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 and 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 (waves and waves) and its relation to Hodge decomposition
- The Weyl law and equipartition of energy between waves and 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}, zbl = {1171.35427}, mrnumber = {2478808}, language = {en}, url = {http://archive.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 SP - 55 EP - 69 VL - 25 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/tsg.247/ DO - 10.5802/tsg.247 LA - en ID - TSG_2006-2007__25__55_0 ER -
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/
[1] J. Bolte & R. Glaser. Quantum ergodicity for Pauli Hamiltonians with spin . 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 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
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