230
5 Wave Evolution in Non-uniform Currents in Deep Water
h
as
Cg 0
1
(
) 2
(
) (
)
ho
= ao
Cg
1 - ! qcs ·
(5.116)
The expression (5.116) for ultimate cases will be analysed. The following
equation is obtained for the deep water case:
( h) 2 1{Vs [([})2 kl- 1 1 2 }- 1
-
- - 2-+2
-
+4ho
- 2
co
koco
ko
X [
(5.117)
There is the following solution for shallow water:
(~)' ~H~)' [ (n:)' +4~~]
+~H~)'n: [J(n:)' +4~n:Jr. (5.118)
It should be noted that in the case of a current velocity with zero shear
[} = 0, the well-known solutions for wave transformation by a depth uniform
current are derived from (5.117) and (5.118).
Wave blocking in a current with linear depth velocity profile. The
wave blocking kinematics by a countercurrent with a non-uniform depth velocity profile will be considered only for the one-dimensional case. The blocking condition can be written in the form:
ow
Cg = ok = o.
(5.119)
The wave blocking condition for the finite depth case can be derived as:
Vs = ~ flch- 2 (kH)
- [~n 2 H th(kH) ch- 2 (kH) + gth(kH) + gkH ch- 2 (kH)]
x [n 2 th 2 (kH) +4gkth(kH)r 112 .
(5.120)