Fourth order envelope equation for nonlinear dispersive gravity waves

Abstract

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Higher-order modulation effects on solitary wave envelopes in deep water

“…This initial disturbance eventually transforms to a solitary wave of the Dysthe equation, having lower peak amplitude and moving with higher speed than that of the original wave, in agreement with the numerical results of Lo & Mei (1985); the increase in wave speed is caused by a downshift in wave frequency, proportional to the square of the wave amplitude, and turns out to be independent of the wave-induced mean flow, to leading order in wave steepness. These asymptotic results are consistent with the experiments of Feir (1967) and Su (1982) and support the explanation of wave-group separation, proposed by Roskes (1977) and Lo & Mei (1985).…”

“…It is noteworthy that, although, as it turns out, Roskes (1977) neglected the interaction of the wave envelope with the induced mean flow, his numerical calculations were capable of reproducing, a t least qualitatively, the group splitting observed by Feir (1967). Later, Dysthe (1979) carried out a formal derivation of an envelope equation which extends the range of validity of the NLS to longer time, comparable with O(E-~) wave periods; in addition to the higher-order terms proposed by Roskes (1977), Dysthe's equation also includes the effect of the wave-induced mean flow. Lo & Mei (1985) conducted a detailed numerical study of the long-time evolution of short wavepackets using the full Dysthe equation, and found good quantitative agreement with the experiments of Su (1982).…”

“…A first attempt to predict the observations of Feir (1967) theoretically was made by Roskes (1977), using the NLS, modified with some additional terms representing higher-order modulation effects. It is noteworthy that, although, as it turns out, Roskes (1977) neglected the interaction of the wave envelope with the induced mean flow, his numerical calculations were capable of reproducing, a t least qualitatively, the group splitting observed by Feir (1967).…”

“…A first attempt to predict the observations of Feir (1967) theoretically was made by Roskes (1977), using the NLS, modified with some additional terms representing higher-order modulation effects. It is noteworthy that, although, as it turns out, Roskes (1977) neglected the interaction of the wave envelope with the induced mean flow, his numerical calculations were capable of reproducing, a t least qualitatively, the group splitting observed by Feir (1967). Later, Dysthe (1979) carried out a formal derivation of an envelope equation which extends the range of validity of the NLS to longer time, comparable with O(E-~) wave periods; in addition to the higher-order terms proposed by Roskes (1977), Dysthe's equation also includes the effect of the wave-induced mean flow.…”

On a fourth-order envelope equation for deep-water waves

“…For a symmetrical spectrum these terms give rise to an asymmetrical energy transfer. It is perhaps of interest to note that Roskes (1977) (although in his equation the Hilbert-transform term is missing) found these fourth-order terms to have a symmetry-breaking effect on the evolution of a single soliton, i.e. the soliton splits up into a large pulse followed by a small one.…”