On the boundary integral equations in the theory of elastic vibrations

Forward seismic problems are solved for elastic media by rigorous methods (i.e., methods with controllable accuracy). Analysis of the current state of research on this subject suggests that the most promising methods are based on integral and integro-differential equations, notwithstanding the rather modest results of their application to solving forward problems in the theory of elastic vibrations. The second Green integral theorem for seismic waves, formulated and proven in the paper, yields a system of two boundary (surface) integral equations for the displacement vector u(M0) and the normal (to the boundary surface) vector component of the stress tensor tn(M0). The integrands of the surface integrals in terms of which the function tn(M0) is expressed on both sides of the interface between the medium and the heterogeneity contain the second derivatives of the Green’s tensor functions Ĝe (M0, M) and Ĝi (M0, M), respectively, which are responsible for a cubic singularity (third-order singularity) if the integration point M coincides with the observation point M0. An original method of eliminating the cubic singularity proposed in the paper involves special tensor normalization of the integrals on the outer and inner sides of the interface and subsequent subtraction of one integral from another in order to construct the second integral equation.