Elastic wave equation

Colin de Verdière, Yves

Séminaire de théorie spectrale et géométrie, Tome 25 (2006-2007), pp. 55-69.

The goal of this talk is to describe the Lamé operator which drives the propagation of linear elastic waves. The main motivation for me is the work I have done in collaboration with Michel Campillo’s group from LGIT (Grenoble) on passive imaging in seismology. From this work, several mathematical problems emerged: equipartition of energy between $S-$ and $P-$waves, high frequency description of surface waves in a stratified medium and related inverse spectral problems.

We discuss the following topics:

• What is the definition of the operator and the natural (free) boundary conditions?
• The polarizations of waves ($S-$waves and $P-$waves) and its relation to Hodge decomposition
• The Weyl law and equipartition of energy between $S-$waves and $P-$waves. We formulate here questions in the spirit of Schnirelman’s Theorem about limits of Wigner measures of eigenmodes and of Schubert’s Theorem about the large time equipartition of an evolved Lagrangian state.
• Rayleigh waves for the half-space: we compute in a rather explicit way the spectral decomposition following the work of Ph. Sécher. Of particular interest are the scattering matrix and the density of states.

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