# CMS-Wave:Wave-current Interaction

Under Construction

## Wave-current Interaction

The characteristic velocities $c_x$, $c_y$, and $c_{\theta}$ are calculated as

 $c_x = c_g \cos \theta + U$ (1)
 $c_y = c_g \sin \theta + V$ (2)
 $c_{\theta} = \frac{\sigma}{\sinh 2 k h} \biggl( \sin \theta \frac{\partial h}{\partial x} - \cos \theta \frac{\partial h}{\partial y } \biggr) + \cos \theta \sin \theta \frac{\partial U}{\partial x} - \cos ^2 \theta \frac{\partial U}{\partial y} + \sin ^2 \theta \frac{\partial V}{\partial x} - \cos \theta \sin \theta \frac{\partial V}{\partial y}$ (3)

The dispersion relationships between the relative angular frequency σ, the absolute angular frequency ω, the wave number vector k, and the current velocity vector U = U2 +V2 are (Jonsson 1990)

Symbol Description
$c$ Wave celerity
$c_g$ Wave group velocity
$\sigma$ Wave frequency
$E$ Spectral wave density
$k$ Wave number
$h$ Total water depth
$U$ Depth-averaged current velocity in x-direction
$V$ Depth-averaged current velocity in y-direction