---
title: "Sabine's formula and good acoustics"
subject: "Engineering Physics"
chapter: 3
key: "semester_1/engineering_physics:sabine-formula-acoustic-hall"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/sabine-formula-acoustic-hall
site: BCT Engineering
---

# Sabine's formula and good acoustics

Engineering Physics · Chapter 3: Acoustics

- Asked in: 2072 Kartik, 5 marks; 2071 Shrawan, 5 marks; 2068 Shrawan, 5 marks; 2067 Asar, 5 marks; 2067 Asoj, 5 marks (asked 5 times)
- Video: [Sabine formula | Acoustics | Reverberation Time | Physics | Btech | BSc | UPSC | NET | CTET | NCERT](https://www.youtube.com/watch?v=PNes04XOpLQ) — Gautam Varde
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/sabine-formula-acoustic-hall
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-3

## Question

Write the features of an acoustically good hall/auditorium (and why reverberation time must be studied before building one). Derive Sabine's formula, the relation for reverberation time in terms of absorption coefficient, volume and surface area of the hall.

## Short answer

A good hall has optimum reverberation time, uniform loudness, no echoes, and low noise; studying reverberation time balances absorption. Sabine's formula T = 0.161 V/Σ aS (seconds) follows from dE/dt =−EcA/4V.

## Answer

#### Features of an acoustically good hall
1. **Optimum reverberation time**: neither too long (speech becomes blurred as echoes overlap) nor too short (sound dull and weak). About 1 to 1.5 s for speech halls, 1.5 to 2.5 s for music halls.
2. **Adequate loudness** at every seat; sound reflected from walls and ceiling adds to the direct sound.
3. **Uniform distribution** of sound, with no dead spots or focal points; avoid concave surfaces or use diffusers.
4. **No echoes** or sound shadows; delayed reflections (more than about 0.1 s) are avoided.
5. **Good diffusion** by irregular wall surfaces, and no resonance of walls or objects.
6. **Low background noise**: sound insulation from outside, quiet ventilation.

**Why reverberation time must be studied first:** it decides the balance of absorbing and reflecting surfaces needed. If the hall is built without calculating it, the speech will be unclear (too long a time) or lifeless (too short), and correcting it afterwards is costly.

#### Sabine's formula
**Reverberation time T** is the time taken for the sound intensity to fall to 10⁻⁶ of its initial value (a fall of 60 dB) after the source stops.

Let V = volume of the hall, S1, S2, … surfaces with absorption coefficients a1, a2, …, and total absorption A = Σ aS. Let E be the sound energy density and c the speed of sound.

1. Energy falling on unit area of a wall per second = Ec/4.
2. Energy absorbed per second by all surfaces = (Ec/4) Σ aS = EcA/4.
3. Rate of change of total energy in the hall: V dE/dt = −EcA/4.
4. Integrating:
```text
dE/E = −(cA/4V) dt   →   E = E0 e^(−cAt/4V)
```
5. Reverberation time: E/E0 = 10⁻⁶ at t = T:
```text
10⁻⁶ = e^(−cAT/4V)
T = (4V ln 10⁶) / (cA) = 4 × 13.82 V / (c A)
```
6. With c = 343 m/s (air at about 20°C), 4 × 13.82/343 = 0.161:
```text
T = 0.161 V / Σ aS = 0.161 V / (ā S)
```
where S is the total surface area and ā = ΣaS/S is the average absorption coefficient. This is Sabine's formula (V in m³, S in m², T in seconds). Larger V increases T; more absorbing surface reduces T.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Reverberation time is defined as the time for sound intensity to
   A. fall to half its value
   B. fall to 10⁻⁶ of its initial value after the source stops
   C. reach its maximum
   D. fall to zero
   Answer: B. T is the time for a 60 dB fall, i.e. intensity down to 10⁻⁶ of the initial value. (Key point: Definition of reverberation time)

2. Equating energy absorbed per second to loss of energy, V dE/dt = -EcA/4, gives
   A. E = E0 e^(cAt/4V)
   B. E = E0 e^(-cAt/4V)
   C. E = E0 e^(-4Vt/cA)
   D. E = E0 - cAt/4V
   Answer: B. Integrating dE/E = -(cA/4V)dt gives an exponential decay. (Key point: Energy decay)

3. Sabine's formula is
   A. T = 0.161 ΣaS / V
   B. T = 0.161 V / ΣaS
   C. T = 0.161 V ΣaS
   D. T = V / (0.161 ΣaS)
   Answer: B. T = 4V ln(10⁶)/(cA) = 0.161 V/ΣaS with c = 343 m/s. (Key point: Sabine's formula)

4. Why must reverberation time be studied before building a hall?
   A. To fix the colour of the walls
   B. To choose the right balance of absorbing and reflecting surfaces, since errors are costly to fix later
   C. To reduce the volume of the hall
   D. To make sure there is no sound at all
   Answer: B. Too long a time blurs speech and too short makes sound dull, so the right absorption must be planned in advance. (Key point: Why study reverberation time)

## Related in Engineering Physics

- [Average absorption coefficient of a hall](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Areverberation-absorption-coefficient-hall) — Engineering Physics, chapter 3
- [Good acoustic building, reverberation time](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aacoustic-measures-reverberation-decrease) — Engineering Physics, chapter 3
- [Reverberation time of small hall](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ahall-average-absorption-reverberation-bicc) — Engineering Physics, chapter 3
- [Number of persons from reverberation time](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Areverberation-persons-present-hall) — Engineering Physics, chapter 3, asked 2×
- [Hall effect, coefficient and mobility](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ahall-effect-coefficient-mobility-conductivity) — Engineering Physics, chapter 8
- [Hall voltage, coefficient and mobility](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Ahall-numerical-copper-strip-thin) — Engineering Physics, chapter 8

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