The results of laboratory experiments and numerical simulations were performed to investigate the interactions between the weakly three-dimensional waves in an ‘X’ configuration, which has a 16-degree approaching angle. In addition, another oblique two-dimensional experiment was also conducted for comparison with the ‘X’ configuration but in one single channel by removing a dummy wall in the interaction region. Our experimental results show that as the wave trains propagate into the interaction region, it is obvious that there is an increase in the wave height which reaches a maximum height of about 1.37H0 for different initial wave steepness at the center of the interaction region, and then decreases thereafter, where H0 is the input wave height. Then wave elevations at different positions downstream of the interaction region were also studied, indicating that the frequency and initial wave steepness were highly correlated with the wave-wave interaction between the weakly three-dimensional waves. For the wave with low frequency (f = 0.8 Hz), a crescent wave surface formed at the beginning of the interaction and then separated into two two-dimensional waves after the interaction, which illustrates that the waves can still keep their initial characteristic and propagate as their initial directions downstream of the interaction region. While the frequency increased (f = 1.2 Hz), three-dimensional effects appeared to dominate the interaction of weakly three-dimensional waves, especially for the large initial steepness, and the wave surfaces were also three-dimensional after interactions. Finally, numerical simulations with larger approaching angles were conducted to further understand the influence of propagation direction on the interactions between the weakly three-dimensional waves. The results suggest that intense interactions and strong three-dimensional characteristics of the wave trains downstream interactions can result from larger approaching angles.
In this work we consider the problem of finding the simplest arrangement of resonant deep-water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wavevectors K1+K2=K3 (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wavepackets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction in a symmetric configuration. Numerical simulations of the governing equations in natural variables using pseudospectral methods require the inclusion of up to 6-wave interactions, which imposes a strong dealiasing cut-off in order to properly resolve the evolving waves. We study the resonance numerically by looking at a target mode in the base triad and showing that the energy transfer to this mode is more efficient when the system is close to satisfying the resonant conditions. We first look at encountering plane waves with base frequencies in the range 1.32–2.35 Hz and steepnesses below 0.1, and show that the time evolution of the target mode’s energy is dramatically changed at the resonance. We then look at a scenario that is closer to experiments: Encountering wavepackets in a 400-m long numerical tank, where the interaction time is reduced with respect to the plane-wave case but the resonance is still observed; by mimicking a probe measurement of surface elevation we obtain efficiencies of up to 10% in frequency space after including near-resonant contributions. Finally, we perform preliminary experiments of encountering wavepackets in a 35-m long tank, which seem to show that the resonance exists physically. The measured efficiencies via probe measurements of surface elevation are relatively small, indicating that a finer search is needed along with longer wave flumes with much larger amplitudes and lower frequency waves. A further analysis of phases generated from probe data via the analytic signal approach (using the Hilbert transform) shows a strong triad phase synchronisation at the resonance, thus providing independent experimental evidence of the resonance.
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