Time-reversal of nonlinear waves: Applicability and limitations

Abstract

Time-reversal (TR) refocusing of waves is one of fundamental principles in wave physics. Using the TR approach, "Time-reversal mirrors" can physically create a time-reversed wave that exactly refocus back, in space and time, to its original source regardless of the complexity of the medium as if time were going backwards. Lately, laboratory experiments proved that this approach can be applied not only in acoustics and electromagnetism but also in the field of linear and nonlinear water waves. Studying the range of validity and limitations of the TR approach may determine and quantify its range of applicability in hydrodynamics. In this context, we report a numerical study of hydrodynamic TR using a uni-directional numerical wave tank, implemented by the nonlinear high-order spectral method, known to accurately model the physical processes at play, beyond physical laboratory restrictions. The applicability of the TR approach is assessed over a variety of hydrodynamic localized and pulsating structures' configurations, pointing out the importance of high-order dispersive and particularly nonlinear effects in the refocusing of hydrodynamic stationary envelope solitons and breathers. We expect that the results may motivate similar experiments in other nonlinear dispersive media and encourage several applications with particular emphasis on the field of ocean engineering.

Experimental reconstruction of extreme sea waves by time reversal principle

“…In this section, we will recall the main results of previous studies (Chabchoub & Fink 2014;Ducrozet et al 2016b) that have demonstrated the possibility of the use of a TR mirror to refocus extreme water waves obtained from analytic solutions to the NLSE in a water-wave facility. The TR procedure is briefly explained, followed by a description of the different rogue wave profiles used in the study.…”

“…At first, we define time-series ramps at the beginning and end of the chosen time window in order to ensure (i) a smooth initialization of the wavemaker's movement and (ii) the periodicity of the signal, which is decomposed on Fourier components. Then, the effective motion is defined using a linear transfer function to initiate the wave profiles (Chabchoub & Fink 2014;Ducrozet et al 2016b). This latter linearization has been identified as a possible source of inaccuracy in the TR procedure.…”

“…The application of the elegant time reversal (TR) principle (Fink 1992) in the context of water waves to refocus nonlinear and extreme waves in a hydrodynamic wave flume has been recently studied experimentally and numerically (Chabchoub & Fink 2014;Ducrozet, Fink & Chabchoub 2016b). Those studies take as reference wave profiles from exact analytic solutions of the nonlinear Schrödinger equation (NLSE) (Zakharov 1968).…”

“…Earlier experiments (Baldock, Swan & Taylor 1996) have shown that the wave profile measured after a large focused wave group is identical to the time-reversed profile of the wave field at the opposite location before the focused wave. The comprehensive numerical study provided in Ducrozet et al (2016b) assessed the applicability of the TR refocusing procedure over a wide range of propagating distances and initial wave steepness for different analytic solutions of the NLSE (stationary envelope soliton and doubly localized Peregrine-type breathers of first and second order (Peregrine 1983;Akhmediev, Eleonskii & Kulagin 1985)), and thus over a wide range of dispersion and nonlinearity parameters. When dealing with pulsating solutions exhibiting large amplifications of the carrier wave, as in the case for breathers (Onorato et al 2013;Dudley et al 2019;Chabchoub, Onorato & Akhmediev 2016), it can be implied that the TR procedure, applied experimentally to properly scaled real-ocean rogue waves in a water-wave flume, will provide an accurate refocusing up to limiting thresholds, such as the characteristic steepness of the considered wave field.…”

“…The present study provides the first validation of the TR procedure for real-ocean rogue waves, measured in different geographic locations. Previous works dealing with the experimental application of the TR method to water waves were limited to the use of reference waves from analytic solutions of the NLSE (Chabchoub & Fink 2014;Ducrozet et al 2016b). These are valid only for unidirectional, weakly nonlinear and narrow-banded wave fields.…”