The use of singular integrals in wave propagation problems; with application to the point source in a semi-infinite elastic medium

Abstract This is an investigation of the field due to a general point source of energy in an isotropic, elastic solid with a free surface. The paper is divided into three parts. In part I we are concerned with the development of new plane wave representations for the fundamental solutions of elastodynamics. There are two types of situation involved; we have the simpler type involved in the case of a steady point source which moves steadily with any constant velocity in an elastic medium, this type involves superposition of plane waves with respect to a single parameter, and we have the more complicated transient problem in which a point source is set up at a given moment, and thereafter moves at constant velocity, without change of strength. In part II we make use of the new representation for the field of a steadily moving source in the calculation of fields and displacements in the presence of a free surface and in part III we do the same for the transient source. We discuss in some detail the application of the new approach to the case of a vertical load, to a horizontal load, and to a couple of arbitrary orientation, and we give a general discussion of the singularities to be expected for the general point source.