6.3 Bottom and a Non-Uniform Current Influence on Waves
271
the expression for the frequency can be written as follows:
(kH) 2
w = ky9ii- - 6 -ky9ii- Vkcos(,B) .
(6.29)
The designation k = kH, V = V cos ({3) / ygH, w = w ..fil79 will be introduced and the ratio for the non-dimensional frequency w can be presented
as:
-
- (
-)
1-3
w = k 1- v - 6k .
(6.30)
The dispersion curves w ( k, V) for different values of the parameter V
(i.e. current velocities for a fixed depth) are shown in Fig. 6.8. The curves
w ( k, V) can be interpreted as the evolution of wave packet parameters with
its propagation in a horizontal non-uniform current. As can be seen (see
Fig. 6.8), several values of the wave number k can correspond to the same
frequency. Thus, there are three values of the wave number k (for V < 1)
having the frequency w with two of them (for w > 0) being positive and one
negative. The latter should not be considered, as it makes no physical sense
and appears as a root of the dispersion equation expansion. Two other values
can be determined using a solution of the cubic equation (6.30), being equal
to:
where
-
u+v u-v. r;;
k1 = - - - + --IV3.
2
2
'
k - _ u+v u-v. r;; 3
2 - - -
2 - - - 2
- l V . l ,
u= {j-3w+J(3w) 2 - [2(1-v)r;
v= V-3w-J(3w) 2 - [2(1-v)r
(6.31)
The values k1 and k2 are merged into one at the point B, in which a local
extreme value for the function w ( k) is achieved at k = J 2 ( 1 - V) . At
this point, the following ratio is valid: w = k 3 /3; VB = ..,jjjH [ 1 - ~ ~] .
The group wave velocity Cgx = ..,jjjH [ 1 - ~ (kH) 2 - V J becomes zero. If the
current velocity were greater than the value VB, there would be no waves at
this point (see Fig. 6.8, curve for V = 0.99).
The point B is a blocking point. If the wave packet propagating opposite
to the current and arriving at the blocking point (see Fig. 6.8, curve V < 0.5)