7.3 REVERBERATION TIME

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When the source of sound in a roomis suddenly turned off, a certain period

of time is required before the sound energy is practically all absorbed by the

surfaces in the room. For many applications, the duration of this time

period is important for effective use of the space.

Let us consider the room shown in Fig. 7-7. The acoustic energy

density associated with the sound after one reflection is given by the following

expression:

D1 ј Doр1 _ __Ю (7-19)

The quantity Do is the original acoustic energy density before the sound

strikes any walls. The acoustic energy density after the second reflection is

found in a similar manner:

D2 ј D1р1 _ __Ю ј Doр1 _ __Ю2 р7-20Ю

By extension, we see that the acoustic energy density in the room after n

reflections is given by the following expression:

Dn ј Doр1 _ __Юn (7-21)

Room Acoustics 281

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The time between one reflection and the next reflection is related to the

speed of sound c:

t1 ј d=c (7-22)

The quantity d is the mean free path of the sound wave, or the average

distance that the sound travels before being reflected. The mean free path

is given by the following expression for a room (Pierce, 1981):

d ј 4V=So (7-23)

The quantity V is the total volume of the room. Making the substitution

from Eq. (7-23) for the mean free path into Eq. (7-22), we obtain the following

relationship for the time between reflections:

t1 ј

4V

Soc

(7-24)

The total time t after n reflections is given by the following:

t ј nt1 ј

4Vn

cSo

(7-25)

The number of reflections n during the time t is found from Eq. (7-25):

n ј

cSot

4V

(7-26)

282 Chapter 7

FIGURE 7-7 Reverberant sound field after the sound source has been turned off for

various numbers of reflections.

Copyright © 2003 Marcel Dekker, Inc.

The expression for the acoustic energy density in the room as a function

of the time after the source of sound has been turned off may be found

by combining Eqs (7-21) and (7-26):

D ј Doр1 _ __ЮрSoc=4VЮt (7-27)

We may write the term involving the average surface absorption coefficient

in the following form:

р1 _ __Ю ј expЅlnр1 _ __Ю_ ј exp _ln

1

1 _ __

_ _ __

(7-28)

The acoustic energy density in Eq. (7-27) may be expressed in the following

alternative form:

D ј Do expр_cat=4VЮ (7-29)

The quantity a is called the number of absorption units and is defined by the

following expression:

a ј So ln

1

1 _ __

_ _

(7-30)

In architectural work, if the units of the surface area are expressed in

ft2, then the units for the absorption are called sabins, in honor of W. C.

Sabine, who conducted the initial work in laying a foundation for the

rational design for architectural acoustics. On the other hand, if the room

surface area is expressed in m2, the absorption units are sometimes called

mks sabins. To avoid any confusion, we will express the absorption units

directly in units of area.

The acoustic pressure and acoustic energy density are related by Eq.

(2-20):

D ј

p2

_oc

and Do ј

p2

o

_oc

(7-31)

The quantity po is the original acoustic pressure before the sound source

is turned off. Equation (7-29) may be written in terms of the acoustic

pressures:

p2

p2

o ј expр_cat=4VЮ ј р p=pref Ю2

р po=pref Ю2 (7-32)

Taking log10 of both sides of Eq. (7-32) and multiplying by 10, we obtain the

following result:

Lp;o _ Lp ј Ѕ10 log10рeЮ_рcat=4VЮ ј 1:0857cat=V (7-33)

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The reverberation time Tr is defined as the time required for the sound

pressure level to decrease by 60 dB. Setting рLp;o _ LpЮ ј 60 dB in Eq.

(7-33), we may solve for the reverberation time, t ј Tr:

Tr ј

55:26V

ca

(7-34)

This expression is called the Norris–Eyring reverberation time (Eyring, 1930),

which is a modification of the expression originally developed by Sabine,

who used a ј So__.

If the walls of a room have significantly different surface absorption

coefficients from those of the floor–ceiling combination, sound waves traveling

between the walls will decay at a different rate from those traveling in

the vertical direction. To take this phenomenon into consideration, it was

proposed that the number of absorption units be expressed as three terms:

for reflections between the side walls, the end walls, and the floor–ceiling

combination (Fitzroy, 1959). The Fitzroy expression for the number of

absorption units is given by the following expression:

1

a ј _

1

So

рSx=SoЮ

lnр1 _ __xЮ ю рSy=SoЮ

lnр1 _ __yЮ ю рSz=SoЮ

lnр1 _ __zЮ

_ _

(7-35)

The quantity Sx is the side-wall surface area, Sy is the end-wall surface area,

Sz is the floor–ceiling area, and So is the total surface area. The quantities

__x, __y, and __z are the surface absorption coefficients for the side-wall surfaces,

the end-wall surfaces, and the ceiling–floor area, respectively. If any of

the three average surface absorption coefficients exceed 0.60, the corresponding

term, lnр1 _ __Ю, is replaced by р___Ю for that surface combination.

The effect of people and furniture in the room may be introduced by

adding the absorption capacity of the empty room ao, given by Eq. (7-35), to

the absorption of the people and furniture:

a ј ao ю _р_SЮpeople (7-36)

It has been shown (Fitzroy, 1959) that the reverberation time calculated

from the Fitzroy equation, Eq. (7-35), is in better agreement with

experimental data for rooms with nonuniform absorption than is Eq. (7-30).

For a room in which the average floor–ceiling absorption coefficient р_zЮ was about 13 times that of the other two combinations, the reverberation

time calculated from Eq. (7-30) was found to be about 25% of the measured

reverberation time, whereas Eq. (7-35) yielded values for the reverberation

time within 3% of the measured values. If the average absorption coefficients

of the surfaces are approximately the same, either relationship will

predict values of reverberation time near that measured.

284 Chapter 7

Copyright © 2003 Marcel Dekker, Inc.

A value for the optimum reverberation time may be needed for design

purposes. The optimum reverberation time depends on the usage of the

space (church, music hall, conference room, etc.). An empirical relationship

for the measured reverberation time in acoustically good concert halls and

opera houses (Beranek, 1962) has been developed:

Tr ј

1

0:1ю5:4рSF=VЮ

(7-37)

The quantity SF is the total floor area (m2) of the audience, orchestra, and

chorus areas, and V is the volume of the space (m3).

For design purposes, the optimum reverberation time for the 500 Hz

octave band may be estimated from the following empirical expression

(Beranek, 1954, p. 425):

Tr;opt ј a ю

blog10 V

log10 e ј aю2:3026 b log10 V (7-38)

The quantityV is the volume of the space (m3). The numerical values for the

constants a and b are given in Table 7-3A. The reverberation time ratio from

Table 7-3B may be used to estimate the optimum reverberation time for

other octave bands for a room used for music. For rooms involving speech

only, it is recommended that the value for reverberation time at 500 Hz be

used for other frequencies.

Example 7-2. Determine the reverberation time for the 500 Hz octave band

for the room given in Example 7-1. The sonic velocity for the air in the room

(at 218C or 708F) is 343.8 m/s (1128 fps).

The average surface absorption coefficient and surface area were

calculated previously:

__ ј 0:2361 and So ј 150:04m2

The number of absorption units calculated from Eq. (7-30) is as

follows:

a ј р150:04Ю ln

1

1 _ 0:2361

_ _

ј 40:41m2 (or, mks sabins)

The volume of the room is as follows:

V ј р6:20Юр6:00Юр3:10Ю ј 115:32m3 р4072 ft3Ю

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Using the Norris–Eyring expression, Eq. (7-34), we may estimate the

reverberation time:

Tr ј р55:26Юр115:32Ю

р343:8Юр40:41Ю ј 0:459 s

Because the floor and ceiling in this example have surface absorption

coefficients that are much different from that of the walls, the Fitzroy relationship,

Eq. (7-35), would probably yield more accurate results for the

reverberation time. The parameters for the side wall (the long wall) are as

follows:

Sx ј р2Юр6:20Юр3:10Ю ј 38:44m2

__x ј 0:06

Sx=So ј р38:44Ю=р150:04Ю ј 0:2562

286 Chapter 7

TABLE 7-3A Values for the Constants in Eq. (7-38) for

the Optimum Reverberation Time at 500 Hz

Type of space a b

Catholic church; organ music ю0:098 1/5

Protestant church; synagogue; concert hall _0:162 1/5

Music studio; opera house _0:352 1/5

Conference room; movie theater _0:101 2/15

Broadcast room for speech _0:192 1/9

TABLE 7-3B Reverberation Time Ratio

ЅTt;optр f Ю=Tr;optр500 HzЮ_ for Music Rooms

Frequency, Hz Ratio Frequency, Hz Ratio

31.5 2.40 1,000 1.05

63 1.93 2,000 1.05

125 1.46 4,000 1.05

250 1.13 8,000 1.05

500 1.00

Source: Beranek (1954, p. 426).

Copyright © 2003 Marcel Dekker, Inc.

The values for the end wall (the shorter wall) are as follows:

Sy ј р2Юр6:00Юр3:10Ю ј 37:20m2

__y ј р0:06Юр37:20 _ 2:64Ю ю р0:05Юр2:64Ю

р37:20Ю ј

2:206

37:20 ј 0:0593

Sy=So ј 0:2479

Finally, the parameters for the floor–ceiling combination are as follows:

Sz ј р2Юр6:20Юр6:00Ю ј 74:40m2

__z ј р0:21Юр37:20Ю ю р0:55Юр37:20Ю

р74:40Ю ј

28:27

74:40 ј 0:3800

Sz=So ј 0:4959

The number of absorption units, without the effect of the people and

furniture, is given by Eq. (7-35):

1

ao ј _

1

р150:04Ю

р0:2562Ю

lnр1 _ 0:06Ю ю р0:2479Ю

lnр1 _ 0:0593Ю ю р0:4959Ю

lnр1 _ 0:3800Ю

_ _

1

ao ј

4:1406 ю 4:0552 ю 1:0373

150:04 ј

9:2332

150:04 ј 0:06154m_2

ao ј 16:250m2

The effect of the people and furniture may be included by using Eq. (7-36):

a ј 16:250 ю 2:64 ј 18:89m2

The reverberation time for the 500 Hz octave band, using the Fitzroy

relationship, is calculated from Eq. (7-34):

Tr ј р55:26Юр115:32Ю

р343:8Юр18:89Ю ј 0:981 s

This value for the reverberation time would probably be more nearly

in agreement with measured values for the room, because there will be

sound waves moving between the walls that are not damped out as rapidly

as the sound waves moving between the floor and ceiling.

Example 7-3. Suppose the room in Example 7-2 is a conference room.

Determine the amount of 1-inch fiberglass formboard that must be added

to the walls to reduce the reverberation time at 500 Hz, according to the

Fitzroy relationship, to the optimum value.

The optimum reverberation time may be determined from Eq. (7-38):

Tt;opt ј _0:101 ю р2=15Юр2:3026Ю log10р115:32Ю ј 0:532 s

Room Acoustics 287

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The number of absorption units required to achieve this reverberation time

is found from Eq. (7-34):

a ј

55:26V

cTr ј р55:26Юр115:32Ю

р343:8Юр0:532Ю ј 34:842m2

The number of absorption units, excluding the people and furniture, is

found from Eq. (7-36):

ao ј 34:842_2:64 ј 32:202m2

There are several approaches that could be used in this case.

Generally, it is better to distribute the sound absorption material than to

concentrate the material on one surface. Let us determine the amount of

formboard required to make the average surface absorption coefficients

for the side walls and the end walls equal. Using the Fitzroy expression,

Eq. (7-35), we obtain the following values:

1

ao ј

1

32:202 ј _

1

р150:04Ю

р0:2562ю0:2479Ю

lnр1__Ю ю р0:4959Ю

lnр1_0:3800Ю

_ _

4:6593 ј _

0:5041

lnр1__Юю1:0373

1__ ј 0:8701 or; __x ј__y ј 0:1299

At 500 Hz, the surface absorption coefficient for fiberglass formboard

is _5 ј 0:79, from Appendix D. The required surface area S5 covered by the

formboard on the side walls may be found from the following expression:

__xSx ј р0:1299Юр38:44Ю ј р0:06Юр38:44 _ S5Ю ю 0:79S5

S5 ј

4:9934 _ 2:3064

р0:79 _ 0:06Ю ј 3:680m2 р39:61 ft2Ю

If the formboard is distributed evenly on both walls, the required surface

area per wall is as follows:

1

2 S5 ј 1:840m2 р19:81 ft2Ю

This is a little less than 10% of each side-wall surface area, so it would be

practical (and possible) to add the formboard to the side walls.

Similarly, the required area of formboard S6 on the end walls to

achieve the desired average surface absorption coefficient is found as

follows:

__ySy ј р0:1299Юр37:20Ю ј р0:06Юр34:56 _ S6Ю ю р0:05Юр2:64Ю ю 0:79S6

S6 ј

4:8323 _ 2:0736 _ 0:1320

р0:79 _ 0:06Ю ј 3:598m2 р38:7 ft2Ю

288 Chapter 7

Copyright © 2003 Marcel Dekker, Inc.

If half of the area is placed on each end wall, the amount of one end wall

covered by the formboard is as follows:

1

2 S6 ј 1:799m2 р19:4 ft2Ю

This area is also about 10% of each end-wall surface area, so it would be

possible to add the acoustic material on the end wall.

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