REFLECTION AND TRANSMISSION COEFFICIENTS FOR 3-DIMENSIONAL PLANE-WAVES IN ELASTIC MEDIA

Problems for transient line and point load sources in a multilayered e lastic medium may be treated by the method of generalized ray. In this method an integral representation of the Laplace-transformed multiply reflected and/or transmitted cylindrical/spherical wave, known as a r ay integral, is constructed by linear superposition of the Laplace-tra nsformed plane waves. The inverse Laplace transform of the ray integra l can be found in closed form by applying the Cagniard method. For pro blems in the Cartesian coordinates for line load sources emitting cyli ndrical waves consistent with either the plane strain conditions or th e antiplane strain conditions and for problems in the cylindrical coor dinates for axisymmetric and asymmetric point load sources emanating s pherical waves, it is well known that: (1) the system of incident, ref lected, and transmitted cylindrical/spherical waves at an interface se parating two dissimilar media can be divided into two independent of e ach other if both present, parts: the coupled P and SV waves, and the SH waves, (2) the reflected and transmitted ray integrals representing the Laplace-transformed reflected and transmitted cylindrical/spheric al waves can be constructed by linear superposition of the Laplace-tra nsformed plane P and SV waves, or the plane SH waves, and (3) the pote ntial reflection and transmission coefficients for the plane P, SV, an d SH waves are basic to such a superposition. In the present paper we treat the asymmetric three-dimensional problem in the Cartesian coordi nates for an arbitrary oriented point force radiating the spherical P and S waves. For this problem all four functions representing the disp lacement potentials are coupled in the boundary conditions at the inte rface, the total wave motion at the interface is composed of the coupl ed spherical P and S waves, and the Laplace-transformed reflected and transmitted spherical waves are therefore constructed by linear superp osition of the three-dimensional coupled plane P and S waves. Since su ch a superposition requires the knowledge of the potential reflection and transmission coefficients for the three-dimensional coupled plane P and S waves, the purpose of the present paper is to derive systemati cally these coefficient formulas.