Sabine Formula

Reverberation time

Reverberation is defined as persistence of audible sound after the source has been switched off.
The time for which sound persist is known as reverberation time.

Assumptions

Distribution of energy is uniform in all parts of the hall.
No interference of sound waves in the hall.
Absorption coefficient does not depend on the intensity of sound.
There is no loss of energy in air.

Derivation

Average energy density within the hall in all direction = u
Energy received by elementary volume dV = u dV

Consider an element of surface area ds of a plane wall.
Elementary volume at a distance r from the centre of ds in the direction θ from normal on ds = dV
Solid angle subtended by surface element at elementary volume, dω = ds cos θ / r²
Area of shaded elementary portion = dr × r dθ
If the whole figure is rotated about the normal by an angle 2π then the elementary volume trace by this portion

dV = elementary area × circumferential path

= dr . rdθ × 2πr sin θ = 2πr² sin θ dr dθ

Sound energy in volume dV

u dV = u × 2πr² sin θ dr dθ

The fraction of this energy incident on area ds

$\frac{d\omega }{4\pi}\times (udV)=\frac{d\omega }{4\pi}\times u\times 2\pi r^{2}sin\theta drd\theta$
$\because \;\;\;\;\; d\omega=\frac{dscos\theta }{r^{2}}$

$\therefore \;\;\;\;\; \frac{d\omega }{4\pi}\times (udV)=\frac{dscos\theta }{4\pi r^{2}}\times u\times 2\pi r^{2}sin\theta drd\theta$

$or \;\;\;\;\; \frac{d\omega }{4\pi}\times (udV)=\frac{udssin2\theta }{4}drd\theta$

Energy received by ds due to whole shell lying between two hemispheres for all values of θ

$\frac{udsdr}{4}\int_{0}^{\pi /2}sin2\theta d\theta =\frac{udsdr}{4}$

Energy received per second by ds

$\frac{uds}{4}\int_{0}^{v}dr =\frac{uvds}{4}$

Here v = speed of sound in air
If a = absorption coefficient of ds, then
Energy absorbed per second by ds

$=a\times \frac{uvds}{4}$

Total energy absorbed per second by hall

$\sum a\times \frac{uvds}{4}= \frac{uv}{4}\sum a\times ds$

Total absorption of all the surface within the enclosure A = Σ a ds
Total energy absorbed per second by hall = uvA/4
Since V is the volume of enclosure, therefore
Total sound energy in the room at any time = uV
Rate of increase of this energy

$E_{1}=\frac{d}{dt}(uV)=V\frac{du}{dt}$

Rate of absorption of energy

$E_{2}=\frac{1}{4}(uvA)$

Rate of emission of energy from the source, E = E₁ + E₂

$or \;\;\;\;\; E=V\frac{du}{dt}+\frac{1}{4}(uvA)$

$or \;\;\;\;\; \frac{E}{V}=\frac{du}{dt}+\frac{1}{4}\frac{uvA}{V}$

Let vA / 4V = β

$\therefore \;\;\;\;\; \frac{du}{dt}=\frac{E}{V}-u\beta$

$or \;\;\;\;\; \frac{du}{(E/V-u\beta )}=dt$

$or \;\;\;\;\; -\frac{log(E/V-u\beta )}{\beta}=t+C$

(By integration)

Growth of sound

When source starts of sounding, t = 0, u = 0

$so \;\;\;\;\; C=-\frac{log(E/V)}{\beta }$

$But \;\;\;\;\; -\frac{log(E/V-u\beta )}{\beta}=t+C$

$\therefore \;\;\;\;\; t=\frac{log(E/V )}{\beta}-\frac{log(E/V-u\beta )}{\beta}$

$\therefore \;\;\;\;\; log(E/V-u\beta)-log(E/V )=-\beta t$

$or\;\;\;\;\; 1-\frac{u\beta }{E/V}=exp(-\beta t)$

$or\;\;\;\;\; \frac{u\beta }{E/V}=1-exp(-\beta t)$

$or\;\;\;\;\; u=\frac{E }{\beta V}[1-exp(-\beta t)]$

$or\;\;\;\;\; u=\frac{4E }{vA}[1-exp(-\beta t)]\;\;\;\;(\because \beta =vA/4V)$

When t = ∞, u = u_max ⇒ u_max = 4E / vA
u = u_max [1 – exp (-βt)]

Decay of sound

When source is cut off, t = 0, u = u_max, E = 0

$so\;\;\;\;\;C=-\frac{log(-u_{max}\beta )}{\beta }$

$But \;\;\;\;\; -\frac{log(E/V-u\beta )}{\beta}=t+C$

$\therefore \;\;\;\;\; t=\frac{log(-u_{max}\beta )}{\beta}-\frac{log(-u\beta )}{\beta}$

$\therefore \;\;\;\;\; log(-u\beta)-log(-u_{max}\beta )=-\beta t$

$or\;\;\;\;\;log(u/u_{max})=-\beta t$

$or\;\;\;\;\; u/u_{max}=exp (-\beta t)$

$or\;\;\;\;\; u=u_{max}exp (-\beta t)$

Reverberation period (T)

u = u_max [1 – exp (-βt)]
If t = T, u = 10^-6 u_max

then 10⁻⁶ u_max = u_max exp (−βT)

or 10⁻⁶ = exp (−βT)

or −βT = − 6 log_e 10

or T = [6 log_e 10] / β = [6 × 2.303] / β

or T = [6 × 2.303] / (vA / 4V) [ ∵ vA / 4V = β]

or T = [6 × 2.303 × 4V] / vA

$\because \;\;\;\;\; d\omega=\frac{dscos\theta }{r^{2}}$

$\therefore \;\;\;\;\; \frac{d\omega }{4\pi}\times (udV)=\frac{dscos\theta }{4\pi r^{2}}\times u\times 2\pi r^{2}sin\theta drd\theta$

$or \;\;\;\;\; \frac{d\omega }{4\pi}\times (udV)=\frac{udssin2\theta }{4}drd\theta$

But v = 340 m/s

$https://latex.codecogs.com/svg.image?T=\frac{0.162V}{A} second$

Above formula is known as Sabine formula.

To know about Sabine formula in detail please click here.

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