encountered in studies of elastic wave propagation in stratified media can be formulated in terms of a finite number of linear, first order, ordinary differential equations with variable coefficients. Volterra (1887) has shown that solutions to such a system of equations are conveniently represented by the product integral, or propagator, of the matrix of coefficients. In this paper we summarize some of the better known properties of propagators plus numerical methods for their computation. When the dispersion relation is some mth order minor of the integral matrix it is possible to deal with mth minor propagators so that the dispersion relation is a single element of the mth minor integral matrix. In this way one of the major sources of loss of numerical accuracy in computing the dispersion relation is avoided. Propagator equations for SH and for P‐SV waves are given for both isotropic and transversely isotropic media. In addition, the second minor propagator equations for P‐SV waves are given. Matrix polynomial approximations to the propagators, obtained from the method of mean coefficients by the Cayley‐Hamilton theorem and the Lagrange‐Sylvester interpolation formula, are derived.