Contributions to a Theory of Separability of the Vector Wave Equation of Elasticity for Inhomogeneous Media

Abstract

In a recent paper, the author introduced a generalization of the Helmholtz theorem for representation of vectors in terms of potential functions. It was shown that use of this generalization leads to separation (in the dependent variables) of the vector wave equation of elasticity for certain inhomogeneous media whose properties are functions of the Cartesian coordinate s. The conditions for separability are a system of nonlinear differential equations for the constitutive parameters. Only a few special solutions of this system were given. An almost complete catalog of the separable cases is developed in the present paper. The vector differential properties of the field for the separable cases are discussed. It is shown that the P and S fields are irrotational and solenoidal, respectively, provided we regard as the fundamental field vectors the quantities f1uP, f2uSV, and uSH rather than the displacement vectors themselves, where f1 and f2 are appropriate weighting functions. The vector differential properties of the displacement vectors themselves are derived and these results compared with similar results obtained by the eikonal method of analysis.