Scattering and diffraction. Group velocity

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GROUP VELOCITY

Taken from Michael Fowler, Univ. of Virginia, USA

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To find out what is happening here, 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. So, 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.)

The wavepacket describing a nonrelativistic electron in quantum mechanics is also like a "single beat" but in this case the group velocity is twice the phase velocity, so as the wave propagates the individual waves disappear at the trailing edge of the packet.

A complete discussion of the quantum case can be found in my lecture on Wavepackets.

When the wave components have similar (but different) speeds, and therefore slightly different wavelengths, ie the medium is dispersive, the pulse velocity (group velocity) is different from the velocities of the individual components, so that the group velocity can be written as:

vg = 2π  Δω ΔK

that is equal to the phase velocity when all components have the same speed.

Depending on the range of frequencies and velocities, the pulse can produce the illusion of moving faster or slower than the individual waves and even it can appear to go backwards.

In the example above, the pulse is generated by superposition of two waves of equal amplitude, with phase velocity equal to 1 and two frequencies (and two wavelengths) very close to each other, so that the total wave can be written as:

2 A cos (K1x - ω1t) cos [ (K2 - K1) x - (ω2 - ω1) t ]

With the upper slider bar you can change the group velocity = (
ω2 - ω1) / (K2 - K1)

With the lower slider bar you can change the frequency =  ω1

It is worth mentioning that Fourier analysis can demonstrate  that the dimension of the pulse Δx is inversely proportional to the range of frequencies used, the source of all uncertainty relations in ondulatory movements.

But let's go back...