Sabine's formula and good acoustics

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:

dE/E = −(cA/4V) dt   →   E = E0 e^(−cAt/4V)

5. Reverberation time: E/E0 = 10⁻⁶ at t = T:

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:

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.