Saramsha

No. 040  ·  26 October 2026

Build a Wave to Kill a Wave

Noise-cancelling headphones don't block sound — a microphone hears it, the electronics flip it upside down, and a speaker plays that flipped copy back so the two cancel in the air just before they reach your ear. Flipping it is the easy part. Timing it is not — drag how late the copy arrives and watch cancellation turn into something else entirely.

the noise arriving the copy, flipped and late what reaches your ear

About this piece

A copy, upside down, slightly late

The idea behind active noise cancellation is almost embarrassingly simple: a tiny microphone on the outside of the earcup listens to the world, the electronics flip that signal upside down — every peak becomes a trough and every trough a peak — and a speaker inside the cup plays the flipped copy back. Where the real sound and its inverted twin meet, right at your eardrum, they add up to nothing. That's not a metaphor; it is ordinary wave addition, the same arithmetic that makes two ripples on a pond cancel where a crest meets a trough.

The catch is that "listen, flip, play back" is not instant. Sound has to reach the microphone, get digitised, get inverted, get converted back to an analogue voltage, and push a speaker diaphragm — and every one of those steps costs time. Call that total delay δ. The headphone isn't playing −noise(t), which would cancel perfectly; it's playing −noise(t − δ), a copy of what the microphone heard a moment ago. For a single tone of frequency f and amplitude A, adding that late, flipped copy to the original gives:

resultant(t) = 2A·sin(πfδ) · cos(2πft − πfδ)

Only the first factor matters for loudness: the amplitude that survives is 2A|sin(πfδ)|. That one expression is the whole story on this page. When fδ is a whole number, sin is zero and the two waves erase each other exactly, no matter how large A was. When fδ lands on a half-integer — 0.5, 1.5, 2.5 — sine hits its peak of 1 and the "cancelling" copy arrives perfectly in step with the original instead of against it, so the two don't subtract, they add: the headphone has just made the sound twice as loud, a genuine +6 dB, by trying to cancel it.

A comb, not a slope

Engineers have a name for this exact shape — a delayed copy added back to a signal produces a comb filter, the same structure behind the swooshing sound of a flanger pedal, and it is "combed" for a reason: plotted against frequency, the response isn't a smooth rolloff, it's a picket fence of alternating nulls and peaks, repeating every 1/δ hertz. A longer delay packs the teeth of that comb closer together, which is exactly what the chart above is showing as you drag the delay slider — more teeth means more places in the audible range where a fixed-delay system doesn't just fail to help, it actively hurts.

This is why noise-cancelling headphones are, in practice, a fight over microseconds. A good ANC chip gets the round trip down to a few tens of microseconds, which pushes the first amplification peak so high in frequency that almost the entire audible range sits inside the comb's first calm notch — which is exactly the "fast chip" preset above. A noticeably laggier path drags that first peak down into the middle of the spectrum, squarely into speech and clicks, which is what the "sluggish" preset shows happening to the 600 Hz and 2,000 Hz presets here. Real engine drones and HVAC hums sit at the very bottom of that spectrum, which is also the safest part of any comb — one reason ANC is so good at erasing exactly that kind of sound.

What real headphones do differently

Nothing here is literally what ships in a pair of headphones. Real systems don't apply one fixed delay to a pure tone; they run an adaptive filter that continually estimates the actual noise spectrum — a mix of many frequencies at once — and reshapes its own response to match it, deliberately rolling off and limiting its gain at high frequencies specifically so it can never land squarely on an amplification peak like the ones this page lets you dial in on purpose. Above a few kilohertz, active cancellation mostly gives up and leaves the job to passive isolation: the simple fact of a cushioned cup sealing against your head, which blocks short wavelengths perfectly well and needs no electronics or timing budget at all.

One of the earliest patents for the idea — a microphone, an inverter and a speaker, filed to quiet noisy headsets — dates to the 1950s, from the engineer Lawrence J. Fogel. It stayed a curiosity for decades, limited by electronics too slow to keep δ small enough to matter. Amar Bose is said to have started thinking seriously about it after a transatlantic flight in 1978, when the airline headphones issued against the jet-engine roar were nearly useless; Bose's aviation headsets reached pilots by the late 1980s, and the first consumer QuietComfort headphones reached shop shelves in 2000 — by which point digital signal processing had finally made the round trip fast enough to be worth the trouble.

The sound is modelled as a single sine wave of amplitude 1; the "copy" is that same wave, delayed by δ and inverted, with no amplitude error assumed — only timing. Loudness is read directly off the surviving amplitude, 2|sin(πfδ)|, converted to decibels. The scope redraws four cycles of whatever frequency is selected; the chart sweeps the same formula across the full 20–4,000 Hz range of the frequency slider. Audio, where your browser allows it, plays the model exactly: one oscillator feeding a direct path and a delayed, inverted path, summed in your headphones for real. ?f= and ?d= set the frequency (Hz) and delay (µs) directly.