The wave radiation stresses ${\displaystyle (S_{ij})}$ are calculated using linear wave theory as (Longuet-Higgins and Stewart 1961; Dean and Dalrymple 1984)

 ${\displaystyle S_{ij}=\int \int E_{w}(f,\theta )\left[n_{g}\omega _{i}\omega _{j}+\delta _{ij}\left(n_{g}-{\frac {1}{2}}\right)\right]dfd\theta }$ (1)

where:

f = the wave frequency [1/s]
${\displaystyle \theta }$ = the wave direction [rad]
${\displaystyle E_{w}}$ = wave energy = ${\displaystyle 1/16\ \rho gH_{s}^{2}}$ [N/m]
${\displaystyle H_{s}}$ = significant wave height [m]
${\displaystyle w_{i}}$ = wave unit vector = ${\displaystyle (cos\ \theta ,sin\ \theta )}$[-]
{\displaystyle \delta _{ij}=\left\{{\begin{aligned}&1\ for\ i=j\\&0\ for\ i\neq j\end{aligned}}\right.}

${\displaystyle n_{g}={\frac {c_{g}}{c}}={\frac {1}{2}}\left(1+{\frac {2kh}{sinh\ 2kn}}\right)[-]}$
${\displaystyle c_{g}}$ = wave group velocity [m/s]
${\displaystyle c}$ = wave celerity [m/s]
${\displaystyle k}$ = wave number [rad/s]

The wave radiation stresses and their gradients are computed within the wave model and interpolated in space and time in the flow model.

# References

• Dean, R. G., and R. A. Dalrymple. 1984. Water wave mechanics for engineers and scientists. Englewood Cliffs, NJ: Prentice-Hall.
• Longuet-Higgins, M. S., and R. W. Stewart. 1961. The changes in amplitude of short gravity waves on steady non-uniform currents. Journal of Fluid Mechanics 10(4):529–549.