3.4 Basic Waves Characteristics
87
Group velocity is a quantity not as easily measured as the phase velocity.
To understand the relationship between phase velocity and group velocity, let
us examine a band of ripples created by tossing a stone into a still pond. The
band gets wider with increasing distance from the original disturbance. Ripples
of greater wavelength progressively travel faster than shorter ones due to the
dispersion effect. More careful observation shows that each individual ripple
travels faster than the band of ripples. The velocity of the band is called the
group velocity, which is about half the phase velocity of the individual ripples
which travel through that band. In order to justify this relation, we will follow
arguments by Barber (1969), and discuss an idealized example of two waves of
the same amplitude but with a relatively small difference in wavelength, say
6.L (see Fig. 3.6a). Superposition of these two waves results in group wave
pattern as shown in Fig. 3.6b.
If we look at two wave crests immediately following each other, one lags by a
distance L, the other by a distance L + 6.L, so that the crest of the longer train
is a distance 6.L behind the crest of the other. The train of longer wavelength
travels at a different speed, say C + 6.C, instead of C. The lag, 6.L, is made
up in a short time 6.t:
6.t = 6.L.
6.C
(3.8)
Both wave trains move forward during this time and the place where the crests
coincide moves back, relative to them, by one whole wavelength. The highest
wave is located in the middle of the wave group, which advances a shorter
distance than the waves themselves. This distance is:
6.x = C 6.t - L.
(3.9)
The first term is the distance travelled by the wave crest during the time 6.t and
the second term is the one wavelength the group has dropped back. Therefore,
the velocity of the wave group becomes:
Cg = ~: = C - ~t = C - L (~~).
For small 6.C and 6.L, Eq. (3.10) becomes:
dC
C =C-L-.
9
dL
(3.10)
(3.11)
Therefore, differentiating and substituting into Eq. (3.11) we finally obtain:
1
Cg =-C
2 '
(3.12)
showing that the groups travel at only half of the speed of the individual waves.
87
Group velocity is a quantity not as easily measured as the phase velocity.
To understand the relationship between phase velocity and group velocity, let
us examine a band of ripples created by tossing a stone into a still pond. The
band gets wider with increasing distance from the original disturbance. Ripples
of greater wavelength progressively travel faster than shorter ones due to the
dispersion effect. More careful observation shows that each individual ripple
travels faster than the band of ripples. The velocity of the band is called the
group velocity, which is about half the phase velocity of the individual ripples
which travel through that band. In order to justify this relation, we will follow
arguments by Barber (1969), and discuss an idealized example of two waves of
the same amplitude but with a relatively small difference in wavelength, say
6.L (see Fig. 3.6a). Superposition of these two waves results in group wave
pattern as shown in Fig. 3.6b.
If we look at two wave crests immediately following each other, one lags by a
distance L, the other by a distance L + 6.L, so that the crest of the longer train
is a distance 6.L behind the crest of the other. The train of longer wavelength
travels at a different speed, say C + 6.C, instead of C. The lag, 6.L, is made
up in a short time 6.t:
6.t = 6.L.
6.C
(3.8)
Both wave trains move forward during this time and the place where the crests
coincide moves back, relative to them, by one whole wavelength. The highest
wave is located in the middle of the wave group, which advances a shorter
distance than the waves themselves. This distance is:
6.x = C 6.t - L.
(3.9)
The first term is the distance travelled by the wave crest during the time 6.t and
the second term is the one wavelength the group has dropped back. Therefore,
the velocity of the wave group becomes:
Cg = ~: = C - ~t = C - L (~~).
For small 6.C and 6.L, Eq. (3.10) becomes:
dC
C =C-L-.
9
dL
(3.10)
(3.11)
Therefore, differentiating and substituting into Eq. (3.11) we finally obtain:
1
Cg =-C
2 '
(3.12)
showing that the groups travel at only half of the speed of the individual waves.