Waves: Cnoidal theory

This is a steady periodic water wave theory designed to be used for long waves in shallow water, in direct contrast to Stokes' 5th. The high-order cnoidal theory of Fenton (1979) was regarded as the standard reference for many years, but it gives unsatisfactory predictions of water particle velocities and was superseded by Fenton in 1990 and 1995.

Fenton's original paper gave formulae for fluid velocities based on a Fourier series expansion about the term $\epsilon = H/d=$ wave height / water depth. In his later works Fenton discovered that much better results could be obtained by expanding about a shallowness parameter $\delta$. We follow this latter approach.

A 5th order stream function representation is used but, rather than terms involving cosines, the Jacobian elliptic function cn is used instead, hence the term cnoidal. The function takes two parameters, $x$ as usual, and an additional $m$ which determines how cusped the function is. In fact, when $m\!=\!0$, cn is simply cos and the Jacobian elliptic functions can be regarded as the standard trigonometric functions. The solitary wave of infinite length corresponds to $m\!=\!1$, and long waves in shallow water have values of $m$ close to 1. Fenton shows that the cnoidal theory is applicable only to such long shallow-water waves.

The initial step of the solution is to determine m from an implicit equation: as in Stokes' theory, this equation is the dispersion relationship. The solution is in this case performed using the bisection method, since the equation shows singular behaviour for $m\!=\!1$ causing derivative methods such as Newton-Raphson to fail.

With $m$ determined, Fenton gives formulae to calculate surface elevation and other wave kinematics. In practice $m$ is always close to 1, and Fenton takes advantage of this to simplify the formulae. He simply takes $m\!=\!1$ in all cases except where $m$ is the argument of an elliptic or Jacobian function. This technique is known as Iwagaki approximation and proves to be highly accurate.