The Cauchy problem for elastic waves in an anisotropic medium

Abstract

The propagation of elastic waves in a homogeneous solid is governed by a hyperbolic system of three linear second-order partial differential equations with constant coefficients. When the solid is also isotropic, the form of these equations is well known and provides the foundation of the conventional theory of elasticity (Love 1944). The explicit solution of the initial value, or Cauchy, problem for the isotropic case was found by Poisson, and in a different way by Stokes (1883). If the initial disturbance is sharp and concentrated, the resulting disturbance at a field point will consist of an initial sharp pressure wave, a continuous wave for a certain period, and a final sharp shear wave. The disturbance then ceases. Here we shall consider a medium which is homogeneous but not isotropic, and will describe, using Fourier transforms, the elastic waves produced by a local initial disturbance. The solution again consists of a continuous wave which lasts for a definite period of time, and a number of sharp waves, but the detailed nature of the waves may, in highly anisotropic media, be very different and much more complicated. The continuous wave may arrive at a field point in advance of the first sharp wave, though it will always terminate with the last sharp wave. The number of the sharp waves may not exceed 75. The solution appears as the sum of three modes, which correspond to the three sheets of a certain wave surface. The geometry of this surface, which may be quite complicated (Musgrave 1954 a ), qualitatively determines the nature of the solution. These calculations may serve as a foundation for the study of time-dependent elastic waves. There is also mathematical interest in this example of a hyperbolic system for which the wave surface may have certain types of singularities not usually considered in the existing general theory of hyperbolic differential equations.