Mathematical analysis of the propagation of elastic guided waves in heterogeneous media. (English) Zbl 0714.35045

This article is concerned with the propagation of elastic waves in isotropic heterogeneous media, invariant under translation in one direction. The authors give a theoretical analysis of the existence of guided waves and of their properties. By definition, a guided wave (or guided mode) is a solution of the elastodynamic equations in the form \[ U_ j(x,t)=\tilde u_ j(x_ 1,x_ 2)\exp i(\omega t-\beta x_ 3),\quad j=1,2,3, \] where \(\omega >0\) is the pulsation of the mode, \(\beta >0\) is the wave number, and \((\tilde u_ j(x_ 1,x_ 2)\), \(j=1,2,3)\) is a complex valued vector field which must satisfy \[ 0<\sum^{3}_{j=1}\int_{R^ 2}| \tilde u_ j(x_ 1,x_ 2)|^ 2 dx_ 1 dx_ 2<+\infty. \] Such solutions can appear in a nonhomogeneous medium, if and only if \(\omega\) and \(\beta\) satisfy a dispersion relation. The problem is reduced to research the eigenvalues \(\omega^ 2\) and the eigenfunctions \(\tilde u\) of a selfadjoint operator \({\mathcal A}(\beta)\) in the Hilbert space \(L^ 2(R^ 2,\rho dx_ 1 dx_ 2)\). All the results stem from the spectral analysis of \({\mathcal A}(\beta)\) and the main tool of the analysis is the spectral theory of selfadjoint operators and more specifically the Max-Min principle.
Reviewer: Y.C.Yang


35L55 Higher-order hyperbolic systems
35P05 General topics in linear spectral theory for PDEs
74H45 Vibrations in dynamical problems in solid mechanics
47B25 Linear symmetric and selfadjoint operators (unbounded)
Full Text: DOI Link


[1] Achenbach, J. D., Wave Propagation in Elastic Solids (1973), North-Holland: North-Holland Amsterdam · Zbl 0268.73005
[2] Achenbach, J. D.; Gautesen, A. K.; McMaken, H., Rays Methods for Waves in Elastic Solids (1982), Pitman: Pitman London · Zbl 0498.73020
[3] Auld, B. A., (Acoustic Fields and Waves in Solids, Vol. I (1973), Wiley-Interscience: Wiley-Interscience New York). (Acoustic Fields and Waves in Solids, Vol. II (1973), Wiley-Interscience: Wiley-Interscience New York)
[4] Bamberger, A.; Bonnet, A. S., Calcul des modes guidés d’une fibre optique. Deuxiè me partie: analyse mathématique, (Rapport interne, Vol. 143 (1986), Centre de Mathématiques Appliquées, École Polytechnique: Centre de Mathématiques Appliquées, École Polytechnique 91 120 Palaiseau, France) · Zbl 0706.35099
[5] Bamberger, A.; Joly, P.; Kern, M., Étude mathématique des modes élastiques guidés par l’extérieur d’une cavité cylindrique de section arbitraire, (Rapport de Recherche, Vol. 650 (mars 1987), INRIA: INRIA Rocquencourt, France)
[6] Bonnet, A-S, Analyse Mathématique de la Propagation de Modes Guidés dans les Fibres Optiques, (Rapport de Recherche, Vol. 229 (août 1988), ENSTA: ENSTA 91 120 Palaiseau, France)
[7] Cagniard, L., Réflexion et Réfraction des Ondes Séismiques Progressives (1939), Gauthier-Villard: Gauthier-Villard Paris, (English translation, McGraw-Hill, New York, 1962) · JFM 65.1490.02
[8] Dermenjian, Y.; Guillot, J. C., Théorie spectrale de la propagation des ondes acoustiques dans un milieu stratifié perturbé, J. Differential Equations, 62, 357-409 (1986) · Zbl 0611.35063
[9] Dermenjian, Y.; Guillot, J. C., Scattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle, Math. Methods Appl. Sci., 10, 87-124 (1988) · Zbl 0647.73011
[10] Dunford, N.; Schwartz, J. T., (Linear Operators. Part II, Spectral Theory (1963), Interscience: Interscience New York) · Zbl 0128.34803
[11] Eringen, A. C.; Suhubi, E. S., (Elastodynamics, Linear Theory, Tome II (1975), Academic Press: Academic Press New York) · Zbl 0344.73036
[12] Guillot, J. C., Complétude des modes TE et TM pour un guide d’ondes optiques planaires, Rapport INRIA, no. 385 (mars 1985)
[13] Guillot, J. C., Existence and uniqueness of a Rayleigh surface wave propagating along the free boundary of a transversally isotropic elastic half space, Math. Methods Appl. Sci., 8, 289-310 (1986) · Zbl 0606.73024
[14] Joly, P., Un nouveau résultat d’ondes guidées en milieu élastique hétérogène, C. R. Acad. Sci Sér. I Math., 309, 793-796 (1989) · Zbl 0685.73020
[15] Miklowitz, J., Elastic waves and waveguides, (Applied Mathematics and Mechanics, Vol. 22 (1978), North-Holland: North-Holland Amsterdam) · Zbl 0109.17901
[16] Reed, M.; Simon, B., Methods of Modern Mathematical Physics I. Functional Analysis (1973), Academic Press: Academic Press San Diego
[17] Reed, M.; Simon, B., Methods of Modern Mathematical Physics IV. Analysis of Operators (1978), Academic Press: Academic Press San Diego · Zbl 0401.47001
[18] Schechter, M., Operator Methods in Quantum Mechanics (1981), Elsevier/North-Holland: Elsevier/North-Holland New York · Zbl 0456.47012
[19] Weder, R., Spectral and scattering theory in perturbed stratified fluids, J. Math. Pures Appl., 64, 149-173 (1985) · Zbl 0597.76074
[20] Weder, R., Absence of eigenvalues of the acoustic propagator in deformed wave guides, Rocky Mountain J. Math., 18, No. 2, 495-503 (1988) · Zbl 0657.76070
[21] Weder, R., Spectral and scattering theory in deformed optical wave guides, J. Reine Angew. Math., 390, 130-169 (1988) · Zbl 0644.35080
[22] Wilcox, C. H., Sound propagation in stratified fluids, Appl. Math. Sci., 50 (1984) · Zbl 0543.76107
[23] Wilcox, C. H., Scattering theory for the d’Alembert equation in exterior domains, (Lectures Notes in Mathematics, Vol. 442 (1975), Springer-Verlag: Springer-Verlag New York) · Zbl 0125.46005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.