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Clearly the length of time it takes for sound to die is a function not only of the absorption of the surfaces in a room but also a function of the length of time between interactions with the surfaces of the room. We can use these facts to derive an equation for the reverberation time in a room. The first thing to determine is the average length of time a sound wave will travel between interactions with the surfaces of the room. This can be found from the mean free path of the room, which is a measure of the average distances between surfaces, assuming all possible angles of incidence and position. For an approximately rectangular box, the mean free path is given by the following equation:

M F P = \frac{4 V}{S}

$M F P = \frac{4 V}{S}$

(A4.1)

where MFP = the mean free path (in m)
V = the volume (in m³)
and S = the surface area (in m²)

The time between surface interactions may be simply calculated from A4.1 by dividing it by the speed of sound to give:

τ = \frac{4 V}{S c}

$τ = \frac{4 V}{S c}$

(A4.2)

where τ = the time between reflections (in s)
and c = the speed of sound (in ms⁻¹, or meters per second)

Equation A4.2 gives us the time between surface interactions, and at each of these interactions α is the proportion of the energy absorbed, where α is the average absorption coefficient discussed earlier. If α of the energy is absorbed at the surface, then (1 − α) is the proportion of the energy reflected back to interact with further surfaces. At each surface a further proportion, α, of energy will be removed so the proportion the original sound energy that is reflected back will go as follows:

Energy_{After one reflection} = Energy_{Before reflection} (1 − α)

Energy_{After two reflections} = Energy_{Before reflections} (1 − α)²

Energy_{After three reflections} = Energy_{Before reflections} (1 − α)³

⋮

Energy_{After n reflections} = Energy_{Before reflections} (1 − α)^ⁿ

(A4.3)

As α is less than 1, (1 − α) will be also. Thus Equation A4.3 shows that the sound energy decays away in an exponential manner. We are interested in the time it takes the sound to decay by a fixed proportion and so need to calculate the number of reflections that have occurred in a given time interval. This is easily calculated by dividing the time interval by the mean time between reflections, calculated using Equation A4.2, to give:

n = \frac{t}{(\frac{4 V}{S c})} = t (\frac{S c}{4 V})

$n = \frac{t}{(\frac{4 V}{S c})} = t (\frac{S c}{4 V})$

(A4.4)

where t = the time interval (in s)

By substituting Equation A4.4 into Equation A4.3, we can get an expression for the remaining energy in the sound after a given time period as:

{Energy}_{After a time interval} = {Energy}_{Initial} {(1 - α)}^{t (\frac{S c}{4 V})}

${Energy}_{After a time interval} = {Energy}_{Initial} {(1 - α)}^{t (\frac{S c}{4 V})}$

(A4.5)

and therefore the ratio that the sound energy has decayed by at that time as:

\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}} = {(1 - α)}^{t (\frac{S c}{4 V})}

$\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}} = {(1 - α)}^{t (\frac{S c}{4 V})}$

(A4.6)

In order to find the time that it takes for the sound to decay by a given ratio, we must take logarithms, to the base (1 − α), on both sides of Equation A4.6 to give:

{log}_{(1 - α)} (\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}}) = t (\frac{S c}{4 V})

${log}_{(1 - α)} (\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}}) = t (\frac{S c}{4 V})$

which can be rearranged to give the time required for a given ratio of sound energy decay as:

t = (\frac{4 V}{S c}) {log}_{(1 - α)} (\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}})

$t = (\frac{4 V}{S c}) {log}_{(1 - α)} (\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}})$

(A4.7)

Unfortunately, Equation A4.7 requires that we take a logarithm to the base (1 − α)! However, we can get around this by remembering that this can be calculated using natural logarithms as:

{log}_{(1 - α)} (\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}}) = \frac{ln (\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}})}{ln(1 - α)}

${log}_{(1 - α)} (\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}}) = \frac{ln (\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}})}{ln(1 - α)}$

So Equation A3.7 becomes:

t = (\frac{4 V}{S c}) \frac{ln (\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}})}{ln(1 - α)}

$t = (\frac{4 V}{S c}) \frac{ln (\frac{{Energy}_{After n reflections}}{{Energy}_{Before reflections}})}{ln(1 - α)}$

(A4.8)

Equation A4.8 gives a relationship between the ratio of sound energy decay and the time it takes and so can be used to calculate this time. There is an infinite number of possible ratios that could be used. However, the most commonly used ratio is that which corresponds to a decrease in sound energy of 60 dB, or 106. When this ratio is substituted into Equation A4.8, we get an equation for the 60 dB reverberation time, known as T₆₀, which is:

\begin{matrix} T_{60} = (\frac{4 V}{S c}) \frac{{ln(10}^{- 6})}{ln(1 - α)} = (\frac{V}{S ln(1 - α)}) \frac{4 × (- 13 .82)}{{344 ms}^{- 1}} \\ = \frac{- 0 .161 V}{S ln(1 - α)} \end{matrix}

$\begin{matrix} T_{60} = (\frac{4 V}{S c}) \frac{{ln(10}^{- 6})}{ln(1 - α)} = (\frac{V}{S ln(1 - α)}) \frac{4 × (- 13 .82)}{{344 ms}^{- 1}} \\ = \frac{- 0 .161 V}{S ln(1 - α)} \end{matrix}$

(A4.9)

where T₆₀ = the 60 dB reverberation time (in s)

Thus the reverberation time is given by:

T_{60} = \frac{- 0 .161 V}{S ln(1 - α)}

$T_{60} = \frac{- 0 .161 V}{S ln(1 - α)}$

(A4.10)

where T₆₀ = the 60 dB reverberation time (in s)

Equation A4.10 is known as the Norris-Eyring reverberation formula, and the negative sign in the numerator compensates for the negative sign that results from the natural logarithm resulting in a reverberation time which is positive. Note that it is possible to calculate the reverberation time for other ratios of decay and that the only difference between these and Equation A4.10 would be the value of the constant. The argument behind the derivation of reverberation time is a statistical one, and so there are some important assumptions behind Equation A4.10. These assumptions are:

that the sound visits all surfaces with equal probability and at all possible angles of incidence. That is, the sound field is diffuse. This is required in order to invoke the concept of an average absorption coefficient for the room. Note that this is a desirable acoustic goal for subjective reasons as well; we prefer to listen to and perform music in rooms with a diffuse field.
that the concept of a mean free path is valid. Again, this is required in order to have an average absorption coefficient, but in addition, it means that the room’s shape must not be too extreme. This means that this analysis is not valid for rooms that resemble long tunnels; however, most real rooms are not too deviant, and the mean free path equation is applicable.

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Table of Contents for
Appendix 4 Deriving the Reverberation Time Equation

Appendix 4: Deriving the Reverberation Time Equation

Table of Contents for Appendix 4 Deriving the Reverberation Time Equation

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Appendix 4: Deriving the Reverberation Time Equation

Table of Contents for
Appendix 4 Deriving the Reverberation Time Equation