How a tube makes a note
Blow across a bottle or into a flute, and the air inside the tube vibrates at particular frequencies. Sound waves travel along the tube, reflect at the ends and reinforce each other only when the tube length fits a whole pattern of the wave. These resonances set the notes of organ pipes, flutes, clarinets, didgeridoos, bottles and even the booming sound of some air ducts.
Open and closed pipes
- Open at both ends: both ends are pressure nodes. The longest wave that fits is twice the length of the pipe: f₁ = v ÷ 2L, and all multiples 2f₁, 3f₁… also resonate.
- Closed at one end: the closed end is a pressure maximum. The longest wave is four times the length: f₁ = v ÷ 4L, and only odd multiples 3f₁, 5f₁… resonate.
A closed pipe therefore sounds an octave lower than an open pipe of the same length, which is how stopped organ pipes save space.
Speed of sound and end correction
The speed of sound in air is about v = 331.3 × √(1 + T ÷ 273.15) m/s, with T in °C: 1,125 ft/s (343 m/s) at 68 °F. The sound wave also extends slightly past each open end, so the effective length is longer than the physical length by about 0.3 × diameter per open end.
Example
An open pipe 2 feet long and 1 inch wide at 68 °F:
- Effective length: 0.625 m (2.05 ft)
- Fundamental: 274.6 Hz, about 8 semitones below A4, close to C♯4
- Second harmonic: 549.3 Hz
- Wavelength of the fundamental: 4.1 ft
Closing one end of the same pipe lowers the fundamental to about 140 Hz.
Musical uses
- To tune a pipe to a note, find its frequency with the note frequency calculator, then adjust the length until the fundamental matches.
- Wind chimes made from tubes use the resonance of the metal, not the air inside, so they follow different formulas.
- A clarinet behaves like a closed pipe, which is why it overblows by a twelfth instead of an octave; a flute behaves like an open pipe.
Limits
The formulas assume a straight cylindrical pipe much longer than its width. Conical bores, such as those of the oboe and saxophone, behave like open pipes with all harmonics. Finger holes, bells and mouthpieces shift the frequencies, so instrument makers always fine-tune by ear and with a tuner.