Elastic wave equation

Yves Colin de Verdière[1]

  • [1] Université Grenoble 1 Institut Fourier — UMR CNRS-UJF 5582 BP 74 38402-Saint Martin d’Hères cedex (France)

Séminaire de théorie spectrale et géométrie (2006-2007)

  • Volume: 25, page 55-69
  • ISSN: 1624-5458

Abstract

top
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 decompositionThe 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.

How to cite

top

Colin de Verdière, Yves. "Elastic wave equation." Séminaire de théorie spectrale et géométrie 25 (2006-2007): 55-69. <http://eudml.org/doc/11230>.

@article{ColindeVerdière2006-2007,
abstract = {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 (waves and waves) and its relation to Hodge decompositionThe 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.},
affiliation = {Université Grenoble 1 Institut Fourier — UMR CNRS-UJF 5582 BP 74 38402-Saint Martin d’Hères cedex (France)},
author = {Colin de Verdière, Yves},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {polarizations of waves; Hodge decomposition; Weyl law; Schnirelman's Theorem; scattering matrix},
language = {eng},
pages = {55-69},
publisher = {Institut Fourier},
title = {Elastic wave equation},
url = {http://eudml.org/doc/11230},
volume = {25},
year = {2006-2007},
}

TY - JOUR
AU - Colin de Verdière, Yves
TI - Elastic wave equation
JO - Séminaire de théorie spectrale et géométrie
PY - 2006-2007
PB - Institut Fourier
VL - 25
SP - 55
EP - 69
AB - 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 (waves and waves) and its relation to Hodge decompositionThe 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.
LA - eng
KW - polarizations of waves; Hodge decomposition; Weyl law; Schnirelman's Theorem; scattering matrix
UR - http://eudml.org/doc/11230
ER -

References

top
  1. J. Bolte & R. Glaser. Quantum ergodicity for Pauli Hamiltonians with spin 1 / 2 . Nonlinearity, 13:1987–2003 (2000). Zbl1041.81529MR1794842
  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). Zbl0253.35058
  3. Yves Colin de Verdière. Ergodicité et fonctions propres du laplacien. Commun. Math. Phys., 102:497–502 (1985). Zbl0592.58050MR818831
  4. F. Faure & S. Nonnenmacher. On the maximal scarring for quantum cat map eigenstates. Commun. Math. Phys., 245:201–214 (2004). Zbl1071.81044MR2036373
  5. P. Gérard & E. Leichtnam. Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J., 71:559–607 (1993). Zbl0788.35103MR1233448
  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). Zbl1098.81034MR2013805
  8. L. Landau & E. Lifchitz. Théorie de l’élasticité. Editions Mir, Moscow, 1990. Zbl0166.43101
  9. L. Margerin, in preparation . 
  10. M. Reed & B. Simon. Methods of Modern Mathematical Physics, vol 1. Academic Press, (1972). Zbl0242.46001MR493419
  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). Zbl0931.35114
  12. A. Schnirelman, Ergodic properties of eigenfunctions. Usp. Math. Nauk., 29:181–182 (1974). MR402834
  13. R. Schubert. Semi-classical Behaviour of Expectation Values in Time Evolved Lagrangian States for Large Times. Commun. Math. Phys., 256:239–254 (2005). Zbl1067.81040MR2134343
  14. Günter Schwarz. Hodge Decomposition–A Method for Solving Boundary Values Problems. Springer LN, 1607 (1995). Zbl0828.58002MR1367287
  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. Zbl0432.73021MR535598
  16. S. Zelditch, Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J., 55:919–941 (1987). Zbl0643.58029MR916129
  17. S. Zelditch & M. Zworski. Ergodicity of eigenfunctions for ergodic billiards. Commun. Math. Phys., 175:673–682 (1996). Zbl0840.58048MR1372814

NotesEmbed ?

top

You must be logged in to post comments.