ABSTRACT
Time reversal (TR) of acoustic, elastic, and electromagnetic waves
has been extensively studied in recent years [1, 2]. In a standard
TR experiment, waves generated by a source are first measured
by an array of antennas positioned around the source and then
time-reversed and simultaneously rebroadcasted by the same
antenna array. Due to the time invariance of the wave process, the
reemitted energy will focus back on the original source, whatever
the complexity of the propagation medium [3, 4]. The present
paper concentrates on the application of TR to the focusing and
manipulation of water waves both in linear and nonlinear regimes.
The problem is not as trivial as that with acoustic waves. Let us cite
Richard Feynman [5], “Water waves that are easily seen by everyone
and which are usually used as an example of waves in elementary
courses are the worst possible example, because they are in no
respects like sound and light; they have all the complications that
waves can have.” Water waves are scalar waves; that refers to the
evolution of small perturbation of the height of fluid under the
action of gravity and surface tension. They are dispersive by nature,
nonlinear when generated with standard wave makers, and they
experience strong damping at the scale of laboratory experiments.
The effect of dispersion on the TR process has already been studied
in TR experiments for guided elastic waves [6, 7]; these waves are
dispersionless in free space, and the dispersion is due only to the
reflection on the boundaries of the waveguide. In the case of water
waves, the dispersion is intrinsic but preserves the TR invariance
(obviously, not taking the damping into account). The effect of
the nonlinearities has been experimentally studied in Ref. [8] for
acoustic waves where it has been shown that the TR invariance is
preserved as long as nonlinearities do not create dissipation, that
is, as long as the propagation distance is smaller than the shock
distance. In the case of water waves, the effect of nonlinearities
has to be treated. The evolution dynamics in time and space of
nonlinear wave trains in deep water can be modeled using the
focusing nonlinear Schro¨dinger equation (NLSE). We will show
the implication of the TR invariance on the NLSE and we will
demonstrate a way to experimentally focus, both in time and space,
rogue waves using the principles of TR mirrors.
