Mechanics of Fluids

At a particular instant the crests a and b coincide, so the maximum combined
amplitude for the group is at this position. A little later the faster waves
(B, say) will have gained a distance δλ relative to the slower waves (A) and
then crests a ′ and b ′ will coincide. Since the relative velocity between trains
A and B is δc, this takes a time δλ/δc. However, while the point where crests
coincide has moved back a distance λ (relative to A), A itself has in this time
interval δλ/δc moved forward a distance c(δλ/δc). Thus the maximum combined
amplitude has moved forward a net distance cδλ/δc − λ. The velocity
with which it does so is therefore
cδλ/δc − λ
δλ/δc
δc
= c − λ
δλ
which, as δc and δλ both tend to zero, becomes
c − λ dc dc
= c + m
dλ dm
(10.63)
This is known as the group velocity cg, and clearly, unless the individual
wave velocity c is independent of the wavelength, the group velocity differs
from c.
Provided that there are only slight variations of c and λ, the result is true
for any number of waves of any type. But, in particular, eqn 10.63 shows
Oscillatory waves 475
Fig. 10.46