In brass instruments and in the didgeridoo the sound starts at the musician's lips, which vibrate against the mouthpiece and close it like a reed. The tube, however, is not a simple cylinder: it widens, gradually or abruptly, up to the bell. In these experiments with TubeLabLight we see how the widening shifts the resonances of the tube. It is best to follow the path on cylindrical pipes first.

The exponential horn

1. Instructions
   • Open TubeLabLight with the Exponential horn project: a metal tube 875 mm long, whose radius grows from 6 to 22 mm following an exponential curve, as in a natural trumpet without valves (but without a mouthpiece). The Mouthpiece is closed by the lips.
   • Read the frequencies of the modes and the ×f₁ column (ratio to the fundamental). Listen with ▶ Listen.
   • Now turn the horn into a cylinder: press Make it cylindrical (the radius of all six points becomes 6 mm). Compare the frequencies of the modes with those of the horn.
2. Remarks
   • The cylinder closed by the lips has its fundamental at 96 Hz and its modes in the ratios 1, 3, 5, 7, ...: the odd series of the pipe closed at one end.
   • In the horn the fundamental rises to 155 Hz and the modes are in the ratios 1; 2.05; 3.25; 4.48; 5.73; ...: they are no longer the odd multiples, and the first ones approach the complete harmonic series. The widening has raised mostly the low modes: low-frequency waves are reflected early, where the tube begins to widen, while high-frequency waves continue as far as the bell.
   • In a real trumpet the shape of the mouthpiece and of the bell is designed so that the resonances are almost exactly harmonic: this is how a natural trumpet can play the series of natural harmonics. Our horn, without a mouthpiece, gets only partly there.
3. Related topics: see How the timbre of the trumpet is formed, The natural harmonics of the trumpet, Impedance matching
4. To find out more: see Mixed pipe, Acoustic impedance, Reflection in a horn (path with the 2D Waves app)

The didgeridoo

1. Instructions
   • Choose the Didgeridoo project from the Project menu: a eucalyptus trunk hollowed out by termites, 1538 mm long, with a radius that grows slowly from 16 to 22 mm and then opens into a bell up to 45 mm. The Mouthpiece is closed by the lips.
   • Listen with ▶ Listen: this is the drone typical of the instrument, on the fundamental.
   • Read the frequencies of the modes and the ×f₁ column, and look at the shape of the first modes by clicking the dots in the chart.
2. Remarks
   • The fundamental is about 71 Hz (the app shows C♯2 +49¢, that is, almost D2). The didgeridoo is almost always played on this note.
   • The following modes are in the ratios 2.57; 4.10; 5.62; 7.16; ...: neither the odd multiples of the closed cylinder (3, 5, 7) nor the complete series (2, 3, 4). The slightly conical bore and the bell shift them downwards compared with the cylinder: their spacing is about 1.5 times the fundamental instead of 2.
   • These "inharmonic" modes are not heard as separate notes, but they colour the sound. The player, by changing the shape of the mouth and throat while playing, reinforces now one, now another region of the spectrum and creates the characteristic "talking" sound of the didgeridoo. TubeLabLight does not simulate this effect: it computes only the sound of the tube.
3. Related topics: see Musical instruments, Sound and resonance, Conical bore
4. To find out more: see Perception of timbre, Human voice

What is the bell of the didgeridoo for?

1. Instructions
   • With the Didgeridoo project loaded, note the frequencies of the first five modes.
   • Press Remove the bell: the radius of the last three points (x = 1310, 1458 and 1538 mm) becomes 22 mm and the bell disappears. Compare the frequencies.
   • Then press Make it cylindrical: the radius of all points becomes 20 mm and the tube becomes a cylinder.
2. Remarks
   • Without the bell the fundamental drops from 71 to 64 Hz, almost two semitones; the second mode also drops considerably, while from the third mode upwards the modes change much less. The bell, as in the horn, raises mostly the low frequencies.
   • The cylinder of the same length has its fundamental at 56 Hz and its modes in the ratios 1, 3, 5, 7: the odd series. The real didgeridoo is "halfway" between a cylinder and a cone.
   • For didgeridoo makers this means that small changes to the bell shift the drone note without changing the high modes much, and vice versa. TubeLab, the full version of this app, was created precisely to design instruments of this kind.
3. Related topics: see Impedance matching, Mixed pipe, Cylindrical bore
4. Other experiments
   • Reload the project and shorten the tube by 5% with the −5% button next to Length: the fundamental rises from 71 to 75 Hz, almost a semitone. Lowering the temperature from 28 °C to 18 °C, on the other hand, changes it by less than a third of a semitone (from 71.3 to 70.1 Hz).