Normal modes of a cylindrical pipe

• Consider a cylindrical pipe. Many wind instruments, such as the flute, the clarinet or the trumpet, are built from a cylindrical pipe, even though they often have end bells of various shapes.
• Its normal modes are characterised by a single integer. It gives directly the number of (pressure) antinodes that the air in the pipe forms as it oscillates. For example, mode {2} means there are 2 antinodes.
• For a given length of the pipe, the natural frequency of each mode is determined only by this number and by the speed of sound in the gas filling the pipe.
• The particular shape of the modes depends on the so-called boundary conditions, that is, on the shape of the ends of the pipe. Let us examine two important cases: a pipe open at both ends, and a pipe closed at one end and open at the other
• The animations show how the sound pressure varies inside the pipe.

Open-open pipe

• In the most ideal case there are no energy losses either at the ends or inside the pipe, so the pressure at the ends equals atmospheric pressure (that is, the sound pressure is zero). This means there is always a node at the ends.
• The pipe can always contain only an even number of quarter wavelengths (or, equivalently, a whole number of half wavelengths)
• The natural frequencies of the pipe are integer multiples of the fundamental frequency.

	corda_freq_1.mp3 mode {1}, ${\displaystyle f=f_0}$
	corda_freq_2.mp3 mode {2}, ${\displaystyle f=2f_0}$
	corda_freq_3.mp3 mode {3}, ${\displaystyle f=3f_0}$
	corda_freq_4.mp3 mode {4}, ${\displaystyle f=4f_0}$

Closed-open pipe

• In this case, still referring to ideal conditions, the sound pressure has a node at the open end and an antinode at the closed end.
• The pipe can always contain only an odd number of quarter wavelengths
• The natural frequencies are odd multiples of the fundamental frequency.
• Consequently a closed-open pipe cannot emit even multiples of the fundamental frequency. This has important consequences for the timbre of instruments based on half-open pipes, such as the clarinet.

	corda_freq_1.mp3 mode {1}, ${\displaystyle f=f_0}$
	corda_freq_3.mp3 mode {2}, ${\displaystyle f=3f_0}$
	corda_freq_5.mp3 mode {3}, ${\displaystyle f=5f_0}$
	corda_freq_7.mp3 mode {4}, ${\displaystyle f=7f_0}$

Links and further reading

• See the section questions and answers for questions and curiosities about normal modes, musical instruments and more.
• Details of the acoustic properties of wind instruments can be found on the pages of the individual instruments, and also in the following sections: acoustics of model wind instruments, cylindrical pipe, modified cylindrical pipe, mixed pipe
• Mathematical details of the wave equation in the various cases are on the page wave equation in a pipe