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Rogue Waves and the Non-Linear S chrödinger Equation

Rogue Waves and the Non-Linear S chrödinger Equation. By Nick Armstrong and Brandan Mantei. Introduction. A rogue wave is generally defined as a wave that has a height that is double that of the significant wave height (SWH).

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Rogue Waves and the Non-Linear S chrödinger Equation

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  1. Rogue Waves and the Non-Linear Schrödinger Equation By Nick Armstrong and Brandan Mantei

  2. Introduction A rogue wave is generally defined as a wave that has a height that is double that of the significant wave height (SWH). The significant wave height is the average height of the largest third of the waves recorded in a given period of time.

  3. Graphical Example

  4. Background Until recently it was believed that rogue waves were extremely rare events that occurred only once every 10,000 years and only lasted for a few brief moments. This belief was based on the idea that oceanic waves followed the linear Schrödinger equation. Numerous first person acounts suggested otherwise.

  5. Draupner Wave The Draupner wave was measured by a wave sensor on one of the Draupner oil rig platforms in the North Sea 100 miles off of Norway. This wave was the first rogue wave to be recorded by a scientific instrument and therefore the first scientific proof of the existence of rogue waves. This wave and the later observation of several other rouge waves led to further research.

  6. A New Explanation Because of the number of rouge waves observed since the Draupner Wave, the linear Schrödinger equation has been dismissed as appropriate for describing rogue waves. The Non-linear Schrodinger Equation (NLS) has since become the most acceptable model.

  7. How The NLS has only analytical solutions but several have been found that simulate rogue waves that occur in varying conditions. Here is one that describes deep water waves: i( Ψt + CgΨ x) − μΨxx − νΨ|Ψ|2 = 0. Where: Cg = ω0/2k0 μ = ω0/8k20 ν = ω0k20/2 And Ψ is a slowly varying envelope (amplitude) function of x and t.

  8. How According to the NLS, a rogue waves starts as a normal sinusoidal wave, but at some point in time starts to rob energy from neighboring waves. The increase in energy leads to an increase in amplitude and ultimately a rogue wave develops.

  9. Modeling Rogue Waves http://www.youtube.com/watch?v=VQsk5l2DV5w 6:45-8:25

  10. Cite • http://www.cems.uvm.edu/~mreardon/Pictures_files/Math%20295C%20Presentation.pdf

  11. Currently As of today, data and models suggest that deep water rouge waves are in fact real and occur on a fairly consistent basis in deep open water. Even as often as once or twice a week on earth as opposed to the previously accepted 1 in 10,000 years. There is no current means of forecasting rogue waves.

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