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Fundamental of a Closed Pipe

f=v4Lf = \frac{v}{4L}
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Worked example: 25 cm closed pipe (v = 343) → f = 343 Hz. Press Try an example to run it live, then adjust anything.

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Where Fundamental of a Closed Pipe fits

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Harmonic FrequenciesWave Speed (v = fλ)Period-Frequency RelationAngular Frequency (ω = 2πf)
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See also

Fundamental of an Open Pipe: The same pipe with both ends open, which sounds an octave higher and keeps its even harmonics.Fundamental Frequency of a String: Both use fundamental frequency.Harmonic Frequencies: Both use fundamental frequency.Doppler Effect (Approaching Observer): Both use speed of sound.Doppler Effect (Approaching Source): Both use speed of sound.Speed of Sound in Air: Both use speed of sound.
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Fundamental of a Closed Pipe explained

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A pipe sealed at one end must have a node (no air motion) at the closed end and an antinode (maximum motion) at the open mouth. The longest wave satisfying both fits just a quarter of a wavelength inside, so the fundamental is v/4L: an octave lower than an open pipe of the same length. Blow across a 20 cm test tube and you get roughly 343/(4 × 0.20) ≈ 429 Hz, close to concert A.

This quarter-wave trick is why a clarinet plays nearly an octave deeper than a flute of the same length: the reed end acts closed while the flute is open at both ends. It also explains the closed pipe's hollow timbre: the boundary conditions only permit odd harmonics (f, 3f, 5f...), stripping out every even overtone. Your own ear canal is a closed pipe about 2.5 cm deep, resonating near 3400 Hz, which is precisely where human hearing is most sensitive, and where audiologists find the most noise damage.

Fundamental of a Closed Pipe formula

f=v4Lf = \frac{v}{4L}
Where
  • ff= Fundamental frequency (Hz)
  • vv= Speed of sound (m/s)
  • LL= Pipe length (m)
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The wave equations and harmonics

7 formulasWave speed, period and frequency, the speed of a wave on a string, and the harmonic series for strings and pipes that decides what note you hear.

Revised 16 October 2026
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formula.expert. (2026, October 16). Fundamental of a Closed Pipe Formula & Calculator. https://formula.expert/solvers/closed-pipe-fundamental

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