This is not a real musical instrument, but rather a sound toy. However, because of the particular way in which the sound is produced, this curious object is worth mentioning.

Structure and acoustic characteristics

This object, which is sold under various trade names, is a flexible plastic tube, open at both ends and corrugated along its whole length. The model shown here is 75 cm long and 2.5 cm in diameter, with ridges 6 mm apart. The tube produces sounds when it is held by one end and whirled rapidly.

These are the most remarkable features of the phenomenon:

1. A tube without corrugations produces no sound.
2. A tube with one end stopped produces no sound.
3. Below a certain limiting speed of rotation the tube produces no sound, while above that speed it emits a very soft sound of well-defined pitch.
4. This "fundamental" note depends on the length of the tube and on the spacing of the corrugations. Not all combinations of these two dimensions, however, produce sounding tubes.
5. As the speed of rotation increases the tube emits higher sounds, but the frequency of the note does not increase continuously with the speed of rotation: it "jumps" to a higher frequency when the speed of rotation exceeds certain critical values.
6. All the frequencies emitted by the tube correspond to its natural frequencies, and are therefore integer multiples of the fundamental frequency.
7. The sound does not depend on the particular shape that the tube takes while rotating.

Let us now see how to make sense of these observations.

How the sound is produced

The discrete frequencies that we hear correspond to the natural frequencies of a cylindrical tube whose length equals the effective length of the tube. There is therefore no doubt that they are due to the standing waves set up in the tube (see the page on the cylindrical pipe). We still have to understand, however, how these standing waves are excited by the rotation.

Points 1 and 2 of the previous section are obviously crucial. Point 1 tells us that the corrugation of the tube wall is necessary to obtain the sound. Point 2 shows that an air flow is needed, which passes through the tube and strikes the corrugations. The air flow is generated by the rotation, while it is clearly prevented if one end of the tube is stopped. Note that in this respect the rotating tube differs from wind instruments. In wind instruments, in fact, the air flow does not need to pass through the whole pipe, as one can easily understand by thinking, for example, of a stopped organ pipe, that is, one closed at one end.

• How a steady air flow is generated by the rotation is easily explained even in the simplest approximation (Bernoulli's theorem).

The rotation makes the speed of the air hitting the tube at the end that is held (close to the centre of rotation) much lower than the speed of the air at the free end (close to the periphery). As a consequence, at the periphery the air pressure will be much lower than the pressure at the held end, and it is this pressure difference that drives the air from the centre towards the periphery.

• Now that we have an air flow running along the tube, let us see how it interacts with the corrugations.

At each corrugation the air flow is perturbed and produces a small vortex and a jump in pressure. The frequency of this perturbation is directly proportional to the speed of the air in the tube and inversely proportional to the spacing of the corrugations.

• At this point we have a fluid-dynamic mechanism (the turbulent flow of the air) that can produce a perturbation at a given frequency, depending on the speed of rotation and on the corrugation, and an acoustic mechanism (the resonance of the tube) that can amplify certain frequencies, which depend essentially on the length of the tube. When the frequency of the perturbation in the fluid coincides with (or is very close to) a resonance frequency of the tube, the perturbation is amplified and we can hear a sound.

• We can therefore conclude that this is a case of resonance in which the periodic external force (or driving force) is not given once and for all, but depends on the motion of the system and on its corrugation.

References

1. Nakamura Y, Fukamachi N. Sound generation in corrugated tubes. Fluid Dynamics Research. 1991;7(5-6):255-261.

2. Cadwell LH. Singing corrugated pipes revisited. American Journal of Physics. 1994;62(3):224.