Waves: Stokes' 5th

The engineering industry's standard reference on 5th order Stokes' wave theory is Skjelbreia and Hendrickson (1961), which presents a 5th order Stokes' theory with expansion term $ak$ where $a$ is the amplitude of the fundamental harmonic and $k=2\pi/L$ is the wave number ($L=$ wavelength). The length a has no physical meaning, and choosing $ak$ as expansion parameter means that convergence for very steep waves cannot be achieved.

Fenton (1985) subsequently presented a 5th order Stokes' theory based instead around an expansion term $kH/2$ ($H=$ wave height), and demonstrated that it is more accurate than Skjelbreia and Hendrickson's original theory. We therefore implement Fenton's theory in OrcaFlex. It is worth noting that the linear theory of Airy is a 1st order Stokes' theory.

Assuming that the wave train data supplied are water depth, wave height and wave period, then the wave number $k$ must be computed before the theory can be applied. This is done by solving a nonlinear implicit equation in terms of $k$, the dispersion relationship, using Newton-Raphson. Once $k$ is known, a number of coefficients are calculated and these are used in power series expansions to find the surface profile and wave kinematics.

Accuracy

Inherent in the 5th order method is a truncation of all terms of order greater than 5. For long waves in shallow water, these discarded terms become significant and make the theory invalid. Essentially this is a deep water, steep wave theory. See ranges of applicability for a discussion of the validity of the different theories.