Beats

Loud, soft, loud

Narrated video. Transcript:

Strike two tuning forks together: one of 256 hertz and one of 260 hertz. You do not hear two separate notes. You hear one note that swells and fades, about four times every second.

This regular rise and fall of loudness is called beats. It happens whenever two sounds of nearly equal frequency are heard together.

Look at the two pressure waves arriving at your ear. At this moment they are in step: compression meets compression. They add up, and the sound is loud.

The 260 hertz wave gains a little on the other every cycle. Just an eighth of a second later, it is half a cycle ahead. Compression meets rarefaction, they cancel, and the sound is soft.

Another eighth of a second, and it has gained a whole cycle. They are back in step, and the sound is loud again.

Now the maths. At one point, s one equals a cos ω one t and s two equals a cos ω two t. Adding them with the cos A plus cos B formula gives s equals 2a cos ω b t, times cos ω a t.

Here ω a is the average, ω one plus ω two over 2, and ω b is half the difference, ω one minus ω two over 2.

Since ω b is much smaller than ω a, we hear a tone at the average frequency, 258 hertz, whose amplitude, 2a cos ω b t, slowly rises and falls.

The sound is loud whenever cos ω b t is plus 1 or minus 1. That happens twice in each of its cycles. So the beat frequency is ν beat equals ν one minus ν two.

For our forks, 260 minus 256 gives 4 beats per second. If the two frequencies differ by more than about ten hertz, the beats come too fast for the ear to follow.

Tuning with beats

Narrated video. Transcript:

Musicians use beats every day. A sitar player plays a string together with a reference note and listens.

While the string is out of tune, the sound wobbles. As she turns the peg, the beats slow down: five a second, then two, then one. When they vanish completely, the two notes match.

But there is a catch. Beats tell you the difference in frequency, not which note is higher. 4 beats a second could mean 4 hertz above or 4 hertz below.

To find out, change one note slightly and watch what the beats do. Here is an example.

Two sitar strings, A and B, both playing the note Ga, are slightly out of tune and give 6 beats per second. String A is at 324 hertz. So B is either 330 or 318 hertz.

Now the tension in A is reduced slightly. Less tension means a smaller wave speed, so A’s frequency goes down a little. The beats drop to 3 per second.

If B were 330 hertz, lowering A would move it further away, and the beats would increase. They decreased, so A moved closer to B.

So B must be 318 hertz. Check: A drops from 324 to 321 hertz, and 321 minus 318 is 3 beats per second.

The same trick works with tuning forks. Sticking a little wax on a prong makes it vibrate more slowly; filing a prong makes it vibrate faster. Watch the beats, and you know which way it moved.

Try it: hear the beats

Interactive simulation. Things to try:

• Tick “Sound on” to hear both notes together (use headphones or turn the volume down).
• In the zoomed view, wave B slides past wave A. Loud when they line up, soft when they are opposite.
• Count the loud moments in the bottom graph in 1 s. It equals |ν_A − ν_B|.
• Set the detune to +4 Hz and then −4 Hz: the beats are the same.

Try it: tune the string

Interactive simulation. Things to try:

• Note A is hidden. Turn on the sound and tune string B.
• As you get closer, the beats slow down. When they stop, you have matched A.
• If the beats speed up, you are tuning the wrong way.

What do you hear?

Two tuning forks of 256 Hz and 260 Hz are sounded together. What do you hear?

1. Two separate notes, 256 Hz and 260 Hz
2. A low hum at 4 Hz
3. A note of about 258 Hz that gets louder and softer 4 times a second
4. A note of 516 Hz

Two sitar strings

Two sitar strings A and B give 6 beats per second. The frequency of A is 324 Hz. When the tension in A is slightly reduced, the beat frequency falls to 3 per second. What is the frequency of B?

1. 330 Hz
2. 327 Hz
3. 321 Hz
4. 318 Hz

Summary

• Sum of two nearby tones: s = 2a cos(ω_b t) cos(ω_a t)
• Average and half-difference: ω_a = (ω₁ + ω₂)/2, ω_b = (ω₁ − ω₂)/2
• Beat frequency: ν_beat = ν₁ − ν₂

• Beats are the periodic rise and fall of loudness when two sounds of slightly different frequency are heard together.
• You hear the average frequency; its loudness peaks |ν₁ − ν₂| times a second.
• Beats are used to tune instruments: when the beats vanish, the frequencies are equal.
• Beats give only the size of the difference. To find which note is higher, change one slightly (tension, wax, filing) and see whether the beats speed up or slow down.