6.3 Bottom and a Non-Uniform Current Influence on Waves
281
The values of wave elements are shown in Fig. 6.12b for the parameter
ii = 5. It should be noted that waves at the initial point can be considered to
be short in comparison with the depth, but the situation can be changed with
wave propagation. This situation occurs in the case of a wave propagating
from deep to shallow water in a current with "' > 1. It should be remembered
that a similar case is considered in the first test calculation, but without
any current. This allows revealing the effect of decreasing depth in its "pure
form". At the initial stage of wave propagation, with the current velocity
being small (v = 10- 3 , see Fig. 6.12b), and the depth variations being important, the wave element transformation is in agreement with ordinary wave
presentations in a coastal area. At first, the wave heights are decreased, and
increased later on. At this stage the periods are not practically changed and
the lengths are decreased. When the current velocity is large, and the wave
velocities are small due to decreasing depth, a decrease of the wave heights
takes place again. For small depths ("! ~ 100), with the prevailing current
effect, a gradual increase of wavelengths is observed (5. rv "!) and the waves
become flatter. With increasing the parameter v, the wave transformation
becomes more influenced by current velocity variations along the flow compared to depth variations. A monotonic height decrease and length increase
are observed.
The graphs of wave element variations with initial basin depth being
less than the wavelength (ii = 1) are presented in Fig. 6.12c. If the initial
current is negligible (v = 10- 3 ), a depth decrease results at first in the wave
height increase and in the wavelength decrease. There is no initial decrease
of the height, as in case of a wave propagating from deep to shallow water
(Fig. 6.12a). The variations of wave heights in shallow water are estimated
using the Green formula: h "' {t'f. However, later on ("! > 15), when the
wavelength becomes shorter and the current velocity is comparable with the
velocity of wave propagation, the role of the current becomes decisive. The
wave height decreases with increasing current velocity. With further depth
decrease and current velocity increase, the wavelength increases ("! > 50). If
the initial current velocity is v = 10- 1 , the monotonic height h decrease and
length 5. increase are observed with increasing current velocity.
The wave element transformations for different initial general directions
show that the described tendencies become weaker with increase of the angle /38 (i.e. the angle between the general wave direction and the current
velocity). At the same time the opposite situation can be revealed at small
depth.
Now, the wave behaviour in a countercurrent will be considered. Two situations can be identified: increasing and decreasing currents (see Fig. 6.13).
It should be noted that in the case of an increasing countercurrent, reverse
waves carried backwards by the flow can appear in a basin area with changing
current velocity. The wave description is limited by consideration of decreasing and increasing countercurrent, excluding the part where the gradients of
current velocity are different from zero.
281
The values of wave elements are shown in Fig. 6.12b for the parameter
ii = 5. It should be noted that waves at the initial point can be considered to
be short in comparison with the depth, but the situation can be changed with
wave propagation. This situation occurs in the case of a wave propagating
from deep to shallow water in a current with "' > 1. It should be remembered
that a similar case is considered in the first test calculation, but without
any current. This allows revealing the effect of decreasing depth in its "pure
form". At the initial stage of wave propagation, with the current velocity
being small (v = 10- 3 , see Fig. 6.12b), and the depth variations being important, the wave element transformation is in agreement with ordinary wave
presentations in a coastal area. At first, the wave heights are decreased, and
increased later on. At this stage the periods are not practically changed and
the lengths are decreased. When the current velocity is large, and the wave
velocities are small due to decreasing depth, a decrease of the wave heights
takes place again. For small depths ("! ~ 100), with the prevailing current
effect, a gradual increase of wavelengths is observed (5. rv "!) and the waves
become flatter. With increasing the parameter v, the wave transformation
becomes more influenced by current velocity variations along the flow compared to depth variations. A monotonic height decrease and length increase
are observed.
The graphs of wave element variations with initial basin depth being
less than the wavelength (ii = 1) are presented in Fig. 6.12c. If the initial
current is negligible (v = 10- 3 ), a depth decrease results at first in the wave
height increase and in the wavelength decrease. There is no initial decrease
of the height, as in case of a wave propagating from deep to shallow water
(Fig. 6.12a). The variations of wave heights in shallow water are estimated
using the Green formula: h "' {t'f. However, later on ("! > 15), when the
wavelength becomes shorter and the current velocity is comparable with the
velocity of wave propagation, the role of the current becomes decisive. The
wave height decreases with increasing current velocity. With further depth
decrease and current velocity increase, the wavelength increases ("! > 50). If
the initial current velocity is v = 10- 1 , the monotonic height h decrease and
length 5. increase are observed with increasing current velocity.
The wave element transformations for different initial general directions
show that the described tendencies become weaker with increase of the angle /38 (i.e. the angle between the general wave direction and the current
velocity). At the same time the opposite situation can be revealed at small
depth.
Now, the wave behaviour in a countercurrent will be considered. Two situations can be identified: increasing and decreasing currents (see Fig. 6.13).
It should be noted that in the case of an increasing countercurrent, reverse
waves carried backwards by the flow can appear in a basin area with changing
current velocity. The wave description is limited by consideration of decreasing and increasing countercurrent, excluding the part where the gradients of
current velocity are different from zero.