Standing waves on a string

A wave meets the end of a string

Tie one end of a string to a wall. Send a pulse along it. When the pulse reaches the wall, it comes back. But it comes back upside down.

Why? The wall cannot move. As the pulse arrives, the string pulls up on the wall. The wall pulls down on the string by the third law, and that sends back an inverted pulse. We say the reflected wave has a phase change of $\pi$.

Now tie the end to a light ring that slides freely on a smooth rod. The end can move. The pulse comes back the same way up, with no change of phase.

• Fixed end: the reflected wave is inverted, a phase change of $\pi$. The end is always at rest.
• Free end: the reflected wave is upright, no change of phase.

Waves that stand still

Send a long wave along the string, not just one pulse. It reflects at the fixed end and comes back. Now two waves of the same amplitude, frequency and wavelength travel along the string in opposite directions:

Add them with the rule for $\sin A + \sin B$:

Look at this result. There is no $kx - \omega t$ in it. The pattern does not move along the string. Every point moves up and down in step, with the amplitude $2a\sin kx$, which depends on where the point is. We call it a standing wave.

Nodes and antinodes

Some points never move at all. There $\sin kx = 0$, so $kx = n\pi$ and

These points are the nodes. Half way between them, $\sin kx = \pm 1$ and the amplitude is largest, $2a$. These points are the antinodes. So:

• next nodes are $\lambda/2$ apart;
• next antinodes are $\lambda/2$ apart;
• a node and the nearest antinode are $\lambda/4$ apart.

A string tied at both ends

A string of a sitar or a violin is tied at both ends. Both ends must be nodes. So the string can hold only a whole number of loops, each half a wavelength long:

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

These are the normal modes of the string. The lowest, $n = 1$, is the fundamental, or the first harmonic. The others are whole multiples of it: the second harmonic $2f_1$, the third $3f_1$, and so on.

The speed of a wave on the string is $v = \sqrt{T/\mu}$, where $T$ is the tension and $\mu$ the mass of each metre. So the fundamental is

A musician tunes a string in three ways. A shorter string gives a higher note. A tighter string gives a higher note. A lighter string gives a higher note. That is why the thin strings of a guitar play the high notes.

An example. A sitar string is $0.90\ \text{m}$ long, with a tension of $81\ \text{N}$ and a mass of $1.0\ \text{g}$ in each metre. Then $v = \sqrt{\dfrac{81\ \text{N}}{0.001\ \text{kg/m}}} = 284.6\ \text{m/s}$, and $f_1 = \dfrac{284.6\ \text{m/s}}{2 \times 0.90\ \text{m}} = 158.1\ \text{Hz}$.

A sonometer in the laboratory is such a string, stretched over two bridges by a hanging load. Moving the bridges changes $L$. Changing the load changes $T$.

Practise with new numbers

Try it in Python

The program works out the first four normal modes of the sitar string in the example.

Common mistakes

• Taking the node spacing as $\lambda$. Next nodes are $\lambda/2$ apart.
• Thinking a standing wave carries energy along the string. The two travelling waves carry equal energy in opposite directions, so none flows on the whole.
• Writing $f \propto T$. The frequency grows as $\sqrt{T}$.
• Mixing up the mass of the string with the mass of each metre. $\mu$ is in kg/m.

Remember these

• Fixed end: reflected wave inverted. Free end: upright.
• Standing wave: $y = 2a\sin kx\cos\omega t$. Nodes and antinodes are each $\lambda/2$ apart.
• String tied at both ends: $L = n\lambda/2$ and $f_n = \dfrac{nv}{2L}$, every harmonic.
• $v = \sqrt{T/\mu}$, so $f_1 = \dfrac{1}{2L}\sqrt{\dfrac{T}{\mu}}$.

Check yourself

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

1. In a standing wave, the distance from one node to the next is $25\ \text{cm}$. What is the wavelength?

2. A string has a fundamental of $120\ \text{Hz}$. What are its second and third harmonics?

3. A string vibrates in three loops. How many nodes does it have, counting the ends?

4. A violinist presses a string down so that the part that vibrates is two thirds as long. What happens to the note?