228
5 Wave Evolution in Non-uniform Currents in Deep Water
1.6
1.4
f.2
0.8
0.6
0.2
O.f
0.3
0.5
-0.5
Contrary
current
Fig. 5.35. Wavelength variation in current for different vorticity
After the equation (5.109) is derived in its transcendental form, it should
be analysed in its ultimate cases, obtaining the solution to the problem in
explicit form.
In the case of deep water the solution of (5.109) can be written in the
form:
k
ko
2Ya + Ya__Q_ + 1 ± [(Ya__Q_ + 1)
2
+4Th]
Co
Co koCo
Co koCo
Co
1/2 '
(5.110)
where the(±) sign in the denominator indicates the ambiguity of determining
the wave number in a current, dependent on its initial value k0 and the current
velocity parameters. Similarly to the case of the vertically uniform current,
the ( +) sign can be referred to direct waves and the (-) sign to reverse waves
reflected by the horizontal non-uniform current.
The direct wave dependence is presented in Fig. 5.35 in accordance with
(5.110) for the different values of non-dimensional vorticity fl/(k0C0 ). The
effect of current vorticity on the wavelength transformation is observed.
As for the shallow water case the following solution of (5.109) can be
obtained:
!5._ = { Vs _ ~ flH + ~ [(flH) 2 + 4 ] 112 }-l
ko
Co
2 Co
2
Co
(5.111)