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The geometry of elastic waves propagating in an anisotropic elastic medium

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Geometric and Quantum Aspects of Integrable Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 424))

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Abstract

We evaluate the fundamental solution of the hyperbolic system describing the generation and propagation of elastic waves in an anisotropic solid by studying the homology of the so-called slowness hypersurface defined by the characteristic equation. Our starting point is the Herglotz-Petrovsky-Leray integral representation of the fundamental solution. We find an explicit decomposition of the latter solution into integrals over vanishing cycles associated with the isolated singularities on the slowness surface. As is well known in the theory of isolated singularities, integrals over vanishing cycles satisfy a system of differential quations known as Picard-Fuchs equations. We discuss a method to obtain these equations explicitly. Subsequently, we use these to analyse the asymptotic behavior of the fundamental solution near wave front singularities in three dimensions. Our work sheds new light on how to compute the so-called Cagniard-De Hoop contour which is used in numerical integration schemes to obtain the full time behaviour of the fundamental solution for a given direction of propagation.

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Authors and Affiliations

  1. Koninklijke/Shell Exploratie en Produktie Laboratorium, Volmerlaan 6, 2288 GD, Rijswijk, The Netherlands

    Dirk-J. Smite

  2. Schlumberger Cambridge Research, High Cross, Madingley Road, CB3 0EL, Cambridge, England

    Maarten V. de Hoop

Authors
  1. Dirk-J. Smite
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  2. Maarten V. de Hoop
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G. F. Helminck

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© 1993 Springer-Verlag

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Smite, DJ., de Hoop, M.V. (1993). The geometry of elastic waves propagating in an anisotropic elastic medium. In: Helminck, G.F. (eds) Geometric and Quantum Aspects of Integrable Systems. Lecture Notes in Physics, vol 424. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0021445

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  • DOI: https://doi.org/10.1007/BFb0021445

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  • Print ISBN: 978-3-540-57365-4

  • Online ISBN: 978-3-540-48090-7

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