How Do Noise-Cancelling Headphones Work?

TL;DR: Noise-cancelling headphones listen to the noise with a tiny microphone, work out its mirror image, and play that mirror image back at you. Where the original wave pushes the air one way, the mirror pushes it the other way by the same amount, and at your eardrum the two add up to almost nothing (physicists call this superposition). It works beautifully on the low, steady drone of a plane or a train, and it falls apart on high, sharp sounds, because tiny timing and position errors throw off high notes far more than low ones.

This is part seven of Lunch Break Physics, the series that has argued a suction cup isn’t sucking, a fridge doesn’t make cold, the sky isn’t blue because it reflects the sea, ice floats because water freezes into a roomy cage, planes fly by throwing air downward, and that the shower curtain attacks you for reasons nobody can fully pin down. This week has a clean answer again, and it’s the first in the series that’s genuinely counter-intuitive. Standard physics below, sources at the bottom, and corrections are welcome in the comments.

The story you were told

Ask someone how noise-cancelling headphones work and you’ll usually get one of two answers. Either they simply block the sound out (a better seal, thicker padding, a wall between your ear and the world), or, in the cleverer-sounding and more wrong version, they detect the noise and play a louder sound over it to bury it.

Blocking is real, but ordinary headphones and earplugs already do it, and it isn’t what the “noise-cancelling” label refers to. Playing something louder over the top is exactly backwards. Add unrelated sound to a room and the room gets louder. So neither story can be right, because these headphones make it quieter by adding sound. How is that even possible?

Two waves in the demo

In the demo below, the blue wave is the outside noise and the orange one is the anti-noise the headphones emit. The green wave underneath is their sum, which is what your ear actually hears. With the two lined up, the green sum just doubles, which is the “play a louder noise over it” idea, and you can see it makes things worse. At 180 degrees the orange wave becomes the blue one’s mirror and the green line goes flat. The slider sets the phase the headphone aims for. The demo also has a small, fixed electronic delay built in, which is why the cancellation falls apart once you push the frequency up.

Interactive demo — view on the website with JavaScript enabled.

The flat green line comes from adding a second wave, which made the total at that spot smaller. Nothing was blocked and nothing was absorbed. The energy still exists. The sound has been rearranged into a quiet zone at your ear [3].

The actual physics

Sound is a pressure wave. The air near your ear gets squeezed a little above normal, then stretched a little below, over and over [1]. Draw that as a wiggly line and the crests are the squeezes, the troughs the stretches. Two sounds arriving at the same spot simply add, moment by moment. If both push, the push gets bigger. One pushing while the other pulls by the same amount leaves nothing. Physicists call the adding superposition. The special case where crest lands on trough is destructive interference, and two identical waves exactly out of step cancel completely, leaving zero [2].

So a noise-cancelling headphone is really a tiny, fast copying machine. A microphone on the outside samples the incoming pressure wave, the electronics flip it upside down so every squeeze becomes a stretch, and the speaker plays that inverted copy into your ear a fraction of a millisecond later. Your eardrum feels the original noise and its mirror image at the same instant. They add to nearly nothing right there.

Two microphones and an adaptive filter

The “copying machine” is a simplification. A real headphone can’t just invert once and hope, because the noise keeps changing and no inversion is ever perfect. Many sets use two microphones (some only one) plus a filter that keeps adjusting itself [3].

The first is a feedforward mic on the outside of the cup. It hears the noise slightly before it works its way to your eardrum, which buys the electronics a head start on their own reaction delay, so the mirror lands better. The second is a feedback mic inside the cup, in the sealed pocket right next to your ear. Its whole job is to listen to what’s left over after cancellation, the residual error, and report it back so the system can nudge the anti-noise until that leftover shrinks toward zero. An adaptive filter does the nudging. It continuously adjusts the amplitude and phase of the mirror wave to drive the inside mic’s reading down, so the headphone tracks the noise as it shifts [4]. The electronics process the sound many thousands of times a second, and how often the filter retunes depends on the design. Sets with both mics are called hybrid [3].

Fit matters too. A leak weakens the seal and changes the sound paths the filter corrects for, and the inside mic can fix some of that but not all of it. The inside mic also sits near your eardrum, a short distance from the spot it’s supposed to quiet [3].

A little bit of math

Take one steady tone in the noise, a wave of height A, and add a second wave of the same height, shifted in phase by an angle φ. There’s a clean trig identity for the sum of two equal sine waves:

A·sin(θ) + A·sin(θ + φ)  =  2A·cos(φ/2)·sin(θ + φ/2)

The amplitude of the resulting wave is the factor out front, 2A·|cos(φ/2)|, and that single factor tells the story for one tone. Real noise is many tones at once, so the filter has to get each one right.

In phase, φ = 0°: cos(0) = 1, so the amplitude is 2A. Twice as tall, which is about 6 decibels higher in sound level at that spot. This is the “play it louder” idea, and the math agrees it’s the wrong direction.
Exactly out of phase, φ = 180°: cos(90°) = 0, so the amplitude is zero. At that spot, that tone is silent.
Halfway, φ = 90°: cos(45°) ≈ 0.71, so about 1.4A, somewhere between doubling and cancelling.

In the simplest version, the headphone plays its inverted copy a tiny bit late. A fixed time delay is a phase error that grows with frequency, because phase error is 360° × frequency × delay, and the frequency slider in the demo lets you watch that happen. For a low 100 Hz rumble the wavelength is about 3.4 metres. A fraction-of-a-millisecond delay is a tiny slice of that, so the mirror still lands almost perfectly and the sound cancels. A 4,000 Hz hiss is different. Its wavelength is only about 8.6 centimetres, so the same tiny delay is now a large slice of the wave. The “mirror” arrives so far out of step that it stops cancelling and starts adding [3]. Higher still, it swings between the two. Clever filters can pre-compensate a steady tone, but high notes stay hard. With a wavelength of centimetres, even the gap between microphone and eardrum matters, and a cup that has shifted changes the path. Sudden sounds are a separate problem, because they give no warning [3].

The same effect elsewhere

When a wave meets a matched, opposite wave, the two sum to less than either alone, and that holds for light as much as for sound. Anti-reflection coatings on glasses and camera lenses use it. A microscopically thin film is layered onto the lens so that light reflecting off its top surface and light reflecting off its bottom surface come back nearly out of step, for a chosen range of colours around green. The two reflections destructively interfere, so less light bounces off and more gets through [6]. Other colours cancel less, which is where the faint coloured sheen comes from.

Some cars, and large ventilation and exhaust systems, have the microphone-and-anti-noise idea built in as an active silencer, aimed squarely at the low-frequency hum that passive baffles can’t easily stop. It doesn’t silence a whole room. Away from the target spot the waves can add as well as cancel [7].

And you can hear it in mid-air. Play the same steady tone through two speakers and walk around. At certain places the two waves arrive crest-to-trough and the sound drops away to almost nothing. Nobody engineered those dead spots. They’re the same destructive interference, laid out across the room instead of built into electronics [2].

Other ways to make it quieter

A few other things reduce noise and so look like they belong in the same story. None of them works by adding an opposite wave.

Passive isolation is the padded cushions themselves. The ear cups and their foam reflect part of the sound and soak up part of it, and only the soaked-up part becomes a whisper of heat in the material. That’s a barrier, and it’s the part that actually handles the high frequencies the electronics give up on [5]. Real headphones use both, so the two are easy to mix up. Earplugs work the same way, foam or silicone that blocks and absorbs the pressure wave before it reaches your eardrum, with no microphone and nothing added.

Playing music or white noise to drown it out is masking, the “play a louder noise over it” idea done deliberately. It adds sound energy so your brain stops attending to the noise underneath, but the noise is still fully there, and the room is genuinely louder.

Everyday consequences

Observation	Why
The engine drone dies but the crying baby cuts through	Low steady frequencies cancel well. The low part of a cry is trimmed, but its fast-changing high part gets through.
Cancellation needs power; earplugs don’t	Sampling, inverting, and playing anti-noise is active work. A dead battery leaves you with only the passive seal.
A poor fit ruins even good headphones	The high end is handled by the physical seal, and a leak also changes the sound paths the filter was tuned for, so the sounds the electronics can’t catch pour straight in [3].
Some people feel a faint “pressure” with ANC on	One explanation is that the low-frequency anti-noise gives the brain a cue it reads as pressure, even though there’s no lasting push on the eardrum like the one you feel in a descending plane [3].
It’s better on a plane than in a café	A jet cabin is a wall of steady low-frequency drone, which is what the electronics handle best. Café clatter and chatter are high-pitched and sudden, the kind they handle worst.
Some sets can make a narrow band of sound slightly worse	Near the edge of what they can cancel, a mistimed mirror adds to the noise, as the demo shows. Well-built sets back off outside the range where it works.

Lunch Break Physics runs Tuesdays at noon. Last week: why the shower curtain attacks you. Next Tuesday: why a curveball curves. Got an everyday-physics puzzle you’d like poked at? The comments are open.

Sources

[1] Understanding Sound — U.S. National Park Service: sound travels through air as a wave described by frequency (pitch, in hertz) and amplitude, whose sound level is expressed in decibels; humans hear roughly 20 Hz to 20,000 Hz, and decibels are logarithmic, so a 10 dB rise means ten times the sound intensity.
[2] Superposition and Interference — OpenStax College Physics 2e (§16.10): when waves overlap their disturbances add (superposition); two identical waves in phase produce constructive interference at twice the amplitude, and two identical waves exactly out of phase produce pure destructive interference in which the resulting amplitude is zero and the waves completely cancel.
[3] How Active Noise Cancellation Works, and What It Cannot Cancel, Sound Control Technologies: feedforward, feedback and hybrid designs; cancellation holds only where the waves meet; a tiny timing error cancels 100 Hz but adds at ~4 kHz; the pressure feeling is a brain cue.
[4] Active Noise Control (reference and error microphones), MathWorks documentation: reference and error microphones drive an adaptive filter.
[5] Active Noise Reduction, FizziQ glossary: works mainly on low frequencies and complements the cushions’ passive attenuation.
[6] Thin Film Interference, OpenStax College Physics 2e (§27.7): non-reflective coatings are tuned to one wavelength (550 nm in the example).
[7] Active noise control, Wikipedia: ducts and vehicles; away from the target spot the waves alternately cancel and reinforce.