Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 18

Sound Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

Devise a rule for finding your distance in kilometers from a lightning flash by counting the seconds from the time you see the flash until you hear the thunder. Assume that the sound travels to you along a straight line.

Prabhu Ramji

Prabhu Ramji

Numerade Educator

Problem 2

You are at a large outdoor concert, seated $300 \mathrm{~m}$ from the speaker system. The concert is also being broadcast live via satellite (at the speed of light, $3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}$ ). Consider a listener $5000 \mathrm{~km}$ away who receives the broadcast. Who hears the music first, you or the listener and by what time difference?

Prabhu Ramji

Numerade Educator

Problem 3

Two spectators at a soccer game in Montjuic Stadium see, and a moment later hear, the ball being kicked on the playing field. The time delay for one spectator is $0.23 \mathrm{~s}$ and for the other $0.12 \mathrm{~s}$. Sight lines from the two spectators to the player kicking the ball meet at an angle of $90^{\circ} .$ (a) How far is each spectator from the player? (b) How far are the spectators from each other?

Prabhu Ramji

Numerade Educator

Problem 4

A column of soldiers, marching at 120 paces per minute, keep in step with the beat of a drummer at the head of the column. It is observed that the soldiers in the rear end of the column are striding forward with the left foot when the drummer is advancing with the right. What is the approximate length of the column?

Prabhu Ramji

Numerade Educator

Problem 5

Earthquakes generate sound waves inside Earth. Unlike a gas, Earth can experience both transverse (S) and longitudinal (P) sound waves. Typically, the speed of S waves is about $4.5 \mathrm{~km} / \mathrm{s}$, and that of $\mathrm{P}$ waves $8.0 \mathrm{~km} / \mathrm{s}$. A seismograph records $\mathrm{P}$ and S waves from an earthquake. The first P waves arrive $3.0$ min before the first S waves (Fig. $18-28)$. Assuming the waves travel in a straight line, how far away does the earthquake occur?

Prabhu Ramji

Numerade Educator

Problem 6

The speed of sound in a certain metal is $v^{\text {metal }}$ One end of a long pipe of that metal of length $L$ is struck a hard blow. A listener at the other end hears two sounds, one from the wave that travels along the pipe and the other from the wave that travels through the air. (a) If $v^{\text {air }}$ is the speed of sound in air, what time interval $\Delta t$ elapses between the arrivals of the two sounds? (b) Suppose that $\Delta t=1.00 \mathrm{~s}$ and the metal is steel. Find the length $L$.

Prabhu Ramji

Numerade Educator

Problem 7

A stone is dropped into a well. The sound of the splash is heard $3.00$ s later. What is the depth of the well?

Prabhu Ramji

Numerade Educator

Problem 8

The audible frequency range for normal hearing is from about $20 \mathrm{~Hz}$ to $20 \mathrm{kHz}$. What are the wavelengths of sound waves at these frequencies?

Prabhu Ramji

Numerade Educator

Problem 9

Diagnostic ultrasound of frequency $4.50 \mathrm{MHz}$ is used to examine tumors in soft tissue. (a) What is the wavelength in air of such a sound wave? (b) If the speed of sound in tissue is $1500 \mathrm{~m} / \mathrm{s}$, what is the wavelength of this wave in tissue?

Keshav Singh

Numerade Educator

Problem 10

The pressure in a traveling sound wave is given by the equation $$\Delta P(x, t)=(1.50 \mathrm{~Pa}) \sin \pi[(0.900 \mathrm{rad} / \mathrm{m}) x-(315 \mathrm{rad} / \mathrm{s}) t]$$ Find the (a) pressure amplitude, (b) frequency, (c) wavelength, and (d) speed of the wave.

Prabhu Ramji

Numerade Educator

Problem 11

In Fig. $18-29$, two loudspeakers, separated by a distance of $2.00 \mathrm{~m}$, are in phase. Assume the amplitudes of the sound from the speakers are approximately the same at the position of a listener, who is $3.75 \mathrm{~m}$ directly in front of one of the speakers. (a) For what frequencies in the audible range $(20 \mathrm{~Hz}$ to $20 \mathrm{kHz})$ does the listener hear a minimum signal? (b) For what frequencies is the signal a maximum?

Prabhu Ramji

Numerade Educator

Problem 12

Two point sources of sound waves of identical wavelength $\lambda$ and amplitude are separated by distance $D=$
$2.0 \lambda$. The sources are in phase. (a) How many points of maximum signal (that is, maximum constructive interference) lie along a large circle around the sources? (b) How many points of minimum signal (destructive interference) lie around the circle?

Prabhu Ramji

Numerade Educator

Problem 13

Two loudspeakers are located $3.55 \mathrm{~m}$ apart on an outdoor stage. A listener is $18.3 \mathrm{~m}$ from one and $19.5 \mathrm{~m}$ from the other. During the sound check, a signal generator drives the two speakers in phase with the same amplitude and frequency. The transmitted frequency is swept through the audible range $(20 \mathrm{~Hz}$ to $20 \mathrm{kHz})$. (a) What are the three lowest frequencies at which the listener will hear a minimum signal because of destructive interference? (b) What are the three lowest frequencies at which the listener will hear a maximum signal?

Prabhu Ramji

Numerade Educator

Problem 14

Two sound waves, from two different sources with the same frequency, $540 \mathrm{~Hz}$, travel in the same direction at $330 \mathrm{~m} / \mathrm{s}$. The sources are in phase. What is the phase difference of the waves at a point that is $4.40 \mathrm{~m}$ from one source and $4.00 \mathrm{~m}$ from the other?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 15

In Fig. $18-30$, sound with a $40.0 \mathrm{~cm}$ wavelength travels rightward from a source and through a tube that consists of a straight portion and a half-circle. Part of the sound wave travels through the half-circle and then rejoins the rest of the wave, which goes directly through the straight portion. This rejoining results in interference. What is the smallest radius $r$ that results in an intensity minimum at the detector?

Prabhu Ramji

Numerade Educator

Problem 16

A $1.0 \mathrm{~W}$ point source emits sound waves isotropically. Assuming that the energy of the waves is conserved, find the intensity (a) $10 \mathrm{~m}$ from the source and (b) $2.5 \mathrm{~m}$ from the source.

Prabhu Ramji

Numerade Educator

Problem 17

A source emits sound waves isotropically. The intensity of the waves $2.50 \mathrm{~m}$ from the source is $1.91 \times 10^{-4} \mathrm{~W} / \mathrm{m}^{2}$ Assuming that the energy of the waves is conserved, find the power of the source.

Prabhu Ramji

Numerade Educator

Problem 18

Two sounds differ in sound level by $1.00 \mathrm{~dB}$. What is the ratio of the greater intensity to the smaller intensity?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 19

A certain sound source is increased in sound level by $30 \mathrm{~dB}$. By what multiple is (a) its intensity increased and (b) its pressure amplitude increased?

Sandro Maludze

Numerade Educator

Problem 20

The source of a sound wave has a power of $1.00 \mu \mathrm{W}$. If it is a point source, (a) what is the intensity $3.00 \mathrm{~m}$ away and (b) what is the sound level in decibels at that distance?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 21

(a) If two sound waves, one in air and one in (fresh) water, are equal in intensity, what is the ratio of the pressure amplitude of the wave in water to that of the wave in air? Assume the water and the air are at $20^{\circ} \mathrm{C}$. (See Table $\left.15-2 .\right)$ (b) If the pressure amplitudes are equal instead, what is the ratio of the intensities of the waves?

Prabhu Ramji

Numerade Educator

Problem 22

Assume that a noisy freight train on a straight track emits a cylindrical, expanding sound wave, and that the air absorbs no energy. How does the amplitude $\Delta P^{\max }$ of the wave depend on the perpendicular distance $r$ from the source?

Prabhu Ramji

Numerade Educator

Problem 23

Find the ratios (greater to smaller) of (a) the intensities. and (b) the pressure amplitudes for two sounds whose sound levels differ by $37 \mathrm{~dB}$.

Prabhu Ramji

Numerade Educator

Problem 24

A point source emits $30.0 \mathrm{~W}$ of sound isotropically. A small microphone intercepts the sound in an area of $0.750 \mathrm{~cm}^{2}, 200 \mathrm{~m}$ from the source. Calculate (a) the sound intensity there and (b) the power intercepted by the microphone.

Sandro Maludze

Numerade Educator

Problem 25

Figure 18-31 shows an air-filled, acoustic interferometer, used to demonstrate the interference of sound waves. Sound source $S$ is an oscillating diaphragm; $D$ is a sound detector, such as the ear or a microphone. Path $S B D$ can be varied in length, but path $S A D$ is fixed. At $D$, the sound wave coming along path $S B D$ interferes with that coming along path $S A D$. In one demonstration, the sound intensity at $D$ has a minimum value of 100 units at one position of the movable arm and continuously climbs to a maximum value of 900 units when that arm is shifted by $1.65 \mathrm{~cm}$. Find (a) the frequency of the sound emitted by the source and (b) the ratio of the amplitude at $D$ of the $S A D$ wave to that of the $S B D$ wave. (c) How can it happen that these waves have different amplitudes, considering that they originate at the same source?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 26

A violin string $15.0 \mathrm{~cm}$ long and fixed at both ends oscillates in its $n=1$ mode. The speed of waves on the string is $250 \mathrm{~m} / \mathrm{s}$, and the speed of sound in air is $348 \mathrm{~m} / \mathrm{s}$. What are (a) the frequency and (b) the wavelength of the emitted sound wave?

Sandro Maludze

Numerade Educator

Problem 27

Organ pipe $A$, with both ends open, had a fundamental frequency of $300 \mathrm{~Hz}$. The third harmonic of organ pipe $B$, with one end open, has the same frequency as the second harmonic of pipe $A$. How long are (a) pipe $A$ and (b) pipe $B$ ?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 28

The water level in a vertical glass tube $1.00 \mathrm{~m}$ long can be adjusted to any position in the tube. A tuning fork vibrating at $686 \mathrm{~Hz}$ is held just over the open top end of the tube, to set up a standing wave of sound in the air-filled top portion of the tube. (That air-filled top portion acts as a tube with one end closed and the other end open.) At what positions of the water level is there resonance?

Prabhu Ramji

Numerade Educator

Problem 29

(a) Find the speed of waves on a violin string of mass $800 \mathrm{mg}$ and length $22.0 \mathrm{~cm}$ if the fundamental frequency is $920 \mathrm{~Hz}$. (b) What is the tension in the string? For the fundamental, what is the wavelength of (c) the waves on the string and (d) the sound waves emitted by the string?

Sandro Maludze

Numerade Educator

Problem 30

A certain violin string is $30 \mathrm{~cm}$ long between its fixed ends and has a mass of $2.0 \mathrm{~g}$. The "open" string (no applied finger) sounds an A note $(440 \mathrm{~Hz})$. (a) To play a $\mathrm{C}$ note $(523$ $\mathrm{Hz}$ ), how far down the string must one place a finger? (b) What is the ratio of the wavelength of the string waves required for an A note to that required for a C note? (c) What is the ratio of the wavelength of the sound wave for an A note to that for a $\mathrm{C}$ note?

Prabhu Ramji

Numerade Educator

Problem 31

Fig. $18-32, S$ is a small loudspeaker driven by an audio oscillator and amplifier, adjustable in frequency from 1000 to $2000 \mathrm{~Hz}$ only. Tube $D$ is a piece of cylindrical sheet-metal pipe $45.7 \mathrm{~cm}$ long and open at both ends. (a) If the speed of sound in air is $344 \mathrm{~m} / \mathrm{s}$ at the existing temperature, at what frequencies will resonance occur in the pipe when the frequency emitted by the speaker is varied from $1000 \mathrm{~Hz}$ to $2000 \mathrm{~Hz}$ ? (b) Sketch the standing wave (using the style of Fig. $18-16 b$ ) for each resonant frequency.

Prabhu Ramji

Numerade Educator

Problem 32

A string on a cello has length $L$, for which the fundamental frequency is $f$. (a) By what length $l$ must the string be shortened by fingering to change the fundamental frequency to $r f ?$ (b) What is $l$ if $L=0.80 \mathrm{~m}$ and $r=1.2 ?$ (c) For $r=1.2$, what is the ratio of the wavelength of the new sound wave emitted by the string to that of the wave emitted before fingering?

Vishal Gupta

Numerade Educator

Problem 33

A well with vertical sides and water at the bottom resonates at $7.00 \mathrm{~Hz}$ and at no lower frequency. (The air-filled portion of the well acts as a tube with one closed end and one open end.) The air in the well has a density of $1.10 \mathrm{~kg} / \mathrm{m}^{3}$ and a bulk modulus of $1.33 \times 10^{5} \mathrm{~Pa}$. How far down in the well is the water surface?

Prabhu Ramji

Numerade Educator

Problem 34

A tube $1.20 \mathrm{~m}$ long is closed at one end. A stretched wire is placed near the open end. The wire is $0.330 \mathrm{~m}$ long and has a mass of $9.60 \mathrm{~g} .$ It is fixed at both ends and oscillates in its fundamental mode. By resonance, it sets the air column in the tube into oscillation at that column's fundamental frequency. Find (a) that frequency and (b) the tension in the wire.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 35

The period of a pulsating variable star may be estimated by considering the star to be executing radial longitudinal pulsations in the fundamental standing wave mode. That is, the star's radius varies periodically with time, with a displacement antinode at the star's surface. (a) Would you expect the center of the star to be a displacement node or antinode? (b) By analogy with a pipe with one open end, show that the period of pulsation $T$ is given by $$T=\frac{4 R}{\langle v\rangle}$$
where $R$ is the equilibrium radius of the star and $\langle v$ ) is the average sound speed in the material of the star. (c) Typical white dwarf stars are composed of material with a bulk modulus of $1.33 \times 10^{22} \mathrm{~Pa}$ and a density of $10^{10} \mathrm{~kg} / \mathrm{m}^{3}$. They have radii equal to $9.0 \times 10^{-3}$ solar radius. What is the approximate pulsation period of a white dwarf?

Chai Santi

Numerade Educator

Problem 36

Pipe $A$, which is $1.2 \mathrm{~m}$ long and open at both ends, oscillates at its third lowest harmonic frequency. It is filled with air for which the speed of sound is $343 \mathrm{~m} / \mathrm{s}$. Pipe $B$, which is closed at one end, oscillates at its second lowest harmonic frequency. These frequencies of pipes $A$ and $B$ happen to match. (a) If an $x$ axis extends along the interior of pipe $A$, with $x=0$ at one end, where along the axis are the displacement nodes? (b) How long is pipe $B ?$ (c) What is the lowest harmonic frequency of pipe $A$ ?

Prabhu Ramji

Numerade Educator

Problem 37

A violin string $30.0 \mathrm{~cm}$ long with linear density $0.650 \mathrm{~g} / \mathrm{m}$ is placed near a loudspeaker that is fed by an audio oscillator of variable frequency. It is found that the string is set into oscillation only at the frequencies 880 and $1320 \mathrm{~Hz}$ as the frequency of the oscillator is varied over the range $500-1500 \mathrm{~Hz}$. What is the tension in the string?

Keshav Singh

Numerade Educator

Problem 38

The A string of a violin is a little too tightly stretched. Four beats per second are heard when the string is sounded together with a tuning fork that is oscillating accurately at concert $\mathrm{A}(440 \mathrm{~Hz}) .$ What is the period of the violin string oscillation?

Prabhu Ramji

Numerade Educator

Problem 39

A tuning fork of unknown frequency makes three beats per second with a standard fork of frequency $384 \mathrm{~Hz}$. The beat frequency decreases when a small piece of wax is put on a prong of the first fork. What is the freauency of this fork?

Prabhu Ramji

Numerade Educator

Problem 40

You have five tuning forks that oscillate at close but different frequencies. What are the (a) maximum and (b) minimum number of different beat frequencies you can produce by sounding the forks two at a time depending on how the frequencies differ?

Prabhu Ramji

Numerade Educator

Problem 41

Two identical piano wires have a fundamental frequency of $600 \mathrm{~Hz}$ when kept under the same tension. What fractional increase in the tension of one wire will lead to the occurrence of 6 beats/s when both wires oscillate simultaneously?

Sandro Maludze

Numerade Educator

Problem 42

Trooper $B$ is chasing speeder $A$ along a straight stretch of road. Both are moving at a speed of $160 \mathrm{~km} / \mathrm{h}$. Trooper $B$, failing to catch up, sounds his siren again. Take the speed of sound in air to be $343 \mathrm{~m} / \mathrm{s}$ and the frequency of the source to be $500 \mathrm{~Hz}$. What is the Doppler shift in the frequency heard by speeder $A$ ?

Prabhu Ramji

Numerade Educator

Problem 43

The $16000 \mathrm{~Hz}$ whine of the turbines in the jet engines of an aircraft moving with speed $200 \mathrm{~m} / \mathrm{s}$ is heard at what frequency by the pilot of a second craft trying to overtake the first at a speed of $250 \mathrm{~m} / \mathrm{s}$ ?

Prabhu Ramji

Numerade Educator

Problem 44

An ambulance with a siren emitting a whine at $1600 \mathrm{~Hz}$ overtakes and passes a cyclist pedaling a bike at $2.44 \mathrm{~m} / \mathrm{s}$. After being passed, the cyclist hears a frequency of $1590 \mathrm{~Hz}$. How fast is the ambulance moving?

Keshav Singh

Numerade Educator

Problem 45

A whistle of frequency 540 Hz moves in a circle of radius $60.0 \mathrm{~cm}$ at a rotational speed of $15.0 \mathrm{rad} / \mathrm{s}$. What are (a) the lowest and (b) the highest frequencies heard by a listener a long distance away, at rest with respect to the center of the circle?

Prabhu Ramji

Numerade Educator

Problem 46

A stationary motion detector sends sound waves of frequency $0.150 \mathrm{MHz}$ toward a truck approaching at a speed of $45.0 \mathrm{~m} / \mathrm{s}$. What is the frequency of the waves reflected back to the detector?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 47

A French submarine and a U.S. submarine move toward each other during maneuvers in motionless water in the North Atlantic (Fig. 18-33). The French sub moves at $50.0 \mathrm{~km} / \mathrm{h}$, and the U.S. sub at $70.0 \mathrm{~km} / \mathrm{h}$. The French sub sends out a sonar signal (sound wave in water) at $1000 \mathrm{~Hz}$. Sonar waves travel at $5470 \mathrm{~km} / \mathrm{h}$. (a) What is the signal's frequency as detected by the U.S. sub? (b) What frequency is detected by the French sub in the signal reflected back to it by the U.S. sub?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 48

A sound source $A$ and a reflecting surface $B$ move directly toward each other. Relative to the air, the speed of source $A$ is $29.9 \mathrm{~m} / \mathrm{s}$, the speed of surface $B$ is $65.8 \mathrm{~m} / \mathrm{s}$, and the speed of sound is $329 \mathrm{~m} / \mathrm{s}$. The source emits waves at frequency $1200 \mathrm{~Hz}$ as measured in the source frame. In the reflector frame, what are (a) the frequency and (b) the wavelength of the arriving sound waves? In the source frame, what are (c) the frequency and (d) the wavelength of the sound waves reflected back to the source?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 49

An acoustic burglar alarm consists of a source emitting waves of frequency $28.0 \mathrm{kHz}$. What is the beat frequency between the source waves and the waves reflected from an intruder walking at an average speed of $0.950 \mathrm{~m} / \mathrm{s}$ directly away from the alarm?

Sandro Maludze

Numerade Educator

Problem 50

A bat is flitting about in a cave, navigating via ultrasonic bleeps. Assume that the sound emission frequency of the bat is $39000 \mathrm{~Hz}$. During one fast swoop directly toward a flat wall surface, the bat is moving at $0.025$ times the speed of sound in air. What frequency does the bat hear reflected off the wall?

Sandro Maludze

Numerade Educator

Problem 51

A girl is sitting near the open window of a train that is moving at a velocity of $10.00 \mathrm{~m} / \mathrm{s}$ to the east. The girl's uncle stands near the tracks and watches the train move away. The locomotive whistle emits sound at frequency $500.0 \mathrm{~Hz}$. The air is still.
(a) What frequency does the uncle hear?
(b) What frequency does the girl hear? A wind begins to blow from the east at $10.00 \mathrm{~m} / \mathrm{s}$.
(c) What frequency does the uncle now hear?
(d) What frequency does the girl now hear?

Sandro Maludze

Numerade Educator

Problem 52

A $2000 \mathrm{~Hz}$ siren and a civil defense official are both at rest with respect to the ground. What frequency does the official hear if the wind is blowing at $12 \mathrm{~m} / \mathrm{s}$ (a) from source to official and (b) from official to source?

Sandro Maludze

Numerade Educator

Problem 53

Two trains are traveling toward each other at $30.5$ $\mathrm{m} / \mathrm{s}$ relative to the ground. One train is blowing a whistle at $500 \mathrm{~Hz}$.
(a) What frequency is heard on the other train in still air? (b) What frequency is heard on the other train if the wind is blowing at $30.5$ $\mathrm{m} / \mathrm{s}$ toward the whistle and away from the listener? (c) What frequency is heard if the wind direction is reversed?

Prabhu Ramji

Numerade Educator

Problem 54

A bullet is fired with a speed of $685 \mathrm{~m} / \mathrm{s}$. Find the half angle made by the shock cone with the line of motion of the bullet.

Prabhu Ramji

Numerade Educator

Problem 55

A jet plane passes over you at a height of $5000 \mathrm{~m}$ and a speed of Mach 1.5. (a) Find the Mach cone half angle. (b) How long after the jet passes directly overhead does the shock wave reach you? Use $331 \mathrm{~m} / \mathrm{s}$ for the speed of sound.

Prabhu Ramji

Numerade Educator

Problem 56

A plane flies at $1.25$ times the speed of sound. Its sonic boom reaches a man on the ground $1.00$ min after the plane passes directly overhead. What is the altitude of the plane? Assume the speed of sound to be $330 \mathrm{~m} / \mathrm{s}$.

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 57

You decide to build a pipe organ in your dormitory room using PVC pipe. Estimate whether you could build an organ that would cover the entire range of human hearing without bending any pipes.

Salamat Ali

Numerade Educator

Problem 58

You have set up two stereo speakers on your back patio railing as shown in the top view diagram in Fig. 18-34. You are worried that at certain positions you will lose frequencies as a result of interference. The coordinate grid on the edge of the picture has its large tick marks separated by 1 meter. For ease of calculation, make the following assumptions:
- Assume that the relevant objects lie on integer or half-integer grid points of the coordinate system.
- Take the speed of sound to be $343 \mathrm{~m} / \mathrm{s}$.
- Ignore the reflection of sound from the house, trees, and so on.
- The speakers are in phase.
(a) What will happen if you are sitting in the middle of the bench?
(b) If you are sitting in the lawn chair on the left, what will be the lowest frequency you will lose to destructive interference?
(c) Can you restore the frequency lost in part (a) by switching the leads to one of the speakers, thereby reversing the phase of that source?
(d) With the leads reversed, what will happen to the sound for a person sitting at the center of the bench?

Mohit Khurana

Texas A&M University

Problem 59

A salesperson claimed that a stereo system had a maximum audio power of $120 \mathrm{~W}$. Testing the system with several speakers set up so as to simulate a point source, the consumer noted that she could get as close as $1.2 \mathrm{~m}$ with the volume full on before the sound hurt her ears. Was the salesperson truthful? Explain your answer with a calculation.

Shoukat Ali

Other Schools

Problem 60

An experimenter wishes to measure the speed of sound in an aluminum rod $10 \mathrm{~cm}$ long by measuring the time it takes for a sound pulse to travel the length of the rod. If results good to four significant figures are desired, how precisely must the length of the rod be known and how closely must the experimenter be able to resolve time intervals?

Mark J

Numerade Educator