Eyring Reverberation Time

Sabine's equation has a flaw that only shows up in the rooms people pay to have treated: it never lets the sound stop. Push the absorption coefficient to 1.0 — every surface a perfect absorber, a sound field that cannot survive its first reflection — and Sabine still returns a finite reverberation time. It has to, because it treats absorption as a steady leak proportional to $\bar{\alpha}$ rather than as something that happens discretely, at each bounce.

Carl Eyring's 1930 paper fixed the accounting. If a fraction $\bar{\alpha}$ of the energy is lost at every reflection, then a fraction $(1-\bar{\alpha})$ survives, and after $n$ reflections the surviving energy is $(1-\bar{\alpha})^n$. That is exponential decay in the number of reflections, and taking its logarithm replaces Sabine's $\bar{\alpha}$ with $-\ln(1-\bar{\alpha})$. The correction is the honest version and Sabine's is its small-absorption approximation, because $-\ln(1-x) \approx x$ when $x$ is small.

How small is small enough is the practical question. At $\bar{\alpha} = 0.1$ the two forms differ by about 5 %, less than the uncertainty in the coefficients themselves. At 0.3 the gap is 19 %, at 0.5 it is 39 %, and at 0.8 Sabine overstates the reverberation by nearly a factor of two. The working rule is Sabine below 0.2 and Eyring above it — and at $\bar{\alpha} = 1$ Eyring correctly returns zero, because $\ln(0)$ diverges and no sound survives the first reflection.

Neither equation rescues a room whose absorption is all in one place. Both assume the sound field is diffuse, and a treated ceiling over a hard floor and hard walls is emphatically not: the horizontal reflections between parallel hard walls persist long after the vertical ones have died, and the measured decay curve bends rather than falling straight. Millington and Sette proposed a refinement that applies the logarithm surface by surface rather than to the average, which behaves better with mixed materials but misbehaves with any surface at $\alpha = 1$. In an awkward room, all of these are estimates, and a ray-tracing model or an in-situ measurement is what settles the argument.