CollegePhysics2e-SSM-Ch17

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OpenStax College Physics 2e Student Solutions Manual Chapter 17 CHAPTER 17: PHYSICS OF HEARING 17.2 SPEED OF SOUND, FREQUENCY, AND WAVELENGTH 1. When poked by a spear, an operatic soprano lets out a 1200-Hz shriek. What is its wavelength if the speed of sound is 345 m/s? Solution , so that 3. Calculate the speed of sound on a day when a 1500 Hz frequency has a wavelength of 0.221 m. Solution 5. Show that the speed of sound in air is 343 m/s, as claimed in the text. Solution 7. Dolphins make sounds in air and water. What is the ratio of the wavelength of a sound in air to its wavelength in seawater? Assume air temperature is . Solution We know (from Table 17.1 ) and at from Problem 17.5 , since and , we know 9. (a) If a submarine’s sonar can measure echo times with a precision of 0.0100 s, what is the smallest difference in distances it can detect? (Assume that the submarine is in the ocean, not in fresh water.) (b) Discuss the limits this time resolution imposes on the ability of the sonar system to detect the size and shape of the object creating the echo. Solution (a) (b) This means that sonar is good for spotting and locating large objects, but it isn’t able to resolve smaller objects, or detect the detailed shapes of objects. Objects like ships or large pieces of airplanes can be found by sonar, while smaller

OpenStax College Physics 2e Student Solutions Manual Chapter 17 pieces must be found by other means. 11. Suppose a bat uses sound echoes to locate its insect prey, 3.00 m away. (See Figure 17.9 .) (a) Calculate the echo times for temperatures of and . (b) What percent uncertainty does this cause for the bat in locating the insect? (c) Discuss the significance of this uncertainty and whether it could cause difficulties for the bat. (In practice, the bat continues to use sound as it closes in, eliminating most of any difficulties imposed by this and other effects, such as motion of the prey.) Solution (a) (b) % uncertainty = (c) This uncertainty could definitely cause difficulties for the bat, if it didn’t continue to use sound as it closed in on its prey. A 5% uncertainty could be the difference between catching the prey around the neck or around the chest, which means that it could miss grabbing its prey. 17.3 SOUND INTENSITY AND SOUND LEVEL 12. What is the intensity in watts per meter squared of 85.0-dB sound? Solution 14. A sound wave traveling in air has a pressure amplitude of 0.5 Pa. What is the intensity of the wave? Solution

OpenStax College Physics 2e Student Solutions Manual Chapter 17 16. What sound intensity level in dB is produced by earphones that create an intensity of ? Solution 18. (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) From Table 17.3 , (b) Thus, 20. (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) One factor of 10 (10.0 dB) and one factor of 5 (7.0 dB) make an overall factor of 50.1 , i.e. (b) Two 10-decibel losses give two factors of , or a total factor of The remaining 3 decibels give an additional factor of . So the ratio is a factor of i.e. 22. 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 A factor of 1000 in intensity corresponds to an increase of 30.0 dB, i.e., . Thus,

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OpenStax College Physics 2e Student Solutions Manual Chapter 17 24. 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 The intensity is proportional to where is the amplitude. We know that 40 dB is equivalent to an increase of a factor of in intensity. 26. 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 28. 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 , and concentrates the sound onto two eardrums with a total area of with an efficiency of 40.0%? Solution 17.4 DOPPLER EFFECT AND SONIC BOOMS 30. (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) Given

OpenStax College Physics 2e Student Solutions Manual Chapter 17 (b) 32. 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 34. 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) (b) 36. 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 The first eagle hears: The second eagle hears:

OpenStax College Physics 2e Student Solutions Manual Chapter 17 17.5 SOUND INTERFERENCE AND RESONANCE: STANDING WAVES IN AIR COLUMNS 38. A “showy” custom-built car has two brass horns that are supposed to produce the same frequency but actually emit 263.8 and 264.5 Hz. What beat frequency is produced? Solution The beat frequency is 40. What beat frequencies result if a piano hammer hits three strings that emit frequencies of 127.8, 128.1, and 128.3 Hz? Solution 42. (a) What is the fundamental frequency of a 0.672-m-long tube, open at both ends, on a day when the speed of sound is 344 m/s? (b) What is the frequency of its second harmonic? Solution (a) (b) 44. What are the first three overtones of a bassoon that has a fundamental frequency of 90.0 Hz? It is open at both ends. (The overtones of a real bassoon are more complex than this example, because its double reed makes it act more like a tube closed at one end.)

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OpenStax College Physics 2e Student Solutions Manual Chapter 17 Solution 46. What length should an oboe have to produce a fundamental frequency of 110 Hz on a day when the speed of sound is 343 m/s? It is open at both ends. Solution 48. (a) Find the length of an organ pipe closed at one end that produces a fundamental frequency of 256 Hz when air temperature is . (b) What is its fundamental frequency at ? Solution (a) (b) 50. The ear canal resonates like a tube closed at one end. (See Figure 17.37 .) If ear canals range in length from 1.80 to 2.60 cm in an average population, what is the range of fundamental resonant frequencies? Take air temperature to be , which is the same as body temperature. How does this result correlate with the intensity versus frequency graph ( Figure 17.35 ) of the human ear?

OpenStax College Physics 2e Student Solutions Manual Chapter 17 Solution For this frequency range, the graph shows a dip in the intensity of sound needed to pass the hearing threshold, presumably because these frequencies are resonances. 52. A crude approximation of voice production is to consider the breathing passages and mouth to be a resonating tube closed at one end. (See Figure 17.29 .) (a) What is the fundamental frequency if the tube is 0.240-m long, by taking air temperature to be ? (b) What would this frequency become if the person replaced the air with helium? Assume the same temperature dependence for helium as for air. Solution (a) (b) 54. What frequencies will a 1.80-m-long tube produce in the audible range at if: (a) The tube is closed at one end? (b) It is open at both ends? Solution (a) Since must be an odd number, is 419 or 421 (depending on the individual).

OpenStax College Physics 2e Student Solutions Manual Chapter 17 (b) 17.6 HEARING 55. The factor of 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? Solution 57. 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. Solution We know that we can discriminate between two sounds if their frequencies differ by at least 0.3%, so the closest frequencies to 500 Hz that we can distinguish are . 59. If your radio is producing an average sound intensity level of 85 dB, what is the next lowest sound intensity level that is clearly less intense? Solution Although 1 dB sound level changes can be discerned, 3 dB changes are the threshold for most people to notice. 61. Based on the graph in Figure 17.34 , what is the threshold of hearing in decibels for frequencies of 60, 400, 1000, 4000, and 15,000 Hz? Note that many AC electrical appliances produce 60 Hz, music is commonly 400 Hz, a reference frequency is 1000 Hz, your maximum sensitivity is near 4000 Hz, and many older TVs produce a 15,750 Hz whine. Solution From Figure 17.34 ,

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OpenStax College Physics 2e Student Solutions Manual Chapter 17 63. What is the approximate sound intensity level in decibels of a 600-Hz tone if it has a loudness of 20 phons? If it has a loudness of 70 phons? Solution From Figure 17.34 : a 600 Hz tone at a loudness of 20 phons has a sound level of about 23 dB , while a 600 Hz tone at a loudness of 70 phons has a sound level of about 70 dB . 65. Suppose a person has a 50-dB hearing loss at all frequencies. By how many factors of 10 will low-intensity sounds need to be amplified to seem normal to this person? Note that smaller amplification is appropriate for more intense sounds to avoid further hearing damage. Solution 50 dB is five factors of 10. 67. (a) What is the intensity in watts per meter squared of a just barely audible 200-Hz sound? (b) What is the intensity in watts per meter squared of a barely audible 4000-Hz sound? Solution (a) At 200 Hz, Using Table 17.3 , 23 dB is twice (2.00) the intensity of 20 dB. From Table 17.2 , (b) At 4000 Hz, –7 dB is the intensity of 0 dB. Thus, 69. 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? Solution From Figure 17.34 , the 0 phons line is normal hearing. So, this person can barely

OpenStax College Physics 2e Student Solutions Manual Chapter 17 hear a 100 Hz sound at 10 dB above normal, requiring a 47 dB sound level ( ). For a 4000 Hz sound, this person requires 50 dB above normal, or a 43 dB sound level ( ) to be audible. So, the 100 Hz tone must be 4 dB higher than the 4000 Hz sound. To calculate the difference in intensity, use the equation and convert the difference in decibels to a ratio of intensities. So the 100 Hz tone must be 2.5 times more intense than the 4000 Hz sound to be audible by this person. 71. What is the ratio of intensities of two sounds of identical frequency if the first is just barely discernible as louder to a person than the second? Solution A 1 dB difference can barely be discerned. 17.7 ULTRASOUND 72. What is the sound intensity level in decibels of ultrasound of intensity , used to pulverize tissue during surgery? Solution By Table 17.2 , 160 dB corresponds to , so has a sound level of 170 dB . 74. Find the sound intensity level in decibels of ultrasound used in medical diagnostics. Solution

OpenStax College Physics 2e Student Solutions Manual Chapter 17 76. In the clinical use of ultrasound, transducers are always coupled to the skin by a thin layer of gel or oil, replacing the air that would otherwise exist between the transducer and the skin. (a) Using the values of acoustic impedance given in Table 17.5 calculate the intensity reflection coefficient between transducer material and air. (b) Calculate the intensity reflection coefficient between transducer material and gel (assuming for this problem that its acoustic impedance is identical to that of water). (c) Based on the results of your calculations, explain why the gel is used. Solution Use Table 17.5 for values and the equation for . (a) , so 100% reflected. (b) (c) Gel is used to facilitate the transmission of the ultrasound between the transducer and the patient's body. 78. (a) Find the size of the smallest detail observable in human tissue with 20.0-MHz ultrasound. (b) Is its effective penetration depth great enough to examine the entire eye (about 3.00 cm is needed)? (c) What is the wavelength of such ultrasound in air? Solution (a) Let = velocity of sound in tissue. (b) Effective penetration depth = Yes , it is enough. (c) 80. (a) How far apart are two layers of tissue that produce echoes having round-trip times (used to measure distances) that differ by ? (b) What minimum frequency must the ultrasound have to see detail this small?

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OpenStax College Physics 2e Student Solutions Manual Chapter 17 Solution (a) (b) 82. A dolphin is able to tell in the dark that the ultrasound echoes received from two sharks come from two different objects only if the sharks are separated by 3.50 m, one being that much farther away than the other. (a) If the ultrasound has a frequency of 100 kHz, show this ability is not limited by its wavelength. (b) If this ability is due to the dolphin’s ability to detect the arrival times of echoes, what is the minimum time difference the dolphin can perceive? Solution (a) Since , the ability to resolve the two sharks is not limited by its wavelength. (b) 84. Ultrasound reflected from an oncoming bloodstream that is moving at 30.0 cm/s is mixed with the original frequency of 2.50 MHz to produce beats. What is the beat frequency? (Assume that the frequency of 2.50 MHz is accurate to seven significant figures.) Solution By Example 17.8 , Note: extra digits were retained in order to show the difference. This file is copyright 2022, Rice University. All Rights Reserved.

OpenStax College Physics 2e Student Solutions Manual Chapter 17

CollegePhysics2e-SSM-Ch17

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