Time reversal by time-dependent perturbations

Guillaume Bal (1) , Mathias Fink (2) , Olivier Pinaud (3)

1 APAM - Department of Applied Physics and Applied Mathematics [New York]
2 Physique des ondes pour la médecine
3 G2ELab - Laboratoire de Génie Electrique de Grenoble

We consider the time-reversal of waves in time-dependent media. The constitu-tive parameters of a wave equation are assumed constant in time during intervals (0, t−τ)∪(t+τ, 2t) for τ t and time-varying during the time interval (t−τ, t+τ) to model a quasi-instantaneous time mirror. We show that under appropriate hypotheses , a time-reversed signal is generated by such time-dependent fluctuations. At time 2t, the time-reversed signal is an appropriate differentiation of the original initial condition when the time-varying fluctuations are independent of space. In the case of spatially varying fluctuations, we show that the time-reversed signal enjoys stronger refocusing properties when propagation occurs in a highly heterogeneous environment, as in the case of classical time-reversal. This paper offers additional theoretical justifications to the experimental results obtained in [1] and proposes potential extensions. We present numerical simulations that also corroborate and quantify the theoretical predictions.

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Guillaume Bal, Mathias Fink, Olivier Pinaud. Time reversal by time-dependent perturbations. 2018. ⟨hal-01881455⟩