Mathematical Problems in Mechanics
A two-scale model for the wave equation with oscillating coefficients and data
Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1439-1442.

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.

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.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2009.10.017
Brassart, Matthieu 1; Lenczner, Michel 2

1 UMR 6623, Laboratoire de mathématiques de Besançon, Université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France
2 FEMTO-ST, Université Technologique de Belfort-Montbeliard, 26, chemin de l'Epitaphe, 25030 Besançon, France
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Brassart, Matthieu; Lenczner, Michel. A two-scale model for the wave equation with oscillating coefficients and data. Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1439-1442. doi : 10.1016/j.crma.2009.10.017. http://www.numdam.org/articles/10.1016/j.crma.2009.10.017/

[1] Allaire, G. Homogenization and two-scale convergence, SIAM J. Math. Anal., Volume 23 (1992) no. 6, pp. 1482-1518

[2] Allaire, G.; Conca, C. Analyse asymptotique spectrale de l'équation des ondes. Complétude du spectre de Bloch, C. R. Acad. Sci. Paris, Ser. I, Volume 321 (1995) no. 5, pp. 557-562

[3] Francfort, G.A.; Murat, F. Oscillations and energy densities in the wave equation, Comm. Partial Differential Equations, Volume 17 (1992) no. 11–12, pp. 1785-1865

[4] M. Kader, Contributions to modeling and control of distributed intelligent systems: Application to beam vibration control, PhD thesis, Université de Franche-Comté, France, 2000

[5] Lenczner, M.; Kader, M.; Perrier, P. Two-scale model of the wave equation with oscillating coefficients, C. R. Acad. Sci. II, Mec. Phys. Astron., Volume 328 (2000) no. 4, pp. 335-340

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