THE GEOMETRY OF THE HYPERBOLIC SYSTEM FOR AN ANISOTROPIC PERFECTLY ELASTIC MEDIUM

We evaluate the fundamental solution of the hyperbolic system describi ng the generation and propagation of elastic waves in an anisotropic s olid by studying the homology of the algebraic hypersurface defined by the characteristic equation, also known as the ''slowness'' surface. Our starting point is the Herglotz-Petrovsky-Leray integral representa tion of the fundamental solution. We find an explicit decomposition of the latter solution into integrals over vanishing cycles associated w ith the isolated singularities on the slowness surface. As is well kno wn in the theory of isolated singularities, integrals over vanishing c ycles satisfy a system of differential equations known as Picard-Fuchs equations. Such equations are linear and can have at most regular sin gular points. We discuss a method to obtain these equations explicitly . Subsequently, we use the monodromy properties around the regular sin gular points to find the asymptotic behavior according to the differen t types of singularities that may appear on a wave front in three dime nsions. This is a method alternative to the one that arises in the Mas lov theory of oscillating integrals. Our work sheds new light on how t o compute and classify the Cagniard-De Hoop contour in the complex rad ial horizontal slowness plane; this contour is used in numerical integ ration schemes to obtain the full time behaviour of the fundamental so lution for a given direction of propagation.