Stability of weakly nonlinear deep-water waves in two and three dimensions

Abstract

The stability of a weakly nonlinear wave train on deep water to two- and three-dimensional modulations is investigated using an improved approximation due to Zakharov (1968). The results are expressible in simple analytical forms, and show good quantitative agreement with available experimental data and exact numerical calculations over a broad range of wave steepness in the unidirectional case.

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

“…The agreement of his theoretical results with experiments of Benjamin & Feir (1967) and Lake et al (1977) is even better. Similar results were obtained by Crawford et al (1981) from the so-called Zakharov equations. On the other hand, Dysthe (1979) gave an improved version of the nonlinear Schrodinger equation by including higher-order effects.…”

“…It is remarkable that even for A 0 = 0·1 the effect of the wave-induced current is considerable. For comparison, we have also shown in figure 2 results of Crawford et al (1981). As a starting point for the stability analysis of a uniform wavetrain, these authors used the Zakharov equation for weakly nonlinear water waves, an equation which retains all the higher-order dispersion effects, but which is correct to third order in amplitude only.…”

“…The procedure to obtain the evolution ofthe action density is well-known (cf. Hasselmann 1962;Davidson 1969Davidson , 1972Longuet-Higgins 1976;Crawford et al 1981), so we omit the details.…”

Interction of a short-wave field with a dominant long wave in deep water: derivation form Zakharov's spectral formulation

“…This formulation was later adopted and expounded in greater detail by Crawford, Saffman & Yuen [2], Crawford, Lake, Saffman & Yuen [3] and Yuen & Lake [6]; see also Craik [1] for a brief account. Not surprisingly, solutions of Zakharov's equations are available only for special cases, but these known solutions often improve upon previous approximations.…”

On Benjamin-Feir instability and evolution of a nonlinear wave with finite-amplitude sidebands

“…The linear instability analysis based on the CSE yields p = 2 1/2 p m = 2 3/2 ε, where the nonlinearity ε = k 0 a 0 represents the carrier wave steepness. The Janssen's (2003) comment regarding the limits of applicability of BFI is based, in part, on the fact that the linear relation between p (or p m ) and ε breaks down when the stability analysis is performed using either the linearized Zakharov equation (Crawford et al, 1981;Stiassnie and Shemer, 1984), or full nonlinear equations (McLean et al, 1981).…”