The method of separation of both the dependent and independent variables of the vector wave equation of elasticity has heretofore been applied almost exclusively to problems of wave propagation in homogeneous media. In this paper, a generalization of this method that extends its applicability to several types of inhomogeneous media is developed. The mechanical properties of the media considered here depend on one Cartesian coordinate, and Poisson's ratio is taken to be constant. Formulations are given for axially symmetric waves in three dimensions and for waves in two dimensions. Three linearly independent vector solutions of the wave equation are expressed in terms of three scalar potential functions. One of these functions represents generalized SH waves and satisfies an independent second-order wave equation in all the problems considered here. Certain functional forms for the mechanical properties are found for which the remaining two potential functions also satisfy independent second-order wave equations; these potentials may be identified with generalized P and SV waves, respectively. The SH waves are solenoidal, but in general the P and SV waves are not purely irrotational and solenoidal, respectively, for the problems treated.
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Joseph F. Hook (1961) studied this question.