Sound Pressure and Intensity
Intensity =
Power
Surface Area
measured in units of
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W
m2
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Intensity of Audible Sound
Audible range depends on frequency
For f = 1kHz:
10-12 W/m2
1 W/m2
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(just barely audible)
(painfully loud)
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1
Inverse Square Law
Inverse square law governs how the sound intensity
decreases with distance from the sound source.
Intensity decreases as
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r2
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Example with Numbers
• Sound source of power 1 W at a
distance of 1 m radiates equally in all
directions
• Sphere with radius 1 m has a surface
area of S = 4πr2 = 12.5 m2
• Sound intensity at 1 m distance is
therefore I = 1 W/12.5 m2 = 0.08 W/m2
• Scientific notation: I=800 x 10-4 W/m2
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2
Example with Numbers
• Now we double our distance:
2 m away from sound source
• Sphere with radius 2 m has a surface
area of S = 4πr2 = 50 m2 (4x as much)
• Sound intensity at 2 m distance is
therefore
I = 1 W/50 m2 = 200 x 10-4 W/m2
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Example with Numbers
• Now we double our distance again:
4 m away from sound source
• Sphere with radius 4 m has a surface
area of S = 4πr2 = 200 m2
• Sound intensity at 4 m distance is
therefore
I = 1 W/201 m2 = 50 x 10-4 W/m2
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3
Inverse Square Law
Distance
[m]
1
Intensity 800
2
4
8
200
50
12.5
1/4
1/16
1/64
[10-4W/m2]
1/r2
1
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Definition of the “Bel”
The log of the ratio of two intensities is expressed in units of Bel
⎛I ⎞
log ⎜ 1 ⎟ = # B
⎝ I2 ⎠
For example:
I1=100 x 10-5 W/m2
I2= 6 x 10-5 W/m2
⎛ 100 × 10 −5 W / m 2 ⎞
⎛ 100 ⎞
log ⎜
= log ⎜
= log17 = 1.2B
⎝ 6 × 10 −5 W / m 2 ⎟⎠
⎝ 6 ⎟⎠
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Definition of the “deciBel”
1 deci stands for “0.1”, therefore 1 deciBel = 0.1 Bel
or 10 deciBel = 1 Bel
⎛I ⎞
10 log ⎜ 1 ⎟ = # dB
⎝ I2 ⎠
−5
⎛ 100 × 10 −5
W / m 22 ⎞
⎛ 100 ⎞
10 log ⎜
= 10 log17 = 1.2
12dB
= 10 log ⎜
B
⎟
−5
2
−5
2
⎝ 6 ⎟⎠
⎝ 6 × 10 W / m ⎠
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Sound Intensity Level
⎛I ⎞
SIL = 10 log ⎜ 1 ⎟
⎝ I2 ⎠
(a number with the unit dB)
The dB-scale is a relative scale
SIL in dB tells how much more intense I1 is compared to I2
Example:example:
Another
What does
What
90 does
dB mean?
0 dB mean?
900 = 10 log(I11/I
/I22)
9 = log(I1/I2)
0
1009 = I1/I2
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or
I1=
=10
I2 9 x I2
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Reference Intensity
The dB-scale is a relative scale, i.e. we must always have
two intensities to compare to each other.
Often (but not always!) the reference intensity is the
threshold intensity
Io = 10-12 W/m2
Example: What is the SIL of the threshold of pain?
(I = 1 W/m2)
⎛ 1W / m 2 ⎞
⎛ 1 ⎞
12
10 log ⎜ −12
=
10
log
⎟
⎜⎝ −12 ⎟⎠ = 10 log(10 ) = 120dB
⎝ 10 W / m 2 ⎠
10
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Using the 10-12 W/m2 Reference
A sound source makes a sound intensity of I=1x10-3W/m2.
What is the SIL?
⎛ 10 −3 ⎞
SIL = 10 log ⎜ −12 ⎟ = 10 log(10 −3 × 1012 ) = 10 log(10 −3+12 )
⎝ 10 ⎠
SIL = 10 log(10 9 ) = 90dB
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Using the 10-12 W/m2 Reference
A sound source makes a sound intensity of I=6x10-3W/m2.
What is the SIL?
⎛ 6 × 10 −3 ⎞
SIL = 10 log ⎜
= 10 log(6 × 10 −3 × 1012 ) = 10 log(6 × 10 9 )
⎝ 10 −12 ⎟⎠
SIL = 10 ( log(6 × 10 9 )) = 10 ( log(6) + log(10 9 )) = 10(0.8 + 9) = 98dB
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What is twice as intense?
A sound source makes a sound intensity of I = 1 x 10-5 W/m2.
A second source makes the same sound intensity.
It is reasonable to say that both together are making a sound
that is twice as intense as one source by itself.
What is the SIL for one and for both sources?
⎛ 10 −5 ⎞
SIL(one) = 10 log ⎜ −12 ⎟ = 10 log(10 7 ) = 70dB
⎝ 10 ⎠
⎛ 2 × 10 −5 ⎞
SIL(both) = 10 log ⎜
= 10 log(2 × 10 7 ) = 10 ( log 2 + log(10 7 ))
⎝ 10 −12 ⎟⎠
= 10 log 2 + 10 log(10 7 ) = 3 + 70 = 73dB
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What is four times as intense?
A sound source makes a sound intensity of I.
Now we turn on three more of these sources to get four times
the sound intensity of one.
What is the SIL for one, two, and for all four sources?
⎛ I⎞
SIL(one) = 10 log ⎜ ⎟ = some number of dB
⎝ I0 ⎠
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What is four times as intense?
What is the SIL for two sources?
⎛ I⎞
SIL(one) = 10 log ⎜ ⎟ = some number of dB
⎝ I0 ⎠
I
I ⎞
⎛2× I⎞
⎛
SIL(two) = 10 log ⎜
=
10
log(2
×
)
=
10
log
2
+
log(
)
⎜⎝
I0
I 0 ⎟⎠
⎝ I 0 ⎟⎠
= 10 log 2 + 10 log(
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I
) = 3dB + whatever the number of dB for one
I0
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What is four times as intense?
What is the SIL for four sources?
⎛ I⎞
SIL(one) = 10 log ⎜ ⎟ = some number of dB
⎝ I0 ⎠
I
I ⎞
⎛4× I⎞
⎛
SIL( four) = 10 log ⎜
= 10 log(4 × ) = 10 ⎜ log 4 + log( )⎟
⎟
I0
I0 ⎠
⎝ I0 ⎠
⎝
= 10 log(2 × 2) + 10 log(
I
I
) = 10 log 2 + 10 log 2 + 10 log( )
I0
I0
= 3dB + 3dB + whatever the number of dB for one
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Sound Pressure Level
⎛p ⎞
SPL = 20 log ⎜ 1 ⎟
⎝ p2 ⎠
Sound intensity is related to sound pressure: p2 = I Zo
Zo is called the characteristic impedance of the medium and is
constant for a given medium (e.g. air) as long as the temperature
and the air pressure remain the same.
p12 I1Z o I1
=
=
p12 = I1Z o and p22 = I 2 Z o
p22 I 2 Z o I 2
⎛ p2 ⎞
⎛I ⎞
⎛p ⎞
SIL = 10 log ⎜ 1 ⎟ = 10 log ⎜ 12 ⎟ = 20 log ⎜ 1 ⎟
⎝ I2 ⎠
⎝ p2 ⎠
⎝ p2 ⎠
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9