Standing Waves in Open and Closed Pipes

In an air column, standing sound waves have displacement nodes at closed ends and displacement antinodes at open ends; an open pipe of length L has frequencies f_n = nv ÷ 2L (n = 1, 2, 3, …, all harmonics), while a closed pipe (one end closed) has f_n = (2n − 1)v ÷ 4L (only odd harmonics). In brief: a closed pipe sounds an octave lower than an open pipe of the same length, since its fundamental wavelength is 4L, compared with 2L. The resonance tube experiment gives the speed of sound from the lengths at which resonance occurs. Because the real antinode lies a little beyond the open end, an end correction of about 0.6r (r is the radius of the pipe) is added to the length.

Organ pipes, flutes, clarinets and the sound of blowing across a bottle all depend on standing waves in air columns. This guide explains the boundary conditions for open and closed ends, derives the harmonics of both types of pipe, discusses the end correction and the resonance tube experiment, compares the pipes in a table, works through numerical examples and corrects common mistakes.

What are the boundary conditions?

A sound wave in a pipe is a longitudinal wave of displacement of air molecules (or pressure variation). At a closed end, the air cannot move along the pipe, so there is a displacement node (and a pressure antinode, where the pressure variation is greatest). At an open end, the air is free to move, so there is a displacement antinode (and a pressure node, since the pressure is close to atmospheric). The sound waves traveling along the pipe are reflected at both ends, even at open ends (partially, because of the sudden change in the conditions as the wave leaves the pipe), and the reflected waves superpose with the incident waves to form standing waves. The resonant frequencies are those at which the boundary conditions at the two ends are satisfied.

What are the harmonics of an open pipe?

With antinodes at both ends, the length of the pipe must be a whole number of half-wavelengths: L = nλ ÷ 2. Therefore

f_n = nv ÷ 2L, n = 1, 2, 3, ….

The fundamental has one node in the middle, and wavelength 2L. All the harmonics (f, 2f, 3f, …) are present, as for a string fixed at both ends. The flute and the organ’s open diapason pipes are open pipes, and they have a bright, rich sound.

What are the harmonics of a closed pipe?

With a node at the closed end and an antinode at the open end, the length is an odd number of quarter-wavelengths: L = (2n − 1)λ ÷ 4. Hence

f_n = (2n − 1)v ÷ 4L, that is, f₁, 3f₁, 5f₁, … with f₁ = v ÷ 4L.

The fundamental wavelength is 4L, so the pipe sounds an octave lower than an open pipe of the same length, and only odd harmonics are present, which gives the hollow, “woody” tone of the clarinet (which behaves approximately as a closed pipe) and a stopped organ pipe. The end correction: the antinode at an open end is slightly outside the pipe, at about 0.6r, so the effective length is L + 0.6r for one open end (L + 1.2r for two). Humans’ vocal tract resonances and the sound of a bottle also come from the air column.

Open and closed pipes compared

Worked example: pipes and the resonance tube

1. Open pipe. L = 0.50 m, v = 343 m/s: f₁ = 343 ÷ (2 × 0.50) = 343 Hz; f₂ = 686 Hz; f₃ = 1,029 Hz.
2. Closed pipe. L = 0.50 m: f₁ = 343 ÷ (4 × 0.50) = 171.5 Hz; the next resonances are 3f₁ = 514.5 Hz and 5f₁ = 857.5 Hz. The even multiples (343 Hz, 686 Hz) are absent.
3. Length required. To produce the note A (440 Hz) as the fundamental in an open pipe: L = v ÷ 2f = 343 ÷ 880 = 0.39 m. In a closed pipe: L = 343 ÷ 1,760 = 0.195 m, half as long, which is why bass instruments use closed or folded pipes.
4. Resonance tube experiment. A tuning fork of 512 Hz is held above a tube that is closed at the bottom by water, and the water level is lowered. The first resonance occurs at an air column length L₁ = λ/4 − e, and the second at L₂ = 3λ/4 − e, where e is the end correction. Then L₂ − L₁ = λ/2, independent of e. With λ = 343 ÷ 512 = 0.670 m: L₁ ≈ 0.1675 − e, L₂ ≈ 0.5025 − e. If the measured lengths are 0.150 m and 0.485 m, then λ = 2 × (0.485 − 0.150) = 0.670 m and v = fλ = 512 × 0.670 = 343 m/s. The end correction e = 0.1675 − 0.150 = 0.0175 m (about 1.8 cm; for a tube of 3 cm radius, 0.6r = 1.8 cm ✓).
5. Temperature. On a cold day, with v = 325 m/s, the pipe in item 1 plays f₁ = 325 Hz, lower by 5 percent. Organs go flat in cold churches.

How were these instruments developed?

Flutes made of bone, more than 40,000 years old, have been found in Germany. The pipe organ was invented by Ctesibius in Alexandria in the third century BCE (the hydraulis). Daniel Bernoulli analyzed the vibrations of air columns in the 1760s, and Hermann von Helmholtz and Lord Rayleigh developed the theory of resonance and the end correction in the 19th century. Kundt’s tube (1866) used dust patterns in a resonating tube to show the nodes and measure the speed of sound in gases.

Which mistakes are common?

Mistake 1: Confusing displacement and pressure nodes

A displacement node at a closed end is a pressure antinode. At an open end the opposite is true.

Mistake 2: Thinking that a closed pipe has the same fundamental as an open pipe of the same length

The closed pipe is an octave lower, since its fundamental wavelength is 4L, not 2L.

Mistake 3: Assuming that all harmonics are present in a closed pipe

Only odd multiples of the fundamental occur.

Mistake 4: Ignoring the end correction in precise work

It adds about 0.6r at each open end, and matters for narrow pipes of short length.