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Deriving the Reverberation Time Equation
Appendix 4: Deriving the Reverberation Time Equation Clearly the length of time that 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 that 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:
MFP
4V S
(A4.1)
where MFP the mean free path (in m) V the volume (in m 3 ) and S the surface area (in m 2 )
The time between surface interactions may be simply calculated from A4.1 by dividing it by the speed of sound to give:
τ
4V Sc
(A4.2)
where τ the time between reflections (in s) and c the speed of sound (in ms1 , or meters per second) Equation A4.2 gives us the time between surface interactions and at each of these interactions is the proportion of the energy absorbed, where is the average absorption coefficient discussed earlier. If of the energy Acoustics and Psychoacoustics Copyright © 2009 Elsevier Ltd. All rights of reproduction in any form reserved.
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452 append ix 4: Deriving the Reverberation Time Equation
is absorbed at the surface, then (1 ) is the proportion of the energy reflected back to interact with further surfaces. At each surface a further proportion, , of energy will be removed so the proportion of 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
3
Energy After three reflections Energy Before reflections (1 1 α)
(A4.3)
Energy After n reflections Energy Before reflections (1 α)n
As is less than 1, (1 ) 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
Sc t t 4 V 4 V Sc
(A4.4)
where t the time interval (in s) By substituting Equation A3.4 into Equation A4.3 we can get an expression for the remaining energy in the sound after a given time period as:
Sc t 4 V
Energy After a time interval Energy Initial (1 α)
(A4.5)
and therefore the ratio that the sound energy has decayed by at that time as: Sc
t Energy After n reflections (1 α) 4 V Energy Before reflections
(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 ), on both sides of Equation A4.6 to give:
Energy After n reflections Sc t log(1α) 4 V Energy Before reflections
Appendix 4: Deriving the Reverberation Time Equation
which can be rearranged to give the time required for a given ratio of sound energy decay as:
Energy After n reflections 4V t log(1α) Sc Energy Befoore reflections
(A4.7)
Unfortunately Equation A4.7 requires that we take a logarithm to the base (1 )! However, we can get round this by remembering that this can be calculated using natural logarithms as:
Energy After n reflections log(1α) Energy Before reflections
Energy After n reflections ln Energy Before reflections ln(1 α)
So Equation A4.7 becomes:
Energy After n reflections ln 4 V Energy Before refflections t Sc 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 are 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:
4 (13.82) 4 V ln(106 ) V T60 S ln(1 α) 344 ms1 Sc ln(1 α) 0.161V S ln(1 α)
(A4.9)
where T60 the 60 dB reverberation time (in s) Thus the reverberation time is given by:
T60
0.161V 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
453
454 append ix 4: Deriving the Reverberation Time Equation
arising from the natural logarithm, resulting in a reverberation time that 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: n
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.
n
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 which resemble long tunnels; however, most real rooms are not too deviant and the mean free path equation is applicable.
MFP
4V S
(A4.1)
where MFP the mean free path (in m) V the volume (in m 3 ) and S the surface area (in m 2 )
The time between surface interactions may be simply calculated from A4.1 by dividing it by the speed of sound to give:
τ
4V Sc
(A4.2)
where τ the time between reflections (in s) and c the speed of sound (in ms1 , or meters per second) Equation A4.2 gives us the time between surface interactions and at each of these interactions is the proportion of the energy absorbed, where is the average absorption coefficient discussed earlier. If of the energy Acoustics and Psychoacoustics Copyright © 2009 Elsevier Ltd. All rights of reproduction in any form reserved.
451
452 append ix 4: Deriving the Reverberation Time Equation
is absorbed at the surface, then (1 ) is the proportion of the energy reflected back to interact with further surfaces. At each surface a further proportion, , of energy will be removed so the proportion of 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
3
Energy After three reflections Energy Before reflections (1 1 α)
(A4.3)
Energy After n reflections Energy Before reflections (1 α)n
As is less than 1, (1 ) 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
Sc t t 4 V 4 V Sc
(A4.4)
where t the time interval (in s) By substituting Equation A3.4 into Equation A4.3 we can get an expression for the remaining energy in the sound after a given time period as:
Sc t 4 V
Energy After a time interval Energy Initial (1 α)
(A4.5)
and therefore the ratio that the sound energy has decayed by at that time as: Sc
t Energy After n reflections (1 α) 4 V Energy Before reflections
(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 ), on both sides of Equation A4.6 to give:
Energy After n reflections Sc t log(1α) 4 V Energy Before reflections
Appendix 4: Deriving the Reverberation Time Equation
which can be rearranged to give the time required for a given ratio of sound energy decay as:
Energy After n reflections 4V t log(1α) Sc Energy Befoore reflections
(A4.7)
Unfortunately Equation A4.7 requires that we take a logarithm to the base (1 )! However, we can get round this by remembering that this can be calculated using natural logarithms as:
Energy After n reflections log(1α) Energy Before reflections
Energy After n reflections ln Energy Before reflections ln(1 α)
So Equation A4.7 becomes:
Energy After n reflections ln 4 V Energy Before refflections t Sc 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 are 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:
4 (13.82) 4 V ln(106 ) V T60 S ln(1 α) 344 ms1 Sc ln(1 α) 0.161V S ln(1 α)
(A4.9)
where T60 the 60 dB reverberation time (in s) Thus the reverberation time is given by:
T60
0.161V 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
453
454 append ix 4: Deriving the Reverberation Time Equation
arising from the natural logarithm, resulting in a reverberation time that 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: n
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.
n
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 which resemble long tunnels; however, most real rooms are not too deviant and the mean free path equation is applicable.