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Beats

Source lecture(s): SC133 Lec 25

Intuition

Play two tones almost in tune — 440 Hz and 442 Hz — and you hear a single note that throbs: wah–wah–wah, twice per second. The waves drift in and out of step, adding when aligned and cancelling when opposed. Beats are interference unfolding in time rather than in space — and the most sensitive tuner nature ever gave a musician's ear.

The mathematics

Add equal-amplitude tones at \(f_1\) and \(f_2\) (product-to-sum identity):

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

You hear the average frequency, modulated by a slow envelope. The loudness peaks twice per envelope cycle, so

\[\boxed{\,f_\text{beat} = |f_1 - f_2|\,}\]

440 & 442 Hz ⇒ a 441 Hz tone beating 2 times per second.

Tuning by beats

Slow beats = nearly in tune. A tuner adjusts until the beats slow to zero — the ear detects sub-hertz mismatches this way, far beyond its ability to compare pitches directly. Piano technicians, guitarists using harmonics, and orchestras tuning to the oboe all listen for the wobble, not the pitch.

Beyond music

  • Superheterodyne radio & radar guns: mix a received signal with a local oscillator; the beat ("intermediate") frequency is easy to measure — police radar literally hears the beat between sent and Doppler-shifted returns.
  • Optical beats between lasers measure tiny frequency differences.
  • Binaural beats (one tone per ear) are constructed by your brain — a perception experiment hiding in headphones.

Worked example: tuning a guitar string

Your A-string against a 110 Hz reference produces beats every 0.5 s. How far off are you?

\(f_\text{beat} = 1/0.5 = 2\,\text{Hz}\) ⇒ the string is at 108 or 112 Hz. Tighten slightly: beats speed up ⇒ you were sharp (112 Hz) — reverse. When the wobble dies, you're tuned.

Try it live

In the wave studio, set the two frequencies a few hertz apart and watch the envelope form — then close the gap and watch the beats slow to nothing.

Common mistakes

  • Halving the beat frequency. The envelope \(\cos(2\pi\frac{\Delta f}{2}t)\) peaks twice per cycle — the audible beat rate is the full \(|f_1 - f_2|\).
  • Expecting beats from far-apart frequencies. Beyond ~15 Hz difference the throb blurs into roughness, then into two separate tones.
  • Confusing beats with standing waves — beats are temporal interference of different frequencies; standing waves are spatial interference of equal frequencies.

Knowledge graph position

Prerequisites: Superposition, Sound waves. Leads to: signal processing, radio engineering (SC134 and beyond).

Quiz

Q1 (computational). A 256 Hz tuning fork with a slightly loaded 260 Hz fork: beat frequency?

Answer

\(|260 - 256| = 4\) Hz — four throbs per second, at an apparent pitch of 258 Hz.

Q2 (conceptual). While tuning, you hear beats at 3 Hz. You tighten the string and beats rise to 5 Hz. Sharp or flat — and what should you do?

Answer

You moved the wrong way: the difference grew, so the string was already sharp. Loosen through 3, 2, 1 Hz until the beating vanishes.

Q3 (multiple choice). Two flutes play "the same note" but listeners hear a slow wobble. The flutes differ in frequency by about: (a) 100 Hz (b) 1 Hz (c) 0 Hz

Answer

(b). A slow, audible throb means a small difference — beats live in the few-hertz regime.