Nonlinear excitation of long-trapped waves by a group of short swells

A nonlinear theory is presented for the resonance of long gravity waves trapped on an uneven bottom when a long packet of short swells is incident. By allowing the trapped wave to be comparable in amplitude to the incident swells, the transient evolution of trapped waves is studied from initial growth through maturity to final decay, for swell packets of finite duration. For a totally submerged ridge, it is found that, while the trapped waves are resonated by second-order periodic modulations of the swell envelope, energy transfer to short swells and nonlinear emission of long waves act as damping mechanisms to limit the amplification. Bottom friction is not qualitatively crucial and affects the results only quantitatively. For a closed beach, breaking of short swells is dealt with empirically but resonant modulation is still the primary factor in exciting surf beats.