### CPVC. The C stands for clarinet, or chalumeau… (part 2)

So let’s measure the speed of sound. And one other thing.

I took some 3/4″ CTS CPVC and cut off three arbitrary lengths, later measured at (piece designated t1) 141, (t2) 265, and (t3) 366 mm. I also have a couple of couplers (hah) to join these pieces together, each adding about 4 mm to the length of the combination.

From my old PVC clarinet I have another CPVC coupler, one I reamed out slightly to accommodate the tenon of a soprano clarinet mouthpiece. So I can put the mouthpiece on the end of a tubing piece, or pieces, and play a note. I have an Android app that’s marketed as a guitar tuner, but it has the feature that it shows the frequency it measures. Here’s what I found, averaging a few measurements for each case:

ID L (mm) f (Hz)
t1 141 399
t2 265 252
t3 366 192
t1+t2 410 176
t1+t3 510 144
t2+t3 635 119
t1+t2+t3 782 98

Now, for a perfect cylindrical bore, stopped at one end, there’s a formula for the resonant frequencies as a function of length. Loosely it’s $f = v/4L$, where $v$ is the speed of sound and $L$ is the tube length. But there are complications. Actually it’s not $L$ in the formula but $L_{eff}$, the effective length, which is about equal to $L+0.6r$; here $r$ is the radius of the bore. For 3/4″ PVC the bore diameter is 0.713″, but there’s another complication: we don’t exactly have a cylinder here, because there’s a mouthpiece on one end. So $L_{eff} = L+0.6r + L_{eff}^M$, where $L_{eff}^M$ is the effective length of the mouthpiece — and there’s no formula for that. It has to be measured.

I didn’t check the temperature in the basement where I was working but it was probably around 16°C, and at that temperature $v$ is about 340.7 m/s. Using the length and frequency for each combination of tubes, solving for $L_{eff}^M$ gives values from 67 to 82 mm. Not very high precision, but there you are.

But we don’t need to use the book value for the speed of sound; we can measure it! The spreadsheet software I’m using can do a least squares fit of the data, the result of which is $v$ = 333.0 m/s — roughly 2% below the book number — and $L_{eff}^M$ = 61 mm. It certainly was not below 5°C in the basement, so that $v$ is too low, and presumably so is $L_{eff}^M$. Is this just due to experimental errors? I don’t know. None of the points looks like an outlier. It really seems like the slope is 2% too low. Odd. Maybe an effect of the non-cylindrical mouthpiece?

I got out my bass clarinet mouthpiece, too, and found it would slip loosely over the tubing, but then the bore taper inside the mouthpiece gave a good enough seal that I could play notes with that, too. I figured I’d try to determine $L_{eff}^M$ for that mouthpiece as well. It gave slightly higher frequencies, implying a slightly smaller $L_{eff}^M$, closer to 60 mm (using the book value for the speed of sound), but only using the shortest tube combinations. For the longer ones I get $L_{eff}^M$ closer to 70 mm again. Or  if I use the least squares method, I get a really low value for the speed of sound, and $L_{eff}^M$ = 47 mm. Ridiculous. Again, I don’t know why, but given how fat the bass clarinet mouthpiece is, the idea that there’s a mouthpiece effect being ignored here would seem to be strengthened.

Still, good enough for the moment. Fairly cool that you can measure the speed of sound even at this level of accuracy just by blowing notes on a tube.