# 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!

August 19, 2009

Stand outside, take a laser pointer, and wiggle the beam across the surface of the moon very fast

But have you met my friend Emily’s dog? If she was on the moon, she would be chasing the laser at whatever speed the spot was moving. Speed of light, smeedoflite.

2. #2 Uncle Al
August 19, 2009

Dispersion plays backwards as chirping. That is what makes an otherwise boring tsunami so intersting when it progressively arrives at shallows. Surf’s up!

http://en.wikipedia.org/wiki/File:2037_Apophis_Path_of_Risk.jpg

Remember the wave tank in physics lab? Could be a big splash for the Pacific rim Friday 13 April 2036.

3. #3 Matthew
August 19, 2009

That dispersion animation is really nice.

4. #4 ppnl
August 19, 2009

Next stop Bell’s inequality? Exactly how do you define information or even causality?

5. #5 mmfiore
August 19, 2009

I like the graphics examples thanks for those good illustrations. You may be interested in my site as it also talks about the nature of light.

As an alternative to Quantum Theory there is a new theory that describes and explains the mysteries of physical reality. While not disrespecting the value of Quantum Mechanics as a tool to explain the role of quanta in our universe. This theory states that there is also a classical explanation for the paradoxes such as EPR and the Wave-Particle Duality. The Theory is called the Theory of Super Relativity and is located at: http://www.superrelativity.org
This theory is a philosophical attempt to reconnect the physical universe to realism and deterministic concepts. It explains the mysterious.

6. #6 rob
August 19, 2009

groan.

7. #7 Watts
August 19, 2009

A question that I always felt a bit silly about, and never dared to ask in physics class:

Where does the requirement for causality to be preserved come from?

It certainly seems like an obvious requirement, at least from our everyday casual experience. However, QM and relativity show that there are things on very small and very large scales which behave very differently from our casual experience. Why do we assume that causes must precede effects, when the definition of “precede” gets tricky due to simultaneity? Or does it not get tricky? Is there a proof which demonstrates that causality must be preserved?

8. #8 Luke
August 20, 2009

Hi, I was just wondering; what is the average increase-in/loss-of lifespan caused by jaywalking? I know there are a bunch of confounding variables, but as a Fermi problem it’s quite interesting. I did a brief analysis, but the numbers are nowhere near precise.

On average, jaywalking takes between 0-10 seconds to get on to the road. Waiting takes between 0 – 60 seconds. So, on average, 25 seconds is saved per crossing.

Conservatively, the average person crosses the road 6 times a day. Just taking into account people between 20 and 70, we consider the data over 50 years.

So, 25*6*365*50 = 2.7375Ms = Approximately 30 days of your life saved by jaywalking.

You have a 1/18535 chance of dying by being hit by a car. Assume that half of people jaywalk, and you will definitely not die if you don’t. Then, given that you jaywalk, you have a 1/9535 chance of dying and losing an average 25 years.

Let’s say that you have a 1/900 chance of being hit by a car, but surviving with a 10% decreased quality of life (losing 2.5 years).

(1/900)*2.5yrs + (1/9535)*25yrs = 2 days.

So, by jaywalking, you live a month longer.

9. #9 Anonymous
August 20, 2009

Stand outside, take a laser pointer, and wiggle the beam across the surface of the moon very fast

That was the same little comparison I worked out 30+ years ago as a semi-precocious 13 year old, proud of my developing math skills who thought he was ready to walk in Einstein’s steps. I never thought I’d seen it before nor since- or has it appeared before?

Nice post.

10. #10 Robert
August 20, 2009

Since this is thought experiment, we can keep the wave at a normal human speed of 12 m/s but reduce the speed of light to 6 m/s.

Then it’s obvious your faster than light wave isn’t gong to look like a proper wave from the point of view of the people performing the wave.