5.7 Wind Wave Transformation in Cross-Velocity Shear Current
211
The solution will be transformed into a more general form by introducing
the following non-dimensional parameters: a = kmHo is the initial effective
depth; vo = Vo Jk;:l9 is the initial effective current velocity; p, = Vmax/Vo is
the maximum current velocity; 1 = H0 / His the ratio of initial depth to the
given value at the point considered; A = V /Vo is the relative current value;
and .X1 = Vl/Vo (where Vi is the velocity at the last point of the current
profile).
The value f = ko/k is defined as the solution of (5.66). Proceeding to
the new variable fP = k/km and non-dimensional parameters, the spectrum
F (y, (3) can be written in the form:
th (fj 2 noh) !- nt5 y--(n+2)
F (Y, (3) = Q (f3o) mo (n + 1) th (fy2ao)
x exp [- n: 1 (J.Y2) -~] '
(5.84)
where Q(f3o) = 3 ~cos 4 ((30 -f38)6>(1-sin 2 (f3)), and 6> is the Heaviside
function.
The aforementioned kinematic conditions allow the solution for the described cases to be written in the following form.
The solution within the first quadrant (0 < (3 < n/2) can be written:
at the point A (see Fig. 5.20) as:
FA=F(Y,(3);
(5.85)
at the point B :
FB = F (iJ, (3) 6> [ vth (fj 2 noh)
- Jth {Y 2 noho sin ((3)) sin ((3) - vo (p,- .X) sin ((3) J .
(5.86)
Within the second quadrant (n/2 < (3 < n) it is defined at the point A
as:
FA = F (Y, (3- n) 6> [ Jth (Y 2 aoho sin ((3)) sin ((3)
- Jth (y 2 noh) + vo (p,- .X) sin ((3) J ;
at the point B:
(5.87)
(5.88)
Within the third quadrant (n < (3 < 3n/2) the solution is written at the
point A as: