with slightly different wavelengths and periods. The équation of the
water surface is given by:
î? =
+ t?2
H
— cos
2
2nx 2nt\
A " Tj
H
2
(2-33)
where
and n2 are the contributions of each of the two components.
They may be summed since superposition of solutions is permissible when
linear wave theory is used. For simplicity, the heights of both wave
components hâve been assumed equal. Since the wavelengths of the two
component waves,
and L2, hâve been assumed slightly different, for
some values of x at a given time, the two components will be in phase
and the wave height observed will be 2H; for some other values of x,
the two waves will be complétély out of phase and the résultant wave
height will be zéro. The surface profile made up of the sum of the two
sinusoidal waves is given by Equation 2-33 and is shown in Figure 2-5.
The waves shown on Figure 2-5 appear to be traveling in groups described
by the équation of the envelope curves:
^envelope
± H cos
(2-34)
It is the speed of these groups, i.e. the velocity of propagation of
the envelope curves, that represents the group velocity. The limiting
speed of the wave groups as they become large, i.e., as the wavelength,
Lx, approaches L2 and consequently the wave period Ty approaches T2
is the group velocity and can be shown to be equal to:
c =----*
2 T
4rrd/L
sinh (47id/L)
(2-35)
= nC ,
where
1
4nd/L
sinh (4rrd/L)
In deep water, the term (4nd/L)/sinh(4ird/L) is approximately zéro and,
S
1 h.
2 T
1
= — CQ (deep water) ,
(2-36)
or the group velocity is
sinh(4ird/L) « 4ird/L and,
one-half the phase velocity. In shallow water,
L
C = -
g
T
C »
gd' (shallow water) t
(2-37)
2-25