This website requires JavaScript.
DOI: 10.1017/jfm.2023.205

On coupled envelope evolution equations in the Hamiltonian theory of nonlinear surface gravity waves

Yan Li
Mar 2023
This paper presents a novel theoretical framework in the Hamiltonian theoryof nonlinear surface gravity waves. The envelope of surface elevation and thevelocity potential on the free water surface are introduced in the framework,which are shown to be a new pair of canonical variables. Using the twoenvelopes as the main unknowns, coupled envelope evolution equations (CEEEs)are derived based on a perturbation expansion. Similar to the High OrderSpectral method, the CEEEs can be derived up to arbitrary order in wavesteepness. In contrast, they have a temporal scale as slow as the rate ofchange of a wave spectrum and allow for the wave fields prescribed on acomputational (spatial) domain with a much larger size and with spacing longerthan the characteristic wavelength at no expense of accuracy and numericalefficiency. The energy balance equation is derived based on the CEEEs. Thenonlinear terms in the CEEEs are in a form of the separation of wave harmonics,due to which an individual term is shown to have clear physical meanings interms of whether or not it is able to force free waves which obey thedispersion relation. Both the nonlinear terms that can only lead to the forcingof bound waves and these which are capable of forcing free waves aredemonstrated, with the latter through the analysis of the quartet and quintetresonant interactions of linear waves. The relations between the CEEEs and twoother existing theoretical frameworks are established, including the theory fora train of Stokes waves up to second order in wave steepness [Fenton, J.waterway, Port, Coast. & Ocean Eng., 111, 2, 1985] and a semi-analyticalframework for three-dimensional weakly nonlinear surface waves with arbitrarybandwidth and large directional spreading by Li & Li [Phys. Fluids, 33, 7,2021].