Built on Facts

The wave. We’ve talked about it before. It’s where people in a stadium stand up and sit down in sequence so that a wave of standing people travels around the circumference of the stands.

They have a typical speed of around 12 m/s, though it is by no means entirely universal. You could certainly imagine a crowd being instructed to go much faster or slower when doing the wave, and though I never done seen it I don’t think it would be very difficult.

If you had a truly enormous imaginary stadium, maybe one with hundreds of thousands of miles of bleachers and an audience to match, you could run some interesting demonstrations. I pose a thought experiment: how fast can you get the wave to move in this stadium? Speed of sound? Speed of light? Faster? If faster than light, is this necessarily a violation of relativity?

Comments

#1 max fagin
August 18, 2009

Assuming it took no time for the audience to stand up and sit down, and assuming you could sync all audience members to the same inertial reference frame (say by placing a clock in the center of the stadium, equidistance from all audience members) then yes, you could get the “wave” to travel as fast as you wanted in the inertial reference frame of the stadium clock.

However, you could only do this by “cheating” and giving each audience member an absolute time in which to stand up and sit down. If, for example, you were to tell the person in seat 1A to stand up at 12:00:00 AM, and the person in seat 1B to stand up 100 picoseconds later, then assuming the two seats were one foot apart, you would have just made the “wave” travel at 10 times the speed of light. But since no information has passed from seat 1A to seat 1B, special relativity has not been violated.

However, if you wanted to make the wave travel in the more traditional way (i.e. each person stands up as soon as they see the person to their left do so) then the wave velocity would be limited to the speed of light, as the information “I have stood up” would have to travel from person to person as the wave propagates.
#2 Zifnab
August 18, 2009

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.
#3 Andrew
August 18, 2009

max fagin is right about lightspeed, but here’s a way you could just about achieve the same effect without giving people specific times:

If you have a really mammoth, circular stadium, you could have a big arrow in the middle, spinning around, telling people when to stand up. As long as the end moves slower than the speed of light, the position in the crowd that its pointing to needn’t.

A person in the crowd sees the crowd opposite them delayed, because the stadium is so vast. There’ll be someone there who they see waving just before they wave. They can react to that, and the wave can continue without the arrow.

Arguably this is several waves bouncing and tracing chords through the circle, but it’d look the same as a wave flowing round it.
#4 rehana
August 18, 2009

I wonder if it would move faster if the seats were farther apart. Might also die out more easily.
#5 Colin
August 18, 2009

I think this is what Andrew is getting at.

You have a circular stadium with diameter D and the time for information to move from one side to the other is D/c but the distance the wave travels is PI*D for a wave speed of PI*c provided your “arrow” got the people in phase.

You could vary the wave speed from 1 to PI times c depending on which chord of the circle you use. Through the center you get PI*c; and adjacent seat you get just c.

Now, what if you cut the stadium in half, and flip it over so the person that was across from you is now D/X feet above your head. I think you then get a wave of X*PI*c, yes?
#6 Colin
August 18, 2009

(I missed a 2 in there: PI*D/2. PI*D would be the whole circumference. Throw in /2 as necessary.)
#7 Eric Lund
August 18, 2009

No, it wouldn’t necessarily be a violation of relativity, as long as it’s the phase velocity and not the group velocity. Indeed, you can have waves with infinite phase velocity: in a plasma you can have cutoff frequencies, where k approaches 0 at a finite frequency and therefore the phase velocity, which is defined as ω/k, approaches infinity.

A real-life version of this problem occurs in some jets originating from black holes at galactic centers. You need (1) a rapidly rotating compact body (2) with a magnetic axis offset from the rotational axis (3) producing a jet which remains coherent out to sufficiently large distances. As the compact body rotates, so does the direction of the jet, which produces a phase front in the wave. At sufficiently large distances, this phase front appears to move faster than light. But it is only a phase front; no matter or energy is actually moving faster than light. Andrew’s big rotating arrow is a good analogy: here the wobbling magnetic axis acts as the arrow.
#8 ppnl
August 18, 2009

Shadows can go faster than light. You can achieve the same effect with people waves. But the thing is no information can be transmitted by such methods.
#9 Excited State
August 18, 2009

I think max fagin said it right.

Another interesting question: can you send human waves through each other? I don’t think so. I think the information would be lost, because there’s no superposition; you’re either standing or not, you can’t stand twice as high if two waves are passing through.

So what would happen? Would one wave swamp the other? Or would they annihilate. I think annihilation.
#10 Colin
August 18, 2009

Binary state is an arbitrary assumption/restriction and Max’s actual restriction was zero-delay in standing up. 1) You don’t have to play Max’s assumption and 2) I see no reason why you have to assume binary state.

It depends on what your trigger is to stand up. If it’s only looking left then you can’t get a wave flowing left. If it’s looking both left & right and “averaging” your response then you certainly can get a superposition of waves.

You could get a superposition of “2″ by jumping but you’ll attenuate above that.
#11 Uncle Al
August 18, 2009

Wasn’t that sort of thing performed with a train of buckets of thiotimoline (so soluble that it dissolved in water up to 1.12 seconds before the water was added)? Isaac Asimov, “The Endochronic Properties of Resublimated Thiotimoline” (1948). It turned out poorly.
#12 Paul Murray
August 18, 2009

If the wave is carrying information (ie: the time at which it was started), then speed of light, natch.

If you synchronise watches, distribute them, and have people stand up and down, then no limit. But there isn’t a meaningful sense in which there is a “thing” that’s “moving” – it’s just a geometrical construct. And there’ll be a frame of reference where the “wave” is moving backward.

BTW: would tachyons look like these “virtual particles” thingos? Real particles have a limited region in space but persist over time. Virtual particles only exist for a brief time, but mediate action-at-a-distance.
#13 Paul Murray
August 18, 2009

And I’d still like to know why a charged sphere in a gravitational field doesn’t emit radiation – it’s accelerating upward with respect to the local inertial frame of reference.
#14 looseinthedeuce
August 20, 2009

The fans at Michigan Stadium have performed some of these waves many many times. The wave begins traditionally, going around a couple of times (12m/s seems about right). Then, it slows to about 1/2 speed (and takes FOREVER to make it around). It speeds up to twice the original speed, and also reverses direction like it’s bouncing off a wall. The fans also perform a “double wave” where the wave goes in both directions, and the two waves do pass through each other and continue on. The 2x speed wave is also done as a double wave. Google “michigan stadium wave video” and you’ll probably find something.
#15 Alex Deam
August 20, 2009

Following on from the “cheating” method of using synchronized watches to generate the wave, I’m wondering:

If you’re a stationary observer looking at all the people standing up and sitting down faster than they would if they actually looked at those next to them and stood up when they received the information that they’ve stood up, then wouldn’t that NOT look like a wave to the stationary observer? Because for a stationary observer to see the wave, he uses light travelling at the speed of light. But the “wavers” aren’t standing up according to the speed of light, so what does he see….?

I stand up on the Sun less than 4 years later, have we violated the speed of light?

No, but you have violated biology as we understand it currently. How are you standing on the Sun???
#16 Tripz
August 21, 2009

I wonder if it would move faster if the seats were farther apart.