The Speed(s) of a Wave

All right, the answer to yesterday's question about the maximum speed of a stadium wave, as many commenters rightly said, is "as fast as you want." The comments went into some depth on this, and I like the way Zifnab put it:

I mean, if you've got two independent agents doing their thing, the "speed" between the two just gets faster the farther apart they are. But if they have no relation to one another... what are we even asking? If our imaginary stadium is the size of the Milky Way Galaxy and the seats are stars and you stand up on Alpha Centari and I stand up on the Sun less than 4 years later, have we violated the speed of light? I mean, that just seems silly.

Exactly, a "wave" in and of itself isn't anything in particular other than an abstract concept. Stand outside, take a laser pointer, and wiggle the beam across the surface of the moon very fast. There's no limit on how fast the spot can travel relative to the surface of the moon. Nothing is actually moving. People do get suspicious of this, and I can understand. If nothing can go faster than light, what exactly counts as a "thing"? Isn't that a little sketchy?

The key is that the statement "nothing can go faster than light" is itself shorthand for a more precise concept. The concept is the relationship between cause and effect, or causality. Under the theory of relativity, we can take the concept of causality to say that whatever happens right here and now can only affect something else at a distance d if d < t/c. That is, whatever you do now only affects other places after enough time has gone by to at least let light get to those places. So if everyone's seated in our fictitious giant stadium and I start the wave in a normal way, it can't go faster than light. What I do can't affect the rest of the stadium any faster than the light travel time. But if I'm clever and pre-arrange standing times throughout the stadium I can get the wave to go faster than light. There's no problem because the my effect (the synchronizing message) propagated more slowly than light. No one is affecting anyone else at a distance greater than t/c. Once the superluminal wave starts there's no way I can change my mind and stop it because my "stop" signal can't travel faster than light.

You'll sometimes hear this stated as "information can't travel faster than light". That's true as far as it goes, but also runs the risk of being vague as to what information is. The causality formulation is probably more clear. Not perfectly clear, but more clear.

Also interesting is the fact that we're talking about waves, and the whole concept of "speed" with regard to waves is just wacky. There's at least three different and important ways of assigning a speed to a wave. Let's take a graphic from Wikipedia:

This is a wave composed of a number (probably two) of different sine waves superimposed on top of each other. They interfere to some extent, and the result of this interference is that the overall wave consists of a train of pulses which are large compared to the wavelengths of the original sine waves. One of the ways of describing the velocity of this wave is the phase velocity. That's the velocity of the original component sine waves. You can track this speed by looking at the red dot.

But you can also look at the speed of the pulses. That's called the group velocity, and you can track that speed by looking at the green dots. The group velocity and the phase velocity are not the same in general. In a vacuum both velocities are just the speed of light, but in a material with a refractive index they usually won't be.

Either of those velocities can be greater than c under certain circumstances. What's not possible is to violate the causality rule. I can construct a faster-than-light group of laser light in material, but only if my laser has been shining long enough to get the "pre-arranged" waves distributed properly throughout the material in the first place. And that I can only do at or below the speed of light. This signal velocity is a third way of defining wave velocity. Signal velocity describes the causal connection of a wave with its environment, and can't be faster than c under any circumstance.

As a last bit of lagniappe, light waves of different wavelengths tend to have different phase velocities when traveling through matter. This has the effect of spreading out a pulse made of different frequencies. Such an effect is called dispersion. Mainly it's an excuse to post this graphic, which I've made as a brief explanatory "look, dispersion!" graphic for a funding presentation my group is doing. (For the record, it's not intended to be technically precise. Units are arbitrary, absorption is glossed over.)

Whew. That's a lot of post. Time for other work now!