Deriving Sabine's and Eyring's Reverberation Time Formulas from First Principles

Reverberation time is the most fundamental parameter in room acoustics. Two formulas dominate its prediction: Sabine's equation (1898) and Eyring's equation (1930). Most practitioners use these formulas as black boxes — plug in volume, surface area, and absorption coefficients, get a number out. This article derives both formulas from first principles so you understand not just what they calculate, but why they work, where they fail, and how they differ.

The mathematics requires nothing beyond algebra, logarithms, and basic calculus. If you can follow an exponential decay, you can follow these derivations.

Prerequisites: The Diffuse Field

Both Sabine's and Eyring's formulas assume a diffuse sound field. This is the single most important assumption in statistical room acoustics, and understanding it is essential before proceeding to the derivations.

A diffuse sound field has two properties:

Property 1: Uniform energy density. The sound energy per unit volume is the same everywhere in the room. There are no "hot spots" or "dead zones." This requires that sound has reflected enough times from room surfaces to distribute energy uniformly throughout the volume.

Property 2: Random incidence. At any point in the room, sound arrives from all directions with equal probability. There is no preferred direction of propagation. This requires that the room surfaces scatter sound efficiently and that no single reflection path dominates.

In practice, neither property is perfectly satisfied. But in rooms that are roughly rectangular with dimensions within a factor of 3 of each other, that have reflective surfaces on most boundaries, and where the sound source has been operating long enough for the field to build up, the diffuse field assumption is a reasonable approximation.

The diffuse field assumption breaks down in:

• Very elongated rooms (corridors, tunnels)
• Rooms with one dominant absorptive surface (e.g., a room where the ceiling absorbs 90% of sound but all other surfaces are reflective)
• Coupled spaces (rooms connected by large openings)
• Very large rooms where the mean free path approaches the room dimensions

Mean Free Path: The Average Distance Between Reflections

In a diffuse field, sound waves travel in straight lines between reflections. The average distance between two successive reflections is called the mean free path, denoted by l (lowercase L).

For a convex room of volume V and total surface area S, the mean free path has been shown (by Kosten, 1960, building on earlier work by Kuttruff) to be:

l = 4V / S

This result is elegant and general — it depends only on the volume and surface area, not on the room shape. For a rectangular room with dimensions L x W x H:

• V = L x W x H
• S = 2(LW + WH + LH)

• V = 144 m3
• S = 2(48 + 18 + 24) = 180 m2
• l = 4 x 144 / 180 = 3.2 m

The time between successive reflections is:

delta_t = l / c = 4V / (cS)

Where c is the speed of sound in air (approximately 343 m/s at 20 degrees Celsius). For the example room:

delta_t = 3.2 / 343 = 0.00933 seconds = 9.33 milliseconds

Sound reflects approximately 107 times per second in this room. Each reflection removes some energy through absorption, and the cumulative effect of these absorptive reflections is what causes reverberation decay.

Deriving Sabine's Formula

Step 1: Energy After One Reflection

Consider a sound wave carrying energy E that strikes a room surface. The surface has an average absorption coefficient alpha_bar. On each reflection, a fraction alpha_bar of the energy is absorbed, and the remaining fraction (1 - alpha_bar) is reflected.

After one reflection, the remaining energy is:

E_1 = E_0 (1 - alpha_bar)

Step 2: Energy After n Reflections

After n reflections, the remaining energy is:

E_n = E_0 (1 - alpha_bar)^n

This is a geometric decay. Each reflection multiplies the energy by the same factor (1 - alpha_bar).

Step 3: Number of Reflections in Time t

The number of reflections that occur in time t is:

n = t / delta_t = t c S / (4V)

Step 4: Energy as a Function of Time

Substituting the expression for n into the energy decay:

E(t) = E_0 (1 - alpha_bar)^(tcS/4V)

This can be rewritten using the identity a^b = e^(b ln a):

E(t) = E_0 exp[ (tcS / 4V) ln(1 - alpha_bar) ]

Since alpha_bar is between 0 and 1, ln(1 - alpha_bar) is negative, and the energy decays exponentially with time. This exponential decay is the physical basis of reverberation.

Step 5: Sabine's Approximation

Here is where Sabine and Eyring diverge. Sabine made a simplifying approximation. For small values of alpha_bar (much less than 1), the natural logarithm can be approximated by its first-order Taylor expansion:

ln(1 - alpha_bar) approximately equals -alpha_bar (when alpha_bar is small)

Substituting this approximation:

E(t) approximately equals E_0 exp[ -tcS alpha_bar / (4V) ]

Since the total absorption A = S x alpha_bar (in square metres Sabine), this becomes:

E(t) approximately equals E_0 exp[ -tcA / (4V) ]

Step 6: Finding T60

Reverberation time T60 is defined as the time for the energy to decay by a factor of 10^6 (which corresponds to a 60 dB decay in sound pressure level, since SPL is proportional to 10 log10 of energy):

E(T60) / E_0 = 10^(-6)

Setting E(T60) / E_0 = 10^(-6):

exp[ -T60 cA / (4V) ] = 10^(-6)

Taking the natural logarithm of both sides:

-T60 cA / (4V) = ln(10^(-6)) = -6 ln(10)

Solving for T60:

T60 = 24V ln(10) / (cA)

Now substitute the numerical values: ln(10) = 2.3026, c = 343 m/s:

T60 = 24 x 2.3026 x V / (343 x A) = 55.26 V / (343 A) = 0.161 V / A

This is Sabine's formula:

T60 = 0.161 V / A (metric, V in m3, A in m2 Sabine)

For imperial units (V in cubic feet, A in ft2 Sabine), the constant changes because c = 1125 ft/s:

T60 = 24 x 2.3026 / 1125 x V / A = 0.049 V / A (imperial)

What the Constant 0.161 Contains

The constant 0.161 is not arbitrary. It encodes three physical quantities:

• The factor 24, which comes from the mean free path formula (4V/S) and the 60 dB decay definition (6 decades of energy)
• The natural logarithm of 10 (2.3026), which converts the 60 dB definition from base-10 to the natural exponential
• The speed of sound in air at 20 degrees Celsius (343 m/s)

Why Sabine Fails at High Absorption

The critical step in Sabine's derivation is the approximation ln(1 - alpha_bar) approximately equals -alpha_bar. Let us examine the error in this approximation:

The error grows rapidly with increasing absorption. At alpha_bar = 0.20 (a room with acoustic ceiling tiles and carpet), the error is already 10%, meaning Sabine's formula overestimates RT60 by 10%. At alpha_bar = 0.50 (a heavily treated room), the error reaches 28%.

The physical absurdity becomes clear at alpha_bar = 1.00. If every surface in the room is a perfect absorber (alpha = 1.00), there should be no reflections and therefore no reverberation — T60 should be zero (or more precisely, the time it takes sound to travel from the source to the nearest surface). But Sabine's formula gives:

T60 = 0.161 V / (S x 1.00) = 0.161 V / S

For the example room (V = 144 m3, S = 180 m2): T60 = 0.161 x 144 / 180 = 0.129 seconds. This is physically impossible — a room with perfectly absorptive surfaces cannot sustain reverberation for 129 milliseconds.

Deriving Eyring's Formula

Carl F. Eyring's contribution (1930) was to avoid Sabine's linearisation approximation and retain the exact logarithmic term.

Starting from the Exact Energy Decay

Returning to Step 4 of the derivation, where the energy decay is:

E(t) = E_0 exp[ (tcS / 4V) ln(1 - alpha_bar) ]

This is exact (within the diffuse field assumption). Setting E(T60) / E_0 = 10^(-6):

exp[ (T60 cS / 4V) ln(1 - alpha_bar) ] = 10^(-6)

Taking natural logarithms:

(T60 cS / 4V) ln(1 - alpha_bar) = -6 ln(10)

Solving for T60:

T60 = -24 ln(10) V / [ cS ln(1 - alpha_bar) ]

Substituting numerical values (ln(10) = 2.3026, c = 343 m/s):

T60 = 0.161 V / [ -S ln(1 - alpha_bar) ]

This is Eyring's formula:

T60 = 0.161 V / [ -S ln(1 - alpha_bar) ] (metric)

Comparing the Denominators

The only difference between Sabine and Eyring is the denominator:

• Sabine: A = S alpha_bar (linear absorption area)
• Eyring: -S ln(1 - alpha_bar) (logarithmic correction)

The alpha = 1 Test

With Eyring's formula, at alpha_bar = 1.00:

T60 = 0.161 V / [ -S ln(1 - 1.00) ] = 0.161 V / [ -S ln(0) ] = 0.161 V / [ -S x (-infinity) ] = 0.161 V / infinity = 0

Eyring's formula correctly predicts T60 = 0 for a room with perfectly absorptive surfaces. The logarithmic singularity at alpha = 1 is physically meaningful: it takes infinitely strong attenuation (per the logarithmic term) to produce zero reverberation, which corresponds to the physical reality that every reflection is completely absorbed.

Numerical Comparison: Sabine vs Eyring

Let us compute RT60 using both formulas for the example room (V = 144 m3, S = 180 m2) across a range of average absorption coefficients:

The pattern is clear:

• Below alpha_bar = 0.10, both formulas agree within 5%. Either is acceptable.
• Between alpha_bar = 0.10 and 0.20, Sabine overestimates by 5-12%. The error is noticeable but may be acceptable for preliminary design.
• Above alpha_bar = 0.20, Sabine overestimates by 12% or more. Eyring should be used.
• Above alpha_bar = 0.40, Sabine's error exceeds 25%. Using Sabine in this range is professionally indefensible.

What This Means in Practice

A modern meeting room might have:

• Acoustic ceiling tiles: alpha = 0.75, area = 40 m2
• Carpet: alpha = 0.25, area = 40 m2
• Plaster walls: alpha = 0.05, area = 100 m2
• Glass windows: alpha = 0.10, area = 20 m2
• Weighted average: alpha_bar = (40 x 0.75 + 40 x 0.25 + 100 x 0.05 + 20 x 0.10) / 200 = (30 + 10 + 5 + 2) / 200 = 0.235

The Millington-Sette Extension

Both Sabine and Eyring use the mean absorption coefficient alpha_bar = A / S, which averages the absorption coefficients across all surfaces. This averaging creates errors when surfaces have very different absorption coefficients — for example, a room with an acoustic ceiling (alpha = 0.85) and bare concrete walls (alpha = 0.03).

The Millington-Sette formula (also called the Norris-Eyring or Millington formula) addresses this by applying the logarithmic correction to each surface individually:

T60 = 0.161 V / [ -sum( S_i ln(1 - alpha_i) ) ]

Where S_i is the area of the i-th surface and alpha_i is its absorption coefficient.

Why Millington-Sette Matters

The difference between Eyring and Millington-Sette becomes significant when the room has both highly absorptive and highly reflective surfaces. Consider the meeting room above:

Eyring calculation (using alpha_bar = 0.235):

• Denominator = -200 x ln(1 - 0.235) = -200 x (-0.268) = 53.6 m2
• T60 = 0.161 x 144 / 53.6 = 0.433 seconds

• Ceiling: -40 x ln(1 - 0.75) = -40 x (-1.386) = 55.45
• Floor: -40 x ln(1 - 0.25) = -40 x (-0.288) = 11.51
• Walls: -100 x ln(1 - 0.05) = -100 x (-0.0513) = 5.13
• Windows: -20 x ln(1 - 0.10) = -20 x (-0.1054) = 2.11
• Total denominator = 55.45 + 11.51 + 5.13 + 2.11 = 74.20 m2
• T60 = 0.161 x 144 / 74.20 = 0.312 seconds

When to Use Each Formula

In practice, Millington-Sette gives the lowest T60 prediction and can underestimate reverberation time in rooms where the diffuse field assumption is not well satisfied — which is precisely the scenario where surfaces have very different absorption. The reality usually lies between Eyring and Millington-Sette.

Air Absorption

Both formulas as derived above assume that sound energy is lost only at surface reflections. In practice, sound is also absorbed by the air itself as it propagates between reflections. Air absorption is caused by viscous losses and molecular relaxation processes in oxygen and nitrogen, and it is significant at high frequencies (above 2000 Hz) and in large rooms.

The air absorption coefficient m (in Nepers per metre) depends on temperature, humidity, and frequency. ISO 9613-1 provides the definitive calculation method. Typical values at 20 degrees Celsius and 50% relative humidity:

Air absorption is incorporated into Sabine's formula by adding an air absorption term to the total absorption:

T60 = 0.161 V / (A + 4mV)

For Eyring, the correction modifies the exponential decay rate:

T60 = 0.161 V / [ -S ln(1 - alpha_bar) + 4mV ]

When Air Absorption Matters

In the example room (V = 144 m3), the air absorption contribution at 4000 Hz is 4mV = 0.0484 x 144 = 6.97 m2. If the surface absorption A at 4000 Hz is 60 m2, the air absorption adds approximately 12% to the total absorption. This is significant enough to affect the predicted T60 at 4000 Hz and should be included.

For frequencies below 1000 Hz in rooms smaller than 500 m3, air absorption is negligible (less than 2% of total absorption) and can be safely ignored.

For large rooms — concert halls (V = 10,000-25,000 m3), sports arenas, and places of worship — air absorption dominates the high-frequency decay. A concert hall with V = 20,000 m3 has an air absorption contribution of 4mV = 0.0484 x 20,000 = 968 m2 at 4000 Hz, which can exceed the surface absorption entirely. This is why large rooms tend to sound "warm" or "dark" — the air itself is a high-frequency absorber.

ISO 3382-2 Guidance

ISO 3382-2:2008, Annex A provides guidance on formula selection for predicting reverberation time in ordinary rooms:

Section A.1 (Sabine): States that Sabine's formula "is applicable when the mean absorption coefficient is less than about 0.2" and that it "tends to overestimate the reverberation time when the mean absorption coefficient is large."

Section A.2 (Eyring): States that Eyring's formula "gives more accurate results than Sabine's formula when the mean absorption coefficient is large" and notes that it correctly predicts T60 = 0 when alpha_bar = 1.

The standard does not explicitly endorse Millington-Sette but acknowledges that "when the absorption coefficients of the various surfaces differ widely, the use of an area-weighted mean absorption coefficient may introduce errors."

Practical Recommendations

Based on ISO 3382-2, the academic literature, and practical experience:

1. Use Sabine only when alpha_bar is below 0.15 and you need a quick estimate. It is acceptable for untreated rooms with hard surfaces (plaster, concrete, glass).

1. Use Eyring as the default for all treated rooms and any room where alpha_bar exceeds 0.15. This covers most modern buildings with acoustic ceiling tiles, carpet, or any absorptive treatment.

1. Use Millington-Sette when surfaces have very different absorption coefficients (spread of more than 0.5 between the lowest and highest alpha values) and you want a lower-bound estimate of T60.

1. Include air absorption at 2000 Hz and above in all rooms, and at all frequencies in rooms larger than 500 m3.

1. Report both Sabine and Eyring in design reports so the client or reviewer can see the range of predictions. If the two values bracket the target (Eyring below, Sabine above), the design is likely to comply but should be verified by measurement after construction.

Summary of Formulas

For reference, the three formulas in their complete form with air absorption:

Sabine: T60 = 0.161 V / (A + 4mV) where A = sum(alpha_i x S_i)

Eyring: T60 = 0.161 V / [ -S ln(1 - alpha_bar) + 4mV ] where alpha_bar = A / S

Millington-Sette: T60 = 0.161 V / [ -sum( S_i ln(1 - alpha_i) ) + 4mV ]

All three share the same physical foundation: energy decays exponentially as sound reflects from partially absorptive surfaces in a diffuse field. The only difference is how they approximate the energy loss per reflection. Sabine linearises the logarithm, Eyring retains it with an averaged argument, and Millington-Sette retains it with surface-specific arguments. Each successive formula is more accurate but requires more detailed input data.

Understanding these derivations transforms the formulas from opaque black boxes into tools with clear physical meaning, known limitations, and predictable failure modes. When a formula gives an answer that seems wrong, the derivation tells you exactly where the assumptions broke down and which formula to use instead.