CMS-Wave: Wave Radiation Stresses

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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


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} =
&1\ for\  i = j \\
&0\ for\  i \neq j

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.


  • 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.

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