Resonantly Forced Gravity--Capillary Lumps on Deep Water

A theoretical study is made of the wave disturbance generated by a locally confined external pressure on the surface of deep water moving with speed $V$ near the minimum gravity--capillary phase speed, $c_{min}$. According to linear inviscid theory, the response when $V$ coincides with $c_{min}$ is unbounded, and the interplay of nonlinear and damping effects is crucial close to this resonance. The analysis is based on an approximate model that combines the linear dispersion relation in the vicinity of $c_{min}$ with quadratic and cubic nonlinearity as well as viscous damping. For $V$ well below $c_{min}$, the transient response from rest approaches the small-amplitude steady state predicted by linear theory, but nonlinear effects come into play at a certain forcing speed, $c_{crit}