# 17.3 Sound intensity and sound level  (Page 4/7)

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Describe how amplitude is related to the loudness of a sound.

Amplitude is directly proportional to the experience of loudness. As amplitude increases, loudness increases.

Identify common sounds at the levels of 10 dB, 50 dB, and 100 dB.

10 dB: Running fingers through your hair.

50 dB: Inside a quiet home with no television or radio.

100 dB: Take-off of a jet plane.

## Section summary

• Intensity is the same for a sound wave as was defined for all waves; it is

$I=\frac{P}{A},$
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 $\Delta p$
$I=\frac{{\left(\Delta p\right)}^{2}}{2{\text{ρv}}_{w}},$
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

$\beta \phantom{\rule{0.25em}{0ex}}\left(\text{dB}\right)=\text{10}\phantom{\rule{0.25em}{0ex}}{\text{log}}_{\text{10}}\left(\frac{I}{{I}_{0}}\right),$
where ${I}_{0}={10}^{–12}\phantom{\rule{0.25em}{0ex}}W/{\text{m}}^{2}$ is the threshold intensity of hearing.

## 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?

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&Exercises

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

$3.16×{10}^{–4}\phantom{\rule{0.25em}{0ex}}{\text{W/m}}^{2}$

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?

A sound wave traveling in $20ºC$ air has a pressure amplitude of 0.5 Pa. What is the intensity of the wave?

$3.04×{10}^{–4}\phantom{\rule{0.25em}{0ex}}{\text{W/m}}^{2}$

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

What sound intensity level in dB is produced by earphones that create an intensity of $4.00×{\text{10}}^{\text{−2}}\phantom{\rule{0.25em}{0ex}}{\text{W/m}}^{\text{2}}$ ?

106 dB

Show that an intensity of ${10}^{–12}\phantom{\rule{0.25em}{0ex}}{W/m}^{2}$ is the same as ${10}^{–16}\phantom{\rule{0.25em}{0ex}}{W/cm}^{2}$ .

(a) 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

(a) What is the intensity of a sound that has a level 7.00 dB lower than a $4.00×{10}^{–9}\phantom{\rule{0.25em}{0ex}}{W/m}^{2}$ sound? (b) What is the intensity of a sound that is 3.00 dB higher than a $4.00×{10}^{–9}\phantom{\rule{0.25em}{0ex}}{W/m}^{2}$ sound?

(a) How much more intense is a sound that has a level 17.0 dB higher than another? (b) If one sound has a level 23.0 dB less than another, what is the ratio of their intensities?

(a) 50.1

(b) $5.01×{10}^{–3}$ or $\frac{1}{200}$

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

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?

70.0 dB

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?

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?

100

If a sound intensity level of 0 dB at 1000 Hz corresponds to a maximum gauge pressure (sound amplitude) of ${10}^{–9}\phantom{\rule{0.25em}{0ex}}\text{atm}$ , what is the maximum gauge pressure in a 60-dB sound? What is the maximum gauge pressure in a 120-dB sound?

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?

$1.45×{10}^{–3}\phantom{\rule{0.25em}{0ex}}\text{J}$

(a) Ear trumpets were never very common, but they did aid people with hearing losses by gathering sound over a large area and concentrating it on the smaller area of the eardrum. What decibel increase does an ear trumpet produce if its sound gathering area is ${900 cm}^{2}$ and the area of the eardrum is ${0.500 cm}^{2}$ , but the trumpet only has an efficiency of 5.00% in transmitting the sound to the eardrum? (b) Comment on the usefulness of the decibel increase found in part (a).

Sound is more effectively transmitted into a stethoscope by direct contact 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 ${15.0 cm}^{2}$ , and concentrates the sound onto two eardrums with a total area of ${0.900 cm}^{2}$ with an efficiency of 40.0%?

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%. (This value is the sound intensity level right at the speaker.)

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