In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–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 Diederik Korteweg and Gustav de Vries, in their 1895 paper in which they also propose their dispersive long-wave equation, now known as the Korteweg–de Vries equation. In the limit of infinite wavelength, the cnoidal wave becomes a solitary wave.
The Benjamin–Bona–Mahony equation has improved short-wavelength behaviour, as compared to the Korteweg–de Vries equation, and is another uni-directional wave equation with cnoidal wave solutions. Further, since the Korteweg–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.
Background
Korteweg–de Vries, and Benjamin–Bona–Mahony equations
The Korteweg–de Vries equation (KdV equation) can be used to describe the uni-directional propagation of weakly nonlinear and long waves—where long wave means: having long wavelengths as compared with the mean water depth—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 λ in excess of about five times the average water depth h, so for λ > 5 h; and for the period τ greater than with g the strength of the gravitational acceleration.Dingemans (1997) pp. 718–721. To envisage the position of the KdV equation within the scope of classical wave approximations, it distinguishes itself in the following ways:
Korteweg–de Vries equation — describes the forward propagation of weakly nonlinear and dispersive waves, for long waves with λ > 7 h. Shallow water equations — are also nonlinear and do have amplitude dispersion, but no frequency dispersion; they are valid for very long waves, λ > 20 h. Boussinesq equations — 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 — 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 — 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 approach—as also applied in the KdV equation—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 Benjamin–Bona–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. 689–691.
Cnoidal waves
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. 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.
The surface elevation η(x,t), as a function of horizontal position x and time t, for a cnoidal wave is given by:
where H is the wave height, λ is the wavelength, c is the phase speed and η2 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.
An important dimensionless parameter for nonlinear long waves (λ ≫ h) is the Ursell parameter:
For small values of U, say U < 5,Due to the way it has been normalised, the Ursell parameter indicates linear theory is applicable when U ≪ 32 π2 / 3 ≈ 100. a linear theory can be used, and at higher values nonlinear theories have to be used, like cnoidal wave theory. The demarcation zone between—third or fifth order—Stokes' and cnoidal wave theories is in the range 10–25 of the Ursell parameter., p. 61. As can be seen from the formula for the Ursell parameter, for a given relative wave height H/h the Ursell parameter—and thus also the nonlinearity—grows quickly with increasing relative wavelength λ/h.
Based on the analysis of the full nonlinear problem of surface gravity waves within potential flow theory, the above cnoidal waves can be considered the lowest-order term in a perturbation series. Higher-order cnoidal wave theories remain valid for shorter and more nonlinear waves. A fifth-order cnoidal wave theory was developed by Fenton in 1979. A detailed description and comparison of fifth-order Stokes' and fifth-order cnoidal wave theories is given in the review article by Fenton.
Cnoidal wave descriptions, through a renormalisation, are also well suited to waves on deep water, even infinite water depth; as found by Clamond. A description of the interactions of cnoidal waves in shallow water, as found in real seas, has been provided by Osborne in 1994.
Surface tension In case surface tension effects are (also) important, these can be included in the cnoidal wave solutions for long waves.
Periodic wave solutions
Korteweg–de Vries equation
The Korteweg–de Vries equation (KdV equation), as used for water waves and in dimensional form, is:Dingemans (1997) pp. 692–693.
where {| border="0" |- | η || : surface elevation, a function of x and t, with the positive direction upwards (opposing gravity), |- | x || : horizontal coordinate, |- | t || : time, |- | g || : the value of Earth's gravity, |- | h || : the mean water depth, and |- | ∂x and ∂t || : partial derivative operators with respect to x and t.
Non-dimensionalisation All quantities can be made dimensionless using the gravitational acceleration g and water depth h:
and
The resulting non-dimensional form of the KdV equation is
In the remainder, the tildes will be dropped for ease of notation.
Relation to a standard form The form
is obtained through the transformation
and
but this form will not be used any further in this derivation.
Fixed-form propagating waves Periodic wave solutions, travelling with phase speed c, are sought. These permanent waves have to be of the following:
with the wave phase:
Consequently, the partial derivatives with respect to space and time become:
and
where denotes the ordinary derivative of η(ξ) with respect to the argument ξ.
Using these in the KdV equation, the following third-order ordinary differential equation is obtained:Dingemans (1997) p. 701.
Integration to a first-order ordinary differential equation This can be integrated once, to obtain:
with r an integration constant. After multiplying with 4 , and integrating once more
with s another integration constant. This is written in the form
The cubic polynomial f(η) becomes negative for large positive values of η, and positive for large negative values of η. Since the surface elevation η is real valued, also the integration constants r and s are real. The polynomial f can be expressed in terms of its roots η1, η2 and η3:Dingemans (1997) pp. 708–715.
Because f(η) is real valued, the three roots η1, η2 and η3 are either all three real, or otherwise one is real and the remaining two are a pair of complex conjugates. In the latter case, with only one real-valued root, there is only one elevation η at which f(η) is zero. And consequently also only one elevation at which the surface slope is zero. However, we are looking for wave like solutions, with two elevations—the wave crest and trough (physics)—where the surface slope is zero. The conclusion is that all three roots of f(η) have to be real valued.
Without loss of generality, it is assumed that the three real roots are ordered as:
Solution of the first-order ordinary-differential equation Now, from equation () it can be seen that only real values for the slope exist if f(η) is positive. This corresponds with η2 ≤ η≤ η1, which therefore is the range between which the surface elevation oscillates, see also the graph of f(η). This condition is satisfied with the following representation of the elevation η(ξ):
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 () and () can be obtained:
Using these in equations () and (), the following ordinary differential equation relating ψ and ξ is obtained, after some manipulations:
with the right hand side still positive, since η1 − η3 ≥ η1 − η2. Without loss of generality, we can assume that ψ(ξ) is a monotone function, since f(η) has no zeros in the interval η2 < η < η1. So the above ordinary differential equation can also be solved in terms of ξ(ψ) being a function of ψ:
with:
and
where m is the so-called elliptic parameter,Abramowitz & Stegun (1965) p. 590.The elliptic parameter m is distinct from the elliptic modulus k: m = k2. See Abramowitz & Stegun (1965) p. 590. satisfying 0 ≤ m ≤ 1 (because η3 ≤ η2 ≤ η1). If ξ = 0 is chosen at the wave crest η(0) = η1 integration gives
with F(ψ|m) the incomplete elliptic integral of the first kind. The Jacobi elliptic functions cn and sn are inverses of F(ψ|m) given by
and
With the use of equation (), the resulting cnoidal-wave solution of the KdV equation is found
What remains, is to determine the parameters: η1, η2, Δ and m.
Relationships between the cnoidal-wave parameters 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:
and so
The cnoidal wave solution can be written as:
Second, the trough is located at ψ = π, so the distance between ξ = 0 and ξ = λ is, with λ the wavelength, from equation ():
giving
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. So