Sabine vs Eyring vs Fitzroy: When to Use Which RT60 Equation

TL;DR

Three equations. Three different answers. Same room. If you have ever plugged numbers into the Sabine equation and wondered why measured reverberation times disagreed, the problem is almost certainly equation selection, not measurement error. Wallace Clement Sabine published his formula in 1898 for the Fogg Art Museum at Harvard — a room with hard plaster walls and modest absorption. Carl Eyring extended the theory in 1930 to handle rooms with higher absorption. David Fitzroy went further in 1959, treating each axis pair independently for rooms with wildly non-uniform surface treatment. Each equation has a domain of validity. Use the wrong one and your RT60 prediction can be off by 40% or more. This article walks through all three equations with a real field story, worked calculations, and a decision table so you never pick the wrong formula again.

The Recording Studio That Broke the Sabine Equation

In 2024, a small post-production facility in Melbourne commissioned acoustic treatment for a 48 m³ voiceover booth. The ceiling was treated with 100 mm thick polyester absorbers covering 95% of the 16 m² ceiling area. Walls were a mix of plywood diffusers and 50 mm fabric panels. The floor was carpeted.

Using the Sabine equation, the consultant predicted an RT60 of 0.18 seconds at 1 kHz. The measured value after installation was 0.31 seconds — 72% higher than predicted. The client was understandably unhappy. The acoustic treatment had cost AUD 28,000, and the room was noticeably more reverberant than promised.

Had the consultant used the Eyring equation, the prediction would have been 0.28 seconds. Still not perfect, but within the ±10% tolerance that ISO 3382-2:2008 considers acceptable for engineering-grade measurements. The Sabine equation failed because it was never designed for rooms where most of the sound energy is absorbed on each reflection.

The Sabine Equation: Where It Works and Where It Breaks

Wallace Clement Sabine derived his formula empirically by measuring organ-pipe decay times in rooms at Harvard. The relationship he established is elegantly simple:

RT60 = 0.161 × V / A

Where V is room volume in cubic metres and A is total absorption area in metric sabins (Σ αᵢ × Sᵢ). Per ISO 3382-2:2008 §A.1, this is the standard reverberation time formula for ordinary rooms.

The Sabine equation assumes:

• Sound energy decays continuously and uniformly
• The sound field is perfectly diffuse (equal energy density everywhere)
• Average absorption coefficient ᾱ is small (below approximately 0.3)

When Sabine Fails

The equation produces physically impossible results when ᾱ approaches 1.0. If every surface were a perfect absorber (α = 1.0 for all surfaces), Sabine still predicts a finite, positive RT60. This is obviously wrong — a room with perfect absorption on every surface would have zero reverberation. The mathematical reason is that Sabine's derivation assumes sound undergoes many reflections before decaying. When surfaces absorb most energy on each reflection, this assumption collapses.

The Eyring Equation: The Correction for High Absorption

Carl F. Eyring published his corrected formula in 1930. Instead of using the absorption coefficient directly, Eyring uses the natural logarithm of (1 - ᾱ), which correctly predicts zero reverberation when ᾱ = 1.0:

RT60 = 0.161 × V / [-S × ln(1 - ᾱ)]

Where S is the total surface area and ᾱ is the area-weighted average absorption coefficient. Per ISO 3382-2:2008 §A.2, this is the recommended formula when absorption is non-negligible.

Worked Example: The Melbourne Voiceover Booth

Room dimensions: 4.0 m × 4.0 m × 3.0 m (V = 48 m³)

Total surface area S = 68.0 m², Total absorption A = 36.78 m²

Average ᾱ = 36.78 / 68.0 = 0.541

Sabine: RT60 = 0.161 × 48 / 36.78 = 0.21 s

Eyring: RT60 = 0.161 × 48 / [-68.0 × ln(1 - 0.541)] = 0.161 × 48 / [-68.0 × (-0.778)] = 0.161 × 48 / 52.90 = 0.146 s

Wait — the Eyring prediction is lower than Sabine, yet the measured value was higher than both? This reveals a subtlety that catches many practitioners: the Eyring equation uses the average absorption coefficient, which assumes absorption is evenly distributed. In this room, it is not. The ceiling is 95% absorptive while the plywood diffusers are 15%. This non-uniformity is precisely the condition that calls for the Fitzroy equation — or at minimum, a frequency-by-frequency analysis rather than a single broadband average.

Try modelling this room yourself with the AcousPlan RT60 calculator →

The Fitzroy Equation: Non-Uniform Absorption

David Fitzroy proposed his equation in 1959 to handle rooms where absorption varies dramatically between axis pairs. Instead of averaging absorption across all surfaces, Fitzroy calculates an effective RT60 by weighting three orthogonal pairs:

1/RT60_Fitzroy = (Sx/S) × [-S × ln(1 - ᾱx)] / (0.161V) + (Sy/S) × [-S × ln(1 - ᾱy)] / (0.161V) + (Sz/S) × [-S × ln(1 - ᾱz)] / (0.161V)

Where:

• Sx, Sy, Sz are the areas of each axis pair (floor+ceiling, front+back walls, left+right walls)
• ᾱx, ᾱy, ᾱz are the average absorption coefficients for each pair
• S is total surface area

When Fitzroy Excels

The Decision Framework

After 15 years of comparing predictions to measurements across hundreds of rooms, the following decision framework has proven reliable:

Step 1: Calculate the area-weighted average absorption coefficient ᾱ.

• If ᾱ < 0.25 → Use Sabine. Stop here.

• If max(ᾱ_pair) / min(ᾱ_pair) < 2.0 → Use Eyring. Stop here.

• Use Fitzroy as a check against Eyring.
• If the two disagree by more than 20%, the room geometry is likely causing non-diffuse conditions and a ray-tracing simulation (ODEON, CATT-Acoustic, or similar) is warranted.

Quick Reference Table

Air Absorption: The Factor Everyone Forgets

All three equations can be extended to include air absorption, which becomes significant at high frequencies (4 kHz and above) and in large rooms (V > 500 m³). Per ISO 3382-2:2008, the corrected Sabine equation becomes:

RT60 = 0.161 × V / (A + 4mV)

Where m is the air absorption coefficient in Nepers per metre, dependent on temperature and relative humidity. At 20°C and 50% RH, m ≈ 0.012 at 4 kHz. For the Melbourne booth (48 m³), the air absorption contribution is negligible (4 × 0.012 × 48 = 2.3 m², compared to 36.8 m² of surface absorption). In a 5,000 m³ concert hall, it adds 240 m² of effective absorption at 4 kHz — enough to reduce high-frequency RT60 by 0.3-0.5 seconds.

Common Mistakes in Practice

Mistake 1: Using manufacturer NRC values in Sabine. NRC is a single-number rating averaged over four frequencies (250, 500, 1000, 2000 Hz). It tells you nothing about low-frequency performance. Always use octave-band absorption coefficients and calculate RT60 at each frequency independently.

Mistake 2: Ignoring furniture and people. A classroom with 30 students has approximately 15 m² of additional absorption at mid-frequencies. Sabine's original work at Harvard explicitly included audience absorption. Design for the occupied condition.

Mistake 3: Applying Sabine to coupled volumes. Open-plan offices, atriums with balconies, and rooms connected by large openings do not behave as single volumes. The Sabine equation assumes a single enclosed space. Coupled volumes require either separate calculations per zone or computational modelling.

Summary

The Sabine equation is not wrong — it is merely limited. For the vast majority of commercial interior projects where absorption is moderate and reasonably uniform, Sabine gives predictions within ±15% of measured values. When absorption is high, switch to Eyring. When absorption is also non-uniform, check against Fitzroy. And when none of the analytical equations give confidence, use computational simulation.

The Melbourne voiceover booth story illustrates the cost of using the wrong equation: AUD 28,000 in treatment that underperformed expectations. Five minutes of equation selection could have prevented the problem entirely.

Calculate RT60 with Sabine and Eyring side by side → AcousPlan RT60 Calculator