Frequenze proprie delle canne en
From "Physics, Waves, Music": a website about the physics of waves and sound, acoustics of musical instruments, musical scales, harmony and music.
| AI-translated from Italian. Original page: Frequenze_proprie_delle_canne |
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: an open end behaves as if the pipe were a little longer, by about 0.6 times the radius of the pipe (at low frequencies)[1] (see the page cylindrical pipe). 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
The higher frequencies are integer multiples of its fundamental frequency f0
with n = 1, 2, 3, …
Half-open pipes
The fundamental frequency of a cylindrical pipe open at one end and closed at the other is given by
The higher frequencies are odd integer multiples of its fundamental frequency f0
with n = 0, 1, 2, …: the even harmonics are therefore missing.
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.
The flute behaves as a pipe open at both ends, because the embouchure is open to the outside. In the clarinet, instead, the reed end, which opens and closes only for brief instants, behaves as a closed end: to a first approximation the clarinet is a half-open pipe and sounds about an octave below a flute of the same length.
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
Check your understanding
Problem. A cylindrical pipe open at both ends is 60 cm long and has a radius of 9.5 mm (c = 343 m/s). Calculate the fundamental frequency without and with the end correction.
Solution. Without correction = 343/1.2 Hz ≈ 286 Hz. With the correction, each of the two open ends lengthens the pipe by about 0.6 · 9.5 mm ≈ 5.7 mm: the effective length is about 61.1 cm and the fundamental drops to 343/1.223 Hz ≈ 281 Hz, about 30 cents lower.Problem. The same 60 cm pipe is closed at one end. What are its first three natural frequencies (without end correction)?
Solution. = 343/2.4 Hz ≈ 143 Hz; the higher frequencies are 3 and 5 times the fundamental: 429 Hz and 715 Hz. The closed pipe sounds an octave below the open one and lacks the even harmonics.Problem. The lowest note of an organ keyboard is C2 (65.4 Hz). How long must an open pipe producing it be? And a closed (stopped) pipe?
Solution. Open pipe: = 343/130.8 m ≈ 2.62 m. This is the so-called "8-foot" pipe (about 2.4 m: the nominal length neglects the end correction and temperature). Stopped pipe: ≈ 1.31 m, half as long: this is why stopped pipes allow organs to produce low notes with shorter pipes.Notes
- ↑ H. Levine, J. Schwinger, On the radiation of sound from an unflanged circular pipe, Phys. Rev. 73, 383–406 (1948), doi:10.1103/PhysRev.73.383.
