Wave Particle Kinematics

In the case of Irregular Waves and Regular Waves (Linear Airy option), water particle kinematics are based on linear Airy theory. The following equations describe the wave particle velocity vector \(\bvector{u}_w=\left(\bvector{u}_{w x}, \bvector{u}_{w y}, \bvector{u}_{w z}\right)\), the corresponding acceleration vector \(\bvector{\dot{u}}_w=\left(\bvector{\dot{u}}_{w x}, \bvector{\dot{u}}_{w y}, \bvector{\dot{u}}_{w z}\right)\), the hydrodynamic component of the pressure \(p\) and the water surface elevation \(\zeta\) for a regular wave of height \(H\) and period \(t\) at the point \((x, y, z)\) in global coordinate system:

\[ \begin{align} \zeta &= \frac{H}{2}\cos(\alpha - \omega t) \\ \bvector {u}_{wx} &= \frac{\sigma\ H}{2\sinh(kd)}\cos\left( \mu_{w} \right)\cosh\left\lbrack k(d + z) \right\rbrack\cos(\alpha - \omega t)\\ \bvector {u}_{wy} &= \frac{- \sigma\ H}{2\sinh(kd)}\sin{(\mu_{w})}\cosh\left\lbrack k(d + z) \right\rbrack\cos(\alpha - \omega t)\\ \bvector {u}_{wz} &= \frac{\sigma\ H}{2\sinh(kd)}\sinh\left\lbrack k(d + z) \right\rbrack\sin(\alpha - \omega t)\\ \bvector{\dot{u}}_{wx} &= \frac{\sigma^{2}H}{2\sinh(kd)}\cos{(\mu_{w})}\cosh\left\lbrack k(d + z) \right\rbrack\sin(\alpha - \omega t)\\ \bvector{\dot{u}}_{wy} &= \frac{- \sigma\omega\ H}{2\sinh(kd)}\sin{(\mu_{w})}\cosh\left\lbrack k(d + z) \right\rbrack\sin(\alpha - \omega t)\\ \bvector{\dot{u}}_{wz} &= \frac{- \sigma\omega\ H}{2\sinh(kd)}\sinh\left\lbrack k(d + z) \right\rbrack\cos(\alpha - \omega t)\\ p &= \frac{\rho gH}{2\cosh(kd)}\cosh\left\lbrack k(d + z) \right\rbrack\cos(\alpha - \omega t)\\ \end{align} \]

where \(\sigma\) and \(\omega = 2\pi f\) are the angular wave frequencies, \(f\) is the wave frequency, \(t\) is time, \(d\) is the water depth (assumed to be constant), \(\rho\) is the water density, \(g\) is the acceleration due to gravity, and

\[ \begin{equation} \alpha = kx\cos\mu_{w} - ky\sin\mu_{w}-\phi \end{equation} \]

where \(\mu_{w}\) is the direction from which waves arrive at the tower and \(\phi\) is wave phase. If wave-current interaction models are considered, then in some models, \(\sigma\) might be different than \(\omega\), i.e. \(\sigma \neq \ \omega\) (see summary of Wave-Current interaction options for more information). The wave number \(k\) is found as the solution to the dispersion relation

\[ \begin{equation} \omega^{2} = gk\tanh{kd} \label{eq:dispersionrelation} \end{equation} \]

In case of wave-current interaction, the correct dispersion relation for each option is given in summary of Wave-Current interaction options.

For the calculation of extreme waves, the above equations are used directly to calculate the wave particle kinematics at each submerged tower station. For fatigue load calculations, however, it is necessary to calculate an irregular (i.e. random, non-repeating) series of waves. Section wave-current interaction considers the effect of wave and current interaction when SEA file is used for computing water particle kinematic, pressure or wave height.

Last updated 30-08-2024