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

 $S_{ij}=\int\int E_w(f,\theta)\left[n_g \omega_i \omega_j + \delta_{ij}\left(n_g - \frac{1}{2}\right) \right]dfd\theta$ (1)

where:

f = the wave frequency [1/s]
$\theta$ = the wave direction [rad]
$E_w$ = wave energy = $1/16\ \rho g H_s^2$ [N/m]
$H_s$ = significant wave height [m]
$w_i$ = wave unit vector = $(cos\ \theta, sin \ \theta)$[-]
\delta_{ij} = \left\{\begin{align} &1\ for\ i = j \\ &0\ for\ i \neq j \end{align}\right.

$n_g = \frac{c_g}{c} = \frac{1}{2}\left(1 + \frac{2kh}{sinh\ 2kn}\right)[-]$
$c_g$ = wave group velocity [m/s]
$c$ = wave celerity [m/s]
$k$ = 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.