Time-reversal of nonlinear waves: Applicability and limitations

Time-reversal (TR) refocusing of waves is one of the 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 backward. Laboratory experiments have 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 time-reversal using a unidirectional 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.

Figure 1

Sketch of the NWT.

Figure 2

Target wave elevation used to define the wave maker's motion: Peregrine breather with $ka=0.09$ and $a=0.003$ m, evaluated at $x=0$.

Figure 3

Peregrine breather surface elevation with $ka=0.09$ and $a=0.003$ m. Shown on top is the probe signal measured at $x_M$ after the propagation on a distance $k{x}_M=270$ and on the bottom is the time-reversed signal, providing a new wave maker's boundary condition.

Figure 4

Comparison of the Peregrine surface profile as generated by the wave maker (dashed line) with the reconstructed and refocused waves after the TR procedure (solid line). The carrier parameters are $ka=0.09$ and $a=0.003$ m, while $k{x}_M=270$.

Figure 5

Accuracy diagram of the time-reversal for the stationary envelope soliton $R_{\text{amp}}$ as a function of initial steepness $ka$ and propagating distance $k{x}_M$.

Figure 6

Free surface elevation of the envelope soliton for an amplitude $a=0.01$ m with a steepness of $ka=0.30$, recorded at the probe after a dimensionless propagation distance of $k{x}_M=330$.

Figure 7

Results of time-reversal for an envelope soliton for an amplitude $a=0.01$ m, steepness $ka=0.30$, and $k{x}_M=330$.

Figure 8

Accuracy diagram of the time-reversal for the doubly localized Peregrine breather $R_{\text{amp}}$ as a function of initial steepness $ka$ and propagating distance $k{x}_M$.

Figure 9

Accuracy diagram of the time-reversal for the doubly localized Akhmediev-Peregrine breather of order two $R_{\text{amp}}$ as a function of initial steepness $ka$ and propagating distance $k{x}_M$.

Figure 10

Accuracy of the TR method for the Peregrine breather. Dashed lines are isolines in terms of the modulation instability process.

Figure 11

Propagation of the Peregrine breather $\left(ka=0.09\right)$ in the wave basin, space-time representation in scaled NLSE variables: comparison of the analytic solution (top) and the HOS NWT numerical simulation (bottom).

Figure 12

Space and time evolution of the Peregrine breather for $ka=0.09$ and $k{x}_M=270$ (dash-dotted line) in the NWT: demodulation (top) and TR refocusing (bottom).

Figure 13

Comparison of the Peregrine surface profile at maximal compression, as initially generated with respect to the NLSE theory (dashed line) and after TR refocusing (solid line), for $ka=0.09$ and $k{x}_M\in \{60,165,270,375,480\}$ (from top to bottom).

Figure 14

Probe signal measured after the propagation of the Peregrine breather for $k{x}_M=270$ for different steepnesses $ka\in \{0.03,0.07,0.10,0.12\}$ (from top to bottom).

Figure 15

Space and time evolution of the Peregrine breather for $ka=0.12$ in the NWT: comparison of the analytic solution (top) and the HOS NWT numerical simulation (bottom).

Figure 16

Comparison of the Peregrine surface profile at maximal compression, as initially generated with respect to the NLSE theory (dashed line) and after TR refocusing (solid line), for $k{x}_M=270$ and $ka\in \{0.03,0.07,0.1,0.12\}$ (from top to bottom).

Figure 17

Space and time evolution of the Peregrine breather for $ka=0.12$ and $k{x}_M=480$ (dash-dotted line) in the NWT: demodulation (top) and TR refocusing (bottom).