1983
DOI: 10.1017/s0022112083000014
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On a fourth-order envelope equation for deep-water waves

Abstract: The ordinary nonlinear Schrodinger equation for deep-water waves (found by a perturbation analysis to 0(€ 3 ) in the wave steepness €) compares unfavourably with the exact calculations of Longuet-Higgins (1978) for € > 0·10. Dysthe (1979) showed that a significant improvement is found by taking the perturbation analysis one step further to 0(€ 4 ). One of the dominant new effects is the wave-induced mean flow. We elaborate the Dysthe approach by investigating the effect of the wave-induced flow on the long-t… Show more

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Cited by 65 publications
(51 citation statements)
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“…Both the maximum growth rate of the instability and wavenumber at marginal stability have been shown graphically for some different values of dimensionless thermocline depth. [14] Fourth-order nonlinear evolution equations for surface gravity waves …”
Section: Resultsmentioning
confidence: 99%
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“…Both the maximum growth rate of the instability and wavenumber at marginal stability have been shown graphically for some different values of dimensionless thermocline depth. [14] Fourth-order nonlinear evolution equations for surface gravity waves …”
Section: Resultsmentioning
confidence: 99%
“…Substituting this expression for dE/d% in (27) we get the following single nonlinear evolution equation, which is same as the equation (2) of Janssen [14] and (2.20) of Hogan [13] with K = 0, that is, in the absence of capillarity.…”
Section: -J-=4h-g-(ss*) (31)mentioning
confidence: 99%
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“…Janssen (1983) showed that the approximation to the Hasselmann equation given by Dungey & Hui (1979) for narrow-band spectra can be derived from the MNLS equations using the same statistical assumptions as Hasselmann. It is perhaps worth noting that the zero-cumulant hypotheses implied by the approximate multivariate complex Gaussian property of the Fourier coefficients, and necessary for the derivation of Hasselmann's equation, may not be realistic in the present case.…”
Section: Discussionmentioning
confidence: 99%
“…The dominant new effect that comes in the fourth order is the influence of wave induced mean flow and this produces a significant deviation in the stability character. Fourth order nonlinear evolution equation for deep water surface gravity waves including different effects were derived and stability analysis was considered by several authors (Stiassnie, 1984;Hogan, 1985;Dhar and Das, 1990;1991;1994;Janssen, 1983).…”
Section: Introductionmentioning
confidence: 99%