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cnoidal wave
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{{short description|A nonlinear and exact periodic wave solution of the Kortewegâ€“de Vries equation}}File:Periodic waves in shallow water.jpg|thumb|right|300px|US Army bombers flying over near-periodic swell in shallow water, close to the PanamaPanama In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Kortewegâ€“de Vries equation. These solutions are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.The cnoidal wave solutions were derived by Korteweg and de Vries, in their 1895 paper in which they also propose their dispersive long-wave equation, now known as the Kortewegâ€“de Vries equation. In the limit of infinite wavelength, the cnoidal wave becomes a solitary wave.The Benjaminâ€“Bonaâ€“Mahony equation has improved short-wavelength behaviour, as compared to the Kortewegâ€“de Vries equation, and is another uni-directional wave equation with cnoidal wave solutions. Further, since the Kortewegâ€“de Vries equation is an approximation to the Boussinesq equations for the case of one-way wave propagation, cnoidal waves are approximate solutions to the Boussinesq equations.Cnoidal wave solutions can appear in other applications than surface gravity waves as well, for instance to describe ion acoustic waves in plasma physics.{{citation |title=Physics of intense beams in plasmas |first1=M.V. |last1=Nezlin |publisher=CRC Press |year=1993 |isbn=978-0-7503-0186-2 |page=205 }}File:Cnoidal wave m=0.9.svg|thumb|right|600px|A cnoidal wave, characterised by sharper crests and flatter troughs than in a sinesineFile:Ile de rÃ©.JPG|thumb|right|300px|Crossing swells, consisting of near-cnoidal wave trains. Photo taken from Phares des Baleines (Whale Lighthouse) at the western point of ÃŽle de RÃ© (Isle of RhÃ©), France, in the Atlantic OceanAtlantic Ocean

## Background

### Kortewegâ€“de Vries, and Benjaminâ€“Bonaâ€“Mahony equations

File:Water wave theories.svg|thumb|right|300px|Validity of several theories for periodic water waves, according to Le MÃ©hautÃ© (1976).{{citation |first=B. |last=Le MÃ©hautÃ© |year=1976 |title=An introduction to hydrodynamics and water waves |publisher=Springer |isbn=978-0-387-07232-6}} The light-blue area gives the range of validity of cnoidal wave theory; light-yellow for Airy wave theory; and the dashed blue lines demarcate between the required order in Stokes' wave theory. The light-gray shading gives the range extension by numerical approximations using fifth-order stream-function theory, for high waves (H > Â¼ Hbreaking).]]The Kortewegâ€“de Vries equation (KdV equation) can be used to describe the uni-directional propagation of weakly nonlinear and long wavesâ€”where long wave means: having long wavelengths as compared with the mean water depthâ€”of surface gravity waves on a fluid layer. The KdV equation is a dispersive wave equation, including both frequency dispersion and amplitude dispersion effects. In its classical use, the KdV equation is applicable for wavelengths Î» in excess of about five times the average water depth h, so for Î» > 5 h; and for the period Ï„ greater than scriptstyle 7 sqrt{h/g} with g the strength of the gravitational acceleration.Dingemans (1997) pp. 718â€“721. To envisage the position of the KdV equation within the scope of classical wave approximations, it distinguishes itself in the following ways:
• Kortewegâ€“de Vries equation â€” describes the forward propagation of weakly nonlinear and dispersive waves, for long waves with Î» > 7 h.
• Shallow water equations â€” are also nonlinear and do have amplitude dispersion, but no frequency dispersion; they are valid for very long waves, Î» > 20 h.
• Boussinesq equations â€” have the same range of validity as the KdV equation (in their classical form), but allow for wave propagation in arbitrary directions, so not only forward-propagating waves. The drawback is that the Boussinesq equations are often more difficult to solve than the KdV equation; and in many applications wave reflections are small and may be neglected.
• Airy wave theory â€” has full frequency dispersion, so valid for arbitrary depth and wavelength, but is a linear theory without amplitude dispersion, limited to low-amplitude waves.
• Stokes' wave theory â€” a perturbation-series approach to the description of weakly nonlinear and dispersive waves, especially successful in deeper water for relative short wavelengths, as compared to the water depth. However, for long waves the Boussinesq approachâ€”as also applied in the KdV equationâ€”is often preferred. This is because in shallow water the Stokes' perturbation series needs many terms before convergence towards the solution, due to the peaked crests and long flat troughs of the nonlinear waves. While the KdV or Boussinesq models give good approximations for these long nonlinear waves.

The KdV equation can be derived from the Boussinesq equations, but additional assumptions are needed to be able to split off the forward wave propagation. For practical applications, the Benjaminâ€“Bonaâ€“Mahony equation (BBM equation) is preferable over the KdV equation, a forward-propagating model similar to KdV but with much better frequency-dispersion behaviour at shorter wavelengths. Further improvements in short-wave performance can be obtained by starting to derive a one-way wave equation from a modern improved Boussinesq model, valid for even shorter wavelengths.Dingemans (1997) pp. 689â€“691.

### Cnoidal waves

(File:Cnoidal wave profiles.svg|thumb|right|400px|Cnoidal wave profiles for three values of the elliptic parameter m. {|
| : m = 0,
| : m = 0.9 and
| : m = 0.99999.
)The cnoidal wave solutions of the KdV equation were presented by Korteweg and de Vries in their 1895 paper, which article is based on the PhD thesis by de Vries in 1894.ARXIV, de Jager, E.M., 2006, math/0602661v1, On the origin of the Kortewegâ€“de Vries equation, Solitary wave solutions for nonlinear and dispersive long waves had been found earlier by Boussinesq in 1872, and Rayleigh in 1876. The search for these solutions was triggered by the observations of this solitary wave (or "wave of translation") by Russell, both in nature and in laboratory experiments. Cnoidal wave solutions of the KdV equation are stable with respect to small perturbations.{{citation |journal=Quarterly Journal of Mechanics and Applied Mathematics |year=1977 |volume=30 |issue=1 |pages=91â€“105 |doi=10.1093/qjmam/30.1.91 |title=On the stability of cnoidal waves |first=P.G. |last=Drazin |authorlink=Philip Drazin}}The surface elevation Î·(x,t), as a function of horizontal position x and time t, for a cnoidal wave is given by:
eta(x,t) = eta_2 + H, operatorname{cn}^2, left( begin{array}{c|c} displaystyle 2, K(m), frac{x-c,t}{lambda} & m end{array} right),
where H is the wave height, Î» is the wavelength, c is the phase speed and Î·2 is the trough elevation. Further cn is one of the Jacobi elliptic functions and K(m) is the complete elliptic integral of the first kind; both are dependent on the elliptic parameter m. The latter, m, determines the shape of the cnoidal wave. For m equal to zero the cnoidal wave becomes a cosine function, while for values close to one the cnoidal wave gets peaked crests and (very) flat troughs. For values of m less than 0.95, the cnoidal function can be approximated with trigonometric functions.{{citation |title=Calculation and approximation of the cnoidal function in cnoidal wave theory |author1=Yunfeng Xu |author2=Xiaohe Xia |author3=Jianhua Wang |year=2012 |journal=Computers & Fluids |volume=68 |pages=244â€“247 |doi=10.1016/j.compfluid.2012.07.012}}An important dimensionless parameter for nonlinear long waves (Î» â‰« h) is the Ursell parameter:
U = frac{H, lambda^2}{h^3} = frac{H}{h}, left( frac{lambda}{h} right)^2.
For small values of U, say U

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