268
6 Wave Transformation in Shallow Water
area, quasi-regular, almost monochromatic waves with a narrow frequency
spectrum and maxima at divisible frequencies are distinguished in the wave
field. If the angular energy distribution function is predominantly narrowed
due to wave refraction with decreasing depth, then the behaviour of the
frequency spectrum can be explained with the help of the non-linear wave
interaction.
6.3 Joint Influence of Bottom and a Non-Uniform
Current on Wave 'Iransformation
Wave transformation in a non-uniform current in deep water or separately in
shallow water without any current have been studied so far. A more general
solution describing the combined effect of an uneven depth and a horizontal
non-uniform current on waves is considered in this section. Among different
variants of the joint variations of water depth and current velocity the case of
depth and current, changing monotonically in the same direction, is chosen.
This situation can take place, for example, in channels. In particular, the case
of the deep water wave transformation in a current considered above follows
from the general problem solution. The second particular solution following
also from the general one and presenting another ultimate case describes
shallow water wave evolution without any current.
Spectral solution of the problem.
The problem formulation should
be considered in the local rectangular coordinate system { x, y, z}. Let the
Oz axis be directed vertically upwards, and the non-disturbed water surface
coincide with the plane xOy. Waves from an area with depth H0 and current
velocity V 0 (at x < 0) are assumed to propagate to the area with current
V and depth H (at x > 0). The velocity Vis assumed to be directed along
the Ox axis and to change monotonically in this direction: V = {V (x); 0}.
In accordance with the mass conservation law a relationship is considered to
exist between the variation of the horizontal current velocity V and the depth
H. This relationship is due to the water flow conservation law: V (x) H (x) =
VoHo. Thus, the depth H is changed in the same direction as the current
velocity (see Fig. 6.7).
The wave evolution problem will be considered within the framework
of the spectral equation of the wave action density balance (5.19)-(5.22).
Although an accurate solution for this equation set is obtained for deep water
only, a similar solution can be assumed to be valid also for the finite depth
water case. The solution can be considered to have the following form at least
for large values of the parameter q (q » 1):
{
No (ko, r, to)
N(k,r,t) =
N= (k)
when No< N=
when N 0 > N=,
(6.27a)
6 Wave Transformation in Shallow Water
area, quasi-regular, almost monochromatic waves with a narrow frequency
spectrum and maxima at divisible frequencies are distinguished in the wave
field. If the angular energy distribution function is predominantly narrowed
due to wave refraction with decreasing depth, then the behaviour of the
frequency spectrum can be explained with the help of the non-linear wave
interaction.
6.3 Joint Influence of Bottom and a Non-Uniform
Current on Wave 'Iransformation
Wave transformation in a non-uniform current in deep water or separately in
shallow water without any current have been studied so far. A more general
solution describing the combined effect of an uneven depth and a horizontal
non-uniform current on waves is considered in this section. Among different
variants of the joint variations of water depth and current velocity the case of
depth and current, changing monotonically in the same direction, is chosen.
This situation can take place, for example, in channels. In particular, the case
of the deep water wave transformation in a current considered above follows
from the general problem solution. The second particular solution following
also from the general one and presenting another ultimate case describes
shallow water wave evolution without any current.
Spectral solution of the problem.
The problem formulation should
be considered in the local rectangular coordinate system { x, y, z}. Let the
Oz axis be directed vertically upwards, and the non-disturbed water surface
coincide with the plane xOy. Waves from an area with depth H0 and current
velocity V 0 (at x < 0) are assumed to propagate to the area with current
V and depth H (at x > 0). The velocity Vis assumed to be directed along
the Ox axis and to change monotonically in this direction: V = {V (x); 0}.
In accordance with the mass conservation law a relationship is considered to
exist between the variation of the horizontal current velocity V and the depth
H. This relationship is due to the water flow conservation law: V (x) H (x) =
VoHo. Thus, the depth H is changed in the same direction as the current
velocity (see Fig. 6.7).
The wave evolution problem will be considered within the framework
of the spectral equation of the wave action density balance (5.19)-(5.22).
Although an accurate solution for this equation set is obtained for deep water
only, a similar solution can be assumed to be valid also for the finite depth
water case. The solution can be considered to have the following form at least
for large values of the parameter q (q » 1):
{
No (ko, r, to)
N(k,r,t) =
N= (k)
when No< N=
when N 0 > N=,
(6.27a)