# Waves: Wave Spectra

## ISSC spectrum

The ISSC spectrum (also known as Bretschneider or modified Pierson-Moskowitz) is defined as

$$S(f) = \frac{5}{16}\Hs^2\ \fm^4\ f^{-5}\ \exp\left[ -\frac{5}{4}\biggl( \frac{f}{\fm} \biggr)^{-4}\right]$$

where $f$ is frequency. The other two parameters, the peak frequency $\fm$ and the significant wave height $\Hs$, are data items.

For more details see Tucker 1991, page 107.

## JONSWAP spectrum

The JONSWAP spectrum is defined as

$$S(f) = \frac{\alpha g^2}{16\pi^4}\ f^{-5}\ \exp\left[ -\frac{5}{4}\biggl( \frac{f}{\fm} \biggr)^{-4}\right] \gamma^b$$

where

\begin{align} b &= \exp\left[ -\frac{1}{2\sigma^2}\bigg( \frac{f}{\fm}-1 \bigg)^2 \right] \\ \sigma &= \begin{cases} \sigma_1 \quad &\text{for $f\le\fm$} \\ \sigma_2 \quad &\text{for $f\gt\fm$} \end{cases} \end{align}

with $g$ the acceleration due to gravity and the remaining parameters $\gamma$, $\alpha$, $\sigma_1$ and $\sigma_2$ being data items.

For more details see

## Ochi-Hubble spectrum

The Ochi-Hubble spectrum allows double-peaked spectra to be set up, enabling you to represent sea states that include both a remotely generated swell and a local wind generated sea. See the Ochi-Hubble paper for full details of the spectral formula.

 Figure: Example Ochi-Hubble spectrum and its components

The Ochi-Hubble spectrum is the sum of two separate component spectra – the example graph shows the two components and their sum. The component with the lower-frequency peak corresponds to the remotely-generated swell; that with the higher-frequency peak corresponds to the local wind-generated sea. This is why the Ochi-Hubble spectrum is often called a two-peaked spectrum; however in practice, the resulting total spectrum typically has only one peak (from the remotely-generated swell) plus a shoulder of energy from the local wind-generated sea.

The component spectra are each determined by a set of three parameters – $H_\mathrm{s1}$, $f_\mathrm{m1}$, $\lambda_1$ for the lower-frequency component and $H_\mathrm{s2}$, $f_\mathrm{m2}$, $\lambda_2$ for the higher. In OrcaFlex you can either specify all these six parameters explicitly, or you can simply give the overall significant wave height $\Hs$ and tell OrcaFlex to automatically select the most probable six parameters for that value of $\Hs$ based on formulae given in the Ochi-Hubble paper (Table 2b).

## Torsethaugen spectrum

The Torsethaugen spectrum is another two-peaked spectrum, more suited to North Sea application than Ochi-Hubble. See the Torsethaugen and Haver paper for details of the spectral formula.

 Warning: The two-peaked Ochi-Hubble and Torsethaugen spectra make no allowance for the directionality of the swell and wind components of the sea state. In reality the separate components frequently come from different directions. However, since an OrcaFlex wave train has a single principal direction, it is more appropriate to model a two-peaked sea state using two separate OrcaFlex wave trains, one for the swell component and one for the local wind generated component.

## Gaussian swell spectrum

The Gaussian swell spectrum is based on the Normal (or Gaussian) probability density function and is defined as

$$S(f) = \left( \frac{\Hs}{4} \right) ^2\ \frac{1}{\sigma\sqrt{2\pi}}\ \exp\left[ -\frac{(f-\fm)^2}{2\sigma^2} \right]$$

where $\Hs$, $\fm$ and $\sigma$ are the input data.