2.3.8. Statistical description of waves

2.3.8.1. Spectrum discretization

Let be \(S(\Omega,α)\) an energy density spectrum describing a sea state in terms of the wave frequency and direction of propagation. The discretization procedure to obtain a stationary and ergodic realization based on monochromatic waves is as follows: Let be the minimum frequency to be considered, the maximum frequency to be considered, the lower direction of propagation to be considered, the larger direction of propagation to be considered, the number of wave frequencies, and the number of wave directions to be considered. Then, the frequency and direction discretization sizes are given by:

(2.604)\[\begin{aligned} & ∆\Omega = \frac{\Omega_{max} - \Omega_{min}}{N^w} \end{aligned}\]

(2.605)\[\begin{aligned} & \Delta α = \frac{α_{max} - α_{min}}{N^α - 1} \end{aligned}\]

then, the wave elevation is given by:

(2.606)\[\begin{aligned} & η = \sum_{i=1}^{N^w} \sum_{j=1}^{N^\alpha} A_{ij} cos \left( k_{ij} cos(\alpha_j) x + k_{ij} sin(\alpha_j) - \Omega_{ij} t + \delta_{ij} \right) \end{aligned}\]

where \(\Omega_{ij}\) is the wave angular velocity and a random variable with uniform distribution in within \([\Omega_i - \Delta \Omega/2, \Omega_i + \Delta \Omega/2]\), \(\Omega_i = \Omega_{min} + (i-1/2)\Delta \Omega\), \(α_j = α_{min} + (j-1)\Delta α\), is a random variable with uniform distribution in \([0, 2\pi]\), \(t\) represents time, and \(x\), \(y\) are the horizontal Cartesian coordinates. The wave number is obtained from the dispersion relationship:

(2.607)\[\begin{aligned} & \Omega^2_{ij} = g \cdot k_{ij} \cdot tanh⁡ \left( k_{ij} H \right) \end{aligned}\]

and the wave amplitude is calculated from the wave energy distribution as:

(2.608)\[\begin{aligned} & A_{ij} = \sqrt{ 2 \Delta \Omega \Delta α S \left(\Omega_i, α_j \right) } \frac{ (1/16) H_S^2}{\sum_{l,m} \sqrt{2 \Delta \Omega \Delta α S \left(\Omega_l,α_m \right)}} \end{aligned}\]

where \(H_s\) is the significant wave height, and \(m_0 = \int_0^\infty \int_{-\pi}^{\pi} S \left( \omega , \alpha \right)\) is zero order moment of the spectrum wave energy.

2.3.8.2. Convergence

Convergence of the discretized spectrum will happen as \(\omega_{min} → 0\) , \(\omega_{max} → \infty\) , \(\Delta \omega → 0\) , \(\alpha_{min} → 0\) , \(\alpha_{max} → 2\pi\) , and \(\Delta \alpha → 0\). The rate of convergence with \(\Delta \omega\) and \(\Delta \alpha\) is that of the rectangle rule of numerical integration.

2.3.8.3. Spectral moments

2.3.8.3.1. Zero order moment

The spectral energy of a wave spectrum is given by:

(2.609)\[\begin{aligned} & m_0 = \int_0^∞ \int_{-π}^π S\left(\Omega,α \right) d\Omega dα = \frac{1}{16} \rho g H_S^2 \end{aligned}\]

Then, the discrete spectrum is scaled such that the spectral moment is conserved. Therefore:

(2.610)\[\begin{aligned} & \sum_{i,j} \frac{1}{2} A_{ij}^2 = \frac{1}{16} H_S^2 \end{aligned}\]

2.3.8.3.2. First order moment

The first order moment of the discrete spectrum is:

(2.611)\[\begin{aligned} & m_1^* = \sum_{i,j} S\left(\Omega_i, α_j \right) \Omega_{i,j} \Delta \Omega \Delta α = \sum_{i,j} S\left(\Omega_i, α_j \right) \Omega_{i,j} \Delta \Omega \Delta α + \sum_{i,j} S\left(\Omega_i, α_j \right) \epsilon _{i,j} \Delta \Omega \Delta α \end{aligned}\]

where \(\epsilon_{ij}\) is uniform distributed between \(\left[-\Delta \omega /2, \Delta \omega /2 \right]\), \(\sum_{i,j} S \left(\Omega_i, α_j \right) \Omega_{i,j} \Delta \Omega \Delta α\) is a deterministic component of the first moment, and \(\sum_{i,j} S\left(\Omega_i, α_j \right) \epsilon _{i,j} \Delta \Omega \Delta α\) is a random component. Assuming that \(\omega_{max} → \infty\), \(\omega_{min} = 0\), \(\alpha_{min} = 0\), \(\alpha_{max} = 2\pi\), the deterministic component converges to:

(2.612)\[\begin{aligned} & lim (\Delta \Omega → 0 \Delta α → 0) ⁡\sum_{i,j} S \left(\Omega_i, α_j \right) \Omega_{i,j} \Delta \Omega \Delta α = \int_0^∞ \int_(-π)^π \Omega S \left(\Omega, α \right) d\Omega dα \end{aligned}\]

On the other hand, for large values of \(N^{\omega}\), the probabilistic component is a random variable with normal distribution. The mean \(\mu\) and variance \(σ^2\) of this distribution are:

(2.613)\[\begin{aligned} & \mu = \sum_{i,j} S(\Omega_i,α_j )\Delta \Omega \Delta α \int_{-\Delta \Omega/2}^{\Delta \Omega/2} \Omega \frac{1}{\Delta \Omega} d\Omega \end{aligned}\]

(2.614)\[\begin{aligned} & σ^2 = \sum_{i,j} S(\Omega_i,α_j ) \Delta \Omega \Delta α \int_{-\Delta \Omega/2}^{\Delta \Omega/2} \Omega^2 \frac{1}{\Delta \Omega} d\Omega = \sum_{i,j} S(\Omega_i,α_j ) \Delta \Omega \Delta α \frac{\Delta \Omega^2}{12} \end{aligned}\]

The probabilistic component converges to a random variable with zero mean and zero variance.

2.3.8.4. Wave spectrums

2.3.8.4.1. Pearson Moskowitz

This is probably the simplest idealized spectrum, obtained by assuming a fully developed sea state, generated by wind blowing steadily for a long time over a large area [3]. The resulting spectrum was [4]:

(2.615)\[\begin{aligned} & S(T) = H_s^2 T_m (0.11/2π)(T_m/T)^{-5} e^{-0.44(Tm/T)^{-4} } \end{aligned}\]

where \(T_m=2πm_0/m_1\), with \(m_0\) and \(m_1\) the zero and first moments of the wave spectrum.

2.3.8.4.2. Jonswap

The JONSWAP spectrum was established during a joint research project, the “JOint North Sea WAve Project” [5]. This is a peak-enhanced Pierson-Moskowitz spectrum given on the form:

(2.616)\[\begin{aligned} & S(T) = \left( \frac{5}{32π} H_s^2 \frac{T^5}{T_p^4} \right) \cdot \epsilon ^\Gamma \cdot e^{-1.25(T_p⁄T)^{-4}} \cdot (1 - 0.287log⁡(\epsilon )) \end{aligned}\]

(2.617)\[\begin{aligned} & \Gamma = e^{-[(0.159 \Omega T_p - 1)/(σ\sqrt{2})]^2} \end{aligned}\]

where \(\Omega=2π/T\), \(σ=0.07\) for \(\Omega ≤ 6.28/T_p\), \(σ=0.09\) for \(\Omega>6.28/T_p\), \(T\) is the wave period; \(H_s\) is the significant wave height, \(T_p\) is the peak wave period and :math:`epsilon ` is the peakedness parameter. An alternative definition of the JONSWAP spectrum is given by [4]:

(2.618)\[\begin{aligned} & S(\Omega) = \left( \frac{155H_s^2}{T_m^4 \Omega^5} \right) \cdot 3.3^\Gamma \cdot e^{-944 T_m^{-4} \Omega^{-4}} \end{aligned}\]

(2.619)\[\begin{aligned} & \Gamma = e^{-[(0.191\Omega T_m^{-1})/(σ\sqrt{2})]^2} \end{aligned}\]

where \(σ=0.07\) for \(\Omega ≤ 5.24/T_m\), \(σ = 0.09\) for \(\Omega > 5.24/T_m\), \(T_m = 2π m_0/m_1\).

2.3.8.4.3. White noise

The white noise spectrum corresponds to a uniform energy distribution within a wave frequency interval, having zero energy outside the prescribed interval. This type of spectrum is used in SeaFEM to carry out Response amplitude operators (RAOs) analyses (see section 11).