28
Integrating a horizontal component of (2.44) vertically over the depth of the water column
from z = -h to z = r¡, using the boundary conditions (2.46-2.48) and then taking time
averages (the same procedure as in appendix A) the following definition of the radiation
stresses is obtained:
ro
Sap = p u'au'pdz + 6ap
J h
[i
Pdz~ +
h
+ y(i')
(2.50)
where the overbar is used to indicate the time averaged values and 6ap is the Kronicker
delta function
tap
for a = P
for a ^ f3
(2.51)
Mei (1972) was able to express Eq. (2.50) in terms of the general complex amplitude
function B by relating it to the velocity potential and the free surface height by
_ to cosh k[h0 + z) iut
a cosh kh
t) = B e~iut
(2.52)
(2.53)
An expression for p is obtained by integrating the vertical component of Eq. (2.44)
dw
dw
dw
dp
(2.54)
Integrating from any depth z to the surface, using Leibnitzs rule for interchanging the
order of differentiation and integration, employing the two surface boundary conditions,
Eqs. (2.46) and (2.47) and the continuity Eq. (2.45), Eq. (2.54) yields:
d fi d H
= pgin ~ z) +pJ wdz + pJ Upwdz pw]
(2.55)
Taking time averages gives
rfl du'-w'
p[x,y,z) = pain z) + pj* gX/9 dz~p)2 (2 56)
Substitution of Eqs. (2.52),(2.53) and (2.56) into (2.50) and making the assumption that
horizontal derivatives of k, h and a can be neglected (this is essentially a mild slope as
sumption), the following is obtained after straightforward but tedious calculations which