Karim Ramdani

SELECTIVE ACOUSTIC FOCUSING USING TIME-HARMONIC REVERSAL MIRRORS ∗

A mathematical study of the focusing properties of acoustic fields obtained by a time-reversal process is presented. The case of time-harmonic waves propagating in a nondissipative medium containing sound-soft obstacles is considered. In this context, the so-called D.O.R.T. method (decomposition of the time-reversal operator in French) was recently proposed to achieve selective focusing by computing the eigenelements of the time-reversal operator. The present paper describes a justification of this technique in the framework of the far field model, i.e., for an ideal time-reversal mirror able to reverse the far field of a scattered wave. Both cases of closed and open mirrors, that is, surrounding completely or partially the scatterers, are dealt with. Selective focusing properties are established by an asymptotic analysis for small and distant obstacles.

Reconstructing the potential for the one-dimensional Schrödinger equation from boundary measurements

We consider the inverse problem of the determining the potential in the dynamical Schrödinger equation on the interval by the measurement on the whole boundary. Provided that source is generic using the Boundary Control method we recover the spectrum of the problem from the observation at either left or right end points. Using the specificity of the one-dimensional situation we recover the spectral function, reducing the problem to the classical one which could be treated by known methods. We adapt the algorithm to the situation when only the finite number of eigenvalues are known and provide the result on the convergence of the method.

Wide Frequency Band Numerical Approaches for Multiple Scattering Problems by Disks

Efficient, robust and accurate algorithms are proposed for solving the multiple scattering problem by M circular obstacles for the whole spectrum of frequency. The representation of the solution is based on an integral equation formulation next solved by using Fourier basis. Numerical examples are provided to show that the approaches are efficient.

Diffraction by an Acoustic Grating Perturbed by a Bounded Obstacle

An original approach to solve 2D time harmonic diffraction problems involving locally perturbed gratings is proposed. The propagation medium is composed of a periodically stratified half-space and a homogeneous half-space containing a bounded obstacle. Using Fourier and Floquet transforms and integral representations, the diffraction problem is formulated as a coupled problem of Fredholm type with two unknowns: the trace of the diffracted field on the interface separating the two half-spaces on one hand, and the restriction of the diffracted field to a bounded domain surrounding the obstacle, on the other hand.

Far Field Modeling of Electromagnetic Time Reversal and Application to Selective Focusing on Small Scatterers

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 decomposition of the time reversal operator (DORT) method is given for axially symmetric scatterers.

Asymptotic Analysis of a Neumann Problem in a Domain with Cusp. Application to the Collision Problem of Rigid Bodies in a Perfect Fluid

We study a two dimensional collision problem for a rigid solid immersed in a cavity filled with a perfect fluid. We are led to investigate the asymptotic behavior of the Dirichlet energy associated with the solution of a Laplace--Neumann problem as the distance

Mathematical analysis of conducting and superconducting transmission lines

\varepsilon>0

NUMERICAL SIMULATION OF ACOUSTIC TIME REVERSAL MIRRORS

between the solid and the cavitys bottom tends to zero. Denoting by

Conformal mapping for cavity inverse problem: an explicit reconstruction formula

\alpha>0

Transmission through a thick periodic slab

the tangency exponent at the contact point, we prove that the solid always reaches the cavity in finite time, but with a nonzero velocity for