5. 7 Wind Wave Transformation in Cross-Velocity Shear Current
203
The current velocity V* at the turning point can be written as:
*
w- J gky th(kyH)
v =
k
.
y
(5. 73)
In case of monotonic depth variations, it is necessary to prescribe the dependence V = V(x) and H = H(x) or V = V(H), and then to solve the transcendental equation relative to the value V* (or H*): J gky th ( kyH ( x*)) +
V (x*) = w. For the existence of the reflected wave solution, the equation x*
should be found within the range x < x* < Xm·
It should be emphasized that a necessary condition for the existence of
wave components in quadrant II is their presence in quadrant I. The former
originate from the latter, as a result of their reflection from the current. In
this case the angle turn takes place, i.e. an angle f3 is changed to the value
7t- {3. The corresponding spectrum can be written as follows:
F1 1 =F(k,n-{3)
X e [ Wm - V) k sin (/3) + J gky th (kyH) - J gk th (kH)] (5.74)
Spectral components should always exist in the quadrant IV in case they
are present at the initial boundary:
(5. 75)
The angular spectrum distribution of the these components becomes narrower due to the influence of current and shallow water (5. 70).
In quadrant III, it is possible to observe spectral components if wave
reflection takes place in the area x > Xm, when the corresponding difference of
the current velocity values is present. The kinematic condition of the existence
of waves in this quadrant is written as follows:
V1 < V* < Vo,
(5.76)
where V1 is the current velocity at the right boundary point of the current
profile. The wave reflection can be increased in the presence of a local coastal
countercurrent (V1 < 0). It should be remembered that the spectral components in quadrant III arise from the components in quadrant IV due to their
reflection. The wave spectrum in this quadrant can be presented as follows:
F1rr = F (k, 1t- {3) 8 [ J gk th (kH) - J gky th (kyH) + (V- Vo) k sin(/3)]
X e [ J gky th (kyH)- J gkth (kH)- (V- Vl) k sin(/3)] . (5.77)
It can be shown that the first of the Heavyside functions 8 is equal to one
for the given spectrum in the current.
203
The current velocity V* at the turning point can be written as:
*
w- J gky th(kyH)
v =
k
.
y
(5. 73)
In case of monotonic depth variations, it is necessary to prescribe the dependence V = V(x) and H = H(x) or V = V(H), and then to solve the transcendental equation relative to the value V* (or H*): J gky th ( kyH ( x*)) +
V (x*) = w. For the existence of the reflected wave solution, the equation x*
should be found within the range x < x* < Xm·
It should be emphasized that a necessary condition for the existence of
wave components in quadrant II is their presence in quadrant I. The former
originate from the latter, as a result of their reflection from the current. In
this case the angle turn takes place, i.e. an angle f3 is changed to the value
7t- {3. The corresponding spectrum can be written as follows:
F1 1 =F(k,n-{3)
X e [ Wm - V) k sin (/3) + J gky th (kyH) - J gk th (kH)] (5.74)
Spectral components should always exist in the quadrant IV in case they
are present at the initial boundary:
(5. 75)
The angular spectrum distribution of the these components becomes narrower due to the influence of current and shallow water (5. 70).
In quadrant III, it is possible to observe spectral components if wave
reflection takes place in the area x > Xm, when the corresponding difference of
the current velocity values is present. The kinematic condition of the existence
of waves in this quadrant is written as follows:
V1 < V* < Vo,
(5.76)
where V1 is the current velocity at the right boundary point of the current
profile. The wave reflection can be increased in the presence of a local coastal
countercurrent (V1 < 0). It should be remembered that the spectral components in quadrant III arise from the components in quadrant IV due to their
reflection. The wave spectrum in this quadrant can be presented as follows:
F1rr = F (k, 1t- {3) 8 [ J gk th (kH) - J gky th (kyH) + (V- Vo) k sin(/3)]
X e [ J gky th (kyH)- J gkth (kH)- (V- Vl) k sin(/3)] . (5.77)
It can be shown that the first of the Heavyside functions 8 is equal to one
for the given spectrum in the current.