Skip to main content
Log in

Approximation by finite differences of the propagation of acoustic waves in stratified media

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Summary

In this paper, we analyze the approximation of acoustic waves in a two layered media by a finite diffrences variational scheme. We examine in particular the approximation of the guided waves. We point out the existence of purely numerical parasitic phenomena and quantify the numerical dispersion relative to guided waves.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achenbach, J.D.: Wave propagation in elastic solids. Amsterdam: North Holland 1973

    Google Scholar 

  2. Alford, R.M., Kelly, R., Boore, D.M.: Accuracy of finite difference modelling of the acoustic wave equation. Geophysics.6, 834–842 (1974)

    Google Scholar 

  3. Bamberger, A., Chavent, G., Lailly, P.: Etude the schémas numérique des équations de lélastodynamique linéaire. Rapport INRIA No. 41, October 1980

  4. Bamberger, A., Guillot, J.C., Joly, P.: Numerical diffraction by a uniform grid. SIAM Numer. Anal.25, 753–783 (1988)

    Google Scholar 

  5. Lions, J.L.: Cours d'Analyse Numérique. Paris: Hermann 1973

    Google Scholar 

  6. Ciarlet, P.G.: The finite element method for elliptic problems. Amsterdam: North Holland 1978

    Google Scholar 

  7. Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York: Mac Graw Hill 1955

    Google Scholar 

  8. Daudier, S., Nicoletis, L.: Study of reflection and transmission properties of the variational scheme with finite differences. Rapport IFP 1984

  9. Dermenjian, Y., Guillot, J.C.: Theorle spectrale de la propagation des ondes acoustiques dans un milieu stratifié perturbé. J. Diff. Eq.62, 357–409 (1986)

    Google Scholar 

  10. Dunford, N., Schwartz, J.J.: Linear Operators, Part II; Spectral Theory. New York: Interscience 1983

    Google Scholar 

  11. Guillot, J.C.: Completude des modes T.E. et T.M. pour un guide d'ondes optique planaire. Rapport INRIA No. 385, Mars 1985

  12. Guillot, J.C., Joly, P.: Approximation par differences finies de la propagation d'ondes acoustique en milieu stratifié(1). Rapport INRIA (in preparation)

  13. Guillot, J.C., Joly, P.: Approximation numérique de la propagation, des ondes acoustiques en milieu stratifié. Note aux C.R. Acad. Sci. Paris, t. 303, Série I, No. 5, 1986

  14. Guillot, J.C., Wilcox, C.H.: Spectral analysis of the Epstein operator. Proc. R. Soc. Edinb. Sect. A80, 85–98 (1978)

    Google Scholar 

  15. Höhn, W.: Finite elements for the eigenvalue problem of differential operators in unbounded intervals. Math. Methods Appl. Sci.7, 1–19 (1985)

    Google Scholar 

  16. Joly, P.: Analyse numérique des ondes de Rayleigh. Thèse de 3ème Cycle Université Paris Dauphine 1983

  17. Kelly, R.: Numerical study of Love waves. Geophysics, Vol. 48, No. 7 (1983)

  18. Miklowitz, J.: The theory of elastic waves and wave-guides. Amsterdam: North-Holland 1980

    Google Scholar 

  19. Nicoletis, L.: Simulation numérique de la propagation d'ondes sismiques. Thèse de Docteur Ingénieur-Université Pierre et Marie Curie 1981

  20. Raviart, P.A., Thomas, J.M.: Analyse numérique des équations aux dérivées partielles. Paris: Masson 1983

    Google Scholar 

  21. Trefethen, L.N.: Group velocity in finite difference schemes. SIAM Review. Vol. 24, No. 2, April 1982

  22. Vichnevetsky, R., Bowles, J.B.: Fourier Analysis of Numerical Approximations of Hyperbolic equations. SIAM Stud. Appl. Math., Philadelphia, 1982

  23. Weder, R.: Spectral and scattering theory in perturbed stratified fluids. J. Math. Pures Appl.64, 149–173 (1985)

    Google Scholar 

  24. Weder, R.: Spectral an scattering theory in perturbed stratified fluids II: transmission problems and exterior domains. J. Differ. Equations64, 109–131 (1986)

    Google Scholar 

  25. Weinberger, H.: Variational Methods for Eigenvalue Approximations. SIAM-Regional Conference Series in Applied Mathematics 1974

  26. Wilcox, C.H.: Spectral analysis of the Pekeris operators. Arch. Ration. Mech. Anal.60, 259, 300 (1976)

    Google Scholar 

  27. Wilcox, C.H.: Sound propagation in Stratified Fluids. Appl. Math. Sci. Vol. 50. Berlin Heidelberg New York: Springer 1984

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guillot, J.C., Joly, P. Approximation by finite differences of the propagation of acoustic waves in stratified media. Numer. Math. 54, 655–702 (1989). https://doi.org/10.1007/BF01396488

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01396488

Subject Classification

Navigation