the decibel scale

1. The Decibel Scale
Brandon Morton PHYS 101 (L2I)

2. The Ear
 The quietest sound the ear can detect has an intensity
of 10-12 W/m2
 The loudest sound the ear can detect has an intensity
of 10 W/m2
 Due to this large range (1013 W/m2), it is convenient to
use a logarithmic scale
 The decibel is a common unit for logarithmic
comparison of power or intensity

3. Reference Point
 When using a logarithmic scale we need to define a
reference point.
 Since the lowest intensity the human ear can detect is
10-12 W/m2, we will set that to 0 decibels (I0 = 10-12
W/m2)
 To express any sound intensity (β) in decibels we use:
β(I) = 0dB + 10 log10 (I/I0)

4. Sample Question
 A mosquito's buzz is often rated with a decibel rating of
40 dB. Normal conversation is often rated at 60 dB.
How many times more intense is normal conversation
compared to a mosquito's buzz?
 a. 2
 b. 20
 c. 100
 d. 200

5. Answer
 Since the decibel scale is logarithmic each time you
increase by 10dB the intensity increases by 10 times.
 In this example, the sound increases by 20 dB which
means the intensity will be 100 times more intense.
 This is because it is 102 = 10 x 10 = 100
 Giving an answer of C. 100

6. Another Question
 Given that prolonged exposure to sounds greater
than 85 decibels can cause permanent hearing
damage, do you need to wear ear protection when
relaxing at the gun range?
 Given that a gunshot from a 0.22 rim-fire rifle has an
intensity of approximately I = (2.5 ×1013)I0

7. Answer
 We need to put the intensity of I = (2.50 ×1013)I0 into
the equation: β(I) = 0dB + 10 log10 (I/I0)
 β(I) = 10 log10 ((2.5 ×1013)I0 /I0)
 = 10 log10 (2.5 ×1013)
 = 133.979 dB
 = 1.34 x 102 dB

8. Answer Continued
 This means that at the rifle range I would be subjected
to a noise level of approximately 134 dB.
 Since this is significantly above the 85 dB level,
Hearing loss will occur.
 Therefore, I should wear the ear protection.

9. Bibliography
 http://www.cardinalspellman.org/ourpages/auto/2014/2/
13/40259160/Answers%2010%20Sound.pdf (accessed
February 21st 2015)
 http://logsanddecibles.blogspot.ca/ (Accessed February
21st 2015)