Sabine vs Eyring Equation — When Sabine Fails Above α 0.4 | Free Calculator | AcousPlan

Wallace Sabine published his reverberation formula in 1898. The rooms he measured had plaster walls, hardwood floors, and velvet seat cushions. The average sound absorption coefficient was somewhere between 0.04 and 0.08. Under those conditions, his formula is excellent. Under the conditions of a modern treated meeting room or acoustic office, it overestimates reverberation time by 15 to 49 percent.

That error is large enough to matter. If you design a meeting room to a Sabine-predicted RT60 of 0.65s expecting to hit the 0.6s WELL v2 target, and Eyring predicts 0.44s, you have two options: you either add treatment you did not need, or you skip the Eyring check and discover the problem at site measurement. The second outcome happens constantly.

Here is the exact boundary where Sabine becomes unreliable, the Eyring correction, the mathematical proof of why the difference grows with absorption, and worked examples showing the error magnitude in real room types.

The Foundation: What Sabine's Formula Assumes

The Derivation

Sabine derived his formula from first principles of acoustic energy balance in a reverberant room. In steady state, the power injected by a source equals the power absorbed by surfaces. When the source is switched off, the remaining energy E decays according to:

E(t) = E₀ × exp(-c × ᾱ × S × t / 4V)

Where c is the speed of sound (343 m/s at 20°C), ᾱ is the mean absorption coefficient, S is total surface area, and V is room volume.

Setting E(t)/E₀ = 10^(-6) (which corresponds to the 60 dB decay used to define T60) and solving for t:

T60 = -60 / (10 × log₁₀(exp(-c ᾱ S / 4V))) = 0.161 V / A

Where A = ᾱ S is the total absorption area in m² Sabine, and 0.161 = 24 ln(10) / c = 24 × 2.303 / 343.

Per ISO 3382-2:2008 Annex A Section A.1, this is the standard Sabine formula. The ISO standard describes it as appropriate for the estimation of reverberation time in rooms that are "not too highly damped" — a phrase that is crucial but rarely noticed.

The Diffuse Field Assumption

Sabine's derivation assumes that sound energy is uniformly distributed throughout the room at all times — the diffuse field. For this to hold, three conditions must be satisfied:

1. The mean free path between reflections must be much smaller than the room dimensions
2. Absorption must be distributed roughly uniformly across all surfaces
3. The absorption coefficient of each surface must be small enough that sound reflects many times before decaying

The specific mathematical issue: when a sound ray strikes a surface with absorption coefficient α, it retains a fraction (1 - α) of its energy. After n reflections, it retains (1 - α)^n. Sabine linearises this as ≈ exp(-n α) = exp(-n α), which equals (1-α)^n only when α is small.

The error in the linearisation:

• α = 0.10: (1-α) = 0.90, exp(-α) = 0.905 → error 0.5%
• α = 0.20: (1-α) = 0.80, exp(-α) = 0.819 → error 2.4%
• α = 0.40: (1-α) = 0.60, exp(-α) = 0.670 → error 11.7%
• α = 0.70: (1-α) = 0.30, exp(-α) = 0.497 → error 65.7%
• α = 1.00: (1-α) = 0.00, exp(-α) = 0.368 → infinite percentage error

The Eyring Correction

The Formula

In 1930, Carl F. Eyring published a corrected formula that replaces the linearised denominator with the exact expression:

T60 = 0.161 V / (-S ln(1 - ᾱ))

Where:

• V = room volume (m³)
• S = total surface area (m²)
• ᾱ = mean absorption coefficient = A / S = Σ(αᵢ Sᵢ) / S
• ln = natural logarithm

Why the Denominator Changes

In Eyring's derivation, the energy retained after one reflection off a surface with mean absorption ᾱ is (1 - ᾱ). After n reflections (where n = c t / l̄, and l̄ = 4V/S is the mean free path):

E(t) = E₀ × (1 - ᾱ)^(ct/l̄) = E₀ × (1 - ᾱ)^(cSt/4V)

Setting E(t)/E₀ = 10^(-6):

ln(10^(-6)) = (cSt/4V) × ln(1 - ᾱ)

-6 ln(10) = (cSt/4V) × ln(1 - ᾱ)

T60 = -6 ln(10) × 4V / (cS × ln(1 - ᾱ)) = 0.161 V / (-S ln(1 - ᾱ))

The Eyring denominator -S ln(1-ᾱ) reduces to Sabine's denominator A = ᾱ S when ᾱ is small, because -ln(1-x) ≈ x for small x (Taylor series). At ᾱ = 0.05, the difference is 2.7%. At ᾱ = 0.30, it is 19%. At ᾱ = 0.60, it is 66%.

The Error Table: Sabine vs. Eyring Across Absorption Levels

For a room with volume 400 m³ and surface area 400 m² (a medium conference room approximately 16m × 10m × 2.5m):

The 0.20 threshold is not arbitrary. Below it, Sabine gives an error under 12% — acceptable for early design estimates. Above it, the error grows rapidly and becomes design-consequential.

Worked Example: The Meeting Room That Would Fail WELL

Room Parameters

A corporate meeting room:

• Dimensions: 8m × 5m × 2.7m = 108 m³
• Total surface area: 2 × (8×5) + 2 × (8×2.7) + 2 × (5×2.7) = 80 + 43.2 + 27 = 150.2 m²

Mean absorption coefficient: ᾱ = 77.9 / 150.2 = 0.519

This is a very well-treated meeting room — ceiling tiles, carpet, and significant wall panel coverage. The mean α of 0.519 is squarely in the range where Sabine becomes highly inaccurate.

Sabine Prediction

T60_Sabine = 0.161 × 108 / 77.9 = 17.39 / 77.9 = 0.223 s

Eyring Prediction

-S ln(1 - ᾱ) = -150.2 × ln(1 - 0.519) = -150.2 × ln(0.481) = -150.2 × (-0.732) = 109.9

T60_Eyring = 0.161 × 108 / 109.9 = 17.39 / 109.9 = 0.158 s

The Sabine prediction is 0.223s. Eyring gives 0.158s. The difference is 0.065s — a 41% overestimate by Sabine.

Now consider the design implication. The WELL v2 Feature 74 recommended RT60 for a small meeting room (< 100 m² floor area) is 0.3–0.5s. A designer using Sabine to target 0.35s would dial back the treatment until Sabine predicted 0.35s. At that treatment level, Eyring would predict approximately 0.24s — the room is significantly over-treated. The result is a meeting room that sounds dead and uncomfortable, with limited voice projection and an unnaturally dry quality.

Alternatively, a designer targeting 0.6s WELL compliance would see Sabine predicting 0.6s and conclude the room is just barely compliant. Eyring would show the actual RT60 is approximately 0.41s — well below the target, meaning the room already has far more treatment than needed.

Full Octave-Band Comparison

Let us run both formulas across all octave bands for this room. The surface absorption coefficients at each frequency:

Total absorption A and mean ᾱ at each band:

Note: absorption coefficients listed as > 1.00 are a known ISO 354 reverberation room artefact for high-performance materials; they are clipped to 1.00 for Eyring calculations.

The mid-frequency Sabine predictions (0.20–0.22s) look acceptable — very short, but within bounds for a small treated room. The Eyring predictions (0.14–0.16s) reveal a room that has been treated to near-anechoic conditions at mid frequencies. In practice, this room would sound uncomfortably dead.

The 125 Hz octave band shows the smallest error (14%) because the lower absorption coefficient at this frequency keeps Sabine closer to its valid range. This is consistent with the general rule: bass treatment is sparse enough that Sabine is more accurate at 125 Hz than at 1000 Hz.

When to Use Each Formula

Decision Framework

If ᾱ < 0.15 (lightly treated, most educational spaces, auditoriums):
    → Use Sabine. Error < 10%.
If 0.15 ≤ ᾱ < 0.25 (moderately treated classrooms, lightly fitted offices):
    → Sabine acceptable for concept design. Use Eyring for detailed design.
    → Error 10–15%.
If ᾱ ≥ 0.25 (treated meeting rooms, offices, control rooms, studios):
    → Use Eyring. Sabine error 15–50%+.
    → ISO 3382-2:2008 §A.2 recommends Eyring for this range.
If non-uniform absorption (one surface much more absorptive than others):
    → Consider Millington-Sette formula or ray-tracing simulation.
    → Eyring underestimates T60 when absorption is concentrated on one surface.

The Millington-Sette Caveat

Even Eyring assumes uniform distribution of absorption. When one surface (typically the ceiling in office acoustics) carries the majority of absorption while walls and floor are reflective, neither Sabine nor Eyring is fully accurate. Millington (1932) and Sette (1933) independently proposed:

T60 = 0.161 V / (-Σ Sᵢ ln(1 - αᵢ))

Where the summation is applied surface-by-surface before summing, rather than to the mean absorption coefficient. This is more accurate when absorption distribution is highly non-uniform.

For the meeting room example: -Σ Sᵢ ln(1 - αᵢ) at 500 Hz = -(40 × ln(0.05)) + (40 × ln(0.65)) + (43.2 × ln(0.95)) + (27 × ln(0.20))

Wait — let us use (1-αᵢ): = -(40 × ln(1-0.95)) - (40 × ln(1-0.35)) - (43.2 × ln(1-0.05)) - (27 × ln(1-0.80)) = -(40 × ln(0.05)) - (40 × ln(0.65)) - (43.2 × ln(0.95)) - (27 × ln(0.20)) = -(40 × (-2.996)) - (40 × (-0.431)) - (43.2 × (-0.051)) - (27 × (-1.609)) = 119.8 + 17.2 + 2.2 + 43.4 = 182.6

T60_Millington = 0.161 × 108 / 182.6 = 0.095 s

The Millington-Sette result (0.095s) is even shorter than Eyring (0.158s). Note the ceiling tile absorption coefficient of 0.95 drives an enormous term because -ln(1-0.95) = -ln(0.05) = 3.0, which dominates the sum. This reflects a real physical phenomenon: surfaces with near-unit absorption devour energy so rapidly that the diffuse field collapses almost immediately.

For practical design work: use Eyring as the default correction. Apply Millington-Sette only when absorption coefficients exceed 0.70 on any single dominant surface.

The Practical Protocol for Acoustic Calculations

ISO 3382-2:2008 §7.3 recommends measuring RT60 at six octave bands: 125, 250, 500, 1000, 2000, and 4000 Hz. The calculation method should match the room condition:

For design predictions:

1. Calculate mean ᾱ at each frequency band
2. If ᾱ < 0.20 at all bands: use Sabine throughout
3. If ᾱ ≥ 0.20 at any band: use Eyring at all bands (consistency is important — mixing methods across bands creates errors)
4. If absorption is highly non-uniform (one surface αᵢ > 0.70, others < 0.20): flag for simulation

• WELL v2 Feature 74: measured on-site per ISO 3382-2, so calculation method is design-stage only
• DIN 18041:2016: design calculations accepted using either method, but measured value governs
• ANSI S12.60:2010: design stage uses ASTM E2235 measurements for acceptance
• BS 8233:2014: calculation using Sabine or Eyring accepted; measured results govern final compliance

The Key Number to Remember

The threshold is ᾱ = 0.20. Below it, Sabine gives an error under 12% — tolerable at design stage. Above it, the error grows rapidly. At ᾱ = 0.30, Sabine overestimates by 17%. At ᾱ = 0.50, by 33%. At ᾱ = 0.70, by 57%.

Modern treated rooms routinely exceed ᾱ = 0.30 at mid frequencies. The floor of any NRC calculation conversation should be: "what is the mean absorption coefficient at 500 Hz?" If the answer is above 0.20, Eyring is the correct tool.

The companion article why your RT60 calculation is probably wrong provides additional worked examples including the specific case where Sabine passes WELL F74 on paper and Eyring reveals the room is over-treated. The acoustic design process guide shows where formula selection sits in the full design workflow.

Summary

1. Sabine's formula is a first-order linearisation of the exact energy decay equation. It is accurate only when mean absorption ᾱ < 0.20.
2. Eyring (ISO 3382-2:2008 §A.2) uses the exact logarithmic term -S ln(1-ᾱ) and is accurate across the full absorption range.
3. At ᾱ = 0.30 — the condition of a typical treated meeting room — Sabine overestimates RT60 by 17%. At ᾱ = 0.50, the overestimate is 33%.
4. Most modern treated office spaces and meeting rooms have ᾱ ≥ 0.25 at mid frequencies, which puts them in the zone where Eyring is mandatory for reliable design predictions.
5. Millington-Sette is the more accurate method when any single surface has αᵢ > 0.70, but Eyring is adequate for most design work.
6. ISO 3382-2:2008 explicitly recommends Eyring for highly damped rooms. Any specification or calculation report that uses Sabine for rooms with NRC 0.85 ceiling tiles and carpet should be questioned.