Phase velocity and group velocity
(c) Zhengqing Yun, 2011-2012
Objective: Observe the difference between phase and group velocity; understand that the
group velocity can be less than, equal to, and greater than the phase velocity and can
exceed the speed of light; the group velocity can even be negative!
1. Phase velocity and group velocity
For a harmonic plane wave, e.g., E( x, t )  A cos( x  t ) , the phase velocity is defined as
vp 

.

This is obtained by assuming the phase is constant, i.e., x  t  C , and we take
differentiation of this equation, obtaining dx  dt  0 . Thus we have
dx 
which is the

dt 
phase velocity.
When two harmonic plane waves with slightly different frequencies are added, a resultant
wave can be represented by, assuming they have the same amplitude, E0,
E( x, t )  E1 ( x, t )  E2 ( x, t )  2E0 cos(   x    t ) cos(   x    t ) .
(1)
where    12 (1   2 ) and    12 (1  2 ) . The two cosine functions in (1) are both
traveling waves and we can define their phase velocities as
v p 



v

and
.
p


(2)
Fig. 1 shows the two original plane waves (green and pink), the two traveling wave in (1)
(blue and red), and the resultant wave (black), i.e., E(x, t) in (1).
Note that v p 

is the phase velocity corresponding to the envelope (modulating) curve

in Fig. 1 (blue); while v p 

is the phase velocity corresponding to the carrier wave

(red). Since the envelope curve encloses a group of wavelets, v p 
velocity which may travel at a different velocity from v p 

is called the group


. More generally, the group

velocity can be defined as
1
vg 
cos(   x   t )
E2
d
d
.
cos(   x   t )
(3)
E = E1+E2
E1
Fig. 1. Two plane waves and the resultant wave (E0=1/2)
It is known that the phase velocity can be greater than the speed of light in free space
without violating the rule of Einstein’s theory of relativity: ‘nothing can travel at a speed
greater than light.’ The reason is that the phase velocity does not represent the signal or
energy velocity.
The group velocity, on the other hand, was used to represent the energy velocity which
should not exceed the speed of light. Recently, lab experiments have shown that the
group velocity can be less than, equal to, and greater than the speed of light, and group
velocity can even be negative! Reasons to these ‘strange’ observations have been
discussed and it is found that the energy is still traveling at a speed less than the light
speed. So Einstein’s theory is still valid.
This lab does not try to explain these advanced problems in physics. Instead, we plan to
observe the phase and group velocity based on the addition of two harmonic waves and
draw some conclusions about the conditions which make the group velocity less than,
equal to, greater than, and opposite to the phase velocity.
2. Lab software
In this lab, we explore under what conditions the group velocity can be less than, equal to,
and greater than the speed of light, and even be negative!
We only observe the group velocity vs. phase velocity defined in (2) and (3). Fig. 2
shows the starting screen shot of the software. The lower part shows the two waves
(purple and green curves; at beginning only the purple curve can be seen since these two
curves are overlapped). The upper part (the white curve) is the sum of the two waves in
the lower part. There are five controllers you can use. The first one is the red sphere
2
which can be dragged on the slider to change the relative amplitudes of the two harmonic
plane waves; the second is the yellow sphere which is used to change the relative
frequencies of the two plane waves; the third is the blue sphere used to change the
relative phase velocities; the fourth is the white sphere which, when right clicked, will
turn on/off the envelope; the last one is the green one which controls the animation speed.
You can click inside the window to start or stop the animation. Use right-click on the
white sphere to toggle the envelope on/off.
Fig. 2. The starting screen of the software.
Fig. 3 is a screen shot when various parameters including the amplitudes, the frequencies,
and the velocities are set, and the envelope is turned on.
3
Resultant wave: E = E1+E2
Envelopes
E2
E1
Fig. 3. A screen shot when parameters are set and the envelope is turned on.
3. Exercises
In the following, the group velocity is the velocity of the envelope(s); the phase velocity
is the velocity of the resultant wave (white curve); the parameters include amplitude, the
frequency, and the velocity for the original two waves, E1 and E2. Note that these
parameters are expressed in a relative manner, i.e., E02 in terms of E01, f2 in terms of f1,
and v2 in terms of v1. Also when we say a velocity is positive, we mean that the
corresponding wave is moving from left to right; when a velocity is negative, the wave is
moving from right to left.
Exercise 3.1. Adjust the parameters E02, f2, v2 so that the group velocity is positive and
less than the phase velocity. Record these parameters. (Note: These parameters are not
unique. Just list the parameters you find.)
Exercise 3.2. Adjust the parameters E02, f2, v2 so that the group velocity is positive and
greater than the phase velocity. Record these parameters. (Note: These parameters are
not unique. Just list the parameters you find.)
4
Exercise 3.3. Adjust the parameters E02, f2, v2 so that the group velocity is positive and
equal to the phase velocity. Record these parameters. (Note: These parameters are not
unique. Just list the parameters you find.)
Exercise 3.4. Adjust the parameters E02, f2, v2 so that the group velocity is zero. (Note:
These parameters are not unique. Just list the parameters you find.)
Exercise 3.5. Adjust the parameters E02, f2, v2 so that the group velocity is negative.
(Note: These parameters are not unique. Just list the parameters you find.)
Exercise 3.6. Based on the experiences you’ve obtained from Exercises 1~5, develop a
theory that tells how to adjust parameters E02, f2, v2 to get the group velocity designated in
Exercises 1~5.
5