Numerical simulation of acoustic time reversal mirrors

We study the time reversal phenomenon in a homogeneous and nondissipative 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 self-adjoint and compact. The DORT 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 behavior 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.