5.9 Vertical Non-Uniform Current Effect
229
Considering the ratios (5.110) and (5.111), one can expect a less significant
effect of velocity depth shear in shallow water compared to the deep water
case.
It is interesting to compare solutions of (5.109)-(5.111) with the case
of wave transformation in a vertically uniform current. The velocity of the
latter is assumed to be equal to the averaged (by depth) non-uniform current
velocity. Thus, this ratio can be written as:
Vs
Vm
1 flH
Co= Co+ 2 Co
(5.112)
It can then be derived from (5.111):
k _ { Vm
1 [(flH)
2
]
112
} -
1
- - -+- -
+4
ko
Co
2
Co
(5.113)
It can be seen from (5.113) that the increase of the modulus of the vorticity (with a constant averaged current velocity Vm) both in a fair and in
a countercurrent leads to a wavelength increase.
Now the determination of the wave height in a current will be considered. The adiabatic invariant conservation (see Chap. 3) for the current with
a linear velocity depth shear in the absence of wave energy dissipation can
be written in the following form:
: {_..!!_cg} =O,
ux arm
(5.114)
where E = (1/16)ph 2 (2g- flcs) (as/am) is the wave energy density; cs is
the phase velocity of waves relative to the surface current velocity; am, as are
the wave frequencies relative to the mean and surface velocities, respectively;
am = w - kVm, as = w - kVs, and Cg is the absolute value of the group
wave velocity. Unlike the equation of ordinary wave action conservation for
the case of a vertically uniform current, the equation (5.114) contains the
dependence of the vertical current vorticity.
The expression for the group velocity is obtained with the help of differentiation of the dispersion dependence (5.108) with respect to the wave
number:
C _ v; g(1 +E) - flEes
g-s+
2
n
cs,
g- J&CS
E= 2kH
sh(2kH)
(5.115)
According to (5.114), the wave height in the area with a current is determined by the ratio: