16.8.1 Sound Intensity Level

Human hearing spans an extremely wide range of intensities, from the threshold of hearing at $\approx 1 \times 10^{- 12} {W/m}^{2}$ (at midrange frequencies) to the threshold of pain at $\approx 10 {W/m}^{2}$ . If we want to make a scale of loudness, it’s convenient and logical to place the zero of our scale at the threshold of hearing. To do so, we define the sound intensity level, expressed in decibels (dB), as

β = (10 dB) \log_{10} (\frac{I}{I_{0}})

(16.61)

where $I_{0} = 1.0 \times 10^{- 12} {W/m}^{2}$ . The symbol $β$ is the Greek letter beta. Notice that $β$ is computed as a base-10 logarithm, not a natural logarithm.

The decibel is named after Alexander Graham Bell, inventor of the telephone. Sound intensity level is actually dimensionless because it’s formed from the ratio of two intensities, so decibels are just a name to remind us that we’re dealing with an intensity level rather than a true intensity.

Right at the threshold of hearing, where $I = I_{0}$ , the sound intensity level is

β = (10 dB) \log_{10} (\frac{I_{0}}{I_{0}}) = (10 dB) \log_{10} (1) = 0 dB

Note that 0 dB doesn’t mean no sound; it means that, for most people, no sound is heard. Dogs have more sensitive hearing than humans, and most dogs can easily perceive a 0 dB sound. The sound intensity level at the pain threshold is

β = (10 dB) \log_{10} (\frac{10 {W/m}^{2}}{10^{- 12} {W/m}^{2}}) = (10 dB) \log_{10} (10^{13}) = 130 dB

The major point to notice is that the sound intensity level increases by 10 dB each time the actual intensity increases by a factor of 10. For example, the sound intensity level increases from 70 dB to 80 dB when the sound intensity increases from $10^{- 5} {W/m}^{2}$ to $10^{- 4} {W/m}^{2}$ . Perception experiments find that sound is perceived as “twice as loud” when the intensity increases by a factor of 10. In terms of decibels, we can say that the perceived loudness of a sound doubles with each increase in the sound intensity level by 10 dB.

TABLE 16.4 gives the sound intensity levels for a number of sounds. Although 130 dB is the threshold of pain, quieter sounds can damage your hearing. A fairly short exposure to 120 dB can cause damage to the hair cells in the ear, but lengthy exposure to sound intensity levels of over 85 dB can produce damage as well.

TABLE 16.4 Sound intensity levels of common sounds

Sound	$β$ (dB)
Threshold of hearing	0
Person breathing, at 3 m	10
A whisper, at 1 m	20
Quiet room	30
Outdoors, no traffic	40
Quiet restaurant	50
Normal conversation, at 1 m	60
Busy traffic	70
Vacuum cleaner, for user	80
Niagara Falls, at viewpoint	90
Snowblower, at 2 m	100
Stereo, at maximum volume	110
Rock concert	120
Threshold of pain	130

EXAMPLE 16.10 Blender noise

The blender making a smoothie produces a sound intensity level of 83 dB. What is the intensity of the sound? What will the sound intensity level be if a second blender is turned on?

SOLVE We can solve Equation 16.61 for the sound intensity, finding $I = I_{0} \times 10^{β / 10 dB}$ . Here we used the fact that 10 raised to a power is an “antilogarithm.” In this case,

I = (1.0 \times 10^{- 12} {W/m}^{2}) \times 10^{8.3} = 2.0 \times 10^{- 4} {W/m}^{2}

A second blender doubles the sound power and thus raises the intensity to $I = 4.0 \times 10^{- 4} {W/m}^{2}$ . The new sound intensity level is

β = (10 dB) \log_{10} (\frac{4.0 \times 10^{- 4} {W/m}^{2}}{1.0 \times 10^{- 12} {W/m}^{2}}) = 86 dB

REVIEW In general, doubling the actual sound intensity increases the decibel level by 3 dB.