# Singing in the Rain

So not only did I miss the Saturday and Sunday posts, but also the Monday and Tuesday ones as well. Gah! It’s honestly quite a bit more difficult to keep to the one-per-day regimen this semester than I though. Nonetheless I’ll try to stick to that schedule to the greatest extent possible. KBO.

Today I’d like to talk about singing in the shower. You will notice, if you’re brave enough to try it, that certain notes resonate with a particular volume. It’s probably why so many people like to sing in the shower – it turns the most humble voice into booming thunder. Or at least an approximation thereof.

Physicists can describe the effect in terms of sound waves. While any wave you care to produce by singing will bounce and clatter around between the walls of the shower, certain notes have just the right wavelength to fit between the walls exactly. Schematically they look like this, from Wikipedia:

Any wave that fits an integer number of half-wavelengths in between the walls will resonate. But the wavelength is related to the frequency by the standard dispersion relationship c = λ*f, where c is the speed of sound, λ is the wavelength, and f is the frequency. A typical frequency for singing might be middle C which is about 261.6 Hz.

If you’re clever and have a sufficiently trained ear, you can in theory calculate the speed of sound in air by singing in the shower. Multiplying the frequency by the wavelength is all you need. The problem is that you don’t know the wavelength of a particular note a priori. It might be the distance between shower walls divided by 1/2 or 1 or 3/2 or 2 or 5/2, etc. You can be tricky and find a way around this. The trick is to find two successive resonances. For our first wave of unknown n (with distance d between shower walls),

Gradually slide up the frequency scale from f1 until you hit the next resonance at some frequency f2. The same equation adjusted to reflect that we’ve gone up one n is:

That’s two equations, so we can eliminate the unknown n:

Game, set, match… assuming you have the ability to generate and recognize the frequencies f1 and f2 that produce the resonances. And assuming you’re only getting the resonances between the front and back of the shower, which isn’t a safe assumption at all. But in theory at least we’re the Pavarotti of science!

1. #1 Uncle Al
April 8, 2009

Box resonance depends upon whether the critical points bounce from or are pinned to the wall. A suitable wave would have different rules for dielectric and conductive walls. Designed showers can focus all the energy back into the singer’s skull, crushing it like a stepped upon grape, re the otherwise ridiculous (i.e., weapons studies) National Ignition Facility singing for its supper.

2. #2 magista
April 9, 2009

The stairwells in our school are like that. They’ve got flat plaster walls and ceilings, large-pane glass windows and doors and tile floors. All three dimensions are different, so they resonate to an incrredible range of frequencies.

I take a couple of students in there at a time and tell them to sing when we get to the sound unit…

3. #3 hendrix
April 10, 2009

I guess it is also cool to input the sound speed and then compute d… afterwards… measure it 🙂

4. #4 Anonymous
November 6, 2009

uuhh ??:S