Wave propagation problems in certain elastic anisotropic half spaces

The Smirnov-Sobolev method of self-similar potentials is used to solve certain wave propagation problems in anisotropic media. The solutions are expressed in terms of analytic functions which are determined from the boundary conditions in a straightforward manner. Two types of problems are considered. The first type concerns the two dimensional case of an orthotropic material under plain-strain conditions subjected to a suddenly applied line force on the surface or in the interior of a half space. The second type treats the three dimensional problem of a point force suddenly applied on the surface of a transversely isotropic half space. This solution is formed, in general, by a rotational superposition of the solutions for a plane strain and for an antiplane problem with appropriately defined boundary conditions. The mapping of the wave fields in the complex domain composed of a four-sheeted Riemann surface is examined in detail. Some of the techniques used in the numerical treatment of certain singularities are briefly discussed. The numerical results are given in the form of the time histories for the displacements and stresses in the two dimensional case and as time histories of only the displacements in the three dimensional case.