text: In this example, a cnoidal wave according to the Korteweg–de Vries (KdV) equation is considered.  The following parameters of the wave are given:
  * mean water depth h = ,
  * wave height H = ,
  * wave period τ = 7 s, and
  * gravitational acceleration g = 9.81 m/s (32 ft/s).
  Instead of the period τ, in other cases the wavelength λ may occur as a quantity known beforehand.
  First, the dimensionless period is computed:
  which is larger than seven, so long enough for cnoidal theory to be valid. The main unknown is the elliptic parameter m. This has to be determined in such a way that the wave period τ, as computed from cnoidal wave theory for the KdV equation:
  :    and  
  is consistent with the given value of τ; here λ is the wavelength and c is the phase speed of the wave. Further, K(m) and E(m) are complete elliptic integrals of the first and second kind, respectively. Searching for the elliptic parameter m can be done by trial and error, or by use of a numerical root-finding algorithm. In this case, starting from an initial guess m = 0.99, by trial and error the answer
  is found. Within the process, the wavelength λ and phase speed c have been computed:
  * wavelength λ = , and
  * phase speed c = .
  The phase speed c can be compared with its value  according to the shallow water equations:
  showing a 3.8% increase due to the effect of nonlinear amplitude dispersion, which wins in this case from the reduction of phase speed by frequency dispersion.
  Now the wavelength is known, the Ursell number can be computed as well:
  which is not small, so linear wave theory is not applicable, but cnoidal wave theory is. Finally, the ratio of wavelength to depth is λ / h = 10.2 > 7, again indicating this wave is long enough to be considered as a cnoidal wave.