Vibrations of Air Columns: Open and Closed Pipes Guide

Blow across a bottle top and the air column inside sings — the air vibrates as a standing wave, and the pipe's ends decide which notes are possible. An open pipe (open at both ends, like a flute) and a closed pipe (stopped at one end, like a clarinet or a bottle) follow different rules, and the difference is written in their harmonics.

These rules built pipe organs, woodwinds, and mufflers. The key: an open end is a displacement antinode (air rushes freely — a pressure node), while a closed end is a displacement node (air can't move — a pressure antinode).

Key Concepts

• Open pipe (both ends open): antinodes at both ends. Allowed: λ_n = 2L/n, so f_n = nv/(2L), n = 1, 2, 3, … — all harmonics present.
• Closed pipe (one end closed): node at the closed end, antinode at the open end. Only odd quarter-wavelengths fit: λ_n = 4L/n (n odd), so f_n = nv/(4L), n = 1, 3, 5, … — only odd harmonics.
• End correction: the antinode sits slightly outside the open end (by e ≈ 0.6r, where r is the tube radius), so use effective length L + e (open pipe: L + 2e).
• Tone colour: missing even harmonics give the closed pipe a hollow tone (clarinet) versus the open pipe's bright tone (flute).

Formulas

• Open pipe: f_n = nv/(2L), n = 1, 2, 3, …
• Closed pipe: f_n = nv/(4L), n = 1, 3, 5, … (odd only)
• Units: f in Hz, L in m, v ≈ 343 m/s in air at 20°C.
• Assumptions and limits: uniform cylindrical tube, diameter much smaller than wavelength, small-amplitude sound, open ends at atmospheric pressure. Conical or flared pipes (oboe, trumpet bell) follow modified rules.

Worked Example

Two pipes, each L = 0.50 m, speed of sound v = 340 m/s.

1. Open pipe: f_1 = v/(2L) = 340/(2 x 0.50) = 340/1.0 = 340 Hz. Harmonics: 680, 1020, 1360 Hz — every multiple.
2. Closed pipe: f_1 = v/(4L) = 340/(4 x 0.50) = 340/2.0 = 170 Hz — exactly one octave lower than the open pipe of the same length.
3. Its next modes: f_3 = 3 x 170 = 510 Hz, f_5 = 850 Hz — the even harmonics (340, 680…) are simply absent.
4. Why clarinets overblow a twelfth, not an octave: jumping to the next available mode (3rd harmonic) multiplies frequency by 3, and a factor of 3 is an octave plus a fifth.

Common Misconceptions

• "A closed pipe has all the harmonics." No — only the odd ones (1, 3, 5…). The node-at-closed-end, antinode-at-open-end condition cannot fit even numbers of quarter wavelengths.
• "A longer pipe gives a higher note." Backwards — longer means lower (f ∛ 1/L). Organ pipes for bass notes are metres tall; piccolo pipes are centimetres.
• "The open end is a pressure antinode." Wrong — at the open end the pressure equalises with the atmosphere (pressure node), while the air displacement is maximal (displacement antinode). Don't mix up the two pictures.

Real-World Applications

• Flutes and piccolos are open pipes (embouchure hole + open end); clarinets are closed pipes (reed end acts closed).
• Pipe organs use stopped (closed) pipes to get deep bass from half the pipe length — saving space and metal.
• Car exhaust mufflers are tuned pipe resonators that cancel engine drone by destructive interference.
• Bottles and jugs (the "bottle band") are closed pipes — less water means a longer air column and a lower note.

To recap: open pipes sing at f_n = nv/(2L) with every harmonic; closed pipes at f_n = nv/(4L) with odd harmonics only — an octave lower for the same length. One boundary rule writes the whole woodwind family.