# 17.3 Sound intensity  (Page 6/14)

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## Summary

• Intensity $I=P\text{/}A$ is the same for a sound wave as was defined for all waves, where P is the power crossing area A . The SI unit for I is watts per meter squared. The intensity of a sound wave is also related to the pressure amplitude $\Delta p\text{:}$
$I=\frac{{\left(\text{Δ}p\right)}^{2}}{2\phantom{\rule{0.2em}{0ex}}\rho v},$

where $\rho$ is the density of the medium in which the sound wave travels and ${v}_{w}$ is the speed of sound in the medium.
• Sound intensity level in units of decibels (dB) is
$\beta \left(\text{dB}\right)=10\phantom{\rule{0.2em}{0ex}}{\text{log}}_{10}\left(\frac{I}{{I}_{0}}\right),$

where ${I}_{0}={10}^{-12}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$ is the threshold intensity of hearing.
• The perception of frequency is pitch. The perception of intensity is loudness and loudness has units of phons.

## Conceptual questions

Six members of a synchronized swim team wear earplugs to protect themselves against water pressure at depths, but they can still hear the music and perform the combinations in the water perfectly. One day, they were asked to leave the pool so the dive team could practice a few dives, and they tried to practice on a mat, but seemed to have a lot more difficulty. Why might this be?

The ear plugs reduce the intensity of the sound both in water and on land, but Navy researchers have found that sound under water is heard through vibrations mastoid, which is the bone behind the ear.

A community is concerned about a plan to bring train service to their downtown from the town’s outskirts. The current sound intensity level, even though the rail yard is blocks away, is 70 dB downtown. The mayor assures the public that there will be a difference of only 30 dB in sound in the downtown area. Should the townspeople be concerned? Why?

## Problems

What is the intensity in watts per meter squared of a 85.0-dB sound?

The warning tag on a lawn mower states that it produces noise at a level of 91.0 dB. What is this in watts per meter squared?

$1.26\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{\text{−}3}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$

A sound wave traveling in air has a pressure amplitude of 0.5 Pa. What is the intensity of the wave?

What intensity level does the sound in the preceding problem correspond to?

85 dB

What sound intensity level in dB is produced by earphones that create an intensity of $4.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$ ?

What is the decibel level of a sound that is twice as intense as a 90.0-dB sound? (b) What is the decibel level of a sound that is one-fifth as intense as a 90.0-dB sound?

a. 93 dB; b. 83 dB

What is the intensity of a sound that has a level 7.00 dB lower than a $4.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-9}{\text{-W/m}}^{2}$ sound? (b) What is the intensity of a sound that is 3.00 dB higher than a $4.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-9}{\text{-W/m}}^{2}$ sound?

People with good hearing can perceive sounds as low as −8.00 dB at a frequency of 3000 Hz. What is the intensity of this sound in watts per meter squared?

$1.58\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{\text{−}13}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$

If a large housefly 3.0 m away from you makes a noise of 40.0 dB, what is the noise level of 1000 flies at that distance, assuming interference has a negligible effect?

Ten cars in a circle at a boom box competition produce a 120-dB sound intensity level at the center of the circle. What is the average sound intensity level produced there by each stereo, assuming interference effects can be neglected?

A decrease of a factor of 10 in intensity corresponds to a reduction of 10 dB in sound level: $120\phantom{\rule{0.2em}{0ex}}\text{dB}-10\phantom{\rule{0.2em}{0ex}}\text{dB}=110\phantom{\rule{0.2em}{0ex}}\text{dB}.$

The amplitude of a sound wave is measured in terms of its maximum gauge pressure. By what factor does the amplitude of a sound wave increase if the sound intensity level goes up by 40.0 dB?

If a sound intensity level of 0 dB at 1000 Hz corresponds to a maximum gauge pressure (sound amplitude) of ${10}^{-9}\phantom{\rule{0.2em}{0ex}}\text{atm}$ , what is the maximum gauge pressure in a 60-dB sound? What is the maximum gauge pressure in a 120-dB sound?

We know that 60 dB corresponds to a factor of ${10}^{6}$ increase in intensity. Therefore,
$I\propto {X}^{2}⇒\frac{{I}_{2}}{{I}_{1}}={\left(\frac{{X}_{2}}{{X}_{1}}\right)}^{2}\text{,}\phantom{\rule{0.2em}{0ex}}\text{so that}\phantom{\rule{0.2em}{0ex}}{X}_{2}={10}^{\text{−}6}\phantom{\rule{0.2em}{0ex}}\text{atm}.$
120 dB corresponds to a factor of ${10}^{12}$ increase $⇒{10}^{\text{−}9}\phantom{\rule{0.2em}{0ex}}\text{atm}{\left({10}^{12}\right)}^{\text{1/2}}={10}^{\text{−}3}\phantom{\rule{0.2em}{0ex}}\text{atm}.$

An 8-hour exposure to a sound intensity level of 90.0 dB may cause hearing damage. What energy in joules falls on a 0.800-cm-diameter eardrum so exposed?

Sound is more effectively transmitted into a stethoscope by direct contact rather than through the air, and it is further intensified by being concentrated on the smaller area of the eardrum. It is reasonable to assume that sound is transmitted into a stethoscope 100 times as effectively compared with transmission though the air. What, then, is the gain in decibels produced by a stethoscope that has a sound gathering area of $15.0\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2}$ , and concentrates the sound onto two eardrums with a total area of $0.900\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2}$ with an efficiency of $40.0\text{%}$ ?

28.2 dB

Loudspeakers can produce intense sounds with surprisingly small energy input in spite of their low efficiencies. Calculate the power input needed to produce a 90.0-dB sound intensity level for a 12.0-cm-diameter speaker that has an efficiency of $1.00\text{%}$ . (This value is the sound intensity level right at the speaker.)

The factor of 10 -12 in the range of intensities to which the ear can respond, from threshold to that causing damage after brief exposure, is truly remarkable. If you could measure distances over the same range with a single instrument and the smallest distance you could measure was 1 mm, what would the largest be?

$1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{6}\phantom{\rule{0.2em}{0ex}}\text{km}$

What are the closest frequencies to 500 Hz that an average person can clearly distinguish as being different in frequency from 500 Hz? The sounds are not present simultaneously.

Can you tell that your roommate turned up the sound on the TV if its average sound intensity level goes from 70 to 73 dB?

$73\phantom{\rule{0.2em}{0ex}}\text{dB}-70\phantom{\rule{0.2em}{0ex}}\text{dB}=3\phantom{\rule{0.2em}{0ex}}\text{dB};$ Such a change in sound level is easily noticed.

If a woman needs an amplification of $5.0\phantom{\rule{0.2em}{0ex}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{5}$ times the threshold intensity to enable her to hear at all frequencies, what is her overall hearing loss in dB? Note that smaller amplification is appropriate for more intense sounds to avoid further damage to her hearing from levels above 90 dB.

A person has a hearing threshold 10 dB above normal at 100 Hz and 50 dB above normal at 4000 Hz. How much more intense must a 100-Hz tone be than a 4000-Hz tone if they are both barely audible to this person?

2.5; The 100-Hz tone must be 2.5 times more intense than the 4000-Hz sound to be audible by this person.

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