The wall made heavy only at one note

The wall that stops sound by its weight found the rule that every builder of quiet rooms works against. Sound arriving at a wall pushes on it, and the wall passes the sound on to the far side by moving; a wall that is hard to move passes little. For a limp wall at low frequencies the only thing making it hard to move is its inertia, its mass per unit area, and the transmission loss rises six decibels for every doubling of the mass or of the frequency. To block low notes, a wall must be heavy. The essay ended with a question about whether a light partition could ever insulate like a heavy one, and pointed at structures that try: panels whose mass, at chosen frequencies, is not what they weigh.

A mass that depends on the frequency #

The mass law treats a wall as a single rigid mass driven by the sound’s pressure. Its resistance to being shaken at angular frequency ω is $\omega m$ per unit area, and the fraction of the sound’s power that gets through, at normal incidence, is

$$\tau = \frac{1}{1 + (\omega m / 2\rho c)^2},$$

with ρc the impedance of air, 413 rayls. What happens where the medium changes found the same arithmetic at every boundary between two impedances: the more the panel’s impedance, ωm, exceeds the air’s, the more is reflected.

Suppose the panel is not a single mass. Hang small masses from it on springs, each set tuned to resonate at one frequency $\omega_0$. Shaken slowly, the hidden masses move with the panel and it behaves as its total weight. Shaken fast, they cannot follow and stay nearly still, and it behaves as its bare weight. Shaken near $\omega_0$, they swing hard, and the force they exert back on the panel through their springs changes the force needed to shake it. Dividing that force by the panel’s acceleration gives a dynamic mass per unit area,

$$m_{\text{eff}}(\omega) = m + \frac{m_r\,\omega_0^2}{\omega_0^2 - \omega^2},$$

which is what the sound feels in place of the panel’s weight, with $m_r$ the resonators’ mass per unit area and damping left out for the moment.

A dynamic mass that goes through zero #

The second figure draws it for the panel of the first: two kilograms per square metre with half as much again hung on springs tuned to 300 hertz. Below the tuning the dynamic mass rises above the panel’s total weight, as the resonators begin to lag and their springs pull back on the panel in step with its own inertia. Near 300 hertz it climbs to fifteen kilograms per square metre, five times what the panel weighs, held down only by the small damping in the springs. Just above the tuning it changes sign. The resonators now swing against the panel’s motion so strongly that they push it along, and the force needed to shake the panel is less than its own inertia would demand, then nothing, then opposite. At $\omega_0\sqrt{1 + m_r/m}$, 367 hertz here, the dynamic mass passes through zero.

A wall with no dynamic mass is not a wall. The sound’s pressure moves it without resistance and passes straight through, and that is the dip in the first figure: at 367 hertz the light panel with resonators transmits more sound than the bare panel would have, five decibels of loss against fifteen. At 300 hertz, by contrast, it blocks thirty-five decibels, which the mass law would give only a wall of twenty-six kilograms per square metre. The same added kilogram, rigidly fixed, would have raised the loss from 13.4 to 16.8 decibels at every frequency alike. To put the comparison in building terms: a sheet of plasterboard twelve and a half millimetres thick weighs about ten kilograms per square metre, so the light panel with its resonators blocks a 300 hertz tone as well as a wall of two and a half layers of plasterboard would, while weighing less than a third of one layer. And because the mass law rises by six decibels for every doubling of mass, matching the resonant panel’s thirty-five decibels at that tone with plain weight would take nearly nine times what it weighs.

This is the mechanism the mass that makes another stand still used to silence a vibrating machine: a tuned absorber, a small mass on a spring that resonates against the machine and holds it still at one frequency, paid for with two new resonances on either side. Here the absorbers are spread over a surface and the thing held still is the panel, which is why sound cannot shake it through.

A negative mass with nothing light in it #

A negative mass sounds like a contradiction, and it is worth seeing why it is not. Nothing in the panel weighs less than nothing. The panel and every resonator have ordinary positive masses, and their total momentum responds to the total force in the ordinary way. What is negative is the ratio of the force applied to the panel’s face to the panel’s own acceleration, at one frequency, in the steady state. Above the tuning the hidden masses move in opposite phase to the panel, and their springs push on the panel in the direction it is already going, more strongly than its own inertia resists; the sound’s pressure then has to pull back on the panel to keep it moving as it does. Measured only by what the face does, the panel behaves as if pushing it made it move the other way.

The same bookkeeping turns up wherever a hidden motion is driven above its resonance. The frequency below which nothing gets in found the electrons of a plasma, driven by a wave below their resonance, giving the plasma a negative permittivity that reflects the wave; the mass a curve decides found an electron near the top of a band responding to a force as though its mass were negative, because the crystal lattice it moves through is doing the pushing. In each case a negative response is the visible face of an internal motion out of step, and it lasts only over the band of frequencies where the internal motion is out of step in the right way.

Bands bought with mass #

The improvement is real and narrow, and its width is set by how much of the panel’s mass is in the resonators. The third figure varies that share. With a tenth of the mass resonating, the peak is tall but so narrow that it never beats a plain panel of the same total weight by ten decibels anywhere; with half, it does so over about thirty hertz around 300; with one and a half times the panel’s own mass in resonators, over about sixty. The pass band moves away as the share grows, from 315 to 474 hertz, because the zero of the dynamic mass sits at $\omega_0\sqrt{1+\mu}$ with μ the mass ratio.

The resonators are therefore not a way round having mass. They are a way of deciding where the mass acts. A resonant panel blocks a chosen band as a heavy wall would, but only by putting a substantial fraction of its weight into the resonators, and outside the band it is no better than its total weight would make it. The mismatch no network can remove found a theorem of this shape for electrical matching: a lossless network can improve a mismatch over a band only at the cost of making it worse elsewhere, and the total improvement is bounded. The passive panel faces an analogous constraint, rooted in the same causality: a structure that responds strongly at one frequency must respond strongly, in the opposite sense, near it.

Damping trades the peak for the dip #

The resonators’ springs lose some energy every cycle, and how much decides the trade between peak and dip. With very little damping the peak is high and sharp, forty-seven decibels in the fourth figure, and the pass band above it is deep: at its bottom the panel transmits almost everything. Heavy damping flattens both. The peak drops to twenty-one decibels, the pass band fills to fourteen, and the panel behaves more like a heavy, lossy wall over a wider range than like a selective one. Designers of resonant panels choose the damping to suit what they need to block and what they can afford to let through, and the pass band is often the constraint: a partition that blocks a machine’s hum but transmits a slightly higher note perfectly may be worse than one that does neither well.

Two tunings, two peaks, two leaks #

The obvious way to widen the protection is to use more than one tuning, and the last figure splits the same resonating mass between two sets, at 250 and 500 hertz. Each makes a peak where it resonates and a leak just above it. The protection now covers two bands, each narrower than one set alone would give with the whole mass, and the panel has two pass bands instead of one. Panels designed for broad bands use many tunings and enough damping to blur the peaks into a plateau, and the more tunings there are, the more the result resembles a wall that is simply heavy — the mass law recovered, with the mass distributed over frequency instead of concentrated in one band.

How the springs are made #

The resonators can be almost anything with a mass and a restoring force. The first locally resonant materials used heavy spheres coated in soft rubber and embedded in a stiff matrix: the sphere is the mass, the rubber the spring, and the frequency is set by the rubber’s thickness and stiffness. Thin stretched membranes carrying small weights do the same with far less material, the membrane’s tension acting as the spring, and can be tuned by moving the weight. Cantilevers cut into a plate and loaded at their tips, plates with heavy inclusions, and small air cavities connected to the outside through necks — Helmholtz resonators, whose air plug is the mass and whose trapped air is the spring — have all been used. Each fixes its resonance by its own geometry rather than by the panel’s, which is what lets a thin, light structure resonate at a low frequency.

Where a single tone is the problem #

Resonant panels are most useful where the noise is a tone, and many of the most intrusive noises are. Transformers hum at twice the frequency of the mains, a hundred or a hundred and twenty hertz, and its harmonics; propellers and fans produce tones at their blade-passing frequency; diesel engines and compressors produce strong low-frequency tones that heavy enclosures would be needed to stop. In aircraft, every kilogram of soundproofing costs fuel, and the cabin noise of propeller aircraft is dominated by the blade-passing tone and its harmonics. A panel that blocks those few frequencies as a heavy wall would, at a fraction of the weight, is worth its pass bands as long as the pass bands fall where the noise is quiet. That is the design problem the first figure states: put the peak on the tone and the leak where there is nothing to leak.

Where the panel is a metamaterial #

Arrays of such resonators are called acoustic metamaterials, and the name is earned at the scale of the wavelength. The resonators in the figures are not spaced to make the panel a grating or a crystal: at 300 hertz the wavelength in air is more than a metre, and the resonators can be centimetres apart. The gap a repeat opens found that a periodic structure blocks waves whose wavelength matches its period, Bragg reflection. A locally resonant structure blocks waves whose frequency matches its resonators, whatever their spacing, and so can block a long wavelength with a thin, finely structured layer. In 2000 Zhengyou Liu and colleagues demonstrated the effect with lead spheres coated in rubber and set in epoxy, a material with band gaps at wavelengths a hundred times its lattice spacing, and membrane panels weighted with small masses, a few hundred grams per square metre, have since been shown to block chosen low-frequency tones better than the mass law allows by tens of decibels.

Where the description holds #

The figures treat the panel as limp, a pure mass with no stiffness of its own, and the sound as arriving straight on. A real panel is stiff and bends, with its own resonances and the coincidence dip that the mass-law essay found, and those interact with the resonators’ effect. Sound arriving at an angle meets a panel whose response depends on the angle. The resonators are treated as a smooth continuum attached to every point of the panel, where real ones are discrete, and the approximation holds only while they are much closer together than the wavelength of the sound and of the panel’s own bending waves. Normal incidence is what a laboratory impedance tube measures, a small sample clamped across a pipe with loudspeakers at one end; a building’s sound arrives from every direction at once, and the loss for that diffuse field is several decibels lower everywhere and smears the resonant peak. And the figures give the loss of an isolated panel between two half-spaces of air, not the loss of a partition in a building, where sound also travels round it through the floor, the ceiling and every crack.

What the figures cannot show #

The figures show transmission at a single frequency at a time. They cannot show what a listener hears through such a panel when the sound is not a pure tone: a broadband noise loses its band near the tuning and keeps the rest, including the pass band just above, so a resonant panel can change the colour of a noise more than its loudness. They cannot show that the resonators’ energy has to go somewhere. At the peak the panel barely moves while the resonators swing hard, and the energy they hold sloshes back and forth with the sound and is dissipated in their damping, which is why a resonant panel at its tuning reflects most of the sound back into the room it came from, as a heavy wall does.

Still open: how light a broadband quiet wall can be #

A partition that blocked a broad band of low frequencies as well as a heavy wall, while weighing a fraction as much, would change how buildings, vehicles and aircraft are made, and the search for one is active. Combinations of membranes, resonators and porous absorbers, panels with spatially varying tunings, and structures that couple their resonators to one another have widened the bands achievable at a given weight, and limits derived from causality — of the kind that relate an absorber’s bandwidth to its thickness — bound how far any passive design can go. Active panels, with sensors and actuators that cancel the sound’s push electronically, escape those bounds at the cost of power and complexity. What the best passive trade between weight, thickness and bandwidth is, and how close practical panels can come to it, is being worked out.

The habit worth carrying away is to ask what a structure’s mass is at the frequency that matters. A panel with resonators hung inside it has a dynamic mass $m + m_r\omega_0^2/(\omega_0^2 - \omega^2)$, far larger than its weight near the tuning and passing through zero at $\omega_0\sqrt{1+\mu}$: a 2 kg/m² panel with 1 kg/m² of resonators tuned to 300 Hz blocks 35 dB there, as a 26 kg/m² wall would, and lets sound through at 367 Hz more easily than the bare panel, with a protected band that widens only as more of its weight resonates. Inertia can be borrowed at one frequency, and paid back at the next.