Sabine Equation Sabine Reverberation Equation

The equation and what it predicts

The Sabine equation in metric units is RT60 ≈ 0.161 × V / A, where RT60 is the reverberation time in seconds, V is the room volume in cubic metres, and A is the total absorption in metric Sabines, computed as the sum of S_i × α_i for each interior surface, with units of m². The 0.161 s/m constant follows from the speed of sound at 20°C, taken as 343 m/s.

RT60 itself is the time, in seconds, for the sound pressure level in a room to drop by 60 dB after a steady-state source is switched off. ISO 3382-1 and 3382-2 standardize the measurement and in practice allow the decay to be measured over 20 dB (T20) or 30 dB (T30) and linearly extrapolated to the full 60 dB drop.

Sabine arrived at the relationship empirically while spending 1895 to 1898 diagnosing Harvard's new Fogg Art Museum lecture room, where a spoken word remained audible for about 5.5 seconds and was unintelligible. He derived the formal expression on 29 October 1898.

Where the 0.161 constant comes from

Sabine's derivation models a diffuse sound field whose energy decays exponentially as it strikes absorbing surfaces at an average rate of cA/(4V). Setting the energy ratio to 10⁻⁶ — the linear equivalent of a 60 dB drop — and solving gives RT60 = (24 ln 10 / c) × V / A; substituting c ≈ 343 m/s at 20°C in dry air yields the familiar metric constant 0.161 s/m.

Total absorption A is the sum, over every interior surface of the room, of that surface's area S_i multiplied by its frequency-dependent absorption coefficient α_i. The result has units of m² and is reported in metric Sabines, where 1 Sabine equals 1 m² of perfect absorber. Furniture, occupants, and air absorption (significant above ~2 kHz in large rooms) are added as their own absorption terms.

Three assumptions underlie the derivation: a diffuse, statistically uniform sound field where energy is equally likely to travel in any direction; absorption that is small to moderate and distributed reasonably uniformly across the boundary; and a room large compared to the wavelengths of interest, so modal behaviour can be ignored. These assumptions hold best in mid-size to large rooms with an average absorption coefficient roughly between 0.1 and 0.4.

Where Sabine breaks down

The clearest failure mode is high absorption. Sabine over-predicts RT60 once the area-weighted average absorption coefficient α_avg exceeds roughly 0.2 to 0.25, with errors of 15 to 30 percent by α_avg ≈ 0.4. The Eyring equation (also called Norris–Eyring), RT60 ≈ −0.161 × V / (S × ln(1 − α_avg)), replaces the linear absorption term with a natural-log term and correctly returns RT60 = 0 in a fully absorbing room (α = 1), where Sabine still gives a non-zero result. Eyring is the preferred default once average absorption is high.

The diffuse-field assumption fails in several common cases: small rooms where modal density is low at bass frequencies; rooms with strongly clustered absorption, such as all the absorption mounted on a single suspended ceiling; and coupled spaces where one room opens into another with a different decay time. In those situations Sabine's single-number RT60 misrepresents the actual decay shape, and ray-tracing or finite-element methods are needed instead.

Even where Sabine applies, a single number across the audible band is misleading because absorption coefficients are strongly frequency-dependent. Porous absorbers — fibreglass, mineral wool, acoustic foam — absorb high frequencies far more efficiently than low frequencies, while membrane and Helmholtz resonators target specific narrow bands. ISO 3382 and engineering practice therefore report RT60 across the standard octave bands from 125 Hz to 4 kHz, and a Sabine calculation must be done independently for each band.

Sizing absorption for a home theatre

Home-theatre rooms typically target a mid-band RT60 of about 0.3 to 0.4 s. For a 4,000 ft³ (~113 m³) basement room aiming at RT60 = 0.4 s, Sabine gives A = 0.161 × 113 / 0.4 ≈ 45 metric Sabines of mid-band absorption — roughly 50 to 60 m² of high-NRC fabric panel, depending on each panel's coefficient.

That panel count is realistic if you know what one panel actually delivers. A 50 mm fabric-wrapped fibreglass or mineral-wool panel typically provides an absorption coefficient of 0.85 to 0.95 at 500 Hz — about 0.85 to 0.95 Sabines per m² — but falls to roughly 0.20 to 0.40 at 125 Hz unless mounted with a 50 to 100 mm air gap behind it, which improves low-frequency absorption by acting as a quarter-wavelength resonator. NRC ratings of 0.90+ on product datasheets describe the 250 to 2000 Hz arithmetic average, not bass performance.

That gap explains why broadband panels alone never tame the low end. Sabine treats absorption as a scalar at each frequency, and the Sabines per m² of a thin porous panel collapse below ~125 Hz because the panel is much thinner than a quarter-wavelength of bass — at 100 Hz, λ is 3.43 m and λ/4 is about 86 cm. Effective bass control therefore requires either very thick porous traps with deep air gaps placed in pressure zones (corners), or membrane and Helmholtz resonators tuned to the offending room modes. Sabine sets the broadband target; modal analysis sets where the bass traps go.

Sabine is not the final word. It sizes a starting amount of broadband absorption from a target RT60, but the design must then be verified — not predicted — with an in-room measurement using a tool such as REW (Room EQ Wizard) and a calibrated microphone. Measurement captures the modal behaviour, decay shape, and frequency-dependent RT (T20/T30 per octave band per ISO 3382-2) that the diffuse-field assumption cannot reproduce in a small living-room-sized space.