On the near resonances of collinear steady-state wave groups in finite water depth

Abstract

In this paper, the role nearly resonant components play in collinear steady-state wave groups in finite water depth is investigated theoretically. Fully nonlinear water wave equations are solved when the resonance conditions are nearly satisfied. Convergent solutions are obtained for steady-state wave groups with multiple near resonances. Spectra analysis shows that, as water depth decreases from deep water to finite water depth, the four-wave nearly resonant interactions play a dominant role in nonlinear steady-state wave groups. Meanwhile the dispersion relation changes with the water depth and some low and high frequency components appear in the spectrum due to the three-wave nearly resonant interactions. Comparison between the linear and nonlinear wave groups are made in finite water depth when the mean rates of mass, momentum and energy fluxes are almost conserved. It is found that wave energy in nonlinear wave groups is redistributed as more components join the resonance when the nonlinearity increases. Besides, both the wave group steepness and magnitude of water particle horizontal velocities near the crests and troughs in nonlinear wave groups increase more rapidly with the nonlinearity. The significance of near resonances in finite water depth for collinear steady-state wave groups is demonstrated.