Waves

1. Regular Waves

Regular waves are given as a list of one or more wave components. Each component has a propagation direction, phase angle, wave period, and amplitude.

2. Irregular Waves

When using Irregular waves the sea state is described using a wave spectrum function. The Spreading Type option may be used to choose between a unidirectional or short crested sea state using a cosine spreading function.

2.1. 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 \(\sigma_b = 0.09\)

2.2. Three parameter JONSWAP spectrum

The spectrum function is the same as for the six parameter JONSWAP spectrum given in Equation \(\eqref{eq-jonswap-6p}\) 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

2.3. 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

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

2.4. Double peaked JONSWAP spectrum

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

2.5. Pierson-Moskowitz spectrum

The Pierson-Moskowitz spectrum is given by

\[S(\omega )=\frac{a}{\omega ^5}\exp(-\frac{b}{\omega ^4}) \label{eq-pm-spec}\]

where

\[a = 0.0081 g^2\\ \\ b = \frac{3.11}{H_s^2}\]

and where

\(g\)

acceleration of gravity

The following parameters are given as input:

\(H_s\)

significant wave height

2.6. Pierson-Moskowitz ISSC spectrum

The spectrum is given by Equation \(\eqref{eq-pm-spec}\) where

\[a = 173.0 \frac{H_s^2}{T_1} \\ \\ b = \frac{691.0}{T_1^4} \label{eq-pm-issc-ab}\]

The following parameters are given as input:

\(H_s\)

significant wave height

\(T_1\)

average period

2.7. Pierson-Moskowitz spectrum with zero-corrsing period

The spectrum is given by Equation \(\eqref{eq-pm-spec}\) and Equation \(\eqref{eq-pm-issc-ab}\) where

\[T_1 = 1.0864 T_z\]

The following parameters are given as input:

\(H_s\)

significant wave height

\(T_z\)

zero upcrossing period

2.8. Ochi-Hubble spectrum

The spectrum is given by

\[\begin{align}\begin{split} S(\omega) = & E_1 G_1 \omega_{n, 1}^{4 - 0.25 \lambda_1} \exp\left[-\left(\lambda_1 + 0.25\right)\omega_{n,1}^{-4}\right] \\ & + E_2 G_2 \omega_{n, 2}^{4 - 0.25 \lambda_2} \exp\left[-\left(\lambda_2 + 0.25\right)\omega_{n,2}^{-4}\right] \end{split}\end{align}\]

where

\[\begin{align}\begin{split} \omega_{n,1} &= \frac{\omega}{w_{p,1}} \\ \omega_{n,2} &= \frac{\omega}{w_{p,2}} \\ E_1 &= \frac{H_{s,1}^2}{16\omega_{p,1}} \\ E_2 &= \frac{H_{s,2}^2}{16\omega_{p,2}} \\ G_1 &= \frac{4(\lambda_1 + 0.25)^{\lambda_1}}{\gamma(\lambda_1)} \\ G_2 &= \frac{4(\lambda_2 + 0.25)^{\lambda_2}}{\gamma(\lambda_2)} \\ \lambda_1 &= 3.0 \\ \lambda_2 &= 1.54 e^{-0.062 H_s} \\ \omega_{p,1} &= 0.7 e^{-0.046 H_s} \\ \omega_{p,2} &= 1.15 e^{-0.039 H_s} \\ H_{s,1} &= 0.84 H_s \\ H_{s,2} &= 0.54 H_s \end{split}\end{align}\]

The following parameters are given as input:

\(H_s\)

significant wave height

2.9. Numerically defined spectrum

This is a wave spectrum specified by the user. The spectrum is defined by spectral values and frequencies. Spectral values outside the given frequency range are assumed to be zero. Linear interpolation is used within the frequency range.