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On the Modulation of Water Waves in the Neighbourhood of $kh$ $\approx $ 1.363

R. S. Johnson


In 1967, T. Brooke Benjamin showed that periodic wave-trains on the surface of water could be unstable. If the undisturbed depth is $h$, and $k$ is the wavenumber of the fundamental, then the Stokes wave is unstable if $kh$ $\geq $ $\sigma _{0}$, where $\sigma $$_{0}$ $\approx $ 1.363. The instability is provided by the growth of waves with a wavenumber close to $k$. This result is associated with an almost resonant quartet wave interaction and can be obtained by examining the cubic nonlinearity in the nonlinear Schrodinger equation for the modulation of harmonic water waves: this term vanishes at $kh$ = $\sigma $$_{0}$. In this paper the multiple-scales technique is adapted in order to derive the appropriate modulation equation for the amplitude of the fundamental when $kh$ is near to $\sigma $$_{0}$. The resulting equation takes the form i$A_{T}$ - $a_{1}A_{\zeta \zeta}$ - $a_{2}A|A|^{2}$ + $a_{3}A|A|^{4}$ + i($a_{4}|A|^{2}A_{\zeta}$ - $a_{5}A(|A|^{2})_{\zeta})$ - $a_{6}A\psi _{T}$ = 0, where $\psi _{\zeta}$ = $|A|^{2}$, and the $a$$_{i}$ are real numbers. [Coefficients $a_{3}$-$a_{6}$ are given on $kh$ $\approx $ 1.363 only.] This equation is uniformly valid in that it reduces to the classical non-linear Schrodinger equation in the appropriate limit and is correct when $a$$_{2}$ = 0, i.e. at $kh$ = $\sigma $$_{0}$. The equation is used to examine the stability of the Stokes wave and the new inequality for stability is derived: this now depends on the wave amplitude. If the wave is unstable then it is expected that solitons will be produced: the simplest form of soliton is therefore examined by constructing the corresponding ordinary differential equation. Some comments are made concerning the phase-plane of this equation, but more analytical details are extracted by treating the new terms as perturbations of the classical Schrodinger soliton. It is shown that the soliton is both flatter (symmetrically) and skewed forward, although the skewing eventually gives way to an oscillation above the mean level.

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