Why do two almost-identical notes make a slow wobble in the sound, and how does that help you tune a guitar?

How to use this simulation

1. Start with the defaults: wave 1 at 440 Hz and wave 2 at 444 Hz. The beat frequency reads 4 Hz, so you get 4 loud moments every second, 0.25 s apart.
2. Look at the top picture. It zooms in on a few milliseconds, so you can see both waves, one in the theme colour and one in amber. Sometimes their crests line up, sometimes a crest meets a trough.
3. Look at the lower picture. It shows a whole second of the two waves added. The grey wave swells and shrinks, and the dashed green envelope outlines it. The green dots below mark the loud moments.
4. Drag f₂ towards f₁. The beats slow down. When the two frequencies match exactly, the beats vanish and you hear one steady note. Drag it far away and the beats speed up until they blur into a rough buzz.
5. Lower the amplitude of wave 2. The quiet moments are no longer silent, because the smaller wave cannot cancel the larger one completely. The readings show the largest and smallest amplitude.
6. Tick Play the sound. It starts a quiet tone only after your click and stops when you untick or leave the page. Listen for the wobble at 4 Hz, then try 1 Hz and 15 Hz.

What actually happens

Imagine two runners on a circular track, one going slightly faster than the other. Start them side by side. For a while they run together, then the fast one pulls ahead, and eventually it laps the slow one and they are side by side again. Waves of slightly different frequencies behave the same way. Sometimes they are in step (crest on crest) and add up to a big wave. Later they are out of step (crest on trough) and cancel. Then they drift back into step.

When the waves are in step, the sound is loud. When they are out of step, it is faint. The loud-faint pulsing you hear is called beats. It happens in every kind of wave, in sound, in light and in radio, but sound is where we notice it easily because our ears are good at hearing slow changes in loudness.

Beats are not a separate kind of wave. They are what superposition looks like when the two frequencies are close together.

The maths, short and sweet

Add two waves of equal size: cos(2π f₁ t) + cos(2π f₂ t). A trigonometric identity turns the sum into a product: 2 cos(π (f₁ − f₂) t) × cos(π (f₁ + f₂) t).

Read it as one wave with a changing size. The fast part, cos(π (f₁ + f₂) t), oscillates at the average frequency (f₁ + f₂)/2. That is the pitch you hear: 442 Hz for the defaults. The slow part, 2 cos(π (f₁ − f₂) t), is the envelope. It sets how big the fast wave is, and it swings from +2 to −2 and back, at the slow rate of |f₁ − f₂|/2 cycles per second.

The ear does not care about the sign of the amplitude, only its size, so loudness peaks twice per envelope cycle. That doubles the rate, and the beat frequency is the full difference, |f₁ − f₂|. With 440 Hz and 444 Hz that is 4 Hz, and a beat every 0.25 s.

Unequal amplitudes

With equal waves the quiet moments are perfectly silent: the waves cancel completely. Real sources are never exactly equal. If one wave is larger, the smaller one can only reduce it, never wipe it out. The envelope then swings between A₁ + A₂ (loudest) and |A₁ − A₂| (quietest).

Take amplitudes 3 and 1, in any unit. The loudest moments have amplitude 4 and the quietest 2. Sound intensity goes with the square of the amplitude, so the loud moments are (4 ÷ 2)² = 4 times as intense as the quiet ones. Beats are still there, but gentler.

In the animation, lower the amplitude slider and watch the dashed envelope pinch less and less at the bottom. The readings panel gives the largest and smallest amplitude.

What the ear makes of the difference

The beat frequency decides what you perceive. A difference of a fraction of a hertz is a slow swell, once every few seconds. At a few hertz you hear a clear wobble. Past roughly 15 Hz the pulses blur into a buzzing roughness, and past about 40 Hz your brain starts to hear two separate notes. These limits are round numbers, and people differ, but the pattern holds.

The table runs from very slow to very fast. The second column is the gap between loud moments, and the third counts the beats in ten seconds of listening.

Tuning a guitar by beats

Beats are how a guitarist tunes without any gadget. The trick: press the low E string at the fifth fret. In equal temperament that note should match the open A string above it. The pressed string sounds at 82.41 Hz × 2^(5/12) = 110.0 Hz.

Play both together. If the A string is at 113 Hz, the difference is 3.0 Hz, so you hear 3.0 beats every second, one every 0.33 s. Slowly turn the tuning peg and the beats slow down. When they disappear, the two strings agree. The last few beats take patience: a beat every four seconds is a difference of a quarter of a hertz.

Why this works so well: the ear can hear a difference of a fraction of a hertz as a slow throb, though it could never judge two near-identical pitches played one after another. A piano tuner uses the same idea.

One practical question remains: which way to turn the peg? Tighten the string slightly. If the beats get faster, the string was already too high (sharp), so loosen it. If they get slower, you are going the right way.

Why an out-of-tune note stands out

A semitone above A at 440 Hz is 466.2 Hz. That is a gap of 26.2 Hz, far past the beat range, so two notes a semitone apart sound rough and tense rather than pulsing. Musicians use that roughness on purpose; it is why a chord with a semitone clash feels uneasy.

Choirs and orchestras listen for the opposite: the slow beats between two voices on the same note. When the beating stops, they are in tune.

Solved problems

Five problems in the exam style, with the same functions as the animation.

Mistakes that cost marks

Check this list before an exam. Each one has taken marks from someone.

• Writing the beat frequency as (f₁ + f₂)/2. That is the pitch you hear, not the beat rate.
• Halving the difference because the envelope is cos(π(f₁ − f₂)t). The loudness peaks at the full difference.
• Using the formula for waves of very different frequencies. Beats only make sense when the two frequencies are close, within about 15 Hz.
• Forgetting that unequal amplitudes leave the quiet moments non-zero, with intensity ratio ((A₁ + A₂)/(A₁ − A₂))².
• Answering a wax or file question without checking which way the beats move. Waxing a fork lowers it; filing a prong raises it.
• Calling beats and resonance the same. Resonance is one object responding to a push; beats are two sources adding.
• Thinking beats happen only inside the ear. They exist in the air, as the superposed wave itself.

Where else beats turn up

Two engines on a plane that are not quite at the same speed make a slow throbbing drone that pilots call a beat. Radio engineers use beats to bring a signal down to a frequency they can handle, and some radar guns measure speed from beats between sent and returned waves.

If you want to see the opposite situation, where two waves of the same frequency running in opposite directions make a still pattern, try the standing-waves animation next.

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