From modulational instability to focusing dam breaks in water waves

We report water wave experiments performed in a long tank where we consider the evolution of nonlinear deep-water surface gravity waves with the envelope in the form of a large-scale rectangular barrier. Our experiments reveal that, for a range of initial parameters, the nonlinear wave packet is not disintegrated by the Benjamin-Feir instability but exhibits a specific, strongly nonlinear modulation, which propagates from the edges of the wave packet toward the center with finite speed. Using numerical tools of nonlinear spectral analysis of experimental data, we identify the observed envelope wave structures with focusing dispersive dam break flows, a peculiar type of dispersive shock waves recently described in the framework of the semiclassical limit of the 1D focusing nonlinear Schrödinger equation (1D-NLSE). Our experimental results are shown to be in a good quantitative agreement with the predictions of the semiclassical 1D-NLSE theory. This is the first observation of the persisting dispersive shock wave dynamics in a modulationally unstable water wave system.

Figure 1

Experimental results showing the nonlinear evolution of several rectangular wave trains along the 1D water tank. The evolution is plotted in the frame of reference moving at the group velocity ${\omega }_0/\left(2k_0\right)$ of the wave packets. (a) A large-scale wave packet of constant amplitude unstable to small perturbations of its envelope is disintegrated by the Benjamin-Feir instability [the wave steepness is $k_0{a}_0\simeq 0.19$, the carrier period is $T_0=0.87\mathrm{s}$ and the wave amplitude is $a_0=3.7\mathrm{cm}$ (see text)]. (b) Three “boxes” of constant identical amplitudes, which are not disintegrated by the Benjamin-Feir instability, undergo some strongly nonlinear modulation which propagates with finite speed from the front and back edges of the wave packet toward the center under the form of counter-propagating dispersive dam break flows with DSW structure [the wave steepness is 0.082, the carrier period is $T_0=0.99\mathrm{s}$ and the wave amplitude is $a=2\mathrm{cm}$ (see text)]. The thick black dashed lines represent the theoretical breaking lines separating the genus 1 region from the genus 0 region; see calculation details in Secs. 3b and 4a.

Figure 2

Numerical simulation of Eq. (2) showing (a) the space time evolution of the wave field having a profile specified by Eq. (3) at $\xi =0$ ($q=1$, $T=1$, $\varepsilon =0.04$). The space-time evolution is separated into three regions of increasing genus $g$ (see text). The genus $g=0$ region corresponds locally to the plane wave solution. The genus $g=1$ region is associated to DSWs that are generated from the edges of the box. The genus $g=2$ region emerges from the collision of the two focusing dam break flows, see also the Supplemental Material Video S2 [63]. Curves plotted in blue lines in (b)–(d) represent the wave amplitude profiles at $\xi =0.25$, $\xi =0.35$, $\xi =0.477$, respectively. Red dashed lines in (b) and (c) represent the amplitudes of the modulated cnoidal waves that are determined from Eqs. (5, 6, 7). (e), (f) Spectral (IST) portraits of isolated structures made at $\xi =0.477$. The spectral portrait in (e) is mostly composed of two complex conjugate bands demonstrating that the analyzed structure has a genus $g=1$ (solitonlike structure, see text). The spectral portrait in (f) is composed of three bands demonstrating that the genus of the analyzed structure is $g=2$ (breatherlike structure).

Figure 3

Modulus of the water wave envelopes with durations (a) $\mathrm{\Delta }{T}_1=30\mathrm{s}$, (b) $\mathrm{\Delta }{T}_2=45\mathrm{s}$, (c) $\mathrm{\Delta }{T}_3=60\mathrm{s}$. The red (respectively, blue) lines represent the experimental envelopes of the signal recorded at $z_1=6\mathrm{m}$ (respectively, $z_{20}=120\mathrm{m}$), close to (respectively, far from) the wave maker. The magenta lines represent the envelopes computed from the numerical simulation of Eq. (1) ($k_0=4.1{\mathrm{m}}^{-1}$, ${\omega }_0=6.34{\mathrm{s}}^{-1}$, $\alpha =0.91$) by taking as initial condition the complex envelope measured by the gauge closest to the wave maker (red lines, $z_1=6\mathrm{m}$).

Figure 4

Experimental results showing the nonlinear evolution of the rectangular wave packet of largest size ($\mathrm{\Delta }{T}_3=60\mathrm{s}$, $a=2\mathrm{cm}$, $k_{0}a=0.082$, $T_0=0.99\mathrm{s}$) in Fig. 1. (a) Signals recorded by the 20 gauges placed all along the tank. All the envelopes superimposed on the carrier wave are computed from solutions given by Eqs. (5, 6, 7) using ${\tau }_0=-0.48$. The breaking lines plotted with full black lines in (a) are computed from Eq. (8). (b) Space-time evolution of the modulus of the envelope of the experimental signals. The white lines in (b) represent the breaking curves computed from Eq. (8).

Figure 5

Discrete IST spectra of the three envelopes measured at $z_1=6\mathrm{m}$ (Left column) and $z_{20}=120\mathrm{m}$ (Right column). (a), (b) $\mathrm{\Delta }{T}_1=30\mathrm{s}$, (c), (d), $\mathrm{\Delta }{T}_2=45\mathrm{s}$, (e), (f) $\mathrm{\Delta }{T}_3=60\mathrm{s}$. Only the upper half part of the complex plane is represented.

Figure 6

Space-time evolution of the envelopes of three rectangular wave packets ($\mathrm{\Delta }{T}_1=30\mathrm{s}$, $\mathrm{\Delta }{T}_2=45\mathrm{s}$, $\mathrm{\Delta }{T}_3=60\mathrm{s}$) in the regime where higher-order nonlinear effects have a perturbative influence. The carrier frequency $f_0=1/T_0$ is 1.28 Hz. Other experimental parameters are $k_0=6.58{\mathrm{m}}^{-1}$, $a=0.014\mathrm{m}$ (the wave steepness is $k_{0}a=0.09$). White lines represent breaking curves that are determined using the methodology described in Sec. 4a.

Figure 7

(a) Envelope of the central wave packet of Fig. 6 measured at $z_1=6\mathrm{m}$, close to the wave maker. (b) Discrete IST spectrum of the wave field plotted in (a). (c) Envelope of the central wave packet of Fig. 6 measured at $z_{20}=120\mathrm{m}$, far from the wave maker. (d) Discrete IST spectrum of the wave field plotted in (c). (e) Local IST spectrum of the coherent structure highlighted in red in (c). (f) Local IST spectrum of the coherent structure highlighted in magenta in (c).