Elastic wave equation
Séminaire de théorie spectrale et géométrie, Volume 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.
DOI: 10.5802/tsg.247
Colin de Verdière, Yves 1

1 Université Grenoble 1 Institut Fourier — UMR CNRS-UJF 5582 BP 74 38402-Saint Martin d’Hères cedex (France)
@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},
     volume = {25},
     year = {2006-2007},
     doi = {10.5802/tsg.247},
     mrnumber = {2478808},
     zbl = {1171.35427},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/tsg.247/}
}
TY  - JOUR
AU  - Colin de Verdière, Yves
TI  - Elastic wave equation
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2006-2007
DA  - 2006-2007///
SP  - 55
EP  - 69
VL  - 25
PB  - Institut Fourier
UR  - http://www.numdam.org/articles/10.5802/tsg.247/
UR  - https://www.ams.org/mathscinet-getitem?mr=2478808
UR  - https://zbmath.org/?q=an%3A1171.35427
UR  - https://doi.org/10.5802/tsg.247
DO  - 10.5802/tsg.247
LA  - en
ID  - TSG_2006-2007__25__55_0
ER  - 
%0 Journal Article
%A Colin de Verdière, Yves
%T Elastic wave equation
%J Séminaire de théorie spectrale et géométrie
%D 2006-2007
%P 55-69
%V 25
%I Institut Fourier
%U https://doi.org/10.5802/tsg.247
%R 10.5802/tsg.247
%G en
%F TSG_2006-2007__25__55_0
Colin de Verdière, Yves. Elastic wave equation. Séminaire de théorie spectrale et géométrie, Volume 25 (2006-2007), pp. 55-69. doi : 10.5802/tsg.247. http://www.numdam.org/articles/10.5802/tsg.247/

[1] J. Bolte & R. Glaser. Quantum ergodicity for Pauli Hamiltonians with spin 1/2. Nonlinearity, 13:1987–2003 (2000). | MR | Zbl

[2] J. Chazarain. Construction de la paramétrix du problème mixte hyperbolique pour l’équation des ondes. C. R. Acad. Sci., Paris, Sér. A 276:1213–1215 (1973). | Zbl

[3] Yves Colin de Verdière. Ergodicité et fonctions propres du laplacien. Commun. Math. Phys., 102:497–502 (1985). | MR | Zbl

[4] F. Faure & S. Nonnenmacher. On the maximal scarring for quantum cat map eigenstates. Commun. Math. Phys., 245:201–214 (2004). | MR | Zbl

[5] P. Gérard & E. Leichtnam. Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J., 71:559–607 (1993). | MR | Zbl

[6] R. Hennino, N. Trégourès, N. M. Shapiro, L. Margerin, M. Campillo, B. A. van Tiggelen & R. L. Weaver. Observation of Equipartition of Seismic Waves. Phys. Rev. Lett., 86:3447–3450 (2001).

[7] J.P. Keating, J. Marklof & B. Winn. Value Distribution of the Eignefunctions and Spectral Determinants of Quantum Star Graphs. Commun. Math. Phys., 241:421–452 (2003). | MR | Zbl

[8] L. Landau & E. Lifchitz. Théorie de l’élasticité. Editions Mir, Moscow, 1990. | Zbl

[9] L. Margerin, in preparation .

[10] M. Reed & B. Simon. Methods of Modern Mathematical Physics, vol 1. Academic Press, (1972). | MR | Zbl

[11] Ph. Sécher. Etude spectrale du système différentiel 2×2 associé à un problème d’élasticité linéaire. Ann. Fac. Sc. Toulouse, 7:699–726 (1998). | Numdam | Zbl

[12] A. Schnirelman, Ergodic properties of eigenfunctions. Usp. Math. Nauk., 29:181–182 (1974). | MR

[13] R. Schubert. Semi-classical Behaviour of Expectation Values in Time Evolved Lagrangian States for Large Times. Commun. Math. Phys., 256:239–254 (2005). | MR | Zbl

[14] Günter Schwarz. Hodge Decomposition–A Method for Solving Boundary Values Problems. Springer LN, 1607 (1995). | MR | Zbl

[15] M. Taylor. Rayleigh waves in linear elasticity as a propagation of singularities phenomenon. Partial differential equations and geometry (Proc. Conf., Park City, Utah, 1977), pp. 273–291, Lecture Notes in Pure and Appl. Math., 48, Dekker, New York, 1979. | MR | Zbl

[16] S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J., 55:919–941 (1987). | MR | Zbl

[17] S. Zelditch & M. Zworski. Ergodicity of eigenfunctions for ergodic billiards. Commun. Math. Phys., 175:673–682 (1996). | MR | Zbl

Cited by Sources: