How long a room rings, and where the formula stops

Wallace Clement Sabine was given a lecture theatre at Harvard in 1895 that nobody could hear anything in, and no theory to fix it with. What he did was measure: an organ pipe, a stopwatch, and the time taken for the sound to become inaudible after the pipe was stopped, with seat cushions carried in from a neighbouring hall in varying quantities.

The result was the first quantitative law in architectural acoustics, and it is one line.

The time for sound to decay by sixty decibels is 0.161 times the volume in cubic metres, divided by the total absorption in square metres — where “total absorption” is each surface’s area multiplied by how much of the sound it swallows.

What the formula knows and does not know #

The remarkable thing about the equation is how little it needs.

It does not know the shape of the room. A cube, a corridor and an irregular hall of the same volume and the same total absorption get the same answer.

It does not know where the absorption is. Carpet on the floor and carpet on the ceiling are interchangeable.

It does not know the frequency, except through the absorption coefficients, which are measured band by band — so a room’s reverberation time is really six or eight numbers rather than one, and quoting a single figure is already a summary.

That last omission is the important one and it is the source of the assumption everything else rests on. Sabine’s derivation treats the sound field as diffuse: energy arriving equally from all directions at every point, decaying by a fixed proportion at each encounter with a surface. Below a certain frequency a room is not like that at all — it is a set of discrete standing waves with large gaps between them — and the boundary between the two regimes has a name and a formula of its own.

Where it goes wrong, and by how much #

The failure is visible in the equation without any experiment at all. Set the average absorption coefficient to one — every surface swallowing everything, which is to say a room with no walls — and Sabine’s formula returns 0.161V/S, a positive number.

A room that absorbs everything has no reverberation. The correct answer is zero.

The reason is in the derivation. Sabine treats absorption as a continuous drain on the energy in the room, which is a good approximation when each reflection removes a small fraction and the sound survives many reflections. When each reflection removes most of the energy, the sound does not survive many reflections and treating the loss as continuous overstates how long it lasts.

Carl Eyring’s correction in 1930 replaces the total absorption with −S·ln(1−ᾱ), which is the same quantity for small ᾱ and goes to infinity as ᾱ approaches one — giving a reverberation time of zero for a fully absorbent room, as it should.

Eighteen per cent is the number worth carrying away, because of which rooms sit on which side of it.

A concert hall with an audience has an average absorption around 0.2 to 0.3 — the audience is most of the absorption, and a full hall and an empty one are acoustically different buildings. A stone church is nearer 0.1. Both are in or near the region where the two formulas agree, which is exactly the region Sabine measured in, and it is why the equation has stood for a century in the field it was built for.

A domestic room with carpet, curtains and soft furniture runs to 0.4 or 0.5. A treated studio control room goes higher still. In those rooms Sabine is wrong by tens of per cent, and they are where most recorded music has been made since about 1960.

Eighteen per cent is where the two curves visibly part on a drawn axis, though, and that is a fact about the drawing. The ratio between the two predictions is $-\ln(1-\bar\alpha)/\bar\alpha$ and it can be given a threshold that means something: the smallest change in reverberation time a listener can hear is about five per cent, which is a published measurement and not a derivation.

The two formulas become audibly different at ᾱ = 0.094, not at 0.18. By the time they separate on the graph they are already twice the audible threshold apart, and a full concert hall — the object the equation was built for, in the region this essay has just called safe — sits at two to four times it.

That is a genuine correction to the reassurance above, and it comes with its own resolution. Sabine did not obtain his absorption coefficients independently and then predict decay times; he measured decay times and derived the coefficients from them, through his own equation. So the published tables carry the correction inside them for rooms of the kind he measured, and Sabine plus Sabine’s coefficients gets a hall right in a way that Eyring plus Sabine’s coefficients would not. The equation and its data are a matched pair, which is why a century of halls built on the wrong formula came out at the length they were meant to, and why the failure shows up in the small absorbent rooms where nobody calibrated anything.

The derivation, in one paragraph #

The equation is short enough to see through, and seeing through it is what makes the assumptions visible rather than merely stated.

Consider a sound field with energy density E filling a room of volume V, so the total energy is EV. In a diffuse field, the rate at which energy strikes the surfaces is EcS/4 per unit time, where c is the speed of sound and S the surface area — the factor of four comes from averaging the angle of incidence over a hemisphere, and it is the one piece of the derivation that requires the field to be diffuse rather than directional. Each encounter removes a fraction ᾱ of what arrives. So the energy falls at a rate proportional to itself, which is exponential decay with a time constant of 4V/(cSᾱ). Requiring a fall of sixty decibels — a factor of a million in energy — and putting in the speed of sound gives 0.161V/(Sᾱ).

Every assumption is now in view. The field is diffuse, which fails below the Schroeder frequency. The absorption acts on arrival, which is where Eyring differs. Energy is removed in the room’s boundaries rather than in its air, which fails at high frequencies in large halls where air absorption is significant and is handled by adding a term. And the decay is exponential, which fails whenever a room is really two coupled rooms.

The constant 0.161 carries the speed of sound, so it is temperature-dependent, and the version quoted here is for about 20 °C.

What the number is for #

Reverberation time is not an aesthetic preference dressed up as a measurement; it is a design target with known values, and the values differ by use.

Speech wants a short time — around 0.6 to 1.0 seconds in a lecture room — because a long one smears consonants into each other and destroys intelligibility. The vowels survive it, since a vowel is a pair of sustained resonances and a resonance is exactly the thing reverberation preserves. The consonants are attack transients, they are short, and reverberation fills the gaps between them with the tails of what came before.

Orchestral music wants a long one, 1.8 to 2.2 seconds in a concert hall, because the blend and the sense of envelopment depend on it and because the repertoire was written in rooms like that. The room is not a container the music sits in; it is part of the instrument, and a piece scored for one reverberation time played in another is a different piece.

Organ music wants longer again, three seconds or more, which is why cathedrals are good for it and why organ writing has the slow harmonic rhythm it does — a chord change every half second in a room with a four-second tail is mud.

One thing the target hides is that a room does nothing whatever to a note’s beginning. The attack of a struck or bowed note is over in a few milliseconds, long before any reflection has returned from a wall twenty metres away, so the onset a listener hears is the onset the instrument produced in every room there is. What the room changes is the tail, entirely. That asymmetry is why reverberation alters the character of sustained music far more than percussive music, why a dry recording of an organ sounds wrong and a dry recording of a woodblock sounds normal, and why the design targets above are targets for repertoire rather than for instruments.

The conflict between speech and music in one building is real and unresolvable by any amount of cleverness, which is why multi-purpose halls have movable absorption and why they are generally regarded as compromised at both.

One number for a thing that has six #

Quoting a room’s reverberation time as a single figure hides the fact that absorption is strongly frequency-dependent and that the materials in a room absorb quite different amounts at different pitches.

Porous absorbers — carpet, curtains, acoustic foam, an audience — work by viscous loss as air moves through them, which requires the air to be moving, which happens away from a boundary. So they are efficient when they are thick compared with a quarter of a wavelength and nearly useless when they are not. A 25 mm foam panel is a good absorber above about 1 kHz and does essentially nothing at 100 Hz.

Panel and membrane absorbers work the other way, resonating at a frequency set by their mass and the air gap behind them, and they are the only practical way to absorb bass in a small room.

And a frequency-dependent decay does not merely shorten the tail; it changes the spectrum of what is in it. A room that absorbs the upper partials faster than the lower ones returns a note whose timbre at the end of its decay is darker than the timbre it started with — the partial list is being filtered as the sound dies, so the instrument’s own spectrum is only what arrives first. That progressive darkening is what a room being called “warm” refers to, and its opposite, a room that keeps its treble longer than its bass, is what “bright” refers to.

The consequence is that a room with a flat reverberation time across frequency is a designed object rather than a default. Untreated domestic rooms are almost always bass-heavy, because the soft furnishings absorb the treble and nothing absorbs the bass, and the reverberation time at 100 Hz can be two or three times what it is at 4 kHz.

That is also why a single number in a specification is a summary of a curve, and why the standard practice is to quote the average of the 500 Hz and 1 kHz bands and to publish the rest.

Where the diffuse assumption breaks entirely #

Above the failure at high absorption sits a second and more fundamental limit: the frequency below which a room has no diffuse field to speak of.

The Schroeder frequency marks it, and it is roughly 2000 times the square root of the reverberation time divided by the volume. For a concert hall of 18,000 cubic metres with a two-second tail that is about 21 Hz — below the audible range, so the whole of the music is in the diffuse regime. For a domestic room of 34 cubic metres with a 0.4-second tail it is about 220 Hz, which is the middle of the bass register.

That is the reason small rooms have bass problems and large ones do not. Below the Schroeder frequency the room is a set of resonances, the response varies wildly from position to position, and the reverberation time is not a meaningful description of anything. Above it the modes overlap and the statistical picture holds.

No amount of absorption fixes this, because the problem is the spacing of the modes rather than their strength, and the spacing is set by the dimensions. It is why studio control rooms are designed around their proportions before anything is put on the walls.

The arithmetic is elementary and unforgiving. The lowest axial mode along a dimension L is at c/2L, so a 5-metre room has one at 34 Hz and a 3.1-metre one at 55 Hz. Below the lower of those the room supports nothing at all, and between them the response is a series of peaks and dips of twenty decibels or more depending on where the listener sits. A reverberation time computed for that region is a number describing a field that does not exist.

The hall that was built on it, and the one that was not #

Sabine’s first application was the Boston Symphony Hall, opened in 1900 and designed with his advice — the first concert hall in the world whose acoustics were calculated rather than copied. It is still among the three or four halls most consistently rated highest by musicians, which is a remarkable outcome for a first attempt at a new engineering discipline.

What is less often said is what the calculation actually did. Sabine did not design the hall’s shape; he took a rectangular plan modelled on the Leipzig Gewandhaus and used the equation to set the volume so that the reverberation time would come out near two seconds with an audience in it. The shape — the shoebox, the coffered ceiling, the shallow balconies — came from a building everybody already admired.

That division of labour is the honest measure of what the formula supplies. It gets the reverberation time right, and reverberation time is one of perhaps a dozen quantities that make a hall good. The ones it says nothing about include lateral reflections, which decide the sense of envelopment; the delay before the first reflection, which decides the sense of intimacy; and the evenness of all of these across the seats, which decides how many of them are worth sitting in.

The counterexample is the Royal Albert Hall, whose reverberation time is unobjectionable and which was for a century notorious for a distinct echo from its domed ceiling — a single strong late reflection, invisible to any calculation of total absorption, and eventually treated by hanging a hundred and thirty-five fibreglass discs from the roof in 1969.

Whose measurement, and how it is made #

Sabine’s own method — listen and time the decay by stopwatch — was superseded but the definition was not. Modern measurement uses Manfred Schroeder’s 1965 backward-integration method: record the response to an impulse, integrate the squared response backwards from the end, and read the decay curve off the result. It gives a smooth curve from a single measurement where the direct method needed many.

One practical detail matters for reading any published figure. A sixty-decibel decay is rarely measurable, because the noise floor of a real room is not sixty decibels below a comfortable source level. So the standard practice is to measure the slope over the first twenty or thirty decibels and extrapolate — the quantities called T20 and T30 — and a quoted reverberation time is almost always one of those rather than a measured T60.

That extrapolation assumes the decay is a straight line on a logarithmic scale, which is what both formulas predict and what real rooms only approximately do. A room with two coupled spaces — a hall with a stage house, a church with side chapels — has a decay with two slopes, and a single number describes neither.

What the picture cannot show #

A reverberation time is also a number about energy and not about audibility. Whether a decaying tail is heard depends on what is sounding over it, and a loud sound raises the threshold for about two hundred milliseconds after it ends — so the first fifth of a second of any decay is partly hidden by the note that produced it, whatever the formula says about its level.

A reverberation time is one number for a whole room, and the thing a listener notices most is not in it: early reflections, the handful of strong echoes arriving in the first fifty milliseconds. Those are what produce the sense of a space’s size and of being enveloped by it, they depend entirely on where the listener is sitting, and they are averaged away by every formula in this essay.

Nor can it show what those early reflections do to the position of the sound, which turns out to be nothing at all: the first wavefront wins and everything arriving in the next thirty-five milliseconds is denied a vote on direction while contributing its full share of loudness and colour. A reverberation time is a statement about the tail; the head of the response is governed by a completely different mechanism with a completely different clock.

The curves also cannot show that absorption coefficients are themselves measurements with substantial uncertainty, made in a standardised reverberation chamber that resembles no real room, and that they routinely exceed one — a physically impossible value produced by edge effects in the test. A prediction built on them is not more accurate than they are.

The ladder from here goes to the part of a sound the room does not touch: the first fifty milliseconds, which arrive before any reflection does and which carry most of what identifies an instrument.