Air Columns & Organ Pipes
A pipe closed at one end plays only odd harmonics; an open pipe plays them all.
Lesson at a glanceShow the key points
- Opposite sound waves give p = 2p₀ cos kx sin ωt: pressure nodes and antinodes λ/2 apart; a pressure node is a displacement antinode and vice versa.
- Closed end: pressure antinode (displacement node). Open end: pressure node (displacement antinode), slightly outside the pipe (end correction).
- Closed pipe: l = (2n + 1)λ/4 (16.14), ν = (2n + 1)v/4l (16.15); fundamental v/4l; only odd harmonics.
- Open pipe: l = nλ/2 (16.17), ν = nv/2l (16.18); fundamental v/2l; all harmonics, so a richer tone.
Blow across the top of a bottle and it hums one particular note. Add water and the note rises. The air inside is a column with its own natural frequencies, just like a string.
Standing sound waves
Two identical sound waves travelling in opposite directions along a tube add to give
Every point oscillates in step, with an amplitude that depends on its position: a standing wave.
- Pressure nodes, where the pressure never changes, are where cos kx = 0. Pressure antinodes, where it changes most, are where |cos kx| = 1. Neighbouring nodes (or antinodes) are λ/2 apart.
- Since pressure and displacement are π/2 out of phase, a pressure node is a displacement antinode and a pressure antinode is a displacement node.
Conditions at the ends
The air can’t move into the wall: displacement node, so pressure antinode.
The air connects to the open atmosphere, so its pressure stays normal: pressure node, displacement antinode.
Strictly, the pressure node at an open end sits a little outside the tube, because the air just beyond the opening also takes part. The extra length is the end correction, about 0.6 times the tube’s radius. We ignore it unless told otherwise.
Closed organ pipe
An organ pipe is closed at one end and open at the other. It needs a pressure antinode at the closed end and a node at the open end. Node to the nearest antinode is λ/4, so the length must be an odd number of quarter-wavelengths:
Open organ pipe
Open at both ends, the pipe has a pressure node at each end. The length must fit a whole number of half-wavelengths:
A pipe closed at one end has its first overtone in unison with a 340 Hz fork. Speed of sound 340 m/s. How long is the pipe?
List the three lowest natural frequencies of a 50 cm pipe open at both ends (v = 340 m/s).
- 1Standing sound wave p = 2p₀ cos kx sin ωt; pressure nodes are displacement antinodes, λ/2 apart.
- 2Closed end: pressure antinode. Open end: pressure node (slightly outside: end correction).
- 3Closed pipe: ν = (2n + 1)v/4l, odd harmonics only.
- 4Open pipe: ν = nv/2l, all harmonics.
Check yourself
Keep exploring
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