OrcaFlex Manual - Orcina

Theory, Environment Theory
156
w
years but it gives unsatisfactory predictions of water particle velocities. This work has been superseded by Fenton
(1990 and 1995).
Fenton's original paper gave formulae for fluid velocities based on a Fourier series expansion about the term
ε = H / d. In his later works Fenton discovered that much better results could be obtained by expanding about a
"shallowness" parameter δ. We follow this approach.
A 5th order stream function representation is used but instead of terms involving cos the Jacobian elliptic function
cn is used, hence the term cnoidal. The function takes two parameters, x as usual, and also m which determines how
cusped the function is. In fact when m = 0, cn is just cos and the Jacobian elliptic functions can be regarded as the
standard trigonometric functions. The solitary wave which has 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 should only be applied for long
waves in shallow water and for such waves m is close to 1.
The initial step of the solution is to determine m and an implicit equation with m buried deep within must be solved.
As in the Stokes' theory this equation is the dispersion relationship. The solution is performed using the bisection
method since the equation shows singular behaviour for m ≈ 1 and derivative methods fail.
After m has been determined Fenton gives formulae to calculate surface elevation and other wave kinematics. In
practice m is close to 1 and Fenton takes advantage of this to simplify the formulae. He simply sets m = 1 in all
formulae except where m is the argument of an elliptic or Jacobian function. This technique is known as Iwagaki
approximation and proves to be very accurate.
Ranges of Applicability
Regular wave trains are specified in OrcaFlex by water depth, wave height and wave period. Which wave theory
should one use for any given wave train? For an infinitesimal wave in deep water then Airy wave theory is accurate.
For finite waves a non-linear theory should be used. In order to decide which wave theory to use one must calculate
the Ursell number given by
U = HL 2 / d 3
See Non-linear Wave Theories for notation conventions used.
If U < 40 then the waves are said to be short and Stokes' 5th may be used. For U > 40 we have long waves and the
cnoidal wave theory can be used. The stream function theory is applicable for any wave. The boundary number 40
should not be considered a hard and fast rule. In fact for Ursell number close to 40 both the Stokes' 5th theory and
the cnoidal theory have inaccuracies and the stream function method is recommended. In regions well away from
Ursell number 40 then the relevant analytic theories (Stokes' 5th or cnoidal) perform very well.
Our recommendations are:
Ursell number Recommended wave theory