Wave spectra

Pierson-Moskowitz spectrum

The Pierson-Moskowitz spectrum \(S_{PM}(\omega)\) is given by :

\[S_{PM}(\omega) = \frac{5}{16} \cdot H_S^2 \omega_p^4 \cdot \omega^{-5} \exp\left[-\frac{5}{4}\left(\frac{\omega}{\omega_p}\right)^{-4}\right]\]

where \(\omega_p = 2\pi / T_p\) is the angular spectral peak frequency and \(H_S\) is the significant wave height.

For more details see [DNV].

JONSWAP spectrum

The JONSWAP wave spectrum is an extension of the Pierson-Moskovitz wave spectrum, for a developing sea state in a fetch limited situation, with an extra peak enhancement factor \(\gamma\). The spreading function is given by

\[S_j(\omega) = \alpha_{\gamma} S_{PM}(\omega) \gamma^{\exp \left[-\frac{(\omega-\omega_p)^2}{2\sigma^2\omega_p^2} \right]}\]

where

  • \(S_{PM}(\omega)\) is the Pierson-Moskowitz spectrum,
  • \(\gamma\) is the non-dimensional peak shape parameter,
  • \(\sigma\) is the spectral width parameter,
\[\begin{split}\sigma &=& 0.07, \omega \leq \omega_p\\ \sigma &=& 0.09, \omega > \omega_p\end{split}\]
  • \(\alpha_{\gamma}= 1 - 0.287\log(\gamma)\) is a normalizing factor.

The JONSWAP spectrum is expected to be a reasonable model for \(3.6 < \frac{T_p}{H_S} < 5\). Default value for \(\gamma = 3.3\) can be changed by the user, but has to be specified between 1 and 10.

For more details see

Directional wave spectra

Directional short-crested wave spectra \(S(\omega,\theta)\) is expressed in terms of uni-directional wave spectra,

\[S(\omega,\theta) = S(\omega)D(\theta)\]

where \(D(\theta)\) is a directional function, \(\theta\) is the angle between the direction of elementary wave trains and the main wave direction of the short-crested wave system. The directional function fulfills the requirement:

\[\int D(\theta) d\theta = 1\]

Cos2s directional function

This directional model, proposed by Longuet-Higgins [LonguetHiggins1963] is an extension of the cosine-squared model. The spreading function is given by:

\[D(\theta) = \frac{2^{2s-1}}{\pi} \frac{\Gamma^2(s+1)}{\Gamma^2(2s+1)} \cos^{2s} \left(\frac{\theta - \theta_0}{2}\right)\]

where

  • \(\Gamma\) is the Gamma function,
  • \(\theta_0\) is the mean wave direction,
  • \(s\) is the spreading parameter.

The spreading parameter is defined constant and can be set by the user.

References

[LonguetHiggins1963]Longuet-Higgins, M.S., et al, Observations of the Directional Spectrum of Sea Waves Using the Motions of a Floating Buoy, Ocean Wave Spectra, Prentice-Hall, Inc., Englewood Cliffs, N. J., pp 111-13, 1963
[KIM20008]Kim, C.H., Nonlinear Waves and Offshore structures, World Scientific Publishing Company, Vol.27, 2008
[MOLIN2002]Molin, B., Hydrodynamique des Structures Offshore, Editions Technip, 2002
[DNV](1, 2) VERITAS, Det Norske. Modelling and analysis of marine operations. Offshore Standard, 2011.