Standing waves in air columns

Music from a column of air

Blow across the top of an empty bottle. You hear a low note. Pour some water in and blow again. The note is higher. A flute and an organ pipe work the same way. The air inside them holds a standing wave.

Sound in air is a longitudinal wave. The air moves to and fro along the pipe. A sound wave reflects at the ends of the pipe, as a wave on a string reflects at its ends. The wave going one way and the reflected wave going back add to a standing wave.

What happens at each end

• At a closed end, the air cannot move. It is a displacement node.
• At an open end, the air moves freely. It is a displacement antinode.

Where the air does not move, it is squeezed and stretched the most. So a displacement node is a pressure antinode, and a displacement antinode is a pressure node. In this lesson "node" means a displacement node.

A pipe closed at one end

The closed end is a node and the open end an antinode. A node and the nearest antinode are $\lambda/4$ apart. So the length holds one quarter wavelength, or three quarters, or five quarters:

We want the frequency, so $f = v/\lambda$ gives

The fundamental is $f_1 = \dfrac{v}{4L}$. The other modes are $3f_1$, $5f_1$ and so on. A closed pipe has only the odd harmonics.

A pipe open at both ends

Both ends are antinodes. Next antinodes are $\lambda/2$ apart, so

An open pipe has every harmonic, $f_1$, $2f_1$, $3f_1$ and so on, like a string. Its fundamental $v/2L$ is twice that of a closed pipe of the same length.

An example. Take two pipes, each $0.425\ \text{m}$ long, with sound at $340\ \text{m/s}$. The open pipe gives $\dfrac{340\ \text{m/s}}{2 \times 0.425\ \text{m}} = 400\ \text{Hz}$, then $800\ \text{Hz}$, $1200\ \text{Hz}$. The closed one gives $\dfrac{340\ \text{m/s}}{4 \times 0.425\ \text{m}} = 200\ \text{Hz}$, then $600\ \text{Hz}$, $1000\ \text{Hz}$.

Why do two instruments that play the same note still sound different? Each one sounds its fundamental with some of its harmonics. The mix of harmonics is different. A closed pipe, with no even harmonics, sounds hollow.

The end correction

The antinode at an open end is not exactly at the end. It sits a little outside, about $0.6r$ beyond it, where $r$ is the radius of the pipe. This small extra length is the end correction. A closed pipe has one open end, so its air column acts as $L + 0.6r$ long. An open pipe has two, so it acts as $L + 1.2r$ long. The resonance tube, in the next course, uses this.

Practise with new numbers

Try it in Python

The program lists the first four modes of an open pipe and of a closed pipe of the same length.

Common mistakes

• Putting an antinode at a closed end. The air cannot move there, so it is a node.
• Giving a closed pipe a second harmonic. It has only odd harmonics. Its first overtone is the third harmonic.
• Mixing up displacement nodes and pressure nodes. They are at opposite places.
• Forgetting that an open pipe has two ends to correct. Its end correction is twice that of a closed pipe.

Remember these

• Closed end: displacement node. Open end: displacement antinode.
• Closed at one end: $f = \dfrac{(2n - 1)v}{4L}$, odd harmonics only.
• Open at both ends: $f_n = \dfrac{nv}{2L}$, every harmonic.
• End correction about $0.6r$ at each open end.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A test tube has $12\ \text{cm}$ of air above the water in it. Treat it as a closed pipe with sound at $340\ \text{m/s}$. What is its lowest note?

2. An open pipe has a fundamental of $260\ \text{Hz}$. What is its first overtone?

3. One end of an open pipe is closed. What happens to its fundamental?

4. Where in a closed pipe is the pressure of the air changing most?