27
in a parabolic equation solution. Only the complex amplitude, which contains the mag
nitude and the phase of the wave, is obtained. Yans (1987) solution to this problem is
to obtain 0 directly from the complex amplitude by assuming that the slope of the water
surface resulting from the waves indicates the direction of wave propagation. This value
for 0 and twice the absolute value of the complex amplitude for the wave height are then
used in Eqs. (2.28-2.30) to obtain the radiation stresses. Hie value thus obtained for 0 is
then also used in the dispersion relationship. For plane waves this works fine; however, in
diffraction zones where short-crested wave are encountered problems may arise, since the
instantaneous slope of the wave at a particular location can be other than in the direction
of wave propagation.
In this report the radiation stresses are obtained directly from the complex amplitude
and its gradients using equations developed by Mei (1972) and repeated in his book (1982).
The basis of the derivation is to use the definition of the velocity potential to substitute for
the wave induced velocities in the definition for the radiation stresses.
For an inviscid incompressible fluid the instantaneous equations of motion are
+ = Vp pgez (2.44)
V q = 0 (2.45)
where q represents the velocity components, p is the pressure and ex is the z or vertical unit
vector. Denoting the horizontal components of q as ua,a = 1,2 and the vertical component
of q as u>, the boundary conditions are
= 0 (2.46)
= tv (2.47)
on z = rj, where r¡ is the instantaneous free surface, and the bottom boundary condition
dh
u, = -, (2.48)
q is assumed to be composed of a time averaged part q and a fluctuating part q'\ i.e.
dr)
dr]
dt +Up'dxp
f= fl*. ,*) + *(*. .*,*)
(2.49)