Room Acoustics

D.N. Manik 1
ME 734 VIBRO-ACOUSTICS
Sound power level and sound pressure
level relationship
2
2
4
rms
o o
p W
I
c r ρ π
= =
2
2 2 2
4
rms
o o ref ref
p W
c p p r ρ π
=
Point source with no reflections
Room acoustics
D.N. Manik 2
ME 734 VIBRO-ACOUSTICS
Sound power level and sound pressure
level relationship
2
2
2
4 log 10 log 10 log 10 r
W
W
p
p
ref ref
rms
π − =
2
1
ref
ref
o o
p
W pW
c ρ
= =
π − − = 4 log 10 log 20 r L L
w p
dB 11 log 20 − − = r L L
w p
Room acoustics
D.N. Manik 3
ME 734 VIBRO-ACOUSTICS
Sound power level and sound pressure
level relationship
1
2
log 20
1 2
r
r
L L
p p
− =
6 dB reduction for doubling
the distance
Room acoustics
D.N. Manik 4
ME 734 VIBRO-ACOUSTICS
Sound power level and sound pressure
level relationship
Lp = Lw – 20 log r – 10 log 2π dB
Lp = Lw – 20 log r – 8 dB
Point source on the ground
1
2
log 20
1 2
r
r
L L
p p
− =
6 dB reduction due to doubling
distance
Room acoustics
D.N. Manik 5
ME 734 VIBRO-ACOUSTICS
DIRECTIVITY INDEX
2
2
) (
s
s
D
p
p
I
I
Q
α
α
α = =
Directivity factor, Q
D
(α αα α) is the ratio of sound intensity I
α αα α
,
at some distance ‘r’ from the source and at an angle α αα α to
a specific axis, of a directional noise source of sound
power W, to the intensity I
s
produced at some distance r
from a uniformly radiating sound source of equal sound
power
Room acoustics
D.N. Manik 6
ME 734 VIBRO-ACOUSTICS
DIRECTIVITY INDEX
DIα = 10 log Q
D
(α) =
s
p p
L L
α
−
) (
2
2
α
α D s
Q p p =
) ( log 10 α
α D ps p
Q L L + =
2
2
2
( )
10log 10log ( ) 10log 4
4
10log 4
D
p w w D
w
Q
L L L Q r
r
L DI r
α
α
α
α π
π
π
= + = + −
= + −
Room acoustics
D.N. Manik 7
ME 734 VIBRO-ACOUSTICS
Position DIRECTIVITY INDEX
surface spherical the of fraction
Q
P
1
=
Room acoustics
D.N. Manik 8
ME 734 VIBRO-ACOUSTICS
DIRECTIVITY INDEX
) (α
α D P
Q Q Q = 2
10log ( ) 10log4
p w P D
L L Q Q r
α
α π = + −
Directivity factor for various mounting positions

Position Directivity factor,
Q
P
Directivity index DI
P
,
10 log Q
P
dB
Free space
Centre of a large flat
surface
Intersection of two
large flat surfaces
Intersection of three
flat surfaces
1
2

4

8
0
3

6

9
Room acoustics
D.N. Manik 9
ME 734 VIBRO-ACOUSTICS
SOUND POWER MEASUREMENT
dB Q r
p
p
L
ref
m
W
, log 10 4 log 10 log 10
2
2
2
α
π − + =
Anechoic room
Room acoustics
D.N. Manik 10
ME 734 VIBRO-ACOUSTICS
SOUND POWER MEASUREMENT
∑
=
=
n
i
i m
p
n
p
1
2 2
1
dB r
p
p
L
ref
m
W
, 3 4 log 10 log 10
2
2
2
− + = π
1 2
D P
Q Q = =
Room acoustics
D.N. Manik 11
ME 734 VIBRO-ACOUSTICS
SOUND POWER MEASUREMENT
m
p p D
L L Q DI − = =
α
α
α) ( log 10
10
( ) 10
m
p p
D
L L
Q
α
α
−
=
) ( 2 ) ( α α
α D D P
Q Q Q Q = =
Room acoustics
D.N. Manik 12
ME 734 VIBRO-ACOUSTICS
Acoustic absorption coefficient Room acoustics
D.N. Manik 13
ME 734 VIBRO-ACOUSTICS
Acoustic absorption coefficient
L = difference in decibels between the maximum and minimum
sound pressure levels
D
1
= Distance from the face of specimen to the nearest minimum
of the standing wave pattern
D
2
= Distance from the first to the second minimum in standing
wave pattern

Room acoustics
D.N. Manik 14
ME 734 VIBRO-ACOUSTICS
Acoustic absorption coefficient
2
0 0
0 0
/ 1
1
/ 1
n
z c
z c
ρ
α
ρ
| |
−
= −
|
+
\ ¹
Normal absorption coefficient
z r jx = +
2
20
20
1 10
1 10
1
|
|
|
¹
|



\
|
+
−
− =
L
L
n
α
Room acoustics
D.N. Manik 15
ME 734 VIBRO-ACOUSTICS
Acoustic absorption coefficient Room acoustics
D.N. Manik 16
ME 734 VIBRO-ACOUSTICS
Acoustic absorption coefficient
( )
0 0 0 0 0 0
cot
z r jx
h A jB
c c c ρ ρ ρ
= + = +
| |
20 / 1
10 cot
L
h A
−
=
|
|
¹
|


\
|
− π =
2
1
2
1
D
D
B
Room acoustics
D.N. Manik 17
ME 734 VIBRO-ACOUSTICS
Acoustic absorption coefficient
1
j η µ κ
ξ
= = +
( )
( )
2
2
cos
cos 4
θ µ κ
θ µ
θ α
+ +
=
(
(
¸
(


¸

|
|
¹
|


\
|
+
−
+
|
|
¹
|


\
|
+
+ − =
−
µ η
κ
κ
µ
η
µ
µ µ α
2
1
2 2
2
stat
tan
1 2
1 ln 1 8
k
Room acoustics
D.N. Manik 18
ME 734 VIBRO-ACOUSTICS
Acoustic absorption coefficient
Statistical Acoustic Absorption Coefficients

Octave Band Center
Frequency, Hz
Size 125 250 500 1000 2000 4000
Fibrous Glass 68 kg/cubic
meter
25.0
mm
0.07 0.23 0.48 0.83 0.88 0.80
50.0
mm
0.20 0.55 0.89 0.97 0.83 0.79
100.0
mm
0.39 0.91 0.99 0.97 0.94 0.89
Polyurethane Foam (open cell)
6.25
mm
0.05 0.07 0.10 0.20 0.45 0.81
12.5
mm
0.05 0.12 0.25 0.57 0.89 0.98
25.0 0.14 0.30 0.63 0.91 0.98 0.91
Room acoustics
D.N. Manik 19
ME 734 VIBRO-ACOUSTICS
AVERAGE ABSORPTION COEFFICIENT
∑
=
α = α
n
i
i ij
T
j
S
S
1
1
Consider an enclosed volume of various
surface areas, S
1
, S
2
…S
n
with respective
absorption coefficients α
1
, α
2
, …α
n
corresponding to the j
th
band

Room acoustics
D.N. Manik 20
ME 734 VIBRO-ACOUSTICS
AVERAGE ABSORPTION COEFFICIENT
Room constant
1
1
≠ α
α −
α
=
j
j
j
T j
S R
Room acoustics
D.N. Manik 21
ME 734 VIBRO-ACOUSTICS
D.N. Manik 22
ME 734 VIBRO-ACOUSTICS
ROOM METHOD OF MEASURING ABSORPTION COEFFICIENT
j
air
j
T
j
S
T
α × + α
=
V 4
V 161 . 0
60
j
T
60 = Reverberation time for j
th
band
V = Room volume
S
T
= Total surface area
j
α
= Average absorption coefficient
j
air
α
= Absorption due to air from Table

Room acoustics
D.N. Manik 23
ME 734 VIBRO-ACOUSTICS
Acoustic absorption due to air at 25 C
Relative Octave-band center frequency, Hz
Humidity 63 125 250 500 1000 2000 4000 8000
Percent
10 0 0 0 0.003 0.006 0.008 0.02 0.045
20 0 0 0 0.001 0.002 0.004 0.015 0.046
30 0 0 0 0 0.0015 0.0028 0.0078 0.0171
40 0 0 0 0 0.001 0.0025 0.0064 0.013
50 0 0 0 0 0.001 0.0024 0.0059 0.0111
60 0 0 0 0 0.001 0.0022 0.0055 0.0102
70 0 0 0 0 0.001 0.0021 0.0052 0.0097
80 0 0 0 0 0.001 0.002 0.005 0.0093
90 0 0 0 0 0.001 0.002 0.005 0.0093


Room acoustics
D.N. Manik 24
ME 734 VIBRO-ACOUSTICS
Energy density
Consider an undisturbed volume of fluid V
0

that
changes to a volume V
1
due to the passage of
an acoustic disturbance. The change in potential
energy due to this change of volume is given by

1
0
V
V
U pdV = −
∫
Room acoustics
2
d m
dV V V
ρ ρ
= − = −
D.N. Manik 25
ME 734 VIBRO-ACOUSTICS
0
0
p dp
d
γ
ρ ρ
=
0
0
p dp d
d dV V
γ ρ ρ
ρ ρ
= −
1
0
V
d
V
U p dV = −
∫
For small changes in volume due to acoustic waves
0
0
d
dp p
dV V
γ
= −
2
0 0
0 0
0
2
d
p
d
d d
V V p
U p d p
p p γ γ
= =
∫
Room acoustics
D.N. Manik 26
ME 734 VIBRO-ACOUSTICS
2
0 0
0 0
0
2
d
p
d
d d
V V p
U p d p
p p γ γ
= =
∫
2
0
2
0 0
2
d
V p
U
c ρ
=
2
0
0
o
p
c
γ
ρ
=
2
0 0
2
V u
T
ρ
=
Kinetic energy change due to passage of acoustic wave
2
0
2
0 0
2
d
V p
T
c ρ
=
For plane waves,
0 0
d
p
c
u
ρ =
Room acoustics
D.N. Manik 27
ME 734 VIBRO-ACOUSTICS
Total energy T+U
2 2 2
0 0 0
2 2 2
0 0 0 0 0 0
2 2
d d d
V p V p V p
c c c ρ ρ ρ
+ =
Room acoustics
Energy per unit volume, D=
2
2
0 0
d
p
c ρ
D.N. Manik 28
ME 734 VIBRO-ACOUSTICS
Reverberant sound
( ) α − = 1 W W
rev
2
2
0
c
rev
rev
o
p V
E
c ρ
=
Reverberant sound power
Energy due to reverberation
0
4
T
R
c
c S
n
V
=
Number of reflections per sec
S
T
surface area, V
c
volume
D.N. Manik 29
ME 734 VIBRO-ACOUSTICS
Reverberant sound
2 2
0
2
0 0
4 4
T c T
rev rev
rev
c o o
c S p V S p
E
V c c
α α
ρ ρ
= =
&
rev
E α
Energy loss
Rate of energy loss
Rate of energy loss=reverberant sound power
( )
2
4 1
4
rev
o
T
W p
W
c R
S
α
ρ
α
| |
−
|
= =
|
\ ¹
( ) α − = 1 W W
rev
D.N. Manik 30
ME 734 VIBRO-ACOUSTICS
Reverberant sound
2
2
4
direct
o
p
WQ
c r
α
ρ π
| |
|
=
|
\ ¹
2 2
2
4
4
direct rev
o o
p p
Q
W
c c r R
α
ρ ρ π
| | | |
¦ ¹
| |
+ = +
´ `
| |
¹ )
\ ¹ \ ¹
10
2
4
10 log
4
p w
Q
L L
r R
α
π
¦ ¹
= + +
´ `
¹ )
Room acoustics