Waves: Ranges of applicability

Regular wave trains are specified in OrcaFlex by water depth, wave height and wave period. Which wave theory should one choose for any given wave train? For a close-to-idealised infinitesimal wave in deep water, Airy wave theory is accurate. For less ideal waves, a nonlinear theory should be used – but which one? Our advice is that you won't go far wrong with Dean's stream function theory. As a numerical best-fit method, it does not suffer from the possible truncation problems of the Stokes' 5th and cnoidal theories which depend on finite power series expansions. The stream function method is robust: in all circumstances it will, if it converges, give sensible results. By contrast, the other two methods may, if misapplied, give misleading results.

If you do wish to use either the Stokes' 5th or cnoidal method, you should check its validity by calculating the Ursell number $U$ given by \begin{equation} U = \frac{H L^2}{d^3} \end{equation} where $H$ is wave height, $L$ is wavelength and $d$ is the water depth at the seabed origin.

If $U{\lt}40$, then the waves are said to be short and Stokes' 5th may be used. For $U{\gt}40$, you have long waves and the cnoidal theory is appropriate. The stream function theory is applicable for any wave. The boundary value 40 should not be considered a hard and fast rule. In fact, for Ursell number close to 40. both Stokes' 5th theory and the cnoidal theory have inaccuracies and Dean's 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 therefore

You should also carry out a further check on the validity of Stokes' 5th or cnoidal by comparing against the stream function theory. This is a very important point. OrcaFlex has no way of telling if a theory has been misapplied, other than giving warnings for obvious abuses – it is up to you, the user, to make sure you are using an applicable theory. By comparing with the stream function theory a reliable check can be made – and if two theories give the same answers then one should be filled with confidence!