214
5 Wave Evolution in Non-uniform Currents in Deep Water
The expression (5.93) can reduced to the following form, omitting intermediate calculations:
cos (f38) (1- dV fij- sin (f38))
4
h2=-r============V==9==========
( 1 - d V ~sin (f38))
4 - sin 2 ({38)
(5.94)
The wave height evolution in a non-uniform current is described by the
relation (5.94), depending on the velocity difference d V, the initial value of
the wave number k0 and the angle {30 . As can be seen, this expression coincides
with the analogous one obtained using monochromatic classical wave theory
(Peregrine, 1976). There is a singularity, arising in the ratio (5.94) at the point
in which the denominator becomes zero. At this point the current velocity
variation is equal to:
dV = {I_1- y'sin({30 )
V k~ sin (f3o) ·
(5.95)
This is a turning point of the wave packet propagation, as soon as the
group velocity component Cgx = ~ .JI ' 7;- becomes equal to zero, i.e. kx = 0.
As shown, this singularity is smoothed by the finite width spectrum, producing a finite value of the spectral average statistical wave height.
But in the case of a monochromatic wave it is necessary to obtain a more
precise description of the solution. That is why a wave field should be considered in detail in a shear current around the turning point. It follows from the
general kinematic relations that the wave packet trajectory can be written
in the form (5.61) and (5.62). The component of the group velocity is equal
to zero: Cgx = 0 at the turning point {3 = n/2 and wave packet propagation
occurs in the opposite direction along the Ox axis. In order to obtain a wave
pattern near the caustic the asymptotic method, described in Sect. 5.6, can
be applied again.
To describe the waves in a shear current with the help of the general
integral presentation (5.50), it is sufficient to use the following kinematic
relations:
w = F (k~, kx, x) = const;
(5.96)
In the vicinity of the turning point the component of the wave vector
is estimated as: kx = ±y'K~ (x*- x), where the (+) sign is referred to the
direct wave and the (-) sign to the reflected one. At the turning point x = x*
the following values are defined: K* = ~ ( ~~ {j:l) -l~x=x*, Cgx = 0, kx =
k; = 0, a* fao = Jkx/ko. Substitution of these relations into the asymptotic
expression (5.50) results in the following free surface elevation:
5 Wave Evolution in Non-uniform Currents in Deep Water
The expression (5.93) can reduced to the following form, omitting intermediate calculations:
cos (f38) (1- dV fij- sin (f38))
4
h2=-r============V==9==========
( 1 - d V ~sin (f38))
4 - sin 2 ({38)
(5.94)
The wave height evolution in a non-uniform current is described by the
relation (5.94), depending on the velocity difference d V, the initial value of
the wave number k0 and the angle {30 . As can be seen, this expression coincides
with the analogous one obtained using monochromatic classical wave theory
(Peregrine, 1976). There is a singularity, arising in the ratio (5.94) at the point
in which the denominator becomes zero. At this point the current velocity
variation is equal to:
dV = {I_1- y'sin({30 )
V k~ sin (f3o) ·
(5.95)
This is a turning point of the wave packet propagation, as soon as the
group velocity component Cgx = ~ .JI ' 7;- becomes equal to zero, i.e. kx = 0.
As shown, this singularity is smoothed by the finite width spectrum, producing a finite value of the spectral average statistical wave height.
But in the case of a monochromatic wave it is necessary to obtain a more
precise description of the solution. That is why a wave field should be considered in detail in a shear current around the turning point. It follows from the
general kinematic relations that the wave packet trajectory can be written
in the form (5.61) and (5.62). The component of the group velocity is equal
to zero: Cgx = 0 at the turning point {3 = n/2 and wave packet propagation
occurs in the opposite direction along the Ox axis. In order to obtain a wave
pattern near the caustic the asymptotic method, described in Sect. 5.6, can
be applied again.
To describe the waves in a shear current with the help of the general
integral presentation (5.50), it is sufficient to use the following kinematic
relations:
w = F (k~, kx, x) = const;
(5.96)
In the vicinity of the turning point the component of the wave vector
is estimated as: kx = ±y'K~ (x*- x), where the (+) sign is referred to the
direct wave and the (-) sign to the reflected one. At the turning point x = x*
the following values are defined: K* = ~ ( ~~ {j:l) -l~x=x*, Cgx = 0, kx =
k; = 0, a* fao = Jkx/ko. Substitution of these relations into the asymptotic
expression (5.50) results in the following free surface elevation: