What are beats?
Play two sounds of nearly equal frequency together. You hear one tone whose loudness goes up and down again and again: loud, soft, loud, soft. This regular rise and fall of loudness is called beats. One loud-to-loud cycle is one beat.
Beats happen because of superposition in time: at one moment the two waves are in step and add (loud); a little later one has got half a cycle ahead and they cancel (soft).
Beat frequency: the maths
Take two waves of equal amplitude at one place:
s₁ = a cos(ω₁t), s₂ = a cos(ω₂t)
Using cos A + cos B = 2 cos((A − B)/2) cos((A + B)/2):
s = [2a cos(ω_b t)] cos(ω_a t), where ω_a = (ω₁ + ω₂)/2 and ω_b = (ω₁ − ω₂)/2.
- cos(ω_a t) is the fast wave we hear, at the average frequency (f₁ + f₂)/2.
- 2a cos(ω_b t) is a slowly changing amplitude (the “envelope”).
- Loudness depends on the size of the amplitude, which peaks twice in each cycle of cos(ω_b t). So the beat frequency is 2 × (f₁ − f₂)/2:
f_beat = |f₁ − f₂|, time for one beat = 1/|f₁ − f₂|.
Our ears can count beats only up to about 10 per second. Faster than that, they blend into a rough sound.
Uses of beats
- Tuning instruments: play a string with a standard note; adjust until the beats disappear.
- Finding an unknown frequency: sound it with a known fork and count beats. Then change one of them a little (load a fork with wax to lower its frequency, or file it to raise it) and see whether beats increase or decrease to decide the sign.
- Detecting small changes, for example in some speed-measuring (Doppler) devices.
The wax and file rule
- Wax on a fork → its frequency falls.
- Filing the prongs → its frequency rises.
Try it: beats with two phones
On two phones open any free tone-generator app. Set one to 440 Hz and the other to 443 Hz and play them side by side. Count the wobbles for 5 seconds: you should get about 15, so 3 per second. Now set the second one to 441 Hz: predict 1 beat per second, then check. In the 3D, step 5 lets you do the same with the sliders.
Key formulas and definitions
- f_beat = |f₁ − f₂|
- Time between beats = 1/|f₁ − f₂|
- s = 2a cos(ω_b t) cos(ω_a t), ω_a = (ω₁ + ω₂)/2, ω_b = (ω₁ − ω₂)/2
- Heard pitch = (f₁ + f₂)/2
- Wax on fork → f falls; filing → f rises
Worked examples
1. Two forks of 384 Hz and 380 Hz are sounded together. How many beats per second?
f_beat = 384 − 380 = 4 beats per second.
2. In the example above, what is the time between two loud sounds, and what pitch is heard?
Time = 1/4 = 0.25 s. Heard pitch = (384 + 380)/2 = 382 Hz.
3. A fork of 256 Hz gives 5 beats/s with an unknown fork. What can the unknown frequency be?
f = 256 ± 5, so 251 Hz or 261 Hz.
4. In the last example, the unknown fork is loaded with wax and the beats become 7 per second. Find its frequency.
Wax lowers the unknown frequency. If it were 261 Hz, lowering would bring it closer to 256 and beats would drop. Beats rose, so it was 251 Hz (now about 249 Hz, 7 beats).
5. Fork A gives 4 beats/s with a 320 Hz fork. When A is filed, the beats become 2 per second. Find A’s original frequency.
A is 316 or 324 Hz. Filing raises A. From 316 Hz it moves towards 320, so beats fall: fits. So A = 316 Hz.
6. Two strings of a sitar should both play 300 Hz, but 30 beats are counted in 10 s. What is the difference in their frequencies?
Beats per second = 30/10 = 3. So the frequencies differ by 3 Hz.
7. Two organ pipes give 5 beats/s when sounded together. One is an open pipe of length 0.5 m (v = 340 m/s). What can the other frequency be?
Open pipe f = v/2L = 340/1 = 340 Hz. The other is 340 ± 5 = 335 Hz or 345 Hz.
8. Wave equations: y₁ = a sin(600πt) and y₂ = a sin(608πt). Find the beat frequency.
f₁ = 600π/2π = 300 Hz, f₂ = 304 Hz. Beats = 4 per second.
Common mistakes
- Writing the beat frequency as (f₁ − f₂)/2. That is the envelope’s frequency; loud moments come twice per envelope cycle, so beats = |f₁ − f₂|.
- Thinking beats need a big frequency difference. They are heard only when the difference is small (under about 10 Hz).
- Mixing up wax and file: wax lowers the frequency, filing raises it.
- Forgetting the ± when finding an unknown frequency from beats. There are two possible answers until you test.