Elastic wave equation
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.
@article{TSG_2006-2007__25__55_0,
     author = {Colin de Verdi\`ere, Yves},
     title = {Elastic wave equation},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {55--69},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {25},
     year = {2006-2007},
     doi = {10.5802/tsg.247},
     mrnumber = {2478808},
     zbl = {1171.35427},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/tsg.247/}
}
Colin de Verdière, Yves. Elastic wave equation. Séminaire de théorie spectrale et géométrie, Tome 25 (2006-2007) , pp. 55-69. doi : 10.5802/tsg.247. http://archive.numdam.org/articles/10.5802/tsg.247/

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