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Numerical Simulation of Acoustic Time Reversal Mirrors

Chokri Ben-Amar 1 Nabil Gmati 1 Christophe Hazard 2 Karim Ramdani 3, 4 
2 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
4 CORIDA - Robust control of infinite dimensional systems and applications
IECN - Institut Élie Cartan de Nancy, LMAM - Laboratoire de Mathématiques et Applications de Metz, Inria Nancy - Grand Est
Abstract : We study the time reversal phenomenon in a homogeneous and non-dissipative medium containing sound-hard obstacles. We propose two mathematical models of time reversal mirrors in the frequency domain. The first one takes into account the interactions between the mirror and the obstacles. The second one provides an approximation of these interactions. We prove, in both cases, that the time reversal operator $T$ is selfadjoint and compact. The D.O.R.T method (french acronym for Decomposition of the Time Reversal Operator) is explored numerically. In particular, we show that selective focusing, which is known to occur for small and distant enough scatterers, holds when the wavelength and the size of these scatterers are of the same order of magnitude (medium frequency situation). Moreover, we present the behaviour of the eigenvalues of $T$ according to the frequency and we show their oscillations due to the interactions between the mirror and the obstacles and between the obstacles themselves.
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Chokri Ben-Amar, Nabil Gmati, Christophe Hazard, Karim Ramdani. Numerical Simulation of Acoustic Time Reversal Mirrors. SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2007, 67 (3), pp.777-791. ⟨10.1137/060654542⟩. ⟨hal-00140349⟩



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