# 17: Physics of Hearing (Exercises)

# Conceptual Questions

Exercise \(\PageIndex{1}\)

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?

Exercise \(\PageIndex{2}\)

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

Exercise \(\PageIndex{1}\)

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

**Solution:**

**\(3.16 \times 10^{-4} \space W/m^2\)**

Exercise \(\PageIndex{2}\)

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?

Exercise \(\PageIndex{3}\)

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

**Solution:**

**\(3.04 \times 10^{-4} \space W/m^2\)**

Exercise \(\PageIndex{4}\)

What sound intensity level in dB is produced by earphones that create an intensity of \(4.00 \times 10^{-2} \space W/m^2\)?

**Solution:**

**106 dB**

Exercise \(\PageIndex{5}\)

Show that an intensity of \(10^{-12} W/m^2\) is the same as \(10^{-16} W/m^2\).

Exercise \(\PageIndex{6}\)

(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?

**Solution:**

**(a) 93 dB**

**(b) 83 dB**

Exercise \(\PageIndex{7}\)

(a) What is the intensity of a sound that has a level 7.00 dB lower than a \(4.00 \times 10^{-9} W/m^2\) sound? (b) What is the intensity of a sound that is 3.00 dB higher than a \(4.00 \times 10^{-9} W/m^2\) sound?

Exercise \(\PageIndex{8}\)

(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?

**Solution:**

**(a) 50.1**

**(b) \(5.01 \times 10^{-3}\) or \(\frac{1}{200}\)**

Exercise \(\PageIndex{9}\)

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?

Exercise \(\PageIndex{10}\)

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?

**Solution:**

**70.0 dB**

Exercise \(\PageIndex{11}\)

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?

Exercise \(\PageIndex{12}\)

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?

**Solution:**

**100**

Exercise \(\PageIndex{13}\)

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

Exercise \(\PageIndex{14}\)

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?

**Solution:**

**\(1.45 \times 10^{-3} J\)**

Exercise \(\PageIndex{15}\)

(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 \(900 \space cm^2\) and the area of the eardrum is \(0.500 \space cm^2\), 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).

Exercise \(\PageIndex{16}\)

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 \(15.0 \space cm^2\) and concentrates the sound onto two eardrums with a total area of \(0.900 \space cm^2\) with an efficiency of 40.0%?

**Soltuion:**

**28.2 dB**

Exercise \(\PageIndex{17}\)

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.)

# Conceptual Questions

Exercise \(\PageIndex{1}\)

Is the Doppler shift real or just a sensory illusion?

Exercise \(\PageIndex{2}\)

Due to efficiency considerations related to its bow wake, the supersonic transport aircraft must maintain a cruising speed that is a constant ratio to the speed of sound (a constant Mach number). If the aircraft flies from warm air into colder air, should it increase or decrease its speed? Explain your answer.

Exercise \(\PageIndex{3}\)

When you hear a sonic boom, you often cannot see the plane that made it. Why is that?

# Problems & Exercises

Exercise \(\PageIndex{1}\)

(a) What frequency is received by a person watching an oncoming ambulance moving at 110 km/h and emitting a steady 800-Hz sound from its siren? The speed of sound on this day is 345 m/s.

(b) What frequency does she receive after the ambulance has passed?

**Solution:**

**(a) 878 Hz**

**(b) 735 Hz**

Exercise \(\PageIndex{2}\)

(a) At an air show a jet flies directly toward the stands at a speed of 1200 km/h, emitting a frequency of 3500 Hz, on a day when the speed of sound is 342 m/s. What frequency is received by the observers?

(b) What frequency do they receive as the plane flies directly away from them?

Exercise \(\PageIndex{3}\)

What frequency is received by a mouse just before being dispatched by a hawk flying at it at 25.0 m/s and emitting a screech of frequency 3500 Hz? Take the speed of sound to be 331 m/s.

**Solution:**

**\(3.79 \times 10^3 \space Hz\)**

Exercise \(\PageIndex{4}\)

A spectator at a parade receives an 888-Hz tone from an oncoming trumpeter who is playing an 880-Hz note. At what speed is the musician approaching if the speed of sound is 338 m/s?

Exercise \(\PageIndex{5}\)

A commuter train blows its 200-Hz horn as it approaches a crossing. The speed of sound is 335 m/s.

(a) An observer waiting at the crossing receives a frequency of 208 Hz. What is the speed of the train?

(b) What frequency does the observer receive as the train moves away?

**Solution:**

**(a) 12.9 m/s**

**(b) 193 Hz**

Exercise \(\PageIndex{6}\)

Can you perceive the shift in frequency produced when you pull a tuning fork toward you at 10.0 m/s on a day when the speed of sound is 344 m/s? To answer this question, calculate the factor by which the frequency shifts and see if it is greater than 0.300%.

Exercise \(\PageIndex{7}\)

Two eagles fly directly toward one another, the first at 15.0 m/s and the second at 20.0 m/s. Both screech, the first one emitting a frequency of 3200 Hz and the second one emitting a frequency of 3800 Hz. What frequencies do they receive if the speed of sound is 330 m/s?

**Solution:**

**First eagle hears \(4.23 \times 10^3 \space Hz\)**

**Second eagle hears \(3.56 \times 10^3 \space Hz\)**

Exercise \(\PageIndex{8}\)

What is the minimum speed at which a source must travel toward you for you to be able to hear that its frequency is Doppler shifted? That is, what speed produces a shift of 0.300% on a day when the speed of sound is 331 m/s?