The wall that stops sound by its weight

What happens where the medium changes found that a wave meeting a change of impedance — of how much push it takes to make the medium move — is partly reflected and partly transmitted, by amounts fixed by the ratio of the two impedances and nothing else. The lever that lets air speak to water found the middle ear solving the problem that ratio sets: air and water differ in impedance by a factor of 3,600, so sound in air bounces almost entirely off water, and the ear’s bones and membranes are a transformer that matches one to the other.

A wall between two rooms is the same problem from the other side. Sound in air arrives at a solid wall, and the aim is not to match the impedances but to mismatch them as badly as possible, so that as little as possible gets through. Most walls do this well, and the reason is almost embarrassingly simple: a wall is heavy. The physics of how heavy, and of the two ways a real wall falls short of its weight, is the physics of a single impedance placed in the path of a wave.

A mass in series with the air #

Sound arriving at a thin wall pushes on it with its pressure. The wall, being free to move a little, moves — back and forth at the sound’s frequency — and as it moves it pushes the air on the far side, radiating sound into the next room. A wall that did not move would transmit nothing; a wall of no mass would move exactly as the air does and transmit everything. Between those, what decides how much the wall moves is the force it takes to accelerate it, and for a wall that is only mass, that force per unit area is the surface mass $m$ times the acceleration. In impedance terms the wall is $i\omega m$, an inertia, placed in series between two bodies of air each of impedance $\rho c$, about 413 rayl.

For sound arriving square-on, the fraction of the energy transmitted is then

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

and the transmission loss, ten times the logarithm of $1/\tau$ in decibels, grows as $20\log_{10}(\omega m/2\rho c)$ once $\omega m$ is much larger than $2\rho c$ — which, for any wall worth the name, it is at every audible frequency.

That is the mass law. Its content is in the logarithm: the loss depends on the product of mass and frequency, so doubling either adds six decibels. A wall of ten kilograms per square metre — a single sheet of plasterboard, a pane of glass four millimetres thick — stops about thirty decibels at five hundred hertz in the ideal case. Doubling it to twenty adds six. Doubling again adds six more. Bass gets through more easily than treble for the same reason: at a lower frequency the same mass moves more for the same push.

Nothing in the law involves absorption. The energy that does not get through is reflected back into the room it came from, as it is at any impedance mismatch. That is why a well-insulated room with hard walls can be very quiet from outside and very loud inside: the sound made in it bounces around rather than escaping.

Why heavier is the only lever #

The mass law is unforgiving. Every ten decibels costs a factor of about three in mass, and the decibel scale is where people’s sense of loudness lives — ten decibels is roughly a halving of perceived loudness.

To stop thirty decibels at five hundred hertz, with sound arriving from every direction as it does in a real room, a single wall needs about fifteen kilograms per square metre — a sheet and a half of plasterboard. Forty decibels needs forty-seven; fifty needs nearly a hundred and fifty, a half-brick wall; sixty needs a slab of concrete twenty centimetres thick. The figures explain the brick and concrete party walls of older terraced houses and blocks of flats, and why lightweight modern partitions, built for speed and cheapness, leak sound unless they are built differently. Building differently means beating the mass law, and two physical effects decide whether a real wall beats it or falls below it.

The angle at which a stiff wall vanishes #

A real wall is not only mass; it is also stiff, and stiffness changes the problem when sound arrives at an angle. Sound arriving obliquely does not push the whole wall in step. Its crests sweep across the wall’s surface, pressing one strip and then the next, and the pattern of pressure travels along the wall at the trace speed, $c/\sin\theta$ — faster than sound, and infinitely fast for sound arriving square-on. The wall responds by bending, and a bending wave in a plate has its own speed, which rises with frequency.

At one frequency for each angle, the bending wave the sound drives travels along the wall exactly as fast as the trace of the sound’s crests. The wall then responds not as a mass being shaken but as a plate in resonance with the sound, and its stiffness exactly cancels its inertia: the wall stops opposing the sound and lets it through almost freely. This is coincidence, and it is the same matching of speeds that makes a supersonic source radiate along the cone it leaves behind, read in reverse: a wave whose trace along a surface matches the surface’s own wave speed couples to it completely.

The lowest coincidence frequency, for sound grazing along the wall, is the critical frequency $f_c = (c^2/2\pi)\sqrt{m/B}$, where $B$ is the plate’s bending stiffness. Below it there is no coincidence at any angle. Above it there is always some angle at which sound arrives in step with the bending, and in a room where sound arrives from every direction, some of it always gets through that way. The depth of the dip is limited only by how much the plate’s own internal friction damps its bending waves, measured by a loss factor of a few hundredths for most building materials.

For a single pane of window glass four millimetres thick, the critical frequency is about three kilohertz, and for a sheet of plasterboard a little lower — in the middle of the range where speech carries its consonants, and where the ear is most sensitive. Just above coincidence a pane’s loss falls more than twenty decibels below its mass law. That is why a single-glazed window lets through the sound of voices so clearly, and why making a pane thicker is less use than its weight suggests: a thicker pane is heavier, but it is also stiffer in proportion to the cube of its thickness, and its critical frequency moves down into a range that matters more. Laminated glass, two panes bonded with a layer of soft plastic, raises the damping and fills in the dip, which is the reason it is sold as acoustic glazing.

The mass law’s other weak point is the bottom of the range. Insulation falls by six decibels for every halving of frequency, so a wall that stops forty decibels of conversation at five hundred hertz stops only twenty-eight of the rumble of a lorry’s engine at a hundred and twenty-five. Low-frequency noise — traffic, a neighbour’s bass, the hum of a building’s ventilation fans — is what gets through walls that stop everything else, and it is the hardest to treat, because the only remedy the mass law offers is weight. The same arithmetic explains why a loud party heard through a wall is mostly its bass line: the wall has filtered the music, taking the top off and leaving the bottom, as any series inertia does.

Two walls, a gap, and a spring #

The way to beat the mass law is to split the wall into two leaves with a gap between them. Below a certain frequency the air in the gap is a stiff spring that couples the two leaves so that they move together, and the pair behaves like a single wall of their combined mass. Above it, the gap decouples them, and sound has to get through two mass laws in succession, each multiplying its loss.

Between those regimes is a resonance. The two leaves are two masses joined by the air spring, and at the frequency at which they bounce against each other on it — the mass–air–mass resonance, at 91 hertz for two plasterboards ten centimetres apart — the pair lets sound through almost freely, worse than a single wall. It is the note a bottle sings, with the bottle’s neck replaced by a second wall, and it is two walls letting more through than one at resonance, as two mirrors do at the frequencies a Fabry–Pérot cavity transmits. Wall designers place it below the range that matters, by making the leaves heavy and the gap wide, and above it the loss climbs at about eighteen decibels per octave — the six of each leaf’s mass law plus the twelve of the spring that separates them.

The computed improvement at a kilohertz, thirty-six decibels over the same mass in one piece, is what an idealised wall gives for sound square-on, with nothing connecting the two leaves but air. Real double walls give far less, for reasons that are the practical content of building acoustics. The leaves are fixed to shared studs, which carry vibration straight across. The gap’s own standing waves, visible as the dips at higher frequencies in the figure, transmit sound at their resonances unless the gap is filled with fibrous material that damps them. And sound goes round a wall through any gap, duct or shared floor. Good party walls separate the leaves on independent studs, fill the cavity with mineral wool and seal every edge, and they then reach fifty to sixty decibels with a fraction of the mass a single wall would need.

A small hole undoes a heavy wall #

Insulation is an average of transmissions, not of losses, and that makes it fragile. If a wall of area $S$ is made of parts with areas $S_i$ and transmission coefficients $\tau_i$, the sound through it is $\sum S_i\tau_i/S$, and a part that transmits everything dominates the sum however small it is. A wall that stops fifty decibels transmits a hundred-thousandth of the energy that falls on it. Leave one per cent of its area open — a gap under a door, an unsealed service duct, an electrical socket back to back with another — and that one per cent transmits a hundredth, a thousand times more than the rest of the wall put together: the wall now stops twenty decibels. A gap of a tenth of a per cent limits it to thirty.

That is why acoustic consultants spend more of their time on gaps than on walls, and why a heavy door with no seal is barely better than a light one. It is the impedance picture again, at its most literal: a hole is a place where the impedance in series with the air is zero, and sound takes it as current takes a short circuit.

Insulation is not absorption #

The difference between stopping sound and soaking it up is the most common confusion in the subject, and the impedance picture makes it sharp. An absorber is designed to match the air’s impedance, so that sound enters it without reflecting and is turned to heat inside: a thick layer of soft, porous material. A barrier is designed to mismatch it, so that sound reflects: a heavy, airtight sheet. Foam tiles glued to a wall reduce the echo in a room and do almost nothing to the sound passing through the wall, because they are light and porous and sound passes through them nearly unimpeded. Mineral wool inside a double wall is a different matter, not because it stops sound itself but because it damps the standing waves in the gap that would otherwise carry sound across.

Laboratory measurements separate the two by putting the wall between two hard, echoing rooms, playing sound in one and measuring the levels in both; the difference, corrected for how much absorption the receiving room has, is the wall’s transmission loss, measured band by band and then condensed into a single number weighted towards the frequencies of speech. A party wall that rates fifty on that scale makes normal speech next door inaudible and a shout faintly audible.

The wall as a filter #

All three effects are one statement about impedance. The wall is a series element between two bodies of air: an inertia at low frequencies, which gives the mass law; an inertia cancelled by stiffness at coincidence, which gives the dip; and, for a double wall, an inertia, a spring and another inertia, a filter with a resonance and a steep slope above it. That is how the matching layers and tapers that remove a reflection are analysed too, with the aim reversed: there the goal was to make a change of medium invisible, and here to make it as visible as possible.

The reversal has a limit worth noticing. No network can remove a mismatch over a wide band, Bode found, and that limit, applied in reverse, says that a wall of given mass cannot be made to reflect perfectly over a wide band either, by any arrangement of its parts: it can trade insulation at one frequency for insulation at another, as the double wall trades a resonance at 91 hertz for steepness above it, but the trade has a budget.

What the pictures cannot show #

The single-wall curves are computed for infinite, uniform panels; real walls have edges, and below a few hundred hertz a real panel’s own vibration modes, set by its size and fixings, make its loss ragged. The field-incidence average assumes sound arrives equally from every direction up to 78° from square-on, a convention that matches laboratory measurements reasonably and real rooms less well. The double wall is computed for sound square-on, which makes its resonance sharper and its advantage larger than a real wall shows.

None of the figures includes flanking: sound that travels round a wall, through the floor and ceiling and along the structure, which in most buildings limits the insulation between rooms well before the wall itself does. A wall is only as good as its weakest path, and the paths are rarely through it.

The domain of the argument is sound in air meeting a thin partition whose thickness is small compared with the wavelength. Inside it, the wall is an impedance, and its mass, its stiffness and how its layers are separated decide how much passes.

Still open: how light a quiet wall can be #

The mass law is a limit on a single, uniform wall, and the double wall beats it at the cost of thickness. Whether a thin, light partition can insulate as well as a heavy one has been attacked with acoustic metamaterials: panels carrying small resonators — masses on springs, membranes with weights attached — tuned so that at chosen frequencies the panel’s effective mass becomes enormous or negative, and it reflects nearly everything. They work, over narrow bands, and they inherit Bode’s budget: a resonator that stops one band passes another. How broad a band a light structure can block, and whether such panels can be made cheaply enough to put in buildings, are being pursued with the aim of quieter lightweight construction, where the mass law is at its most expensive.

The law itself is old and short. A wall between two rooms is a mass in series with the air, so the fraction of sound it passes square-on is 1/(1 + (ωm/2ρc)²) — six more decibels for every doubling of mass or frequency, 37.6 dB at 500 Hz for 20 kg/m² — and a real wall falls short of it where its bending wave matches the sound’s trace, about 3 kHz for a window pane, and beats it above the resonance of two leaves bouncing on the air between them. Quiet is bought with weight, unless the weight is split and the air between is made to work.