Wave Radiation Stresses

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 gH_{s}^{2}

[N/m]

H_{s}

= significant wave height [m]

w_{i}

= wave unit vector =

(cos\ \theta ,sin\ \theta )

[-]

\delta _{ij}=\left\{{\begin{aligned}&1\ for\ i=j\\&0\ for\ i\neq j\end{aligned}}\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.