Experimental Evidence of Nonlinear Focusing in Standing Water Waves

Nonlinear wave focusing originating from the universal modulation instability (MI) is responsible for the formation of strong wave localizations on the water surface and in nonlinear wave guides, such as optical Kerr media and plasma. Such extreme wave dynamics can be described by breather solutions of the nonlinear Schrödinger equation (NLSE) like by way of example the famed doubly-localized Peregrine breathers (PB), which typify particular cases of MI. On the other hand, it has been suggested that the MI relevance weakens when the wave field becomes broadband or directional. Here, we provide experimental evidence of nonlinear and distinct PB-type focusing in standing water waves describing the scenario of two counterpropagating wave trains. The collected collinear wave measurements are in excellent agreement with the hydrodynamic coupled NLSE (CNLSE) and suggest that MI can undisturbedly prevail during the interplay of several wave systems and emphasize the potential role of exact NLSE solutions in extreme wave formation beyond the formal narrow band and unidirectional limits. Our work may inspire further experimental investigations in various nonlinear wave guides governed by CNLSE frameworks as well as theoretical progress to predict strong wave coherence in directional fields.

Figure 1

Sketch of the wave flume setup as installed at the University of Sydney after removing the artificial beach installation to enable full wave reflection. The piston-type wave generator on the right end generates long-crested unidirectional waves. The glass wall at the opposing end acts as a wave reflector.

Figure 2

Top (a) and (b) Experimental observation of two standing PBs as measured by the wave gauges. Bottom (c) and (d) Results of numerical CNLSE simulations starting from the same boundary conditions as the laboratory experiments. Left (a) and (c) Wave parameters $a=0.01\mathrm{m}$ and $ak=0.09$. Right (b) and (d) Wave parameters $a=0.01\mathrm{m}$ and $ak=0.10$.

Figure 3

Spectral evolution of the standing Peregrine breathers, reported in Fig. 2. (a) Computed for the measured surface elevation for the carrier parameters $a=0.01\mathrm{m}$ and $ak=0.09$. (c) Incident PB wave dynamics isolated from (a). (e) Opposing wave train as isolated from (a). (b) Computed for the measured surface elevation for the carrier wave parameters $a=0.01\mathrm{m}$ and $ak=0.10$. (d) Incident PB wave dynamics isolated from (b). (f) Opposing wave train as isolated from (b).

Figure 4

(a) Evolution of a standing degenerate soliton for $a=0.01\mathrm{m}$ and $ak=0.11$. (b) Corresponding CNLSE simulations. (c) Comparison of the maximal temporal surface profile (top line) with the CNLSE prediction (bottom line). (d) Comparison of the maximal interpolated spatial surface profile (top line) with the CNLSE prediction (bottom line). The dashed line corresponds to the respective wave envelope profile, which has been extracted from the experimental data using the Hilbert transform.

Figure 5

Evolution of a PB in standing waves with the carrier steepness $ak=0.09$ as simulated by the HOSM by arranging two dozen phases allowing for the reconstruction of a wave envelope. (a) The surface displacement envelope $k\eta$ is shown by the pseudocolor. The surface displacement profile of the highest focused wave is displayed in black while the dashed black lines point out the location of maximal envelope compression, both according to the analytic PB solution. The magenta parallelogram shows the area where the maximum wave amplitude is evaluated. The coordinate and time are normalized with the dominant wave length $\lambda$ and period $T$, respectively. (b) Evolution of the maximum wave amplitude in the HOSM simulations and according to the analytic PB solution of the NLSE. The filled areas correspond to the intervals between crest and trough amplitudes, while the solid curves depict their simple means.