Beats

3 min read#waves

Two tones of nearly equal frequency drift in and out of step, producing a slow throb whose rate is the difference of the two frequencies.

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Beats

Beats are interference unfolding in time. Sound two notes of almost the same pitch — two guitar strings a hair out of tune — and you hear neither a clean chord nor a clash, but a single tone that pulses: loud, soft, loud, soft. That slow throb is the two frequencies repeatedly drifting into step (reinforcing) and out of step (cancelling), exactly the Interference rule applied to overlap in time rather than space.

The envelope at the average, the throb at the difference

Add two equal-amplitude tones at f_1 and f_2 and a product-to-sum identity rearranges the sum into a fast carrier riding inside a slow envelope:

\cos(2\pi f_1 t)+\cos(2\pi f_2 t)=\underbrace{2\cos\!\big(2\pi \tfrac{f_1-f_2}{2}\,t\big)}_{\text{slow envelope}}\;\underbrace{\cos\!\big(2\pi \tfrac{f_1+f_2}{2}\,t\big)}_{\text{fast carrier}}.

The ear hears the carrier at the average frequency (f_1+f_2)/2, swelling and fading under the envelope. Because loudness peaks twice per envelope cycle (at both the positive and negative bulge), the audible beat frequency is the full difference:

f_{\text{beat}}=|f_1-f_2|
(1)

Tune one string toward the other and the beats slow down; when they vanish entirely, the strings are in unison. Piano tuners do exactly this by ear.

Top two traces: two pure tones at nearby frequencies. Bottom trace: their sum, with the slow beat envelope drawn faintly. The sum swells and collapses where the tones fall in and out of phase. Drag left/right to detune the second tone — more detuning, faster beats.

The same physics as coupled pendulums

If beats feel familiar, they should: two weakly Coupled Oscillators do exactly this. Their two normal modes sit at slightly different frequencies, and energy sloshes fully from one pendulum to the other and back at the difference frequency — a beat you can watch instead of hear. Near Resonance, the difference between drive and natural frequency similarly controls how the response builds and ebbs.

See also

Related topics

  • InterferenceTwo overlapping waves reinforce where their crests align and cancel where crest meets trough, painting a fixed pattern of bright and dark fringes.
  • ResonanceDrive an oscillator near its natural frequency and the response explodes — the principle behind tuning radios, shattering glass, and toppling bridges.
  • WavesFrom a single swinging oscillator to ripples that travel, interfere, and carry frequency through space — the bridge between Oscillations and Fourier Analysis.
  • Coupled OscillatorsConnect two oscillators and they stop acting alone — exchanging energy through beats, splitting into normal modes, and ultimately synchronizing.
  • Superposition PrincipleWhen two waves meet, the medium's displacement is simply the sum of what each wave would do alone — the linearity that underlies all of wave physics.
  • Standing WaveTrap a wave between two fixed ends and it stops traveling — freezing into a pattern of motionless nodes and violently swinging antinodes.

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