Definitions and approximations

In pipes the vibrating medium is the gas contained in the pipe (usually air). Let c be the speed of sound in the gas.

Elastic vibrations in a gas are only longitudinal, which brings us back to the case of longitudinal vibrations in bars.

The formulas below hold if we assume that the pipe does not lose acoustic energy from its ends. In other words, we assume that the sound wave stays trapped in the pipe, without being radiated outwards. This case clearly does not represent a useful real musical instrument, since the sound of such a pipe could not be heard outside. However, this model case is an important key to understanding a model instrument, which approximates the resonance frequencies well in many real cases. In other cases it is a first approximation, which can however be refined, if we are interested only in the fundamental frequency, by applying an effective-length correction. In general, however, the correction depends on frequency, so an accurate simulation of a real instrument requires much more detailed calculations.

Cylindrical pipes

Open pipes

The fundamental frequency of a cylindrical pipe open at both ends is given by

${\displaystyle f_{0}={\frac {c}{2L}}}$

The higher frequencies are integer multiples of its fundamental frequency f₀

${\displaystyle f_{n}=nf_{0}\;}$

Half-open pipes

The fundamental frequency of a cylindrical pipe open at one end and closed at the other is given by

${\displaystyle f_{0}={\frac {c}{4L}}}$

The higher frequencies are odd integer multiples of its fundamental frequency f₀

${\displaystyle f_{n}=(2n+1)f_{0}\;}$

Examples of instruments using cylindrical pipes: the flute. The trumpet and the clarinet have a mainly cylindrical bore, but with a final bell that modifies some of its acoustic properties.

Conical pipe

Conical or truncated conical pipes have the same natural frequencies as an open-open cylindrical pipe whose length equals the height of the cone (or of the truncated cone)

Links

Besides the pages on the individual instruments, for an approach based on the concept of acoustic impedance see the simulations of model pipes of various shapes, in particular:

• Acoustics of model wind instruments
• Cylindrical pipe
• Conical pipe
• Modified cylindrical pipe
• Mixed pipe