When group velocity is an optical illusion

Physicists love to talk about waves.  And with good reason.  They are abundant in nature, easy to visualize, and have fairly simple mathematical descriptions (sines and cosines, mostly).  Most importantly, just about any shape in the world can be described by the right combination of waves (this was Joseph Fourier’s most important contribution to mathematics).

Physicists love waves so much, that they have a wave description for every particle in the universe (electrons, atoms, photons, phonons, etc.).  Something like an electron is said to have a “wavelength” and an “angular frequency”, even though it seems funny to think of an electron as a combination of waves.  Nonetheless, in the hands of physicists matter becomes a “matter wave“, and the description works so well that it forces us to reconsider the way we think about the way the universe is constructed.

But these are profound, philosophical issues.  The purpose of this post is to discuss something much more superficial: the ways that wave behavior can be deceiving.  In particular, I want to talk about a concept that every undergraduate physics major is forced to grapple with: group velocity.  Group velocity, in short, is the speed at which a high-amplitude “wave packet” moves.  Here’s a nice illustration from Wikipedia showing how the “wave packet” made by combining two different waves can move at a speed slower than the constituent waves:

The red dot moves at the speed of an individual component wave. The green dots move at the speed of the "packets" -- the group velocity.

In physics, a given “packet” usually represents a particle of some kind (or at least something that we talk about as if it were a particle).  And the speed of the packet is the group velocity.  So physics majors gradually become indoctrinated with the idea that the group velocity is more important (more “physical”, we say) than the the speed of individual waves (called the “phase velocity”).  Further, we get used to the idea that while individual waves may move quickly, particles and information can only move forward as fast as the group velocity.  And the group velocity must be slower than the phase velocity.

But are these “doctrines” true?  Does the group velocity of a combination of waves always move at a reasonable speed, slower than the phase velocity?

The answer is no.  If the only rules of the game are “individual waves come together to form groups”, then we will see that you can form packets with any group velocity at all — as fast or as slow as you want.  In these cases the group velocity becomes completely meaningless for our “particles as groups of waves” description, and all our wave-related “doctrines” become little more than optical illusions.

Now that I’ve talked a bit about physics, we can move on to the fun part: wave-based optical illusions.  First, I’m going to abandon the sines and cosines in favor of an even simpler version of waves.  From here out, my waves will just be black and white stripes.  The distance from one black stripe to the next is the “wavelength”, and the speed at which black and white stripes move forward is their velocity.  It seems a little crude, but all the important behaviors of waves will still be there.  Here’s my “blockified” version of the group velocity demo from above:

A "well-behaved" group of waves

The rectangles at the top are a little thicker (longer wavelength) and a little faster (higher velocity).  Where they overlap in the middle you get groups.  Do you see them?  It helps to squint a little bit.  If the animation has stopped, refresh the page and it should start again.  In this particular example, the group velocity is smaller than either of the component waves.

But if you start playing with the relative velocities of the two component waves, all sorts of funny things can happen.  For example, here’s one where the “groups” aren’t moving at all, even though the component waves are:

The waves are moving to the right, but the groups are staying put.

When the short-wavelength waves (thin bars) are allowed to move faster than the long-wavelength waves, we get all sorts of “indecent” behavior.  Here, for example, is an example of a group velocity that is much faster than the speed of component waves:

Here the groups move faster than their component waves

This sort of behavior actually happens for light in certain materials, where the index of refraction increases at higher wavelengths.  You can imagine the surprise of the scientists when they found that they had group velocities (“particles”!) moving faster than the speed of light.

Finally, you can get even get left-moving groups from right-moving waves:

negative group velocity!

So in the end, what was the point of this post, other than to show off some dizzying images?  The only message is that the behavior of a “wave group” is not always meaningful for describing the movement of particles or information.  There is no such thing as a particle that goes faster than the speed of light, or a process that sends information backwards when you try to make it go forwards.  So we need to think about group velocity more carefully, in a way that is more nuanced than “the group velocity is the speed at which actual particles and information are transferred.”  You can tell that something is going to go wrong with your standard ideas of group velocity when smaller wavelengths move faster.  Are there other circumstances where group velocity becomes meaningless?  Probably.  If you know, I’d love to hear about it.  Maybe we can make more funny movies.

Credits

I should give some credit where it is due:

• I was first inspired to dabble with this topic when I stumbled across this research description on the Duke physics website.  It is quite clearly written, and probably more instructive than I am.
• Greg Egan created a great little applet you can play with that makes things a bit easier to understand.
• Given my limited programming skills, I never would have been able to make the pictures here without Kedar Patwardhan’s avi2gif script for matlab.