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Chapter 12

SOUND

Educators


Problem 1

A hiker determines the length of a lake by listening for the echo of her shout reflected by a cliff at the far end of the lake. She hears the echo 2.5 s after shouting. Estimate the length of the lake.

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Problem 2

A sailor strikes the side of his ship just below the waterline. He hears the echo of the sound reflected from the ocean floor directly below 2.0 s later. How deep is the ocean at this point? Assume the speed of sound in sea water is 1560 m$/$s (Table 12$-$1) and does not vary significantly with depth.

Abhishek J.
Numerade Educator

Problem 3

($a$) Calculate the wavelengths in air at 20$^\circ$C for sounds in the maximum range of human hearing, 20 Hz to 20,000 Hz. ($b$) What is the wavelength of an 18-MHz ultrasonic wave?

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Problem 4

On a warm summer day (31$^\circ$C), it takes 4.80 s for an echo to return from a cliff across a lake. On a winter day, it takes 5.20 s. What is the temperature on the winter day?

Abhishek J.
Numerade Educator

Problem 5

An ocean fishing boat is drifting just above a school of tuna on a foggy day. Without warning, an engine backfire occurs on another boat 1.55 km away (Fig. 12$-$33). How much time elapses before the backfire is heard ($a$) by the fish, and ($b$) by the fishermen?

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Problem 6

A person, with his ear to the ground, sees a huge stone strike the concrete pavement. A moment later two sounds are heard from the impact: one travels in the air and the other in the concrete, and they are 0.80 s apart. How far away did the impact occur? See Table 12$-$1.

Abhishek J.
Numerade Educator

Problem 7

A stone is dropped from the top of a cliff. The splash it makes when striking the water below is heard 2.7 s later. How high is the cliff?

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Problem 8

What is the intensity of a sound at the pain level of 120 dB? Compare it to that of a whisper at 20 dB.

Abhishek J.
Numerade Educator

Problem 9

What is the sound level of a sound whose intensity is 1.5 $\times$ 10$^{-6}$ W$/m^2$?

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Problem 10

You are trying to decide between two new stereo amplifiers. One is rated at 75 W per channel and the other is rated at 120 W per channel. In terms of dB, how much louder will the more powerful amplifier be when both are producing sound at their maximum levels?

Abhishek J.
Numerade Educator

Problem 11

If two firecrackers produce a combined sound level of 85 dB when fired simultaneously at a certain place, what will be the sound level if only one is exploded? [$Hint$: Add intensities, not dBs.

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Problem 12

A person standing a certain distance from an airplane with four equally noisy jet engines is experiencing a sound level of 140 dB. What sound level would this person experience if the captain shut down all but one engine? [$Hint$: Add intensities, not dBs.]

Abhishek J.
Numerade Educator

Problem 13

One CD player is said to have a signal-to-noise ratio of 82 dB, whereas for a second CD player it is 98 dB. What is the ratio of intensities of the signal and the background noise for each device?

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Problem 14

A 55-dB sound wave strikes an eardrum whose area is 5.0 $\times$ 10$-$5 m$^2$. ($a$) How much energy is received by the eardrum per second? ($b$) At this rate, how long would it take your eardrum to receive a total energy of 1.0 J?

Abhishek J.
Numerade Educator

Problem 15

At a rock concert, a dB meter registered 130 dB when placed 2.5 m in front of a loudspeaker on stage. ($a$) What was the power output of the speaker, assuming uniform spherical spreading of the sound and neglecting absorption in the air? ($b$) How far away would the sound level be 85 dB?

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Problem 16

A fireworks shell explodes 100 m above the ground, creating colorful sparks. How much greater is the sound level of the explosion for a person at a point directly below the explosion than for a person a horizontal distance of 200 m away (Fig. 12$-$34)?

Abhishek J.
Numerade Educator

Problem 17

If the amplitude of a sound wave is made 3.5 times greater, (a) by what factor will the intensity increase? (b) By how many dB will the sound level increase?

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Problem 18

Two sound waves have equal displacement amplitudes, but one has 2.2 times the frequency of the other. What is the ratio of their intensities?

Abhishek J.
Numerade Educator

Problem 19

What would be the sound level (in dB) of a sound wave in air that corresponds to a displacement amplitude of vibrating air molecules of 0.13 mm at 440 Hz?

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Problem 20

($a$) Estimate the power output of sound from a person speaking in normal conversation. Use Table 12$-$2. Assume the sound spreads roughly uniformly over a sphere centered on the mouth. ($b$) How many people would it take to produce a total sound output of 60 W of ordinary conversation? [$Hint$: Add intensities, not dBs.]

Abhishek J.
Numerade Educator

Problem 21

Expensive amplifier A is rated at 220W, while the more modest amplifier B is rated at 45W. ($a$) Estimate the sound level in decibels you would expect at a point 3.5 m from a loudspeaker connected in turn to each amp. ($b$) Will the expensive amp sound twice as loud as the cheaper one?

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Problem 22

A 5000-Hz tone must have what sound level to seem as loud as a 100-Hz tone that has a 50-dB sound level? (See Fig. 12$-$6.)

Abhishek J.
Numerade Educator

Problem 23

What are the lowest and highest frequencies that an ear can detect when the sound level is 40 dB? (See Fig. 12$-$6.)

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Problem 24

Your ears can accommodate a huge range of sound levels. What is the ratio of highest to lowest intensity at ($a$) 100 Hz, ($b$) 5000 Hz? (See Fig. 12$-$6.)

Abhishek J.
Numerade Educator

Problem 25

Estimate the number of octaves in the human audible range, 20Hz to 20 kHz.

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Problem 26

What would you estimate for the length of a bass clarinet, assuming that it is modeled as a closed tube and that the lowest note that it can play is a D$^\flat$ whose frequency is 69 Hz?

Abhishek J.
Numerade Educator

Problem 27

The A string on a violin has a fundamental frequency of 440 Hz. The length of the vibrating portion is 32 cm, and it has mass 0.35 g. Under what tension must the string be placed?

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Problem 28

An organ pipe is 116 cm long. Determine the fundamental and first three audible overtones if the pipe is ($a$) closed at one end, and ($b$) open at both ends.

Abhishek J.
Numerade Educator

Problem 29

($a$) What resonant frequency would you expect from blowing across the top of an empty soda bottle that is 24 cm deep, if you assumed it was a closed tube? ($b$) How would that change if it was one-third full of soda?

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Problem 30

If you were to build a pipe organ with open-tube pipes spanning the range of human hearing (20 Hz to 20 kHz), what would be the range of the lengths of pipes required?

Abhishek J.
Numerade Educator

Problem 31

A tight guitar string has a frequency of 540 Hz as its third harmonic. What will be its fundamental frequency if it is fingered at a length of only 70$\%$ of its original length?

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Problem 32

Estimate the frequency of the "sound of the ocean" when you put your ear very near a 15-cm-diameter seashell (Fig. 12$-$35).

Abhishek J.
Numerade Educator

Problem 33

An unfingered guitar string is 0.68 m long and is tuned to play E above middle C (330 Hz). ($a$) How far from the end of this string must a fret (and your finger) be placed to play A above middle C (440 Hz)? ($b$) What is the wavelength on the string of this 440-Hz wave? ($c$)What are the frequency and wavelength of the sound wave produced in air at 22$^\circ$C by this fingered string?

Ben N.
Numerade Educator

Problem 34

($a$) Determine the length of an open organ pipe that emits middle C (262 Hz) when the temperature is 18$^\circ$C. ($b$) What are the wavelength and frequency of the fundamental standing wave in the tube? ($c$) What are $\lambda$ and $f$ in the traveling sound wave produced in the outside air?

Abhishek J.
Numerade Educator

Problem 35

An organ is in tune at 22.0$^\circ$C. By what percent will the frequency be off at 11$^\circ$C?

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Problem 36

How far from the mouthpiece of the flute in Example 12$-$11 should the hole be that must be uncovered to play F above middle C at 349 Hz?

Abhishek J.
Numerade Educator

Problem 37

($a$) At $T =$ 22$^\circ$C, how long must an open organ pipe be to have a fundamental frequency of 294 Hz? ($b$) If this pipe is filled with helium, what is its fundamental frequency?

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Problem 38

A particular organ pipe can resonate at 264 Hz, 440 Hz, and 616 Hz, but not at any other frequencies in
between. ($a$) Show why this is an open or a closed pipe. ($b$) What is the fundamental frequency of this pipe?

Abhishek J.
Numerade Educator

Problem 39

A uniform narrow tube 1.70 m long is open at both ends. It resonates at two successive harmonics of frequencies 275 Hz and 330 Hz.What is ($a$) the fundamental frequency, and ($b$) the speed of sound in the gas in the tube?

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Problem 40

A pipe in air at 23.0$^\circ$C is to be designed to produce two successive harmonics at 280 Hz and 320 Hz. How long must the pipe be, and is it open or closed?

Abhishek J.
Numerade Educator

Problem 41

How many overtones are present within the audible range for a 2.18-m-long organ pipe at 20$^\circ$C ($a$) if it is open, and ($b$) if it is closed?

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Problem 42

Determine the fundamental and first overtone frequencies when you are in a 9.0-m-long hallway with all doors closed. Model the hallway as a tube closed at both ends.

Abhishek J.
Numerade Educator

Problem 43

When a player's finger presses a guitar string down onto a fret, the length of the vibrating portion of the string is shortened, thereby increasing the string's fundamental frequency (see Fig. 12$-$36). The string's tension and mass per unit length remain unchanged. If the unfingered length of the string is $\ell =$ 75.0 cm, determine the positions $x$ of the first six frets, if each fret raises the pitch of the fundamental by one musical note compared to the neighboring fret. On the equally tempered chromatic scale, the ratio of frequencies of neighboring notes is 2$^{1/12}$.

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Problem 44

The human ear canal is approximately 2.5 cm long. It is open to the outside and is closed at the other end by the eardrum. Estimate the frequencies (in the audible range) of the standing waves in the ear canal. What is the relationship of your answer to the information in the graph of Fig. 12$-$6?

Abhishek J.
Numerade Educator

Problem 45

Approximately what are the intensities of the first two overtones of a violin compared to the fundamental? How many decibels softer than the fundamental are the first and second overtones? (See Fig. 12$-$15.)

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Problem 46

A piano tuner hears one beat every 2.0 s when trying to adjust two strings, one of which is sounding 350 Hz. How far off in frequency is the other string?

Abhishek J.
Numerade Educator

Problem 47

A certain dog whistle operates at 23.5 kHz, while another (brand X) operates at an unknown frequency. If humans can hear neither whistle when played separately, but a shrill whine of frequency 5000 Hz occurs when they are played simultaneously, estimate the operating frequency of brand X.

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Problem 48

What is the beat frequency if middle C (262 Hz) and C$^\#$ (277 Hz) are played together? What if each is played two octaves lower (each frequency reduced by a factor of 4)?

Abhishek J.
Numerade Educator

Problem 49

A guitar string produces 3 beats$/$s when sounded with a 350-Hz tuning fork and 8 beats$/$s when sounded with a 355-Hz tuning fork.What is the vibrational frequency of the string? Explain your reasoning.

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Problem 50

Two violin strings are tuned to the same frequency, 294 Hz. The tension in one string is then decreased by 2.5$\%$. What will be the beat frequency heard when the two strings are played together? [$Hint$: Recall Eq. 11$-$13.]

Abhishek J.
Numerade Educator

Problem 51

The two sources of sound in Fig. 12$-$16 face each other and emit sounds of equal amplitude and equal frequency (305 Hz) but 180$^\circ$ out of phase. For what minimum separation of the two speakers will there be some point at which ($a$) complete constructive interference occurs and ($b$) complete destructive interference occurs. (Assume $T =$ 20$^\circ$C.)

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Problem 52

Two piano strings are supposed to be vibrating at 220 Hz, but a piano tuner hears three beats every 2.5 s when they are played together. ($a$) If one is vibrating at 220.0 Hz, what must be the frequency of the other (is there only one answer)? ($b$) By how much (in percent) must the tension be increased or decreased to bring them in tune?

Abhishek J.
Numerade Educator

Problem 53

Two loudspeakers are 1.60 m apart. A person stands 3.00 m from one speaker and 3.50 m from the other. ($a$)What is the lowest frequency at which destructive interference will occur at this point if the speakers are in phase? ($b$) Calculate two other frequencies that also result in destructive interference
at this point (give the next two highest). Let $T =$ 20$^\circ$C.

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Problem 54

Two loudspeakers are placed 3.00 m apart, as shown in Fig. 12$-$37. They emit 474-Hz sounds, in phase. A microphone is placed 3.20 m distant from a point midway between the two speakers, where an intensity maximum is recorded. $(a)$ How far must the microphone be moved to the right to find the first intensity minimum? $(b)$ Suppose the speakers are reconnected so that the 474-Hz sounds they emit are exactly out of phase.At what positions are the intensity maximum and minimum now?

Abhishek J.
Numerade Educator

Problem 55

A source emits sound of wavelengths 2.54 m and 2.72 m in air. ($a)$ How many beats per second will be heard? (Assume $T =$ 20$^\circ$C. ) ($b$) How far apart in space are the regions of maximum intensity?

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Problem 56

The predominant frequency of a certain fire truck's siren is 1650 Hz when at rest.What frequency do you detect if you move with a speed of 30.0 m$/$s (a) toward the fire truck, and (b) away from it?

Abhishek J.
Numerade Educator

Problem 57

A bat at rest sends out ultrasonic sound waves at 50.0 kHz and receives them returned from an object moving directly away from it at 27.5 m$/$s. What is the received sound frequency?

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Problem 58

Two automobiles are equipped with the same singlefrequency horn. When one is at rest and the other is moving toward the first at 18 m$/$s, the driver at rest hears a beat frequency of 4.5 Hz. What is the frequency the horns emit? Assume $T =$ 20$^\circ$C.

Abhishek J.
Numerade Educator

Problem 59

As a bat flies toward a wall at a speed of 6.0 m$/$s, the bat emits an ultrasonic sound wave with frequency 30.0 kHz. What frequency does the bat hear in the reflected wave?

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Problem 60

In one of the original Doppler experiments, a tuba was played at a frequency of 75 Hz on a moving flat train car, and a second identical tuba played the same tone while at rest in the railway station. What beat frequency was heard in the station if the train car approached the station at a speed of 14.0 m$/$s?

Abhishek J.
Numerade Educator

Problem 61

A wave on the ocean surface with wavelength 44 m travels east at a speed of 18 m$/$s relative to the ocean floor. If, on this stretch of ocean, a powerboat is moving at 14 m$/$s (relative to the ocean floor), how often does the boat encounter a wave crest, if the boat is traveling ($a$) west, and ($b$) east?

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Problem 62

A police car sounding a siren with a frequency of 1580 Hz is traveling at 120.0 km$/$h. ($a$) What frequencies does an observer standing next to the road hear as the car approaches and as it recedes? ($b$) What frequencies are heard in a car traveling at 90.0 km$/$h in the opposite direction before and after passing the police car? ($c$) The police car passes a car traveling in the same direction at 80.0 km$?$h. What two frequencies are heard in this car?

Abhishek J.
Numerade Educator

Problem 63

The Doppler effect using ultrasonic waves of frequency 2.25 $\times$ 10$^6$ Hz is used to monitor the heartbeat of a fetus. A (maximum) beat frequency of 240 Hz is observed. Assuming that the speed of sound in tissue is 1540 m$/$s, calculate the maximum velocity of the surface of the beating heart.

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Problem 64

($a$) How fast is an object moving on land if its speed at 24$^\circ$C is Mach 0.33? ($b$) A high-flying jet cruising at 3000 km$/$h displays a Mach number of 3.1 on a screen. What is the speed of sound at that altitude?

Abhishek J.
Numerade Educator

Problem 65

The wake of a speedboat is 12$^\circ$ in a lake where the speed of
the water wave is 2.2 km$/$h. What is the speed of the boat?

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Problem 66

An airplane travels at Mach 2.1 where the speed of sound is 310 m$/$s. ($a$) What is the angle the shock wave makes with the direction of the airplane's motion? ($b$) If the plane is flying at a height of 6500 m, how long after it is directly overhead will a person on the ground hear the shock wave?

Abhishek J.
Numerade Educator

Problem 67

A space probe enters the thin atmosphere of a planet where the speed of sound is only about 42 m$/$s. $(a)$ What is the probe's Mach number if its initial speed is 15,000 km$/$h? $(b)$ What is the angle of the shock wave relative to the direction of motion?

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Problem 68

A meteorite traveling 9200 m$/$s strikes the ocean. Determine the shock wave angle it produces ($a$) in the air just before entering the ocean, and ($b$) in the water just after entering. Assume $T =$ 20$^\circ$C.

Abhishek J.
Numerade Educator

Problem 69

You look directly overhead and see a plane exactly 1.45 km above the ground flying faster than the speed of sound. By the time you hear the sonic boom, the plane has traveled a horizontal distance of 2.0 km. See Fig. 12$/$38. Determine ($a$) the angle of the shock cone, $\theta$, and ($b$) the speed of the plane and its Mach number. Assume the speed of sound is 330 m$/$s.

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Problem 70

A supersonic jet traveling at Mach 2.0 at an altitude of 9500 m passes directly over an observer on the ground. Where will the plane be relative to the observer when the latter hears the sonic boom? (See Fig. 12$/$39.)

Abhishek J.
Numerade Educator

Problem 71

A fish finder uses a sonar device that sends 20,000-Hz sound pulses downward from the bottom of the boat, and then detects echoes. If the maximum depth for which it is designed to work is 85 m, what is the minimum time between pulses (in fresh water)?

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Problem 72

A single mosquito 5.0 m from a person makes a sound close to the threshold of human hearing (0 dB). What will be the sound level of 200 such mosquitoes?

Abhishek J.
Numerade Educator

Problem 73

What is the resultant sound level when an 81-dB sound and an 87-dB sound are heard simultaneously?

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Problem 74

The sound level 8.25 m from a loudspeaker, placed in the open, is 115 dB. What is the acoustic power output (W) of the speaker, assuming it radiates equally in all directions?

Abhishek J.
Numerade Educator

Problem 75

A stereo amplifier is rated at 225 W output at 1000 Hz. The power output drops by 12 dB at 15 kHz. What is the power output in watts at 15 kHz?

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Problem 76

Workers around jet aircraft typically wear protective devices over their ears. Assume that the sound level of a jet airplane engine, at a distance of 30 m, is 130 dB, and that the average human ear has an effective radius of 2.0 cm.What would be the power intercepted by an unprotected ear at a distance of 30 m from a jet airplane engine?

Abhishek J.
Numerade Educator

Problem 77

In audio and communications systems, the $\textbf{gain}$, $\beta$, in decibels is defined for an amplifier as $$\beta = 10 \space log \bigg(\frac {P_{out}}{ P_{in}}\bigg),$$ where $P_{in}$ is the power input to the system and $P_{out}$ is the power output. ($a$) A particular amplifier puts out 135 W of power for an input of 1.0mW. What is its gain in dB? ($b$) If a signal-to-noise ratio of 93 dB is specified, what is the

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Problem 78

Manufacturers typically offer a particular guitar string in a choice of diameters so that players can tune their instruments with a preferred string tension. For example, a nylon high-E string is available in a low- and high-tension model with diameter 0.699 mm and 0.724 mm, respectively. Assuming the density $\rho$ of nylon is the same for each model, compare (as a ratio) the tension in a tuned high- and low-tension string.

Abhishek J.
Numerade Educator

Problem 79

A tuning fork is set into vibration above a vertical open tube filled with water (Fig. 12$-$40). The water level is allowed to drop slowly.As it does so, the air in the tube above the water level is heard to resonate with the tuning fork when the distance from the tube opening to the water level is 0.125 m and again at 0.395 m. What is the frequency of the tuning fork?

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Problem 80

Two identical tubes, each closed at one end, have a fundamental frequency of 349 Hz at 25.0$^\circ$CC. The air temperature is increased to 31.0$^\circ$CC in one tube. If the two pipes are now sounded together, what beat frequency results? noise power if the output signal is 10 W?

Abhishek J.
Numerade Educator

Problem 81

Each string on a violin is tuned to a frequency 1$\frac{1}{2}$ times that of its neighbor. The four equal-length strings are to be placed under the same tension; what must be the mass per unit length of each string relative to that of the lowest string?

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Problem 82

A particular whistle produces sound by setting up the fundamental standing wave in an air column 7.10 cm long. The tube is closed at one end. The whistle blower is riding in a car moving away from you at 25 m$/$s What frequency do you hear?

Abhishek J.
Numerade Educator

Problem 83

The diameter $D$ of a tube does affect the node at the open end of a tube. The end correction can be roughly approximated as adding $D/3$ to $\ell$ give us an effective length for the tube in calculations. For a closed tube of length 0.55 m and diameter 3.0 cm, what are the frequencies of the first four harmonics, taking the end correction into consideration?

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Problem 84

The frequency of a steam train whistle as it approaches you is 565 Hz. After it passes you, its frequency is measured as 486 Hz. How fast was the train moving (assume constant velocity)?

Abhishek J.
Numerade Educator

Problem 85

Two trains emit 508-Hz whistles. One train is stationary. The conductor on the stationary train hears a 3.5-Hz beat frequency when the other train approaches. What is the speed of the moving train?

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Problem 86

Two loudspeakers are at opposite ends of a railroad car as it moves past a stationary observer at 12.0 m$/$s, as shown in Fig. 12$-$41. If the speakers have identical sound frequencies of 348 Hz, what is the beat frequency heard by the observer when ($a$) he listens from position A, in front of the car, ($b$) he is between the speakers, at B, and ($c$) he hears the speakers after they have passed him, at C?

Abhishek J.
Numerade Educator

Problem 87

Two open organ pipes, sounding together, produce a beat frequency of 6.0 Hz. The shorter one is 2.40 m long. How long is the other?

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Problem 88

A bat flies toward a moth at speed 7.8 m$/$s while the moth is flying toward the bat at speed 5.0 m$/$s. The bat emits a sound wave of 51.35 kHz.What is the frequency of the wave detected by the bat after that wave reflects off the moth?

Abhishek J.
Numerade Educator

Problem 89

A bat emits a series of high-frequency sound pulses as it approaches a moth. The pulses are approximately 70.0 ms apart, and each is about 3.0 ms long. How far away can the moth be detected by the bat so that the echo from one pulse returns before the next pulse is emitted?

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Problem 90

Two loudspeakers face each other at opposite ends of a long corridor. They are connected to the same source which produces a pure tone of 282 Hz. A person walks from one speaker toward the other at a speed of 1.6 m$/$s. What "beat" frequency does the person hear?

Abhishek J.
Numerade Educator

Problem 91

A sound-insulating door reduces the sound level by 30 dB. What fraction of the sound intensity passes through this door?

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Problem 92

The "alpenhorn" (Fig. 12$-$42) was once used to send signals from one Alpine village to another. Since lower frequency sounds are less susceptible to intensity loss, long horns were used to create deep sounds. When played as a musical instrument, the alpenhorn must be blown in such a way that only one of the overtones is resonating. The most popular alpenhorn is about 3.4 m long, and it is called the F$^\#$ (or G$^\flat$ ) horn. What is the fundamental frequency of this horn, and which overtone is close to F$^\#$ ? (See Table 12$-$3.) Model as a tube open at both ends.

Abhishek J.
Numerade Educator

Problem 93

Room acoustics for stereo listening can be compromised by the presence of standing waves, which can cause acoustic "dead spots" at the locations of the pressure nodes. Consider a living room 4.7 m long, 3.6 m wide, and 2.8 m high. Calculate the fundamental frequencies for the standing waves in this room.

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Problem 94

A dramatic demonstration, called "singing rods," involves a long, slender aluminum rod held in the hand near the rod's midpoint. The rod is stroked with the other hand. With a little practice, the rod can be made to "sing," or emit a clear, loud, ringing sound. For an 80-cm-long rod, ($a$) what is the fundamental frequency of the sound? ($b$) What is its wavelength in the rod, and ($c$) what is the traveling wavelength of the sound in air at 20$^\circ$C?

Abhishek J.
Numerade Educator

Problem 95

The intensity at the threshold of hearing for the human ear at a frequency of about 1000 Hz is $I_0 =$ 1.0 $\times$ 10$^{-12}$ W$/$m$^2$, for which $\beta$, the sound level, is 0 dB. The threshold of pain at the same frequency is about 120 dB, or $I =$ 1.0 W$m^2$, corresponding to an increase of intensity by a factor of 10$^{12}$. By what factor does the displacement amplitude, $A$, vary?

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Problem 96

A $\textbf{Doppler flow meter}$ uses ultrasound waves to measure blood-flow speeds. Suppose the device emits sound at 3.5 MHz, and the speed of sound in human tissue is about 1540 m$/$s. What is the expected beat frequency if blood is flowing in large leg arteries at 3.0 cm$/$s directly away from
the sound source?

Abhishek J.
Numerade Educator