Diagrams.ink / Waves at work

Noise-cancelling: sound plus anti-sound

A microphone hears the hum, the speaker plays its mirror image, and the two meet at your ear. Drag the phase to see when they cancel and when they add.

Fig 1 · Ear cup cutawayAt your ear:
Noise Earcanal Speaker 180° Mic Chip Eardrum 1 Incoming noise 2 Anti-noise from the speaker 3 What reaches your ear = 1 + 2 same scale on all rows
180°

120 Hz

Incoming noiseAnti-noiseSum at the ear: green quiet, orange same, red louder

The idealized low-frequency model

Sound is a pressure wave. A steady low hum has a very long wavelength, much longer than the ear cup, so the pressure is nearly the same all through the small space by your ear. The outside mic samples it, the chip flips it by 180°, and the speaker adds that flipped pressure.

With equal size, the ear hears |1 + eiφ| = 2|cos(φ/2)| times the noise. At φ = 180° that is zero. At φ = 0° the waves line up and the sum doubles (+6 dB). Around 120° and 240° you get back where you started.

Where real headphones fall short

  • Delay. Mic, chip and speaker take time. For higher pitches that delay is a bigger slice of a cycle, so “opposite” turns into “off”. Try Real-world delay and raise the pitch.
  • A small quiet zone. The match holds best close to the ear. Shorter waves change across that space.
  • Unsteady sound. Steady rumble is easy to predict. Speech and clatter are not, so cups, cushions and a good seal handle those passively.
  • Fit and level. Leaks and a mismatched anti-noise level leave a residual.

A teaching model: one pure tone, equal amplitudes, an illustrative delay and made-up sizes. It is not a product specification.