Precession resonance in water waves

Abstract

We describe the theory and present numerical evidence for a new type of nonlinear resonant interaction between gravity waves on the surface of deep water. The resonance constitutes a generalisation of the usual 'exact' resonance as we show that exchanges of energy between the waves can be enhanced when the interaction is three-wave rather than four and the linear frequency mismatch, or detuning, is non-zero i.e. $\omega_1\pm\omega_2\pm\omega_3 \neq0.$ This is possible because the resonance condition is now a match between the so-called 'precession frequency' of a given $\textit{triad interaction}$ and an existent nonlinear frequency in the system. In the limit of weak nonlinearity this precession frequency is simply due to the linear 'drift' of the triad phase; therefore, it tends toward the detuning. This means precession resonance of this type can occur at finite amplitudes, with nonlinear corrections contributing to the resonance. We report energy transfer efficiencies of up to 40%, depending on the model options. To the authors' knowledge this represents the first new type of nonlinear resonance in surface gravity waves since the seminal work of Benjamin & Feir (1967).