Four-wave resonant interaction of surface gravity waves in finite water depth

In this study, we investigated the four-wave resonant and quasiresonant interactions in a special degenerated case, wherein bichromatic mother waves are generated to give birth to a daughter wave. One of the mother waves was counted twice to satisfy the four-wave resonant conditions. Particular attention is paid to the effect of finite water depth. Theo-retical analyses based on the Zakharov equation and direct numerical simulations using a higher-order spectral (HOS) method were performed and compared. The present results revealed that both resonant and quasiresonant four-wave interactions were suppressed by the finite depth and eventually attenuated to zero for sufficiently shallow water. It is found that the corresponding critical depth depends on the crossing angle of the initial mother waves. For the two mother waves with a crossing angle theta = 25 degrees, four-wave resonance survives up to k1h similar to 0.57, where k1 denotes the wave number of the twice-counted mother wave and h is the water depth. Furthermore, it is found that the four-wave resonant interactions for different values of theta survive up to a global threshold value of k1h similar to 0.4. In addition, through three-dimensional (3D) Fourier analyses of the results by direct numerical simulations, it is found that the bound wave effects are enhanced, and more harmonics are generated as the water depth decreases.