Closed Pipes Have Odd Harmonics Only; Open Pipes Have All
Air columns resonate when the wave fits the pipe with the right end conditions: a node at a closed end and an antinode at an open end. That one fact gives every formula for open pipes, closed pipes and resonance-tube experiments.
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Open and closed pipes
An open pipe has antinodes at both ends and resonates at fₙ = n v/(2L) for every whole number n.
If three successive resonances are 425, 595 and 765 Hz, the gaps of 170 Hz equal 2 f1 and the ratio 5 : 7 : 9 is odd, so the pipe is closed with f1 = 85 Hz and L = v/(4 f1) = 340/340 = 1 m. For a closed tube with neighbouring harmonics 460 and 500 Hz, the 40 Hz gap equals 2 f1, so the fundamental is 20 Hz.
Resonance-tube experiments
In a tube with an adjustable air column closed at one end, successive resonances are λ/2 apart.
The smallest closed-tube resonance is λ/4 and the next is 3λ/4, so if the shortest resonant length is 40 cm, the next is 120 cm.
For a 120 cm tube with a 330 Hz fork and v = 330 m/s (λ = 1 m), filling it with water leaves a closed air column of resonant lengths 25 cm and 75 cm, so the smallest water column is 120 - 75 = 45 cm.
Comparisons and changes
The second overtone of an open pipe is its third harmonic, 3v/(2 L_open); the first overtone of a closed pipe is also its third harmonic, 3v/(4 L_closed). Equating them for a closed pipe of 50 cm gives an open pipe of 1 m. An open pipe with half its length dipped in water behaves as a closed pipe of half the length, whose fundamental v/(4 x L/2) = v/(2L) equals the original fundamental n. When a closed tube is heated, both its length (L + l) and the speed of sound (V + v) increase, so the resonant frequency becomes (V + v)/[4(L + l)].
OlympIQ exam lens
- ✓Open pipe: fₙ = n v/(2L), all harmonics; closed pipe: f = (2n - 1) v/(4L), odd harmonics only
- ✓Successive odd harmonics of a closed pipe differ by 2 f1
- ✓Successive resonance lengths in a tube differ by λ/2; the first is λ/4 for a closed tube
- ✓Second overtone of an open pipe = third harmonic; first overtone of a closed pipe = third harmonic
- ✓An open pipe half dipped in water acts as a closed pipe of half the length with the same fundamental
Do not number the overtones the same way for open and closed pipes. A closed pipe has no even harmonics, so its first overtone is the third harmonic, whereas an open pipe's first overtone is the second harmonic.
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