# Wave spectra¶

## Pierson-Moskowitz spectrum¶

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

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

where

- \(S_{PM}(\omega)\) is the Pierson-Moskowitz spectrum,
- \(\gamma\) is the non-dimensional peak shape parameter,
- \(\sigma\) is the spectral width parameter,

- \(\alpha_{\gamma}= 1 - 0.287\log(\gamma)\) 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,

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:

### 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:

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. |