The silence that fits in a tenth of a wavelength

When two waves meet found that waves in a linear medium pass through each other unchanged and simply add, point by point, and that two equal waves of opposite phase add to nothing. Air carrying ordinary sounds is linear to a very good approximation — sound pressures are a few millionths of the air’s own — and so the idea of cancelling a noise by adding its inverse is as old as the idea that sound is a wave. Paul Lueg patented it in 1936: a microphone picks up the noise, an amplifier inverts it, a loudspeaker plays it back, and the two pressures sum to zero.

At the microphone they do. The interesting question is everywhere else, and the answer explains why the commonest successful use of the idea is a pair of headphones and not a quiet room, a quiet street or a quiet aircraft cabin. Superposition makes silence a property of a point, and the size of the region around that point is fixed by the wavelength.

Silence at a point #

A loudspeaker driven to cancel the sound at one microphone produces its own wave, spreading out from the speaker, and the cancelling pressure at the microphone is that wave’s pressure there. Everywhere else the speaker’s wave has a different strength — it falls off with distance from the speaker — and a different phase — it has travelled a different distance — while the noise it is cancelling has its own pattern of strength and phase. The two match only at the microphone.

How fast they come apart depends on how the noise arrives. In a room, a car or a cabin, sound arrives from every direction at once, bounced off every surface, and the noise at one point is only partly correlated with the noise a short distance away: for a diffuse field the correlation between two points falls off as $\sin(kr)/kr$, with $r$ their separation and $k$ the wavenumber. The cancelling speaker’s wave is perfectly correlated with itself everywhere. So the cancellation, perfect at the microphone, decays at the rate the noise decorrelates.

The sinc function is worth a sentence, because it is the whole reason for the bubble. A diffuse field is a superposition of plane waves arriving from every direction with random phases, the decomposition the fan of plane waves inside every beam used for a beam made of many directions. Each plane wave has the same phase at two points a distance $r$ apart only if it travels across the line joining them; one travelling along it has advanced by $kr$. Averaged over all directions, the shared part of the pressure at the two points is $\sin(kr)/kr$, which falls to zero at half a wavelength and stays small. Two points a few tenths of a wavelength apart are hearing nearly independent noises, and no single speaker can cancel two independent noises at once.

The result is a small bubble of quiet. At 500 hertz the region where the noise is at least ten decibels lower — a tenfold reduction in power, about half as loud to the ear — is twelve centimetres across. Outside a slightly larger contour the sum is no quieter than the noise alone, and near the speaker it is much louder, since there the speaker’s own wave dominates the noise it was meant to cancel. Averaged over the room, the speaker has added sound.

At 100 hertz, with a wavelength of 3.4 metres, the same arrangement produces a bubble more than half a metre across, big enough to hold a head. The shape is the same; only the scale has changed. The quiet zone is a fixed fraction of a wavelength, because the only length in the problem, apart from the distance to the speaker, is the wavelength.

The size of the bubble #

The measured rule, established by Stephen Elliott, Philip Nelson and their colleagues at Southampton in the 1980s, is that a cancelling speaker far from its microphone produces a quiet zone of the order of a tenth to a fifth of a wavelength. At a hundred hertz that is tens of centimetres; at a kilohertz, a few; at the frequencies of speech and of most of what makes noise annoying, a centimetre or less. A canceller in the headrest of a car seat, with its microphone beside the listener’s ear, quietens the low rumble of the tyres for a head that keeps still, and fails as soon as the head moves.

The other case on the chart is the speaker very close to the microphone. Then the speaker’s own field, falling steeply with distance from it, dominates the shape of the zone, and the quiet region is a few centimetres across whatever the frequency. That is the arrangement inside a headphone: the speaker, the error microphone and the ear canal all lie within a couple of centimetres of each other, so the region that has to be quiet — the entrance of the ear canal — is inside the bubble at every frequency the canceller handles.

Fifty years from patent to product #

Lueg’s patent described exactly the system in the figures and it did not work, for the reasons this essay is about. The noise at an ear is not a single tone from a single direction; the loudspeakers of the 1930s could not reproduce a waveform faithfully; and the amplifiers had no way to learn the acoustic path from speaker to microphone, which changes with every movement of the head. Harry Olson and Everett May built an “electronic sound absorber” in 1953, a loudspeaker and microphone close together in a hollow behind a listener’s head, and got about ten decibels below a few hundred hertz, the quiet zone a few centimetres across, just as the arithmetic predicts.

What turned the idea into products in the 1980s was the digital adaptive filter: a processor that continuously adjusts its estimate of the inverse noise to minimise what the error microphone hears, tracking changes in the noise and the acoustic path as they happen. The first commercial headsets, for aircraft pilots, appeared in 1989; consumer headphones followed a decade later. Throughout, the physics was the same superposition Lueg drew, and the limits were the two drawn here: the bubble and the delay.

Silence everywhere needs the second source beside the first #

The bubble can be avoided altogether if the cancelling source is put next to the noise source instead of next to the listener. Then, instead of cancelling the noise at a point, the two sources can be made to cancel each other’s radiation in every direction.

Two point sources close together, with equal strengths and opposite phase, form a dipole, and a dipole radiates far less than either source alone when the separation is small compared with the wavelength: each source’s pressure is nearly cancelled by the other’s everywhere in the far field. The best the second source can do, adjusting its strength and phase freely, leaves a fraction $1 - \text{sinc}^2(kd)$ of the first source’s power. Within a tenth of a wavelength the reduction is close to ten decibels; by a quarter of a wavelength it has nearly gone.

That makes global cancellation practical only for low-frequency noise from small sources. A transformer humming at 100 hertz, with a wavelength of 3.4 metres, can be quietened throughout a room by loudspeakers mounted within about 30 centimetres of it, and such systems have been installed beside substations. A transformer’s hum suits the method unusually well: it comes from magnetostriction, the core’s steel lengthening and shortening twice in every cycle of the mains, so it sits at exactly twice the supply frequency and its harmonics, steady for years, and a canceller can learn it once and keep it cancelled. The same trick at 2 kilohertz would need the speakers within two centimetres of every part of the noisy surface.

The energy has not gone anywhere strange. What adding does to the energy found that when two waves cancel somewhere, the energy they would have carried there appears somewhere else or was never emitted. Here it was never emitted: the second source changes the pressure at the face of the first, so that the first, pushing on air that is moving with it, does less work. A loudspeaker cancelling a nearby source does not absorb its sound; it stops the source from producing it.

The delay that limits the band #

Even at a point, cancellation has a second limit, and it is time. The microphone must hear the noise, the electronics must compute the inverse, and the loudspeaker must produce it. Each step takes time, and the anti-noise arrives late by some delay $\tau$.

A fixed delay is a phase error that grows with frequency. At a hundred hertz, fifty microseconds is a two-hundredth of a cycle and the cancellation is good to thirty decibels; at a kilohertz it is a twentieth of a cycle and the cancellation is only ten decibels; at a little over three kilohertz it is a sixth of a cycle, the late anti-noise is as likely to add as to subtract, and the canceller achieves nothing. Above that it makes things worse.

For a steady tone the delay can be dodged. A tone repeats, so the canceller can play now the inverse of what it heard one cycle ago, or several, and the delay disappears into the periodicity; engine noise and propeller noise, dominated by a fundamental and its harmonics, are cancelled this way in some cars and aircraft. Unpredictable noise — a voice, a passing truck, the hiss of air — cannot be predicted, and no causal system can cancel what it has not yet heard. The answer that cannot come first found the same constraint in every response of a physical system to a force: a response cannot precede its cause, and that single rule fixes how far any filter can be pushed. A noise canceller is such a filter, and the rule leaves it the low frequencies.

How a headphone divides the work #

A good noise-cancelling headphone therefore does two different things with two different pieces of physics. Its cups are sealed against the head and made heavy, and they block sound the way the wall that stops sound by its weight does: poorly at low frequencies, where the cup and its cushion move together like a mass on a spring, and increasingly well above that resonance, where the cup’s inertia holds it still. Inside the cup, a microphone and the headphone’s own driver cancel what gets through, well at low frequencies where the wavelength is long and the delay is a small fraction of a cycle, and not at all at high frequencies, where the cup has already done the work.

Even inside the cup the bubble matters. The error microphone sits beside the driver, not at the eardrum, and the canceller silences the microphone’s position. At low frequencies the whole cavity between driver and eardrum is a small fraction of a wavelength, the pressure is nearly uniform across it, and silencing one point silences the eardrum too. Towards a kilohertz the cavity’s few centimetres become a noticeable fraction of a wavelength, the pressure at the eardrum departs from the pressure at the microphone, and the cancellation the listener hears falls below what the microphone measures. Designers place the microphone as close to the ear canal as the cup allows, for exactly the reason the first figure shows.

Neither alone is quiet across the band. Together they are, because the passive and active methods fail in opposite places. Mass blocks short waves; a short delay cancels long ones. The engine drone of an aircraft cabin, at a hundred hertz, is the kind of noise the cup alone barely touches and the canceller removes almost entirely, which is why that is where the headphones are worn.

Cancelling vibration instead of sound #

The same arithmetic applies to any linear wave, and it works better where the wavelengths are long. A car engine shakes its mountings at a few tens of hertz, and active engine mounts — actuators in the mounts driven to cancel the vibration passing into the body — remove most of it, because the relevant wavelengths in the steel structure are metres long and a few actuators control a structure much smaller than a wavelength. Helicopters, whose rotors shake the cabin at a fixed frequency, carry active vibration control for the same reason: a tone, at low frequency, in a structure small compared with its wavelength, is the ideal case on every count.

The opposite operation is just as linear. The mirror that sends a wave back to where it began recorded a wave arriving at an array and replayed it reversed in time, and the replayed wave converged on its source. A canceller replays the arriving wave inverted instead of reversed, and the replayed wave destroys rather than refocuses. Both depend on the medium adding waves exactly, and both are limited by how finely the array samples the wavefield — half a wavelength between elements to do it properly.

What superposition assumes #

Linearity. Sound adds exactly only at small amplitudes. At the levels near a jet engine or inside a combustion chamber the air’s response is measurably nonlinear, waves steepen, and their inverses no longer cancel them cleanly; the one medium that was supposed to add exactly found that only empty space carries waves without any such limit.

A fixed acoustic path. The canceller’s filter models how sound travels from its loudspeaker to its microphone, and adapts slowly as that path changes. A headphone pushed against the head, a window opened in a car, or a passenger moving in a seat changes the path, and an adaptive canceller has to relearn it; a fast change briefly makes things worse.

One error point, or a few. Each microphone gives one point of silence. More microphones and more speakers enlarge the quiet region, but slowly: to quieten a volume several wavelengths across needs a number of channels that grows as the cube of the size in wavelengths, which is why active control of large spaces at high frequencies is not attempted.

Still open: how far a quiet zone can be stretched #

The theory has an elegant limiting case that has never been built at scale. Every front is a source found that the field inside any closed surface is fixed by the field on that surface, as if the surface were covered with sources. Turn that around and it says that a surface covered with suitably driven loudspeakers and microphones can cancel any sound arriving from outside throughout the volume it encloses — or make the volume look, from outside, as if it were not there. Mark Jessel and George Malyuzhinets worked out the idea in the 1960s and 1970s, and versions of it are studied for quiet zones in vehicles and for making objects acoustically invisible.

The practical obstacles are the ones above, multiplied. The surface must be sampled at intervals of less than half a wavelength, so a sphere a metre across, quiet up to a kilohertz, needs hundreds of channels, each with a delay small compared with the time sound takes to cross it. How far adaptive arrays of that kind can be pushed with present processors, and whether a quiet zone big enough for a person to move around in, at the frequencies of speech, is achievable at all, has not been settled. The superposition the method depends on is not in question; what is in question is whether enough independent sources can be made to agree, quickly enough, on what to cancel.

The bubble of silence a single speaker makes is the honest reading of superposition. Two waves cancel where they are equal and opposite, and no other law makes them equal and opposite anywhere else. Over a distance short compared with a wavelength, nearly everything is equal to its value at a point, and so the silence extends a little way. Further than that, it has no reason to.