Time reversal and scattering theory for time-dependent acoustic waves in a homogeneous medium

We consider the scattering of time-dependent acoustic waves by some bounded obstacles located in a homogeneous and non-dissipative medium. We propose a far-field model of time reversal mirror that reverses the asymptotic wave profile of a scattered wave. Following the idea developed for time-harmonic waves in the so-called decomposition of the time reversal operator in French (DORT) method, we define a time reversal operator (TRO) in the time domain, which is related to the scattering operator involved in scattering theory. We show how this operator is related to the family of single frequency TROs involved in the DORT method, which clarifies the link between time-dependent and time-harmonic waves in the time reversal process. As an application of this relation, we prove that iterative time reversal generally converges to a single frequency time-harmonic wave.