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Comptes Rendus Mathématique
Volume 347, n° 23-24
pages 1439-1442 (décembre 2009)
Doi : 10.1016/j.crma.2009.10.017
Received : 4 Mars 2009 ;  accepted : 21 October 2009
A two-scale model for the wave equation with oscillating coefficients and data
Un modèle à deux échelles pour l’équation des ondes à coefficients et données oscillants
 

Matthieu Brassart a , Michel Lenczner b
a UMR 6623, Laboratoire de mathématiques de Besançon, Université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France 
b FEMTO-ST, Université Technologique de Belfort-Montbeliard, 26, chemin de l’Epitaphe, 25030 Besançon, France 

Abstract

We introduce a time-space two-scale transform designed to capture the high and low frequency waves in the asymptotics of the periodic homogenization of the wave equation. The asymptotical solution is the sum of the solution of known homogenized equations and of Bloch waves. We also derive the transport equations satisfied by the Bloch wave coefficients. To cite this article: M. Brassart, M. Lenczner, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.
Résumé

Nous introduisons une transformation à deux échelles en espace et en temps destinée à capturer à la fois les basses fréquences et les ondes de Bloch qui apparaissent lors du processus asymptotique d’homogénéisation de l’équation des ondes à coefficients périodiques. La solution du modèle qui en résulte comprend les ondes de Bloch et une contribution basse fréquence qui est la solution du modèle homogénéisé de l’équation des ondes. On établit aussi les équations de transport vérifiées par les coefficients des ondes de Bloch. Pour citer cet article : M. Brassart, M. Lenczner, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The full text of this article is available in PDF format.


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