Imagine a round stadium with a crowd of 50000 persons. In the center of the stadium is an artist giving a show. At the end of the show everyone applauds and…
My question: Is it possible that during the applauds, considering the speed of sound and all the laws of accoustics, everyone applauding get in sync for one clap so the artist in the center of the stadium hears a big shock wave or something like that. When I say ‘in sync’, I understand that perfection is not required to hear the bang; I mean in sync enough for the combination of claps to create an impressive bang.
I have tried to solve this problem in my stats class in college but I was just not strong enough to come-up with an answer.
You do realize that each individual’s clapping is not an independent event, right? It’s not like rolling a die 50,000 times. If enough people get “in sync,” the rest will gravitate toward their rhythm. Also, if the situation is something like John Mellencamp singing “Jack and Diane,” you could very easily have everyone clapping with the beat because any other clapping would be quite silly.
That said, I highly doubt a performer would ever experience a shock wave from such an event, but I’ll leave the physics for others more knowledgeable in the area.
You’re ignoing the speed of sound problem. It’s not possible to get in sync by ear across an entire statdium. Each person could clap at the appropriate time (people farther away first, closer ones later) to create a single mega-clap. For it to happen randomly with such a large number of people is possible but highly improbable.
I indeed thought about the speed of sound problem. That’s why I am talking about a round stadium and the bang being heard in the middle of it only. That means people would have to clap slightly before those in the front.
I can’t help imagining the signer in the middle going, ‘Holy s####!! What the hell was that!?’
If anyone could come up with a reasonable approximation of the probability, I would love.
I was thinking about this problem and I keep coming back to the idea of the least common multiple. If you have just two people both clapping at a constant rate, say one every three seconds and the other every four seconds, they’ll clap in sync every twelve seconds. You compute that by finding the least common multiple of the rates.
What this means, I think, is that no matter how many people you have clapping, if you wait long enough and no one speeds up or slows down, they will eventually all clap at the same time. The question then becomes what is the probability that this simul-clap will occur during the 60 seconds or so that the applause is happening.
Figuring out what combinations of rates will yield the lowest lcm is a task best suited to a computer algorithm. If there are 50,000 people, though, with randomly distributed rates of clapping, the odds of a sync taking place within a minute or two are going to be really, really small.
Like Opus1 said, you have to take into account that people are applauding at very similar rates in general. Like he said also, this is especially true if people are clapping to the beat of a music. That somehow increases the probability I think.
The factors are I think;
the span of applauding rates (which is relatively narrow I think)
what accousticaly is necessary to create a bang effect (within 2/100 sec for example)
a few factor concerning the behaviors or the people in that sort of circumnstance
the relative position of people that could influence their perception of the applauds
the speed of sound obviously
Anyone has stats & prob calculation experience?
One thing is for sure, if you ever experiment such thing, please let me know… (recordings would be appreciated ;-))
This isn’t hard to achieve at all, because people can see each other clapping. For that matter all you need to do is pulse a light on the stage and have people clap when they see it.
This happens all the time. It’s just a non-event. There’s not enough energy in a clap for it to do much, even when 10,000 people all do it together. It just sounds like one really loud clap.
However, in order for the clapping of the outside rows to reinforce and augment the clapping of the inside rows, the clapping has to be timed so that the outside clap arrives at the inside row at the time of the inside clap. That’s a lot harder to set up.
I imagine that the energy, if coordinated exactly, could be enough to knock someone down, but it would be very hard to set up. Maybe simulated–but they also do such things all the time with for instance lithotripsy.
Assume the loud shock of a handclap is about 50mS long, and the average person claps 2 times a second (this makes the math easier). Each audience member, then, produces a clap wavefront 10% of the time, and so two clappers beating their hands randomly will sync up on 10% of their claps (yes, this is an extremely simplified model, but it will illustrate an order of magnitude). Three plauditers will meet up on 1% of claps, and so on. Thus, the probability of every person in a 50,000-seat crowd randomly producing a clap at the same time (or, equivalently, have the claps perceived by the performer on stage as simultaneous) would be 1e-50000 on any single clap.
However, suppose the problem is limited to, say, the odds that only 1/5 of the crowd happens to sychronize. Since there are (50000!/40000!*10000!)= 1.26e+10866 ways to select 10,000 from 50,000 (finally, I’ve found a use for Sitrling’s formula!), and each of these has a 1e-10000 chance of being synchronized, the chances of this happening are no longer trivial.
This leads me to the conclusion that handclaps are indeed synchronized for a large group of applauders in a large-enough crowd. My guess then is that either the cumulative effect of synchronized applause is dampened by distance, or there is a “tighter” window required for synchronization that the 50mS width of the clap pulse (reducing this to, say, 5mS increases the exponent in the synchronization probabilities above by a factor of 10).
As a related example of the cumulative effects of hand clapping in the right resonant environment, some researchers (seriously) content that the Mayan pyramids were built in sucha a way that they produce resonant sound effects. In particular, one pyramid is thought to mimic the sound of a bird chiring when a person claps near the base
Of course you always get the brainstorm right after you his “Submit”…
If, per the Stirling’s formula calculation, there are 1.26e+10866 ways to choose 10,000 people out of 50,000, and the probability of 10,000 people synchronizing their claps randomly is indeed 1e-10000, then the probability that there is NO collection of 10,000 in the 50,000-seat crowd that is synchronized is:
Remember, this is the probability that NO group of 10,000 is synchronized, so it is nearly certain that on every clap (i.e. twice a second) 10,000 people in the stadium happen to be clapping at the same time (i.e. within 50mS of each other).
Given the enormous exponents involved, I’d say the graph of the probability P that X people out of a group of 50,000 clapped at he same time is a hard step curve: 1 for all values of X below a critical number C, then effectively zero when X>C. I’d also suspect the position of the step–the value C–is highly dependent on what you consider to be “at the same time” (here, when it’s equivalent to “within 50mS”, we see that C>10,000).
Yeah, but by the pigeonhole principle, if each person is clapping 10% of the time, then with 11 people, there will necessarily be two people lined up. With 50,000 people, at least 5,000 of them will be matched up.
Okay, if we’re going to stipulate multiple rows, how about everyone get a light in front of their seat. Each light is individually timed by setting a time delay equal to the propagation time from the seat to the ‘focal point’. So now all everyone has to do is synchronize their own claps to the light in front of their seat, and all the clap sounds will happen at the same time with respect to the focal point.
However if there are multiple rows then you’ll want to space them apart with a distance such that the clap sounds are in phase as they pass over each row, otherwise they’ll cancel each other out. Then you have a problem with the sounds travelling laterally and creating interference patterns.
Come to think of it, probably the best way to set this up is to arrange the seats, get everyone clapping at the same time, then determine where the focal point is by going around with a sound level meter and looking for peaks and nulls. Some places might be pretty quiet because the clap energy cancels out there, and other places will be very loud. Look for the standing waves.
You might even be able to model this on a computer before setting it up.
Very interesting answers! It’s my first time on this site; I never thought I could find people as crazy as myself to be interested in that sort of problems. Great! Thanks!
The reasonning behind the ‘set-up’ to make that event happen is great. Now my original question was ‘Can that happen randomly and if yes, what is the probability?’. I am more than satisfied with the answers you came up with so far and I understand that the ‘accident’ is very unlikely to happen, although a guy named Murphy may argue the other way…
Unless two people have exactly the same rate, but are out of phase. For example, imagine if someone is counting steadily, 1, 2, 3, 4, 5… and one person claps on every odd number, and another on every even number.
My question is, how much energy is released in a single clap? Even if that is multiplied by 50,000, I have a feeling it won’t be THAT much.
Screw all that human crap…
Suppose you have 50,000 cyborgs precisely programmed to clap in time such that their individual clap reaches the center at the same instance…
Then what?