Physics For Scientists And Engineers 6E - part 135

Problems

537

16. The tensile stress in a thick copper bar is 99.5% of its elas-

tic breaking point of 13.0 & 10

N/m

. If a 500-Hz sound

wave is transmitted through the material, (a) what dis-
placement amplitude will cause the bar to break?
(b) What is the maximum speed of the elements of copper
at this moment? (c) What is the sound intensity in the bar?

17.

Prove  that  sound  waves  propagate  with  a  speed  given  by
Equation 17.1. Proceed as follows. In Figure 17.3, consider a
thin  cylindrical  layer  of  air  in  the  cylinder,  with  face  area A
and thickness 'x. Draw a free-body diagram of this thin layer.
Show  that

implies that *[+('P)/+x]A 'x #

A 'x(+

s/+t

). By substituting 'P # *B(+s1+x), obtain the

wave equation for sound, (B/!)(+

s/+x

) # (+

s/+t

). 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

max

cos(kx * )t). Show that this function satisfies the

wave equation provided that

This result reveals

that sound waves exist provided that they move with the
speed

Section 17.3 Intensity of Periodic Sound Waves

18. The area of a typical eardrum is about 5.00 & 10

Calculate the sound power incident on an eardrum at
(a) the threshold of hearing and (b) the threshold of pain.
Calculate the sound level in decibels of a sound wave that
has an intensity of 4.00 "W/m

20. A vacuum cleaner produces sound with a measured sound

level of 70.0 dB. (a) What is the intensity of this sound in
W/m

? (b) What is the pressure amplitude of the sound?

21. The intensity of a sound wave at a fixed distance from a

speaker vibrating at 1.00 kHz is 0.600 W/m

. (a) Deter-

mine the intensity if the frequency is increased to 2.50 kHz
while a constant displacement amplitude is maintained.
(b) Calculate the intensity if the frequency is reduced to
0.500 kHz and the displacement amplitude is doubled.

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 if the frequency is increased to f . while a constant
displacement  amplitude  is  maintained.  (b)  Calculate  the
intensity  if  the  frequency  is  reduced  to  f/2  and  the  dis-
placement amplitude is doubled.

19.

v # f

( #

(2,f

)((/2,) # )/k #

√B/!.

)

/k #

√B/!.

23.

The  most  soaring  vocal  melody  is  in  Johann  Sebastian
Bach’s Mass in B minor. A portion of the score for the Credo
section, number 9, bars 25 to 33, appears in Figure P17.23.
The  repeating  syllable  O  in  the  phrase  “resurrectionem
mortuorum”  (the  resurrection  of  the  dead)  is  seamlessly
passed from basses to tenors to altos to first sopranos, like a
baton in a relay. Each voice carries the melody up in a run
of an octave or more. Together they carry it from D below
middle  C  to  A  above  a  tenor’s  high  C.  In  concert  pitch,
these notes are now assigned frequencies of 146.8 Hz and
880.0 Hz. (a) Find the wavelengths of the initial and final
notes.  (b)  Assume  that  the  choir  sings  the  melody  with  a
uniform  sound  level  of  75.0 dB.  Find  the  pressure  ampli-
tudes  of  the  initial  and  final  notes.  (c)  Find  the  displace-
ment amplitudes of the initial and final notes. (d) What If?
In Bach’s time, before the invention of the tuning fork, fre-
quencies were assigned to notes as a matter of immediate
local convenience. Assume that the rising melody was sung
starting from 134.3 Hz and ending at 804.9 Hz. How would
the answers to parts (a) through (c) change?

24. The tube depicted in Figure 17.2 is filled with air at 20°C

and equilibrium pressure 1 atm. The diameter of the tube
is 8.00 cm. The piston is driven at a frequency of 600 Hz
with an amplitude of 0.120 cm. What power must be sup-
plied to maintain the oscillation of the piston?

A family ice show is held at an enclosed arena. The

skaters  perform  to  music  with  level  80.0 dB.  This  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?

26.

Consider sinusoidal sound waves propagating in these
three different media: air at 0°C, water, and iron. Use den-
sities and speeds from Tables 14.1 and 17.1. Each wave has
the same intensity I

and the same angular frequency )

(a) Compare the values of the wavelength in the three me-
dia.  (b)  Compare  the  values  of  the  displacement  ampli-
tude  in  the  three  media.  (c)  Compare  the  values  of  the
pressure  amplitude  in  the  three  media.  (d)  For  values  of
)

2 000 , rad/s and I

1.00 & 10

W/m

, evaluate

the wavelength, displacement amplitude, and pressure am-
plitude in each of the three media.

27. 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?

25.

resurrecti - o - - - -

resurrecti - o - - -

nem mortuorum

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - rum

nemmortu o

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - rum

nem mortu o

- - - rum

Figure P17.23 Bass (blue), tenor (green), alto (brown), and first soprano (red) parts

for a portion of Bach’s Mass in B minor. For emphasis, the line we choose to call the

melody is printed in black. Parts for the second soprano, violins, viola, flutes, oboes,

and continuo are omitted. The tenor part is written as it is sung.

538

C H A P T E R 17 • Sound Waves

28.

Show that the difference between decibel levels -

and -

of a sound is related to the ratio of the distances r

and r

from the sound source by

A  firework  charge  is  detonated  many  meters  above  the
ground.  At  a  distance  of  400 m  from  the  explosion,  the
acoustic  pressure  reaches  a  maximum  of  10.0 N/m

Assume  that  the  speed  of  sound  is  constant  at  343 m/s
throughout  the  atmosphere  over  the  region  considered,
that  the  ground  absorbs  all  the  sound  falling  on  it,  and
that the air absorbs sound energy as described by the rate
7.00 dB/km.  What  is  the  sound  level  (in  dB)  at  4.00 km
from the explosion?

30.

A  loudspeaker  is  placed  between  two  observers  who  are
110 m  apart,  along  the  line  connecting  them.  If  one  ob-
server  records  a  sound  level  of  60.0 dB  and  the  other
records  a  sound  level  of  80.0 dB,  how  far  is  the  speaker
from each observer?

31.

Two small speakers emit sound waves of different frequen-
cies. Speaker A has an output of 1.00 mW, and speaker B
has an output of 1.50 mW. Determine the sound level (in
dB) at point C (Fig. P17.31) if (a) only speaker A emits
sound, (b) only speaker B emits sound, and (c) both
speakers emit sound.

29.

log

34.

A fireworks rocket explodes at a height of 100 m above the
ground. An observer on the ground directly under the
explosion experiences an average sound intensity of
7.00 & 10

W/m

for 0.200 s. (a) What is the total sound

energy of the explosion? (b) What is the sound level in
decibels heard by the observer?

35. As the people sing in church, the sound level everywhere

inside is 101 dB. No sound is transmitted through the mas-
sive walls, but all the windows and doors are open on a
summer morning. Their total area is 22.0 m

. (a) How

much sound energy is radiated in 20.0 min? (b) Suppose
the ground is a good reflector and sound radiates uni-
formly in all horizontal and upward directions. Find the
sound level 1 km away.

36.

The smallest change in sound level that a person can dis-
tinguish  is  approximately  1 dB.  When  you  are  standing
next  to  your  power  lawnmower  as  it  is  running,  can  you
hear  the  steady  roar  of  your  neighbor’s  lawnmower?  Per-
form  an  order-of-magnitude  calculation  to  substantiate
your answer, stating the data you measure or estimate.

Section 17.4 The Doppler Effect

37. A train is moving parallel to a highway with a constant

speed of 20.0 m/s. A car is traveling in the same direction
as the train with a speed of 40.0 m/s. The car horn sounds
at a frequency of 510 Hz, and the train whistle sounds at a
frequency of 320 Hz. (a) When the car is behind the train,
what frequency does an occupant of the car observe for
the train whistle? (b) After the car passes and is in front of
the train, what frequency does a train passenger observe
for the car horn?

38.

Expectant parents are thrilled to hear their unborn baby’s
heartbeat, revealed by an ultrasonic motion detector. Sup-
pose the fetus’s ventricular wall moves in simple harmonic
motion with an amplitude of 1.80 mm and a frequency of
115 per minute. (a) Find the maximum linear speed of the
heart  wall.  Suppose  the  motion  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 frequency at which sound arrives at the wall of
the  baby’s  heart.  (c)  Find  the  maximum  frequency  at
which reflected sound is received by the motion detector.
By electronically “listening” for echoes at a frequency dif-
ferent from the broadcast frequency, the motion detector
can  produce  beeps  of  audible  sound  in  synchronization
with the fetal heartbeat.

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.

40.

A block with a speaker bolted to it is connected to a spring
having spring constant k # 20.0 N/m as in Figure P17.40.
The total mass of the block and speaker is 5.00 kg, and the
amplitude of this unit’s motion is 0.500 m. (a) If the
speaker emits sound waves of frequency 440 Hz, determine
the highest and lowest frequencies heard by the person to
the right of the speaker. (b) If the maximum sound level
heard by the person is 60.0 dB when he is closest to the

39.

3.00 m

2.00 m

4.00 m

Figure P17.31

32. A jackhammer, operated continuously at a construction

site, behaves as a point source of spherical sound waves. A
construction  supervisor  stands  50.0 m  due  north  of  this
sound  source  and  begins  to  walk  due  west.  How  far  does
she  have  to  walk  in  order  for  the  amplitude  of  the  wave
function to drop by a factor of 2.00?
The  sound  level  at  a  distance  of  3.00 m  from  a  source  is
120 dB. At what distance will the sound level be (a) 100 dB
and (b) 10.0 dB?

33.

Problems

539

speaker, 1.00 m away, what is the minimum sound level
heard by the observer? Assume that the speed of sound is
343 m/s.

46. The loop of a circus ringmaster’s whip travels at Mach 1.38

(that is, v

/v # 1.38). What angle does the shock wave

make with the direction of the whip’s motion?

A supersonic jet traveling at Mach 3.00 at an altitude

of  20 000 m  is  directly  over  a  person  at  time  t # 0  as  in
Figure  P17.47.  (a)  How  long  will  it  be  before  the  person
encounters  the  shock  wave?  (b)  Where  will  the  plane  be
when it is finally heard? (Assume the speed of sound in air
is 335 m/s.)

47.

Figure P17.40

A tuning fork vibrating at 512 Hz falls from rest and
accelerates at 9.80 m/s

. How far below the point of re-

lease is the tuning fork when waves of frequency 485 Hz
reach the release point? Take the speed of sound in air to
be 340 m/s.

42.

At  the  Winter  Olympics,  an  athlete  rides  her  luge  down
the track while a bell just above the wall of the chute rings
continuously. When her sled passes the bell, she hears the
frequency  of  the  bell  fall  by  the  musical  interval  called  a
minor third. That is, the frequency she hears drops to five
sixths of its original value. (a) Find the speed of sound in
air  at  the  ambient  temperature *10.0°C.  (b)  Find  the
speed of the athlete.

43.

A siren mounted on the roof of a firehouse emits sound at
a  frequency  of  900 Hz.  A  steady  wind  is  blowing  with  a
speed of 15.0 m/s. Taking the speed of sound in calm air
to  be  343 m/s,  find  the  wavelength  of  the  sound  (a)  up-
wind of the siren and (b) downwind of the siren. Firefight-
ers  are  approaching  the  siren  from  various  directions  at
15.0 m/s. What frequency does a firefighter hear (c) if he
or she is approaching from an upwind position, so that he
or she is moving in the direction in which the wind is blow-
ing? (d) if he or she is approaching from a downwind posi-
tion and moving against the wind?

44. The Concorde can fly at Mach 1.50, which means the

speed of the plane is 1.50 times the speed of sound in air.
What is the angle between the direction of propagation of
the shock wave and the direction of the plane’s velocity?

45. When high-energy charged particles move through a trans-

parent  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 wa-
ter, 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.  In  a  particular  case,  the
Cerenkov  radiation  produces  a  wave  front  with  an  apex
half-angle of 53.0°. Calculate the speed of the electrons in
the water. (The speed of light in water is 2.25 & 10

m/s.)

41.

(a)

(b)

Observer hears

the ‘boom’

t = 0

Observer

Figure P17.47

Section 17.5 Digital Sound Recording
Section 17.6 Motion Picture Sound

48.

This problem represents a possible (but not recom-
mended) way to code instantaneous pressures in a sound
wave into 16-bit digital words. Example 17.2 mentions that
the pressure amplitude of a 120-dB sound is 28.7 N/m

Let  this  pressure  variation  be  represented  by  the  digital
code 65 536. Let zero pressure variation be represented on
the recording by the digital word 0. Let other intermediate
pressures be represented by digital words of intermediate
size, in direct proportion to the pressure. (a) What digital
word  would  represent  the  maximum  pressure  in  a  40 dB
sound? (b) Explain why this scheme works poorly for soft
sounds. (c) Explain how this coding scheme would clip off
half  of  the  waveform  of  any  sound,  ignoring  the  actual
shape of the wave and turning it into a string of zeros. By
introducing  sharp  corners  into  every  recorded  waveform,
this  coding  scheme  would  make  everything  sound  like  a
buzzer or a kazoo.

49.

Only two recording channels are required to give the illu-
sion of sound coming from any point located between two
speakers  of  a  stereophonic  sound  system.  If  the  same  sig-
nal  is  recorded  in  both  channels,  a  listener  will  hear  it
coming  from  a  single  direction  halfway  between  the  two
speakers. This “phantom orchestra” illusion can be heard
in the two-channel original Broadway cast recording of the
song  “Do-Re-Mi”  from  The  Sound  of  Music (Columbia
Records  KOS  2020).  Each  of  the  eight  singers  can  be
heard at a different location between the loudspeakers. All
listeners with normal hearing will agree on their locations.
The brain can sense the direction of sound by noting how

540

C H A P T E R 17 • Sound Waves

much earlier a sound is heard in one ear than in the other.
Model your ears as two sensors 19.0 cm apart in a flat
screen. If a click from a distant source is heard 210 "s ear-
lier in the left ear than in the right, from what direction
does it appear to originate?

50. Assume that a 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.  (b)  If  the  salesperson  claims  to  be  giving
you  150 W  per  channel,  he  is  referring  to  the  electrical
power  input  to  the  speaker.  Find  the  efficiency  of  the
speaker—that  is,  the  fraction  of  input  power  that  is  con-
verted into useful output power.

Additional Problems

51.

A large set of unoccupied football bleachers has solid seats
and risers. You stand on the field in front of the bleachers
and  fire  a  starter’s  pistol  or  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  fre-
quency and may remind you of a short toot on a trumpet,
or of a buzzer or kazoo. Account for this sound. Compute
order-of-magnitude estimates for its frequency, wavelength,
and duration, on the basis of data you specify.

52. Many artists sing very high notes in ad lib ornaments and ca-

denzas. 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) Sup-
pose  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?  Because  of  complaints,  Strauss  later
transposed the note down to F above high C, 1.397 kHz. By
what increment did the wavelength change?

53. A sound wave in a cylinder is described by Equations 17.2

through 17.4. Show that

54. On a Saturday morning, pickup trucks and sport utility

vehicles  carrying  garbage  to  the  town  dump  form  a
nearly steady procession on a country road, all traveling
at  19.7 m/s.  From  one  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 him? (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 noisy, smelly, inefficient,
garbage-dripping,  roadhogging  trucks  whiz  past  him
now?

55. The ocean floor is underlain by a layer of basalt that con-

stitutes the crust, or uppermost layer, of the Earth in that
region. Below this crust is found denser periodotite rock,
which  forms  the  Earth’s  mantle.  The  boundary  between
these  two  layers  is  called  the  Mohorovicic  discontinuity
(“Moho” for short). If an explosive charge is set off at the
surface of the basalt, it generates a seismic wave that is re-
flected back out at the Moho. If the speed of this wave in
basalt  is  6.50 km/s  and  the  two-way  travel  time  is  1.85 s,
what is the thickness of this oceanic crust?

P # 2!v)

√

max

56.

For a certain type of steel, stress is always proportional to
strain with Young’s modulus as shown in Table 12.1. The
steel has the density listed for iron in Table 14.1. It will fail
by bending permanently if subjected to compressive stress
greater than its yield strength 3

400 MPa. A rod

80.0 cm  long,  made  of  this  steel,  is  fired  at  12.0 m/s
straight  at  a  very  hard  wall,  or  at  another  identical  rod
moving in the opposite direction. (a) The speed of a one-
dimensional  compressional  wave  moving  along  the  rod
is  given  by

, where ! is the density and Y is Young’s

modulus  for  the  rod.  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. How much time elapses be-
fore  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?  Find  (d)  the  strain  in  the  rod  and
(e) the stress. (f) If it is not to fail, show that the maximum
impact  speed  a  rod  can  have  is  given  by  the  expression

To permit measurement of 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 am-
plified sound he receives. Assume that the air is calm and
that the sound speed is 343 m/s, 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 re-
flected from the ground. What frequency does she receive?

58.

A train whistle (f # 400 Hz) sounds higher or lower in fre-
quency depending on whether it approaches or recedes.
(a) Prove that the difference in frequency between the ap-
proaching and receding train whistle is

where  u is  the  speed  of  the  train  and  v is  the  speed  of
sound. (b) Calculate this difference for a train moving at a
speed of 130 km/h. Take the speed of sound in air to be
340 m/s.
Two  ships  are  moving  along  a  line  due  east.  The  trailing
vessel  has  a  speed  relative  to  a  land-based  observation
point  of  64.0 km/h,  and  the  leading  ship  has  a  speed  of
45.0 km/h relative to that point. The two ships are in a re-
gion of the ocean where the current is moving uniformly
due west at 10.0 km/h. The trailing ship transmits a sonar
signal  at  a  frequency  of  1 200.0 Hz.  What  frequency  is
monitored  by  the  leading  ship?  (Use  1 520 m/s  as  the
speed of sound in ocean water.)

60.

A bat, moving at 5.00 m/s, is chasing a flying insect
(Fig. P17.7). If the bat emits a 40.0 kHz chirp and receives
back an echo at 40.4 kHz, at what speed is the insect mov-
ing toward or away from the bat? (Take the speed of sound
in air to be v # 340 m/s.)

61.

A supersonic aircraft is flying parallel to the ground. When
the aircraft is directly overhead, an observer sees a rocket
fired from the aircraft. Ten seconds later the observer

59.

f #

2u/v

1 * u

57.

√!Y

√

Y/!

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