The Decibel

1
The Decibel

• or, everything you ever wanted to know about
  acoustic amplitude, but were too sane to ask

2
Intensity
amount of energy transmitted per second over an
area of one square meter

• Absolute vs Relative Scale

length
temperature
100o
212o
373o
Boiling point of water
Celsius
Fahrenheit
Kelvin
0o
32o
273o
Freezing point of water
Absolute scale
Relative scale
3
Intensity
amount of energy transmitted per second over an
area of one square meter

• Absolute vs Relative Scale

• Ratio
• Intensity reference 10-12 watts/m2

• Logarithmic

Ix
10-12 w/m2
102 w/m2
(loudest)
log
Ir
10-12 w/m2
(softest)
100,000,000,000,000
Range 1 to
4
Intensity
amount of energy transmitted per second over an
area of one square meter

• Linear vs. Logarithmic scale

1000
200
100
20
10
5
4
3
2
1
2
5
3
4
8
10
11
12
13
14
1
15
7
6
9
5
Intensity
amount of energy transmitted per second over an
area of one square meter

• Absolute vs Relative Scale

• Ratio
• Intensity reference 10-12 watts/m2

• Logarithmic

Ix
10-12 w/m2
102 w/m2
(loudest)
log
Bels
Ir
10-12 w/m2
(softest)
100,000,000,000,000
Range 1 to
14
0 to
Bels
6
Intensity
amount of energy transmitted per second over an
area of one square meter

• Absolute vs Relative Scale

• Ratio
• Intensity reference 10-12 watts/m2

• Logarithmic

Ix
102 w/m2
(loudest)
log
10
dB IL
Ir
10-12 w/m2
(softest)
100,000,000,000,000
Range 1 to
14
0 to
Bels
0 to 140
deciBels
7
Intensity
amount of energy transmitted per second over an
area of one square meter

• A sound is measured to have 10-11 w/m2 of sound
  intensity. How many dB IL is that?

10-11
10-12
dB IL 10 x 1
10 dB IL
8
Intensity
amount of energy transmitted per second over an
area of one square meter

• A sound is measured to have 10-10 w/m2 of sound
  intensity. How many dB IL is that?

10-10
10-12
dB IL 10 x 2
20 dB IL
9
Intensity
amount of energy transmitted per second over an
area of one square meter

• A sound is measured to have 2x10-10 w/m2 of sound
  intensity. How many dB IL is that?

2x10-10
10-12
dB IL 10 (log 2 log 100)
10 (0.3 2) 10 (2.3) 23 dB IL
10
Intensity
amount of energy transmitted per second over an
area of one square meter

• A sound is measured to have 8x10-5 w/m2 of sound
  intensity. How many dB IL is that?

8x10-5
10-12
dB IL 10 x 7.9
79 dB IL
11
Intensity
amount of energy transmitted per second over an
area of one square meter

• A sound is measured at 10 dB IL. How many
  watts/m2 is that?

12
Intensity
amount of energy transmitted per second over an
area of one square meter

• A sound is measured at 83 dB IL. How many
  watts/m2 is that?

Note 2x10-4 10-3.7 0.0002
13
Intensity (types of questions)
amount of energy transmitted per second over an
area of one square meter

• A sound has 4.75 x 10-8 watts/m2 of sound
  intensity. How many dB IL is that?
• A sound is measured at 42.8 dB IL. How many
  watts/m2 is that?
• A sound has 1000 times the intensity of the
  reference. How many dB IL is that?

14
Pressure
amount of force per unit area

• Absolute versus Relative
• Ratio
• Pressure reference 0.00002 Pascals (20 mPa)
• Logarithmic

I P2
log ab b log a
2
dB SPL
15
Pressure
amount of force per unit area

• Absolute versus Relative
• Ratio
• Pressure reference 0.00002 Pascals (20 mPa)
• Logarithmic

Px
200 Pa
(loudest)
log
20
dB SPL
Pr
20 mPa
(softest)
10,000,000
Range 1 to
7
0 to
Bels
0 to 140
deciBels
16
Pressure
amount of force per unit area

• A sound is measured to have 0.0002 Pa of sound
  pressure. How many dB SPL is that?

dB SPL 20 x 1
20 dB SPL
17
Pressure
amount of force per unit area

• A sound is measured to have 5 mPa of sound
  pressure. How many dB SPL is that?

dB SPL 20 x 2.4
48 dB SPL
18
Pressure
amount of force per unit area

• A sound is measured at 76 dB SPL. How many
  Pascals is that?

19
Pressure (types of questions)
amount of force per unit area

• A sound has 4.75 mPa of sound pressure. How many
  dB SPL is that?
• A sound is measured at 42.8 dB SPL. How many
  Pascals is that?
• A sound has 1000 times the pressure of the
  reference. How many dB SPL is that?

20
Intensity and Pressure

I P2
dB IL dB SPL
Therefore
watts/m2 m pascals
but
21
Adding Decibels

• Cannot directly add dB
• e.g., 72 dB 72 dB is NOT 144 dB
• Can directly add intensities (w/m2) or pressures
  (Pa)
• e.g., 5 Pa 5 Pa IS 10 Pa
• and, 10-7 w/m2 10-7 w/m2 IS 2 x 10-7 w/m2
• Therefore, when adding, convert dB into w/m2 or
  Pa, and then convert back to dB

So, how do you know whether to add intensities
(w/m2) or pressures (Pa)?
22
Adding Pressure or Intensity?

• If two sounds are phase-locked (same frequency
  and starting phase), then add Pressure.
• e.g., Perfect Summation

23
Adding Pressure or Intensity?

• If two sounds are NOT phase-locked (different
  frequency and/or starting phase), then add
  Intensity.
• e.g., Imperfect Summation

24
Adding Decibels (of equal pressure)
(0.02 Pa)

• Sound A is 60 dB SPL
• Double the pressure of sound A ( add same
  amplitude, frequency, phase)
• What is the combined dB SPL when pressure is
  doubled?

Doubling of pressure 6 dB increase
dB SPL 20 x 3.3
66 dB SPL
25
Adding Pressure or Intensity?

• If two sounds are identical (same amplitude,
  frequency and starting phase), then double
  Pressure.
• e.g., Perfect Summation

6 dB
6 dB
26
Adding Decibels (of equal intensity)
(10-6 w/m2)

• Sound A is 60 dB IL
• Sound B is 60 dB IL
• What is the combined dB IL when both sounds are
  on?

Intensity is doubled
(10-6 w/m2)
Doubling of intensity 3 dB increase
dB IL 10 x 6.3
63 dB IL
27
Adding Pressure or Intensity?

• If two sounds are the same amplitude but NOT
  phase-locked (different frequency and/or starting
  phase), then double Intensity.
• e.g., Imperfect Summation

3 dB
3 dB
28
Adding Decibels (of equal amplitude)

• Sound A is 60 dB SPL
• Sound B is 60 dB SPL
• What is the combined dB SPL (when both sounds are
  on)?

63 dB IL
Doubling of intensity

• Notes
• dB IL dB SPL
• Are they phase-locked?

29
Adding Decibels (of unequal amplitude)

• Sound A has 9 x 10-7 w/m2 of intensity (59.5 dB
  IL).
• Sound B has 6.4 x 10-7 w/m2 of intensity (58.1 dB
  IL).
• What is the combined dB IL?

dB IL 10 x 6.2
61.9 dB IL
30
Adding Decibels (of unequal amplitude)

• Sound A has 0.09 Pa of pressure (73 dB SPL).
• Sound B has 0.13 Pa of pressure (76 dB SPL).
• What is the combined dB SPL if pressures add
  (i.e., sound A and sound B are phase-locked)?

dB SPL 20 x 4.04
81 dB SPL
31
Adding Decibels (of unequal amplitude)

• Sound A has 4 x 10-4 w/m2 of intensity (86 dB
  IL).
• Sound B has 7 x 10-7 w/m2 of intensity (58.5 dB
  IL).
• What is the combined dB SPL?

dB IL 10 x 8.6
86 dB IL 86 dB SPL
32
Adding Decibels (of unequal amplitude)

• Sound A is measured at 45.5 dB SPL.
• Sound B is measured at 43.2 dB SPL.
• What is the combined dB SPL?

A
B
AB
dB IL 10 log (5.64x104)
dB IL 10 x 4.75
47.5 dB IL
10-12 x 104.55 Ix 3.55 x 10-8w/m2
10-12 x 104.32 Ix 2.09 x 10-8w/m2
33
Adding Decibels (of unequal amplitude)

• Sound A has 0.09 Pa of pressure.
• Sound B has 0.13 Pa of pressure.
• What is the combined dB SPL?

A
B
AB
0.13 Pa 76 dB SPL
0.09 Pa 73 dB SPL
76 dB SPL 76 dB IL
73 dB SPL 73 dB IL
77.8 dB IL 77.8 dB SPL
76 dB IL 4.0 x 10-5 w/m2
73 dB IL 2.0 x 10-5 w/m2
34
Adding Decibels

• The JMU music school has a brass ensemble that
  consists of a trumpet (83.4 dB SPL), french horn
  (78.1 dB SPL), trombone (85.0 dB SPL), and a tuba
  (86.3 dB SPL). What is the level of the music
  when the entire ensemble is playing together?

trumpet
All together
83.4 dB SPL 2.19 x 10-4 w/m2
1.03 x 10-3 w/m2
french horn
90.1 dB SPL
78.1 dB SPL 6.46 x 10-5 w/m2
trombone
85.0 dB SPL 3.16 x 10-4 w/m2
tuba
86.3 dB SPL 4.27 x 10-4 w/m2
35
Adding Decibels

• You are attending an outdoor lecture held in a
  temporary shelter with a tin roof. Of course,
  you take your sound level meter with you. You
  measure the speaker at 77.3 dB SPL. It then
  starts to rain, which makes a terrible racket.
  You measure the speaker together with the rain at
  78.9 dB SPL. What was the level of the rain?

rain alone
speaker
77.3 dB SPL 5.37 x 10-5 w/m2
2.39 x 10-5 w/m2
speaker rain
73.8 dB SPL
78.9 dB SPL 7.76 x 10-5 w/m2
36
Modes of Vibration
37
Modes of Vibration
38
Inverse Square Law

• Sound Intensity varies inversely with the square
  of the distance from the sound.

39
Inverse Square Law

• Marv is standing 8 feet from a sound source and
  measures a sound to be 80 dB SPL. Del is
  standing 32 feet from the sound source. How many
  dB SPL would the sound be when it arrives at Del?

Del
32 feet
? dB SPL
8 feet
Marv
80 dB SPL
40
Inverse Square Law
2 miles
200 feet
80 dB SPL
? dB SPL