Normal modes of a conical pipe

• Consider a conical pipe. Many wind instruments, such as the oboe, the bassoon or the saxophone, are built from a conical pipe.
• The normal modes of the conical pipe are characterised by a single integer, as for the cylindrical pipe, but their shape is different, since the pipe is asymmetric.
• 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 (truncated cone), and a pipe closed at one end and open at the other (complete cone).
• The following animations show how the sound pressure varies inside the pipe.

Truncated conical pipe (open-open)

• 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.
• Unlike in the cylindrical pipe, the shape of the pressure wave is not periodic.
• 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}, ${\displaystyle f=4f_0}$

Complete conical pipe (closed-open)

• 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 natural frequencies are still integer multiples of the fundamental frequency, unlike what happens for the closed-open cylindrical pipe. This is only apparently paradoxical (for further details see the page wave equation in a pipe)
• Consequently a complete (closed-open) conical pipe can emit all the multiples of the fundamental frequency, not only the odd ones. This has important consequences for the timbre of instruments based on half-open pipes, such as the oboe or the saxophone.

	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}, ${\displaystyle f=4f_0}$

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, conical pipe, mixed pipe
• Mathematical details of the wave equation in the various cases are on the page wave equation in a pipe