116
term will be proportional to the depth. This would mean that deep water non-shoaling and
non-dissipative waves in varying depths would produce gradients in the radiation stresses.
For plane waves this is not a problem, since the complex amplitude terms are easily shown
to be zero. This is seen by substitution of B = ae,kx and B* = ae~tkx where a = |B|.
The fourth term is straight forward
igk psinh k(h0 + z) _iut igk sinh k(h0 + z) ,
cosh kh
+ B'
+z) B.^, B.e,^
4 a2 cosh2 kh >
7-7T2 1 g2k2 sinh2 k(h + z) n gk sinh2 k{h0 + z)
( ] ~4 a2 cosh2 k[h + rj) ~ sinh2Jkh BB
(B.21)
(B.22)
(B.23)
- p(w')2dz = -pgk\B\2
* J ftp
2 sinh2 k(h0 + z)
sinh2h ~ 4 1 1 V* sinh21fc/i>
The fifth term needs no evaluation and is seen to cancel exactly the second term.
The sixth and last term of Eq. (B.6) is r¡')2 and is evaluated easily
r¡' = 1 + BV")
(rj')2 = I (Be-W + 2BB* + B*e2,wt)
SI
W = ~2BB'= 2
which cancels part of the result from the fourth term.
Adding the above results for all the terms of Eq. (B.6) yields
P91
(B.25)
(B.26)
(B.27)
(B.28)
c -P9~(9BdB*\ 1 ^ 2kh \
4 I \5ia dip y A:2 V sinh2kh)
+
^af)
I |22kh
1 1 sinh 2kh
dxa dip
(|Vfc£|2 k2\B\2)^(2khcoth 2kh 1)] |
(B.29)