By Zencalcy Editorial TeamLast reviewed About us
A tube supports standing waves when reflections at its ends line up. An open pipe has pressure nodes at both ends, giving every harmonic, while a closed pipe has a node at the open end and an antinode at the closed end, giving only odd harmonics. This is why clarinets and stopped organ pipes sound hollow compared with flutes.
Standing-wave frequencies for open and closed pipes.
Open: f_n = n v / 2L Closed: f_n = (2n − 1) v / 4L
n counts the harmonic of the open tube, or the resonant mode of the closed tube. The closed fundamental is half the open fundamental for the same length.
| Symbol | Meaning |
|---|---|
| f_n | Resonant frequency. |
| n | Harmonic or mode number, starting at 1. |
| v | Speed of sound in the tube. |
| L | Effective tube length. |
Open at both ends, the fundamental is 343 / (2 × 0.5) = 343 Hz. Closed at one end, it is 343 / (4 × 0.5) = 171.5 Hz, with the next resonance at 514.5 Hz.
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The closed end forces a pressure antinode and the open end a node. Only odd multiples of a quarter wavelength fit that pattern, so the even harmonics are absent.
Results are computed from the standard equations of classical mechanics. Unit conversions use the SI definitions published by NIST. Air resistance is ignored unless the calculator names a drag model.
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Content updated: September 2026