Every reverberation calculation anyone has ever done starts in the same place. Wallace Clement Sabine, working at Harvard through the 1890s with an organ pipe, a stopwatch and cushions borrowed from a nearby lecture theatre, found that the time a room takes to fall quiet is proportional to its volume and inversely proportional to the absorption in it. In metric units the relationship is RT60 = 0.161 V / A. It is a hundred and thirty years old, it is on the first page of every textbook, and it is wrong in a way that matters as soon as you build something quiet.
What the derivation assumes
Sabine’s equation comes out of a picture of sound as a diffuse field losing energy smoothly and continuously. Imagine a bath draining: the level falls at a rate set by the size of the plughole, and it falls smoothly. In that picture, absorption is a rate, and the total absorption A is the sum of every surface’s area multiplied by its absorption coefficient.
The physical reality is different. Sound does not lose energy continuously. It travels in straight lines at 343 m/s and loses a fixed fraction of its energy each time it strikes a surface. Between strikes it loses almost nothing. The decay is a staircase, not a ramp, and the average distance between the steps — the mean free path — is 4V/S for any convex room, which is a lovely result and the only piece of Sabine’s picture that survives intact.
While the room is fairly reflective, the staircase has small steps and the ramp is an excellent approximation. A hall whose surfaces average 0.05 absorption gives back ninety-five per cent of the energy at each reflection; the smoothing hides nothing. But as the surfaces get more absorptive the steps get bigger, and the smooth approximation starts predicting a decay slower than the real one.
Where it breaks
The place to see it is the limit. Consider a room with no surfaces at all — the open air, or a perfect anechoic chamber where every coefficient is 1.0. Physically, sound leaves and never comes back: the reverberation time is zero. Put the same room through Sabine’s equation and A = S × 1.0, a finite number, so RT60 = 0.161V/S, a finite time. The equation says a room with no walls rings. It cannot be repaired by being careful; the assumption has simply run out.
That is the extreme, but the error arrives long before it. As a rule of thumb, once the mean absorption coefficient of a room passes about 0.2 the discrepancy is worth caring about, and by 0.4 it is a third. A control room designed by Sabine’s equation alone will measure substantially shorter than its calculation, which sounds like good news and is not: it means the designer did not know what they were building, and the same error in the other direction — a room that measured longer than predicted — would be a defect.
What Eyring does
Carl Eyring, in 1930, kept the mean free path and replaced the rate with the arithmetic of repeated reflections. If a fraction a-bar of the energy is removed at each strike, the fraction remaining after n strikes is (1 – a-bar) to the power n, and the number of strikes per second is c·S/4V. Work that decay back to a 60 dB drop and you get
RT60 = 0.161 V / ( -S · ln(1 – a-bar) )
where a-bar is A/S, the area-weighted mean coefficient. The structure is the same and the denominator has changed from S·a-bar to -S·ln(1 – a-bar). For small a-bar those two are nearly identical, because ln(1 – x) ≈ -x, which is exactly why Sabine worked so well in the rooms he had access to. As a-bar approaches 1, ln(1 – a-bar) runs away to negative infinity, the denominator becomes enormous, and RT60 correctly goes to zero. The anechoic chamber stops ringing.
Why we publish both
We give the Sabine and the Eyring figure on every project page, and the reason is not scrupulousness for its own sake. The gap between them is a reading in its own right. If they agree within a few per cent, the room is reverberant, the statistical assumptions are safe, and you may go on using every rule of thumb you know. If they are twenty per cent apart, the room has become absorptive enough that the diffuse-field model is under strain, and a series of other things you were relying on are also less reliable than they look: the assumption that the decay is a straight line on a log plot, the assumption that position within the room does not much matter, and the assumption that the measured reverberation time is a property of the room rather than of where you put the microphone.
A single number would hide all of that. Two numbers and their difference tell a reader, in one glance, how far the model has been stretched to produce them. It is the difference between reporting a result and reporting a result with its error bar, and in a discipline whose findings are routinely quoted back years later by people who were not in the room, the error bar is the more useful half.
Neither equation, it should be said, is good below the Schroeder frequency, where a room stops having a reverberation time in any useful sense and starts having a handful of resonances. Both are equally wrong there, and no amount of choosing between them will help. That is a different article.
