Cnoidal Waves from Korteweg-de Vries Equation

​
η
2
3.
H
2.3
m
0.9
x
0.26
c
1.
λ
0.46
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)=
η
2
+H
2
cn(2/λK(m)(x-ct)m)
,
where
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.

References

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

External Links

Wavelength (ScienceWorld)
Surface Wave (ScienceWorld)
Korteweg-de Vries Equation (Wolfram MathWorld)
Jacobi Elliptic Functions (Wolfram MathWorld)
Complete Elliptic Integral of the First Kind (Wolfram MathWorld)

Permanent Citation

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