C HAPTER
18
Acoustics of Buildings
Acoustics is a branch of physics that deals with the process of generation, transmission and
reception of sound in a room or in a hall. The area of architectural acoustics dealing with the
design and construction of buildings and halls is an important factor in the phenomena. In this
chapter, we would discuss certain aspects of architectural acoustics so that the listeners can
experience best sound.
18.1 ACOUSTICS OF BUILDINGS – BASIC REQUIREMENTS
Sabine (1911) was the first one to study the acoustics of hall scientifically, and proposed a
formula which is the backbone of present day acoustics. Following are some of the requirements
for good acoustics in a hall:
(i) Sound must reach every part of the hall with an audible level of loudness and without
echoes
(ii) The quality of the music and speech should reach everyone
(iii) The reverberation must be optimum
(iv) The sound should be distributed uniformly, throughout the audience. There should not
be undesired focussing of sounds or dead spots in the hall
(v) There should not be any overlapping of syllable or musical notes
(vi) Resonance in the hall should be avoided
(vii) Extraneous noise should not reach the hall
(viii) Sufficient windows and ventillators
(ix) The source should generate sound of adequate intensity. This can be achieved using
loud speakers, sound reflectors, etc.
354
Acoustics of Buildings 355
18.2 SOUND INTENSITY
The intensity of a sound wave I means the rate of flow of sound energy through unit area parallel
to a wave front. As energy per unit time is power, the intensity is measured in units of W/m2.
Though the basic unit is W/m2, it is the practice to specify it in a logarithmic scale. The unit
is bel or more commonly, the decibel (dB) which is 0.1 bel. Often the intensity level b is
measured with respect to a standard zero level intensity I0, which is the lower threshold of
hearing equal to 10–12 W/m2 at 1000 Hz frequency:
I
b(dB) = 10 log (18.1)
I0
where the logarithm is to the base 10. Thus the intensity level of a sound whose intensity
I = 10–5 W/m2 will be
Ê 10 -5 W m 2 ˆ
b = 10 log Á -12 2˜
= 10 log10 7 (18.2)
Ë 10 W m ¯
= 10 ¥ 7 = 70 dB
The intensities and intensity level for number of common sounds are listed in Table 18.1.
Table 18.1 Intensity of various types of sound
Source of the sound Intensity level Intensity
(dB) (W/m2)
Threshold of hearing 0 1× 10–12
Whisper 20 1× 10–10
Quiet radio 40 1 × 10–8
Ordinary conversation, at 50 cm 65 3 × 10–6
Busy street traffic 70 1 × 10–5
Siren at 30 m 100 1 × 10–2
Loud indoor rock concert 120 1
Jet plane at 30m 140 100
18.3 ABSORPTION COEFFICIENT
Different surfaces absorb sound energy differently. An open window passes all the sound waves
falling on it, and hence a open window is taken as a perfect absorber. The extent of absorption
by a surface is expressed in terms of the quantity absorption coefficient. As the open window
is a perfect absorber, absorption coefficient of all substances are measured in terms of open
window unit (OWU). The absorption coefficient a of a surface is defined as the ratio of sound
energy absorbed by the surface to the sound energy absorbed by an equal area of a perfect
absorber given as
Sound energy absorbed by the surface
a=
Sound energy absorbed by an equal area of a perfect absorber
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Sound absorption coefficient of some materials are listed in Table 18.2.
Table 18.2 Sound absorption coefficient of certain
materials at 500 Hz frequency range
Material Absorption coefficient
(OWU)
Open window 1.0
Marble 0.01
Concrete 0.17
Asbestos 0.26
Carpet 0.30
Fibre board 0.50
Heavy curtains 0.50
Fibre glass 0.75
Human body 4.5
18.4 REVERBERATION
The sound produced by a source in a hall suffers successive reflections from the wall, floor,
ceiling and other reflecting materials in the hall. Hence, in addition to the sound of the theater or
a song or a movie, the listeners hear a series of sound waves. This gives the listener a persistence
of sound even after the original sound has ceased. This is called reverberation. We can feel the
reverberation of sound in a hall even after the source producing the sound is turned off.
The intensity of sound produced in a hall decays exponentially to zero with time. The decay
time will be less if sound absorbing materials like audience, windows, curtains, etc. are present
in the hall. Consequently, in such situations the reverberation will be less.
18.4.1 Reverberation Time
The interval of time required for the intensity to drop to one millionth of its original value is
called reverberation time. It can be expressed in terms of sound level in dB. If the incident
intensity is Ii , then the final intensity If is one millionth of Ii . That is,
Ii
I f = 10 -6 Ii or = 10 6 (18.3)
If
In terms of decibels, we get
Ii
dBi = 10 log
I (Standard)
If
dB f = 10 log
I (Standard)
I
dBi - dB f = 10 log i = 10 log 10 6
If
= 10 ¥ 6 = 60 (18.4)
Acoustics of Buildings 357
In other words, reverberation time for a hall is the time required for the intensity to drop by
60 dB.
18.5 SABINE’S FORMULA FOR REVERBERATION TIME
The relation connecting reverberation time, the volume of the hall (V), the area (S) and the
absorption coefficient (a) is known as the Sabine’s formula. Let a1, a2, a3 ...., be the absorption
coefficients of the materials in the hall whose surface areas exposed to sound be s1, s2, s3, ....,
respectively. Then the average value of the absorption coefficient is given as
a s + a 2 s2 + a3 s3 + ... +
Âa s i i
a= 1 1 = i
s1 + s2 + s3 + ... S
Âa s i i = aS (18.5)
i
where S is the total surface area of all surfaces exposed to sound. By statistical method, Jaeger
has shown that sound travels an average distance of 4V/S, where V is the volume of the hall,
between two successive reflections. This distance is known as mean free path.
4V
Time taken between two successive reflections =
Sv
where v is the velocity of sound.
The average number of reflections per second is
Sv
n= (18.6)
4V
Let It be the average intensity per unit volume of the hall at any given instant and dI the drop
in intensity in a small interval of time dt. Then
The number of reflections in time dt = ndt
Drop in intensity per reflection = a I
Drop in intensity for ndt reflections is dI = - a I n dt
The negative sign is to indicate the drop in intensity. Substituting the value of n, we get
-aISv dt
dI =
4V
dI aSv dt
=-
I 4V
Integrating, we get
aSv t
ln I t = - + Constant
4V
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Denoting the initial intensity by I0, we obtain
ÊI ˆ aSv t
ln Á t ˜ = -
Ë 0¯
I 4V
aSv t
It -
=e 4V
I0
Using Eq. (18.3), we obtain
a Sv T
-
e 4V = 10-6 where T is the reverberation time.
aSv T
fi = 6 ln 10 = 6 ¥ 2.3026
4V
Taking the velocity of sound at room temperature as 330 ms–1, we get
6 ¥ 2.3026 ¥ 4V
T=
(330 ms-1 ) aS
0.167V 0.167V
T= = (18.7)
aS Â ai si
i
This is Sabine’s formula for reverberation time which is in fairly good agreement with
experimental values. From Eq. (18.7), it is obvious that the reverberation time is:
(i) Directly proportional to the volume of the hall
(ii) Inversely proportional to the total absorption
(iii) Inversely proportional to the total area of sound absorbing surfaces.
18.6 ARCHITECTURAL ACOUSTICS – FACTORS AFFECTING AND REMEDIES
In a hall that is acoustically well-designed, the sound produced at one place, whether speech
or music, should reach every point in the hall with the required audibility, and then it should
die away at the appropriate time. Therefore, while designing the acoustics of a hall the following
factors should be taken into consideration:
18.6.1 Reverberation
For good acoustics, the reverberation time should neither be too small nor too large. If it is too
small, the loudness will be inadequate and the sound dies away in a very short time. This gives
the hall a dead effect. When the reverberation time is too large, the sound persists for a longer
time resulting in the overlap of successive sounds. This results in loss of clarity which makes
the sound unintelligible. Therefore, the reverberation time must have an appropriate value so
that everyone in the hall hears the sound clearly. This optimum value is referred to as optimum
reverberation time. The suitable reverberation time can be evaluated using Sabine’s formula,
given in Eq. (18.7)
Acoustics of Buildings 359
0.167 V
T=
aS
Experimentally, it is found that for distinct hearing in a hall of volume 300 m3 the optimum
value of reverberation time is about 1.03 s for speech. As the frequency involved is higher for
music, the time would be slightly larger. The reverberation time of a hall can be controlled by
the following factors:
(i) The first and foremost factor is having sufficient windows and ventillators. These are
important as we can keep the windows open or close so that the reverberation time
is optimum.
(ii) Having audience as per the capacity of the hall, as human beings are good absorbers.
(iii) Lining the walls and roof with good absorbers like fibre boards, glass wool, felt, etc.
is another factor.
(iv) Using thick curtains
(v) Covering the floors with carpets
(vi) Providing suitable sound absorbers on the walls
The presence of the above items reduces reverberation in a hall.
18.6.2 Focussing Surfaces
The presence of focussing surfaces such as concave, spherical, cylindrical, and parabolic helps
in the concentration of sound in certain regions, which causes less sound or no sound in certain
other regions. If reflecting surfaces are present, the reflected sound waves may combine with
the direct sound waves and produce stationary waves which make the sound intensity non-
uniform. Hence, for uniform distribution of sound energy:
(i) Curved surfaces should be avoided or at least minimized
(ii) Ceiling should be low
(iii) A parabolic surface may be arranged with the speaker at its focus. This sends out
uniform sound energy in the entire hall, as shown in Fig. 18.1.
Sound source Audience
Fig. 18.1 Parabolic surface with the speaker at the focus.
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18.6.3 Sufficient Loudness
An average person would be able to detect a 30 dB sound at 1000 Hz as reasonably loud.
However, a 30 dB sound at 50 Hz would not be heard at all. Therefore, for satisfactory hearing,
sufficient loudness throughout the hall is a necessity. The following arrangements would
increase the loudness:
(i) By setting up loud speakers at different positions in the hall, the additional sound
energy required can be produced.
(ii) Low ceiling of suitable shape can also serve the purpose by reflecting sound energy
towards the audience (Fig. 18.2).
(iii) By keeping large sounding boards behind the speaker and facing the audience.
Ceiling
Audience
Sound source
Fig. 18.2 Hall with low ceiling of suitable shape.
18.6.4 Absence of Echoes
When direct and reflected sound waves coming from the same source reach the listener in a
time interval greater than (1/7) s, echo is heard, which causes confusion. Hence, for good
hearing, echo must be avoided. This can be achieved by covering walls at distance and the high
ceiling with sound absorbers.
18.6.5 Resonance
Items including wall that are not rigid can cause forced oscillation and resonance. Due to the
interference from the sound thus created, the original sound may be distorted. Locked up air
can also create resonance. The resonant vibrations must be suitably damped.
18.6.6 Echelon Effect
Regular spacing of reflecting surfaces and railings may produce additional notes due to the
regular succession of echoes. This makes the original sound unintelligible. Such surfaces should
be avoided or covered with absorbers.
Acoustics of Buildings 361
SOLVED EXAMPLES
Example 18.1 A point sound source is generating sound at a point. At a distance of 200 m
away, the sound reduces to a level to 60 dB. Calculate the output power of the sound source
in W/m2.
Solution: Let I be the sound intensity at a distance of 200 m. For the intensity, we have
I
dB = 10 log10
I0
I I
60 = 10 log or log =6
I0 I0
I
= 10 6
I0
I = (10 -12 W m 2 ) 10 6 = 10 -6 W m 2
Power of the source = 4 pr 2 I
= 4 p(200 m)2 ¥ 10 -6 W m 2
= 0.5 W
Example 18.2 If the intensity of sound in W/m2 is doubled, what is the change in sound level
in dB?
Solution: Let dB1 be the sound level in dB when the sound intensity is I1 W/m2 and dB2 is
the sound level when the intensity is I2 W/m2.
I1 I2
dB1 = 10 log and dB2 = 10 log
I0 I0
I2 I
dB2 - dB1 = 10 log - 10 log 1
I0 I0
I2
= 10 log = 10 log 2
I1
= 10 ¥ 0.301 = 3.01
The sound intensity level increased by 3 dB.
Example 18.3 The reverberation time of a hall of volume 8000 m3 is 1.5 s. Calculate the total
absorption in the hall.
Solution: From Sabine’s formula, we have
0.167 V 0.167 V
T= or Âa s =
Âa s
i i
i i i
T
i
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0.167 ¥ 8000 m 3
Âa s
i
i i =
1.5s
= 890.7
The total absorption = 890.7 OWU.
Example 18.4 A hall has dimensions of 25 m × 20 m × 8 m. The reverberation time is 4s.
Determine the average absorption coefficient of the surfaces.
Solution: Volume of the hall = 25 × 20 × 8 m3 = 4000 m3
The total area S = 2[(25 ¥ 20) + (25 ¥ 8) + (20 ¥ 8)] m 2
= 1720 m2
From Sabine’s formula, we obtain
0.167V 0.167V
T= or a=
aS TS
Average absorption coefficient is given by
0.167 ¥ 4000 m 3
a= = 0.097 ms -1
4 s ¥ 1720 m 2
= 0.097 OWU m -2
Example 18.5 In a hall the area of the floor and ceiling is 100 m2 each. The area of wall
is 200m2. The absorption coefficients of the wall, ceiling and floor are 0.025, 0.02 and 0.55,
respectively. If the volume of the hall is 475m3, calculate the reverberation time for the hall.
Solution: From Sabine’s formula, we have
0.167 V
T=
Âa s
i
i i
Âa s
i
i i = (200 ¥ 0.025 + 100 ¥ 0.02 + 100 ¥ 0.55)
= 62
0.167 ¥ 475
Reverberation time T=
62
= 1.28 s.
Example 18.6 What is the resultant sound level when a 80 dB sound is added to a 70 dB
level?
Solution: For the 80 dB sound
I1 I1
80 dB = 10 log or 8 = log
I0 I0
Acoustics of Buildings 363
I1 = 108 I 0
For the 70 dB sound
I 2 = 10 7 I 0
Adding the two, we get
I = I1 + I 2 = 108 I 0 + 10 7 I 0 = 11 ¥ 10 7 I 0
The intensity level of the sum in dB is
I
dB = 10 log = 10 log(10 7 ¥ 11)
I0
= 10(7 log10 + log11) = 10[7 + 1.041]
= 80.041
The resultant intensity is 80.041 dB.
REVIEW QUESTIONS
18.1 Explain intensity of sound. What is its unit?
18.2 What is reverberation? Define reverberation time.
18.3 Why is an optimum value of reverberation time needed for maintaining the acoustics of
a hall?
18.4 Define absorption coefficient of a material. What is its unit?
18.5 Explain the factors affecting the acoustic quality of a building.
18.6 State Sabine’s formula for reverberation time and explain the quantities involved.
18.7 State Sabine’s formula. How can it be used for determining the absorbing power of
surfaces involved?
18.8 Give an account of the factors affecting the acoustics of a hall. How are they remedied?
18.9 Derive Sabine’s formula for reverberation time and explain its importance.
PROBLEMS
1. If the sound level in dB increases from 10 dB to 20 dB, what is the ratio of increase
in sound intensity in W/m2?
2. A point source emits sound of output power 50 W. Determine the distance at which sound
reduces to a level of 30 dB.
3. The intensity of sound at a place is 50 dB. What should be the intensity of an additional
sound source to get a resultant intensity of 60 dB?
