# CMS-Wave:Wave-current Interaction

## Wave-current Interaction

The characteristic velocities ${\displaystyle c_{x}}$, ${\displaystyle c_{y}}$, and ${\displaystyle c_{\theta }}$ are calculated as

 ${\displaystyle c_{x}=c_{g}\cos \theta +U}$ (1)
 ${\displaystyle c_{y}=c_{g}\sin \theta +V}$ (2)
 ${\displaystyle c_{\theta }={\frac {\sigma }{\sinh 2kh}}{\biggl (}\sin \theta {\frac {\partial h}{\partial x}}-\cos \theta {\frac {\partial h}{\partial y}}{\biggr )}+\cos \theta \sin \theta {\frac {\partial U}{\partial x}}-\cos ^{2}\theta {\frac {\partial U}{\partial y}}+\sin ^{2}\theta {\frac {\partial V}{\partial x}}-\cos \theta \sin \theta {\frac {\partial V}{\partial y}}}$ (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
${\displaystyle c}$ Wave celerity
${\displaystyle c_{g}}$ Wave group velocity
${\displaystyle \sigma }$ Wave frequency
${\displaystyle E}$ Spectral wave density
${\displaystyle k}$ Wave number
${\displaystyle h}$ Total water depth
${\displaystyle U}$ Depth-averaged current velocity in x-direction
${\displaystyle V}$ Depth-averaged current velocity in y-direction