Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 17

Sound Waves - all with Video Answers

Educators

PP

Chapter Questions

Problem 1

Write an expression that describes the pressure variation as a function of position and time for a sinusoidal sound wave in air. Assume the speed of sound is $343 \mathrm{m} / \mathrm{s}, \lambda=$ $0.100 \mathrm{m},$ and $\Delta P_{\max }=0.200 \mathrm{Pa}$.

Keshav Singh

Keshav Singh

Numerade Educator

Problem 2

As a certain sound wave travels through the air, it produces pressure variations (above and below atmospheric pressure) given by $\Delta P=1.27 \sin (\pi x-340 \pi t)$ in SI units. Find (a) the amplitude of the pressure variations, (b) the frequency, (c) the wavelength in air, and (d) the speed of the sound wave.

Keshav Singh

Numerade Educator

Problem 3

A sinusoidal sound wave moves through a medium and is described by the displacement wave function $$s(x, t)=2.00 \cos (15.7 x-858 t)$$
where s is in micrometers, $x$ is in meters, and t is in seconds. Find (a) the amplitude, (b) the wavelength, and (c) the speed of this wave. (d) Determine the instantaneous dis- placement from equilibrium of the elements of the medium at the position $x=0.0500 \mathrm{m}$ at $t=3.00 \mathrm{ms}$ . (e) Determine the maximum speed of the element's oscillatory motion.

Keshav Singh

Numerade Educator

Problem 4

An experimenter wishes to generate in air a sound wave that has a displacement amplitude of $5.50 \times 10^{-6} \mathrm{m} .$ The pressure amplitude is to be limited to 0.840 $\mathrm{Pa}$ . What is the minimum wavelength the sound wave can have?

Keshav Singh

Numerade Educator

Problem 5

Suppose you hear a clap of thunder 16.2 s after seeing the associated lightning strike. The speed of light in air is $3.00 \times 10^{8} \mathrm{m} / \mathrm{s}$ . (a) How far are you from the lightning strike? (b) Do you need to know the value of the speed of light to answer? Explain.

Keshav Singh

Numerade Educator

Problem 6

Earthquakes at fault lines in the Earth’s crust create seismic waves, which are longitudinal (P waves) or transverse (S waves). The $\mathrm{P}$ waves have a speed of about 7 $\mathrm{km} / \mathrm{s}$ . Estimate the average bulk modulus of the Earth's crust given that the density of rock is about 2500 $\mathrm{kg} / \mathrm{m}^{3}$ .

Keshav Singh

Numerade Educator

Problem 7

A dolphin (Fig. Pl7.7) in seawater at a temperature of $25^{\circ}$ C emits a sound wave directed toward the ocean floor 150 m below. How much time passes before it hears an echo?

Keshav Singh

Numerade Educator

Problem 8

A sound wave propagates in air at $27^{\circ} \mathrm{C}$ with frequency 4.00 $\mathrm{kHz}$ . It passes through a region where the temperature gradually changes and then moves through air at $0^{\circ} \mathrm{C}$. Give numerical answers to the following questions to the extent possible and state your reasoning about what happens to the wave physically. (a) What happens to the speed of the wave? (b) What happens to its frequency? (c) What happens to its wavelength?

Keshav Singh

Numerade Educator

Problem 9

Ultrasound is used in medicine both for diagnostic imaging (Fig. P17.9) and for therapy. For diagnosis, short pulses of ultrasound are passed through the patient’s body. An echo reflected from a structure of interest is recorded, and the distance to the structure can be determined from the time delay for the echo’s return. To reveal detail, the wavelength of the reflected ultrasound must be small compared to the size of the object reflecting the wave. The speed of ultrasound in human tissue is about 1 500 m/s (nearly the same as the speed of sound in water). (a) What is the wavelength of ultrasound with a frequency of 2.40 MHz? (b) In the whole set of imaging techniques, frequencies in the range 1.00 MHz to 20.0 MHz are used. What is the range of wavelengths corresponding to this range of frequencies?

Keshav Singh

Numerade Educator

Problem 10

A sound wave in air has a pressure amplitude equal to $4.00 \times 10^{-3}$ Pa. Calculate the displacement amplitude of the wave at a frequency of 10.0 $\mathrm{kHz}$ .

Keshav Singh

Numerade Educator

Problem 11

A flowerpot is knocked off a window ledge from a height $d=$ 20.0 $\mathrm{m}$ above the sidewalk as shown in Figure P17.11. It falls toward an unsuspecting man of height $h=1.75 \mathrm{m}$ who is standing below. Assume the man requires a time interval of $\Delta t=0.300$ s to respond to the warning. How close to the sidewalk can the flowerpot fall before it is too late for a warning shouted from the balcony to reach the man in time?

Keshav Singh

Numerade Educator

Problem 12

A flowerpot is knocked off a balcony from a height $d$ above the sidewalk as shown in Figure P17.11. It falls toward an unsuspecting man of height $h$ who is standing below. Assume the man requires a time interval of $\Delta t$ to respond to the warning. How close to the sidewalk can the flowerpot fall before it is too late for a warning shouted from the balcony to reach the man in time? Use the symbol $v$ for the speed of sound.

Keshav Singh

Numerade Educator

Problem 13

The speed of sound in air (in meters per second) depends on temperature according to the approximate expression
$$v=331.5+0.607 T_{\mathrm{C}}$$
where $T_{\mathrm{C}}$ is the Celsius temperature. In dry air, the temperature decreases about $1^{\circ} \mathrm{C}$ for every 150 $\mathrm{-m}$ rise in altitude. (a) Assume this change is constant up to an altitude of 9000 $\mathrm{m}$ . What time interval is required for the sound from an airplane flying at 9000 $\mathrm{m}$ to reach the ground on a day when the ground temperature is $30^{\circ} \mathrm{C}$ ? (b) What If? Compare your answer with the time interval required if the air were uniformly at $30^{\circ} \mathrm{C} .$ Which time interval is longer?

Keshav Singh

Numerade Educator

Problem 14

A rescue plane flies horizontally at a constant speed searching for a disabled boat. When the plane is directly above the boat, the boat’s crew blows a loud horn. By the time the plane’s sound detector receives the horn’s sound, the plane has traveled a distance equal to half its altitude above the ocean. Assuming it takes the sound 2.00 s to reach the plane, determine (a) the speed of the plane and (b) its altitude.

Keshav Singh

Numerade Educator

Problem 15

A cowboy stands on horizontal ground between two parallel, vertical cliffs. He is not midway between the cliffs. He fires a shot and hears its echoes. The second echo arrives 1.92 s after the first and 1.47 s before the third. Consider only the sound traveling parallel to the ground and reflecting from the cliffs. (a) What is the distance between the cliffs? (b) What If? If he can hear a fourth echo, how long after the third echo does it arrive?

Vishal Gupta

Numerade Educator

Problem 16

sound wave moves down a cylinder as in Active Figure 17.2. Show that the pressure variation of the wave is described by $\Delta P=\pm \rho v \omega \sqrt{s_{\max }^{2}}-s^{2},$ where $s=s(x, t)$ is given by Equation $17.1 .$

Keshav Singh

Numerade Educator

Problem 17

A hammer strikes one end of a thick iron rail of length 8.50 m. A microphone located at the opposite end of the rail detects two pulses of sound, one that travels through the air and a longitudinal wave that travels through the rail. (a) Which pulse reaches the microphone first? (b) Find the separation in time between the arrivals of the two pulses.

Keshav Singh

Numerade Educator

Problem 18

The area of a typical eardrum is about $5.00 \times 10^{-5} \mathrm{m}^{2}$. (a) Calculate the average sound power incident on an eardrum at the threshold of pain, which corresponds to an intensity of $1.00 \mathrm{W} / \mathrm{m}^{2} .$ (b) How much energy is transferred to the eardrum exposed to this sound for 1.00 $\mathrm{min}$ ?

Keshav Singh

Numerade Educator

Problem 19

Calculate the sound level (in decibels) of a sound wave that has an intensity of $4.00 \mu \mathrm{W} / \mathrm{m}^{2} .$

Keshav Singh

Numerade Educator

Problem 20

The sound intensity at a distance of 16 $\mathrm{m}$ from a noisy generator is measured to be $0.25 \mathrm{W} / \mathrm{m}^{2} .$ What is the sound intensity at a distance of 28 $\mathrm{m}$ from the generator?

Keshav Singh

Numerade Educator

Problem 21

The intensity of a sound wave at a fixed distance from a speaker vibrating at 1.00 $\mathrm{kHz}$ is $0.600 \mathrm{W} / \mathrm{m}^{2} .$ (a) Determine the intensity that results if the frequency is increased to 2.50 $\mathrm{kHz}$ while a constant displacement amplitude is maintained. (b) Calculate the intensity if the frequency is reduced to 0.500 $\mathrm{kHz}$ and the displacement amplitude is doubled.

Keshav Singh

Numerade Educator

Problem 22

The intensity of a sound wave at a fixed distance from a speaker vibrating at a frequency $f$ is $I$ . (a) Determine the intensity that results if the frequency is increased to $f^{\prime}$ while a constant displacement amplitude is maintained. (b) Calculate the intensity if the frequency is reduced to $f / 2$ and the displacement amplitude is doubled.

Keshav Singh

Numerade Educator

Problem 23

A person wears a hearing aid that uniformly increases the sound level of all audible frequencies of sound by 30.0 $\mathrm{dB}$ . The hearing aid picks up sound having a frequency of 250 $\mathrm{Hz}$ at an intensity of $3.0 \times 10^{-11} \mathrm{W} / \mathrm{m}^{2} .$ What is the intensity delivered to the eardrum?

Keshav Singh

Numerade Educator

Problem 24

A sound wave from a police siren has an intensity of 100.0 $\mathrm{W} / \mathrm{m}^{2}$ at a certain point; a second sound wave from a nearby ambulance has an intensity level that is 10 $\mathrm{db}$ greater than the police siren's sound wave at the same point. What is the sound level of the sound wave due to the ambulance?

Keshav Singh

Numerade Educator

Problem 25

The power output of a certain public-address speaker is 6.00 W. Suppose it broadcasts equally in all directions. (a) Within what distance from the speaker would the sound be painful to the ear? (b) At what distance from the speaker would the sound be barely audible?

Keshav Singh

Numerade Educator

Problem 26

As the people sing in church, the sound level everywhere inside is 101 $\mathrm{dB}$ . No sound is transmitted through the massive walls, but all the windows and doors are open on a summer morning. Their total area is $22.0 \mathrm{m}^{2} .$ (a) How much sound energy is radiated through the windows and doors in 20.0 $\mathrm{min}$ ? (b) Suppose the ground is a good reflector and sound radiates from the church uniformly in all horizontal and upward directions. Find the sound level 1.00 $\mathrm{km}$ away.

Keshav Singh

Numerade Educator

Problem 27

The most soaring vocal melody is in Johann Sebastian Bach’s Mass in B Minor. In one section, the basses, tenors, altos, and sopranos carry the melody from a low D to a high A. In concert pitch, these notes are now assigned frequencies of 146.8 Hz and 880.0 Hz. Find the wavelengths of (a) the initial note and (b) the final note. Assume the chorus sings the melody with a uniform sound level of 75.0 dB.
Find the pressure amplitudes of (c) the initial note and (d) the final note. Find the displacement amplitudes of (e) the initial note and (f) the final note.

Jeff Vermeire

Numerade Educator

Problem 28

Show that the difference between decibel levels $\beta_{1}$ and $\beta_{2}$ of a sound is related to the ratio of the distances $r_{1}$ and $r_{2}$ from the sound source by
$$\beta_{2}-\beta_{1}=20 \log \left(\frac{r_{1}}{r_{2}}\right)$$

Keshav Singh

Numerade Educator

Problem 29

A family ice show is held at an enclosed arena. The skaters perform to music with level 80.0 dB. This level is too loud for your baby, who yells at 75.0 dB. (a) What total sound intensity engulfs you? (b) What is the combined sound level?

Keshav Singh

Numerade Educator

Problem 30

Two small speakers emit sound waves of different frequencies equally in all directions. Speaker $A$ has an output of 1.00 $\mathrm{mW}$ , and speaker $B$ has an output of 1.50 $\mathrm{mW}$ . Determine the sound level (in decibels) at point $C$ in Figure $\mathrm{P} 17.30$ assuming (a) only speaker $A$ emits sound, (b) only speaker $B$ emits sound, and (c) both speakers emit sound.

Keshav Singh

Numerade Educator

Problem 31

A firework charge is detonated many meters above the ground. At a distance of $d_{1}=500 \mathrm{m}$ from the explosion, the acoustic pressure reaches a maximum of $\Delta P_{\max }=$ 10.0 Pa (Fig. Pl7.31). Assume the speed of sound is constant at 343 $\mathrm{m} / \mathrm{s}$ throughout the atmosphere over the region considered, the ground absorbs all the sound falling on it, and the air absorbs sound energy as described by the rate $7.00 \mathrm{dB} / \mathrm{km} .$ What is the sound level (in decibels) at a distance of $d_{2}=4.00 \times 10^{3} \mathrm{m}$ from the explosion?

Keshav Singh

Numerade Educator

Problem 32

A fireworks rocket explodes at a height of 100 $\mathrm{m}$ above the ground. An observer on the ground directly under the explosion experiences an average sound intensity of $7.00 \times 10^{-2} \mathrm{W} / \mathrm{m}^{2}$ for 0.200 $\mathrm{s}$ . (a) What is the total amount of energy transferred away from the explosion by sound? (b) What is the sound level (in decibels) heard by the observer?

Keshav Singh

Numerade Educator

Problem 33

The sound level at a distance of 3.00 m from a source is 120 dB. At what distance is the sound level (a) 100 dB and (b) 10.0 dB?

Keshav Singh

Numerade Educator

Problem 34

Why is the following situation impossible? It is early on a Saturday morning, and much to your displeasure your next-door neighbor starts mowing his lawn. As you try to get back to sleep, your next-door neighbor on the other side of your house also begins to mow the lawn with an identical mower the same distance away. This situation annoys you greatly because the total sound now has twice the loudness it had when only one neighbor was mowing.

Keshav Singh

Numerade Educator

Problem 35

A driver travels northbound on a highway at a speed of 25.0 m/s. A police car, traveling southbound at a speed of 40.0 m/s, approaches with its siren producing sound at a frequency of 2 500 Hz. (a) What frequency does the driver observe as the police car approaches? (b) What frequency does the driver detect after the police car passes him? (c) Repeat parts (a) and (b) for the case when the police car is behind the driver and travels northbound.

Keshav Singh

Numerade Educator

Problem 36

Submarine A travels horizontally at 11.0 m/s through ocean water. It emits a sonar signal of frequency $f=$ $5.27 \times 10^{3} \mathrm{Hz}$ in the forward direction. Submarine $\mathrm{B}$ is in front of submarine $\mathrm{A}$ and traveling at 3.00 $\mathrm{m} / \mathrm{s}$ relative to the water in the same direction as submarine A. A crewman in submarine B uses his equipment to detect the sound waves ('pings") from submarine A. We wish to determine what is heard by the crewman in submarine B. (a) An observer on which submarine detects a frequency $f^{\prime}$ as described by Equation 17.19? (b) In Equation 17.19, should the sign of $v_{S}$ be positive or negative? (c) In Equation $17.19,$ should the sign of $v_{O}$ be positive or negative? (d) In Equation $17.19,$ what speed of sound should be used? (e) Find the frequency of the sound detected by the crewman on submarine $\mathrm{B}$ .

Keshav Singh

Numerade Educator

Problem 37

An ambulance moving at 42 m/s sounds its siren whose frequency is 450 Hz. A car is moving in the same direction as the ambulance at 25 m/s. What frequency does a person in the car hear (a) as the ambulance approaches the car? (b) After the ambulance passes the car?

Keshav Singh

Numerade Educator

Problem 38

When high-energy charged particles move through a transparent medium with a speed greater than the speed of light in that medium, a shock wave, or bow wave, of light is produced. This phenomenon is called the Cerenkov effect. When a nuclear reactor is shielded by a large pool of water, Cerenkov radiation can be seen as a blue glow in the vicinity of the reactor core due to high-speed electrons moving through the water (Fig. 17.38). In a particular case, the Cerenkov radiation produces a wave front with an apex half-angle of $53.0^{\circ} .$ Calculate the speed of the electrons in the water. The speed of light in water is $2.25 \times$ $10^{8} \mathrm{m} / \mathrm{s}$.

Keshav Singh

Numerade Educator

Problem 39

A block with a speaker bolted to it is connected to a spring having spring constant $k=20.0 \mathrm{N} / \mathrm{m}$ and oscillates as shown in Figure $\mathrm{P} 17.39 .$ The total mass of the block and speaker is 5.00 $\mathrm{kg}$ , and the amplitude of this unit's motion is 0.500 $\mathrm{m}$ . The speaker emits sound waves of frequency 440 $\mathrm{Hz}$ . Determine (a) the highest and (b) the lowest frequencies heard by the person to the right of the speaker. (c) If the maximum sound level heard by the person is 60.0 $\mathrm{dB}$ when the speaker is at its closest distance $d=$ 1.00 $\mathrm{m}$ from him, what is the minimum sound level heard by the observer?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 40

A block with a speaker bolted to it is connected to a spring having spring constant $k$ and oscillates as shown in Figure $\mathrm{P} 17.39$ . The total mass of the block and speaker is $m,$ and the amplitude of this unit's motion is $A$ . The speaker emits sound waves of frequency $f .$ Determine nected to a spring having spring constant $k$ and oscillates as shown in Figure $\mathrm{P} 17.39$ . The total mass of the block and speaker is $m,$ and the amplitude of this unit's motion is $A$ . The speaker emits sound waves of frequency $f .$ Determine

Keshav Singh

Numerade Educator

Problem 41

Expectant parents are thrilled to hear their unborn baby’s heartbeat, revealed by an ultrasonic detector that produces beeps of audible sound in synchronization with the fetal heartbeat. Suppose the fetus’s ventricular wall moves in simple harmonic motion with an amplitude of 1.80 mm and a frequency of 115 beats per minute. (a) Find the maximum linear speed of the heart wall. Suppose a source mounted on the detector in contact with the mother’s abdomen produces sound at 2 000 000.0 Hz, which travels through tissue at 1.50 km/s. (b) Find the maximum change in frequency between the sound that arrives at the wall of the baby’s heart and the sound emitted by the source. (c) Find the maximum change in frequency between the reflected sound received by the detector and that emitted by the source.

Dominador Tan

Numerade Educator

Problem 42

Why is the following situation impossible? At the Summer Olympics, an athlete runs at a constant speed down a straight track while a spectator near the edge of the track blows a note on a horn with a fixed frequency. When the athlete passes the horn, she hears the frequency of the horn fall by the musical interval called a minor third. That is, the frequency she hears drops to five-sixths its original value.

Keshav Singh

Numerade Educator

Problem 43

Standing at a crosswalk, you hear a frequency of 560 Hz from the siren of an approaching ambulance. After the ambulance passes, the observed frequency of the siren is 480 Hz. Determine the ambulance’s speed from these observations.

Keshav Singh

Numerade Educator

Problem 44

Review. A tuning fork vibrating at 512 Hz falls from rest and accelerates at 9.80 $\mathrm{m} / \mathrm{s}^{2}$ . How far below the point of release is the tuning fork when waves of frequency 485 $\mathrm{Hz}$ reach the release point?

Averell Hause

Carnegie Mellon University

Problem 45

A supersonic jet traveling at Mach 3.00 at an altitude of $h=20000 \mathrm{m}$ is directly over a person at time $t=0$ as shown in Figure $\mathrm{P} 17.45 .$ Assume the average speed of sound in air is 335 $\mathrm{m} / \mathrm{s}$ over the path of the sound. (a) At what time will the person encounter the shock wave due to the sound emitted at $t=0 ?$ (b) Where will the plane be when this shock wave is heard?

Keshav Singh

Numerade Educator

Problem 46

The highest note written for a singer in a published score was F-sharp above high C, 1.480 kHz, for Zerbinetta in the original version of Richard Strauss’s opera Ariadne auf Naxos. (a) Find the wavelength of this sound in air. (b) Suppose people in the fourth row of seats hear this note with level 81.0 dB. Find the displacement amplitude of the sound. (c) What If? In response to complaints, Strauss later transposed the note down to F above high C, 1.397 kHz. By what increment did the wavelength change?

Keshav Singh

Numerade Educator

Problem 47

Trucks carrying garbage to the town dump form a nearly steady procession on a country road, all traveling at 19.7 m/s in the same direction. Two trucks arrive at the dump every 3 min. A bicyclist is also traveling toward the dump, at 4.47 m/s. (a) With what frequency do the trucks pass the cyclist? (b) What If? A hill does not slow down the trucks, but makes the out-of-shape cyclist’s speed drop to 1.56 m/s. How often do the trucks whiz past the cyclist now?

Keshav Singh

Numerade Educator

Problem 48

Assume a 150-W loudspeaker broadcasts sound equally in all directions and produces sound with a level of 103 dB at a distance of 1.60 m from its center. (a) Find its sound power output. If a salesman claims the speaker is rated at 150 W, he is referring to the maximum electrical power input to the speaker. (b) Find the efficiency of the speaker, that is, the fraction of input power that is converted into useful output power.

Keshav Singh

Numerade Educator

Problem 49

An interstate highway has been built through a neighborhood in a city. In the afternoon, the sound level in an apartment in the neighborhood is 80.0 dB as 100 cars pass outside the window every minute. Late at night, the traffic flow is only five cars per minute. What is the average late-night sound level?

Keshav Singh

Numerade Educator

Problem 50

The tensile stress in a thick copper bar is 99.5$\%$ of its elastic breaking point of $13.0 \times 10^{10} \mathrm{N} / \mathrm{m}^{2} .$ If a $500-\mathrm{Hz}$ sound wave is transmitted through the material, (a) what displacement amplitude will cause the bar to break? (b) What is the maximum speed of the clements of copper at this moment? (c) What is the sound intensity in the bar?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

A 150 -glider moves at $v_{1}=2.30 \mathrm{m} / \mathrm{s}$ on an air track toward an originally stationary 200-g glider as shown in Figure P17.51. The gliders undergo a completely inelastic collision and latch together over a time interval of 7.00 ms. A student suggests roughly half the decrease in mechanical energy of the two-glider system is transferred to the environment by sound. Is this suggestion reasonable? To evaluate the idea, find the implied sound level at a position 0.800 m from the gliders. If the student’s idea is unreasonable, suggest a better idea.

Victor Salazar

Numerade Educator

Problem 52

Consider the following wave function in SI units:
$$\Delta P(r, t)=\left(\frac{25.0}{r}\right) \sin (1.36 r-2030 t)$$
Explain how this wave function can apply to a wave radiating from a small source, with r being the radial distance from the center of the source to any point outside the source. Give the most detailed description of the wave that you can. Include answers to such questions as the following and give representative values for any quantities that can be evaluated. (a) Does the wave move more toward the right or the left? (b) As it moves away from the source, what happens to its amplitude? (c) Its speed? (d) Its frequency? (e) Its wavelength? (f) Its power? (g) Its intensity?

Keshav Singh

Numerade Educator

Problem 53

For a certain type of steel, stress is always proportional to strain with Young's modulus $20 \times 10^{10} \mathrm{N} / \mathrm{m}^{2} .$ The steel has density $7.86 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3} .$ It will fail by bending permanently if subjected to compressive stress greater than its yield strength $\sigma_{y}=400 \mathrm{MPa}$ . A rod 80.0 $\mathrm{cm}$ long, made of this steel, is fired at 12.0 $\mathrm{m} / \mathrm{s}$ straight at a very hard wall. (a) The speed of a one-dimensional compressional wave moving along the rod is given by $v=\sqrt{Y / \rho},$ where $Y$ Young's modulus for the rod and $\rho$ is the density. Calculate this speed. (b) After the front end of the rod hits the wall and stops, the back end of the rod keeps moving as described by Newton’s first law until it is stopped by excess pressure in a sound wave moving back through the rod. What time interval elapses before the back end of the rod receives the message that it should stop? (c) How far has the back end of the rod moved in this time interval? Find (d) the strain and (e) the stress in the rod. (f) If it is not to fail, what is the maximum impact speed a rod can have in terms of $\sigma_{y}, Y,$ and $\rho ?$

Keshav Singh

Numerade Educator

Problem 54

A large set of unoccupied football bleachers has solid seats and risers. You stand on the field in front of the bleachers and sharply clap two wooden boards together once. The sound pulse you produce has no definite frequency and no wavelength. The sound you hear reflected from the bleachers has an identifiable frequency and may remind you of a short toot on a trumpet, buzzer, or kazoo. (a) Explain what accounts for this sound. Compute order-of-magnitude estimates for (b) the frequency, (c) the wave-length, and (d) the duration of the sound on the basis of data you specify.

Victor Salazar

Numerade Educator

Problem 55

To measure her speed, a skydiver carries a buzzer emitting a steady tone at 1 800 Hz. A friend on the ground at the landing site directly below listens to the amplified sound he receives. Assume the air is calm and the speed of sound is independent of altitude. While the skydiver is falling at terminal speed, her friend on the ground receives waves of frequency 2 150 Hz. (a) What is the skydiver’s speed of descent? (b) What If? Suppose the skydiver can hear the sound of the buzzer reflected from the ground. What frequency does she receive?

Keshav Singh

Numerade Educator

Problem 56

Spherical waves of wavelength 45.0 cm propagate outward from a point source. (a) Explain how the intensity at a distance of 240 cm compares with the intensity at a distance of 60.0 cm. (b) Explain how the amplitude at a distance of 240 cm compares with the amplitude at a distance of 60.0 cm. (c) Explain how the phase of the wave at a distance of 240 cm compares with the phase at 60.0 cm at the same moment.

Keshav Singh

Numerade Educator

Problem 57

A bat, moving at 5.00 m/s, is chasing a flying insect. If the bat emits a 40.0-kHz chirp and receives back an echo at 40.4 kHz, (a) what is the speed of the insect? (b) Will the bat be able to catch the insect? Explain.

Keshav Singh

Numerade Educator

Problem 58

Two ships are moving along a line due east (Fig. P17.58). The trailing vessel has a speed relative to a land-based observation point of $v_{1}=64.0 \mathrm{km} / \mathrm{h},$ and the leading ship has a speed of $v_{2}=45.0 \mathrm{km} / \mathrm{h}$ relative to that point. The two ships are in a region of the ocean where the current is moving uniformly due west at $v_{\text { current }}=10.0 \mathrm{km} / \mathrm{h}$ . The trailing ship transmits a sonar signal at a frequency of 1200.0 Hz through the water. What frequency is monitored by the leading ship?

Keshav Singh

Numerade Educator

Problem 59

A police car is traveling east at 40.0 m/s along a straight road, overtaking a car ahead of it moving east at 30.0 m/s. The police car has a malfunctioning siren that is stuck at 1 000 Hz. (a) What would be the wavelength in air of the siren sound if the police car were at rest? (b) What is the wavelength in front of the police car? (c) What is it behind the police car? (d) What is the frequency heard by the driver being chased?

Keshav Singh

Numerade Educator

Problem 60

The speed of a one-dimensional compressional wave traveling along a thin copper rod is 3.56 km/s. The rod is given a sharp hammer blow at one end. A listener at the far end of the rod hears the sound twice, transmitted through the metal and through air, with a time interval $\Delta t$ between the two pulses. (a) Which sound arrives first? (b) Find the length of the rod as a function of $\Delta t .$ (c) Find the length of the rod if $\Delta t=127 \mathrm{ms}$ . (d) Imagine that the copper rod is replaced by another material through which the speed of sound is $v_{r}$ . What is the length of the rod in terms of $t$ and $v_{r}^{2}$ (e) Would the answer to part (d) go to a well-defined limit as the speed of sound in the rod goes to infinity? Explain your answer.

Keshav Singh

Numerade Educator

Problem 61

A large meteoroid enters the Earth’s atmosphere at a speed of 20.0 km/s and is not significantly slowed before entering the ocean. (a) What is the Mach angle of the shock wave from the meteoroid in the lower atmosphere? (b) If we assume the meteoroid survives the impact with the ocean surface, what is the (initial) Mach angle of the shock wave the meteoroid produces in the water?

Keshav Singh

Numerade Educator

Problem 62

Three metal rods are located relative to each other as shown in Figure P17.62, where $L_{1}+$
$L_{2}=L_{3} .$ The speed of sound in a rod is given by $v=\sqrt{Y / \rho},$ where $Y$ is Young's modulus for the rod and $\rho$ is the density. Values of density and Young's modulus for the three materials are $\rho_{1}=2.70 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}, Y_{1}=7.00 \times 10^{10} \mathrm{N} / \mathrm{m}^{2}, \rho_{2}=11.3 \times$ $10^{3} \mathrm{kg} / \mathrm{m}^{3}, Y_{2}=1.60 \times 10^{10} \mathrm{N} / \mathrm{m}^{2}, \rho_{3}=8.80 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ $Y_{3}=11.0 \times 10^{10} \mathrm{N} / \mathrm{m}^{2} .$ If $L_{3}=1.50 \mathrm{m},$ what must the ratio $L_{1} / L_{2}$ be if a sound wave is to travel the length of rods 1 and 2 in the same time interval required for the wave to travel the length of rod 3$?$

Keshav Singh

Numerade Educator

Problem 63

With particular experimental methods, it is possible to produce and observe in a long, thin rod both a transverse wave whose speed depends primarily on tension in the rod and a longitudinal wave whose speed is determined by Young’s modulus and the density of the material according to the expression $v=\sqrt{Y / \rho} .$ The transverse wave can be modeled as a wave in a stretched string. A particular metal rod is 150 $\mathrm{cm}$ long and has a radius of 0.200 $\mathrm{cm}$ and a mass of 50.9 $\mathrm{g}$ . Young's modulus for the material is $6.80 \times 10^{10} \mathrm{N} / \mathrm{m}^{2}$ . What must the tension in the rod be if the ratio of the speed of longitudinal waves to the speed of transverse waves is 8.00$?$

Keshav Singh

Numerade Educator

Problem 64

Equation 17.13 states that at distance r away from a point source with power (Power) avg, the wave intensity is
$$I=\frac{(\text {Power})_{\text { arg }}}{4 \pi r^{2}}$$
Study Active Figure 17.10 and prove that at distance $r$ straight in front of a point source with power (Power) arg moving with constant speed $v_{S}$ the wave intensity is
$$I=\frac{(\text {Power})_{\text { avg }}}{4 \pi r^{2}}\left(\frac{v-v_{S}}{v}\right)$$

Victor Salazar

Numerade Educator

Problem 65

The Doppler equation presented in the text is valid when the motion between the observer and the source occurs on a straight line so that the source and observer are moving either directly toward or directly away from each other. If this restriction is relaxed, one must use the more general Doppler equation
$$f^{\prime}=\left(\frac{v+v_{O} \cos \theta_{O}}{v-v_{S} \cos \theta_{S}}\right) f$$
where $\theta_{O}$ and $\theta_{S}$ are defined in Figure $\mathrm{P} 17.65$ a. Use the preceding equation to solve the following problem. A train moves at a constant speed of $v=25.0 \mathrm{m} / \mathrm{s}$ toward the intersection shown in Figure $\mathrm{P} 17.65 \mathrm{b}$ . A car is stopped near the crossing, 30.0 m from the tracks. The train’s horn emits a frequency of 500 Hz when the train is 40.0 m from the intersection. (a) What is the frequency heard by the passengers in the car? (b) If the train emits this sound continuously and the car is stationary at this position long before the train arrives until long after it leaves, what range of frequencies do passengers in the car hear? (c) Suppose the car is foolishly trying to beat the train to the intersection and is traveling at 40.0 m/s toward the tracks. When the car is 30.0 m from the tracks and the train is 40.0 m from the intersection, what is the frequency heard by the passengers in the car now?

Dominador Tan

Numerade Educator

Problem 66

In Section 17.2, we derived the speed of sound in a gas using the impulse–momentum theorem applied to the cylinder of gas in Figure 17.5. Let us find the speed of sound in a gas using a different approach based on the element of gas in Figure 17.3. Proceed as follows. (a) Draw a force diagram for this element showing the forces exerted on the left and right surfaces due to the pressure of the gas on either side of the element. (b) By applying Newton’s second law to the element, show that
$$-\frac{\partial(\Delta P)}{\partial x} A \Delta x=\rho A \Delta x \frac{\partial^{2} s}{\partial t^{2}}$$
(c) By substituting $\Delta P=-(B \partial s / \partial x)(\text { Eq. } 17.3)$ , derive the following wave equation for sound:
$$\frac{B}{\rho} \frac{\partial^{2} s}{\partial x^{2}}=\frac{\partial^{2} s}{\partial t^{2}}$$
(d) To a mathematical physicist, this equation demonstrates the existence of sound waves and determines their speed. As a physics student, you must take another step or two. Substitute into the wave equation the trial solution $s(x, t)=s_{\text { max }} \cos (k x-\omega t)$ . Show that this function satisfies the wave equation, provided $\omega / k=v=\sqrt{B / \rho} .$

Sheh Lit Chang

University of Washington