A spectral study of an infinite axisymmetric elastic layer

Lahcène Chorfi

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 5, page 849-863
  • ISSN: 0764-583X

Abstract

top
We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators A n , n ∈ ℕ , in a suitable Hilbert space. We show that the essential spectrum of A n is an interval of type [ γ , + ∞ [ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.

How to cite

top

Chorfi, Lahcène. "A spectral study of an infinite axisymmetric elastic layer." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.5 (2001): 849-863. <http://eudml.org/doc/194076>.

@article{Chorfi2001,
abstract = {We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators $A_n$, $n\in \mathbb \{N\}$, in a suitable Hilbert space. We show that the essential spectrum of $A_n$ is an interval of type $[\gamma ,+\infty [$ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.},
author = {Chorfi, Lahcène},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {elasticity; axisymmetry; eigenmodes; min-max principle; axisymmetric layer; waves; spectral theory; eigenvalues; Hilbert spaces},
language = {eng},
number = {5},
pages = {849-863},
publisher = {EDP-Sciences},
title = {A spectral study of an infinite axisymmetric elastic layer},
url = {http://eudml.org/doc/194076},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Chorfi, Lahcène
TI - A spectral study of an infinite axisymmetric elastic layer
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 5
SP - 849
EP - 863
AB - We present here a theoretical study of eigenmodes in axisymmetric elastic layers. The mathematical modelling allows us to bring this problem to a spectral study of a sequence of unbounded self-adjoint operators $A_n$, $n\in \mathbb {N}$, in a suitable Hilbert space. We show that the essential spectrum of $A_n$ is an interval of type $[\gamma ,+\infty [$ and that, under certain conditions on the coefficients of the medium, the discrete spectrum is non empty.
LA - eng
KW - elasticity; axisymmetry; eigenmodes; min-max principle; axisymmetric layer; waves; spectral theory; eigenvalues; Hilbert spaces
UR - http://eudml.org/doc/194076
ER -

References

top
  1. [0] A. Bamberger, Y. Dermenjian and P. Joly, Mathematical analysis of the propagation of elastic guided waves in heterogeneous media. J. Differential Equations 88 (1990) 113–154. Zbl0714.35045
  2. [0] A. Bamberger, P. Joly and M. Kern, Propagation of elastic surface waves along a cylindrical cavity of arbitrary cross section. RAIRO Modél. Math. Anal. Numér. 25 (1991) 1–30. Zbl0739.73009
  3. [0] M. Bouchon and D.P. Schmitt, Full-wave acoustic logging in an irregular borehole. Geophysics 54 (1989) 758–765. 
  4. [0] L. Chorfi, Étude mathématique des modes guidés dans un milieu élastique à symétrie de révolution. RAIRO Modél. Math. Anal. Numér. 30 (1996) 299–342. Zbl0852.73021
  5. [0] D.J. Duterte, A.S. Bonnet-Ben Dhia and P. Joly, Mathematical analysis of elastic surface waves in topographic waveguides. M 3 AS (Math. Models Methods Appl. Sci.) 9 (1999) 755–798. Zbl0946.74034
  6. [0] G. Duvaut, Mécanique des milieux continus. Masson, Paris (1990). 
  7. [0] T. Kato, Perturbation Theory for Linear Operators. 2nd edn., Springer-Verlag, New York (1976). Zbl0342.47009MR407617
  8. [0] J. Miklowitz, The Theory of Elastic Waves and Wave Guides. North-Holland Publishing Company, Amsterdam, New York, Oxford (1980). Zbl0565.73025
  9. [0] J.A. Nitsche, On Korn’s second inequality. RAIRO Anal. Numér. 15 (1981) 237–248. Zbl0467.35019
  10. [0] B. Nkemzi and B. Heinrish, Partial Fourier approximation of the Lamé equation in axisymmetric domains. Math. Methods Appl. Sci. 22 (1999) 1017–1041. Zbl0932.65117
  11. [0] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV Analysis of Operators. Academic Press, New York, San Francisco, London (1978). Zbl0401.47001MR493421
  12. [0] M. Schechter, Operator Methods in Quantum Mechanics. North-Holland Publishing Company, Amsterdam, New York, Oxford (1981). Zbl0456.47012MR597895
  13. [0] G. A. Winbow, Seismic sources in open cased boreholes. Geophysics 56 (1991) 1040–1050. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.