Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments

Dan Lucas, Keele University
Marc Perlin, Texas A&M University
Dian-Yong Liu, Dalian Maritime University
Shane Walsh, University College Dublin, Ireland
Rossen Ivanov, Technological University Dublin
Miguel D. Bustamante, University College Dublin

Document Type Article

Fluids, 2021,6, 205.


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, numericaland experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on5-wave resonances. The simplest arrangement is based on a triad of wavevectorsK1+K2=K3(satisfying specific ratios) along with their negatives, corresponding to a scenario of encounteringwavepackets, amenable to experiments and numerical simulations. The normal-form equations forthese encountering waves in resonance are shown to be non-integrable, but they admit an integrablereduction in a symmetric configuration. Numerical simulations of the governing equations in naturalvariables using pseudospectral methods require the inclusion of up to6-wave interactions, whichimposes a strong dealiasing cut-off in order to properly resolve the evolving waves. We study theresonance numerically by looking at a target mode in the base triad and showing that the energytransfer 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 andsteepnesses below0.1, and show that the time evolution of the target mode’s energy is dramaticallychanged at the resonance. We then look at a scenario that is closer to experiments: Encounteringwavepackets in a400-m long numerical tank, where the interaction time is reduced with respectto the plane-wave case but the resonance is still observed; by mimicking a probe measurement ofsurface elevation we obtain efficiencies of up to10%in frequency space after including near-resonantcontributions. Finally, we perform preliminary experiments of encountering wavepackets in a35-mlong tank, which seem to show that the resonance exists physically. The measured efficiencies viaprobe measurements of surface elevation are relatively small, indicating that a finer search is neededalong with longer wave flumes with much larger amplitudes and lower frequency waves. A furtheranalysis of phases generated from probe data via the analytic signal approach (using the Hilberttransform) shows a strong triad phase synchronisation at the resonance, thus providing independentexperimental evidence of the resonance.