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Wave Directional Spreading

For a given wave spectrum \(S(\omega)\) defined by its average direction \(\theta_p\), the energy direction spreading is :

\[ S(\omega,\theta)=S(\omega)D(\theta) \tag{1} \]

With : $$ D(\theta)=\frac{\Gamma(1+n/2)}{\sqrt{\pi}\Gamma(1/2+n/2)}cos^n(\theta-\theta_p) $$

Where \(\Gamma\) is the Gamma function and \(|\theta-\theta_p| \leq \frac{\pi}{2}\)

As a consequence, the direction spreading is defined by the following input values: * The number of directions to be taken into account,

  • The exponent “n” of the cosine function in the following expression (note that the exponent is often denoted n=2)

The idea is to consider the original wave spectrum as a combination of several wave spectra with different directions; for every direction, the spectrum is related to the same reference spectrum \(S_{ref}(\omega)\) by the formula \((1)\). The reference spectrum is the spectrum when no direction spreading is accounted for.

The key fact is to keep the same global energy, eg :

\[ \int_\Omega\int_\Theta S(\omega,\theta) d\theta d\omega=\int_\Omega S_{ref}(\omega)d\omega \]
\[ \int_\Omega\int_\Theta S(\omega,\theta) d\theta d\omega=\int_\Omega\int_\Theta S(\omega) D(\theta)d\theta d\omega \]

Let’s set that :

\[ S_{ref}(\omega)=\sum_{i_{dir}}S^{i_{dir}}(\omega)=\sum_{i_{dir}}S_{ref}(\omega)D(\theta_{i_{dir}})\Delta\theta \]

Every sub-spectrum may be generated following the same procedure as any wave spectrum. This sub-spectrum is defined as such:

  • Its peak period is the reference spectrum peak period,

  • Its heading is \(\theta_{i_{dir}}\),

  • Its significant height is \(H_s^{i_{dir}}=H_s^{ref}\sqrt{D(\theta)\Delta\theta}\)