Lesson 3 - Logs and Levels
Math Prereqs
y  10 then log10  y   x
x
log  xy  = log  x  + log  y 
x
log    log  x   log  y 
y
   n10 log x
10 log x
n
Examples
• Without using your calculator, find the following:
(log10(2) = 0.30)
• log10(10-3) =
• log10(1 x 1012) =
• log10(2 x 1012) =
• log10(200) =
• log10(200) - log10(10) =
• log10(210) =
-3, 12, 3.6, 2.3, 1.3, 3
Pitch is frequency
Audible
20 Hz – 20000 Hz
Infrasonic
< 20 Hz
Ultrasonic
>20000 Hz
Middle C on the piano has a frequency of 262 Hz.
What is the wavelength (in air)?
1.3 m
Intensity of sound
p2
2
p max
I 

c
2 c
• Loudness – intensity of the wave. Energy
transported by a wave per unit time across a unit
area perpendicular to the energy flow.
Source
Intensity (W/m2)
Sound Level
Jet Plane
100
140
Pain Threshold
1
120
Siren
1x10-2
100
Busy Traffic
1x10-5
70
Conversation
3x10-6
65
Whisper
1x10-10
20
Rustle of leaves
1x10-11
10
Hearing Threshold 1x10-12
1
Sound Level - Decibel
 I 
L  10log  
 I0 
I0  1x10
12
W
m2
Why the decibel?
• Ears judge loudness on a logarithmic vice
linear scale
• Alexander Graham Bell
• deci =
1
10
• 1 bel = 10 decibel
I
"bel"  log
I0
 I 
L(in dB)  10log  
 I0 
Reference Level Conventions
p02
I0 =
c
Location
Reference
Intensity
Reference
Pressure
Air
1 x 10-12 W/m2
20 mPa
Water
6.67 x 10-19 W/m2 1 uPa
Historical Reference
• 1 microbar
• 1 bar = 1 x 105 Pa
• 1 mbar = 1 x 105 mPa
 105 m Pa 
20log 
  100 dB
 1 m Pa 
• So to convert from intensity levels referenced to
1 mbar to intensity levels referenced to 1 mPa,
simply add 100 dB
Sound Pressure Level
Mean Squared Quantities:
Power, Energy, Intensity
 I 
L  10log  
 I0 
“Intensity Level”
Root Mean Squared Quantities:
Voltage, Current, Pressure
 p2
L  20 log 
 p 0


  20 log p rms

po

“Sound Pressure Level”
Subtracting Intensity Levels
I2
I1
L2  L1  10log
 10log
I0
I0
I2
L 2  L1  10 log
I1
L2  L1  20 log
p2
p1
Two Submarines
• If a noisy sub was
emitting a source level of
140 dB and a quiet sub
was emitting a source
level of 80 dB,
• What is the difference in
noise levels?
• what does this mean in
terms of relative intensity
and acoustic pressure?
Adding Levels
I1
L1 = 10log
I0
I2
and L 2 = 10log
I0
I tot
"L1  L 2 "  10 log
I0
but I tot  I1  I 2


Ltot  L1  L2  10log 10  10 


L1
10
L2
10
Total Noise
• On a particular day,
noise from shipping is
53 dB and noise from
rain and biologics is
50 dB. What is the
total noise level from
the two sources?
Adding Equal Noise Levels
• Two transducers are both transmitting a
source intensity level of 90 dB. What is the
total source intensity level?
If L1  L2 then LTotal  L1  L2  L1  3dB
Two Submarines
• If a noisy sub was
emitting a source level
of 140 dB and a quiet
sub was emitting a
source level of 80 dB,
what is the total noise
from the two
submarines?
Backups
Reference Values
p02
I0 =
c
 20 mPa 
kg m3   343
2
Air:
Water:
I0 
1.2
m s
 1x1012 W
m2
2

1 mPa 
19 W
I0 

6
.
67
x
10
m2
1000 kg m3 1500 m s
Sound Pressure Level
p2
2
I
c
L  10log
 10log 2  10log 2
p0
I0
p0
c
p
 p2
L  20 log 
 p 0

 p2
 10log 
 p0


  20 log p rms

po





2
Subtracting Intensity Levels
I2
I1
L 2  L1  10 log
 10 log
I0
I0
L 2  L1  10 log I 2  10 log I0  10 log I1  10 log I 0 
L 2  L1  10 log I 2  10 log I1
L 2  L1  10 log
I2
I1
p 22
L 2  L1  10 log
c
p12
c
L 2  L1  10 log
L 2  L1  20 log
p 22
p12
p2
p1
Addition
I tot
"L1  L 2 "  10 log
I0
but I tot  I1  I 2
I1
L1
 log
10
I0
L1
10
10 
I1
I0
similarly, 10
L2
10
I2

I0
therefore I1  I010
L1
10
and I 2  I010
L2
L1

 10
10
so I tot  I0 10  10 


 I tot 
L tot  L1  L 2  10 log 

 I0 
L2
L1

 10
10
L tot  10 log 10  10 


L2
10