Long Waves due to Interactions beneath Wave Groups

Abstract

Long waves coupled to and forced by groups of primary waves in water of uniform, but arbitrary depth are investigated. Stream function wave theory is used to represent the long waves, which are driven by a virtual pressure and a virtual vertical velocity associated with the groups of first-order incoming waves. Results are presented for the case of a wave group formed by two primary waves with equal amplitudes and slightly different frequencies. This method provides more realistic results than those of Longuet-Higgins and Stewart, especially in shallow water, where their approach predicts long-wave heights that are too large. The interpretation for the smaller wave heights obtained herein, especially in shallow water, is that inclusion of the nonlinearities in the long-wave representation causes its celerity to differ from that of the wave groups, thereby “detuning” the system.