490

THE B

IGID

EA

........ VIBRA

TION

S A

ND

WAV

ES

All around

us we see thing

s that wig

gle and

jigg

le. Even thing

s too small to see, such as atom

s, are constantly w

igg

ling and

jigg

ling. A

repeating

, back-and

-forth m

otion about an eq

uilibrium

position is a vibration.

A vib

ration cannot exist in one instant. It needs

time to m

ove back and

forth. Strike a bell and

the vib

rations will continue for som

e time

before they d

ie dow

n.A

disturb

ance that is transmitted

pro-

gressively from

one place to the next

with no actual transp

ort of matter is

a w

ave. A w

ave cannot exist in one p

lace but m

ust extend from

one place

to another. Light and

sound are b

oth form

s of energy that m

ove through

space as w

aves. This chapter is ab

out vib

rations and w

aves, and the follow

-ing

chapters continue w

ith the study of

sound and

light.

Waves transm

it energy through space and tim

e.

5What A

re Standing

Waves?

1.Fill a foam

cup nearly to the top with w

ater. Place the cup on a sm

ooth, dry surface.2.

While applying a m

oderate downw

ard pres-sure, drag the cup across the surface.

3.A

djust the downw

ard pressure on the cup until a pattern of w

aves, called standing w

aves, appears on the surface of the water.

4.N

ow try to change the pattern by altering

both the speed of the cup and the downw

ard pressure.

Analyze and

Conclud

e1.

Observing D

escribe the patterns that you produced on the surface of the w

ater.2.

Predicting What do you think m

ight happen if you w

ere to drag the cup on a different kind of surface?

3.M

aking Generalizations D

o you think stand-ing w

aves can be produced in other media?

Explain.

disco

ver!

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49

0

V

IBR

ATIO

NS

AN

D W

AV

ESO

bje

ctive

s• D

escribe th

e perio

d o

f a p

end

ulu

m. (25.1)

• Describ

e the ch

aracteristics and

p

rop

erties of w

aves. (25.2)

• Describ

e wave m

otio

n. (25.3)

• Describ

e ho

w to

calculate th

e sp

eed o

f a wave. (25.4)

• Give exam

ples o

f transverse

waves. (25.5)

• Give an

examp

le of a

lon

gitu

din

al wave. (25.6)

• Explain

wh

at causes

interferen

ce pattern

s. (25.7)

• Describ

e ho

w a stan

din

g w

ave o

ccurs. (25.8)

• Describ

e ho

w th

e app

arent

frequ

ency o

f waves ch

ang

e as a w

ave sou

rce mo

ves. (25.9)

• Describ

e bo

w w

aves. (25.10)

• Describ

e son

ic bo

om

s. (25.11)

disco

ver!M

ATE

RIA

LS foam

cup

, water

EX

PE

CTE

D OU

TCO

ME R

egio

ns o

f still w

ater, no

des, an

d reg

ion

s o

f cho

pp

y water, an

tino

des,

sho

uld

be o

bservab

le. This

pattern

is the resu

lt of th

e in

terference o

f traveling

w

aves reflecting

from

the

vibratin

g w

alls of th

e cup

.

AN

ALY

ZE A

ND C

ON

CLU

DE

Stud

ents sh

ou

ld o

bserve

regio

ns o

f still water an

d

regio

ns o

f cho

pp

y water.

The p

attern ch

ang

es b

ecause th

e cup

vibrates

differen

tly on

differen

t su

rfaces.

Yes, b

ecause w

aves travel in

all med

ia and

interferen

ce is a ch

aracteristic of w

aves.

1.2.3.

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25.1 Vibration of a Pendulum

Suspend a stone at the end of a string and you have a simple pen-

dulum. Pendulum

s like the one in Figure 25.1 swing back and forth

with such regularity that they have long been used to control the

motion of clocks. G

alileo dis covered that the time a pendulum

takes to sw

ing back and forth through small angles depends only on the

length of the pendulum—

the mass has no effect. T

he time of a

back-and-forth swing of the pendulum

is called the period. T

he period of the pendulum depends only on the length of a

pendulum and the acceleration of gravity. 25.1

A long pendulum

has a longer period than a shorter pendulum;

that is, it swings back and forth m

ore slowly—

less frequently—than a

short pendulum. W

hen walking, w

e allow our legs to sw

ing with the

help of gravity, like a pendulum. In the sam

e way that a long pendu-

lum has a greater period, a person w

ith long legs tends to walk w

ith a slow

er stride than a person with short legs. T

his is most noticeable

in long-legged animals such as giraffes and horses, w

hich run with a

slower gait than do short-legged anim

als such as hamsters and m

ice.

CON

CEPT

CHECK

......What d

etermines the p

eriod of a p

endulum

?

25.2 Wave D

escriptionT

he back-and-forth vibratory motion (often called oscillatory

motion) of a sw

inging pendulum is called sim

ple harmonic

motion. 25.2 T

he pendulum bob filled w

ith sand in Figure 25.2 exhibits sim

ple harmonic m

otion above a conveyor belt. When the

conveyor belt is stationary, the sand traces out a straight line. More

interestingly, when the conveyor belt is m

oving at constant speed, the sand traces out a special curve know

n as a sine curve. A sine curve

is a pictorial representation of a wave.

The source of all w

aves is som

ething that vibrates.

FIGU

RE 25.1 !

Two pendulum

s of the same

length have the same period

regardless of mass.

" FIG

URE 25.2

Frank Oppenheim

er, founder of the Exploratorium

® science m

useum in San Francisco, dem

on-strates that a pendulum

swinging

back and forth traces out a straight line over a stationary surface and a sine curve w

hen the surface moves

at constant speed.

What is the frequency in

vibrations per second of a 100-H

z wave?

Answer: 25.2.1

thin

k!

CH

APTER 25

VIBRATION

S AN

D W

AVES 491

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491

25.1 Vibration of a

PendulumK

ey

Term

sp

eriod

, vibratio

n, w

aves

# Teach

ing

Tip D

isting

uish

b

etween

a simp

le pen

du

lum

(the

bo

b is very sm

all com

pared

to th

e len

gth

of strin

g) an

d a p

hysical

pen

du

lum

(the stick m

akes up

a sig

nifican

t part o

f the m

ass). Exp

lain th

at their ro

tation

al in

ertias are differen

t.

Ask W

hat principle of m

echanics accounts for the different periods of pendulum

s of different lengths? R

otatio

nal

inertia

Attach

a small h

eavy weig

ht

to th

e end

of a p

iece of strin

g

abo

ut 1 m

lon

g. Sw

ing

it to

and

fro: th

is is a simp

le p

end

ulu

m. Id

entify freq

uen

cy an

d p

eriod

. Time h

ow

lon

g

the p

end

ulu

m takes to

make

10 com

plete cycles. R

epeat

to sh

ow

that th

e result d

oes

no

t chan

ge fro

m trial to

trial. D

ivide th

e time b

y 10 to g

et th

e perio

d. A

dd

mo

re mass to

th

e end

of th

e string

with

ou

t ch

ang

ing

the o

verall leng

th o

f th

e pen

du

lum

. Time 10 m

ore

cycles to sh

ow

that w

eigh

t d

oes n

ot affect th

e perio

d.

Th

e perio

d o

f the

pen

du

lum

dep

end

s o

nly o

n th

e leng

th o

f a pen

du

lum

an

d th

e acceleration

of g

ravity.

Te

ac

hin

g R

es

ou

rc

es

• Problem-Solving Exercises in

Physics 12-1, 12-2• Laboratory M

anual 68, 69• Probew

are Lab Manual 13

Dem

on

stratio

nD

em

on

stratio

n

CON

CEPT

CHECK

......

CON

CEPT

CHECK

......

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492

The Parts of a Wave

A w

eight attached to a spring undergoes vertical sim

ple harmonic m

otion as shown in Figure 25.3. A

marking

pen attached to the bob traces a sine curve on a sheet of paper that is m

oving horizontally at constant speed. Like a water w

ave, the high points on a w

ave are called crests. The low

points on a wave are

called troughs. T

he straight dashed line represents the “home” posi-

tion, or midpoint of the vibration. T

he term am

plitude refers to the distance from

the midpoint to the crest (or trough) of the w

ave. So the am

plitude equals the maxim

um displacem

ent from equilibrium

.T

he w

avelength of a wave is the distance from

the top of one crest to the top of the next one. O

r equivalently, the wavelength is the

distance between successive identical parts of the w

ave. The w

ave-lengths of w

aves at the beach are measured in m

eters, the wavelengths

of ripples in a pond in centimeters, and the w

avelengths of light in billionths of a m

eter (nanometers).

Frequency T

he number of vibrations an object m

akes in a unit of tim

e is an object’s frequency. The frequency of a vibrating pendu-

lum, or object on a spring, specifies the num

ber of back-and-forth vibrations it m

akes in a given time (usually one second). A

complete

back-and-forth vibration is one cycle. If it occurs in one second, the frequency is one vibration per second or one cycle per second. If tw

o vibrations occur in one second, the frequency is tw

o vibrations or tw

o cycles per second. The frequency of the vibrating source and the

frequency of the wave it produces are the sam

e.T

he unit of frequency is called the hertz (Hz). A

frequency of one cycle per second is 1 hertz, tw

o cycles per second is 2 hertz, and so on. H

igher frequencies are measured in kilohertz (kH

z—thou-

sands of hertz), and still higher frequencies in megahertz (M

Hz—

millions of hertz) or gigahertz (G

Hz—

billions of hertz). AM

radio w

aves are broadcast in kilohertz, while FM

radio waves are broadcast

in megahertz; radar and m

icrowave ovens operate at gigahertz. A

station at 960 kH

z broadcasts radio waves that have a frequency of

960,000 hertz. A station at 101 M

Hz broadcasts radio w

aves with a

frequency of 101,000,000 hertz. As Figure 25.4 show

s, these radio-w

ave frequencies are the frequencies at which electrons vibrate in the

transmitting antenna of a radio station.

FIGU

RE 25.3 !

A sine curve is a pictorial

representation of a wave.

FIGU

RE 25.4 "Electrons in the transm

itting antenna of a radio station at 960 kH

z on the AM

dial vibrate 960,000 tim

es each second and produce 960-kH

z radio waves.

Be clear about the distinction betw

een frequency and speed.H

ow frequently a w

ave vibrates is altogether different from

how fast

it moves from

one loca-tion to another.

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49

2

25.2 Wave

Description

Ke

y Te

rms

simp

le harm

on

ic mo

tion

, sine

curve, crest, tro

ug

h, am

plitu

de,

wavelen

gth

, frequ

ency, h

ertz

! Teach

ing

Tip B

egin

by

tapp

ing

you

r lecture tab

le or th

e ch

alkbo

ard. C

all attentio

n to

ho

w

frequ

ently yo

u tap

and

relate th

is to th

e term freq

uen

cy. Call

attentio

n to

the tim

e interval

betw

een tap

s and

relate this

to th

e perio

d. Estab

lish th

e recip

rocal relatio

nsh

ip b

etween

freq

uen

cy and

perio

d.

! Teach

ing

Tip M

ove a p

iece of

chalk u

p an

d d

ow

n o

n th

e bo

ard,

tracing

and

retracing

a vertical straig

ht lin

e. Call atten

tion

to

ho

w “freq

uen

tly” you

oscillate

the ch

alk, again

tying

this to

th

e defin

ition

of freq

uen

cy. D

iscuss th

e idea o

f disp

lacemen

t an

d am

plitu

de (m

aximu

m

disp

lacemen

t). With

app

rop

riate m

otio

ns, sh

ow

differen

t freq

uen

cies and

differen

t am

plitu

des. Th

en d

o th

e same

wh

ile walkin

g acro

ss the fro

nt

of th

e bo

ard tracin

g o

ut a sin

e w

ave. Rep

eat sho

win

g w

aves of

differen

t wavelen

gth

s.

! Teach

ing

Tip Po

int o

ut th

at sin

ce a vibratio

n is also

called a

cycle, on

e hertz is also

on

e cycle p

er secon

d.

(1 kHz 5

103 cycles/s;

1 MH

z 5 10

6 cycles/s)

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If the frequency of a vibrating object is known, its period can be

calculated, and vice versa. Suppose, for example, that a pendulum

m

akes two vibrations in one second. Its frequency is 2 H

z. The tim

e needed to com

plete one vibration—that is, the period of vibration—

is 1/2 second. Or if the vibration period is 3 H

z, then the period is 1/3 second. A

s you can see below, frequency and period are inverses of

each other:

��

or periodfrequency

1period

1frequency

CON

CEPT

CHECK

......What is the source of all w

aves?

25.3 Wave M

otionM

ost of the information around us gets to us in som

e form of w

ave. Sound is energy that travels to our ears in the form

of a wave. Light is

energy that comes to our eyes in the form

of a different kind of wave

(an electromagnetic w

ave). The signals that reach our radio and tele-

vision sets also travel in the form of electrom

agnetic waves.

When energy is transferred by a w

ave from a vibrating source to a

distant receiver, there is no transfer of matter betw

een the two points.

To see this, think about the very simple w

ave produced when one end

of a horizontally stretched string is shaken up and down as show

n in Figure 25.5. A

fter the end of the string is shaken, a rhythmic dis-

turbance travels along the string. Each part of the string moves up

and down w

hile the disturbance moves horizontally along the length

of the string. It is the disturbance that moves along the length of the

string, not parts of the string itself.

FIGU

RE 25.5 !

When the string is shaken

up and down, a disturbance

moves along the string.

Noisy Bugs Big bum

blebees flap their w

ings at about 130 flaps per second, and produce sound of 130 H

z. A honeybee flaps its w

ings at 225 flaps per second and produces a higher-pitched sound of 225 H

z. The annoying high-pitched w

hine of a mosquito

results from its w

ings flapping at 600 Hz. These sounds are produced

by pressure variations in the air caused by vibrating wings.

Link

to E

NTO

MO

LOG

Y

CH

APTER 25

VIBRATION

S AN

D W

AVES 493

The Sears Tower in

Chicago sw

ays back and forth at a frequency of about 0.1 H

z. What is

its period of vibration? Answ

er: 25.2.2

thin

k!

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493

Th

e sou

rce of all

waves is so

meth

ing

th

at vibrates.

Te

ac

hin

g R

es

ou

rc

es

• Reading and Study W

orkbook • Transparency 50• Presentation

EXPR

ESS

• Interactive Textbook

25.3 Wave M

otionC

om

mo

n M

iscon

ceptio

nW

hen a wave travels in a m

edium,

the medium

moves w

ith the wave.

FAC

T As a w

ave travels thro

ug

h

a med

ium

, there is n

o tran

sfer of

matter.

" Teach

ing

Tip Po

int o

ut th

at if a leaf is flo

ating

in a p

on

d as

a wave p

asses, the leaf w

ill mo

ve u

p an

d d

ow

n w

ith th

e water

bu

t will n

ot m

ove alo

ng

with

th

e wave.

Have a stu

den

t ho

ld o

ne

end

of a stretch

ed sp

ring

or

a Slinky w

hile yo

u h

old

the

oth

er. Send

transverse p

ulses

alon

g it, stressin

g th

e idea

that th

e distu

rban

ce rather

than

the m

ediu

m m

oves alo

ng

th

e sprin

g. Sh

ake the sp

ring

an

d p

rod

uce a sin

e wave. Th

en

send

a stretch an

d sq

ueeze

(elon

gatio

n an

d co

mp

ression

) d

ow

n th

e sprin

g, sh

ow

ing

a lo

ng

itud

inal p

ulse. Sen

d a

sequ

ence o

f pu

lses and

you

h

ave a wave. A

fter som

e d

iscussio

n, p

rod

uce stan

din

g

waves.

CON

CEPT

CHECK

......

CON

CEPT

CHECK

......

Dem

on

stratio

nD

em

on

stratio

n

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494

Drop a stone in a quiet pond and you’ll produce a w

ave that m

oves out from the center in an expanding circle as show

n in Figure 25.6. It is the disturbance that m

oves, not the water, for after the dis-

turbance passes, the water is w

here it was before the w

ave passed.W

hen someone speaks to you from

across the room, the sound w

ave is a disturbance in the air that travels across the room

. The air m

olecules them

selves do not move along, as they w

ould in a wind. T

he air, like the rope and the w

ater in the previous examples, is the m

edium through

which w

ave energy travels. T

he energy transferred by a wave from

a vibrating source to a receiver is carried by a disturbance in a m

edium. Energy is not transferred by m

atter moving from

one place to another w

ithin the medium

.

CON

CEPT

CHECK

......How

does a w

ave transfer energy?

FIGU

RE 25.6 !A

circular water w

ave in a still pond m

oves out from the center

in an expanding circle.

Making

Waves

Part 11.

Oscillate a m

arking pen back and forth across a piece of paper as you slow

ly pull the paper in a direction perpendicular to your oscillation.

2.Repeat Step 1, but pull the paper faster this tim

e.3.

Think What happens to the w

avelength of the curves when you

pull the paper faster?

Part 21.

Repeatedly dip your finger into a wide pan of w

ater to make

circular waves on the surface.

2.Repeat Step 1, but dip your finger m

ore frequently.3.

Think What happens to the w

avelength of the waves w

hen you dip your finger m

ore frequently?

disco

ver!

For:Visit:W

eb Code: –

Links on wave m

otion

ww

w.SciLinks.org

csn

2503

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49

4

disco

ver!M

ATE

RIA

LS pen

, pap

er, wid

e p

an, w

ater

EX

PE

CTE

D OU

TCO

ME In

Part 1, stu

den

ts will create a p

ictorial

represen

tation

of a w

ave. Th

ey will o

bserve th

e same

pattern

as in Fig

ure 25.2. In

Part 2, stu

den

ts will actu

ally m

ake waves.

THIN

K, PA

RT 1 Th

e wavelen

gth

in

creases.

THIN

K, PA

RT 2 Th

e wavelen

gth

d

ecreases.

Th

e energ

y tran

sferred b

y a w

ave from

a vibratin

g so

urce

to a receiver is carried

by a

distu

rban

ce in a m

ediu

m.

Te

ac

hin

g R

es

ou

rc

es

• Reading and Study W

orkbook • Problem

-Solving Exercises in Physics 13-1

• PresentationEX

PRESS

• Interactive Textbook

CO

NCEP

TCH

ECK

......

CO

NCEP

TCH

ECK

......

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25.4 Wave Speed

The speed of a w

ave depends on the medium

through which the w

ave m

oves. Sound waves, for exam

ple, move at speeds of about

330 m/s to 350 m

/s in air (depending on temperature), and about four

times faster in w

ater. Whatever the m

edium, the speed, w

avelength, and frequency of the w

ave are related. Consider the sim

ple case of w

ater waves, as show

n in Figure 25.7. Imagine that you fix your eyes

at a stationary point on the surface of water and observe the w

aves passing by this point. If you observe the distance betw

een crests (the w

avelength) and also count the number of crests that pass each

second (the frequency), then you can calculate the horizontal distance a particular crest m

oves each second. For example, in

Figure 25.7, one crest passes by the bird every second. The w

aves therefore m

ove at 1 meter per second.

You can calculate the speed of a wave by m

ultiplying the w

avelength by the frequency. For example, if the w

avelength is 3 m

eters and if two crests pass a stationary point each second, then

3 meters !

2 waves pass by in 1 second. T

he waves therefore m

ove at 6 m

eters per second. In equation form, this relationship is w

ritten as

v�%f

where v

is wave speed, l (G

reek letter lambda) is w

avelength, and fis w

ave frequency. This relationship holds for all kinds of w

aves, w

hether they are water w

aves, sound waves, radio w

aves, or light waves.

FIGU

RE 25.7 !If the w

avelength is 1 meter, and one

wavelength per second passes the pole,

then the speed of the wave is 1 m

/s.

The equationv

�G

fm

akes sense: During

each vibration, a wave

travels a distance of one w

avelength.

CH

APTER 25

VIBRATION

S AN

D W

AVES 495

thin

k!

If a water w

ave vibrates up and down tw

o times each second and the

distance between w

ave crests is 1.5 m, w

hat is the frequency of the wave?

What is its w

avelength? What is its speed? Answ

er: 25.4.1

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25.4 Wave Speed

" Teach

ing

Tip Exp

lain th

at the

frequ

ency o

f a vibratin

g so

urce is

the sam

e as the freq

uen

cy of th

e w

ave it pro

du

ces.

" Teach

ing

Tip Exp

lain o

r d

erive wave sp

eed: Sp

eed 5

w

aveleng

th 3

frequ

ency.

Sup

po

rt this w

ith th

e freigh

t car exam

ple.

" Teach

ing

Tip H

ave stud

ents

calculate th

e wavelen

gth

s of

their favo

rite local rad

io statio

ns.

Wavelen

gth

5 sp

eed/freq

uen

cy. For exam

ple, 1000-kHz w

aves have w

avelengths 5 (3 3

108 m

/s)/(10

6 Hz) 5

300 m. Su

rprisin

gly

lon

g!

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496

Table 25.1 shows som

e wavelengths and corresponding frequen-

cies of sound in air at the same tem

perature. Notice that the product of

wavelength and frequency is the sam

e for each example—

340 m/s in this

case. During a concert, you do not hear the high notes in a chord before

you hear the low notes. T

he sounds of all instruments reach you at the

same tim

e. Notice that long w

avelengths have low frequencies, and short

wavelengths have high frequencies. W

avelength and frequency vary inversely to produce the sam

e wave speed for all sounds.

CON

CEPT

CHECK

......How

do you calculate the sp

eed of a w

ave?

If a train of freight cars, each 10 m

long, rolls b

y you at the rate of 2 cars each second

, what is the sp

eed of the train?

You can look at this problem in tw

o ways, the C

hapter 4 way and the

Chapter 25 w

ay.

From C

hapter 4 recall:

20 m/s

v�

dt2

�10 m

1 s�

�

Note that d is the length of that part of the train that passes you in

time

t.

Here in C

hapter 25 we com

pare the train to wave m

otion, w

here the wavelength corresponds to 10 m

, and the frequency is 2 H

z. Then

wave speed

�w

avelength�

frequency�

�(10 m

)�

(2 Hz)

�20 m

/s

O

ne of the nice things about physics is that different ways of

looking at things produce the same answ

er. When this doesn’t hap-

pen, and there is no error in computation, then the validity of one (or

both!) of those ways is suspect.

do

the

ma

th!

What is the w

avelength of a 340-H

z sound wave

when the speed of sound

in air is 340 m/s?

Answ

er: 25.4.2

thin

k!

Table 2

5.1

SoundW

aves

Wavelength (m

)Frequency (H

z)W

ave Speed (m/s)

2.13

1.29

0.86

0.64

160

264

396

528

340

340

340

340

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49

6

Be sure to distinguish electrom

agnetic waves from

longitudinal sound waves. Consider discussing Chapter 27 and Chapter 37 m

aterial here to lead into the fam

ily of electrom

agnetic waves. Show how electrom

agnetic waves are grouped according to their wavelengths and frequencies.

Y

ou

can calcu

late the

speed

of a w

ave by

mu

ltiplyin

g th

e wavelen

gth

by

the freq

uen

cy.

Te

ac

hin

g R

es

ou

rc

es

• Reading and Study W

orkbook • Presentation

EXPR

ESS

• Interactive Textbook

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NCEP

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ECK

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25.5 Transverse Waves

Suppose you create a wave along a rope by shaking the free end up

and down, as show

n in Figure 25.8. The m

otion of the rope is at right angles to the direction in w

hich the wave is m

oving. Whenever the

motion of the m

edium is at right angles to the direction in w

hich a w

ave travels, the wave is a transverse w

ave. W

aves in the stretched strings of m

usical instruments and the electrom

agnetic w

aves that make up radio w

aves and light are transverse.

CON

CEPT

CHECK

......What are som

e examp

les of transverse waves?

25.6 Longitudinal Waves

Not all w

aves are transverse. Sometim

es the particles of the m

edium m

ove back and forth in the same direction in w

hich the w

ave travels. When the particles oscillate parallel to or along the

direction of the wave rather than at right angles to it, the w

ave is a

longitudinal wave.

Sound waves are longitudinal w

aves.B

oth transverse and longitudinal waves can be dem

onstrated with

a loosely-coiled spring, as shown in Figure 25.9. A

transverse wave is

demonstrated by shaking the end of a coiled spring up and dow

n. A

longitudinal wave is dem

onstrated by shaking the end of the coiled spring in and out. In this case w

e see that the medium

vibrates paral-lel to the direction of energy transfer.

CON

CEPT

CHECK

......What is an exam

ple of a long

itudinal w

ave?

! FIG

URE 25.9

Transverse and longitudi-nal w

aves transfer energy from

left to right. a. W

hen the end of a coiled spring is shaken up and dow

n, a transverse w

ave is produced. b.

When it is shaken in

and out, a longitudinal w

ave is produced.

! FIG

URE 25.8

A person creates a trans-

verse wave by shaking

the free end of a rope up and dow

n. The arrows

represent the motion of

the rope.

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APTER 25

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497

25.5 Transverse W

avesK

ey

Term

transverse w

ave

W

aves in th

e stretch

ed strin

gs

of m

usical in

strum

ents an

d th

e electro

mag

netic w

aves that

make u

p rad

io w

aves and

ligh

t are tran

sverse.

25.6 Longitudinal W

avesK

ey

Term

lon

gitu

din

al wave

" Teach

ing

Tip A

llow

stu

den

ts to p

lay with

large

sprin

gs o

r Slinkys u

ntil th

ey can

dem

on

strate and

explain

the

differen

ce betw

een tran

sverse an

d lo

ng

itud

inal w

aves.

Ask W

ith respect to the direction of the w

ave’s motion,

how do the directions of

vibrations differ for transverse and longitudinal w

aves? Perp

end

icular fo

r transverse;

parallel fo

r lon

gitu

din

al

So

un

d w

aves are lo

ng

itud

inal w

aves.

Te

ac

hin

g R

es

ou

rc

es

• Reading and Study W

orkbook • Transparency 51• Presentation

EXPR

ESS

• Interactive Textbook

CON

CEPT

CHECK

......

CON

CEPT

CHECK

......

CON

CEPT

CHECK

......

CON

CEPT

CHECK

......

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498

25.7 InterferenceA

material object such as a rock w

ill not share its space with another

rock. But m

ore than one vibration or wave can exist at the sam

e tim

e in the same space. If you drop tw

o rocks in water, the w

aves produced by each can overlap and form

an interference pattern. A

n interference pattern is a regular arrangement of places w

here w

ave effects are increased, decreased, or neutralized. Interference

patterns occur when w

aves from different sources arrive at the

same point—

at the same tim

e.In constructive interference, the crest of one w

ave overlaps the crest of another and their individual effects add together. T

he result is a w

ave of increased amplitude. A

s Figure 25.10a shows, this is

called reinforcement. In destructive interference, the crest of one

wave overlaps the trough of another and their individual effects are

reduced. The high part of one w

ave simply fills in the low

part of another. A

s Figure 25.10b shows, this is called cancellation.

FIGU

RE 25.10 !There are tw

o types of wave

interference. a. In construc-tive interference, the w

aves reinforce each other to pro-duce a w

ave of increased am

plitude.b. In destructive interference, the w

aves can-cel each other and no w

ave is produced.

Sound, a longitudinal w

ave, requires a medium

. It can’t travel in a vacuum

because there’s nothing to com

press and stretch.

Seismologist

When an earthquake occurs, the sudden release of energy produces

waves. Seis m

ologists study and interpret those waves in order to

determine the strength and location of the earthquake. They com

pare the speed, am

plitude, and reception of primary longitudinal w

aves w

ith secondary transverse waves. Because they understand how

waves

travel and the materials through w

hich they pass, seismologists are

able to describe earthquakes, learn about the composition of Earth,

and possibly predict future earthquakes. Seismologists conduct

research from university and governm

ent facilities, such as the National

Earthquake Information Service (N

EIS) in Colorado.

Ph

ysics o

n th

e Jo

b

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8

25.7 InterferenceK

ey

Term

sin

terference p

attern,

con

structive in

terference,

destru

ctive interferen

ce, ou

t of

ph

ase, in p

hase

! Teach

ing

Tip D

escribe

interferen

ce by d

rawin

g

Figu

re 25.10 on

the b

oard

. If yo

u h

ave a ripp

le tank, sh

ow

th

e overlap

pin

g o

f water w

aves an

d in

terference.

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Wave interference is easiest to see in w

ater. Figure 25.11a shows

the interference pattern made w

hen two vibrating objects touch the

surface of water. T

he gray “spokes” are regions where w

aves cancel each other out. A

t points along these regions, the waves from

the two

objects arrive “out of step,” or out of phase, with one another. W

hen w

aves are out of phase, the crests of one wave overlap the troughs

of another to produce regions of zero amplitude. T

he dark and light-striped regions are w

here the waves are “in step,” or in phase, w

ith each other. W

hen waves are in phase, the crests of one w

ave overlap the crests of the other, and the troughs overlap as w

ell.Interference patterns are nicely illustrated by

the overlapping of concentric circles printed on a pair of clear sheets, as show

n in Figures 25.11b and 25.12. W

hen the sheets overlap with their centers

slightly apart, a so-called moiré pattern is form

ed that is very sim

ilar to the interference pattern of w

ater waves (or any kind of w

aves). A slight shift

in either of the sheets produces noticeably differ-ent patterns. If a pair of such sheets is available, be sure to try this and see the variety of patterns for yourself.

Interference is characteristic of all wave

motion, w

hether the waves are w

ater waves, sound

waves, or light w

aves. The interference of sound is

discussed in the next chapter, and the interference of light in C

hapter 31.

CON

CEPT

CHECK

......What causes interference p

atterns?

! FIG

URE 25.11

a. Two overlapping w

ater w

aves produce an interfer-ence pattern. b. O

verlapping concentric circles produce a pictorial representation of an interfer-ence pattern.

FIGU

RE 25.12 "

A m

oiré pattern is very similar

to an interference pattern.

ab

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499

# Teach

ing

Tip M

ake a pair

of tran

sparen

cies of co

ncen

tric circles. Su

perim

po

se them

on

yo

ur o

verhead

pro

jector an

d

sho

w th

e variety of in

terference

pattern

s that resu

lt wh

en th

eir cen

ters are disp

laced. O

ne

examp

le is sho

wn

in Fig

ure 25.12.

Ask Can w

aves overlap in such a w

ay as to produce a zero am

plitude? Yes, th

at is th

e destru

ctive interferen

ce ch

aracteristic of all w

aves.

In

terference p

atterns

occu

r wh

en w

aves fro

m d

ifferent so

urces arrive at

the sam

e po

int—

at the sam

e tim

e.

Te

ac

hin

g R

es

ou

rc

es

• Reading and Study W

orkbook • Laboratory M

anual 71• Presentation

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ESS

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CEPT

CHECK

......

CON

CEPT

CHECK

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500

25.8 Standing Waves

If you tie a rope to a wall and shake the free end up and dow

n, you w

ill produce a wave in the rope. T

he wall is too rigid to shake, so

the wave is reflected back along the rope to you. B

y shaking the rope just right, you can cause the incident (original) and reflected w

aves to form

a standing wave. A

standing wave is a w

ave that appears to stay in one place—

it does not seem to m

ove through the medium

. C

ertain parts of a standing wave rem

ain stationary. Nodes are the

stationary points on a standing wave.

Interestingly enough, you could hold your fingers on either side of the rope at a node, and the rope w

ould not touch them. O

ther parts of the rope w

ould make contact w

ith your fingers. The posi-

tions on a standing wave w

ith the largest amplitudes are know

n as

antinodes. Antinodes occur halfw

ay between nodes.

Standing waves are the result of interference. W

hen two w

aves of equal am

plitude and wavelength pass through each other in opposite

directions, the waves are alw

ays out of phase at the nodes. As Figure

25.13 shows, the nodes are stable regions of destructive interference.

FIGU

RE 25.13 !The incident and reflected w

aves interfere to produce a standing w

ave. The nodes are places that rem

ain stationary.

thin

k!

Is it possible for one wave

to cancel another wave so

that the combined am

pli-tude is zero? Explain your answ

er. Answ

er: 25.8

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0

25.8 Standing W

avesK

ey

Term

sstan

din

g w

ave, no

de, an

tino

de

" Teach

ing

Tip Em

ph

asize that

a stand

ing

wave is th

e result o

f in

terference.

" Teach

ing

Tip U

se a lon

g th

in

sprin

g o

r a rop

e to d

emo

nstrate

stand

ing

waves. H

ave stud

ents

iden

tify the n

od

es and

com

e up

clo

se to in

spect th

em. C

han

ge

the freq

uen

cy and

sho

w th

at on

ly sp

ecific frequ

encies allo

w th

e creatio

n o

f stand

ing

waves.

" Teach

ing

Tidb

it Figu

re 25.14a sh

ow

s the lo

west

frequ

ency o

f vibratio

n o

f a stan

din

g w

ave—th

e fun

dam

ental

frequ

ency.

" Teach

ing

Tip Po

int o

ut th

at fo

r a string

free at on

e end

and

a tu

be o

pen

at on

e end

and

closed

at th

e oth

er end

, stand

ing

waves

form

wh

en o

dd

integ

er mu

ltiples

of q

uarter w

aveleng

ths fit in

to

the vib

rating

med

ium

. A so

da-

po

p b

ottle is an

examp

le of a

tub

e op

en at o

ne en

d an

d clo

sed

at the o

ther en

d.

A

stand

ing

wave

form

s on

ly if half a

wavelen

gth

or a m

ultip

le of h

alf a w

aveleng

th fits exactly in

to th

e len

gth

of th

e vibratin

g m

ediu

m.

Te

ac

hin

g R

es

ou

rc

es

• Reading and Study W

orkbook • Problem

-Solving Exercises in Physics 13-3

• Transparency 52• Presentation

EXPR

ESS

• Interactive Textbook• N

ext-Time Q

uestion 25-1

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NCEP

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ECK

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NCEP

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ECK

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You can produce a variety of standing waves by shaking the rope

at different frequencies. Once you find a frequency that produces a

standing wave, doubling or tripling the frequency w

ill also produce a standing w

ave. A

standing wave form

s only if half a wavelength

or a multiple of half a w

avelength fits exactly into the length of the vibrating m

edium. In Figure 25.14a, the rope length equals half a

wavelength. In Figure 25.14b, the rope length equals one w

avelength. In Figure 25.14c, the rope length equals one and one-half w

ave-lengths. If you keep increasing the frequency, you’ll produce m

ore interesting w

aves.

Standing waves are set up in the strings of m

usical instruments that

are struck. They are set up in the air in an organ pipe and the air of a

soda-pop bottle when air is blow

n over the top. Standing waves can be

produced in either transverse or longitudinal waves.

CON

CEPT

CHECK

......At w

hat waveleng

ths can a standing

w

ave form in a vib

rating m

edium

?

25.9 The Doppler Effect

Imagine a bug jiggling its legs and bobbing up and dow

n in the m

iddle of a quiet puddle, as shown in Figure 25.15. Suppose the bug

is not going anywhere but is m

erely treading water in a fixed posi-

tion. The crests of the w

ave it makes are concentric circles, because

the wave speed is the sam

e in all directions. If the bug bobs in the w

ater at a constant frequency, the distance between w

ave crests (the w

avelength) will be the sam

e for all successive waves. W

aves encounter point A

as frequently as they encounter point B. T

his means that the

frequency of wave m

otion is the same at points A

and B, or anyw

here in the vicinity of the bug. T

his wave frequency is the sam

e as the bob-bing frequency of the bug.

! FIG

URE 25.14

You can produce a variety of standing w

aves. a. Shake the rope until you set up a standing w

ave of ! 12 ! w

avelength. b. Shake w

ith twice the frequency

and produce a standing wave of

1 wavelength.

c. Shake with three tim

es the fre-quency and produce a standing w

ave of 1! 12 ! w

avelengths.

CH

APTER 25

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S AN

D W

AVES 501

For:Visit:W

eb Code: –

Doppler Effect activity

ww

w.PHSchool.com

csp

4259

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25.9 The Doppler

EffectK

ey

Term

sD

op

pler effect, b

lue sh

ift, red

shift

Co

mm

on

Misco

ncep

tion

Changes in w

ave speed cause the D

oppler effect.

FAC

T The D

op

pler effect is an

ap

paren

t chan

ge in

frequ

ency

du

e to th

e mo

tion

of th

e sou

rce.

" Teach

ing

Tip Place an

electro

nic w

histle th

at emits a

sou

nd

of ab

ou

t 3000 Hz in

to a

spo

ng

e, rub

ber, o

r foam

ball.

Intro

du

ce the D

op

pler effect b

y th

row

ing

the b

all arou

nd

the

roo

m. A

sk stud

ents to

describ

e w

hat th

ey hear as th

e ball m

oves

thro

ug

h th

e air. Then

ask if the

frequ

ency o

f the so

un

d th

at the

wh

istle emits actu

ally chan

ges.

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502

Suppose the jiggling bug moves across the w

ater at a speed less than the w

ave speed. In effect, the bug chases part of the crests it has produced. T

he wave pattern is distorted and is no longer concentric,

as shown in Figure 25.16. T

he center of the outer crest was m

ade w

hen the bug was at the center of that circle. T

he center of the next sm

aller crest was m

ade when the bug w

as at the center of that circle, and so forth. T

he centers of the circular crests move in the direction

of the swim

ming bug. A

lthough the bug maintains the sam

e bob-bing frequency as before, an observer at B

would encounter the crests

more often. T

he observer would encounter a higher frequency. T

his is because each successive crest has a shorter distance to travel so they arrive at B

more frequently than if the bug w

ere not moving tow

ard B.

An observer at A

, on the other hand, encounters a lower fre-

quency because of the longer time betw

een wave-crest arrivals. To

reach A, each crest has to travel farther than the one ahead of it due

to the bug’s motion.

As a w

ave source approaches, an observer encounters w

aves with a higher frequency. A

s the wave source

moves aw

ay, an observer encounters waves w

ith a lower frequency.

This apparent change in frequency due to the m

otion of the source (or receiver) is called the D

oppler effect (after the Austrian scientist

Christian D

oppler, 1803–1853). The greater the speed of the source,

the greater will be the D

oppler effect.W

ater waves spread over the flat surface of the w

ater. Sound and light w

aves, on the other hand, travel in three-dimensional space in

all directions like an expanding balloon. Just as circular wave crests

are closer together in front of the swim

ming bug, spherical sound or

light wave crests ahead of a m

oving source are closer together than those behind the source and encounter a receiver m

ore frequently.

FIGU

RE 25.15 !

A stationary bug jiggling

in still water produces a

circular water w

ave.

FIGU

RE 25.16 !

A bug sw

imm

ing in still w

ater produces a wave

pattern that is no longer concentric.

Police Offi cer

Police officers are responsible for protecting people. While that

involves catching criminals and solving crim

es, it also requires that police officers prevent drivers from

speeding. In this way, police

officers protect pedestrians and people in vehicles. One w

ay that police officers prevent speeding is by using radar equipm

ent. Radar equipm

ent sends waves tow

ard a moving vehicle and uses the

Doppler effect to determ

ine the speed of the vehicle. By knowing

how to operate the device, police officers can determ

ine when a

driver is not obeying the speed limit.

Ph

ysics o

n th

e Jo

b

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50

2

" Teach

ing

Tip D

escribe th

e p

attern th

at a station

ary bu

g

jigg

ling

in still w

ater makes as

sho

wn

in Fig

ure 25.15. D

raw

circles to sh

ow

the to

p view

of

circular rip

ples m

ade b

y a bu

g

bo

bb

ing

in th

e water. Stress

that w

ave speed

, wavelen

gth

, an

d freq

uen

cy are the sam

e in

all directio

ns, as sh

ow

n b

y the

circular sh

ape.

" Teach

ing

Tip N

ow

con

sider

a mo

ving

bu

g an

d th

e pattern

it m

akes (Figu

re 25.16). Explain

h

ow

the freq

uen

cy of w

aves is in

creased in

fron

t of th

e bu

g;

waves w

ou

ld b

e enco

un

tered

mo

re often

(mo

re frequ

ently) b

y yo

ur h

and

placed

in th

e water in

fro

nt o

f the b

ug

. (The o

bserver

wo

uld

also en

cou

nter a sh

orter

wavelen

gth

; since v is a co

nstan

t fo

r a given

med

ium

, then

as f in

creases, l decreases.) Sim

ilarly w

aves wo

uld

be en

cou

ntered

less o

ften (less freq

uen

tly) beh

ind

th

e bu

g.

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Bats hunt moths in

darkness by echo loca-tion and the D

oppler effect. Som

e moths

are protected by a thick covering of fuzzy scales that deaden the echoes.

! FIG

URE 25.17

The pitch of sound is higher w

hen the source m

oves toward

you, and lower w

hen the source m

oves aw

ay.

Sound T

he Doppler effect is evident w

hen you hear the changing pitch of a siren as a firetruck passes you. Look at Figure 25.17. W

hen the firetruck approaches, the pitch sounds higher than norm

al. This

occurs because the sound wave crests are encountering you m

ore fre-quently. W

hen the firetruck passes and moves aw

ay, you hear a drop in pitch because the w

ave crests are encountering you less frequently.Police m

ake use of the Doppler effect of radar w

aves in measur-

ing the speeds of cars on the highway. R

adar waves are electrom

ag-netic w

aves, lower in frequency than light and higher in frequency

than radio waves. Police bounce them

off moving cars as show

n in Figure 25.18. A

computer built into the radar system

calculates the speed of the car relative to the radar unit by com

paring the frequency of the radar w

ith the frequency of the reflected waves.

Light The D

oppler effect also occurs for light. When a light source

approaches, there is an increase in its measured frequency, and

when it recedes, there is a decrease in its frequency. A

n increase in frequency is called a blue shift, because the increase is tow

ard the high-frequency, or blue, end of the color spectrum

. A decrease in

frequency is called a red shift, referring to the low-frequency, or

red, end of the color spectrum. D

istant galaxies, for example, show

a red shift in the light they em

it. A m

easurement of this shift enables

astronomers to calculate their speeds of recession. A

rapidly spinning star show

s a red shift on the side turning away from

us and a blue shift on the side turning tow

ard us. This enables a calculation of the

star’s spin rate.

CON

CEPT

CHECK

......How

does the ap

parent freq

uency of waves chang

e as a w

ave source moves?

CH

APTER 25

VIBRATION

S AN

D W

AVES 503

! FIG

URE 25.18

The police calculate a car’s speed by m

easur-ing the D

oppler effect of radar w

aves.

thin

k!

When a source m

oves tow

ard you, do you m

easure an increase or decrease in w

ave speed? Answ

er: 25.9

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503

" Teach

ing

Tip R

elate the

con

cept o

f the m

ovin

g b

ug

to th

e w

aves from

the m

ovin

g so

urces

in Fig

ures 25.17 an

d 25.18.

Ask The w

aves are more

crowded in front of the

swim

ming bug and m

ore spread out behind. Is the w

ave speed greater in front of the bug (and less behind the bug)? N

o!

Frequ

ency, n

ot sp

eed, is g

reater in

fron

t of th

e bu

g an

d less

beh

ind

.

" Teach

ing

Tip Em

ph

asize the

distin

ction

betw

een w

ave speed

an

d w

ave frequ

ency.

" Teach

ing

Tip Sw

ing

a sou

nd

so

urce at th

e end

of a strin

g in

a h

orizo

ntal circle. R

elate this

to th

e siren o

f a fire eng

ine

and

the rad

ar of th

e hig

hw

ay p

atrol (Fig

ures 25.17 an

d 25.18).

(Men

tion

that so

un

d req

uires a

med

ium

; radar d

oesn

’t.)

" Teach

ing

Tip Po

int o

ut

that lig

ht, rad

ar, TV, an

d rad

io

waves are all electro

mag

netic

in n

ature. Th

e waves d

iffer o

nly in

frequ

ency (an

d h

ence

wavelen

gth

) and

energ

y per

ph

oto

n.

" Teach

ing

Tip R

elate the p

itch

of so

un

d to

the co

lor o

f ligh

t. B

oth

dep

end

on

frequ

ency.

A

s a wave so

urce

app

roach

es, an

ob

server enco

un

ters waves w

ith

a hig

her freq

uen

cy. As th

e wave

sou

rce mo

ves away, an

ob

server en

cou

nters w

aves with

a lo

wer freq

uen

cy.

Te

ac

hin

g R

es

ou

rc

es

• Concept-Developm

ent Practice Book 25-1

• Problem-Solving Exercises

in Physics 13-2• Laboratory M

anual 70

CON

CEPT

CHECK

......

CON

CEPT

CHECK

......

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504

25.10 Bow

Waves

When the speed of the source in a m

edium is as great as the speed

of the waves it produces, som

ething interesting happens. The w

aves pile up. C

onsider the bug in the previous example w

hen it swim

s as fast as the w

ave speed. Can you see that the bug w

ill keep up with the

wave crests it produces? Instead of the crests getting ahead of the bug,

they pile up or superimpose on one another directly in front of the

bug, as suggested in Figure 25.19. The bug m

oves right along with the

leading edge of the waves it is producing.

The sam

e thing happens when an aircraft travels at the speed of

sound. In the early days of jet aircraft, it was believed that this pileup

of sound waves in front of the airplane im

posed a “sound barrier” and that to go faster than the speed of sound, the plane w

ould have to “break the sound barrier.” W

hat actually happens is that the overlapping w

ave crests disrupt the flow of air over the w

ings, so that it is harder to control the plane w

hen it is flying close to the speed of sound. B

ut the barrier is not real. Just as a boat can easily travel faster than the speed of w

ater waves, an airplane w

ith sufficient pow

er can easily travel faster than the speed of sound. Then w

e say that it is supersonic—

faster than sound. A supersonic airplane flies

into smooth, undisturbed air because no sound w

ave can propagate out in front of it. Sim

ilarly, a bug swim

ming faster than the speed

of water w

aves finds itself always entering into w

ater with a sm

ooth, unrippled surface.

When the bug sw

ims faster than w

ave speed, ideally it produces a w

ave pattern as shown in Figure 25.20. It outruns the w

ave crests it produces. T

he crests overlap at the edges, and the pattern made

by these overlapping crests is a V shape, called a bow

wave, w

hich appears to be dragging behind the bug.

A bow

wave occurs w

hen a w

ave source moves faster than the w

aves it produces. The fam

iliar bow

wave generated by a speedboat knifing through the w

ater is pro-duced by the overlapping of m

any circular wave crests.

FIGU

RE 25.19 !

A bug sw

imm

ing at the w

ave speed “keeps up” w

ith the wave crests it

produces.

FIGU

RE 25.20 "A

bug swim

ming faster

than the wave speed

produces a wave pattern

in which the w

ave crests overlap at the edges.

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50

4

25.10 Bow W

avesK

ey

Term

bo

w w

ave

" Teach

ing

Tip A

sk the class

to co

nsid

er the w

aves mad

e by

two

ston

es thro

wn

in th

e water.

Sketch th

e overlap

pin

g w

aves as sh

ow

n b

elow

.

Ask w

here th

e water is h

igh

est ab

ove th

e no

rmal w

ater level, an

d th

en in

dicate th

e two

p

laces wh

ere the w

aves overlap

w

ith X

’s. This is co

nstru

ctive in

terference. Exten

d th

e sw

imm

ing

bu

g co

ncep

t to sp

eeds

greater th

an w

ave speed

s and

sh

ow

the reg

ion

s of o

verlap th

at p

rod

uce th

e bo

w w

ave (sketchin

g

Figu

res 25.15, 25.16, and

25.19). Sh

ow

ho

w a series o

f overlap

s m

akes up

the V

-shap

ed en

velop

e sh

ow

n in

Figu

re 25.21.

" Teach

ing

Tip Exp

lain th

at th

e form

ation

of th

e bo

w

wave in

Figu

re 25.20 is ano

ther

examp

le of co

nstru

ctive in

terference, w

ith an

app

reciable

resultin

g am

plitu

de.

A

bo

w w

ave occu

rs w

hen

a wave so

urce

mo

ves faster than

the w

aves it p

rod

uces.

Te

ac

hin

g R

es

ou

rc

es

• Reading and Study W

orkbook • Transparency 53• Presentation

EXPR

ESS

• Interactive Textbook

CO

NCEP

TCH

ECK

......

CO

NCEP

TCH

ECK

......

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Don’t confuse super-

sonic with ultrasonic.

Supersonic has to do w

ith speed—faster

than sound. Ultrasonic

involves frequency—higher than w

e can hear.

FIGU

RE 25.21 !The w

ave patterns made

by a bug swim

ming at suc-

cessively greater speeds change. O

verlapping at the edges occurs only w

hen the source travels faster than w

ave speed.

Figure 25.21 shows som

e wave patterns m

ade by sources mov-

ing at various speeds. After the speed of the source exceeds the w

ave speed, increased speed produces a bow

wave w

ith a narrower V

shape.

CON

CEPT

CHECK

......What causes a b

ow w

ave?

25.11 Shock Waves

A speedboat knifing through the w

ater generates a two-dim

ensional bow

wave. A

supersonic aircraft similarly generates a shock w

ave. A

shock wave is a three-dim

ensional wave that consists of overlap-

ping spheres that form a cone.

A shock w

ave occurs when an

object moves faster than the speed of sound. Just as the bow

wave

of a speedboat spreads until it reaches the shore of a lake, the conical shock w

ave generated by a supersonic craft spreads until it reaches the ground, as show

n in Figure 25.22.T

he bow w

ave of a speedboat that passes by can splash and douse you if you are at the w

ater’s edge. In a sense, you can say that you are hit by a “w

ater boom.” In the sam

e way, a conical shell of com

pressed air sw

eeps behind a supersonic aircraft. The sharp crack heard w

hen the shock w

ave that sweeps behind a supersonic aircraft reaches the

listeners is called a sonic boom.

We don’t hear a sonic boom

from a slow

er-than-sound, or sub-sonic, aircraft, because the sound w

ave crests reach our ears one at a tim

e and are perceived as a continuous tone. Only w

hen the craft m

oves faster than sound do the crests overlap and encounter the lis-tener in a single burst. T

he sudden increase in pressure has much the

same effect as the sudden expansion of air produced by an explosion.

Both processes direct a burst of high-pressure air to the listener. T

he ear cannot distinguish betw

een the high pressure from an explosion

and the high pressure from m

any overlapping wave crests.

CH

APTER 25

VIBRATION

S AN

D W

AVES 505

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505

25.11 Shock Waves

Ke

y Te

rms

sho

ck wave, so

nic b

oo

m

Co

mm

on

Misco

ncep

tion

A sonic boom

is a mom

entary burst of high pressure produced w

hen som

ething exceeds the speed of sound.

FAC

T A so

nic b

oo

m is actu

ally a co

ntin

uo

us fro

nt o

f hig

h p

ressure

gen

erated b

y faster-than

-sou

nd

so

urces.

" Teach

ing

Tip Q

uestio

ns

raised b

y stud

ents ab

ou

t sho

ck w

aves and

the so

nic b

oo

m can

b

e effectively answ

ered b

y su

bstitu

ting

the exam

ple o

f an

aircraft in th

e air for th

e examp

le o

f a speed

bo

at knifin

g th

rou

gh

th

e water. If yo

u’re en

joyin

g

a picn

ic lun

ch at th

e edg

e of a

river wh

en a sp

eedb

oat co

mes

by an

d d

rench

es you

, you

wo

n’t

attribu

te this to

the id

ea that

the sp

eedb

oat ju

st exceeded

the

speed

of th

e water w

aves. Yo

u

kno

w th

e bo

at is gen

erating

a co

ntin

uo

us b

ow

wave so

lon

g

as it travels faster than

waves in

w

ater. Likewise fo

r aircraft.

Ask W

hy can’t a subsonic aircraft, no m

atter how loud it

may be, produce a shock w

ave or sonic boom

? There w

ill be n

o

overlap

pin

g o

f sph

erical waves

to fo

rm a co

ne u

nless th

e aircraft m

oves faster th

an th

e waves

it gen

erates.

The analogy between bow waves in water and shock waves in air is useful when discussing the shock waves produced by supersonic aircraft.

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506

A com

mon m

isconception is that sonic booms are produced at

the mom

ent that an aircraft flies through the “sound barrier”—that

is, just as the aircraft surpasses the speed of sound. This is equivalent

to saying that a boat produces a bow w

ave only when it first over-

takes its own w

aves. This is not so. T

he fact is that a shock wave and

its resulting sonic boom are sw

ept continuously behind an aircraft traveling faster than sound, just as a bow

wave is sw

ept continuously behind a speedboat. In Figure 25.23, listener B

is in the process of hearing a sonic boom

. Listener A has already heard it, and listener C

w

ill hear it shortly. The aircraft that generated this shock w

ave may

have broken through the sound barrier hours ago!It is not necessary that the m

oving source emit sound for it to

produce a shock wave. O

nce an object is moving faster than the speed

of sound, it will m

ake sound. A supersonic bullet passing overhead

produces a crack, which is a sm

all sonic boom. If the bullet w

ere larger and disturbed m

ore air in its path, the crack would be m

ore boom

like. When a lion tam

er cracks a circus whip, the cracking

sound is actually a sonic boom produced by the tip of the w

hip when

it travels faster than the speed of sound. Snap a towel and the end can

exceed the speed of sound and produce a mini sonic boom

. The bul-

let, whip, and tow

el are not in themselves sound sources, but w

hen traveling at supersonic speeds they produce their ow

n sound as waves

of air are generated to the sides of the moving objects.

On the m

atter of sound in general: You know that you’ll dam

-age your eyes if you stare at the sun. W

hat many people don’t know

is that you’ll sim

ilarly damage your ears if you overexpose them

to loud sounds. D

o as your author does when in a room

with very loud

music—

leave. If for any reason you don’t want to leave—

really enjoy-able m

usic or good camaraderie w

ith friends—stay, but use ear plugs

of some kind! You’re not being a w

imp w

hen you give the same care

to your ears that you give to your eyes.

CON

CEPT

CHECK

......What causes a shock w

ave?

Watch for the advent

of newly designed air-

craft that fly 1.8 times

the speed of sound and produce sonic boom

s only one-hundredth the strength of the super-sonic Concorde, w

hich w

as grounded following

a fatal accident in 2000.

FIGU

RE 25.22 !A

shock wave is sw

ept continuously behind a supersonic aircraft.

FIGU

RE 25.23 "

The shock wave has not yet

encountered listener C, but

is now encountering listener

B, and has already passed

listener A.

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50

6

! Teach

ing

Tip C

on

struct a

sho

ck wave o

n th

e bo

ard as

follo

ws: First p

lace you

r chalk

anyw

here o

n th

e bo

ard to

sign

ify tim

e zero. D

raw a 1-m

-lon

g

ho

rizon

tal line to

the rig

ht to

rep

resent h

ow

far an aircraft h

as m

oved

in a certain

time. Su

pp

ose

it mo

ves at twice th

e speed

of

sou

nd

(Mach

2).

Du

ring

the tim

e it mo

ves 1 m, th

e so

un

d it in

itially mad

e has m

oved

h

alf this d

istance, w

hich

you

m

ark on

the m

idp

oin

t of yo

ur

line. Exp

lain th

at the in

itial sou

nd

h

as expan

ded

sph

erically, wh

ich

you

represen

t two

-dim

ensio

nally

by d

rawin

g a circle. Exp

lain th

at th

is circle represen

ts on

ly on

e of

the n

early infin

ite nu

mb

er of

circles that m

ake up

the sh

ock

wave, w

hich

you

draw

by m

aking

tan

gen

ts from

the en

d p

oin

t to

the circle. Th

e sho

ck wave sh

ou

ld

be a 60

8 wed

ge (30

8 abo

ve you

r h

orizo

ntal lin

e, and

308 b

elow

). M

ove th

e center 10 cm

at a time

in th

e directio

n o

f travel and

d

raw circles (red

uce th

e radiu

s each

time) w

ithin

the tw

o

tang

ents. Exp

lain h

ow

the sp

eed

of th

e craft is simp

ly the ratio

of

the h

orizo

ntal lin

e (1 m) to

the

radial d

istance (0.5 m

) of th

e big

circle (an

d likew

ise the resp

ective h

orizo

ntal lin

es to rad

ii of

smaller circles).

A

sho

ck wave o

ccurs

wh

en an

ob

ject m

oves faster th

an th

e speed

o

f sou

nd

.

Te

ac

hin

g R

es

ou

rc

es

• Concept-Developm

ent Practice Book 25-2, 25-3

• Next-Tim

e Question 25-2

CO

NCEP

TCH

ECK

......

CO

NCEP

TCH

ECK

......

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REVIEW

For:Visit:W

eb Code: –

Self-Assessment

PHSchool.com

csa 2500

Conce

pt Su

mm

ary

••

•••

•

•

The period of a pendulum

depends only on the length of the pendulum

and the acceleration of gravity.

•

The source of all w

aves is a vibration.

•

The energy in w

aves is carried by a dis-turbance in a m

edium.

•

Calculate the w

ave speed by multiplying

the wavelength and the frequency.

•

Waves in the stretched strings of m

usical instrum

ents and electromagnetic w

aves are transverse. Sound w

aves are longitu-dinal.

•

Interference patterns occur when w

aves from

different sources arrive at the same

point—at the sam

e time.

•

A standing w

ave forms if a m

ultiple of half a w

avelength fits into the length of the m

edium.

•

As a w

ave source approaches, an observer encounters w

aves with a higher fre-

quency. As a w

ave source moves aw

ay, an observer encounters w

aves with a low

er frequency.

•

A bow

wave occurs w

hen a wave source

moves faster than the w

aves it produces.

•

A shock w

ave occurs when an object

moves faster than the speed of sound.

5crest (p. 492)

trough (p. 492)

amplitude (p. 492)

wavelength (p. 492)

frequency (p. 492)

hertz (p. 492)

transverse wave

(p. 497)

longitudinal wave

(p. 497)

interference pattern

(p. 498)

constructive interference(p. 498)

destructive interference (p. 498)

out of phase (p. 499)

in phase (p. 499)

standing wave

(p. 500)

node (p. 500)

antinodes (p. 500)

Doppler effect(p. 502)

blue shift (p. 503)

red shift (p. 503)

bow w

ave (p. 504)

shock wave (p. 505)

sonic boom (p. 505)

CH

APTER 25

VIBRATION

S AN

D W

AVES 507

25.2.1 A

100-Hz w

ave vibrates 100 times/s.

25.2.110 s.

The period is �

�1 vib

0.1 Hz

1 vib0.1 vib/s

�

25.4.1 T

he frequency of the wave is 2 H

z; its w

avelength is 1.5 m; and its w

ave speed is %

�ƒ

�(1.5 m

)�

(2 Hz)

�3 m

/s.

25.4.2 T

he wavelength m

ust be 1 m. T

hen wave

speed�

(1 m)

�(340 H

z)�

340 m/s.

25.8 Yes. T

his is called destructive interference. In a standing w

ave, for example, parts of

the wave have no am

plitude—the nodes.

25.9 N

either! It is the frequency of a wave that

undergoes a change, not the wave speed.

thin

k!

Answ

ers

Key Te

rms

••

•••

•

vibration (p. 490)

wave (p. 490)

period (p. 491)

simple harm

onic m

otion (p. 491)

sine curve (p. 491)

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507

R

EVIEW

Te

ac

hin

g R

es

ou

rc

es

• TeacherEXPR

ESS

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508

Check

Conce

pts

••

••

••

Section 25.0 1. D

oes a vibration or a wave spread out

through space?

Section 25.1 2. W

hat is the period of a pendulum that takes

one second to make a com

plete back-and-forth vibration?

3. Suppose that a pendulum has a period of

1.5 seconds. How

long does it take to make

a complete back-and-forth vibration?

4. Is a pendulum w

ith a 1.5-second period longer or shorter in length than a pendulum

w

ith a 1-second period?

Section 25.2 5. H

ow is a sine curve related to a w

ave?

6. Distinguish am

ong these different parts of a w

ave: amplitude, crest, trough, and w

ave-length.

7. Distinguish betw

een the period and the frequency of a vibration or a w

ave. How

do they relate to one another?

Section 25.3 8. D

oes the medium

in which a w

ave travels m

ove along with the w

ave itself? Defend

your answer.

Section 25.4 9. H

ow does the speed of a w

ave relate to its w

avelength and frequency?

10. As the frequency of sound is increased, does

the wavelength increase or decrease? G

ive an exam

ple.

Sections 25.5 and 25.6 11. D

istinguish between a transverse w

ave and a longitudinal w

ave.

Section 25.7 12. D

istinguish between constructive interfer-

ence and destructive interference.

13. Is interference a property of only some

types of waves or of all types of w

aves?

Section 25.8 14. W

hat causes a standing wave?

Section 25.9 15. W

hen a wave source m

oves toward a receiv-

er, does the receiver encounter an increase in w

ave frequency, wave speed, or both?

16. Does the D

oppler effect occur for only some

types of waves or all types of w

aves?

ASSESS

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50

8

A

SSESS

Check

Conce

pts

1. A

wave spreads out through

space.

2. 1 s

3. 1.5 s

4. longer

5. A

sine curve is a pictorial representation of a w

ave.

6. A

mplitude—

maxim

um

displacement; crest—

point of greatest positive displacem

ent; trough—point of greatest negative displacem

ent; wavelength—

distance from one crest to

the next

7. Period—

time to com

plete one cycle; frequency—

how m

any cycles occur in a given tim

e

8. N

o. The disturbance, not the m

aterial itself, moves.

9. Speed 5

wavelength 3

frequency

10. Decreases; sm

aller musical

instruments produce higher

frequency sounds.

11. Transverse—m

edium m

oves perpendicular to w

ave direction; longitudinal—m

edium m

oves back and

forth parallel to wave

direction.

12. Constructive—causes an

additive effect; destructive—canceling effect

13. All. It is a prim

e test for wave

properties.

14. Interference of original wave

with reflected w

ave

15. Increase in frequency only 16. A

ll

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Section 25.10 17. C

ompared w

ith the speed of water w

aves how

fast must a bug sw

im to keep up w

ith the w

aves it produces? How

fast must a boat

move to produce a bow

wave?

18. Distinguish a bow

wave from

a shock wave.

Section 25.11 19. a. W

hat is a sonic boom?

b. H

ow fast m

ust an aircraft fly in order to produce a sonic boom

?

20. If you encounter a sonic boom, is that

evidence that an aircraft just exceeded the speed of sound to becom

e supersonic?

Thin

k a

nd R

ank •

•••

••

Rank each of the follow

ing sets of scenarios in order of the quantity or property involved. List them

from left to right. If scenarios have equal

rankings, then separate them w

ith an equal sign. (e.g., A

= B

)

21. A fire engine’s siren em

its a certain frequency. R

ank from greatest to least the

apparent frequency heard by the stationary listener in each scenario.

(A) T

he fire engine is traveling toward a

listener at 30 m/s.

(B) T

he fire engine is traveling away from

a listener at 5 m

/s. (C

) The fire engine is traveling tow

ard a listener at 5 m

/s. (D

) The fire engine is traveling aw

ay from a

listener at 30 m/s.

22. Shown below

are four different pairs of transverse w

ave pulses that move tow

ard each other. A

t some point in tim

e the pulses m

eet and interact (interfere) with each

other. Rank the four cases from

greatest to least on the basis of the height of the peak that results w

hen the centers of the pairs coincide.

23. All the w

aves below have the sam

e speed in the sam

e medium

. Use a ruler and rank

these waves from

greatest to least according to am

plitude, wavelength, frequency, and

period.

C

HA

PTER 25 VIBRATIO

NS A

ND

WA

VES 509

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509

17. As fast as the w

aves move;

faster than the waves m

ove

18. Bow—

a 2-D “V

” on the water

surface; shock—a 3-D

cone in

the air

19. a. Incident shock wave

b. Faster than sound 20. N

o; it could have been any tim

e ago. It depends on

speed, not time.

Thin

k a

nd R

ank

21. A, C, B, D

22. A, B, D

, C

23. Am

plitude: D, B, A

, C

Wavelength: D

, A, B, C

Frequency: C, B, A

, D

Period: D, A

, B, C

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510

24. The four sets of w

aves below are a top view

of circular w

ave patterns made by a bug

jiggling on the surface of water. R

ank them

from greatest to least based on the speed of

the bug.

25. The shock w

aves depicted below are pro-

duced by supersonic aircraft. Rank them

from

greatest to least based on the speed of the aircraft.

Plu

g a

nd Ch

ug •

••

••

•

26. A nurse counts 76 heartbeats in one m

inute. W

hat are the period and frequency of the heart’s oscillations?

27. New

York’s 300-m high C

iticorp® Tower

oscillates in the wind w

ith a period of 6.80 s. C

alculate its frequency of vibration.

28. Calculate the speed of w

aves in a puddle that are 0.15 m

apart and made by tapping

the water surface tw

ice each second.

29. Calculate the speed of w

aves in water that

are 0.4 m apart and have a frequency of 2 H

z.

30. The low

est frequency we can hear is about

20 Hz. C

alculate the wavelength associated

with this frequency for sound that travels at

340 m/s. H

ow long is this in feet?

Thin

k a

nd Ex

pla

in ••

••

••

31. Does the period of a pendulum

depend of the m

ass of the bob? On the length of the

string?

32. If a pendulum is shortened, does the fre-

quency increase or decrease? What about its

period?

33. Carm

elita swings to and fro in a sitting posi-

tion on a playground swing. W

illiam says

that if she stands while sw

inging, a longer tim

e will occur betw

een back-and-forth sw

ings. Carlos says no, that the to-and-fro

time of the sw

ing will be unaffected. W

ho, if either, do you agree w

ith?

34. You dip your finger repeatedly into a puddle of w

ater and make w

aves. What happens to

the wavelength if you dip your finger m

ore frequently?

35. If you double the frequency of a vibrat-ing object, w

hat happens to its period?

36. How

does the frequency of vibration of a sm

all object floating in water com

pare to the num

ber of waves passing it each second?

37. If you triple the frequency of a vibrating object, w

hat will happen to its period?

5

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51

0

24. D, C, B, A

25. A, C, B

Plu

g a

nd Ch

ug

26. T 5 (1/76) m

in; f 5 76/m

in

27. f 5 1/(6.80 s) 5

0.15 Hz

28. v 5 lf 5

(0.15 m)(2/s) 5

0.3 m/s

29. v 5 lf 5

(0.4 m)(2/s) 5

0.8 m/s

30. l 5

v/f 5 (340 m

/s)/(20/s) 5

17 m or 56 ft

Thin

k a

nd Ex

pla

in 31. Pendulum

period depends on

its length, but not its mass.

32. Shorter pendulum, higher

frequency, shorter period

33. Disagree w

ith both. CG of

“bob” is closer to pivot, so

shorter pendulum has shorter

period.

34. Higher frequency of dip

m

akes shorter wavelength.

35. f = 1/T, and T = 1/f. Double

one, then other is half. So 2f gives 1/2 T.

36. Both are the same.

37. f and T are reciprocals of each other, so tripling the frequency results in one third

the period.

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38. Red light has a longer w

avelength than vio-let light. W

hich has the greater frequency?

39. How

far, in terms of w

avelength, does a w

ave travel in one period?

40. If a wave vibrates up and dow

n twice each

second and travels a distance of 20 m each

second, what is its frequency? Its w

ave speed? (W

hy is this question best answered

by careful reading of the question rather than searching for a form

ula?)

41. The w

ave patterns seen in Figure 25.6 are com

posed of circles. What does this tell you

about the speed of the waves in different

directions?

42. Sound from Source A

has a frequency twice

as great as the frequency of sound from

Source B. C

ompare the w

avelengths of sound from

the two sources.

43. What kind of m

otion should you impart to

a stretched coiled spring to produce a trans-verse w

ave? A longitudinal w

ave?

44. Would it be correct to say that the D

oppler effect is the apparent change in the speed of a w

ave due to the motion of the source?

(Why is this question a test of reading

comprehension as w

ell as a test of physics know

ledge?)

45. In the Doppler effect, does frequency

change? Does w

avelength change? Does

wave speed change?

46. Can the D

oppler effect be observed with

longitudinal waves, w

ith transverse waves,

or with both?

47. A railroad locom

otive is at rest with its

whistle shrieking, and then it starts m

oving tow

ard you.

a. Does the frequency that you hear increase,

decrease, or stay the same?

b. D

oes the wavelength that reaches your ear

increase, decrease, or stay the same?

c. H

ow about the speed of sound in the air

between you and the locom

otive?

48. When a driver blow

s his horn while ap-

proaching a stationary listener, the listener hears an increase in the frequency of the horn. W

ould the listener hear an increase in the frequency of the horn if she w

ere also in a car traveling at the sam

e speed in the same

direction as the first driver? Explain.

49. Astronom

ers find that light coming from

point A

at the edge of the sun has a slightly higher frequency than light from

point B at

the opposite side. What do these m

easure-m

ents tell us about the sun’s motion?

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511

38. The speeds are the same, so

the w

ave with the shorter

wavelength has the greater

frequency—violet.

39. A w

ave takes a time equal to

one period to travel a distance of one w

avelength. D

istance 5 speed 3

time 5

(w

avelength 3 frequency) 3

period 5

(wavelength 3

1/period) 3

period 5

wavelength

40. 2 Hz; 20 m

/s

41. The waves travel at the sam

e speed in all directions.

42. Source A has half w

avelength

of Source B sound.

43. Transverse wave shakes back-

and-forth perpendicular to

coiled spring. Longitudinal w

ave shakes back-and-forth

along length.

44. No, it is the change in the

observed frequency of a w

ave due to motion of the

observer with respect to the

source. There is no change in w

ave speed when the

source moves.

45. Frequency and wavelength

change; not w

ave speed.

46. Both 47. a. Frequency increases.

b. W

avelength decreases.

c. N

o change in speed.

48. No. There is no relative

motion betw

een source and

listener.

49. The sun is spinning, since point A

must be m

oving

toward the observer and

point B m

ust be moving aw

ay.

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(continued)

512

50. Does a boat m

oving through the water

always produce a bow

wave? D

efend your answ

er.

51. Whenever you w

atch a high-flying aircraft overhead, it seem

s that its sound comes

from behind the craft rather than from

w

here you see it. Why is this?

52. How

does the angle of the V shape of a

bow w

ave depend on the speed of the w

ave source?

53. Why is it that a subsonic aircraft, no

matter how

loud it may be, cannot

produce a sonic boom?

54. True or false: A sonic boom

occurs only w

hen an aircraft is breaking through the sound barrier. D

efend your answer.

55. Consider an earthquake caused by a single

disturbance, which sends out both trans-

verse and longitudinal waves that travel w

ith distinctly different speeds in the ground. H

ow can earth scientists in different loca-

tions determine the earthquake origin?

Thin

k a

nd So

lve •

••

••

•

56. The period of a sim

ple pendulum is given

by Lg

�T

�2)

, where g

is the acceleration of gravity and L is the length of the pendulum

. In a lab, you w

ant to double the period of a certain pendulum

. Your friend says you’ll have to m

ake the pendulum tw

ice as long. D

o you agree with your friend?

57. Maria show

s her friends a simple

31-cm-long pendulum

. Her teacher, look-

ing on, asks if she can predict the period of the pendulum

before she demonstrates it.

What’s your prediction?

58. The Foucault pendulum

in the rotunda of the G

riffith Observatory in Los A

ngeles has a 110-kg brass ball at the end of a 12.2-m

- long cable. W

hat is the period of this pen-dulum

?

59. You are looking through your grandparents’ w

indow and notice a hum

mingbird feeder

hanging by a rope. You can’t see the top of the rope, but you notice that in a gentle breeze the feeder m

oves back and forth with

a period of 4.0 seconds. You make a calcula-

tion and announce to your grandparents that the rope is 4 m

long. Your grandparents go outside and m

easure the rope. Should they be im

pressed with you?

60. For your science fair project you decide to m

ake a simple pendulum

for a grandfather clock, such that the period of the pendulum

is 2.00 seconds. Show

that the length of your pendulum

should be just slightly less than the length of a m

eterstick. (Use g =

9.8 m/s

2

here.)

5

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51

2

50. If the speed of the boat exceeds w

ave speed, yes. If slow

er, no.

51. It takes negligible time for

light to get to you from

the airplane, but it takes a noticeable tim

e for sound to

reach you. When sound

from

a fast-moving source

reaches you, the source is farther along.

52. The narrower the angle, the

faster the source

53. At subsonic speeds, there

is no overlapping of waves

to produce high-pressure regions; w

here there is no

shock wave, there is no

sonic boom

.

54. False; a sonic boom occurs

continuously for supersonic source.

55. From difference in arrival

times, each scientist calculates

distance, and draws a circle

of possible sources. Origin

of quake is w

here 3 such

circles drawn by different

scientists overlap.

Thin

k a

nd So

lve

56. No, don’t agree. T ~√

L so

L ~ T2. To double T, L m

ust be 4 tim

es as long.

57. T 5 2

p √

L/g 5

2p√

(0.31 m)/(10 m

/s 2) 5 1.1 s

58. T 5 2

p √

L/g 5

2p√

(12.2 m)/(10 m

/s 2) 5 6.9 s

(or 7.0 s using g 5 9.8 m

/s 2)

59. Yes. From

T 5 2

p √

L/g, L 5

gT2/4

p2 5

[(10 m

/s 2) 3 (4.0 s) 2]/4

p2 5

4.0 m

60. From T 5

2p

√L/g,

L 5 gT

2/4p

2 5 [(9.8 m

/s 2) 3

(2.00 s) 2]/4p

2 5 0.99 m

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61. Melanie is new

to the nursing program.

With a patient she counts 84 heartbeats in

one minute. She calculates that the period

and frequency of the heartbeats are 0.71 s and 1.4 H

z respectively. Is she correct?

62. A design engineer figures that a proposed

new skyscraper w

ill swing to and fro in

strong winds at a frequency of 0.15 H

z. A

new assistant asks how

much tim

e a person in the skyscraper w

ill experience during each com

plete swing. W

hat’s your answer?

63. In lab you strike a tuning fork that has a frequency of 340 H

z. For a speed of sound of 340 m

/s, how does the w

avelength of the resulting sound w

ave compare w

ith the length of a m

eter stick?

64. If a wave vibrates back and forth three tim

es each second, and its w

avelength is 2 meters,

what is its frequency? Its period? Its speed?

65. While w

atching ocean waves at the dock

of the bay, Otis notices that 10 w

aves pass beneath him

in 30 seconds. He also notices

that the crests of successive waves exactly

coincide with the posts that are 5 m

eters apart. W

hat are the period, frequency, wave-

length, and speed of the ocean waves?

66. The crests on a long surface w

ater wave are

20 m apart, and in 1 m

inute 10 crests pass by. W

hat is the speed of this wave?

67. Radio w

aves are electromagnetic w

aves that travel at the speed of light, 300,000 kilo-m

eters per second. What is the w

avelength of FM

radio waves received at 100 m

ega-hertz on your radio dial?

68. The w

avelength of red light is about 700 nanom

eters, or 7 ! 10

–7 m. T

he fre-quency of the red light reflected from

a m

etal surface and the frequency of the vibrating electron that produces it are the sam

e. What is this frequency?

69. The half-angle of the shock-w

ave cone generated by a supersonic aircraft is 45°. W

hat is the speed of the plane relative to the speed of sound?

Activ

ity•

••

••

•

70. Tie a rubber tube, a spring, or a rope to a

fixed support and produce standing waves,

as Figure 25.14 suggests. How

many nodes

can you produce? How

can you change the num

ber of nodes?

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-Solving PracticeA

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513

61. She is correct. f = (84 beats)/(60 s) 5

1.4 Hz.; T 5

1/f 5

1/(1.4 s 21) 5

0.71 s

62. T 5 1/f 5

1/(0.15 s 21) 5

6.7 s

63. The same; from

v 5 lf,

l 5 v/f 5

(340 m/s)/(340 H

z) 5

1.0 m.

64. f 5 3 H

z; T 5 1/3 s; v 5

lf 5

(2 m)(3/s) 5

6 m/s

65. T 5 (30 s)/10 5

3 s; f 5

1/3 Hz; l 5

5 m; v 5

lf 5

(5 m)(1/3 H

z) 5 1.67 m

/s

66. v 5 lf 5

(20 m)(10/m

in) 5

200 m/m

in, or 3.3 m/s

67. l 5 v/f 5

(300,000 km/s) 4

(100,000,000/s) 5

0.003 km,

or 3 m

68. f 5 v/l 5

(3 3 10

8 m/s) 4

(7 3

102

7 m) 5

4.3 3 10

14 Hz,

an extraordinarily high

frequency by ordinary standards

69. The plane’s speed is 1.41 tim

es the speed of sound. In

right triangle, the distance A

B is √2 or 1.41 tim

es the distance A

C.

Activ

ity 70. Check students’ w

ork. The frequency of the incident w

ave determines the num

ber of nodes produced.

Te

ac

hin

g R

es

ou

rc

es

• Computer Test Bank

• Chapter and Unit Tests

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