A New Coastal Wave Model. Part III: Nonlinear Wave–Wave Interaction

Abstract

Resonant wave?wave interaction processes are studied with the nonlinear dispersion relationship for shallow water. The formulation was derived based on a Hamiltonian representation first reported by Zakharov. Results show that four waves are needed for resonant interactions at all depths. Furthermore, when the nonlinear dispersion relationship for waves in intermediate water is considered, two interaction modes can result depending on the water depth and the nonlinearity: (i) in deep water the classic Phillips interactions dominate, involving four waves of comparable wavelengths, whereas (ii) in shallow water the dominant interactions still consist of four waves, but with one component of vanishingly small wavenumber. As an approximate asymptotic limit, the latter become triadic shoaling wave interactions.