Open and Closed Pipe Resonance Calculator
Find the fundamental frequency and harmonics of an air column in a pipe open at both ends or closed at one end (resonance tube), with end correction, speed of sound from temperature and resonance lengths for a tuning fork.
How it works
speed of sound v = 331.3 √(1 + T/273.15) m/s · closed at one end: fₙ = n v ÷ 4(L + 0.6r), n = 1, 3, 5… · open at both ends: fₙ = n v ÷ 2(L + 1.2r), n = 1, 2, 3… · resonance lengths for frequency f (closed): L = λ/4 − 0.6r, 3λ/4 − 0.6r, with λ = v ÷ f
An air column resonates when a whole number of quarter-wavelengths (closed pipe) or half-wavelengths (open pipe) fits in it. A closed end is a displacement node and an open end an antinode, which is why a pipe closed at one end sounds an octave lower than an open pipe of the same length and has only odd harmonics. The antinode actually forms a little outside an open end, so about 0.6 × the radius is added for each open end (Rayleigh’s end correction). In a school resonance-tube experiment the difference between the first two resonance lengths is exactly half a wavelength, which cancels the end correction.
Worked example
A tube closed at one end, 0.5 m long and 4 cm wide, in air at 20 °C: v = 331.3 √(1 + 20/273.15) = 343.21 m/s, the end correction is 0.6 × 2 cm = 1.20 cm, and f₁ = 343.21 ÷ (4 × 0.512) = 167.6 Hz, with harmonics at 3, 5, 7… times that. A 512 Hz tuning fork (λ = 0.6703 m) resonates when the air column is 0.1556 m and 0.4908 m long.
Sources: HyperPhysics — Open and closed air columns · UNSW Physics — Open vs closed pipes
Frequently asked questions
What is the fundamental frequency of a closed pipe?
f₁ = v/(4L), because a quarter wavelength fits between the closed end (node) and the open end (antinode). Adding the end correction, use L + 0.6r.
Why does a closed pipe have only odd harmonics?
The closed end must always be a node and the open end an antinode, so only 1, 3, 5… quarter-wavelengths fit. An open pipe has antinodes at both ends and supports every harmonic.
What is the end correction?
The pressure node at an open end sits slightly beyond the pipe, so the pipe acts longer than it is — by about 0.6 times its radius for each open end (0.61r for an unflanged pipe).
How do I find the speed of sound with a resonance tube?
Find two successive resonance lengths L₁ and L₂ for a tuning fork of frequency f. Then λ = 2(L₂ − L₁) and v = fλ; the end correction cancels out.
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This calculator is for educational purposes. Double-check important results before acting on them.