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Natural Hazards and Earth System Sciences An interactive open-access journal of the European Geosciences Union
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Volume 10, issue 11
Nat. Hazards Earth Syst. Sci., 10, 2421–2427, 2010
https://doi.org/10.5194/nhess-10-2421-2010
© Author(s) 2010. This work is distributed under
the Creative Commons Attribution 3.0 License.

Special issue: Extreme and rogue waves

Nat. Hazards Earth Syst. Sci., 10, 2421–2427, 2010
https://doi.org/10.5194/nhess-10-2421-2010
© Author(s) 2010. This work is distributed under
the Creative Commons Attribution 3.0 License.

Research article 30 Nov 2010

Research article | 30 Nov 2010

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

L. Shemer L. Shemer
  • School of Mechanical Engineering, Tel-Aviv University, Tel-Aviv, Israel

Abstract. In the past decade it became customary to relate the probability of appearance of extremely steep (the so-called freak, or rogue waves) to the value of the Benjamin-Feir Index (BFI) that represents the ratio of wave nonlinearity to the spectral width. This ratio appears naturally in the cubic Schrödinger equation that describes evolution of unidirectional narrow-banded wave field. The notion of this index stems from the Benjamin-Feir linear stability analysis of Stokes wave. The application of BFI to evaluate the evolution of wave fields, with non-vanishing amplitudes of sideband disturbances, is investigated using the Zakharov equation as the theoretical model. The present analysis considers a 3-wave system for which the exact analytical solution of the model equations is available.

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