Re: Physics of Tubas


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Posted by Eric Bamberg on March 01, 2004 at 22:53:01:

In Reply to: Physics of Tubas posted by Cook on March 01, 2004 at 09:53:15:

The acoustical behavior of tapered pipes it pretty darn difficult to understand, probably because you can't actually see what the air is doing in there. Trying to explain what exactly the sound waves are doing is less important than finding out what the variation in shape does to the tuning of the harmonics. Say you have a cylindrical pipe and a saxophone-like cone each with some sort of sound source closing off one end such as buzzing lips. The pipe will resonate at frequencies 1,3,5,7,9 while the cone will resonate at frequencies 2,4,6,8,10. If you could magically vary the shape of the tube from cylinder to cone you would see each harmonic gradually rise in frequency as the flare widens. 1 rises to become 2, and 9 rises to become 10. The higher the harmonic number, the less rise in pitch the flare causes.

The common explanation given is that the taper causes the low frequency waves to reflect sooner, and that's a useful enough think to think about, but remember that a low C sound wave is a gradual variation in pressure 18 feet long, so while the center of the wave may reflect sooner, there is still a significant portion of it that makes it to the open end of the instrument.

Brass instruments are made with pipes designed so that their harmonics occur in the ratio 2,3,4,5,6,7,8, etc. If you were to take a conical resonator and gradually squeeze it in the middle to form a more tuba-like shape, interesting things would happen. The 2nd harmonic would stay pretty much the same frequency, but the fundamental would rise drastically while the harmonics above the 2nd would lower in frequency a much smaller amount. After squeezing that cone into a tuba shape, you'd notice that the 1st and 2nd harmonic that were an octave apart before are now a 5th apart and the 3rd harmonic that was previously a 12th is now an octave. Continue to squeeze that thing beyond tuba shape and the fundamental would go sharp.

That was dealing with ideal shapes, of course, and no brass instrument is ideal. There are minute variations in the shape of various parts of horns that affect both the strength and tuning of the various harmonics. For instance, a rapid flare in the leadpipe causes the strength of the low harmonics to increase and high harmonics to decrease, but it also causes the pitch of the high harmonics to go flat. You could compensate for that by making the flare of the horn more conical. Designing an instrument is a set of trade-offs.

When a piece of tubing is bent, the sound wave tends to take a shorter path through it than if it were straight. This means that as the sound wave sees it, the bent piece of tubing is both shorter and fatter, and the tighter the bend the shorter and fatter it seems. If you wanted to make a straight tuba with the same taper as a folded up tuba, you would have to physically shorten and enlarge the sections where the bends normally occur. Constrictions tend to raise the resonance freqency of high harmonics and lower the frequency of low harmonics, while bulges tend to do the opposite. Long discontinuities tend to affect the lower harmonics and vice versa. Harmonics are most sensitive to discontinuities at their pressure nodes, and are more sensitive to discontinuities toward the input end where the horn is smaller and the energy is more concentrated.

The final flare of the tuba has very little to do with the intonation, that's why a sousaphone plays fine with no bell. The purpose of that big bell flare is to radiate high frequencies so that they dont reflect back to the mouthpiece. The mouthpiece cup itself does a similar job, keeping high frequencies from your lips. Those high frequencies interact with your lips and keeps them closed longer so that they can put as much energy into the low frequencies. The sound ends up thin and blatty with either a shallow mouthpiece or no bell. A more gradual bell flare or a deeper mouthpiece both let you put more energy into the lowest harmonic. The bell flare can add a certain color to the sound since it causes an increase in the radiation efficiency of the tuba centered at one particular frequency. In the case of a very gradual bell flare like a King sousaphone, that can be as low as 120Hz. It kills the high register because most of the high harmonics never reflect back to the mouthpiece.

Antennas are usually not designed as harmonic resonators, they are designed for a certain range of frequencies. Lasers and masers are the only electromagnetic harmonic resonators I can think of. Maybe someone will make a maser from a tuba, the wavelengths are similar.

-Eric




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