184
5 Wave Evolution in Non-uniform Currents in Deep Water
where h0 is the initial wave height; h is the wave height in the current V; and
c and eo are the corresponding wave phase velocities. Using the frequency
preservation conditions, valid for the stationary current velocity, the phase
velocity c can be written in the following form:
~ = ~ [1 ± y'I + 4Vfeo].
co
2
(5.38)
The (±) sign in this relation indicates the ambiguity of determining
the wave parameters in the current V. The variations of the height ratio
h/ho and the length ratio >..j>..0 are shown in Fig. 5.14. The change of wave
parameters corresponding to the ( +) sign is designated as J1. and to the
(-) sign as J2 . The evolution of the so-called straight wave with height h ( +),
increasing monotonically with countercurrent velocity, is described by the
J
I
I
\
Jf
50 \
\
\
\
20
\
\ \
tO - - - . - - - - - -
'
I
5
..... ,_J!....,"'
0.2
0.1
0.05
0.02
Fig. 5.14. Variation of height h/ho and length >-./>-.o of a monochromatic wave
depending on the counterflow velocity: J1 is the transformation of straight wave
components; h is the transformation of reversed wave components; V /eo is the
current velocity normalized by the initial phase speed