R
1
2
^ ga^2
2
c
^1 (14.24)
or
RECg
where
Cg
2
c
1 . (14.25)
Cgis called the group velocity. It can be seen from equation (14.25) that in
deep water (kd→∞), the group velocity Cg→c/2, and that in shallow water
(kd→0),Cg→c.
14.2.6 Waves on currents
The properties of waves in the presence of currents are different from
those in still water. A frame of reference moving with the component of
current velocity Vin the direction of wave propagation is chosen; the
problem is then reduced to wave propagation appropriate to still water. In
the following equations, subscripts aandrdenote the absolute and moving
frames of reference.
ca crV
LfaLfrV.
From equation (14.13)
cr^2
k
g
tanh(kd)
14.3 Range of validity of linear theory
For engineering purposes it is important to establish when linear theory
ceases to be applicable, and when it becomes necessary to apply non-linear
deep- and shallow-water wave theories, as forces acting on structures cal-
culated from linear wave theory can be underestimated.
The ranges over which linear and non-linear theories are applicable
are presented in terms of H/dandd/L(Komar, 1976). The lines that delin-
2 kd
sinh(2kd)
2 kd
sinh(2kd)
584 WAVES AND OFFSHORE ENGINEERING
