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.

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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!

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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.

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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.
May 21, 2009 2:51 pm

This is really fascinating, Mr. G&L. It finally makes sense why car wheels seem to rotate backwards in the movies.

May 22, 2009 3:15 pm

The phenomenon of car wheels going backward in movies is pretty cool, and a little easier to understand than group velocity. It has to do with the frequency with which you’re getting new images of the car wheel. Remember that a movie is just a series of individual images shown in rapid succession (like a flip book). If each successive image is taken after the wheel has completed, say, 9/10 of a rotation, then your brain will misinterpret the information. Instead of thinking that the wheel has rotated 9/10 forward, it will think that it has rotated 1/10 backward.

Here’s a somewhat dramatic demonstration: http://www.youtube.com/watch?v=_eoDVpC67Rc

They actually use this effect to measure frequencies of fast-moving objects. The put a strobe light on a rotating object, and when the frequency of the flashing matches the frequency of rotation the object will appear to be stationary.

It’s actually a pretty common “optical illusion”. If you watch a car driving past a picket fence, its wheels will often appear to be spinning the wrong way. I’ve also seen it done with a pinwheel spinning in front of a computer screen (the old vacuum-tube computer screens usually had a 60 hertz refresh rate).

May 27, 2009 6:12 pm

There is a very interesting analogy of this effect in information processing called aliasing. Check it out here:

http://en.wikipedia.org/wiki/Aliasing

It is a very important phenomenon for all experimental physicists, engineers and computer science folk! It basically provides an upper limit to the frequency one can distinguish given a certain sampling frequency (the frequency at which one “observes” something). It provides another way of thinking about the wagon wheel example above – it is too lengthy to explain in more detail in a comment (you could do a whole post on it!) but I though I should at least mention it.

May 27, 2009 7:07 pm

Thanks Lee. Aliasing is definitely at the heart of the “backwards spinning wheel” phenomenon.

September 22, 2009 3:08 am

very nice page…let me add my 2 cents to this blog by sharing a really good gallery of optical illusions

4. April 20, 2012 6:30 pm

It seems that you have drawn your conclussions first and then built your argument to support them. Nothing wrong with this logic, but it does narrow your imagination.

I am led to understand that you are saying that phase and group velocity are the same thing? That phase and group velocity always propagate in the same direction at the same rate? That any other observations are illusions?

“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.”

I think there is some physics that governs this, not of mass particles thats obvious, but of energy yes. I think there is limits to the gradients and curl rates of change of group and phase velocities in maxwells equations (wish I had the disipline to do the math) that effects the limits of relationships between phase and group velocities (maybe reactive and resistive fields themself), but the simulations can be whatever you want, illusions if you like.

The propagation of energy is clearly established with many experiments to substaintially prove it and supporting math (maxwells, wave equation etc.) to make the logic rigorous. In closed systems (adiabatic) with no pumps or generators to add, the higher energy resorvoir will always (boundary) propagate energy to the lower resorvoir at rates that are specific to geometric and mass buckling (permitivity, permiability, shape of emitter dipole etc, reflector etc). If its a sphere and the sorounding medium is empty space, if the generation is isotropic and homegeneous, the energy density at any place from the center of the point of the sphere is equal.

If for some reason there is a channel that changes the permitivity or permeability energy can flow unequally from the sphere. A channel, or beam, or conduit is created. It makes no differnce how this is done. The lower energy resivoir can easily establish this, the reciever if you want to give it a name, which actually is a transiever transmiting a group velocity, and recieving a phase velocity.

Your matlab(?) generates your illussions, easy to understand. It real condition EM propagates, absorbs, reflects and mixes, as fourier and mandelbrot expanded on. Furthermore the E and M part are not always the same, reactive and resitive, near and far field. The relationship of groups and phase of frequencies and energies is constantly fluctuating and changing.

My question, and contention is that there maybe a way to create a conduit from a controlled beam from a lower energy part of the transmiter of a transciever to and from higher energy part of a transmitter of a transciever. It has been my belief that one possible way to do this is with a coherent controlled phase and group velocity wave.