General solution for strong wave-wave interaction in deep water

A general solution for the interaction among three distinctive free-wave components up to third order in wave steepnesses is derived using both conventional mode-coupling (Stokes) and Zakharov equation methods. It is proved that the solutions derived using these two methods are identical. Since the solution involves numerous terms, the identical match of the solution provides an excellent check to our derivation. Because the linear phases are used in both mode-coupling and Zakharov equation methods, their solutions encounter convergence difficulties in certain wave parametric ranges, such as the interactions among two long waves and one short wave or one long wave and two short waves in addition to the convergence difficulty due to disparate wavelength scales of the short wave and long wave. In these parametric ranges, the solution is derived using a phase-modulation method and the convergence difficulty is resolved. It is found that the two solutions derived respectively by mode-coupling method and phase-modulation method are identical when both solutions are convergent in a parametric range. Based on these findings, a third-order hybrid wave model can be established for studying irregular ocean waves, which may have important implication to variety ocean and coastal engineering applications.