Wave Kinematics

The velocity and acceleration profile over the water depth may be derived from the velocity potentials (finite and infinite depth). For simplicity, the distinction between uni- and multi-directional wave fields is neglected in this section. In the case of a uni-directional wave field, the first summation term becomes redundant. In the case of infinite depth (for most waves of interest this represents a depth greater than 100m), the velocity profiles are defined by:

\[\begin{align} V_x = \sum_{i}^{N_i}\sum_{j}^{N_j} A_{ij}\omega_i cos(\Theta_j)E_c(z)sin(k_i X_j - \omega_i t+\epsilon_{ij}, \end{align}\]
\[\begin{align} V_y = \sum_{i}^{N_i}\sum_{j}^{N_j} A_{ij}\omega_i sin(\Theta_j)E_c(z)sin(k_i X_j - \omega_i t+\epsilon_{ij}, \end{align}\]
\[\begin{align} V_z = \sum_{i}^{N_i}\sum_{j}^{N_j} -A_{ij}\omega_i E_s(z)cos(k_i X_j - \omega_i t+\epsilon_{ij}. \end{align}\]

Hence, the acceleration may be derived:

\[\begin{align} a_x = \sum_{i}^{N_i}\sum_{j}^{N_j} -A_{ij}\omega_i^2 cos(\Theta_j)E_c(z)cos(k_i X_j - \omega_i t+\epsilon_{ij}, \end{align}\]
\[\begin{align} a_y = \sum_{i}^{N_i}\sum_{j}^{N_j} -A_{ij}\omega_i^2 sin(\Theta_j)E_c(z)cos(k_i X_j - \omega_i t+\epsilon_{ij}, \end{align}\]
\[\begin{align} a_z = \sum_{i}^{N_i}\sum_{j}^{N_j} -A_{ij}\omega_i^2 E_s(z)sin(k_i X_j - \omega_i t+\epsilon_{ij}. \end{align}\]

\(E_c\) and \(E_s\) are depth scaling factors that depending on the case are defined as:

\[\begin{align} E_c(z) = \frac{cosh(k_i(z+h))}{sinh(k_ih)} \end{align}\]
\[\begin{align} E_s(z) = \frac{sinh(k_i(z+h))}{sinh(k_ih)} \end{align}\]

for finite depth and:

\[\begin{align} E(z) = e^{k_iz} \end{align}\]

for infinite depth.