Answers to Got the Concept? Questions 575
Answer to What do you think? Question
(b) We learn in Section 13-3 that the speed of waves on a string
(or a flagellum) does not depend on the frequency of the wave.
If the frequency increases, the speed remains the same but the
wavelength (the distance from one crest to the next) decreases.
Answers to Got the Concept? Questions
13-1 (c) In the transverse wave shown in Figure 13-2a the
individual pieces of rope (the wave medium) move vertically.
So the restoring force that influences their motion must also
act vertically. In the same way, since the pieces of the spring in
Figure 13-2b move horizontally, the restoring force that acts on
them must be in the horizontal direction.
13-2 (e) The propagation speed vp of sound waves does not
depend on the frequency f. The frequency and the wavelength
l are related by Equation 13-2, vp = fl, so increasing the frequency makes the wavelength shorter but has no effect on the
propagation speed.
13-3 (a) The speed of light is so great that you see the batter swing a tiny fraction of a second after the swing took
place. The speed of sound is relatively slow by comparison.
From the relationship d = vsound t with distance d, propagation speed vsound, and time t, the distance that sound travels in
t = 0.20 s is d = 1343 m>s2 10.20 s2 = 69 m. Note that the
time for light to travel this same distance is tlight = d>vlight =
169 m2 > 13.0 * 108 m>s2 = 2.3 * 10-7 s, or about 1>4 of a
microsecond (1 microsecond = 1 ms = 1026 s).
13-4 (b) Increasing the frequency decreases the wavelength.
The wavelength sets the scale for the entire interference pattern (the positions of destructive and constructive interference
depend on the wavelength), so decreasing the wavelength will
make all of the distances smaller and the interference pattern
will shrink inward toward the two fingers.
13-5 (e) From Equation 13-18 the wavelength of the n = 1 stand­
ing wave mode on a string of length L is given by L = l>2, so
l = 2L. This does not depend on the tension in the string, so the
wavelength remains the same.
13-6 (b) From Equation 13-19 the frequency of each standing
wave mode on the string is proportional to the square root of
the tension F. So doubling the tension will cause the frequency
to increase not by a factor of 2, but by a factor of 12 = 1.41.
13-7 (a) The bottle is like an air-filled pipe that is open at the
top and closed on the bottom (at the surface of the water). The
fundamental frequency of this pipe is inversely proportional
to the length of the pipe (see Equation 13-22). As you fill the
bottle, the length of the air-filled pipe decreases so that the fundamental frequency goes up. The pitch of the sound from the
bottle depends on the fundamental frequency, so the pitch goes
up as well.
13-8 (e) The beat frequency equals the difference between the
frequencies of the two waves that interfere. The beat frequency
here is 3 Hz, so the second tuning fork has a frequency of either
440 Hz + 3 Hz = 443 Hz or 440 Hz 2 3 Hz = 437 Hz. Without
more information we can’t tell which of these two is correct.
13-9 (d) The sound waves from each fan carry the same power,
so at any distance from the fans the intensity from each fan is
the same. Thus the total intensity doubles when both fans are
running. We saw in Example 13-11 that doubling the intensity
corresponds to adding 3.0 dB to the sound intensity level. So the
sound intensity level due to both fans is 60 dB + 3.0 dB = 63 dB.
13-10 (a) The frequency you hear is higher while you’re running
compared to when you’re standing still. Because you are moving
toward the source of the sound, you will intercept more wave
crests in a given amount of time than you would were you standing still. The qualitative approach we took using Figure 13-24 for
a moving source of sound and a stationary listener applies equally
well for a stationary source and a moving listener.
13-11 (c) No matter how a source of sound waves is moving, the waves spread outward at the speed of sound from the
point where they were emitted. We used this idea in drawing
Figure 13-26.
Questions and Problems
In a few problems, you are given more data than you actually
need; in a few other problems, you are required to supply data
from your general knowledge, outside sources, or informed estimate. Interpret as significant all digits in numerical values that
have trailing zeros and no decimal points.
For all problems, use g = 9.80 m>s2 for the free-fall acceleration due to gravity. Neglect friction and air resistance unless
instructed to do otherwise.
• Basic, single-concept problem
•• Intermediate-level problem, may require synthesis of concepts and multiple steps
••• Challenging problem
SSM Solution is in Student Solutions Manual
Conceptual Questions
1. •Explain the difference between longitudinal waves and
transverse waves and give two examples of each.
Freed_c13_524-581_st_hr1.indd 575
2. •Discuss several ways that the human body creates or
r­ esponds to waves.
3. •When you talk to your friend, are the air molecules that
reach his ear the same ones that were in your lungs? Explain
your answer. SSM
4. •Draw a sketch of a transverse wave and label the amplitude,
wavelength, a crest, and a trough.
5. •If a tree falls in the forest and no humans are there to hear
it, was any sound produced?
6. •Are water waves longitudinal or transverse? Explain your
answer.
7. •A sound wave passes from air into water. Give a qualitative
explanation of how these properties change: (a) wave speed,
(b) frequency, and (c) wavelength of the wave. SSM
8. •Two solid rods have the same Young’s modulus, but one
has larger density than the other. In which rod will the speed
4/18/13 5:45 PM
576 Chapter 13 Waves
of longitudinal waves be greater? Explain your answer, making
reference to the variables that affect the wave speed.
9. •Earthquakes produce several types of wave. The most significant are the primary wave (or P wave) and the secondary
wave (or S wave). The primary wave is a longitudinal wave that
can travel through liquids and solids. The secondary wave is a
transverse wave that can only travel through solids. The speed
of a P wave usually falls between 1000 m>s and 8000 m>s; the
speed of an S wave is around 60–70% of the P wave speeds. For
any given seismic event, discuss the damage that might be due
to P waves and how that would differ when compared to the
damage due to S waves.
10. •(a) What is a transverse standing wave? (b) For a string
stretched between two fixed points, describe how a disturbance
on the string might lead to a standing wave.
11. •Explain the differences and similarities between the concepts of frequency f and angular frequency v. SSM
12. •Search the Internet for the words rarefaction, phonon, and
compression in the context of longitudinal waves. ­Describe
how a longitudinal wave is made up of phonons and explain
the connection between rarefaction and compression in such
a wave.
13. •If you stand beside a railroad track as a train sounding
its whistle moves past, you will experience the Doppler effect.
­Describe any changes in the perceived sound that a person riding on the train will hear.
14. •(a) Describe in words the nature of a sonic boom. (b) Now,
referring to the formula for the Doppler shift, explain the
phenomenon.
15. •Explain the concept of phase difference, f, and predict
the outcome of two identical waves interfering when f = 0°,
f = 90°, f = 180°, f = 270°, and f = 300°.
16. •Two pianists sit down to play two identical pianos.
However, a string is out of tune on Elizabeth’s piano. The G #3
key (208 Hz) appears to be the problem. When Greg plays the
note on his piano and Elizabeth plays hers, a beat frequency
of 6 Hz is heard. Luckily, a piano tuner is present, and she is
ready to correct the problem. However, in all the confusion,
she inadvertently increases the tension in Greg’s G #3 string
by a factor of 1.058. Now, both pianos are out of tune! Yet
oddly, when Elizabeth plays her G #3 note and Greg plays his
G #3, there is no beat frequency. Explain what happened.
17. •How does the length of an organ pipe determine the fundamental frequency?
18. •Why do the sounds emitted by organ pipes that are closed
on one end and open on the other not have even harmonics?
Include in your explanation a sketch of the resonant waves that
are formed in the pipe.
Multiple-Choice Questions
21. •A visible disturbance propagates around a crowded soccer
stadium when fans, section by section, jump up and then sit
back down. What type of wave is this?
A. longitudinal wave
B. transverse wave
C. polarized wave
D. spherical wave
E. polarized spherical wave
22. •Two point sources produce waves of the same wavelength
that are in phase. At a point midway between the sources, you
would expect to observe
A. constructive interference.
B. destructive interference.
C. alternating constructive and destructive interference.
D. constructive or destructive interference depending on
the wavelength.
E. no interference.
23. •Standing waves are set up on a string that is fixed at both
ends so that the ends are nodes. How many nodes are there in
the fourth mode?
A. 2
B. 3
C. 4
D. 5
E. 6 SSM
24. •Which of the following frequencies are higher harmonics
of a string with fundamental frequency of 80 Hz?
A. 80 Hz
B. 120 Hz
C. 160 Hz
D. 200 Hz
E. 220 Hz
25. •A trombone has a variable length. When a musician
blows air into the mouthpiece and causes air in the tube of the
horn to vibrate, the waves set up by the vibrations reflect back
and forth in the horn to create standing waves. As the length
of the horn is made shorter, what happens to the frequency?
A. The frequency remains the same.
B. The frequency will increase.
C. The frequency will decrease.
D. The frequency will increase or decrease depending on
how hard the horn player blows.
E. The frequency will increase or decrease depending on
the diameter of the horn.
19. •(a) Explain the differences and similarities among the
concepts of intensity, sound level, and power. (b) What happens to intensity as the source of sound moves closer to the
observer? (c) What happens to sound level? (d) What happens
to power? SSM
26. •Two tuning forks of frequency 480 Hz and 484 Hz are
struck simultaneously. What is the beat frequency resulting
from the two sound waves?
A. 964 Hz
B. 482 Hz
C. 4 Hz
D. 2 Hz
E. 0 Hz
20. •A car radio is tuned to receive a signal from a particular
radio station. While the car slows to a stop at a traffic signal, the
reception of the radio seems to fade in and out. Use the concept
of interference to explain the phenomenon. Hint: In broadcast
technology, the phenomenon is known as multipathing.
27. •If a source radiates sound uniformly in all directions, and
you triple your distance from the sound source, what happens
to the sound intensity at your new position?
A. The sound intensity increases to three times its
original value.
Freed_c13_524-581_st_hr1.indd 576
4/18/13 5:45 PM
Questions and Problems 577
B. The sound intensity does not change.
C. The sound intensity drops to 1>3 its original value.
D. The sound intensity drops to 1>9 its original value.
E. The sound intensity drops to 1>27 its original
value. SSM
28. •If the amplitude of a sound wave is tripled, the intensity will
A. decrease by a factor of 3.
B. increase by a factor of 3.
C. remain the same.
D. decrease by a factor of 9.
E. increase by a factor of 9.
29. •A certain sound level is increased by an additional 20 dB.
By how much does the intensity increase?
A. The intensity increases by a factor of 2.
B. The intensity increases by a factor of 20.
C. The intensity increases by a factor of 100.
D. The intensity increases by a factor of 200.
E. The intensity does not increase.
30. •A person sitting in a parked car hears an approaching ambulance siren at a frequency f1, and as it passes him and moves
away he hears a frequency f2. The actual frequency f of the
source is
A. f 7 f1
B. f 6 f2
C. f2 6 f 6 f1
D. f = f2 + f1
E. f = f2 – f1
Estimation/Numerical Analysis
31. •Estimate the size of the human voice box using the
fact that the human ear is most sensitive to sounds of about
3000 Hz. SSM
32. •(a) Estimate the distance that sound waves travel in 10 s.
(b) Explain why you might get {5% of the calculated distance
on any given day that you perform an experiment to measure
the speed of sound.
33. •Estimate the number of beats per second of a hummingbird’s wings.
34. •(a) Estimate the speed of a human wave like those seen at
a large sports venue. (b) How would you define the corresponding concepts of wavelength, frequency, and amplitude for the
human wave?
35. •Describe how you might estimate the speed of the transverse waves on a violin string. SSM
36. •Biology Peristalsis is the rhythmic, wavelike contraction of smooth muscles to propel material through the
digestive tract. A typical peristaltic wave will only last for
a few ­seconds in the small intestine, traveling at only a few
centimeters per second. Estimate the wavelength of the digestive wave.
37. •Estimate the difference in time required for the sound of
the “crack” of the bat to reach the left field bleacher seats on a
calm day compared to a blustery day with a stiff breeze blowing out to left field.
38. •(a) Estimate the amplitude, frequency, and wavelength of
a typical ocean wave, far at sea, away from coastlines. (b) What
is the typical speed of ocean waves?
39. •(a) Using a graphing calculator, spreadsheet, or graphing
program, graph the wave function, y(x), that describes the trav-
Freed_c13_524-581_st_hr1.indd 577
eling wave described by the data
shown. Assume the data were
taken at time t = 0 s. (b) Write
a mathematical function that
­describes the traveling wave if
the period of the motion is 2 s.
x (m) y (m) x (m) y (m)
0
1
2
3
4
5
6
0
4.5
7.8
9
7.8
4.5
0
7
8
9
10
11
12
24.5
27.8
29
27.8
24.5
0
Problems
13-1 Waves are disturbances that travel from place to place
13-2 Mechanical waves can be transverse, longitudinal, or a
combination of these
13-3 Sinusoidal waves are related to simple harmonic motion
40. •The period of a sound wave is 0.01 s. Calculate the frequency f and the angular frequency v.
41. •A wave on a string propagates at 22 m>s. If the frequency is 24 Hz, calculate the wavelength and angular wave number. SSM
42. ••A transverse wave on a string has an amplitude of 20 cm,
a wavelength of 35 cm, and a frequency of 2.0 Hz. Write the
mathematical description of the displacement from equilibrium
for the wave if (a) at t = 0, x = 0 and y = 0; (b) at t = 0, x = 0
and y = +20 cm; (c) at t = 0, x = 0 and y = 220 cm; and (d) at
t = 0, x = 0 and y = 12 cm.
43. •Show that the dimensions of speed (distance>time) are
consistent with both versions of the expression for the propav
gation speed of a wave: vp =
and vp = lf.
k
44. ••Write the wave equation for a periodic transverse wave
traveling in the positive x direction at a speed of 20 m>s if it
has a frequency of 10 Hz and an amplitude in the y direction
of 0.10 m.
45. •A wave on a string is described by the equation y 1x,t2 =
10.5 m2 cos 3 11.0 rad>m2x - 110 rad>s2t 4 . What are (a) the
frequency, (b) wavelength, and (c) speed of the wave? SSM
46. ••The pressure wave that travels along the inside of an organ
pipe is given in terms of distance and time by the following function p1x,t2 = 11.0 atm2 sin 3 16.0 rad>m2x - 14.0 rad>s2t 4 .
(a) What is the pressure amplitude of the wave? (b) What is
the wave number of the wave? (c) What is the frequency of
the wave? (d) What is the speed of the wave? (e) What is the
­corresponding spatial displacement, s(x,t), from equilibrium
for the pressure wave, assuming the maximum displacement
is 2.0 cm?
47. ••The equation for a particular wave is y(x,t) = (0.1 m)cos
(kx 2 vt). If the frequency of the wave is 2.0 Hz, what is the
value of y at x = 0 when t = 4.0 s?
48. •Using the graph in
Figure 13-29, write the
mathematical description of the wave if the
period of the motion is
4 s and the wave moves
to the right (toward the
positive x direction).
t = 0 s:
y
+2 m
x
–2 m
2m
Figure 13-29 ​Problem 48
4/18/13 5:45 PM
t = 0 s:
578 Chapter 13 Waves
y
+10 cm
x
49. •Write a mathematical description of the wave using
the
20 cm
–10 cm
graphs in Figure 13-30. SSM
t = 0 s:
x = 0 m:
y
+10 cm
y
+10 cm
x
20 cm
–10 cm
t
–10 cm
2s
Figure
x = 0 m: 13-30 ​Problem 49
y
59. •In Figure 13-32,
rectangular waveforms
are approaching each
other. For each case, use
the ideas of interference
to predict the superposed
wave that results when
the two waves are coincident. Describe the superposed wave graphically
and with text. SSM
(a)
A1
(b)
A1
(c)
2s
–10
50.
•Ifcmthe Young’s modulus for liquid A is twice that of liquid B,
and the density of liquid A is one-half of the density of liquid B,
what does the ratio of the speeds of sound in the two liquids
1vA >vB 2 equal?
53. ••The violin is a four-stringed instrument tuned so that the
ratio of the frequencies of adjacent strings is 3 to 2. (This is the
ratio when taken as high frequency to lower frequency.) If the
diameter of the E string (the highest frequency) on a violin is
0.25 mm, find the diameters of the remaining strings (A, D, and
G), assuming they are tuned as indicated (what musicians call
intervals of a perfect fifth), they are made of the same material,
and they all have the same tension.
54. •At room temperature the bulk modulus of glycerine is
about 4.35 * 109 N>m2 and the density of glycerine is about
1260 kg>m3. Calculate the speed of sound in glycerin.
55. •The Young’s modulus of water is 2.2 * 109 N>m2. The
density of water is 1000 kg>m3. Calculate the speed of sound
in water. SSM
56. •When sound travels through the ocean, where the Young’s
modulus is 2.34 * 109 N>m2, the wavelength associated with
1000 Hz waves is 1.51 m. Calculate the density of seawater.
58. •Two waves interfere at the point x in Figure 13-31. The
resultant wave is shown. Draw the three possible shapes for the
two waves that interfere to produce this outcome. In each case
one wave should head toward the right and one wave should
head toward the left.
x
Freed_c13_524-581_st_hr1.indd 578
Figure 13-31 ​Problem 58
A2 = 1.5A1
(d)
A1
A1 = 1.5A2
A2
Figure 13-32 ​Problem 59
60. •Construct the
resultant wave that is
formed when the two
waves shown in each
case occupy the same
space and interfere
(Figure 13-33).
(a)
(b)
(c)
A
A
1
–A
2
A
A
A
(d)
1
–A
2
A
Figure 13-33 ​Problem 60
61. ••Two identical speakers (1 and 2) are playing a tone with a
frequency of 171.5 Hz, in phase (Figure 13-34). The speakers are
located 6 m apart. Determine what points (A, B, C, D, or E, all
separated by 1 m) will experience constructive interference along
the line that is 6 m in front of the speakers. Point A is directly in
front of speaker 1. The speed of sound is 343 m>s. SSM
6m
57. •What is the speed of sound in gasoline? The Young’s modulus for gasoline is 1.3 * 109 N>m2. The density of gasoline is
0.74 kg>L.
13-5 When two waves are present simultaneously, the total
disturbance is the sum of the individual waves
A2
A1
+10 cm
52. •A long rope is shaken up and down by a rodeo contestant.
The transverse waves travel 12.8 m in a time of 2.1 s. If the
tension in the rope is 80 N, calculate the mass per unit length
for the rope.
A1 = A2
A2
13-4 The propagation speed of a wave depends on the
t
properties of the wave medium
51. •A string that has a mass of 5.0 g and a length of 2.2 m
is pulled taut with a tension of 78 N. Calculate the speed of
transverse waves on the string. SSM
A2 A = A
1
2
1
2
6m
A
B
C
D
E
Figure 13-34 ​Problem 61
13-6 A standing wave is caused by interference between waves
traveling in opposite directions
62. •A string is tied at both ends, and a standing wave is established. The length of the string is 2 m, and it vibrates in the
4/18/13 5:45 PM
Questions and Problems 579
fundamental mode (n = 1). If the speed of waves on the string
is 60 m>s, calculate the frequency and wavelength of the waves.
63. •A string is fixed on both ends with a standing wave vibrating in the fourth harmonic. Draw the shape of the wave and
label the location of antinodes (A) and nodes (N).
64. •A 2.35-m-long string is tied at both ends and it vibrates
with a fundamental frequency of 24 Hz. Find the frequencies
and make a sketch of the next four harmonics. What is the
speed of waves on the string?
65. •A string that is 1.25 m long has a mass of 0.0548 kg and
a tension of 200 N. The string is tied at both ends and vibrated
at various frequencies. (a) What frequencies would you need to
apply to excite the first four harmonics? (b) Make a sketch of
the first four harmonics for these standing waves. SSM
66. ••A string of length L is tied at both ends, and a harmonic
mode is created with a frequency of 40 Hz. If the next successive harmonic is at 48 Hz and the speed of transverse waves on
the string is 56 m>s, find the length of the string. Note that the
fundamental frequency is not necessarily 40 Hz.
67. ••An object of mass M is used to provide tension in a
4.5-m-long string that has a mass of 0.252 kg, as shown in
Figure 13-35. A standing wave that has a wavelength equal to
1.5 m is produced by a source that vibrates at 30 Hz. Calculate
the value of M.
M
Figure 13-35 ​Problem 67
68. ••A guitar string has a mass per unit length of 2.35 g>m.
If the string is vibrating between points that are 60 cm apart,
find the tension when the string is designed to play a note of
440 Hz (A4).
69. ••An E string on a violin has a diameter of 0.25 mm and is
made from steel (density of 7800 kg>m3). The string is designed
to sound a fundamental note of 660 Hz, and its unstretched
length is 32.5 cm. Calculate the tension in the string. SSM
13-7 Wind instruments, the human voice, and the human ear
use standing sound waves
70. •An organ pipe of length L sounds its fundamental tone at
40 Hz. The pipe is open on both ends, and the speed of sound
in air is 343 m>s. Calculate (a) the length of the pipe and (b) the
frequency of the first four harmonics.
71. •An organ pipe sounds two successive tones at 228.6 Hz
and 274.3 Hz. Determine whether the pipe is open at both ends
or open at one end and closed at the other. SSM
72. •A narrow glass tube is 0.40 m long and sealed on the bottom end. It is held under a loudspeaker that sounds a tone at
220 Hz, causing the tube to radically resonate in its first harmonic. Find the speed of sound in the room.
73. •The third harmonic of an organ pipe that is open at both
ends and is 2.25 m long excites the fourth harmonic in another
organ pipe. Determine the length of the other pipe and whether
it is open at both ends or open at one end and closed at the
other. Assume the speed of sound is 343 m>s in air.
74. •Biology If the human ear canal with a typical length of
about 2.8 cm is regarded as a tube open at one end and closed at
Freed_c13_524-581_st_hr1.indd 579
the eardrum, what is the fundamental frequency that we should
hear best? Assume that the speed of sound is 343 m>s in air.
75. •Biology A male alligator emits a subsonic mating call that
has a frequency of 18 Hz by taking a large breath into his chest
cavity and then releasing it. If the hollow chest cavity of an alligator behaves approximately as a pipe open at only one end,
estimate its length. Is your answer consistent with the typical
size of an alligator?
76. •Biology The trunk of a very large elephant may extend up
to 3 m! It acts much like an organ pipe only open at one end
when the elephant blows air through it. (a) Calculate the fundamental frequency of the sound the elephant can create with its
trunk. (b) If the elephant blows even harder, the next harmonic
may be sounded. What is the frequency of the first overtone?
77. •The longest pipe in the Mormon Tabernacle Organ in
Salt Lake City has a “speaking length” of 9.75 m; the smallest pipe is 1.91 cm long. Assuming that the speed of sound is
343 m>s, determine the range of frequencies that the organ
can produce if both open and closed pipes are used with these
lengths. SSM
13-8 Two sound waves of slightly different frequencies
produce beats
78. •The sound from a tuning fork of 440 Hz produces beats
against the unknown emissions from a vibrating string. If beats
are heard at a frequency of 4 Hz, what is the vibrational frequency of the string?
79. •A guitar string is “in tune” at 440 Hz. When a standardized tuning fork rated at 440 Hz is simultaneously sounded
with the guitar string, a beat frequency of 5 Hz is heard. (a)
How far out of tune is the string? (b) What should you do to
correct this?
80. •Two tuning forks are both rated at 256 Hz, but when they
are struck at the same time, a beat frequency of 4 Hz is created.
If you know that one of the tuning forks is in tune (but you are
not sure which one), what are the possible values of the “outof-tune” fork?
81. •A guitar string has a tension of 100 N and is supposed
to have a frequency of 110 Hz. When a standard tone of that
value is sounded while the string is plucked, a beat frequency
of 2 Hz is heard. The peg holding the string is loosened
­(decreasing the tension), and the beat frequency increases.
What should the tension in the string be in order to achieve
perfect pitch? SSM
82. ••Two identical guitar strings under 200 N of tension produce sound with frequencies of 290 Hz. If the tension in one
string drops to 196 N, what will be the frequency of beats when
the two strings are plucked at the same time?
13-9 The intensity of a wave equals the power that it delivers
per square meter
83. •By what factor should you move away from a point source
of sound waves in order to lower the intensity by (a) a factor
of 10? (b) A factor of 3? (c) A factor of 2? (d) Discuss how this
problem would change if it were a point source that was radiating into a hemisphere instead of a sphere.
84. •Calculate the ratio of the acoustic power generated by a
blue whale (190 dB) compared to the sound energy generated
by a jackhammer (105 dB). Assume that the receiver of the
sound is the same distance from the two sources of sound.
4/18/13 5:45 PM
580 Chapter 13 Waves
85. ••The steady drone of rush hour traffic persists on a stretch
of freeway for 4 h each day. A nearby resident undertakes a plan
to harness the wasted sound energy and use it for her home. She
mounts a 1-m2 “microphone” that absorbs 30% of the sound
that hits it. If the ambient sound level is 100 dB at the microphone, calculate the amount of sound energy that is collected. Do
you think her plan is “sound”? How could you improve it? SSM
86. •A longitudinal wave has a measured sound level of 85 dB
at a distance of 3 m from the speaker that created it. (a) Calculate the intensity of the sound at that point. (b) If the speaker
were a point source of sound energy, find its power output.
87. •Biology A single goose sounds a loud warning when an intruder enters the farmyard. Some distance from the goose, you
measure the sound level of the warning to be 88 dB. If a gaggle
of 30 identical geese is present, and they are all approximately
the same distance from you, what will the collective sound level
be if they all sound off simultaneously? Neglect any interference effects.
88. •Two sound levels, b1 and b2, can be compared relative to
each other (rather than to some standardized intensity threshold). Find a formula for this by calculating  b = b2 – b1. You
will need to employ your knowledge of basic logarithm operations to derive the formula.
89. •By what factor must you increase the intensity of a sound
in order to hear (a) a 1-dB rise in the sound level? (b) What
about a 20-dB rise? SSM
90. •Biology The intensity of a certain sound at your eardrum
is 0.003 W>m2. (a) Calculate the rate at which sound energy
hits your eardrum. Assume that the area of your eardrum is
about 55 mm2. (b) What power output is required from a point
source that is 2 m away in order to create that intensity?
91. •Biology The area of a typical eardrum is 5.0 * 1025 m2.
Find the sound power incident on an eardrum at the threshold
of pain.
13-10 The frequency of a sound depends on the motion of the
source and the listener
92. •A fire engine’s siren is 1600 Hz when at rest. What frequency do you detect if you move with a speed of 28 m>s (a)
toward the fire engine, and (b) away from it? Assume that the
speed of sound is 343 m>s in air.
93. •Medical An ultrasound machine can measure blood flow
speeds. Assume the machine emits acoustic energy at a frequency of 2.0 MHz, and the speed of the wave in human tissue is
taken to be 1500 m>s. What is the beat frequency between the
transmitted and reflected waves if blood is flowing in large arteries at 0.020 m>s directly away from the sound source?
94. ••A car sounding its horn (rated by the manufacturer at
600 Hz) is moving at 20 m>s toward the east. A stationary
observer is standing due east of the oncoming car. (a) What
frequency will he hear assuming that the speed of sound is
343 m>s? (b) What if the observer is standing due west of the
car as it drives away?
95. ••A bicyclist is moving toward a sheer wall while holding
a tuning fork rated at 484 Hz. If the bicyclist detects a beat
frequency of 6 Hz (between the waves coming directly from the
tuning fork and the echo waves coming from the sheer wall),
calculate the speed of the bicycle. Assume the speed of sound
is 343 m>s. SSM
Freed_c13_524-581_st_hr1.indd 580
96. ••Medical An ultrasonic scan uses the echo waves coming
from something moving (such as the beating heart of a fetus)
inside the body and the waves that are directly received from
the transmitter to form a measurable beat frequency. This allows the speed of the internal structure to be isolated and analyzed. What is the beat frequency detected when waves with a
frequency of 5 MHz are used to scan a fetal heartbeat (moving
at a speed of {10 cm>s)? The speed of the ultrasound waves in
tissue is about 1540 m>s.
97. ••Biology A bat emits a high-pitched squeal at 50,000 Hz
as it approaches an insect at 10 m>s. The insect flies away from
the bat, and the reflected wave that echoes off the insect returns
to the bat at a frequency of 50,050 Hz. Calculate the speed of
the insect as it tries to avoid being the bat’s next meal. (A bat
can eat over 3000 mosquitoes in one night!) SSM
General Problems
98. ••A large volcanic eruption triggers a tsunami. At a seismic
station 250 km away, the instruments record that the time difference between the arrival of the tidal wave and the arrival
of the sound of the explosion is 9.25 min. Tsunamis typically
travel at approximately 800 km>h. (a) Which sound arrives
first, the sound in the air or in the water? Prove your answer
numerically. (b) How long after the explosion does it take for
the first sound wave to reach the seismic station? (c) How long
after the explosion does it take for the tsunami to reach the
seismic station?
99. ••A transverse wave is propagating according to the following wave function:
y 1x,t2 = 11.25 m2 cos 3 15.00 rad>m2x - 14.00 rad>s2t 4 (a)
Plot a graph of y versus x when t = 0 s. (b) Repeat when
t = 1 s.
100. •Medical A diagnostic sonogram produces a picture of
internal organs by passing ultrasound through the tissue. In one
application, it is used to find the size, location, and shape of
the prostate in preparation for surgery or other treatment. The
speed of sound in the prostate is 1540 m>s, and a diagnostic
sonogram uses ultrasound of frequency 2.00. MHz. The density of the prostate is 1060 kg>m3. (a) What is the wavelength
of the sonogram ultrasound? (b) What is Young’s modulus for
the prostate gland?
101. ••You may have seen a demonstration in which a person
inhales some helium and suddenly speaks in a high-pitched
voice. Let’s investigate the reason for the change in pitch. At
0°C, the density of air is 1.40 kg>m3, the density of helium is
0.1786 kg>m3, the speed of sound in helium is 972 m>s, and the
speed of sound in air is 331 m>s. (a) What is the bulk modulus
of helium at 0°C? (b) If a person produces a sound of frequency
0.500 kHz while speaking with his lungs full of air, what frequency sound will that person produce if his respiratory tract is
filled with helium instead of air? (c) Use your result to explain
why the person sounds strange when he breathes in helium. SSM
102. •Biology When an insect ventures onto a spiderweb,
a slight vibration is set up alerting the spider. The density of
spider silk is approximately 1.3 g>cm3, and its diameter varies
considerably depending on the type of spider, but 3.0 mm is
typical. If the web is under a tension of 0.50 N when a small
beetle crawls onto it 25 cm from the spider, how long will it
take for the spider to receive the first waves from the beetle?
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Questions and Problems 581
103. ••If two musical notes are an octave apart, the frequency
of the higher note is twice that of the lower note. The note
concert A usually has a frequency of 440 Hz (although there is
some variation). (a) What is the frequency of a note that is two
octaves above (higher than) concert A in pitch? (b) If a certain
string on a viola is tuned to middle C by adjusting its tension
to T, what should be the tension (in terms of T) of the string so
that it plays a note one octave below middle C? SSM
104. •••In Western music, the octave is divided into 12 notes
as follows: C, C # >D b, D, D # >E b, E, F, F # >Gb, G, G# >Ab, A,
A# >B b, C9. Note that some of the notes are the same, such as C#
and Db. Each of the 12 notes is called a semitone. In the ideal
tempered scale, the ratio of the frequency of any semitone to
the frequency of the note below it is the same for all pairs of
adjacent notes. So, for example, fD >fC# is the same as fA# >fA.
(a) Show that the ratio of the frequency of any semitone to the
frequency of the note just below it is 21>12. (b) If A is 440 Hz,
what is the frequency of F# in the tempered scale? (c) If you
want to tune a string from Bb to B by changing only its tension, by what ratio should you change the tension? Should you
­increase or decrease the tension?
105. •Find the temperature in an organ loft in Vancouver, British Columbia, if the 5th overtone associated with the pipe that
is resonating corresponds to 1500 Hz. The pipe is 0.7 m long,
and its type (open–open or open–closed) is not specified. The
speed of sound in air depends on the centigrade temperature
(T) according to the following:
v1T2 = 2109,700 + 402T.
106. •Biology An adult female ring-necked duck is typically
16 in long, and the length of her bill plus neck is about 5.0 cm
long. (a) Calculate the expected fundamental frequency of the
quack of the duck. For a rough but reasonable approximation,
assume that the sound is only produced in the neck and bill.
(b) An adult male ring-necked duck is typically 18 in long. If
its other linear dimensions are scaled up in the same ratio from
those of the female, what would be the fundamental frequency
of its quack? (c) Which would produce a higher-pitch quack,
the male or female?
107. •When a worker puts on earplugs, the sound level of a
jackhammer decreases from 105 dB to 75 dB. Then the worker
moves twice as far from the sound. Determine the sound level
at the new location with the earplugs.
108. ••Medical High-intensity focused ultrasound (HIFU) is
one treatment for certain types of cancer. During the procedure, a narrow beam of high-intensity ultrasound is focused
on the tumor, raising its temperature to nearly 90°C and killing it. A range of frequencies and intensities can be used, but
in one treatment a beam of frequency 4.0 MHz produced an
intensity of 1500 W>cm2. The energy was delivered in short
pulses for a total time of 2.5 s over an area measuring 1.4 mm
by 5.6 mm. The speed of sound in the soft tissue was 1540 m>s,
and the density of that tissue was 1058 kg>m3. (a) What was
the wavelength of the ultrasound beam? (b) How much energy
was delivered to the tissue during the 2.5-s treatment? (c) What
was the maximum displacement of the molecules in the tissue
as the beam passed through?
109. •Many natural phenomena produce very high-energy,
but inaudible, sound waves at frequencies below 20 Hz
Freed_c13_524-581_st_hr1.indd 581
­(infrasound). During the 2003 eruption of the Fuego volcano
in Guatemala, sound waves of frequency 10 Hz (and even less)
with a sound level of 120 dB were recorded. (a) What was the
maximum displacement of the air molecules produced by the
waves? (b) How much energy would such a wave deliver to a
2.0 m by 3.0 m wall in 1.0 min? Assume the density of air is
1.2 kg>m3. SSM
110. ••Two identical 375-g speakers are mounted on parallel springs, each having a spring constant of 50.0 N>cm. Both
speakers face in the same direction and produce a steady tone
of 1.00 kHz. Both sounds have an amplitude of 35.0 cm, but
they oscillate 180° out of phase with each other. What is the
highest frequency of the beat that a person will hear if she
stands in front of the speakers?
111. •A jogger hears a car alarm and decides to investigate.
While running toward the car, she hears an alarm frequency
of 869.5 Hz. After passing the car, she hears the alarm at a frequency of 854.5 Hz. If the speed of sound is 343 m>s, calculate
the speed of the jogger.
112. •••A rescuer in an all-terrain vehicle (ATV) is tracking
two injured hikers in the desert, each of whom has an emergency locator transmitter (ELT) stored in his backpack (Figure
13-36). The beacons give off radio signals at 121.5 MHz, in
phase, and there is a receiver in the ATV that is tuned to that
frequency. The speed of the radio waves is 300,000,000 m>s.
The ATV is traveling due east, 200 m north of the hikers, and
the hikers are 100 m apart. If the ATV moves at a constant
speed of 15 m>s, how many times per second will the driver
detect constructive interference between the two signals?
15 m/s
200 m
100 m
ELT # 1
ELT # 2
Figure 13-36 ​Problem 112
113. •••Speaker A is sounding a single tone in phase with
speaker B (Figure 13-37). The two speakers are separated by
8 m. What intensity level will you hear if you are located 6 m in
front of speaker A and the frequency of the tone is 857.5 Hz?
Assume that the 40-W speakers can be approximated as point
sources and the speed of sound is 343 m>s. SSM
8m
A
B
6m
Figure 13-37 ​Problem 113
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