RB-05Room & Building Acoustics
Pipe & Duct Resonance
Standing-wave frequencies of open and closed tubes with end corrections.
Tube
End conditions
m
cm
°C
Show mode
Mode 1 frequency
253.1Hz
Fundamental
253.1Hz
Effective length
0.678m
Cross-mode cut-on
6,700Hz
Standing wave inside the tube
PressureParticle velocity
Resonance seriesall harmonics
§ Theory of operation
Standing waves in tubes
A tube resonates when waves reflected from both ends reinforce each other. The boundary conditions decide which wavelengths fit: an open end is a pressure node (p ≈ 0, maximum air movement), a closed end is a pressure antinode (no air movement).
\begin{aligned} \text{open–open} &:\quad f_n = n\,c\,/\,2L_\mathrm{eff} \qquad n = 1,\,2,\,3\,\ldots\ \text{(all harmonics)} \\ \text{closed–open} &:\quad f_n = (2n-1)\,c\,/\,4L_\mathrm{eff} \qquad \text{odd harmonics only} \\ \text{closed–closed} &:\quad f_n = n\,c\,/\,2L \qquad \text{all harmonics} \end{aligned}
End correction
Air just beyond an open end still participates in the motion, so the acoustic length is longer than the physical length by about 0.6\cdot r per unflanged open end (0.85\cdot r if flanged). Closed ends need no correction.
Where you meet it
- Wind instruments: a flute is open–open; a clarinet behaves as closed–open, which is why it lacks even harmonics and overblows a twelfth.
- Ducts and exhausts: a side branch or stub tuned to \lambda/4 (closed–open) cancels a tonal frequency, the quarter-wave resonator silencer.
- Pipework: gas pulsations in compressor piping excite acoustic resonances that can fatigue pipes and supports.
- Room modes are the three-dimensional version of the closed–closed tube.
The plane-wave assumption holds while the diameter is less than about 0.58\,\lambda (first cross-mode cut-on, f \approx 1.84\,c/\pi d).