Physics For Scientists And Engineers 6E - part 134

channel” of 5.1, carries very low frequencies for dramatic sound from explosions,
earthquakes, and the like.

Current motion pictures are produced with three systems of digital sound recording:

Dolby Digital; In this format, 5.1 channels of digital sound are optically stored
between the sprocket holes of the film. There is an analog optical backup in case the
digital system fails. The first film to use this technique was Batman Returns (1992).

DTS (Digital Theater Sound); 5.1 channels of sound are stored on a separate CD-
ROM which is synchronized to the film print by time codes on the film. There is an
analog optical backup in case the digital system fails. The first film to use this tech-
nique was Jurassic Park (1993).

SDDS (Sony Dynamic Digital Sound); Eight full channels of digital sound are optically
stored outside the sprocket holes on both sides of film. There is an analog optical
backup in case the digital system fails. The first film to use this technique was Last
Action Hero (1993). The existence of information on both sides of the tape is a system
of redundancy—in case one side is damaged, the system will still operate. SDDS em-
ploys a full-spectrum LFE channel and two additional channels (left center and right
center behind the screen). In Figure 17.16, showing a section of SDDS film, both the
analog optical soundtrack and the dual digital soundtracks can be seen.

S E C T I O N 17. 6 • Motion Picture Sound

533

film image

sprocket holes

stereo optical soundtrack

digital soundtrack

Figure 17.16 The layout of information on motion picture film using the SDDS digital

sound system.

Courtesy of Sony Cinema Products Corporation

written permission is prohibited.

534

C H A P T E R 17 • Sound Waves

Sound waves are longitudinal and travel through a compressible medium with a speed
that depends on the elastic and inertial properties of that medium. The speed of
sound in a liquid or gas having a bulk modulus B and density ! is

(17.1)

For sinusoidal sound waves, the variation in the position of an element of the

medium is given by

(17.2)

and the variation in pressure from the equilibrium value is

(17.3)

where 'P

max

is the

pressure amplitude. The pressure wave is 90° out of phase with

the displacement wave. The relationship between s

max

and 'P

max

is given by

(17.4)

The intensity of a periodic sound wave, which is the power per unit area, is

(17.5, 17.6)

The sound level of a sound wave, in decibels, is given by

(17.8)

The constant I

is a reference intensity, usually taken to be at the threshold of hearing

(1.00 & 10

W/m

), and I is the intensity of the sound wave in watts per square meter.

The change in frequency heard by an observer whenever there is relative motion

between a source of sound waves and the observer is called the

Doppler effect. The

observed frequency is

(17.13)

In this expression, the signs for the values substituted for v

and v

depend on the di-

rection of the velocity. A positive value for the velocity of the observer or source is sub-
stituted if the velocity of one is toward the other, while a negative value represents a ve-
locity of one away from the other.

In digital recording of sound, the sound waveform is sampled 44 100 times per sec-

ond. The pressure of the wave for each sampling is measured and converted to a bi-
nary number. In playback, these binary numbers are read and used to build the origi-
nal waveform.

f . #

v $ v

v * v

$ 10 log

max

2!v

max

v)s

max

P # 'P

max

sin(kx * )t)

s(x, t) # s

max

cos(kx * )t)

v #

√

S U M M A R Y

Take a practice test for

this chapter by clicking on
the Practice Test link at
http://www.pse6.com.

1. Why are sound waves characterized as longitudinal?

If an alarm clock is placed in a good vacuum and then acti-
vated, no sound is heard. Explain.

3. A sonic ranger is a device that determines the distance to

an object by sending out an ultrasonic sound pulse and
measuring how long it takes for the wave to return after it
reflects from the object. Typically these devices cannot

reliably detect an object that is less than half a meter from
the sensor. Why is that?

4. A friend sitting in her car far down the road waves to you

and beeps her horn at the same time. How far away must
she be for you to calculate the speed of sound to two sig-
nificant figures by measuring the time it takes for the
sound to reach you?

Q U E S T I O N S

Problems

535

5. If the wavelength of sound is reduced by a factor of 2, what

happens to its frequency? Its speed?

6. By listening to a band or orchestra, how can you de-

termine that the speed of sound is the same for all
frequencies?

7. In Example 17.3 we found that a point source with a

power output of 80 W produces sound with an intensity of
1.00 & 10

W/m

, which corresponds to 40 dB, at a dis-

tance of about 16 miles. Why do you suppose you cannot
normally hear a rock concert that is going on 16 miles
away? (See Table 17.2.)
If the distance from a point source is tripled, by what
factor does the intensity decrease?

9. The Tunguska Event. On June 30, 1908, a meteor burned

up and exploded in the atmosphere above the Tunguska
River valley in Siberia. It knocked down trees over thou-
sands  of  square  kilometers  and  started  a  forest  fire,  but
apparently caused no human casualties. A witness sitting
on his doorstep outside the zone of falling trees recalled
events in the following sequence: He saw a moving light
in the sky, brighter than the sun and descending at a low
angle to the horizon. He felt his face become warm. He
felt the ground shake. An invisible agent picked him up
and  immediately  dropped  him  about  a  meter  farther
away  from  where  the  light  had  been.  He  heard  a  very
loud  protracted  rumbling.  Suggest  an  explanation  for
these  observations  and  for  the  order  in  which  they
happened.

10. Explain how the Doppler effect with microwaves is used to

determine the speed of an automobile.

11. Explain what happens to the frequency of the echo of your

car horn as you move in a vehicle toward the wall of a
canyon. What happens to the frequency as you move away
from the wall?

12. Of the following sounds, which is most likely to have a

sound level of 60 dB: a rock concert, the turning of a page

in this textbook, normal conversation, or a cheering crowd
at a football game?

13. Estimate the decibel level of each of the sounds in the pre-

vious question.

14. A binary star system consists of two stars revolving about

their common center of mass. If we observe the light
reaching us from one of these stars as it makes one com-
plete revolution, what does the Doppler effect predict will
happen to this light?
How can an object move with respect to an observer so
that the sound from it is not shifted in frequency?

16. Suppose the wind blows. Does this cause a Doppler effect

for sound propagating through the air? Is it like a moving
source or a moving observer?

17. Why is it not possible to use sonar (sound waves) to deter-

mine the speed of an object traveling faster than the speed
of sound?

18. Why is it so quiet after a snowfall?
19. Why is the intensity of an echo less than that of the origi-

nal sound?

20. A loudspeaker built into the exterior wall of an airplane

produces  a  large-amplitude  burst  of  vibration  at  200 Hz,
then  a  burst  at  300 Hz,  and  then  a  burst  at  400 Hz
(Boop . .  .  baap  .  .  .  beep),  all  while  the  plane  is  flying
faster than the speed of sound. Describe qualitatively what
an observer hears if she is in front of the plane, close to its
flight path. What If? What will the observer hear if the pi-
lot uses the loudspeaker to say, “How are you?”

21. In several cases, a nearby star has been found to have a

large planet orbiting about it, although the planet could
not be seen. Using the ideas of a system rotating about its
center of mass and of the Doppler shift for light (which is
in several ways similar to the Doppler effect for sound), ex-
plain how an astronomer could determine the presence of
the invisible planet.

15.

Section 17.1 Speed of Sound Waves

Suppose that you hear a clap of thunder 16.2 s after seeing
the associated lightning stroke. The speed of sound waves
in air is 343 m/s, and the speed of light is 3.00 & 10

m/s.

How far are you from the lightning stroke?

2. Find the speed of sound in mercury, which has a bulk

modulus of approximately 2.80 & 10

N/m

and a density

of 13 600 kg/m

A flowerpot is knocked off a balcony 20.0 m above the
sidewalk and falls toward an unsuspecting 1.75-m-tall man
who is standing below. How close to the sidewalk can the
flower pot fall before it is too late for a warning shouted

from the balcony to reach the man in time? Assume that
the man below requires 0.300 s to respond to the warning.

The speed of sound in air (in m/s) depends on tempera-
ture according to the approximate expression

where T

is the Celsius temperature. In dry air the tem-

perature decreases about 1°C for every 150 m rise in alti-
tude. (a) Assuming this change is constant up to an alti-
tude of 9 000 m, how long will it take the sound from an
airplane flying at 9 000 m to reach the ground on a day
when the ground temperature is 30°C? (b) What If? Com-

v # 331.5 $ 0.607T

= straightforward, intermediate, challenging

= full solution available in the Student Solutions Manual and Study Guide

= coached solution with hints available at http://www.pse6.com

= computer useful in solving problem

= paired numerical and symbolic problems

P R O B L E M S

536

C H A P T E R 17 • Sound Waves

pare this to the time interval required if the air were a con-
stant 30°C. Which time interval is longer?

A cowboy stands on horizontal ground between two paral-
lel 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 reflect-
ing from the cliffs. Take the speed of sound as 340 m/s.
(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?

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  perceives  the  horn’s
sound, the plane has traveled a distance equal to half its al-
titude above the ocean. If it takes the sound 2.00 s to reach
the plane, determine (a) the speed of the plane and (b) its
altitude. Take the speed of sound to be 343 m/s.

Section 17.2 Periodic Sound Waves

Note: Use the following values as needed unless otherwise
specified: the equilibrium density of air at 20°C is
! #

1.20 kg/m

. The speed of sound in air is v #

343 m/s. Pressure variations 'P are measured relative
to atmospheric pressure, 1.013 & 10

N/m

Problem 70 in Chapter 2 can also be assigned with this
section.

7. A bat (Fig. P17.7) can detect very small objects, such as

an  insect  whose  length  is  approximately  equal  to  one
wavelength  of  the  sound  the  bat  makes.  If  a  bat  emits
chirps  at  a  frequency  of  60.0 kHz,  and  if  the  speed  of
sound in air is 340 m/s, what is the smallest insect the bat
can detect?

An  ultrasonic  tape  measure  uses  frequencies  above
20 MHz  to  determine  dimensions  of  structures  such  as
buildings.  It  does  this  by  emitting  a  pulse  of  ultrasound
into air and then measuring the time for an echo to return
from a reflecting surface whose distance away is to be mea-
sured. The distance is displayed as a digital read-out. For a
tape  measure  that  emits  a  pulse  of  ultrasound  with  a  fre-
quency of 22.0 MHz, (a) What is the distance to an object
from which the echo pulse returns after 24.0 ms when the
air temperature is 26°C? (b) What should be the duration
of the emitted pulse if it is to include 10 cycles of the ultra-
sonic wave? (c) What is the spatial length of such a pulse?

9. Ultrasound is used in medicine both for diagnostic imaging

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 from the time
delay for the return of the echo the distance to the struc-
ture can be determined. A single transducer emits and de-
tects the ultrasound. An image of the structure is obtained
by  reducing  the  data  with  a  computer.  With  sound  of  low
intensity,  this  technique  is  noninvasive  and  harmless.  It  is
used  to  examine  fetuses,  tumors,  aneurysms,  gallstones,
and  many  other  structures.  A  Doppler  ultrasound  unit  is
used to study blood flow and functioning of the heart. To
reveal  detail,  the  wavelength  of  the  reflected  ultrasound
must be small compared to the size of the object reflecting
the wave. For this reason, frequencies in the range 1.00 to
20.0 MHz are used. What is the range of wavelengths corre-
sponding to this range of frequencies? The speed of ultra-
sound in human tissue is about 1 500 m/s (nearly the same
as the speed of sound in water).

10. A sound wave in air has a pressure amplitude equal to

4.00 & 10

N/m

. Calculate the displacement amplitude

of the wave at a frequency of 10.0 kHz.

11. A sinusoidal sound wave is described by the displacement

wave function

(a)  Find  the  amplitude,  wavelength,  and  speed  of  this
wave. (b) Determine the instantaneous displacement from
equilibrium  of  the  elements  of  air  at  the  position
x # 0.050 0 m  at  t # 3.00 ms.  (c)  Determine  the  maxi-
mum speed of the element’s oscillatory motion.

12. As a certain sound wave travels through the air, it produces

pressure  variations  (above  and  below  atmospheric  pres-
sure) given by 'P # 1.27 sin(,x * 340,t) in SI units. Find
(a)  the  amplitude  of  the  pressure  variations,  (b)  the  fre-
quency, (c) the wavelength in air, and (d) the speed of the
sound wave.
Write an expression that describes the pressure variation as
a  function  of  position  and  time  for  a  sinusoidal  sound
wave in air, if ( # 0.100 m and 'P

max

0.200 N/m

14. Write the function that describes the displacement wave

corresponding to the pressure wave in Problem 13.

An experimenter wishes to generate in air a sound

wave that has a displacement amplitude of 5.50 & 10

The pressure amplitude is to be limited to 0.840 N/m

What is the minimum wavelength the sound wave can
have?

15.

13.

s(x, t) # (2.00 "m)

cos[(15.7 m

)x * (858 s

)t]

Figure P17.7 Problems 7 and 60.

Joe McDonald/V

isuals Unlimited

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