24
2.2 Radiation Stresses
In a series of papers in the early 1960s Longuet-Higgins and Stewart (1962, 1963, 1964)
introduced and developed the mathematics of the radiation stresses. These stresses which
can be conceptually thought of as the flux of excess momentum can be obtained through
integration of the momentum equations over the water column as is shown in Appendix A.
Using linear wave theory the radiation stress components are found to be
Szz = E n(cos2 6 + 1) ^
At
Svv = E n(sin2 0 + 1) ^
At m
Szv = En cos 0 sin 6
(2.28)
(2.29)
(2.30)
where E is the wave energy equal to |pgH2, n is the ratio of the group velocity to the wave
celerity and 6 is the angle the wave rays make with the x axis.
If the wave height and wave angle are known at each grid point, the radiation stresses
and their gradients are readily determined for use in the governing equations. To determine
the wave height and angle at each grid point Birkemeier and Dalrymple (1976), Ebersole
and Dalrymple (1979) and Vemulakonda (1984) each used the following scheme which is
based upon the irrotationality of the wave number and an equation for the conservation of
wave energy. The irrotationality of the wave number determines changes in the wave angle
due to refraction while the energy equation determines changes in the wave height due to
shoaling and refraction. This scheme was first introduced by Noda et al. (1974).
The irrotationality of the wave number
V x = 0 (2.31)
where
k = \k\ cos Oi + |fc| sin Oj
(2.32)
is expanded in Cartesian coordinates to obtain
d0 .36
cos 6-I- sin 6 + sin 0
ox ay
|fc| dx
cos 6
1*1 dy
(2.33)