Speed of sound in football stadium

Ok - I like Alabama football (sorry, but it's true). There is a clip on youtube of the end of the Alabama-Auburn game where the fans sing the traditional "Rammer Jammer". If you don't know what that is, don't worry. I am not sure I completely approve of the cheer, but it is a tradition. If you are curious, here it is:

So, what is cool about this? Notice that the entire crowd is yelling the same thing, but they are out of sync. Can this be used to estimate the size of the stadium?

I made an audio file from that youtube clip and looked at it with Audacity. Although I am completely unfamiliar with audio editing stuff, I think I managed to get the time delay. Here is a shot of the clip:

After listening to this segment about a billion times, I am pretty sure that the highlighted segment is the time difference you would hear from the location of the video maker (in the clip). It is about 0.5 seconds. Why would there be a time delay? Well, there are a couple of reasons. First, when the people at the other end of the stadium say stuff, it takes time for that sound to go across the stadium. The other reason they are off is because if they are farther from the band, they will receive their "music cue" at a different time. I am not sure if you can tell, but the band is much closer to the location of the videographer in this case.

Clearly, I am going to need the speed of sound. A typical value used in the textbooks is 343 m/s. This speed does change with both temperature and humidity. For that day, it was around 16 degrees Celsius and about 95% humidity. This online calculator takes into account air pressure, temperature, and relative humidity. Essentially, the speed doesn't change enough for me to worry about it.

Now for the complicated stuff. The more I think about this, the more I realize I have to make some assumptions. I will proceed anyway with reckless abandon. Here is a diagram:

In this picture, the band makes some sound. This is the signal which the people in the stand know what and when to say. I have actually labeled 2 signals (s₁ and s₂). S₁ is the signal going to the location of the video camera person. S₂ is from the same source (the band) but it goes to some other part of the stadium. In this case, s₂ goes a further distance. This means that those people get the signal to yell at a slightly later time than the video camera guy. If the band is in the middle, these two locations would get the signal at the same time. As a first approximation, I will assume this is true (and then see if I can correct for it, or if it is even worth correcting). If that is the case, the time difference would be only due to the time for the yelling from one side of the stadium to go to the other side. If this distance is d₁ and the speed of the sound is v, then the following would be true:

That distance seems possible. But what if I want to take into account where the band is? Let me squish the diagram into one dimension.

The important thing is the difference in distances. This is also related to the total distance (which is the sum). I will call ?t the total time difference (0.5 seconds). so:

The first term is the sound from the yellers on the other side of the stadium to get to the video camera. The second terms is the time difference between how much later the other yellers started to yell. A couple of things to notice:

• According to this, the delay does not depend on how far away the video camera is from the band but only how far the other people are from the band. Oddly, this makes sense. Suppose the video camera were a 1000 meters away (nose bleed section). The sound from the band would take almost 3 seconds to get to them. If it were just the band and the camera, there would be no sync issues (except for the band - if they could hear the people yell, they would notice them out of sync). If these far away video camera people could hear the other people they also would be delayed a little more than the band. Essentially the signal from the band must travel to the other people and then to the video camera.
• Yes. I know I made some assumptions. The stadium is not composed of a band and two groups of yellers.
• If the band was in the middle of the stadium and the two groups at the ends, the first way I did this would work. Both video camera and the other people would get their signal at the same time so that the delay would essentially be the time for the sound to go across the stadium. If the band is in the middle, 2d_p would be the total distance across.

So, from this, I can calculate the distance from the band to the other end of the stadium:

Is this a reasonable answer? I think so. 86 meters is about 94 yards. I converted to yards because a full football field is 120 yards. If the other yellers are twice as far away from the band as the video camera, that would make the distance between the two yellers at 141 yards apart. I wonder if the googles has the exact dimensions. I was going to be clever and use an image from google maps to measure the distance, but it was crazy. It gave too large a number. To check, I measured the size of the field and got over 200 yards. Oh well, I will just say it is reasonable.

Question: Is this a problem and if so, how do you fix it?

The answer to this depends on what you think the purpose of yelling is. Is it to just yell? If that is the case, there is no problem. Is the purpose so that people on the field all hear stuff in sync? That could mostly be fixed by putting the band in the middle of the stadium (or close to it). Another option would be to have some sort of light signal that everyone could use to time their yelling.