Last updated: 2026-06-16

Pipe resonance

A pipe filled with air resonates at certain frequencies, the same way a guitar string resonates at certain pitches. When a sound wave reflects back and forth inside the air column and reinforces itself, a standing wave forms and the pipe sounds a clear musical note. The frequencies at which this happens depend on the length of the pipe, the speed of sound in the air, and whether the pipe is open at both ends or closed at one end. This calculator returns the resonant frequency for a chosen harmonic, its wavelength, and the fundamental frequency of the pipe.

Open and closed pipes

The boundary conditions at the ends of the pipe decide which standing waves can fit. At an open end the air is free to move, producing a displacement antinode; at a closed end the air cannot move, producing a displacement node.

• A pipe open at both ends has an antinode at each end and fits half a wavelength at its fundamental. It supports every harmonic, so its resonances are $f_n = \frac{n\,c}{2L}$ for $n = 1, 2, 3, \dots$
• A pipe closed at one end has a node at the closed end and an antinode at the open end, fitting only a quarter of a wavelength. It supports only the odd harmonics, so its resonances are $f_n = \frac{(2n-1)\,c}{4L}$.

Because a closed pipe needs only a quarter wavelength where an open pipe needs a half, a closed pipe of the same length sounds one octave lower.

Formula

The fundamental is the first harmonic $f_1$, and the wavelength of any resonance is $\lambda = c / f_n$.

Worked example

Consider an organ flue pipe open at both ends, 0.50 m long, in air where the speed of sound is 343 m/s. Its fundamental frequency is

which is close to the F above middle C. The second harmonic sounds at 686 Hz, the third at 1029 Hz, and so on. If the same pipe were closed at one end, its fundamental would drop to $\frac{343}{4 \times 0.50} = 171.5$ Hz — an octave lower — and only the odd harmonics would sound.

Limits of the model

The formulas treat the pipe as an ideal one-dimensional air column. In a real pipe the air just beyond an open end also vibrates, so the effective length is slightly greater than the physical length. This end correction, roughly 0.6 times the pipe radius per open end, lowers the resonant frequencies a little and matters for accurate instrument tuning. The speed of sound itself depends on temperature and on the gas in the pipe, so a warm pipe sounds sharper than a cold one.