# Brass instrument lengths

I assume (perhaps wrong), that the fundamental frequency of brass instruments can be calculated with the wave equation:

c = lambda * f,

WHERE:

• c = the speed of sound. I calculated this to be 349 m/s at 30 degrees C (temperature of a warm instrument). I think this is right because two control conditions (0C and 20C) gave me correct answers (331 and 343 m/s).
• f = Fundamental frequency of the instrument.
• lambda = length of the instrument.

Given the fundamental frequency of 87.31 Hz (F2) the length of this F-tuba should be 4.0m.

However, Wikipedia's article on tubas says an Eb-tuba is 4.0m and an F-tuba is 3.7m. I took a tape measure and measured my F-tuba along the center of the bore to the edge of the bell, and I measured 3.72m. It seems Wikipedia is right!

Did I use an incorrect equation to calculate the length of a brass tuba or was there another problem?

My guess is that there's a problem in the boundry condition at the bell. c=Lf applies to a non-flared pipe like a Didgeridoo and the flare of the bell effectvely lengthens the instrument. In the sketch below, I measured the long-dashed line to the X where I should have measured to the ?. But is there a way to know the distance between those two marks?

I don't think the problem is related to the speed of sound. If I use the known 331 m/s at 0 deg.C, the F tuba is 3.79m, still longer than my measured 3.72.

My mouth-piece is not 30cm long, so that also doesn't account for it.

Another thing it could be is the extra length added when reflecting from wall to wall as the sound travels around curves. A straight tuba without curves could be closer to the theoretical length I calculated. But I don't know if this is true. Any ideas would be welcome.

Thanks to Elements in Space for adding the "acoustics" tag to the question. I browsed "acoustics" and find this nice answer about boomwhackers:

https://music.stackexchange.com/a/124754/72714

The answer is End Correction. It seems that the pressure waves are nicely aligned as they come out of the tube, and take some distance before mixing with the ambient air. That means a correction can be should applied to the physical tube to match its "effective" length in generating sound.

The Tuba I measured has a 44cm (diameter) bell. As brass instruments are open on both ends, the correction (delta-L) is dL = 0.6D, or 26.4cm. Adding 26.4 cm to the measured 3.72m gets us quite 3.98m which is very close to the theoretical 4.00m.

I think that is within the tolerance of my tape measuring technique.

Still, it's possible that this answer is wrong. That formula applies to a cylindrical tube, while the tuba is conical. 44cm isn't the diameter of the entire instrument! I was happy to get an answer that was close, but it might have been a coincidence. I'm not sure if it's a good, general solution. I'm happy to accept any better answers.

• Another thing to be aware of is that the speed of sound is not a constant because the density of the air inside the instrument is not constant (density of a medium is directly related to the speed of sound in that medium). Both temperature and humidity affect the density, with very audible impact. Regarding pitch sense, exact frequencies and pitch do not always match exactly. Combining those facts with some other acoustic properties of real instruments means we shouldn’t expect the ideal dimensions generated by math to match the real world dimensions. Commented Sep 9, 2023 at 14:16
• If you think that's weird, look up the acoustics of reed instruments. For example, the saxophone has a conical bore, and the effective length goes way "backwards" to a convergence point well in advance of the mouthpiece. Commented Sep 9, 2023 at 21:04
• @CarlWitthoft is that the case for all conical instruments? There’s at least a few conical bore brass instruments too Commented Sep 10, 2023 at 5:30
• @fyrepenguin I think so, but I bet you can track down some research papers for any given instrument. Commented Sep 10, 2023 at 10:22