On the steady-state exactly resonant, nearly resonant, and non-resonant waves and their relationships

Abstract

The steady-state exactly resonant, nearly resonant, and non-resonant waves in infinite water depth are investigated, and their relationships are revealed. In the framework of homotopy analysis method (HAM), the two primary wave components' amplitudes in the initial guess of the velocity potential are fixed and the actual frequencies of the primary waves are unknown. For different wavenumber ratio (k2/k1) values, three groups of steady-state wave systems are obtained with the proper auxiliary linear operator and the initial guess. It is found that when the third-order resonance occurs accurately, the energy of each wave group is mainly concentrated in the primary and third-order resonant wave components. When the value of the wavenumber ratio (k2/k1) moves away from the exact resonance, the energy of the whole wave system is either gradually transferred to the two primary or one resonant wave components that finally evolves into the trivial non-resonant wave system, or the energy is more evenly distributed among more wave components that evolves into multiple nearly resonant wave systems. In addition, the results obtained based on HAM are verified and confirmed by means of the Zakharov equation. This work illustrate that the steady-state wave systems are continuous in wavevector space, the normal non-resonant solution on either side of the resonance point comes from the different third-order resonant solutions, and the occurrence of multiple near resonances can significantly increase the nonlinearity of the wave system.