The Mule is travelling and, while he contemplates possible posts by the sea, regular guest contributor James Glover has stepped in with an analysis of applause.

It may seem indulgent (and possibly non productive) but Friday Afternoon Physics (FAP) like Friday Afternoon Maths is one of my favourite activities. It is also the holiday season and as the owner of this blog is busy counting echidnas down south I wanted to share what is so far my favourite FAP for 2013.

The question is: why isn’t it the case that the more people are clapping in an auditorium that the volume actually decreases?

Now this may seem like a dumb question. Surely the more people clapping = more noise = louder. Right? Well except that it isn’t true. Noise cancelling headphones work by detecting the incoming noise signal and producing a signal of exactly the opposite shape. Detected sound (by our auditory system – ears + brain) is a variation in the ambient pressure. So noise cancelling headphones actually produce, on average, a slightly higher pressure as they carry energy from both the original signal and the noise cancelling signal. Your ear drums don’t care what the overall pressure is though (up to a point) just the differences. An electrical device can’t easily produce an exactly cancelling sound but it can produce a sound with the opposite signal at the same average pressure. So if a lot of people are clapping randomly then while this increases the overall pressure the differential contributions tend, on average, to cancel each other out.

This just doesn’t just seem counterintuitive but experience suggests that the clapping in a room with more people is louder than a room with less people. But it would appear mathematically to be correct. Just as the average of a sequence of random variables has a smaller average amplitude from the mean (the standard deviation) so it must be the case that the more people in a room clapping will have a higher average pressure (not detectable by our auditory system) but also a lower variation (which is what we detect as volume).

Our solution: people nearby are less likely to average out than people far away. If we divided the room into people nearby (say under 25m) and those further out it would appear that the smaller contribution from those further out is not just because they are further way but because there contribution actually evens out. It raises the overall pressure but not so much the apparent volume. In fact in the simplest models it would actually decrease the more people there were present!

My insight is this: next time you are in a concert hall with say 1000 people compare the apparent volume with say a hall with 10,000 people. It would be deafening if it scaled up with distance alone. I am not sure about this but it appears to be true.

My final piece of evidence for this is the following: modern concert halls try to reduce unnecessary echoes, but total reduction of echo is bad. They used to do this by adding partial noise absorbing materials to the roof and walls. That is a disaster. A concert hall without echoes is a soulless place. Modern concert halls add random topographical features (usually in the ceiling at eg. random heights) that produce decoherence. Decoherence means the sound waves reflected back have different phases and hence quickly (but not too quickly) are undetectable as they tend to cancel each other out. The refurbishment of Hamer Hall in Melbourne did exactly this. So the solo horn in Brahms First Piano Concerto reflects but doesn’t reverb. Something similar happens with those noise “reduction” walls you see along freeways. They don’t absorb noise but all those cockatoos and gum leaves act to randomise the noise signal from the highway and even it out – the ear doesn’t notice the increase in average pressure but enjoys the decrease in variation.

Friday Afternoon Physics is good. It doesn’t lead to Nobel (or IgNobel) prizes but occasionally leads to Back of the Beer Coaster Calculations. Just prior to our discussion of this question we also worked out that 2-3 tankers a day could supply Melbourne with fresh water from Antarctica. But that’s another post.

Editors note: the echidna count so far has been zero.

Possibly Related Posts (automatically generated):

• Pressure Drop (27 March 2012)
• Where Music Lives (12 June 2009)
• Randomness (6 April 2014)
• Probability Paradoxes (7 June 2010)