We investigate the acoustic selective focusing properties of time reversal in a two-dimensional acoustic waveguide. A far-field model of the problem is proposed in the time-harmonic case. In order to tackle the question of selective focusing, we derive an asymptotic model for small scatterers. We show that in the framework of this limit problem, approximate eigenvectors of the time reversal operator can be obtained when the number of propagating modes of the waveguide is large enough. This result provides in particular a mathematical justification of the selective focusing properties observed experimentally. Some numerical experiments of selective focusing are presented.

Articulated solid bodies are shape-changing bodies made of rigid solids linked together by means of holonomic constraints prescribed as functions of time. In this paper we study the locomotion of such swimming devices in an ideal fluid. Our study ranges over a wide class of problems: any number of immersed bodies are involved (without being hydrodynamically decoupled), the system fluid-bodies can be partially of totally confined and circulation, buoyancy and collisions between bodies are taken into account. We determine the Euler-Lagrange equation governing the dynamic of the system, study its well-posedness and describe a numerical scheme used in a Matlab toolbox (Biohydrodynamics Toolbox) that has been designed to realize easily related numerical simulations.

Abstract. A time harmonic far field model for closed electromagnetic time reversal mirrors is proposed. Then, a limit model corresponding to small perfectly conducting scatterers is derived. This asymptotic model is used to prove the selective focusing properties of the time reversal operator. In particular, a mathematical justification of the DORT method (Decomposition of the Time Reversal Operator method) is given for axially symmetric scatterers.Key words. electromagnetic scattering, time reversal, far field, small obstacles AMS subject classifications. 35B40, 35P25, 45A05, 74J20,78M351. Introduction. In the last decade, acoustic time reversal has definitely demonstrated its efficiency in target characterization by wave focusing in complex media (see the review papers [13,15]). In particular, it has been shown that selective focusing can be achieved using the eigenvectors (resp. eigenfunctions) of the so-called time reversal matrix (resp. operator). Known as the DORT method (french acronym for Diagonalization of the Time Reversal Operator, cf. [14,32,26,31,16,25,18]), this technique involves three steps. First, an incident wave is emitted in the medium containing the scatterers by the time reversal mirror (TRM). The scattered field is then measured by the mirror and time-reversed (or phase-conjugated in the time harmonic case). Finally, the obtained signal is then reemitted in the medium. By definition, the time reversal operator T is the operator describing two successive cycles Emission/Reception/Time-Reversal. If the propagation medium is non dissipative, the operator T is hermitian, since T = F * F, where F denotes the far field operator. The DORT method can thus be seen as a singular value decomposition of F. Moreover, in a particular range of frequencies (for which the scatterers can be considered as point-like scatterers), T has as many significant eigenvalues as there are scatterers in the medium, and the corresponding eigenfunctions generate incident waves that selectively focus on the scatterers. From the mathematical point of view, a detailed analysis of this problem has been proposed for the acoustic scattering problem by small scatterers in the free space in [19] and in a two-dimensional straight waveguide in [29]. Let us emphasize that time reversal has also been intensively studied in the context of random media (cf.[17] and the references therein).Recently, electromagnetic focusing using time reversal has been demonstrated experimentally [23] and used for imaging applications [24]. One of the first works dealing with mathematical and numerical aspects of electromagnetic time reversal is the paper [34]. The authors analyze therein the DORT method in the case of a homogeneous medium containing perfectly conducting or dielectric objects of particular shapes (circular rods and spheres). Their method is based on a low frequency approximation of a multipole expansion of the scattered field (i.e. a Fourier-Bessel series involving Hankel functions for circular rods and vector spherical functions for s...

We present an existence result for the stationary Vlasov–Poisson system in a bounded domain of ℝN, with more general hypotheses than considered so far in the literature. In particular, we prove the equivalence of the kinetic approach (which consists in looking for the equilibrium distribution function) and the potential approach (where the unknown is the electrostatic potential at equilibrium). We study the dependence of the solution on parameters such as the total mass of the distribution, or those entering in the boundary conditions of the potential. Focusing on the case of a plane polygon, we study the singular behavior of the solution near the re-entrant corners, and examine the dependence of the singularity coefficients on the parameters of the problem. Numerical experiments illustrate and confirm the analysis.