Cnoidal Waves from Korteweg-de Vries Equation

Manipulate[Plot[H ((JacobiCN[2EllipticK[m]((x-c t)/λ),m])^2),{t,0,1},Filling->Bottom,FillingStyle->RGBColor[0.271473, 0.440467, 0.8],Background->LightBlue,ImageSize->{500,320},PlotRange->{Automatic,{0,Automatic}},PlotStyle->{Orange,Thick}],
{{H,2.3,Style["H",Italic]},.1,10.,Appearance->"Labeled"},{{m,.9,Style["m",Italic]},.0,.9999,Appearance->"Labeled"},{{x,.26,Style["x",Italic]},.1,5.,Appearance->"Labeled"},{{c,1.,Style["c",Italic]},.1,2.,Appearance->"Labeled"},{{λ,.46},.1,2.25,Appearance->"Labeled"}]

CAPTION

A cnoidal wave is an exact periodic traveling-wave solution of the Korteweg–de Vries (KdV) equation, first derived by them in 1895. Such a wave describes surface waves whose wavelength is large compared to the water depth.

DETAILS

The surface elevation of a cnoidal wave takes the form

η(x,t)=(Subscript[η, 2])+H ((JacobiCN[2/λ (EllipticK[m]) (x-c t), m])^2),

where (Subscript[η, 2]) is the elevation, H is the wave height, c is the phase velocity (the rate at which the phase of the wave propagates), λ is the wavelength, and cn is the Jacobi elliptic function (hence the name cnoidal). K(m) is the complete elliptic integral of the first kind, with m being the elliptic parameter that for large values produces smoother troughs and more pronounced crests than in the case of a sine wave.

Reference

[1] Wikipedia. "Cnoidal Wave." (May 16, 2013) en.wikipedia.org/wiki/Cnoidal_wave.

THIS NOTEBOOK IS THE SOURCE CODE FROM

Enrique Zeleny

"Cnoidal Waves from Korteweg-de Vries Equation"
 http://demonstrations.wolfram.com/CnoidalWavesFromKortewegDeVriesEquation/
 Wolfram Demonstrations Project
 Published: May 28, 2013

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