On weakly nonlinear modulation of waves on deep water

Abstract

We propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrödinger ͑NLS͒ equation for deep-water waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear four-wave resonance ͑the ''figure 8'' of Phillips͒ even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to high-wave-number modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.

Freak Waves in Random Oceanic Sea States

“…As pointed out in a number of papers [9,10] the main defect in the NLS equation concerning the narrow-band approximation, arises from the fact that linear dispersion is not at high enough order. Reference [10] proposes an equation that includes all the terms in the linear dispersion relation. The equation (eq.…”

“…In our numerical simulations with the Pierson Moskowitz spectrum (γ = 1, α = 0.0081 ) we have used both the NLS equation and its modified form (eq. (1) in [10]). In this specific case the two equations give basically the same results: nonlinearities are weak (Ursell number=0.03, see Fig.…”

Stability of plane waves on deep water with dissipation

“…• Similar computations can be applied to the generalization of NLS derived by Trulsen et al (2000). For this problem, the bound in (25) limits the growth of any perturbation.…”

“…In each plot, the regions between the curves correspond to Λ's that have positive real part. The t = 0 (zero dissipation) case of these (or equivalent) plots were obtained by Dysthe (1979), Trulsen & Dysthe (1996) and Trulsen et al (2000).…”

Периодические двухфазные “волны-убийцы”

“…параллелограммом периодов является прямоугольник. Из фор-мул (14), (16) вытекает, что каждой точке кривых Γ ± 2 соответствуют две точки кри-вой Γ 2 , а на рисунках 4-7 видно, что на каждом прямоугольнике периодов находятся два максимума амплитуды решения. Следовательно, чтобы получать двухфазные "волны-убийцы" с большим числом пиков на параллелограмме периодов решения, необходимо рассматривать накрытия с большим числом прообразов.…”