Modes of Vibration in an Open Pipe

Open-pipe boundary condition

Let a vibrating tuning fork be held near one end of an open pipe of length L. Because both ends are open, air molecules can vibrate freely at both ends. Each open end is therefore a displacement antinode.

Fundamental frequency or first harmonic

The lowest-frequency pattern has one displacement node at the centre and an antinode at each end. The pipe length contains one half-wavelength.

Using v=fλ:

Second harmonic

The next allowed pattern contains two half-wavelengths, or one complete wavelength, along the pipe. It has antinodes at both ends and at the centre.

Third harmonic

The third allowed pattern contains three half-wavelengths along the pipe.

General nth harmonic

In the nth mode, n half-wavelengths fit into the pipe length. The mode number n can be any positive whole number.

Mode comparison

All harmonics are present

Because n may take every positive integer value, an open pipe supports f₁, 2f₁, 3f₁, 4f₁ and so on. Thus all harmonics are present.

Common mistakes

Check your understanding

Quick revision

• Both ends of an open pipe are displacement antinodes.
• In the fundamental mode, half a wavelength fits in the pipe: L=λ₁/2.
• The fundamental wavelength is λ₁=2L and the fundamental frequency is f₁=v2L.
• The second harmonic contains one complete wavelength: λ₂=L and f₂=2f₁.
• The third harmonic contains three half-wavelengths: λ₃=2L/3 and f₃=3f₁.
• For the nth harmonic, L=nλₙ/2 and λₙ=2L/n.
• The general frequency is fₙ=nv/(2L)=nf₁, where n=1,2,3,….
• All harmonics—first, second, third and higher—are present in an open pipe.