In this series of experiments we use TubeLabLight to study the simplest of all wind instruments: a cylindrical tube. We will see how its resonance frequencies depend on the mouthpiece (open or closed), on the length, on the width, on the temperature of the air and on the walls. If you have never used the app, first read the short guide on the page Guided paths for the TubeLabLight app.

Pipe open at both ends

1. Instructions
   • Open TubeLabLight with the Flute project: a cylindrical tube 644 mm long with a radius of 9.5 mm. The Mouthpiece is open, as in the flute.
   • Read in the table of modes the frequencies of the first modes (M1, M2, M3, ...) and the ×f₁ column (ratio to the fundamental).
   • Click the dots of the first three modes in the chart one at a time and look at the blue curve that appears in the drawing of the tube.
   • Press ▶ Listen to listen to the sound.
2. Remarks
   • The fundamental is about 265 Hz (the app shows C4) and the following modes are, to a very good approximation, whole multiples of the fundamental: 2, 3, 4, ... times f₁. This is the complete harmonic series, the same as for a vibrating string.
   • The blue curve is the amplitude of the sound pressure along the tube. It vanishes near the two open ends (where the pressure stays at atmospheric pressure: a pressure node) and mode number n has n−1 nodes inside the tube: these are the standing waves of the air column.
   • With the formula for the ideal open-open pipe, ${\displaystyle f_{1}=c/2L}$, with c = 348 m/s (the speed of sound at 28 °C) and L = 0.644 m, one finds 270 Hz. The frequency computed by the app is a little lower because at the open end the wave is not reflected exactly at the rim but slightly beyond it: the tube behaves as if it were a little longer (end correction).
3. Related topics: see Cylindrical bore, Normal modes of a cylindrical pipe, Natural frequencies of pipes, Standing waves in 1D
4. To find out more: see Effect of friction and radiation, Wave equation in a pipe, Flute

Pipe closed at one end

1. Instructions
   • With the Flute project loaded, change the Mouthpiece to closed (reed or lips): now the mouthpiece is closed, as if the tube were played with a reed. The tube does not change.
   • Read the frequencies of the modes and the ×f₁ column again, and look at the shape of the first modes by clicking the dots.
   • Listen with ▶ Listen and compare with the sound of the open pipe (just go back to open (as in the flute)).
2. Remarks
   • The fundamental drops to about 132 Hz: an octave below the open pipe of the same length. The formula for the ideal closed-open pipe is in fact ${\displaystyle f_{1}=c/4L}$.
   • The following modes are the odd multiples of the fundamental: 3, 5, 7, ... times f₁. The even multiples are missing.
   • At the closed end the blue curve has a maximum (a pressure anti-node): the air cannot move against the closed wall, and the pressure oscillates as much as possible. At the open end there is always a node.
   • The even harmonics of the sound fall halfway between two resonances, where the tube responds weakly. For a cylindrical reed tube, moreover, the app assumes that the reed opens and closes like a square wave, which contains only odd harmonics. This is, to a good approximation, what happens in the low register of the clarinet, which has a nearly cylindrical bore closed by the reed.
3. Related topics: see Ideal semi-open bore, Normal modes of a cylindrical pipe, Natural frequencies of pipes
4. To find out more: see Clarinet, Path: clarinet and oboe

Length and width of the tube

1. Instructions
   • Reload the Flute project from the Project menu.
   • Press the +5% button next to Length three times: each time the tube gets 5% longer, up to about 746 mm. Note the fundamental; then undo the changes with Undo.
   • Now press the +5% button next to Width five times: the radius goes from 9.5 to about 12 mm. Look at the fundamental and at the Q value of the first modes. Also try making the tube narrower with −5%.
2. Remarks
   • Making the tube 16% longer lowers the fundamental from 265 to about 229 Hz, that is by 14% (265/1.16 = 229): the frequencies are inversely proportional to the length. This is why lower-pitched instruments are longer, and why open holes, which "shorten" the tube, raise the note.
   • Widening the tube leaves the frequencies practically unchanged: the pitch depends on the length, not on the width (as long as the tube stays narrow compared with the wavelength).
   • The quality factor does change: for the first mode Q goes from 47 to about 59. In a wider tube a smaller fraction of the air is close to the walls, where energy is lost through friction, and the resonances are narrower. Making the tube narrower decreases Q.
3. Related topics: see Natural frequencies of pipes, Cylindrical bore, Resonance
4. To find out more: see Reflection in a tube with holes (path with the 2D Waves app), Modified cylindrical pipe

Temperature and walls

1. Instructions
   • Reload the Flute project.
   • Move the Temperature slider from 28 °C to 18 °C, then to 8 °C, and see how the fundamental changes (and the cents shown next to the note).
   • Set the temperature back to 28 °C and change the Walls to ideal (no friction), then to rough (wood) and back to smooth (metal). Look at the Q column and at the shape of the peaks in the chart.
2. Remarks
   • From 28 °C to 18 °C the fundamental drops from 264.8 to 260.5 Hz, about 29 cents (3 cents per degree); at 8 °C it drops to 256 Hz. The frequencies are proportional to the speed of sound, which increases with the temperature of the air. This is why wind players "warm up" the instrument by blowing into it before tuning.
   • In the ideal frictionless tube the peaks become very tall and very narrow: Q rises to thousands, because the only remaining energy loss is the sound radiated from the open end. With real walls Q falls to a few tens.
   • Friction at the walls also lowers the frequencies slightly (about 3 Hz for the fundamental, compared with the ideal tube). Between metal and wood the differences are small: in a smooth tube the material matters little.
3. Related topics: see Speed of sound, Resonance, Effect of friction and radiation
4. To find out more: see Acoustic impedance, Speed of sound in different media