128
5 Acoustics in Biology and Medicine
The loudness scale is often given in ‘decibels’. Because β is an exponent,
loudness has no physical units. Rather, the name decibel is a reminder of how the
sound level is defined. The originally appended name for the loudness intervals of
β/10 were called ‘bels’, named after Alexander Graham Bell for his work on the
audiometer, a hearing-test instrument. By including the factor of ten, the loudness
β has a convenient range from about 0 dB to about 120 dB, which is between barely
audible and a level which can produce damage to the ears.
The threshold sound intensity measured and averaged over undamaged and
healthy young ears comes to about I th = 10 −12 W/m 2 for frequencies of 1000 Hz in
air at 25 ◦ C and 1 atm. This is the intensity adopted as the threshold of hearing.
An average size mosquito makes this sound intensity at 3 m from the ear, with
wing vibrations at about 600 Hz plus harmonics. 13 Now, to find to what pressure
the threshold intensity corresponds, use Eq. (5.33) to form the time averages
I =
1
2ρ 0 v
(δp)
2
which gives a root-mean-square (RMS) threshold pressure for human hearing as
p th ≡
(δp) 2 =
2ρ 0 vI =
ρ 0 vI th ,
(5.36)
where we have used the fact that for a sinusoidal wave, the maximum value of the
intensity is twice its time average. The density of air at 25 ◦ C and 1 atm is ρ 0 =
1.184 kg/m 3 and the speed of sound at this temperature and density is v = 346 m/s.
This makes
p th = 20.24 × 10
−6 N/m
2
≈ 20 μPa .
A comparison of sound wave amplitude, pressure, intensity, and loudness is given
in Table 5.3. Note: If your living room has a background sound level of 45 dB, you
Table 5.3 Conversion of sound amplitude, pressure, intensity, and loudness
Amplitude (cm) Pressure (dyne/cm 2 ) Intensity (W/m 2 ) Loudness (dB)
Example
0.07
200
1
120
Jet exhaust
0.007
20
10 −2
100
Siren
7×10 −4
2
1 0 −4
80
Loud radio
7×10 −5
0.2
10 −6
60
Loud speech
7×10 −6
0.02
10 −8
40
Soft speech
7×10 −7
0.002
10 −10
20
Country home
7×10 −8
0.0002
10 −12
0
Q u i e t
13 Male and female mosquitoes find each other by adjusting these harmonics. Humans can do this
too, but not as well.
5 Acoustics in Biology and Medicine
The loudness scale is often given in ‘decibels’. Because β is an exponent,
loudness has no physical units. Rather, the name decibel is a reminder of how the
sound level is defined. The originally appended name for the loudness intervals of
β/10 were called ‘bels’, named after Alexander Graham Bell for his work on the
audiometer, a hearing-test instrument. By including the factor of ten, the loudness
β has a convenient range from about 0 dB to about 120 dB, which is between barely
audible and a level which can produce damage to the ears.
The threshold sound intensity measured and averaged over undamaged and
healthy young ears comes to about I th = 10 −12 W/m 2 for frequencies of 1000 Hz in
air at 25 ◦ C and 1 atm. This is the intensity adopted as the threshold of hearing.
An average size mosquito makes this sound intensity at 3 m from the ear, with
wing vibrations at about 600 Hz plus harmonics. 13 Now, to find to what pressure
the threshold intensity corresponds, use Eq. (5.33) to form the time averages
I =
1
2ρ 0 v
(δp)
2
which gives a root-mean-square (RMS) threshold pressure for human hearing as
p th ≡
(δp) 2 =
2ρ 0 vI =
ρ 0 vI th ,
(5.36)
where we have used the fact that for a sinusoidal wave, the maximum value of the
intensity is twice its time average. The density of air at 25 ◦ C and 1 atm is ρ 0 =
1.184 kg/m 3 and the speed of sound at this temperature and density is v = 346 m/s.
This makes
p th = 20.24 × 10
−6 N/m
2
≈ 20 μPa .
A comparison of sound wave amplitude, pressure, intensity, and loudness is given
in Table 5.3. Note: If your living room has a background sound level of 45 dB, you
Table 5.3 Conversion of sound amplitude, pressure, intensity, and loudness
Amplitude (cm) Pressure (dyne/cm 2 ) Intensity (W/m 2 ) Loudness (dB)
Example
0.07
200
1
120
Jet exhaust
0.007
20
10 −2
100
Siren
7×10 −4
2
1 0 −4
80
Loud radio
7×10 −5
0.2
10 −6
60
Loud speech
7×10 −6
0.02
10 −8
40
Soft speech
7×10 −7
0.002
10 −10
20
Country home
7×10 −8
0.0002
10 −12
0
Q u i e t
13 Male and female mosquitoes find each other by adjusting these harmonics. Humans can do this
too, but not as well.