Nonlinear resonance mechanisms and cross-order harmonic coupling of deep-water waves excited by a moving pressure field

Within the framework of the homotopy analysis method, the method of directly defining inverse mapping is applied to investigate the strongly nonlinear resonance and cross-order harmonic coupling of deep-water waves excited by a moving pressure field. By directly defining new inverse operators, this analytical framework offers a practical way to handle the small-divisor difficulty encountered in traditional perturbation theories. It does not require the weak-nonlinearity assumption or the scale-separation procedure used in methods such as multiple scales. In the primary resonance region, the nonlinear dispersion of deep-water waves causes the forced-response branches to bend rightward and develop fold bifurcations. An empirical relation is obtained for the moderate-to-strong nonlinear regime to characterize cross-order coupling in the discrete spectrum. It suggests that the higher-order harmonics of the deviation velocity potential are strongly influenced by adjacent lower-order background-wave components, which act as an important source of parametric excitation through nonlinear steepening. This study elucidates the regularities of discrete spectra and the cross-order evolution mechanisms of deep-water forced waves under resonance modes.