Explain the odd harmonics of a closed-open tube

Give the two ends different jobs

At the rigid closed left end, the air cannot move longitudinally through the wall. It is a displacement node and, for an ideal single mode, a pressure antinode. At the open right end, pressure perturbation is zero and displacement has an antinode. These conditions apply to different quantities at the same positions.

The browser shows pressure and displacement profiles separately, each normalized to its own amplitude. The plotted vertical directions are graph values, not transverse air motion. A pressure value of zero means no departure from ambient pressure; it does not mean a vacuum. The selected phase may show positive or negative extrema, but the node locations are phase-independent.

Fit a quarter wavelength first

The fundamental must go from a displacement node at the left to the nearest displacement antinode at the right. That distance is one quarter wavelength. Therefore L = λ/4, λ = 4L and f1 = c/(4L).

For a 1 m column and c = 340 m/s, the fundamental wavelength is 4 m and frequency is 85 Hz. The displacement node is at x = 0 and the pressure node at x = 1 m. The left-end pressure is a normalized extreme even though displacement there is zero.

A same-length open-open column has a 170 Hz fundamental under the same ideal speed assumption. Closing one end changes the boundary condition and halves the fundamental in this comparison; it does not halve sound speed. A real change in geometry can introduce additional effects excluded here.

Why the next mode skips an even harmonic

Starting from the closed-end displacement node, the right endpoint must still land on a displacement antinode. The next allowed span is three quarter-wavelengths, then five quarter-wavelengths, and so on. An even number of quarter-wavelengths would put a displacement node at the open end and violate the required condition.

If m counts allowed modes starting at 1, the harmonic number is h = 2m − 1. The allowed wavelengths and frequencies are λ = 4L/h and f = hc/(4L). Mode index and harmonic number are therefore different.

At L = 1 m the first five allowed modes have harmonic numbers 1, 3, 5, 7, 9 and frequencies 85, 255, 425, 595, 765 Hz. The second allowed mode is the third harmonic and the first overtone. The third allowed mode is the fifth harmonic and second overtone. Use the table’s separate columns to avoid confusing those labels.

For the second allowed mode, displacement nodes lie at x = 0 and 2/3 m. Pressure nodes lie at x = 1/3 and 1 m. The interior alternation between the two kinds of node reflects the spatial derivative relation between pressure and displacement. The open-end condition still holds even though the profile contains additional internal structure.

Read the chosen profile and its limits

For a mode with odd harmonic number h, one displacement shape is sin(hπx/(2L)); the corresponding normalized pressure profile is −cos(hπx/(2L)). Both endpoint conditions follow from these expressions. A negative pressure extreme at the closed end is a rarefaction at the selected phase, not an error in the boundary condition. Half a period later its sign would reverse while its antinode status remains.

The model gives ideal natural frequencies and relative spatial profiles. It does not calculate how strongly a driver excites each mode, the width of a resonance peak, radiation from the opening, wall losses or the sound heard by a listener. The absence of even natural harmonics in this ideal one-dimensional family is not a guarantee that every real instrument or microphone recording contains no even-frequency components.

Open-end corrections can shift measured frequencies because the effective acoustic boundary is not exactly at the geometric opening. The browser omits this correction and holds sound speed fixed; an accurate experiment must report geometry, environmental conditions and uncertainty rather than treating the displayed decimals as measured values.

Transfer using a measured fundamental

An idealized closed-open column has fundamental frequency 100 Hz. Predict its next two allowed frequencies and its length if c = 340 m/s. Then explain what a 200 Hz peak would require you to reconsider.

Check the closed-column transfer

The next allowed frequencies are 300 and 500 Hz. The inferred ideal length is 340/(4 × 100) = 0.85 m. A 200 Hz peak is not an allowed natural frequency of this ideal family; it calls for checking the source, recording, boundary condition or model assumptions. It should not be silently relabeled as the second allowed mode.

OpenStax normal modes of a standing sound wave develops the closed-open air-column series. The browser’s five-mode selection is bounded; the ideal sequence continues beyond it.