JONSWAP Spectra

1. Six parameter JONSWAP spectrum

The six parameter JONSWAP spectrum is given by

\[S(\omega )=\frac{\alpha g^2}{\omega ^5}\exp(-\beta (\frac{\omega _p}{\omega })^4)\gamma ^{\exp(-(\displaystyle \frac{(\frac{\omega }{\omega _p}-1)^2}{2\sigma ^2}))} \\ \\ \sigma = \begin{cases} \sigma_a, & \omega \leq \omega_p \\ \\ \sigma_b, & \omega > \omega_p \end{cases} \\ \\ \omega_p = \frac{2\pi}{T_p} \label{eq-jonswap-6p}\]

where the following parameters are input:

\(\alpha\)

spectral parameter

\(T_p\)

peak period

\(\gamma\)

peakedness parameter

\(\beta\)

form parameter, default value \(\beta = 1.25\)

\(\sigma_a\)

spectral parameter a, default value \(\sigma_a = 0.07\)

\(\sigma_b\)

spectral parameter b, default value \(\sigma_b = 0.09\)

2. Three parameter JONSWAP spectrum

The spectrum function is the same as for the six parameter JONSWAP spectrum with the following relations:

\[\alpha =5.061\frac{H_s^2}{T_p^4}(1-0.287\ln(\gamma )) \\ \beta = 1.25 \\ \sigma_a = 0.07 \\ \sigma_b = 0.09 \label{eq-jonswap-3p}\]

The following values are given as input:

\(H_s\)

significant wave height

\(T_p\)

peak period

\(\gamma\)

peakedness parameter

3. Two parameter JONSWAP spectrum

The two parameter JONSWAP spectrum is given by Equation \(\eqref{eq-jonswap-6p}\), Equation \(\eqref{eq-jonswap-3p}\) and the following relation to determine \(\gamma\):

\[\gamma = \begin{cases} 5.0, & T_p \leq 3.6 \sqrt{H_s} \\ \\ \exp\left[3.484 \left(1-0.1975~\delta \frac{T_p^4}{H_s^2}\right)\right], & 3.6 \sqrt{H_s} < T_p < 5.0 \sqrt{H_s} \\ \\ 1.0, & T_p \geq 5.0 \sqrt{H_s} \end{cases}\]

where

\[\delta = 0.036-0.0056\frac{T_p}{\sqrt{H_s}}\]

4. Jonswap Double Peaked

The spectrum function is given in (Torsethaugen, 1996).