Preprint A159/2002
Time-reversed refocusing of surface water waves

Andre Nachbin | Fouque, Jean-Pierre

**Keywords: **
water waves | inhomogeneous media | asymptotic theory

A time-reversal mirror is, roughly speaking, a device which is capable
of receiving a signal in time,
keeping it in memory and sending it back into the medium in the
reversed direction of time. A brief mathematical review of the
time-reversal theory is presented
in the context of the linear shallow water equations. In particular
an explicit expression is given for the refocused pulse in the simplest time-reversal case.
The explicit expression for the power spectral density of the reflection process
is
used to construct the highpass filter which controls the refocusing
process. Time-reversal numerical
experiments in the (effectively) linear regime are used to validate the nonlinear shallow water code.
The numerically refocused pulse is compared with the theoretical predicted shape.
Further numerical experiments illustrate the robustness of the theory. In particular the
time-reversal refocusing with smaller cutoff windows, the self-averaging property
and finally refocusing when the nonlinear term is small but not negligible.