Cnoidal wave

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.

[4] Cnoidal wave solutions of the KdV equation are stable with respect to small perturbations.

[13][14] A description of the interactions of cnoidal waves in shallow water, as found in real seas, has been provided by Osborne in 1994.

This condition is satisfied with the following representation of the elevation η(ξ):[7] in agreement with the periodic character of the sought wave solutions and with ψ(ξ) the phase of the trigonometric functions sin and cos. From this form, the following descriptions of various terms in equations (A) and (B) can be obtained: Using these in equations (A) and (B), the following ordinary differential equation relating ψ and ξ is obtained, after some manipulations:[7] with the right hand side still positive, since η1 − η3 ≥ η1 − η2.

The Jacobi elliptic functions cn and sn are inverses of F(ψ|m) given by With the use of equation (C), the resulting cnoidal-wave solution of the KdV equation is found[7] What remains, is to determine the parameters: η1, η2, Δ and m. First, since η1 is the crest elevation and η2 is the trough elevation, it is convenient to introduce the wave height, defined as H = η1 − η2.

Consequently, we find for m and for Δ: The cnoidal wave solution can be written as: Second, the trough is located at ψ = ⁠1/2⁠ π, so the distance between ξ = 0 and ξ = ⁠1/2⁠ λ is, with λ the wavelength, from equation (D): where K(m) is the complete elliptic integral of the first kind.

Third, since the wave oscillates around the mean water depth, the average value of η(ξ) has to be zero.

The following expressions for η1, η2 and η3 as a function of the elliptic parameter m and wave height H result:[7] Fourth, from equations (A) and (B) a relationship can be established between the phase speed c and the roots η1, η2 and η3:[7] The relative phase-speed changes are depicted in the figure below.

The nonlinear change in the phase speed, for fixed m, is proportional to the wave height H. Note that the phase speed c is related to the wavelength λ and period τ as: All quantities here will be given in their dimensional forms, as valid for surface gravity waves before non-dimensionalisation.

The cnoidal-wave solution of the KdV equation is:[7] with H the wave height—the difference between crest and trough elevation, η2 the trough elevation, m the elliptic parameter, c the phase speed and cn one of the Jacobi elliptic functions.

The wavelength λ, phase speed c and wave period τ are related to H, h and m by:[7] with g the Earth's gravity.

Then the above relations for λ, c and τ are used to find the elliptic parameter m. This requires numerical solution by some iterative method.

This results in a different formulation for Δ as found for the KdV equation: The relation of the wavelength λ, as a function of H and m, is affected by this change in

[22] For practical applications, usually the water depth h, wave height H, gravitational acceleration g, and either the wavelength λ, or—most often—the period (physics) τ are provided.

The following parameters of the wave are given: Instead of the period τ, in other cases the wavelength λ may occur as a quantity known beforehand.

In this case, starting from an initial guess minit = 0.99, by trial and error the answer is found.

They are related through: K′(m) = K(1−m)[27] Since the interest here is in small wave height, corresponding with small parameter m ≪ 1, it is convenient to consider the Maclaurin series for the relevant parameters, to start with the complete elliptic integrals K and E:[28][29] Then the hyperbolic-cosine terms, appearing in the Fourier series, can be expanded for small m ≪ 1 as follows:[26] The nome q has the following behaviour for small m:[30] Consequently, the amplitudes of the first terms in the Fourier series are: So, for m ≪ 1 the Jacobi elliptic function has the first Fourier series terms: And its square is The free surface η(x,t) of the cnoidal wave will be expressed in its Fourier series, for small values of the elliptic parameter m. First, note that the argument of the cn function is ξ/Δ, and that the wavelength λ = 2 Δ K(m), so: Further, the mean free-surface elevation is zero.

Therefore, the surface elevation of small amplitude waves is Also the wavelength λ can be expanded into a Maclaurin series of the elliptic parameter m, differently for the KdV and the BBM equation, but this is not necessary for the present purpose.

This can be done by using the equation for the wavelength, which is different for the KdV and BBM equation:[7][22] Introducing the relative wavenumber κh: and using the above equations for the phase speed and wavelength, the factor H / m in the phase speed can be replaced by κh and m. The resulting phase speeds are: The limiting behaviour for small m can be analysed through the use of the Maclaurin series for K(m) and E(m),[28] resulting in the following expression for the common factor in both formulas for c: so in the limit m → 0, the factor γ → −⁠1/6⁠.

These phase speeds are in full agreement with the result obtained by directly searching for sine-wave solutions of the linearised KdV and BBM equations.

On the other hand, the phase speed of the linearised KdV equation changes sign for short waves with κh >

In this steady flow, the discharge Q through each vertical cross section is a constant independent of ξ, and because of the horizontal bed also the horizontal momentum flux S, divided by the density ρ, through each vertical cross section is conserved.

They are defined as:[34] For fairly long waves, assuming the water depth ζ is small compared to the wavelength λ, the following relation is obtained between the water depth ζ(ξ) and the three invariants Q, R and S:[34] This nonlinear and first-order ordinary differential equation has cnoidal wave solutions.

This relation will be used to replace the bed velocity ub by Q and ζ in the momentum flux S. The following terms can be derived from it: Consequently, the momentum flux S becomes, again retaining only terms up to proportional to ζ3:[34] Which can directly be recast in the form of equation (E).

[35] This can be seen by multiplying the KdV equation with the surface elevation η(x,t); after repeated use of the chain rule the result is: which is in conservation form, and is an invariant after integration over the interval of periodicity—the wavelength for a cnoidal wave.

Note that η2 = −(1/λ) 0∫λ H cn2(ξ/Δ|m) dx, cn(ξ/Δ|m)  = cos ψ(ξ) and λ = 2 Δ K(m), so[37] The potential energy, both for the KdV and the BBM equation, is subsequently found to be[37] The infinitesimal wave-height limit (m → 0) of the potential energy is Epot = ⁠1/16⁠ ρ g H2, which is in agreement with Airy wave theory.