Simulation on Group Velocity and Phase Velocity
The phase velocity is fixed at 1.0 (arbitrary units).
Understanding the Pattern:
To find out what is happening here follow the simple steps -
(1) First set the group velocity equal to one using the slider bar.
You will now see a frozen pattern moving
steadily to the right on the screen. This represents the amplitude of, say, a sound wave
propagating to the right. Now increase the frequency using the lower slider bar. You will
see the peaks crowding closer together. But
why are they coming in these bunches? The
applet is actually plotting the wave generated by two pure notes which are very close
together, so beats are generated, a waa
waa waa sound, each waa
corresponding to one bunch of peaks, or one wavepacket as we shall call it.
As long as the group velocity is set equal to one, the wavepackets move at the same
speed as the individual waves. This is true for ordinary sound and light waves.
But other kinds of waves, such as surface water waves, and quantum electron waves, have
more interesting behavior. The bow wave of a moving ship, viewed where it has fanned out
some distance from the ship, looks like a single wavepacket, which
typically includes a few individual wave peaks. If you look closely, you will see that the
wave peaks move relative to the wavepacket, in fact they go at twice the speed!
In our applet, the speed of the individual waves, called the phase velocity,
is always set equal to
one. The group velocity is the speed of the wavepacket.
(2) Now to get a picture of how the
individual waves behave in the bow wave wavepacket from a ship, set the group
velocity equal to 0.5. You will see individual waves
disappearing at the leading edge of the wavepacket. (Of course, in our applet they move
into the next packet, but the bow wavepacket is a single packet, which is constructed by beating
many close notes together rather than just two.). In this case the the phase velocity
is twice the group velocity, so as the wave propagates the individual
waves disappear at the trailing edge of the packet.
The wavepacket describing a nonrelativistic electron in quantum mechanics also behaves similarly.
[ This applet and the text was written by Dr. Michael Fowler of University of Virginia and can be found at http://www.phys.virginia.edu/classes/109N/more_stuff/Applets/sines/GroupVelocity.html ]