## 08 May 2009

### The physics of singing in the shower

I was singing in the shower, as I do.

I noticed that certain notes seemed to resonate with the shower more than others.

These were, in ascending order, Eb2, G2, C3, and G3, where C4 is middle C. (These may not be exactly right; I don't have perfect pitch. The intervals are right, though.)

Exercise for the reader: how large is my shower?

Anonymous said...

why did you write in C B F G terms and other scary ones? why not just give frequencies :(

Veky said...

That's not the problem. The frequencies (Hz) and wavelenghts (m) are given as:
Eb2 77.78 4.44
G2 98 3.52
C3 130.81 2.64
G3 196 1.76
But I don't know what to do with them... LCM of all four wavelenghts, maybe%-?

Michael Lugo said...

Veky, the LCM of those would be quite large. I can't afford an apartment that large. But you're on the right track.

Jay said...

Do they all resonate equally strongly? Presumably all four of those notes have at least one dimension of your shower in the overtone series. I think I was about to make this unnecessarily hard until I remembered that showers don't have to be square--I'd assume in one direction is was a G, presumably a G4 which would be about .45 meters. The easiest note that's in the overtone series of Eb2 and C3 is probably some Eb, at least Eb4. That would give you the other dimension of about .55 meters. With, of course, either of those dimensions flexible by an integer factor--if you have a large shower it could be .9 meters by 1.1.

AA said...

Interesting "problem" :-)

Is it roughly 0.90*1.52*2 ?
(all in meters)

The formula that returns the standing waves that can be supported through a medium (air, a string, etc) between two endpoints is ftone=Vs/4*Ltone, where ftone is the frequency of the tone in Hz, Vs the velocity of sound in m/s and Ltone is the distance between the endpoints in meters.

Solving for L we get 4 numbers. L1,L2,L3,L4 corresponding to Eb2, G2, C3, G3 in this order. L4/L2 equals 2. This means that these two tones are supported by the same endpoint in the room.

Therefore, the room has dimensions that are integer multiples of L1*L3*(L4/L2)=0.90*1.52*2 and they look about right for an integer multiple of 1.
(This is all for dry air at around 20 C.)

Aaron said...

@AA: If that's true, Michael has a pretty large shower!

Your answer could be right, but the way you found it is wrong, because the equation you're using is equivalent to the statement, "the wavelength of a standing wave must be a multiple of 4L", where L is the distance between the endpoints. This isn't true for any simple physical medium, because it allows for two kinds of standing waves: waves with zero displacement at both ends, and waves with zero displacement at one end and zero pressure at the other end.

A long pipe with hard stoppers at both ends can only support standing waves with zero displacement at both ends, because the stoppers prevent the air at the end from moving. (See here.) Therefore, you should (I think) be using the formula "the wavelength of a standing wave must be a multiple of 2L."

Aaron said...

@ Jay:

When I calculate the pipe lengths corresponding to the notes, I find that they're all integer multiples of a single length. This suggests to me that all of the notes are resonating with the same dimension of the shower, because it seems unlikely that both dimensions would have overlapping overtone series. (Unless the shower is a specially designed musical shower! ^_^)

If I'm right, the most we can say about Michael's shower is this: one of the dimensions is an integer multiple of 0.246 fathoms.

My gut feeling is that if the shower weren't roughly square, there would be some resonances in between the notes Michael listed. Maybe there are, and Michael just isn't telling us about them! But where I live, the showers are roughly square, with sides very close to 0.492 fathoms. This is an integer multiple of 0.246 fathoms, so it fits the pitches we were given quite nicely, and it could be a standard shower size.

So, my bold guess: Michael's shower is square, with a side length of about 0.492 fathoms!

Jay said...

Aaron: You're right, of course. I was looking for a note that had all those notes in its overtone series (which would be rather unreasonably low), when I instead should have been looking for a note in the overtone series of all those notes. The lowest such note is a G4, which is an octave above G3 (2:1), a twelfth above C3 (3:1), a fifteenth above G2 (4:1), and a major seventeenth above Eb2 (5:1); The next note down would be a C2, which almost no one can sing properly. It has a frequency of 392 Hz and a wavelength of .88 meters. Which is your .492 fathoms.

Out of curiousity, why did you say .296 fathoms (a G5)? That is, of course, also in the overtone series of all four notes, but then so is, say, .148 fathoms, or .074 fathoms. If I'm doing it right this time.

AA said...

Hello Aaron

Thanks for the reminder. I worked this out by memory, so perhaps i need a revision.

The standing wave formula where both ends are nodal points is the one you give, which makes the figures i arrived at, twice their supposedly correct values.

The reason i said the figures look about right is because i assume that it's the whole bathroom that the problem is referred to, not just the open-top shower "cubicle" (for which the formula i am using would be the right one but i am playing fair here. It was the wrong one for what i had in mind :-) ).

But let's not get into things like "acoustically coupled spaces"...This was a nice "Sunday morning problem" for me, i do not want to see it getting transformed to a "Monday morning large scale simulation" :-D

Aaron said...

@ Jay:

Any note that resonates in a closed-at-both-ends pipe of length L will also resonate in a similar pipe of length L/2 or L/4, but the shorter pipes will have extra notes between the ones that resonate in the long pipe. So, upon reflection, I realize that I was implicitly assuming that there were no resonances between the ones we were given.

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