Tubes

Standing air modes in open and closed tubes. Theory, formulas, worked examples and interactive graphs.

Conceptual context

An acoustic tube is an air column that can sustain longitudinal standing waves. What oscillates is not a visible string, but the air: some regions alternate compression and rarefaction, while others act as reference points for the pattern. Tubes are therefore the classical example of resonance in wind instruments, organs, bottles, and sound cavities.

The key point of this leaf is that end type changes the frequency series. A tube open at both ends does not select the same modes as a tube open at one end and closed at the other. The effective length $L_{eff}$, sound speed $v$, and modal index $n$ or $m$ determine the frequency $f_n$.

🟢 Essential level

At an open tube end, air can move easily. That region behaves as a displacement antinode. At a closed end, the air next to the wall cannot move longitudinally, so a displacement node appears. This boundary difference completely changes which patterns fit.

The essential idea is to distinguish open-open tube and open-closed tube before calculating. The first admits a complete harmonic series. The second selects only odd patterns, because it must join an antinode at the open mouth with a node at the closed end. The physical length $L$ is useful, but real resonance often depends on $L_{eff}$.

A flute and a clarinet illustrate the difference. The flute behaves as an open-open tube and produces the full harmonic series. The clarinet, closed at the mouthpiece, behaves as open-closed and only excites odd harmonics, giving it a distinctive timbre even when the tube has a similar length.

[!NOTE] In tube acoustics, "open" and "closed" describe air-motion conditions, not only visual appearance.

🔵 Formal level

Basic formulas

For a tube open at both ends, the air column has displacement antinodes at both mouths. The effective length contains an integer number of half-wavelengths, and the modal frequency is:

Open-open tube frequency

The wavelength associated with the same mode is:

Wavelength in an open tube

Here $n$ takes consecutive integer values. Therefore the ideal open tube produces a complete harmonic series. If $L_{eff}$ remains fixed, doubling the modal index doubles the frequency, because the spatial pattern is compressed without changing the speed $v$.

This modal proportionality only holds after the open boundary has been correctly fixed.

Real student questions

Why is an open end a displacement antinode? Because air at the mouth can move outward and inward with little mechanical restriction.

Why is a closed end a displacement node? Because the wall prevents air from advancing longitudinally at that point.

Do all tubes have all harmonics? No. An ideal open-closed tube selects only the odd series.

Why use $L_{eff}$ instead of always using $L$? Because the acoustic boundary of an open mouth does not exactly coincide with the geometric edge.

Final synthesis

Tubes select frequencies through effective length, sound speed, and end type. An open tube uses consecutive integer modes; an open-closed tube uses odd factors. Understanding $f_n$ requires reading the acoustic boundary first and then applying the correct modal series.