Numerical simulations of collinear finite amplitude steady-state resonant waves in deep water

Abstract

In this work, numerical simulations of steady-state resonant wave system have been conducted by high-order spectral method (HOS) to study its evolution mechanism in deep water. Convergent high-order series solutions of steady-state resonant waves are first obtained by homotopy analysis method (HAM). Theoretical solutions together with disturbances of different orders of magnitude are then served as the initial solutions in HOS. It is found that as more accurate wave components are generated at the initial stage of the numerical simulation, the time that amplitude of the two largest wave components keeps unchanged increases. Steady-state resonant waves with time-independent spectra can be obtained if sufficient number of wave components are generated at the initial stage. At the end of simulation, additional new wave components that have not been considered at the initial stage appear in the spectra due to four-wave resonant interactions. For steady-state resonant waves with random disturbances, the numerical simulations confirm again the existence of steady-state resonant waves. Besides, energy transfer among different components is more remarkable for steady-state resonant waves with small disturbances before the wave breaking.