7.6 Modes of a 1-D tube

A cylindrical tube of length $L$, much narrower than the acoustic wavelength, supports plane-wave sound propagating along its axis. The reflections at the two ends are governed by the boundary conditions there, and the steady standing-wave patterns are the modes of the tube. The allowed frequencies depend on which ends are closed (rigid wall — pressure antinode) and which are open (radiating into free air — pressure node).

Boundary conditions

• Closed end: velocity is zero ($v' = 0$). Pressure is maximum (antinode).
• Open end: pressure is approximately zero ($p' = 0$, the radiation is back into the much larger volume of free air). Velocity is maximum (antinode).

(The “approximately” in the open-end condition reflects an end correction — a small effective extension of the tube length by $\sim 0.6\, a$ for a flanged open end of radius $a$, because the radiation field needs a small near-field region to look right. For tubes with $a \ll L$ this correction is negligible; for short tubes it matters.)

Three configurations, three mode patterns

1. Closed–closed tube (both ends rigid). Pressure antinodes at both ends, so the mode shapes are $p'_n(x) = \cos(n\pi x/L)$ for $n = 1, 2, 3, \ldots$ The allowed frequencies are

Fundamental $f_1 = c/2L$, harmonics at integer multiples.

2. Open–open tube (both ends open). Pressure nodes at both ends; mode shapes $p'_n(x) = \sin(n\pi x/L)$. Same frequency formula:

Same as closed–closed: complete integer harmonic series.

3. Closed–open tube (one end rigid, one end open). Pressure antinode at closed end, node at open end. Mode shapes $p'_n(x) = \cos\!\big((2n-1)\pi x / (2L)\big)$. Allowed frequencies:

Fundamental $f_1 = c/4L$ — half the frequency of the open-open tube of the same length. Only odd harmonics: $f_1, 3f_1, 5f_1, \ldots$

Musical consequences

The fundamental frequency depends linearly on $1/L$ and the boundary conditions:

• An open-open flute is a tube of length $L$ with fundamental $f_1 = c/2L$. For $L = 30$ cm, $f_1 = 343/0.6 \approx 570$ Hz.
• A closed-open clarinet is a tube of length $L$ with fundamental $f_1 = c/4L$ — half as high. For $L = 60$ cm, $f_1 = 343/2.4 \approx 143$ Hz.

The harmonic content also differs sharply. An open-open instrument has all harmonics; a closed-open instrument has only odd harmonics. This is why a clarinet sounds different from a flute even at the same pitch and loudness: the spectral structure of the standing wave is fundamentally different.

The interactive

Slide the tube length and select the boundary configuration. Watch the standing-wave pattern and the resonance frequencies update. Notice especially the difference between open-open and closed-open: same length, very different fundamentals.

Vocal tract preview

The human vocal tract is approximately a closed-open tube (closed at the larynx, open at the lips), of length about 17 cm in adult males. Its fundamental and harmonics — at roughly $500, 1500, 2500, 3500\,\text{Hz}$ ($c/4L, 3c/4L, \ldots$) — are the formants of speech, and they’re what tongue and lip position modulate when producing different vowels. We’ll see this in the hearing book.

The 3-D version: cavities

A 1-D tube is the simplest mode-bearing structure. The next lesson generalises to 3-D rectangular cavities, where modes are labelled by three integers and the mode density grows with frequency. That’s where room acoustics begins.