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J1l~~ t ISmslOOf -0

~~k~f~ WJflJl1 ~ l+)

~~Yf~ ~11t ~ (-)

( 1 (VI plusmn )I)jI ) M o~ =(rll) t (i ~-)Lgty) (IJ 4~

( ~j -)() l iexclq] oee ()C)~~ o~ =(q I)~

1(t1 +J) =)(

~=t t

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1 ~ It~ y ~~1a ~r~1 1Q

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J1l~~ t ISmslOOf -0

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( 1 (VI plusmn )I)jI ) M o~ =(rll) t (i ~-)Lgty) (IJ 4~

( ~j -)() l iexclq] oee ()C)~~ o~ =(q I)~

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~=t t

yjW fEiexcl ha fryQ

1=- 4= t

j1~t ~ e~ l-=~ ~1~1- 1 lt1= lfi ~ l

i+=lI J

~) o

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~ 7f PlI1l1raquo11Y) t1) t ~~

1~ N (iexclJ) =f

j1J S~ ~1 If 11~~~

r~~1 5~~

11 iexcl ffrIII 1 11 r1~Q ~ t1gl1~l

-vp-t1 VI 1 VW Ot-f)o1~tI1~ ~ yO viexcl P ~~flqf~

VJjt111 1VSJ1JVDt vro

~7W) l 7f J~

(~f t7V iexcl-)3 i =)~ 1 ~

Ypdm 7-bt1 T Of-ver-m ~~~t TY4deg~ ~~~

(T (1-JI 1l~CJ V~~ Jt l1 ~1tJ1~4411$ )(

(1 (I-~)1S I1V tIr ~1 1 D 1

1(11) (11) T ~ (7) lV

n~-~))qt) V =~

X9~ -Y~ exV ntllJ -f ~11 YJ_~~ _~1~

J~~141 7f ~~~l44f~t ~ ~~ ~kiexcl fYJ 1 ~ YgtWJ 1f t~11III)JI0W -o] rff1 t

vp-pYJ iexcliexcl oiexcl-e~ oVJ y ~141-7jjgtiquestl~lO I -I(11J1 I 141lf4Vr vn1 YJ 41~ YfJ7N) ~ (JWt14TfJ ~1fOJ

rllY) vnt VJ ~a vYiexcl tf _tmi

1ft1 t~OW--l~ 1-ttIt1 61 -viexcl~Q l1Iw 1l~-1Al1 1~ Y-riexcl j ~ ~~fv Gtf4Ygt1J~ ltJ

XV ti rI vi ~ 1i3 t Z

~11~raquo1A~ ~ -Y~ v _~7113

(17t llf ~1~f ~~1411 f ~~ Ij

(T(II-~h)t)S V -rV7~ j=-nI +J 7 =1V

YflMJ v( f 1~ W1 7f 1~t v_~1

V~1~ ~~J4~ o=t iJu =~ ltfa u 11--) ~~rO iexcl ~~Vmiddot (~

1l~ ~~~a~AIIS In t-v o~~ 1 f ~tf ~a i41~V (J ~1LlY1 4S~

~V)ha1M 4 t 11117 l01

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(jf~4raquo) 1~~v

(1~-f)tf~

(11 +~f(l-xty) co (( Q)~ VI] =

[C~t+~~-)~) ((gtJ +(J-J t 1(l-1lgtiexcl) C7 ] =1 ~t~t

[llo) t]lt17 [iacuteJ+~)i ]OJZ = ~ 4 +O~ tt~~ -vt)

raquo Jn~ ~4JL ~~i middot~lJ Wf V)lfmiddot1Q

(J~+l(11-)(gty)~ v Cl~ ( i~ -4-1-~ -)l ~) ~ tf ~

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frAacute~ lV ~i~)]~ ~vr~7f ~3 (1a~)

~N)~Q ~a~ ~ f 143- D) ~ 1l)JI ~~ lt (~141~~)

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11411iA o 1ltro ~ ~~JI~fi~~O ~~ middot ~Jon1i1p

~vysr ~1 1~t ~f-lt7VJ1Iamp1

f(gtI~ (~(1)cPJ Vl ~g~~ffl t ~ _-___~~

(1~ +X)) lt11 V = (1(1-)(~-)~V 1~

(iexcl(gt7-11W) CIJ uacute = ~

~~~~~YI01 X7 ~ ~~~ (qfJ

-- ~ )= bJ(-tgt t ~

f

I -

1~f J~jt)rIJ ~~~ tJf laquo1 ~ tmiddot1~~ 5ff] ~O ~ 41 ~~1(

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11 =(

~r s~~r vf 14~

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x Y-it r~ t 1~

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1-~+~~ Jx

11 -( =y~

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~~orb~~7f ~ ~ Jll5r -p el ~ rj 1~

GfI~1 -viexcl ~VI1 ti

1mlt-y vy ~46t OV111 ~ Yiexcl(J f lt~llfl

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O~)1

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1~ - ~~

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gtt 4WI ~7JWJ ~~O middotJCfMl-JII IWI

r~7v l ~1H or~~ -yrr ~1Y 3

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f+1] 14fk ~ ~~t tJS~ l1 vf lYffJlJ ~1 ~~01 plt~

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1V~ ~~~~ ~1~ YfI1f rtOS 11 OfYJWJ SlfY bull~~7 Y(WO ~~x ~~

bull

(a)

Transparency 65 TIpler Physics lor Scientists and Engineers Figure 16-11 page 488 Figure 16 15 page ~2 Fourth EditionVolume 1

Copyright copy1999 W H Freeman and CompanyStanding waves on a string fiacutexed at bota elles (left) alld on a string fixed at one end (right)

A A------ ~~--~---~~ -~----~~~~-I~_____ ___tnJoIII ~I~~J n=l

bullbull

~-------~~~~-~ shy ~~~ffsect~sect~~~~~~~middot-middotmiddot-~Fundamental first harmonic Fundamental first harmoniacutec A A

--raquo

~=~~~~~~~~~~~~~sect~~~~ A A

(b) n 2

Third harmonic

(e) n=3

Fifth harmoniacutec A A A A

(d) n=4

A

A A

=3jiIAi~~~~~~~~~ Fourth harmonic Seventh harmonic

A A A A A

(e) n=5

I INinth harmonisI~ L bull oC L bull

A A A A

Fifth hannonic I

~7W) l 7f J~

(~f t7V iexcl-)3 i =)~ 1 ~

Ypdm 7-bt1 T Of-ver-m ~~~t TY4deg~ ~~~

(T (1-JI 1l~CJ V~~ Jt l1 ~1tJ1~4411$ )(

(1 (I-~)1S I1V tIr ~1 1 D 1

1(11) (11) T ~ (7) lV

n~-~))qt) V =~

X9~ -Y~ exV ntllJ -f ~11 YJ_~~ _~1~

J~~141 7f ~~~l44f~t ~ ~~ ~kiexcl fYJ 1 ~ YgtWJ 1f t~11III)JI0W -o] rff1 t

vp-pYJ iexcliexcl oiexcl-e~ oVJ y ~141-7jjgtiquestl~lO I -I(11J1 I 141lf4Vr vn1 YJ 41~ YfJ7N) ~ (JWt14TfJ ~1fOJ

rllY) vnt VJ ~a vYiexcl tf _tmi

1ft1 t~OW--l~ 1-ttIt1 61 -viexcl~Q l1Iw 1l~-1Al1 1~ Y-riexcl j ~ ~~fv Gtf4Ygt1J~ ltJ

XV ti rI vi ~ 1i3 t Z

~11~raquo1A~ ~ -Y~ v _~7113

(17t llf ~1~f ~~1411 f ~~ Ij

(T(II-~h)t)S V -rV7~ j=-nI +J 7 =1V

YflMJ v( f 1~ W1 7f 1~t v_~1

V~1~ ~~J4~ o=t iJu =~ ltfa u 11--) ~~rO iexcl ~~Vmiddot (~

1l~ ~~~a~AIIS In t-v o~~ 1 f ~tf ~a i41~V (J ~1LlY1 4S~

~V)ha1M 4 t 11117 l01

V)rgtj~~ VftS VltltW f] 4IfHP ~7f ~O~~Q~~ltJ MI1Joj

(jf~4raquo) 1~~v

(1~-f)tf~

(11 +~f(l-xty) co (( Q)~ VI] =

[C~t+~~-)~) ((gtJ +(J-J t 1(l-1lgtiexcl) C7 ] =1 ~t~t

[llo) t]lt17 [iacuteJ+~)i ]OJZ = ~ 4 +O~ tt~~ -vt)

raquo Jn~ ~4JL ~~i middot~lJ Wf V)lfmiddot1Q

(J~+l(11-)(gty)~ v Cl~ ( i~ -4-1-~ -)l ~) ~ tf ~

YJ~ ~-w1 mItg-l ~~~a~usoJnJY1S

~1flIo~ I1fO middot ~g ~ lIJ~O~lt-wrs

O~lTI()~1~ ~o

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11411iA o 1ltro ~ ~~JI~fi~~O ~~ middot ~Jon1i1p

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f(gtI~ (~(1)cPJ Vl ~g~~ffl t ~ _-___~~

(1~ +X)) lt11 V = (1(1-)(~-)~V 1~

(iexcl(gt7-11W) CIJ uacute = ~

~~~~~YI01 X7 ~ ~~~ (qfJ

-- ~ )= bJ(-tgt t ~

f

I -

1~f J~jt)rIJ ~~~ tJf laquo1 ~ tmiddot1~~ 5ff] ~O ~ 41 ~~1(

bull I

11 =(

~r s~~r vf 14~

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~y11 ~iexcl-2m11 X ~)l)f +

x Y-it r~ t 1~

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~~~ J ~1 1 ~~vtLotJ 11~~1I I

7ln~ ~J7JJ~ (Y

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GfI~1 -viexcl ~VI1 ti

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1V~ ~~~~ ~1~ YfI1f rtOS 11 OfYJWJ SlfY bull~~7 Y(WO ~~x ~~

bull

(a)

Transparency 65 TIpler Physics lor Scientists and Engineers Figure 16-11 page 488 Figure 16 15 page ~2 Fourth EditionVolume 1

Copyright copy1999 W H Freeman and CompanyStanding waves on a string fiacutexed at bota elles (left) alld on a string fixed at one end (right)

A A------ ~~--~---~~ -~----~~~~-I~_____ ___tnJoIII ~I~~J n=l

bullbull

~-------~~~~-~ shy ~~~ffsect~sect~~~~~~~middot-middotmiddot-~Fundamental first harmonic Fundamental first harmoniacutec A A

--raquo

~=~~~~~~~~~~~~~sect~~~~ A A

(b) n 2

Third harmonic

(e) n=3

Fifth harmoniacutec A A A A

(d) n=4

A

A A

=3jiIAi~~~~~~~~~ Fourth harmonic Seventh harmonic

A A A A A

(e) n=5

I INinth harmonisI~ L bull oC L bull

A A A A

Fifth hannonic I

1ft1 t~OW--l~ 1-ttIt1 61 -viexcl~Q l1Iw 1l~-1Al1 1~ Y-riexcl j ~ ~~fv Gtf4Ygt1J~ ltJ

XV ti rI vi ~ 1i3 t Z

~11~raquo1A~ ~ -Y~ v _~7113

(17t llf ~1~f ~~1411 f ~~ Ij

(T(II-~h)t)S V -rV7~ j=-nI +J 7 =1V

YflMJ v( f 1~ W1 7f 1~t v_~1

V~1~ ~~J4~ o=t iJu =~ ltfa u 11--) ~~rO iexcl ~~Vmiddot (~

1l~ ~~~a~AIIS In t-v o~~ 1 f ~tf ~a i41~V (J ~1LlY1 4S~

~V)ha1M 4 t 11117 l01

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(jf~4raquo) 1~~v

(1~-f)tf~

(11 +~f(l-xty) co (( Q)~ VI] =

[C~t+~~-)~) ((gtJ +(J-J t 1(l-1lgtiexcl) C7 ] =1 ~t~t

[llo) t]lt17 [iacuteJ+~)i ]OJZ = ~ 4 +O~ tt~~ -vt)

raquo Jn~ ~4JL ~~i middot~lJ Wf V)lfmiddot1Q

(J~+l(11-)(gty)~ v Cl~ ( i~ -4-1-~ -)l ~) ~ tf ~

YJ~ ~-w1 mItg-l ~~~a~usoJnJY1S

~1flIo~ I1fO middot ~g ~ lIJ~O~lt-wrs

O~lTI()~1~ ~o

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frAacute~ lV ~i~)]~ ~vr~7f ~3 (1a~)

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f(gtI~ (~(1)cPJ Vl ~g~~ffl t ~ _-___~~

(1~ +X)) lt11 V = (1(1-)(~-)~V 1~

(iexcl(gt7-11W) CIJ uacute = ~

~~~~~YI01 X7 ~ ~~~ (qfJ

-- ~ )= bJ(-tgt t ~

f

I -

1~f J~jt)rIJ ~~~ tJf laquo1 ~ tmiddot1~~ 5ff] ~O ~ 41 ~~1(

bull I

11 =(

~r s~~r vf 14~

-rv~~ ~Jtl~ 7f ~ (y

~y11 ~iexcl-2m11 X ~)l)f +

x Y-it r~ t 1~

y~ ~ Yr~ttj ~ v2 -Y-YIv J1lf1 ~ J ~+~ =Jtt iI ~Yrf WVJJ1]l1 1jv ~11~

~~~ J ~1 1 ~~vtLotJ 11~~1I I

7ln~ ~J7JJ~ (Y

nr4~4fi3 ~~ o

YJY1rI) S ~t Tf ~l or~~~ YII7II6 ~~ ~)flSIt Y AgtIJ ~t ~~~~s (J

~ gt11 7 ~~f J1IW1lJf111 1t7-1 ~ Y1(O

_nfJ~lt[ 4nJ3

Ij] (1 i -t -1 _1 ~l v -v l--l J ~ 1 -~ -7~ -u -

~~~ ~ ~~wiexcl ~flf J J ~f~ f1W4Jwa 15 ~VY( ~1f1111 ~IQ~

1-~+~~ Jx

11 -( =y~

Of~~ J -fprlJI1 J53 ~IIJ ~~4 ~

21 uacutexw 111degIW lA raquotw1 7)I middotVq ~J -r 1

~~orb~~7f ~ ~ Jll5r -p el ~ rj 1~

GfI~1 -viexcl ~VI1 ti

1mlt-y vy ~46t OV111 ~ Yiexcl(J f lt~llfl

~ --------shy

7J shy

)1 MI 11p1111 S ~M1 -p t ~~~ ~777(I o~JtY JS~ (

P$-I 11OPt 1ItYltf4f11~ ~ ~~ I (~ l~) ~ l 1-1OIOf Yl I1-0fJ -viexcl 101~5~11lt1 ) S )111 15

O~)1

- d Jt1lJlfJY Y1~Y) -J ~~1-~ (~) p( ol 4~ ~ JIQjVV ~~111 fj 1

1~ - ~~

J-~ ~4_~O__~__v___~___~__________

_1J plusmni

1 ~ t- 1 ~yx6 shy-2J1111 JS ~ + -

~

1 =Vi ~1f~~~ ~ v~ middotl-W ~ yO 171 ~~s

JIJYftIt1-t1J tiP 1111

gtt 4WI ~7JWJ ~~O middotJCfMl-JII IWI

r~7v l ~1H or~~ -yrr ~1Y 3

~IIp f Yf4 r1 Yf

f+1] 14fk ~ ~~t tJS~ l1 vf lYffJlJ ~1 ~~01 plt~

~W~ Ir ( iquest ~ ~~(iacute~~~ JO1S~ I gv f6~ l 1

1V~ ~~~~ ~1~ YfI1f rtOS 11 OfYJWJ SlfY bull~~7 Y(WO ~~x ~~

bull

(a)

Transparency 65 TIpler Physics lor Scientists and Engineers Figure 16-11 page 488 Figure 16 15 page ~2 Fourth EditionVolume 1

Copyright copy1999 W H Freeman and CompanyStanding waves on a string fiacutexed at bota elles (left) alld on a string fixed at one end (right)

A A------ ~~--~---~~ -~----~~~~-I~_____ ___tnJoIII ~I~~J n=l

bullbull

~-------~~~~-~ shy ~~~ffsect~sect~~~~~~~middot-middotmiddot-~Fundamental first harmonic Fundamental first harmoniacutec A A

--raquo

~=~~~~~~~~~~~~~sect~~~~ A A

(b) n 2

Third harmonic

(e) n=3

Fifth harmoniacutec A A A A

(d) n=4

A

A A

=3jiIAi~~~~~~~~~ Fourth harmonic Seventh harmonic

A A A A A

(e) n=5

I INinth harmonisI~ L bull oC L bull

A A A A

Fifth hannonic I

V~1~ ~~J4~ o=t iJu =~ ltfa u 11--) ~~rO iexcl ~~Vmiddot (~

1l~ ~~~a~AIIS In t-v o~~ 1 f ~tf ~a i41~V (J ~1LlY1 4S~

~V)ha1M 4 t 11117 l01

V)rgtj~~ VftS VltltW f] 4IfHP ~7f ~O~~Q~~ltJ MI1Joj

(jf~4raquo) 1~~v

(1~-f)tf~

(11 +~f(l-xty) co (( Q)~ VI] =

[C~t+~~-)~) ((gtJ +(J-J t 1(l-1lgtiexcl) C7 ] =1 ~t~t

[llo) t]lt17 [iacuteJ+~)i ]OJZ = ~ 4 +O~ tt~~ -vt)

raquo Jn~ ~4JL ~~i middot~lJ Wf V)lfmiddot1Q

(J~+l(11-)(gty)~ v Cl~ ( i~ -4-1-~ -)l ~) ~ tf ~

YJ~ ~-w1 mItg-l ~~~a~usoJnJY1S

~1flIo~ I1fO middot ~g ~ lIJ~O~lt-wrs

O~lTI()~1~ ~o

9vev4J 1f ~~ J~if t J(7f1 WId ~1tf ~11 Yf0711 ~a middotvJiJ~) ~ ~~7 ~

frAacute~ lV ~i~)]~ ~vr~7f ~3 (1a~)

~N)~Q ~a~ ~ f 143- D) ~ 1l)JI ~~ lt (~141~~)

)M1 gti)1Y 1I~sa (11 -y lttiexclJ$ noII gtti - lI1 fA

~t ltn~ -o ~VlIrI ~~~~ +--J~ ~~f~MS~l5 ~ IlfJ~W J114 1T1

11411iA o 1ltro ~ ~~JI~fi~~O ~~ middot ~Jon1i1p

~vysr ~1 1~t ~f-lt7VJ1Iamp1

f(gtI~ (~(1)cPJ Vl ~g~~ffl t ~ _-___~~

(1~ +X)) lt11 V = (1(1-)(~-)~V 1~

(iexcl(gt7-11W) CIJ uacute = ~

~~~~~YI01 X7 ~ ~~~ (qfJ

-- ~ )= bJ(-tgt t ~

f

I -

1~f J~jt)rIJ ~~~ tJf laquo1 ~ tmiddot1~~ 5ff] ~O ~ 41 ~~1(

bull I

11 =(

~r s~~r vf 14~

-rv~~ ~Jtl~ 7f ~ (y

~y11 ~iexcl-2m11 X ~)l)f +

x Y-it r~ t 1~

y~ ~ Yr~ttj ~ v2 -Y-YIv J1lf1 ~ J ~+~ =Jtt iI ~Yrf WVJJ1]l1 1jv ~11~

~~~ J ~1 1 ~~vtLotJ 11~~1I I

7ln~ ~J7JJ~ (Y

nr4~4fi3 ~~ o

YJY1rI) S ~t Tf ~l or~~~ YII7II6 ~~ ~)flSIt Y AgtIJ ~t ~~~~s (J

~ gt11 7 ~~f J1IW1lJf111 1t7-1 ~ Y1(O

_nfJ~lt[ 4nJ3

Ij] (1 i -t -1 _1 ~l v -v l--l J ~ 1 -~ -7~ -u -

~~~ ~ ~~wiexcl ~flf J J ~f~ f1W4Jwa 15 ~VY( ~1f1111 ~IQ~

1-~+~~ Jx

11 -( =y~

Of~~ J -fprlJI1 J53 ~IIJ ~~4 ~

21 uacutexw 111degIW lA raquotw1 7)I middotVq ~J -r 1

~~orb~~7f ~ ~ Jll5r -p el ~ rj 1~

GfI~1 -viexcl ~VI1 ti

1mlt-y vy ~46t OV111 ~ Yiexcl(J f lt~llfl

~ --------shy

7J shy

)1 MI 11p1111 S ~M1 -p t ~~~ ~777(I o~JtY JS~ (

P$-I 11OPt 1ItYltf4f11~ ~ ~~ I (~ l~) ~ l 1-1OIOf Yl I1-0fJ -viexcl 101~5~11lt1 ) S )111 15

O~)1

- d Jt1lJlfJY Y1~Y) -J ~~1-~ (~) p( ol 4~ ~ JIQjVV ~~111 fj 1

1~ - ~~

J-~ ~4_~O__~__v___~___~__________

_1J plusmni

1 ~ t- 1 ~yx6 shy-2J1111 JS ~ + -

~

1 =Vi ~1f~~~ ~ v~ middotl-W ~ yO 171 ~~s

JIJYftIt1-t1J tiP 1111

gtt 4WI ~7JWJ ~~O middotJCfMl-JII IWI

r~7v l ~1H or~~ -yrr ~1Y 3

~IIp f Yf4 r1 Yf

f+1] 14fk ~ ~~t tJS~ l1 vf lYffJlJ ~1 ~~01 plt~

~W~ Ir ( iquest ~ ~~(iacute~~~ JO1S~ I gv f6~ l 1

1V~ ~~~~ ~1~ YfI1f rtOS 11 OfYJWJ SlfY bull~~7 Y(WO ~~x ~~

bull

(a)

Transparency 65 TIpler Physics lor Scientists and Engineers Figure 16-11 page 488 Figure 16 15 page ~2 Fourth EditionVolume 1

Copyright copy1999 W H Freeman and CompanyStanding waves on a string fiacutexed at bota elles (left) alld on a string fixed at one end (right)

A A------ ~~--~---~~ -~----~~~~-I~_____ ___tnJoIII ~I~~J n=l

bullbull

~-------~~~~-~ shy ~~~ffsect~sect~~~~~~~middot-middotmiddot-~Fundamental first harmonic Fundamental first harmoniacutec A A

--raquo

~=~~~~~~~~~~~~~sect~~~~ A A

(b) n 2

Third harmonic

(e) n=3

Fifth harmoniacutec A A A A

(d) n=4

A

A A

=3jiIAi~~~~~~~~~ Fourth harmonic Seventh harmonic

A A A A A

(e) n=5

I INinth harmonisI~ L bull oC L bull

A A A A

Fifth hannonic I

O~lTI()~1~ ~o

9vev4J 1f ~~ J~if t J(7f1 WId ~1tf ~11 Yf0711 ~a middotvJiJ~) ~ ~~7 ~

frAacute~ lV ~i~)]~ ~vr~7f ~3 (1a~)

~N)~Q ~a~ ~ f 143- D) ~ 1l)JI ~~ lt (~141~~)

)M1 gti)1Y 1I~sa (11 -y lttiexclJ$ noII gtti - lI1 fA

~t ltn~ -o ~VlIrI ~~~~ +--J~ ~~f~MS~l5 ~ IlfJ~W J114 1T1

11411iA o 1ltro ~ ~~JI~fi~~O ~~ middot ~Jon1i1p

~vysr ~1 1~t ~f-lt7VJ1Iamp1

f(gtI~ (~(1)cPJ Vl ~g~~ffl t ~ _-___~~

(1~ +X)) lt11 V = (1(1-)(~-)~V 1~

(iexcl(gt7-11W) CIJ uacute = ~

~~~~~YI01 X7 ~ ~~~ (qfJ

-- ~ )= bJ(-tgt t ~

f

I -

1~f J~jt)rIJ ~~~ tJf laquo1 ~ tmiddot1~~ 5ff] ~O ~ 41 ~~1(

bull I

11 =(

~r s~~r vf 14~

-rv~~ ~Jtl~ 7f ~ (y

~y11 ~iexcl-2m11 X ~)l)f +

x Y-it r~ t 1~

y~ ~ Yr~ttj ~ v2 -Y-YIv J1lf1 ~ J ~+~ =Jtt iI ~Yrf WVJJ1]l1 1jv ~11~

~~~ J ~1 1 ~~vtLotJ 11~~1I I

7ln~ ~J7JJ~ (Y

nr4~4fi3 ~~ o

YJY1rI) S ~t Tf ~l or~~~ YII7II6 ~~ ~)flSIt Y AgtIJ ~t ~~~~s (J

~ gt11 7 ~~f J1IW1lJf111 1t7-1 ~ Y1(O

_nfJ~lt[ 4nJ3

Ij] (1 i -t -1 _1 ~l v -v l--l J ~ 1 -~ -7~ -u -

~~~ ~ ~~wiexcl ~flf J J ~f~ f1W4Jwa 15 ~VY( ~1f1111 ~IQ~

1-~+~~ Jx

11 -( =y~

Of~~ J -fprlJI1 J53 ~IIJ ~~4 ~

21 uacutexw 111degIW lA raquotw1 7)I middotVq ~J -r 1

~~orb~~7f ~ ~ Jll5r -p el ~ rj 1~

GfI~1 -viexcl ~VI1 ti

1mlt-y vy ~46t OV111 ~ Yiexcl(J f lt~llfl

~ --------shy

7J shy

)1 MI 11p1111 S ~M1 -p t ~~~ ~777(I o~JtY JS~ (

P$-I 11OPt 1ItYltf4f11~ ~ ~~ I (~ l~) ~ l 1-1OIOf Yl I1-0fJ -viexcl 101~5~11lt1 ) S )111 15

O~)1

- d Jt1lJlfJY Y1~Y) -J ~~1-~ (~) p( ol 4~ ~ JIQjVV ~~111 fj 1

1~ - ~~

J-~ ~4_~O__~__v___~___~__________

_1J plusmni

1 ~ t- 1 ~yx6 shy-2J1111 JS ~ + -

~

1 =Vi ~1f~~~ ~ v~ middotl-W ~ yO 171 ~~s

JIJYftIt1-t1J tiP 1111

gtt 4WI ~7JWJ ~~O middotJCfMl-JII IWI

r~7v l ~1H or~~ -yrr ~1Y 3

~IIp f Yf4 r1 Yf

f+1] 14fk ~ ~~t tJS~ l1 vf lYffJlJ ~1 ~~01 plt~

~W~ Ir ( iquest ~ ~~(iacute~~~ JO1S~ I gv f6~ l 1

1V~ ~~~~ ~1~ YfI1f rtOS 11 OfYJWJ SlfY bull~~7 Y(WO ~~x ~~

bull

(a)

Transparency 65 TIpler Physics lor Scientists and Engineers Figure 16-11 page 488 Figure 16 15 page ~2 Fourth EditionVolume 1

Copyright copy1999 W H Freeman and CompanyStanding waves on a string fiacutexed at bota elles (left) alld on a string fixed at one end (right)

A A------ ~~--~---~~ -~----~~~~-I~_____ ___tnJoIII ~I~~J n=l

bullbull

~-------~~~~-~ shy ~~~ffsect~sect~~~~~~~middot-middotmiddot-~Fundamental first harmonic Fundamental first harmoniacutec A A

--raquo

~=~~~~~~~~~~~~~sect~~~~ A A

(b) n 2

Third harmonic

(e) n=3

Fifth harmoniacutec A A A A

(d) n=4

A

A A

=3jiIAi~~~~~~~~~ Fourth harmonic Seventh harmonic

A A A A A

(e) n=5

I INinth harmonisI~ L bull oC L bull

A A A A

Fifth hannonic I

-- ~ )= bJ(-tgt t ~

f

I -

1~f J~jt)rIJ ~~~ tJf laquo1 ~ tmiddot1~~ 5ff] ~O ~ 41 ~~1(

bull I

11 =(

~r s~~r vf 14~

-rv~~ ~Jtl~ 7f ~ (y

~y11 ~iexcl-2m11 X ~)l)f +

x Y-it r~ t 1~

y~ ~ Yr~ttj ~ v2 -Y-YIv J1lf1 ~ J ~+~ =Jtt iI ~Yrf WVJJ1]l1 1jv ~11~

~~~ J ~1 1 ~~vtLotJ 11~~1I I

7ln~ ~J7JJ~ (Y

nr4~4fi3 ~~ o

YJY1rI) S ~t Tf ~l or~~~ YII7II6 ~~ ~)flSIt Y AgtIJ ~t ~~~~s (J

~ gt11 7 ~~f J1IW1lJf111 1t7-1 ~ Y1(O

_nfJ~lt[ 4nJ3

Ij] (1 i -t -1 _1 ~l v -v l--l J ~ 1 -~ -7~ -u -

~~~ ~ ~~wiexcl ~flf J J ~f~ f1W4Jwa 15 ~VY( ~1f1111 ~IQ~

1-~+~~ Jx

11 -( =y~

Of~~ J -fprlJI1 J53 ~IIJ ~~4 ~

21 uacutexw 111degIW lA raquotw1 7)I middotVq ~J -r 1

~~orb~~7f ~ ~ Jll5r -p el ~ rj 1~

GfI~1 -viexcl ~VI1 ti

1mlt-y vy ~46t OV111 ~ Yiexcl(J f lt~llfl

~ --------shy

7J shy

)1 MI 11p1111 S ~M1 -p t ~~~ ~777(I o~JtY JS~ (

P$-I 11OPt 1ItYltf4f11~ ~ ~~ I (~ l~) ~ l 1-1OIOf Yl I1-0fJ -viexcl 101~5~11lt1 ) S )111 15

O~)1

- d Jt1lJlfJY Y1~Y) -J ~~1-~ (~) p( ol 4~ ~ JIQjVV ~~111 fj 1

1~ - ~~

J-~ ~4_~O__~__v___~___~__________

_1J plusmni

1 ~ t- 1 ~yx6 shy-2J1111 JS ~ + -

~

1 =Vi ~1f~~~ ~ v~ middotl-W ~ yO 171 ~~s

JIJYftIt1-t1J tiP 1111

gtt 4WI ~7JWJ ~~O middotJCfMl-JII IWI

r~7v l ~1H or~~ -yrr ~1Y 3

~IIp f Yf4 r1 Yf

f+1] 14fk ~ ~~t tJS~ l1 vf lYffJlJ ~1 ~~01 plt~

~W~ Ir ( iquest ~ ~~(iacute~~~ JO1S~ I gv f6~ l 1

1V~ ~~~~ ~1~ YfI1f rtOS 11 OfYJWJ SlfY bull~~7 Y(WO ~~x ~~

bull

(a)

Transparency 65 TIpler Physics lor Scientists and Engineers Figure 16-11 page 488 Figure 16 15 page ~2 Fourth EditionVolume 1

Copyright copy1999 W H Freeman and CompanyStanding waves on a string fiacutexed at bota elles (left) alld on a string fixed at one end (right)

A A------ ~~--~---~~ -~----~~~~-I~_____ ___tnJoIII ~I~~J n=l

bullbull

~-------~~~~-~ shy ~~~ffsect~sect~~~~~~~middot-middotmiddot-~Fundamental first harmonic Fundamental first harmoniacutec A A

--raquo

~=~~~~~~~~~~~~~sect~~~~ A A

(b) n 2

Third harmonic

(e) n=3

Fifth harmoniacutec A A A A

(d) n=4

A

A A

=3jiIAi~~~~~~~~~ Fourth harmonic Seventh harmonic

A A A A A

(e) n=5

I INinth harmonisI~ L bull oC L bull

A A A A

Fifth hannonic I

~y11 ~iexcl-2m11 X ~)l)f +

x Y-it r~ t 1~

y~ ~ Yr~ttj ~ v2 -Y-YIv J1lf1 ~ J ~+~ =Jtt iI ~Yrf WVJJ1]l1 1jv ~11~

~~~ J ~1 1 ~~vtLotJ 11~~1I I

7ln~ ~J7JJ~ (Y

nr4~4fi3 ~~ o

YJY1rI) S ~t Tf ~l or~~~ YII7II6 ~~ ~)flSIt Y AgtIJ ~t ~~~~s (J

~ gt11 7 ~~f J1IW1lJf111 1t7-1 ~ Y1(O

_nfJ~lt[ 4nJ3

Ij] (1 i -t -1 _1 ~l v -v l--l J ~ 1 -~ -7~ -u -

~~~ ~ ~~wiexcl ~flf J J ~f~ f1W4Jwa 15 ~VY( ~1f1111 ~IQ~

1-~+~~ Jx

11 -( =y~

Of~~ J -fprlJI1 J53 ~IIJ ~~4 ~

21 uacutexw 111degIW lA raquotw1 7)I middotVq ~J -r 1

~~orb~~7f ~ ~ Jll5r -p el ~ rj 1~

GfI~1 -viexcl ~VI1 ti

1mlt-y vy ~46t OV111 ~ Yiexcl(J f lt~llfl

~ --------shy

7J shy

)1 MI 11p1111 S ~M1 -p t ~~~ ~777(I o~JtY JS~ (

P$-I 11OPt 1ItYltf4f11~ ~ ~~ I (~ l~) ~ l 1-1OIOf Yl I1-0fJ -viexcl 101~5~11lt1 ) S )111 15

O~)1

- d Jt1lJlfJY Y1~Y) -J ~~1-~ (~) p( ol 4~ ~ JIQjVV ~~111 fj 1

1~ - ~~

J-~ ~4_~O__~__v___~___~__________

_1J plusmni

1 ~ t- 1 ~yx6 shy-2J1111 JS ~ + -

~

1 =Vi ~1f~~~ ~ v~ middotl-W ~ yO 171 ~~s

JIJYftIt1-t1J tiP 1111

gtt 4WI ~7JWJ ~~O middotJCfMl-JII IWI

r~7v l ~1H or~~ -yrr ~1Y 3

~IIp f Yf4 r1 Yf

f+1] 14fk ~ ~~t tJS~ l1 vf lYffJlJ ~1 ~~01 plt~

~W~ Ir ( iquest ~ ~~(iacute~~~ JO1S~ I gv f6~ l 1

1V~ ~~~~ ~1~ YfI1f rtOS 11 OfYJWJ SlfY bull~~7 Y(WO ~~x ~~

bull

(a)

Transparency 65 TIpler Physics lor Scientists and Engineers Figure 16-11 page 488 Figure 16 15 page ~2 Fourth EditionVolume 1

Copyright copy1999 W H Freeman and CompanyStanding waves on a string fiacutexed at bota elles (left) alld on a string fixed at one end (right)

A A------ ~~--~---~~ -~----~~~~-I~_____ ___tnJoIII ~I~~J n=l

bullbull

~-------~~~~-~ shy ~~~ffsect~sect~~~~~~~middot-middotmiddot-~Fundamental first harmonic Fundamental first harmoniacutec A A

--raquo

~=~~~~~~~~~~~~~sect~~~~ A A

(b) n 2

Third harmonic

(e) n=3

Fifth harmoniacutec A A A A

(d) n=4

A

A A

=3jiIAi~~~~~~~~~ Fourth harmonic Seventh harmonic

A A A A A

(e) n=5

I INinth harmonisI~ L bull oC L bull

A A A A

Fifth hannonic I

Ij] (1 i -t -1 _1 ~l v -v l--l J ~ 1 -~ -7~ -u -

~~~ ~ ~~wiexcl ~flf J J ~f~ f1W4Jwa 15 ~VY( ~1f1111 ~IQ~

1-~+~~ Jx

11 -( =y~

Of~~ J -fprlJI1 J53 ~IIJ ~~4 ~

21 uacutexw 111degIW lA raquotw1 7)I middotVq ~J -r 1

~~orb~~7f ~ ~ Jll5r -p el ~ rj 1~

GfI~1 -viexcl ~VI1 ti

1mlt-y vy ~46t OV111 ~ Yiexcl(J f lt~llfl

~ --------shy

7J shy

)1 MI 11p1111 S ~M1 -p t ~~~ ~777(I o~JtY JS~ (

P$-I 11OPt 1ItYltf4f11~ ~ ~~ I (~ l~) ~ l 1-1OIOf Yl I1-0fJ -viexcl 101~5~11lt1 ) S )111 15

O~)1

- d Jt1lJlfJY Y1~Y) -J ~~1-~ (~) p( ol 4~ ~ JIQjVV ~~111 fj 1

1~ - ~~

J-~ ~4_~O__~__v___~___~__________

_1J plusmni

1 ~ t- 1 ~yx6 shy-2J1111 JS ~ + -

~

1 =Vi ~1f~~~ ~ v~ middotl-W ~ yO 171 ~~s

JIJYftIt1-t1J tiP 1111

gtt 4WI ~7JWJ ~~O middotJCfMl-JII IWI

r~7v l ~1H or~~ -yrr ~1Y 3

~IIp f Yf4 r1 Yf

f+1] 14fk ~ ~~t tJS~ l1 vf lYffJlJ ~1 ~~01 plt~

~W~ Ir ( iquest ~ ~~(iacute~~~ JO1S~ I gv f6~ l 1

1V~ ~~~~ ~1~ YfI1f rtOS 11 OfYJWJ SlfY bull~~7 Y(WO ~~x ~~

bull

(a)

Transparency 65 TIpler Physics lor Scientists and Engineers Figure 16-11 page 488 Figure 16 15 page ~2 Fourth EditionVolume 1

Copyright copy1999 W H Freeman and CompanyStanding waves on a string fiacutexed at bota elles (left) alld on a string fixed at one end (right)

A A------ ~~--~---~~ -~----~~~~-I~_____ ___tnJoIII ~I~~J n=l

bullbull

~-------~~~~-~ shy ~~~ffsect~sect~~~~~~~middot-middotmiddot-~Fundamental first harmonic Fundamental first harmoniacutec A A

--raquo

~=~~~~~~~~~~~~~sect~~~~ A A

(b) n 2

Third harmonic

(e) n=3

Fifth harmoniacutec A A A A

(d) n=4

A

A A

=3jiIAi~~~~~~~~~ Fourth harmonic Seventh harmonic

A A A A A

(e) n=5

I INinth harmonisI~ L bull oC L bull

A A A A

Fifth hannonic I

P$-I 11OPt 1ItYltf4f11~ ~ ~~ I (~ l~) ~ l 1-1OIOf Yl I1-0fJ -viexcl 101~5~11lt1 ) S )111 15

O~)1

- d Jt1lJlfJY Y1~Y) -J ~~1-~ (~) p( ol 4~ ~ JIQjVV ~~111 fj 1

1~ - ~~

J-~ ~4_~O__~__v___~___~__________

_1J plusmni

1 ~ t- 1 ~yx6 shy-2J1111 JS ~ + -

~

1 =Vi ~1f~~~ ~ v~ middotl-W ~ yO 171 ~~s

JIJYftIt1-t1J tiP 1111

gtt 4WI ~7JWJ ~~O middotJCfMl-JII IWI

r~7v l ~1H or~~ -yrr ~1Y 3

~IIp f Yf4 r1 Yf

f+1] 14fk ~ ~~t tJS~ l1 vf lYffJlJ ~1 ~~01 plt~

~W~ Ir ( iquest ~ ~~(iacute~~~ JO1S~ I gv f6~ l 1

1V~ ~~~~ ~1~ YfI1f rtOS 11 OfYJWJ SlfY bull~~7 Y(WO ~~x ~~

bull

(a)

Transparency 65 TIpler Physics lor Scientists and Engineers Figure 16-11 page 488 Figure 16 15 page ~2 Fourth EditionVolume 1

Copyright copy1999 W H Freeman and CompanyStanding waves on a string fiacutexed at bota elles (left) alld on a string fixed at one end (right)

A A------ ~~--~---~~ -~----~~~~-I~_____ ___tnJoIII ~I~~J n=l

bullbull

~-------~~~~-~ shy ~~~ffsect~sect~~~~~~~middot-middotmiddot-~Fundamental first harmonic Fundamental first harmoniacutec A A

--raquo

~=~~~~~~~~~~~~~sect~~~~ A A

(b) n 2

Third harmonic

(e) n=3

Fifth harmoniacutec A A A A

(d) n=4

A

A A

=3jiIAi~~~~~~~~~ Fourth harmonic Seventh harmonic

A A A A A

(e) n=5

I INinth harmonisI~ L bull oC L bull

A A A A

Fifth hannonic I

~

1 =Vi ~1f~~~ ~ v~ middotl-W ~ yO 171 ~~s

JIJYftIt1-t1J tiP 1111

gtt 4WI ~7JWJ ~~O middotJCfMl-JII IWI

r~7v l ~1H or~~ -yrr ~1Y 3

~IIp f Yf4 r1 Yf

f+1] 14fk ~ ~~t tJS~ l1 vf lYffJlJ ~1 ~~01 plt~

~W~ Ir ( iquest ~ ~~(iacute~~~ JO1S~ I gv f6~ l 1

1V~ ~~~~ ~1~ YfI1f rtOS 11 OfYJWJ SlfY bull~~7 Y(WO ~~x ~~

bull

(a)

Transparency 65 TIpler Physics lor Scientists and Engineers Figure 16-11 page 488 Figure 16 15 page ~2 Fourth EditionVolume 1

Copyright copy1999 W H Freeman and CompanyStanding waves on a string fiacutexed at bota elles (left) alld on a string fixed at one end (right)

A A------ ~~--~---~~ -~----~~~~-I~_____ ___tnJoIII ~I~~J n=l

bullbull

~-------~~~~-~ shy ~~~ffsect~sect~~~~~~~middot-middotmiddot-~Fundamental first harmonic Fundamental first harmoniacutec A A

--raquo

~=~~~~~~~~~~~~~sect~~~~ A A

(b) n 2

Third harmonic

(e) n=3

Fifth harmoniacutec A A A A

(d) n=4

A

A A

=3jiIAi~~~~~~~~~ Fourth harmonic Seventh harmonic

A A A A A

(e) n=5

I INinth harmonisI~ L bull oC L bull

A A A A

Fifth hannonic I

(a)

Transparency 65 TIpler Physics lor Scientists and Engineers Figure 16-11 page 488 Figure 16 15 page ~2 Fourth EditionVolume 1

Copyright copy1999 W H Freeman and CompanyStanding waves on a string fiacutexed at bota elles (left) alld on a string fixed at one end (right)

A A------ ~~--~---~~ -~----~~~~-I~_____ ___tnJoIII ~I~~J n=l

bullbull

~-------~~~~-~ shy ~~~ffsect~sect~~~~~~~middot-middotmiddot-~Fundamental first harmonic Fundamental first harmoniacutec A A

--raquo

~=~~~~~~~~~~~~~sect~~~~ A A

(b) n 2

Third harmonic

(e) n=3

Fifth harmoniacutec A A A A

(d) n=4

A

A A

=3jiIAi~~~~~~~~~ Fourth harmonic Seventh harmonic

A A A A A

(e) n=5

I INinth harmonisI~ L bull oC L bull

A A A A

Fifth hannonic I