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.