Beats

A sound that swells and fades

Two friends tune their guitars to play together. They pluck the same string on both. The note does not sound steady. It goes loud, soft, loud, soft, a few times each second. Then one friend turns a peg a little, and the wobble slows down. When it stops, the two strings are in tune.

That wobble is called beats. In this lesson we see where beats come from and how to use them.

Two waves in and out of step

Take two sound waves of the same amplitude, with frequencies $f_1$ and $f_2$ that are close. Both reach your ear together. By the principle of superposition, the air at your ear moves by the sum of the two.

At some moment the two waves are in step. Crest meets crest, and the sum is large. The sound is loud. One wave is a little faster, so it slowly pulls ahead. After a while a crest of one meets a trough of the other. They cancel, and the sound is quiet. Then the faster wave pulls ahead by a whole wave, and they are in step again.

How often does that happen? In one second, wave 1 makes $f_1$ waves and wave 2 makes $f_2$. Wave 2 gains $f_2 - f_1$ waves on wave 1. Each wave it gains brings one more loud moment. So the number of loud moments each second is

This is the beat frequency. The time from one loud moment to the next is the beat period, $T_b = \dfrac{1}{f_b}$.

The maths of the sum

For the same amplitude $a$, we can add the two waves with a sine rule:

Read it in two parts.

• The fast part, $\sin\left(\pi(f_1 + f_2)t\right)$, has the frequency $\dfrac{f_1 + f_2}{2}$. That is the pitch you hear, half way between the two.
• The slow part, $2a\cos\left(\pi(f_1 - f_2)t\right)$, sets how large the swings are. It reaches its biggest size twice in each of its own cycles, so the loudness peaks $|f_1 - f_2|$ times each second.

A worked example. Two forks of $440\ \text{Hz}$ and $444\ \text{Hz}$ sound together.

• Beat frequency: $f_b = 444\ \text{Hz} - 440\ \text{Hz} = 4\ \text{Hz}$, so 4 loud moments each second.
• Beat period: $T_b = \dfrac{1}{4\ \text{Hz}} = 0.25\ \text{s}$.
• Pitch heard: $\dfrac{440\ \text{Hz} + 444\ \text{Hz}}{2} = 442\ \text{Hz}$.

Our ear can follow up to about 10 beats each second. With a larger difference, the beats run together, and we hear a rough sound or two separate notes.

Make the difference $\Delta f$ smaller. The loud moments move further apart. At $\Delta f = 0.25\ \text{Hz}$ there is only one loud moment in $4\ \text{s}$.

Tuning by beats

A piano tuner strikes a key and a tuning fork of the right frequency together. If he hears beats, the string is out of tune. He turns the pin. If the beats get slower, he is going the right way. When the beats stop, the two frequencies are equal.

Finding an unknown frequency

Beats tell us the difference, but not which note is higher. Suppose an unknown fork gives 4 beats each second with a fork of $256\ \text{Hz}$. It could be $252\ \text{Hz}$ or $260\ \text{Hz}$. We need one more test.

• Put a little wax on a prong of the unknown fork. The prong is heavier and vibrates more slowly, so the frequency falls.
• File a prong of the unknown fork. The prong is lighter, so the frequency rises.
• Tighten a string. The tension rises, so the frequency rises.

Then watch the beats. Say we wax the unknown fork and the beats get faster. Its frequency fell and moved further from $256\ \text{Hz}$. So it was already below: $252\ \text{Hz}$. If the beats had got slower, it would have been $260\ \text{Hz}$.

Practise with new numbers

Try it in Python

The program adds two waves, sample by sample, for $2\ \text{s}$. It counts the loud moments, as your ear would.

Common mistakes

• Adding the frequencies. The beat frequency is the difference, not the sum.
• Halving the difference. The loudness peaks $|f_1 - f_2|$ times each second, not half as often.
• Mixing up the pitch and the beats. You hear the average frequency, and it gets loud and soft at the beat frequency.
• Forgetting the sign test. Beats alone give two possible answers. Wax, filing or tightening picks one.

Remember these

• Beat frequency: $f_b = |f_1 - f_2|$. Beat period: $T_b = 1/f_b$.
• The pitch you hear is $\dfrac{f_1 + f_2}{2}$.
• Wax lowers a fork's frequency. Filing raises it. Tightening a string raises it.
• Beats faster after a change means the two frequencies moved apart.

Check yourself

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

1. Two forks of $510\ \text{Hz}$ and $513\ \text{Hz}$ sound together. How many beats do you hear in $5\ \text{s}$?

2. Why do two forks of exactly the same frequency give no beats?

3. A string gives 5 beats each second with a fork. The string is tightened a little, and the beats become 7 each second. Was the string above or below the fork at first?

4. Two flutes play $600\ \text{Hz}$ and $640\ \text{Hz}$. Will a listener hear clear beats?