Étude de deux problèmes de propagation d'ondes transitoires : 1) Focalisation spatio-temporelle en acoustique; 2) Transmission entre un diélectrique et un métamatériau.

Maxence Cassier 1
1 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
Abstract : This thesis consists of two independent parts. In the first one, we consider a reference medium, containing some scatterers whose positions are unknown. Using an array of transducers, our goal is to generate an acoustic wave that would focus in space and in time near one of these scatterers. The technique proposed here is based on DORT method (French acronym for Decomposition of the Time Reversal Operator) which leads to space focusing properties in the frequency domain. It consists in a singular value decomposition (SVD) of the scattering operator, that is, the operator which maps the input signals sent to the transducers to the measure of the scattered wave. In the context of the Foldy-Lax approximation (which is justified here in 2D), by introducing a particular SVD related to the symmetry of the scattering operator, we show how to synchronize the time-harmonic signals derived from the DORT method to achieve space-time focusing. Theoretical properties and a numerical study justify our approach. The second part is devoted to a transmission problem between a dielectric and a metamaterial. The question we consider is the following : does the limiting amplitude principle hold in such a medium? This principle defines the stationary regime as the large time asymptotic behavior of a system subject to a periodic excitation. An answer is proposed here in the case of a two-layered medium composed of a dielectric and a particular metamaterial (Drude model). In this context, we reformulate the time-dependent Maxwell's equations as a Schrödinger equation and perform its complete spectral analysis. As an application of this study, we show finally that the limiting amplitude principle holds except for a particular fequency, called the plasmonic frequency, characterised by a ratio of permittivities and permeabilities equal to -1 across the interface. This frequency is a resonance of the system and the response to this excitation blows up linearly in time.
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Maxence Cassier. Étude de deux problèmes de propagation d'ondes transitoires : 1) Focalisation spatio-temporelle en acoustique; 2) Transmission entre un diélectrique et un métamatériau.. Equations aux dérivées partielles [math.AP]. Ecole Polytechnique X, 2014. Français. ⟨pastel-01023289⟩

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