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Intensity and Pitch

The intensity of a wave is a measure of the amount of energy a wave transports. If there is a stereo speaker producing music on one side of the room, then sound waves transport energy across the room. On the other side of the room we hear the sound with a certain intensity, so that a certain amount of energy per time falls on an ear. A person with bigger ears would have a proportionally greater energy per time falling on them, so a sensible definition of intensity is energy per time per area:

The units are [W/m2]. (See Figure 12-2.)

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Figure 12-2

The human ear can hear sounds from the barely perceptible rush of air at intensity 10–12 W/m2 to the painful roar at intensity 1 W/m2. In order to make these numbers correspond more closely to our perception of sound, we often convert intensity into decibels:
where I0 is the intensity 10–12 W/m2.
Note that an increase by a factor of 10 in intensity I corresponds to adding 10 to β, which is in decibels.


The chart shows some sample calculations.
Description I(W/m2) I/I0  Log I/I0 β (decibels)
rush of air 10–12  1 0 0
wind 10–9 103 3 30
conversation 10–6  106   6 60
water fall 10–3   109  9 90
pain   1     1012  12 120



A loud argument takes place in the next room, and you hear 70 dB. How much energy lands on one ear in one second? (An ear is about 0.05 m by 0.03 m.)



The intensity is given by



= 10-5 W/m2 (0.05 0.03 m)(1 s)
1.5 x 10–8 J.


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Figure 12-3


As you go further from a source of sound, the intensity of the sound decreases. In order to figure out how much it decreases, think of an alarm clock in the center of two concentric spheres (Figure 12-3). The alarm clock produces a certain amount of energy each second. That same amount of energy flows out of sphere A each second, and the same amount flows out of sphere B each second. Thus we have
[power going through surface A] = [power going through surface B], PA = PB
If there is a woman listening at radius A, the intensity she experiences is


because the surface area of the sphere is 4πrA2. Similarly, a man at radius rB experiences intensity


Putting this all together yields
or, in general,
In words, the intensity decreases as the square of the radius (inverse square law).
Memorize the formula, but also understand the reasoning that led to the formula.


Jack and Jill are in a field, and Jill is playing a violin. If the intensity of the sound Jack hears is β0 (in decibels) when he is 63.2 m away, how much louder does she sound when he is 2 m away?



If Jack moves from 63.2 m to 2 m, then the radius decreases by a factor of 63.2/2 = 31.6. Then equation (3) indicates that I increases by a factor of (31.6)2 = 1000. Three factors of 10 is equivalent to adding 10 to β three times, so β = β0 + 30. The violin sounds 30 decibels louder.

The pitch you hear depends on the frequency of the sound wave; the lower the frequency, the lower the note. For instance, the wave in Figure 12-4, with period T = 0.8 ms, corresponds to D#6. The wave in Figure 12-5, with period double the first one, corresponds to D#7, the same note one octave higher.

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Figure 12-4
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Figure 12-5

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