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The geometry of the hyperbolic system for an anisotropic perfectly elastic medium. (English) Zbl 0819.73006

This paper is concerned with the linear theory of dynamic elasticity for anisotropic and homogeneous bodies. The authors evaluate the fundamental solution by studying the homology of the algebraic hypersurface defined by the characteristic equation. First, the Herglotz-Petrovsky-Leray representation of the fundamental solution is derived explicitly by using Gelfand’s plane-wave expansion. It is shown that this formula reduces to an integral over a \((n-2)\)-dimensional cycle on the slowness surface, which can be associated with the Cagniard-de Hoop contour. Then the authors establish a decomposition of the fundamental solution into integrals over so-called vanishing cycles associated with the singularities. Finally, a Picard-Fuchs differential equation for the integrals over vanishing cycles is derived, and it is shown how the asymptotic behaviour in the high-frequency limit follows from the monodromy properties around the regular singular points.
Reviewer: D.Iesan (Iaşi)

MSC:

74E10 Anisotropy in solid mechanics
74J99 Waves in solid mechanics
35L67 Shocks and singularities for hyperbolic equations
35A08 Fundamental solutions to PDEs
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