5.5 Wave Element Transformations in Current, Varying Along its Direction
189
There is a second singularity (as V--+ 0) in the solution with the reverse
wave heights, attaining an infinitely great value, with their lengths tending
to zero. The spectral approach implementation to the solution does not eliminate this singularity as soon as it occurs for all reverse waves at one point,
when at V = 0. The wave height cannot be infinitely increased in a current,
because the waves are to be broken down. The breaking can be taken into account indirectly, assuming that the wave spectrum does not exceed the wave
spectrum equilibrium interval. This condition is fulfilled with the wave transformation in a current taking place sufficiently slowly compared with the time
of the establishment of the wave spectrum under breaking. As mentioned in
Sects. 5.3-5.4, the wave breaking solution (5.33) can be described, assuming
the existence of the equilibrium interval. Its expression can be written as:
-
(3v4
Foo(Y) = 2--5 4 Q(f3)'
g Y (}"max
(5.44)
where
and 8 = ~ is the mean initial wave steepness. Now the wave spectrum
solution is also dependent on the parameter 8, i.e. on the mean steepness of
the initial waves.
Taking the reverse waves into account the area of integration over the
variables {y, (3} is changed, depending on the parameter 1 (r < 1). The
integration area is increased with decreasing 1 from one to smaller values
(see Fig. 5.10) and the mean wave parameters are changed accordingly. As
shown by numerical calculations, changes in the wave element practically
stop at 1 < 0.67, if the current velocity is V < 0.67Vrnax at the given point;
the mean wave elements are not effected by the point Vrnax in the current
velocity profile. The relative contribution of reverse waves is decreased with
increasing the parameter 1 from 0.667 to 1.00. At the same time the change
of mean parameters hjh0 , A.j)..0 and T/To approaches the earlier described
ones at 1 = 1.
The results of calculations of the mean wave height and mean wavelength
transformation with n = 5 at 1 = 0.67 in increasing countercurrent are
shown in Fig. 5.16. The nomograms of the element changes are given for
various mean initial wave steepness. Thus, the wave height increases to the
maximum value and then decreases similarly to the previous case. The maximum wave heights increase with decreasing steepness 8, i.e. the smaller the
initial steepness, the smaller is the probability of wave breaking, and a larger
relative increase of the wave heights is observed.
It should be noted that due to the presence of reverse waves, the mean
heights in the transition current area (0 < 1 < 1) can significantly exceed
the wave heights in the maximum current velocity area ( 1 = 1). At first
189
There is a second singularity (as V--+ 0) in the solution with the reverse
wave heights, attaining an infinitely great value, with their lengths tending
to zero. The spectral approach implementation to the solution does not eliminate this singularity as soon as it occurs for all reverse waves at one point,
when at V = 0. The wave height cannot be infinitely increased in a current,
because the waves are to be broken down. The breaking can be taken into account indirectly, assuming that the wave spectrum does not exceed the wave
spectrum equilibrium interval. This condition is fulfilled with the wave transformation in a current taking place sufficiently slowly compared with the time
of the establishment of the wave spectrum under breaking. As mentioned in
Sects. 5.3-5.4, the wave breaking solution (5.33) can be described, assuming
the existence of the equilibrium interval. Its expression can be written as:
-
(3v4
Foo(Y) = 2--5 4 Q(f3)'
g Y (}"max
(5.44)
where
and 8 = ~ is the mean initial wave steepness. Now the wave spectrum
solution is also dependent on the parameter 8, i.e. on the mean steepness of
the initial waves.
Taking the reverse waves into account the area of integration over the
variables {y, (3} is changed, depending on the parameter 1 (r < 1). The
integration area is increased with decreasing 1 from one to smaller values
(see Fig. 5.10) and the mean wave parameters are changed accordingly. As
shown by numerical calculations, changes in the wave element practically
stop at 1 < 0.67, if the current velocity is V < 0.67Vrnax at the given point;
the mean wave elements are not effected by the point Vrnax in the current
velocity profile. The relative contribution of reverse waves is decreased with
increasing the parameter 1 from 0.667 to 1.00. At the same time the change
of mean parameters hjh0 , A.j)..0 and T/To approaches the earlier described
ones at 1 = 1.
The results of calculations of the mean wave height and mean wavelength
transformation with n = 5 at 1 = 0.67 in increasing countercurrent are
shown in Fig. 5.16. The nomograms of the element changes are given for
various mean initial wave steepness. Thus, the wave height increases to the
maximum value and then decreases similarly to the previous case. The maximum wave heights increase with decreasing steepness 8, i.e. the smaller the
initial steepness, the smaller is the probability of wave breaking, and a larger
relative increase of the wave heights is observed.
It should be noted that due to the presence of reverse waves, the mean
heights in the transition current area (0 < 1 < 1) can significantly exceed
the wave heights in the maximum current velocity area ( 1 = 1). At first