What is the relationship between phase and group velocities in wave propagation?

[Yoni]

I guess this is a common question: What is the difference between phase and group velocities?

When are they equal? Can phase velocity actually be higher than the speed of light in that medium, or even higher than light speed in vacuum?

Can anyone shed some light on this subject?

 

[Andy Resnick]

The phase velocity is defined as the frequency divided by the 'propogation constant', which looks like the wavevector in a linearly dispersive medium. The phase velocity is the apparent velocity of a single wave within a medium. The group velocity is the apparent velocity of a envelope of waves (the pulse envelope), and is defined in terms of the dispersion of a material.

Think of it this way- the reason to hit a gong softly, on the edge, with a big mallet is to excite a broad range of frequencies, thus taking full advantage of the dispersive properties of the gong- the result is a very low group velocity, leading to a very slow crescendo as the "pulse" travels from the edge of the gong to the center.

For all those FTL pulses/ traveling backwards-in-time pulses, etc., all that is in terms of the group velocity. The phase velocity of an electromagnetic wave can be no higher than c_0.

 

[Phrak]

Hi Andy,Yoni

You're somewhat misspoken about the phase--or wave velocity in that last comment, I'm afraid.

gw = c^2. The group velocity times the wave velocity is constant.

In a wave guide, for example, the group velocity is less than c, and the wave velocity is greater. A low energy electron has a small group velocity, and a larger-than-c wave velocity.

There is nothing wrong with all this, no information is propagated at greater than c. It's the group velocity that conveys information.

For a pulse of EMF down a wave guide, the leading edge of the pulse is limited to the group velocity as the waves attenuate into the leading edge. You can see this same sort of effect by dropping a rock in a smooth pond. Watching carefully you will notice the ripples travel outward from the center at a rate faster than the radius of the over-all disturbance.

 

[Andy Resnick]

Yep, you are right.

I blame the lack of coffee.

 

[pam]

"gw = c^2. The group velocity times the wave velocity is constant."
That is true in a wave guide in vacuum.
In a dispersive medium, where n depends on the frequency,
$$v_{group}=\frac{dk}{d\omega}$$ and
$$v_{phase}=\frac{\omega}{k}=c/n$$.
Then their product is not a constant.

 

[rahuldandekar]

Griffiths, in Introduction to Quantum Mechanics, has given a nice derivation of the formula dk/dw for the group velocity which may help you in understanding where it came from.

The essential point is, we need superposition of two or more different frequencies to actually convey information, since a pure sine wave has no information. The information is carried in "groups" or "pulses" which travel with a different velocity than the pure sine waves forming them. Take the example of stationary waves. The groups, here the nodes, don't travel at all, so it seems the wave does not travel at all!

A nice applet that illustrates how we can superpose different frequencies to generate groups which travel at a different velocity: http://gregegan.customer.netspace.net.au/APPLETS/20/20.html .