Nonlinear and exact periodic wave solution of the Korteweg–de Vries equation
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 "cn"oidal 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 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 formula_1 with "g" the strength of the gravitational acceleration. To envisage the position of the KdV equation within the scope of classical wave approximations, it distinguishes itself in the following ways:
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.
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:
formula_2
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:
formula_3
For small values of "U", say "U"  ; 5, 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. 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:
formula_4
where
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All quantities can be made dimensionless using the gravitational acceleration "g" and water depth "h":
formula_5   formula_6   and   formula_7
The resulting non-dimensional form of the KdV equation is
formula_8
In the remainder, the tildes will be dropped for ease of notation.
The form
formula_9
is obtained through the transformation
formula_10   formula_11   and   formula_12
but this form will not be used any further in this derivation.
Periodic wave solutions, travelling with phase speed "c", are sought. These permanent waves have to be of the following:
formula_13   with formula_14 the wave phase:   formula_15
Consequently, the partial derivatives with respect to space and time become:
formula_16   and   formula_17
where "η′" 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:
formula_18
This can be integrated once, to obtain:
formula_19
with "r" an integration constant. After multiplying with 4 "η′", and integrating once more
formula_20
with "s" another integration constant. This is written in the form
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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":
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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 "η′" 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:
formula_21
Now, from equation (A) 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"):
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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:
formula_22
Using these in equations (A) and (B), the following ordinary differential equation relating "ψ" and "ξ" is obtained, after some manipulations:
formula_23
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":
formula_24
with:
formula_25   and   formula_26
where "m" is the so-called elliptic parameter, 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
formula_27   and   formula_28
With the use of equation (C), the resulting cnoidal-wave solution of the KdV equation is found
formula_29
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":
formula_30   and   formula_31   so   formula_32
The cnoidal wave solution can be written as:
formula_33
Second, the trough is located at "ψ" =  "π", so the distance between "ξ" = 0 and "ξ" =  "λ" is, with "λ" the wavelength, from equation (D):
formula_34   giving   formula_35
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
formula_36
where "E"("m") is the complete elliptic integral of the second kind. The following expressions for "η1", "η2" and "η3" as a function of the elliptic parameter "m" and wave height "H" result:
formula_37   formula_38   and   formula_39
Fourth, from equations (A) and (B) a relationship can be established between the phase speed "c" and the roots "η1", "η2" and "η3":
formula_40
The relative phase-speed changes are depicted in the figure below. As can be seen, for "m"  ; 0.96 (so for 1 − "m"  ; 0.04) the phase speed increases with increasing wave height "H". This corresponds with the longer and more nonlinear waves. 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:
formula_41
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:
formula_42
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 trough level "η2" and width parameter "Δ" can be expressed in terms of "H", "h" and "m":
formula_43   and   formula_44
with "K"("m") the complete elliptic integral of the first kind and "E"("m") the complete elliptic integral of the second kind. Note that "K"("m") and "E"("m") are denoted here as a function of the elliptic parameter "m" and not as a function of the elliptic modulus "k", with "m" = "k2".
The wavelength "λ", phase speed "c" and wave period "τ" are related to "H", "h" and "m" by:
formula_45   formula_46   and   formula_47
with "g" the Earth's gravity.
Most often, the known wave parameters are the wave height "H", mean water depth "h", gravitational acceleration "g", and either the wavelength "λ" or else the period "τ". 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.
Benjamin–Bona–Mahony equation.
The Benjamin–Bona–Mahony equation (BBM equation), or regularised long wave (RLW) equation, is in dimensional form given by:
formula_48
All quantities have the same meaning as for the KdV equation. The BBM equation is often preferred over the KdV equation because it has a better short-wave behaviour.
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The derivation is analogous to the one for the KdV equation. The dimensionless BBM equation is, non-dimensionalised using mean water depth "h" and gravitational acceleration "g":
formula_49
This can be brought into the standard form
formula_50
through the transformation:
formula_51   formula_52   and   formula_53
but this standard form will not be used here.
Analogue to the derivation of the cnoidal wave solution for the KdV equation, periodic wave solutions "η"("ξ"), with "ξ" = "x"−"ct" are considered Then the BBM equation becomes a third-order ordinary differential equation, which can be integrated twice, to obtain:
formula_54   with   formula_55
Which only differs from the equation for the KdV equation through the factor "c" in front of ("η′")2 in the left hand side. Through a coordinate transformation "β" = "ξ" / formula_56 the factor "c" may be removed, resulting in the same first-order ordinary