# 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.