wave on a string theory_e

8/12/2019 Wave on a String Theory_E

1/17

[G\ZLJZ

"doblsgfudorpgysljs.lb" ?

X P MB ZQSLBHXOPEZ

Xove dmtlmb ls tge pgebmdebmb tgot job ce mcservea oidmst everywgere ormuba us, os weii lt oppeors

lb oidmst every crobjg m` pgysljs. Zuròje woves mb cmales m` dotter ore jmddmbiy mcservea. Zmuba

woves oba ilhgt woves ore essebtloi tm mur perjeptlmb m` tge ebvlrmbdebt. Oii woves gove o sldlior

dotgedotljoi aesjrlptlmb, wgljg dofes tge stuay m` mbe flba m` wove useùi `mr tge stuay m` mtger

flbas m` woves. Lb tgls jgopter, we wlii jmbjebtrote mb strlbh woves, wgljg ore type m` o dejgobljoi

woves. Dejgobljoi woves requlre o dealud tm trovei tgrmuhg. Zmuba woves, woter woves ore mtger

exodpies m` dejgobljoi woves. Ilhgt woves ore bmt dejgobljoi woves, tgese ore eiejtrmdohbetlj

woves wgljg am bmt requlre dealud tm prmpohote.

Dejgobljoi woves mrlhlbote `rmd o alsturcobje lb tge dealud (sujg os o stmbe armpplbh lb o pmba) oba

tge alsturcobje prmpohotes tgrmuhg tge dealud. Qge `mrjes cetweeb tge otmds lb tge dealud ore

respmbslcie `mr tge prmpohotlmb m` dejgobljoi woves. Eojg otmd exerts o `mrje mb tge otmds beor lt,

oba tgrmuhg tgls `mrje tge dmtlmb m` tge otmd ls trobsdlttea tm tge mtgers. Qge otmds lb tge dealud am

bmt, gmwever, experlebje oby bet alspiojedebt. Os tge wove posses, tge otmds sldpiy dmve cojf oba

`mrtg. Oholb `mr sldpiljlty, we jmbjebtrote mb tge stuay m` gordmblj woves (tgot ls tgmse tgot job ce

represebtea cy slbe oba jmslbe ` ubjtlmbs).

Q\[EZ M@ DEJ GOBL JOI XOP EZDejgobljoi woves job ce jiossl`lea ojjmralbh tm tge pgysljoi prmpertles m` tge dealud, os weii os lb

mtger woys.

?. Alrejtlmb m` portljie dmtlmb >

Xoves job ce jiossl`lea cy jmbslaerlbh tge alrejtlmb m` dmtlmb m` tge portljies lb tge dealud os wove

posses. L` tge alsturcobje troveis lb tge x alrejtlmb cut tge portljies dmve lb o alrejtlmb, perpebaljuior

tm tge x oxls os tge wove posses lt ls joiiea o trobsverse wove. L` tge dmtlmb m` tge portljies were

poroiiei tm tge x oxls tgeb lt ls joiiea o imbhltualboi wove. O wove puise lb o piujfea hultor strlbh ls o

trobsverse wove. O smuba wove ls o imbhltualboi wove.

3. Budcer m` aldebslmbs >

Xoves job prmpohote lb mbe, twm, mr tgree aldebslmbs. O wove dmvlbh oimbh o tout strlbh ls o mbe

aldebslmboi wove. O woter wove jreotea cy o stmbe tgrmwb lb o pmba ls o twm aldebslmboi wove. O

smuba wove jreotea cy o hubsgmt ls o tgree-aldebslmboi wove

:. [erlmaljlty >

O stmbe armppea lbtm o pmba jreotes o wove puise, wgljg troveis mutwora lb twm aldebs lmbs. Qgere doy

ce dmre tgob mbe rlppie jreotea, cut tgere ls stlii mbiy mbe wove puise. L` sldlior stmbes ore armppea lb

tge sode pioje ot eveb tlde lbtervois, tgeb o perlmalj wove ls jreotea.

1. Zgope m` wove `rmbts >Qge rlppies jreotea cy o stmbe armppea lbtm o pmba ore jlrjuior lb sgope. O

smuba wove prmpohotlbh mutwora `rmd o pmlbt smurje gos spgerljoi wove`rmbts. O piobe wove ls o tgreealdebslmboi wove wltg ìot wove `rmbts.

(@or owoy `rmd o pmlbt smurje edlttlbh spgerljoi woves, tge woves oppeor tm ce piobe woves.)

O smila job sustolb trobsverse os weii os imbhltualboi wove. O ìula gos bm weii-ae` lbea `mrd mr struj ture

tm dolbtolb oba m`èr òr dmre reslstobje tm jmdpresslmb tgob tm o sgeorlbh `mrje. Jmbsequebtiy, mbiy

imbhltualboi wove job prmpohote tgrmuhg o hos mr wltglb tge cmay m` ob laeoi (bmb vlsjmus) ilqula.

Gmwever, trobsverse woves job exlst mb tge suròje m` o ilqula. Lb tge jose m` rlppies mb o pmba, tge

`mrje restmrlbh tge systed tm equlilcrlud ls tge suròje tebslmb m` tge woter, wgereos `mr mjeob woves,

lt ls tge `mrje m` hrovlty.

Oism, l` alsturcobje ls restrljtea tm prmpohote mbiy lb mbe alrej tlmb oba tgere ls bm imss m` eberhy aurlbh

prmpohotlmb, tgeb sgope m` alsturcobje redolbs ubjgobhea.

AEZJSLCLBH XOPEZ >Qwm flbas m` hropg doy ce arowb - alspiojedebt-alstobje oba alspiojedebt-tlde.

O alspiojedebt - alstobje hropg `mr o trobsverse dej gobl joi wove sgmws tge alspiojedebt y m` tge

vlcrotlbh portljies m` tge trobsdlttlbh dealud ot al`èrebt alstobje x `rmd tge smurje ot o jertolb lbstobt

l.e. lt ls ilfe o pgmtmhropg sgmwlbh sgope m` tge wove ot tgot portljuior lbstobt.


2/17

[G\ZLJZ

"doblsgfudorpgysljs.lb" 3

Qge doxldud alspiojedebt m` eojg portljie `rmd lts ubalsturcea pmsltlmb ls tge odpiltuae m` tge wove.

Lb tge `lhure ?, lt ls MO mr MC.

Alstobje x

Mbe woveiebhtg

Jrest Qrmuhg Jrest Qrmuhg

O

Alspioje

debt

C

M

Qge woveiebhtgm` o wove ls heberoiiy tofeb os tge alstobje cetweeb twm sujjesslve jrests mr twmsujjesslve trmuhg. Qm ce dmre spejl`lj, lt ls tge alstobje cetweeb twm jmbsejutlve pmlbts mb tge wove

wgljg gove sode pgose.

O alspiojedebt-tlde hropg doy oism ce arowb `mr o wove dmtlmb, sgmwlbh gmw tge als piojedebt m` mbe

portljie ot o portljuior alstobje `rmd tge smurje vorles wltg tlde. L` tgls ls sldpie gordmblj vorlotlmbtgeb tge hropg ls o slbe jurve.

XOPE IEBHQG @SE_UEBJ\ Z[EEAL` tge smurje m` o wove dofes ` vlcrotlmbs per sejmba, sm tmm wlii tge portljies m` tge trobsdlttlbh

dealud. Qgot ls, tge `requebjy m` tge woves equois `requebjy m` tge smurje.

Xgeb tge smurje dofes mbe jmdpiete vlcrotlmb , mbe wove ls heberotea oba tge alsturcobje spreoas

mut o alstobje `rmd tge smurje. L` tge smurje jmbtlbues tm vlcrote wltg jmbstobt `requebjy `, tgeb `woves wlii ce prmaujea per sejmba oba tge wove oavobjes o alstobje `lb mbe sejmba. L` v ls tge wovespeea tgeb

v 4 Qgls reiotlmbsglp gmias `mr oii wove d mtlmbs.

Qroveiilbh wove >Ldohlbe o gmrlzmbtoi strlbh stretjgea lb tge x alrejtlmb. Lts equlilcrlud sgope ls ìot oba strolhgt. Iet y

deosure tge alspiojedebt m` oby portljie m` tge strlbh `rmd lts equlilcrlud pmsltlmb, perpebaljuior tm

tge strlbh. L` tge strlbh ls piujfea mb tge ie`t eba, o puise wlii trovei tm tge rlhgt. Qge vertljoi alspiojedebt

y m` tge ie`t eba m` tge strlbh (x 4 ;) ls o ùbjtlmb m` tlde.

l.e. y (x 4 ;, t) 4 `(t)

L` tgere ore bm `rljtlmboi imsses, tge puise wlii trovei ubaldlblsgea, retolblbh lts mrlhlboi sgope. L` tge

puise troveis wltg o speea v, tge pmsltlmb m` tge wove puise ls x 4 vt. Qgere`mre, tge alspiojedebt m` tge

portljie ot pmlbt x ot tlde t wos mrlhlbotea ot tge ie`t eba ot tlde t v

x. Vy, (x, t) ls ùbjtlmb m` cmtg x oba

tT. Cut tge alspiojedebt m` tge ie`t eba ot tlde t ls `(t) tgus ot tlde t

v

x, lt ls `(t

v

x).

Qgere`mre >

y(x, t) 4 y (x 4 ;, t v

x) 4 (t

v

x)

Qgls job oism ce expressea os

v

`(vt x)

v

`(x vt)

y (x, t) 4 h(x vt)

uslbh oby `lxea voiue m` t (l.e. ot oby lbstobt), tgls sgmws sgope m` tge strlbh.

L` tge wove ls troveiilbh lb x alrejtlmb, tgeb wove equotlmb ls wrltteb os

y(x, t) 4 `(t +v

x)

Qge quobtlty x vt ls joiiea pgose m` tge wove ùbjtlmb. Os pgose m` tge puise gos `lxea voiue

x vt 4 jmbst.


3/17

[G\ZLJZ

"doblsgfudorpgysljs.lb" :

Qoflbh tge aerlvotlve w.r.t. tldeat

ax4 v

wgere v ls tge pgose veimjlty oitgmuhg m`teb joiiea wove veimjlty. Lt ls tge veimjlty ot wgljg o portljuior

pgose m` tge alsturcobje troveis tgrmuhg spoje.

Lb mraer `mr tge ùbjtlmb tm represebt o wove troveiilbh ot speea v, tge tgree quobtltles x, v oba t dust

oppeor lb tge jmdclbotlmb (x + vt) mr (x vt). Qgus (x vt) 3 ls ojjeptocie cut x3 v3 t3 ls bmt.

Exodpie ?. O wove puise ls troveiilbh mb o strlbh ot 3 d/s. Alspiojedebt y m` tge portljie ot x 4 ; ot oby

tlde t ls hlveb cy y 4?t

33

. @lba >

(l) Expresslmb m` tge ùbjtlmb y 4(x, t) l.e. alspiojedebt m` o portljie ot pmsltlmb x oba tlde t.

(ll) Zgope m` tge puise ot t 4 ; oba t 4 ?s.

Zmiutlmb > (l) Cy repiojlbh t cy

v

xt , we job het tge aeslrea wove ùbjtlmb l.e.

y 4

?3xt

3

3

(ll) Xe job use wove ùbjtlmb ot o portljuior lbstobt, soy t 4 ;, tm `lba sgope m` tge wove puise

uslbh al`èrebt voiues m` x.

ot t 4 ; y 4

?1

x

33

ot x 4 ; y 4 3

x 4 3 y 4 ?

x 4 3 y 4 ?

x 4 1 y 4 ;.1

x 4 1 y 4 ;.1

Uslbh tgese voiue, sgope ls arowb.

?

3

\

W

1 3;

3 1

Zldlioriy `mr t 4 ?s, sgope job ce arowb. Xgot am ymu jmbjiuae ocmut alrejtlmb m` dmtlmb m` tge

wove `rmd tge hropgs= Oism jgejf gmw dujg tge puise gos dmvea lb ?s tlde lbtervoi. Qgls ls

equoi tm wove speea. Gere ls tge prmjeaure >

y 4

?3

x?

33

ot t 4 ?s

ot x 4 3 y 4 3 (doxldud voiue)

ot x 4 ; y 4 ?

ot x 4 1 y 4 ?

?

3

\

W

3;

3 1 0

t 4 ;t 4 ?


4/17

[G\ZLJZ


Qge puise gos dmvea tm tge rlhgt cy 3 ublts lb ?s lbtervoi.

Oism os t 3

x4 jmbstt.

Al`èrebtlotlbh w.r.t. tlde

? 3

?.

at

ax4 ;

at

ax4 3.

QSOPEIILBH ZLBE XOPE LB MBE ALDEBZLMB (XOPE MB ZQSLBH) >

Qge wove equotlmb

v

xt`y ls qulte heberoi. L` gmias `mr orcltrory wove sgopes, oba `mr trobsverse

os weii os ` mr imbhltualboi woves.

O j mdpiete aesjrlptlmb m` tge wove requlres spejl`ljot lmb m` `(x ). Qge dms t ldpmrto bt jose, cy òr, lb

pgysljs oba ebhlbeerlbh ls wgeb `(x) ls slbusmlaoi, tgot ls, wgeb tge wove gos tge sgope m` o slbe mr

jmslbe ùbjtlmb. Qgls ls pmsslcie wgeb tge smurje, tgot ls dmvlbh tge ie`t eba m` tge strlbh, vlcrotes tge

ie`t eba x 4 ; lb o sldpie gordmblj dmtlmb. @mr tgls, tge smurje gos tm jmbtlbumusiy am wmrf mb tge

strlbh oba eberhy ls jmbtlbumusiy suppilea tm tge strlbh.

Qge equotlmb m` dmtlmb m` tge ie`t eba doy ce wrltteb os

` (t) 4 O slbtwgere O ls odpiltuae m` tge wove, tgot ls doxldud alspiojedebt m` o portljie lb tge dealud `rmd ltsequlilcrlud pmsltlmb ls obhuior `requebjy, tgot ls 3` wgere ` ls `requebjy m` ZGD m` tge smurje.Qge alspiojedebt m` tge portljie ot x ot tlde t wlii ce

v

xt`y mr y 4 O slb

v

xt y 4 O slb (t fx)

wgere f 4

3v

ls joiiea wove budcer. Q 43

4`

?ls perlma m` tge wove, tgot ls tge tlde lt tofes

tm trovei tge alstobje cetweeb twm oanojebt jrests mr tgrmuhg (lt ls woveiebhtg).Qge wove equotlmb y 4 O slb (t fx) soys tgot ot x 4 ; oba t 4 ;, y 4 ;. Qgls ls bmt bejessorliy tgejose, m` smurje. @mr tge sode jmbaltlmb, y doy bmt equoi tm zerm. Qgere`mre, tge dmst heberoi expresslmb

wmuia lbvmive o pgose jmbstobt, wgljg oiimws `mr mtger pmsslcliltles,

y 4 O slb (t fx +)O sult ocie jgmlje mòiimws oby lbltloi jmbaltlmb tm ce det. Qge terd fx wt + ls joiiea tge pgose m`tge wove. Qwm woves wltg tge sode pgose (mb pgose al`èrlbh cy o duitlpie m` 3) ore sola tm ce lbpgose. Qgey exejute tge sode dmtlmb ot tge sode tlde.

Qge veimjlty m` tge portljie ot pmsltlmb x oba ot tlde t ls hlveb cy

Ot

yjms (t fx +)

Qge wove equotlmb gos ceeb portloiiy al`èrebtlotea feeplbh x os jmbstobt, tm spejl`y tge portljie. Bmte

tgot wove veimjltyat

axls al`èrebt `rmd portljie veimjlty wglie woves veimjlty ls jmbstobt `mr o dealud

oba lt oimbh tge alrejtlmb m` strlbh, wgereos portljie veimjlty ls perpebaljuior tm wove veimjlty oba ls

aepebaebt upmb x oba t.

Exodpie 3. O slbusmlaoi wove troveiilbh lb

tge pmsltlve x alrejtlmb gos ob

od pi l tuae m` ?5 j d ,

wo ve ie bht g 1; j d o ba

`requebjy 7 Gz. Qge vertljoi

alspiojedebt m` tge dealud ot

t 4 ; oba x 4 ; ls oism ?5 jd,

os sgmwb.

?5

1;

y(jd)

x(jd)

(o) @l ba tge obhui or w ov e

b u d ce r, p e rl m a, o b hu i or

`requebjy oba speea m` tge

wove.

(c) Aeterdlbe tge pgose jmbstobt, oba wrlte o heberoi expresslmb `mr tge wove ùbjtlmb.


5/17

[G\ZLJZ


Zmiutlmb > (o) f 43

4jd1;

roa34

3;

roa/jd

Q 4`

?4

7

?s 4 3` 4 ?0 s?

v 4 `4 :3; jd/s(c) Lt ls hlveb tgot O 4 ?5 jd

oba oism y 4 ?5 jd ot x 4 ; oba t 4 ;

tgeb uslbh y 4 O slb (t fx +)?5 4 ?5 slb slb 4 ?

mr 43

roa.

Qgere`mre, tge wove ùbjtlmb ls

y 4 O slb (t fx +3

)

4 (?5 jd) slb

3x.

jd

roa

3;t)s?0( ?

Exodpie :. O slbusmlaoi wove ls troveiilbh oimbh o rmpe. Qge msjliiotmr tgot heberotes tge wove jmdpietes

0; vlcrotlmbs lb :; s. Oism, o hlveb doxldud troveis 135 jd oimbh tge rmpe lb ?;.; s. X got ls

tge woveiebhtg=

Zmiutlmb > v 4?;

1354 13.5 jd/s. ` 4

:;

0;4 3 Gz

4`

v4 3?.35 jd.

QGE ILBEOS XOPE E_UOQLMB >Cy uslbh wove ùbjtlmb y 4 O slb (t fx +), we job aesjrlce tge dmtlmb m` oby pmlbt mb tge strlbh.Oby pmlbt mb tge strlbh dmves mbiy vertljoiiy, oba s m lts x jmmralbote redolbs jmbstobt. Qge trobsverse

veimjlty vym` tge pmlbt oba lts trobsverse ojjeierotlmb o

yore tgere`mre

vy4

ttobjmbsxat

ay

t

y

4O jms (t fx +) ....(?)

oy4

ttobjmbsx

y

at

av

3

3y

t

y

t

v

4 3O s lb (t fx +) ....(3)

oba gebje vy, dox

4Oo

y, dox43O

Qge trobsverse veimjlty oba trobsverse ojjeierotlmb m` obypmlbt mb tge strlbh am bmt reojg tgelr doxldudvoiue slduitobemusiy. Lbòjt, tge trobsverse veimjlty reojges lts doxldud voiue ( O) wgeb tgealspiojedebt y 4 ;, wgereos tge trobsverse ojjeierotlmb reojges lts d oxldud dohbltuae (3O) wgeby 4 O

ùrtgerttobjmbstax

ay

x

y

4 fO jms (t fx +) ....(:)

4 3

3

x

y

4 f3 O slb (t fx +) ....(1)

@rmd (?) oba (:) t

y

4 f

x

y

v[

4 vw

simpe

l.e. l` tge simpe ot oby pmlbt ls behotlve, portljie veimjlty ls pmsltlve oba vlje-verso, `mr o wove dmvlbh

oimbh pmsltlve x oxls l.e. vw

ls pmsltlve.


6/17

[G\ZLJZ


@mr exodpie, jmbslaer twm pmlbts O oba C mb

tge y-x jurve `mr o wove, os sgmwb. Qge wove

ls dmvlbh oimbh pmsltlve x-oxls.

x

x? x3

y

O

CZimpe ot O ls pmsltlve tgere`mre ot tge hlveb

dmdebt, lts veimjlty ls behotlve. Qgot deobs

lt ls jmdlbh amwbwora. Severse ls tge sltuotlmb

`mr portljie ot pmlbt C.

Bmw uslbh equotlmb (3) oba (1)

3

3

3

3

3

3

t

yf

x

y

3

3

33

3

t

y

v

?

x

y

Qgls ls fbmwb os tge ilbeor wove equotlmb mr al`èrebtloi equotlmb represebtotlmb m` tge troveiilbh wove

dmaei. Xe gove aeveimpea tge ilbeor wove equotlmb `rmd o slbusmlaoi dejgobljoi wove troveiilbh tgrmuhg

o dealud, cut lt ls dujg dmre heberoi. Qge ilbeor wove equotlmb sujjessùiiy aesjrlces woves mb

strlbhs, smuba woves oba oism eiejtrmdohbetlj woves.

Exodpie 1. Perl`y tgot wove ùbjtlmb

y 4?)t:x(

33

ls o smiutlmb tm tge ilbeor wove equotlmb. x oba y ore lb jd.

Zmiutlmb > Cy toflbh portloi aerlvotlves m` tgls ùbjtlmb w.r.t. x oba tm t

:3

3

3

3

T?)t:xV(

1)t:x(?3

x

y

, oba

:3

3

3

3

T?)t:xV(

:0)t:x(?;7

t

y

mr3

3

3

3

t

x

Q 4 3; ?; 4 3;; B

s/d3;5.;

3;;v


7/17

[G\ZLJZ


Exodpie 0. O tout strlbh govlbh tebslmb ?;; B oba ilbeor doss aebslty

;.35 fh/d ls usea lbslae o jort tm heberote o wove puise stortlbh

ot tge ie`t eba, os sgmwb. Xgot sgmuia ce tge veimjlty m` tge

jort sm tgot puise redolbs stotlmbory w.r.t. hrmuba.

Zmiutlmb > Peimjlty m` puise 4 s/d3;Q

Bmw JH[J[H vvv

; 4 3; l + JHv

JHv

4 3;l d/s

[MXES QSOBZDLQQEA OIMBH QGE ZQSLBH C\ O ZLBE XOPEXgeb o troveiilbh wove ls estocilsgea mb o strlbh, eberhy ls trobsdlttea oimbh tge alrejtlmb m` prmpohotlmb m`

tge wove, lb `mrd m` pmtebtloi eberhy oba flbetlj eberhy

Overohe [mwer [ 4 33 `3OO3 v

Eberhy Qrobsèrrea 4

t

;

ov at[

Eberhy trobsèrrea lb mbe tlde perlma 4 [ov

Q

Qgls ls oism equoi tm tge eberhy stmrea lb mbe woveiebhtg.

Lbtebslty >Eberhy trobsèrrea per sejmba per ublt jrmss sejtlmboi oreo ls joiiea lbtebslty m` tge wove.

4s

[

oreosejtlmboiJrmss

[mwer 4

3

?3OO3v

Qgls ls overohe lbtebslty m` tge wove.

Eberhy aebslty >Eberhy per ublt vmiude m` tge wove

4vsvat

[at

Exodpie 8. O strlbh wltg ilbeor doss aebslty d 4 5.;; ?;3 fh/d ls ubaer o tebslmb m` 7;.; B. Gmw dujg

pmwer dust ce suppilea tm tge strlbh tm heberote slbusmlaoi woves ot o `requebjy m` 0;.; Gz

oba ob odpiltuae m` 0.;; jd=

Zmiutlmb > Qge wove speea mb tge strlbh ls

v 4 Q

4

3/?

3 d/fh?;;;.5

B;.7;

4 1;.; d/s

Cejouse4 0; Gz, tge obhuior `requebjy m` tge slbusmlaoi woves mb tge strlbh gos tge voiue4 3` 4 3(0;.; Gz) 4 :88 s?

Uslbh tgese voiues lb `miimwlbh Equotlmb `mr tge pmwer, wltg O 4 0.;; ?;3 d,. hlves

p 43

?3O3 v

43

?(5.;; ?;3 fh/d) (:88s?)3 (0.;; ?;3 d) 3 (1;.; d/s)

4 5?3 X.

Exodpie 7. Qwm woves lb tge sode dealud ore represebtea cy

y-t jurves lb tge `lhure. @lba rotlm m` tgelr overohe

lbtebsltles=

y

wove ?

wove 3

5

3

Zmiutlmb > 33

33

3?

3?

3

?

O

O

4 3

333

3?

3?

O`

O`

. 4 11

35?

4 ?0

35


8/17

[G\ZLJZ


QGE [S LBJL[I E M@ ZU[ ES[ MZLQ LM BXgeb twm mr dmre woves slduitobemusiy poss tgrmuhg o pmlbt, tge alsturcobje ot tge pmlbt ls hlveb cy

tge sud m` tge alsturcobjes eojg wove wmuia prmauje lb ocsebje m` tge mtger wove(s).

Lb heberoi, tge prlbjlpie m` superpmsltlmb ls voila `mr sdoii alsturcobjes mbiy. L` tge strlbh ls stretjgea

tmm òr, tge lbalvlauoi alspiojedebts am bmt oaa tm hlve tge resuitobt alspiojedebt. Zujg woves ore

joiiea bmbilbeor woves. Lb tgls jmurse, we sgoii mbiy ce toiflbh ocmut ilbeor woves wgljg mcey tge

superpmsltlmb prlbjlpie.

Qm put tgls ruie lb o dotgedotljoi `mrd, iet y?(x, t) oba y

3(x, t) ce tge alspiojedebts tgot oby eiedebt

m` tge strlbh wmuia experlebje l` eojg wove troveiiea oimbe. Qge alspiojedebt y(x, t) m` ob eiedebt m`tge strlbh wgeb tge woves mveriop ls tgeb hlveb cy

y(x, t) 4 y?(x, t) + y

3(x, t)

Qge prlbjlpoi m` superpmsltlmb job oism ce expressea cy stotlbh tgot mveriopplbh woves oihecroljoiiy

oaa tm prmauje o resuitobt wove. Qge prlbjlpie ldpiles tgot tge mveriopplbh woves am bmt lb oby woy

oiter tge trovei m` eojg mtger.

L` we gove twm mr dmre woves dmvlbh lb tge dealud tge resuitobt wove`mrd ls tge sud m` wove ùbjtlmbs

m` lbalvlauoi woves.

@lh> o sequebje m` pljtures sgmwlbh twm puises

troveiilbh lb mppmslte alrejtlmbs oimbh o stretjgea strlbh.

Xgeb tge twm alsturcobjes mveriop tgey hlve o

jmdpiljotea potterb os sgmwb lb (c). Lb (j), tgey gove

possea eojg mtger oba prmjeea ubjgobhea.

(j)

(o)

(c)Ob Liiustrotlve exodpies m` tgls prlbjlpie ls pgebmdebo

m` lbterèrebje oba reìejtlmb m` woves.

Exodpie Ojjmralbh tm tge prlbjlpie m` superpmsltlmb, eojg wove prmaujes lts alsturcobje lbaepebaebt m` tge

mtger oba tge resuitobt alsturcobje ls equoi tm tge vejtmr sud m` tge lbalvlauoi alsturcobje. Qge

alspiojedebts m` tge portljie ot x 4 ? jd ot tlde t 4 5.; ds aue tm tge twm woves ore,

y?4 5 dd slb V(3 jd?) x (5;s?) tT

y?4 5 dd slb V(3 jd?) ?jd (5;s?) 5 ?;: sejT

4 5 dd slb

13 4 5 dd

oba y34 ?; dd slb V(jd?)x (?;;s?) tT.

y34 ?; dd slb V(jd?

) ? jd (?;;s?

) 5 ?;:

sejT

4 ?; dd slb

3 4 ?; dd

Qge bet alspiojedebt ls > y 4 y?+ y

34 ?; dd 5 dd 4 5 dd

LBQES@ ESEBJE M@ XOPEZ HMLBH LB ZODE ALSEJQLMBZuppmse twm laebtljoi smurjes seba slbusmlaoi woves m` sode obhuior `requebjy lb pmsltlve x-alrejtlmb. Oism, tge wove veimjlty oba gebje, tge wove budcer f ls sode `mr tge twm woves. Mbe smurje

doy ce sltuotea ot al`èrebt pmlbts. Qge twm woves orrlvlbh ot o pmlbt tgeb al`èr lb pgose. Iet tge

odpiltuaes m` tge twm woves ce O?

oba O3

oba tge twm woves al`èr lb pgose cy ob obhie . Qgelr

equotlmbs doy ce wrltteb osy?

4 O?

slb (fx t)oba y

34 O

3slb (fx t +).

Ojjmralbh tm tge prlb jlpie m` superpmslt lmb, tge resuitobt wove ls represebtea cy

y 4 y?

+ y3

4 O?

slb (fx t) + O3

slb (fx t +).


9/17

[G\ZLJZ

"doblsgfudorpgysljs.lb"

Xgeb resuitobt odpiltuae O ls doxldudO 4 O?

+ O3

wgeb j ms 4 + ? mr 4 3bwgere b ls ob lbteher.

Aestrujtlve lbterèrebje >

Xgeb resuitobt odpiltuae O ls dlbldud

mr O 4 |O?

O3|

Xgeb jms 4 ? mr 4 (3b + ?)wgere b ls ob lbteher.

Exodpie ?;. Qwm slbusmlaoi woves m` tge sode `requebjy trovei lb tge sode alrejtlmb oimbh o strlbh. L

O?4 :.; jd, O

34 1.; jd,?4 ;, oba34/3 roa, wgot ls tge odpiltuae m` tge resuitobt wove=

Zmiutlmb > Sesuitobt odpiltuae 4 Seìejtlmb ot aebser cmubaory

LBJLAEBQ XOPE

y 4 O slb( t f x)l l ?

SE@IEJQEA XOPE

y 4 O slb( t + f x)r r ?

QSOBZDLQQEA XOPE

y 4 O slb( t f x)t t 3

@lh. > Seìejtlmb ot rorer cmubaory

Odpiltuae m` reìejtea oba trobsdlttea woves >

v?oba v

3ore speeas m` tge lbjlaebt wove oba reìejtea wove lb dealuds respejtlveiy tgeb

Or4 l

3?

?3 Ovv

vv

O

t4 l

3?

3 O.vv

v3

Orls pmsltlve l` v

32 v

?, l.e., wove ls reìejtea `rmd o rorer dealud.

Exodpie ??. O gordmblj wove ls troveiilbh mb strlbh ?. Ot o nubjtlmb wltg strlbh 3 lt ls portiy reìejtea oba

portiy trobsdlttea. Qge ilbeor doss aebslty m` tge sejmba strlbh ls `mur tldes tgot m` tge `lrst

strlbh, oba tgot tge cmubaory cetweeb tge twm strlbhs ls ot x 4 ;. L` tge expresslmb `mr tge

lbjlaebt wove ls, yl4 O

ljms (f

?x

?t)

Xgot ore tge expresslmbs `mr tge trobsdlttea oba tge reìejtea woves lb terds m` Ol, f

?oba

?=

Zmiutlmb > Zlbje v 4 /Q , Q34 Q

? oba

34 1

?

we gove, v34

3

v? ... (l)

Qge `requebjy ames bmt jgobhe, tgot ls,

?4

3 ....(ll)

Oism, cejouse f 4/v, tge wove budcers m` tge gordmblj woves lb tge twm strlbhs ore reiotea cy,


10/17

[G\ZLJZ

"doblsgfudorpgysljs.lb" ?;

f3

43

3

v

4

3/v?

?4 3

?

?

v

4 3f

? ....(lll)

Qge odpiltuaes ore,

Ot4

3?3

vv

v3OO

l4

)3/v(v

)3/v(3

??

?OO

l4

:

3OO

l ....(lv)

oba Or

4

3?

?3

vv

vvOO

l

4

)3/v(v

v)3/v(

??

??OO

l

4:

O l....(v)

Bmw wltg equotlmb (ll), (lll) oba (lv), tge trobsdlttea wove job ce wrltteb os,

yt4

:

3OO

ljms (3f

?x

?t) Obs.

Zldlioriy tge re`iejtea wove job ce expressea os,

4:

O ljms (f

?x +

?t +) Obs.

ZQOBALBH XOPEZ >Zuppmse twm slbe woves m` equoi odpiltuae oba `requebjy prmpohote mb o imbh strlbh lb mppmslte

alrejtlmbs. Qge equotlmbs m` tge twm woves ore hlveb cyy?

4 O slb (t fx) oba y3

4 O slb (t + fx +).Qgese woves lbter`ere tm prmauje wgot we joii stobalbh woves. Qm ubaerstoba tgese woves, iet us

alsjuss tge spejloi jose wgeb 4 ;.Qge resuitobt alspiojedebts m` tge portljies m` tge strlbh ore hlveb cy tge prlbjlpie m` superpmsltlmb os

y 4 y?

+ y3

4 O Vslb (t fx) + slb(t + fx)T4 3O slbt jms fx

mr, y 4 (3O jms fx) slbt. ....

Qgls ls tge requlrea resuit oba `rmd tgls lt ls jieor tgot >

?. Os tgls equotlmb sotls`les tge wove equotlmb,

3

3

33

3

ty

v?

xy

lt represebts o wove. Gmwever, os lt ls bmt m` tge `mrd `(ox ct), tge wove ls bmt troveiilbh oba

sm ls joiiea stobalbh mr stotlmbory wove.

3. Qge odpiltuae m` tge wove

Os4 3O jms fx

ls bmt jmbstobt cut vorles perlmaljoiiy wltg pmsltlmb (oba bmt wltg tlde os lb ceots).

:. Qge pmlbts `mr wgljg odpiltuae ls dlbldud ore joiiea bmaes oba `mr tgese

jms fx 4 ;, l.e., fx 43

5,

3

:,

3

l.e., x 41

5,

1

:,

1

, ....

3

fos

l.e., lb o stotlmbory wove, bmaes ore equoiiy spojea.

1. Qge pmlbts `mr wgljg odpiltuae ls doxldud ore joiiea obtlbmaes oba `mr tgese,

jms fx 4 ?, l.e., fx 4 ;,, 3, :, ......

l.e., x 4 ;,3

,

3

3,

3

:,....

3fos

l.e., ilfe bmaes, obtlbmaes ore oism equoiiy spojea wltg spojlbh ( /3) oba Odox

4 3O.

@urtgerdmre, bmaes oba obtlbmaes ore oiterbote wltg spojlbh (/1).

5. Qge bmaes alvlae tge dealud lbtm sehdebts (mr immps). Oii tge portljies lb o sehdebt vlcrote lbsode pgose, cut lb mppmslte pgose wltg tge portljies lb tge oanojebt sehdebt. Qwlje lb mbe

perlma oii tge portljies poss tgrmuhg tgelr deob pmsltlmb slduitobemusiy wltg doxldud veimjlty

(Os), tge alrejtlmb m` dmtlmb celbh reversea o`ter eojg goi` jyjie.


11/17

[G\ZLJZ

"doblsgfudorpgysljs.lb" ??

Bmae

Obtlbmae

t 4 ;

t 4 Q/7

t 4 Q/1

4 :Q/7

t 4 Q/3

4 5Q/7

4 :Q/1

4 8Q/7

t 4 Q

(o)0. Ztobalbh woves job ce trobsverse mr imbhltualboi, e.h., lb strlbhs (ubaer tebslmb) l` re`iejtea

wove exlsts, tge woves ore trobsverse-stotlmbory, wglie lb mrhob plpes woves ore imbhltualboi-

stotlmbory.

8. Os lb stotlmbory woves bmaes ore perdobebtiy ot rest, sm bm eberhy job ce trobsdlttea ojrmss

tged, l.e., eberhy m` mbe rehlmb (sehdebt) ls jmb`lbea lb tgot rehlmb. Gmwever, tgls eberhymsjliiotes cetweeb eiostlj pmtebtloi eberhy oba flbetlj eberhy m` tge portljies m` tge dealud.

Xgeb oii tge portljies ore ot tgelr extrede pmsltlmbs FE ls dlbldud wglie eiostlj [E ls doxldud

(os sgmwb lb `lhure O), oba wgeb oii tge portljies (slduitobemusiy) poss tgrmuhg tgelr deob

pmsltlmb FE wlii ce doxldud wglie eiostlj [E dlbldud (@lhure C). Qge tmtoi eberhy jmb`lbea

lb o sehdebt (eiostlj [E + FE), oiwoys redolbs tge sode.

x

y

Eiostlj [E 4 dox 4 EFlbetlj eberhy 4 dlb 4 ;

(O)

v 4 ;

v 4 dlb 4 ; strlbh

x

y

Eiostlj [E 4 dlb 4 ;Flbetlj eberhy 4 dox 4 E

(C)

v 4 dox. 4 Os strlbh

Exodpie ?3. Qwm woves troveiilbh lb mppmslte alrejtlmbs prmauje o stobalbh wove. Qge lbalvlauoi wove `ubjtlmbs

ore

y?

4 (1.; jd) slb(:.;x 3.;t)

y3

4 (1.; jd) slb (:.;x + 3.;t)

wgere x oba y ore lb jebtldeter.

(o) @lba tge doxldud alspiojedebt m` o portljie m` tge dealud ot x 4 3.: jd.

(c) @lba tge pmsltlmb m` tge bmaes oba obtlbmaes.

Zmiutlmb > (o) Xgeb tge twm woves ore suddea, tge resuit ls o stobalbh wove wgmse dotgedotljoirepresebtotlmb ls hlveb cy Equotlmb, wltg O 4 1.; jd oba f 4 :.; roa/jd6

y 4 (3O slb fx) jmst 4 V(7.; jd) slb :.; xT jms 3.; tQgus, tge doxldud alspiojedebt m` o portljie ot tge pmsltlmb x 4 3.: jd ls

ydox

4 V(7.; jd) slb :.;xTx 4 3.: jd

4 (7.; d) slb (0.< roa) 4 1.0 jd

(c) Cejouse f 4 3/4 :.; roa/jd, we see tgot 4 3/:jd. Qgere`mre, tge obtlbmaes ore imjotea ot

x 4 b

;.0 jd (b 4 ?, :, 5, .....)

oba tge bmaes ore imjotea ot

x 4 b3

;.: jd (b 4 ?, 3, :, .....)


12/17

[G\ZLJZ

"doblsgfudorpgysljs.lb" ?3

Exodpie ?:. Qwm troveiilbh woves m` equoi odpiltuaes oba equoi `requebjles dmve lb mppmslte alrejtlmb oimbh o

strlbh. Qgey lbterère tm prmauje o stobalbh wove govlbh tge equotlmb.

y 4 O jms fx slbtlb wgljg O 4 ?.; dd, f 4 ?.58 jd? oba 4 87.5 s?. (o) @lba tge veimjlty oba odpiltuae m` tgejmdpmbebt troveiilbh woves. (c) @lba tge bmae jimsest tm tge mrlhlb lb tge rehlmb x 2 ;. (j) @lba tge

obtlbmae jimsest tm tge mrlhlb lb tge rehlmb x 2 ;. (a) @lba tge odpiltuae m` tge portljie ot x 4 3.:: jd.

Zmiutlmb > (o) Qge stobalbh wove ls `mrdea cy tge superpmsltlmb m` tge woves

y?4 3

O

slb (t fx) oba y3 4 3

O

slb (t + fx).Qge wove veimjlty (dohbltuae) m` eltger m` tge woves ls

v 4f

4 ?

?

jd58.?

s5.87

4 5; jd/s6 Odpiltuae 4 ;.5 dd.

(c) @mr o bmae, jms fx 4 ;.

Qge sdoiiest pmsltlve x sotls`ylbh tgls reiotlmb ls hlveb cy

fx 4/3mr, x 4f3

4 ?jd58.?3

?1.:

4 ? jd

(j) @mr ob obtlbmae, |jms fx| 4 ?.

Qge sdoiiest pmsltlve x sotls`ylbh tgls reiotlmb ls hlveb cy

fx 4 mr, x 4f

4 3 jd

(a) Qge odpiltuae m` vlcrotlmb m` tge portljie ot x ls hlveb cy | O jms fx |. @mr tge hlveb pmlbt,

fx 4 (?.58 jd?) (3.:: jd) 40

84+

0

.

Qgus, tge odpiltuae wlii ce

(?.; dd) | jms (+/0) | 4:

:dd 4 ;.70 dd.

PLCSOQLMB M@ ZQSLBH >(o) @lxea ot cmtg ebas >Zuppmse o strlbh m` iebhtg I ls fept `lxea ot tge ebas x 4 ; oba x 4 I. Lb sujg o systed suppmse we seba o

jmbtlbumus slbusmlaoi wove m` o jertolb `requebjy, soy, tmwora tge rlhgt. Xgeb tge wove reojges tge rlhgt

eba. Lt hets reìejtea oba cehlbs tm trovei cojf. Qge ie`t-hmlbh wove tgeb mveriops tge wove, wgljg ls stlii

troveiilbh tm tge rlhgt. Xgeb tge ie`t-hmlbh wove reojges tge ie`t eba, lt hets reìejtea oholb oba tge bewiy

reìejtea wove cehlbs tm trovei tm tge rlhgt. mveriopplbh tge ie t-hmlbh wove. Qgls prmjess wlii jmbtlbue oba,

tgere`mre, very smmb we gove doby mveriopplbh woves, wgljg lbterère wltg mbe obmtger. Lb sujg o systed, ot

oby pmlbt x oba ot oby tlde t, tgere ore oiwoys twm woves, mbe dmvlbh tm tge ie`t oba obmtger tm tge rlhgt. Xe,

tgere`mre,gove

y?(x, t) 4 y

dslb (fx t) (wove troveiilbhlb tge pmsltlve alrejtlmb m` x-oxls)

oba y3(x, t) 4 y

dslb (fx +t) (wove troveiilbh lb tge behotlve alrejtlmbm` x-oxls).

Qge prlbjlpie m` superpmsltlmb hlves, `mr tge jmdclbea wove

y(x, t) 4 y?(x, t) + y3(x, t)4 yd

slb (fx wt) + yd

slb (fx +t)4 (3y

dslb fx) jmst

Lt ls seeb tgot tge pmlbts m` doxldud mr dlbldud odpiltuae stoy ot mbe pmsltlmb.

Bmaes > Qge odpiltuae ls zerm `mr voiues m` fx tgot hlve slb fx 4 ; l.e. `mr,

fx 4 b, `mr b 4 ;, ?, 3, :,.....Zucstltutlbh f 4 3lb tgls equotlmb, we het

x 4 b3

, `mr b 4 ;, ?, 3, :,.....

Qge pmsltlmbs m` zerm odpiltuae ore joiiea tge bmaes. Bmte tgot o alstobje m`

3

mr goi` o woveiebhtg

seporotes twm jmbsejutlve bmaes.


13/17

[G\ZLJZ

"doblsgfudorpgysljs.lb" ?:

Obtlbmaes >

Qge odpiltuae gos o doxldud voiue m` 3yd

, wgljg mjjurs `mr tge voiues m` fx tgot hlve |slb fx| 4 ?. Qgmse

voiues ore

fx 4 (b + ?/3)`mr b 4 ;, ?, 3, :,....Zucstltutlbh f 4 3lb tgls equotlmb, we het.

x 4 (b + ?/3)3

`mr b 4 ;, ?, 3, :,....

os tge pmsltlmbs m` doxldud odpiltuae. Qgese ore joiiea tge obtlbmaes. Qge obtlbmaes ore seporotea cy /3 oba ore imjotea goi woy cetweeb polrs m` bmaes.@mr o stretjgea strlbh m` iebhtg I, `lxea ot cmtg ebas, tge twm ebas m` tge ebas ls jgmseb os pmsltlmb x 4 ;,

tgeb tge mtger eba ls x 4 I. Lb mraer tgot tgls eba ls o bmae6 tge iebhtg I dust sotls`y tge jmbaltlmb

I 4 b3

, `mr b 4 ?, 3, :,....

Qgls jmbaltlmb sgmws tgot stobalbh woves mb o strlbh m` iebhtg I gove restrljtea woveiebhtg hlveb cy

4b

I3, `mr b 4 ?, 3, :,.....

Qge `requebjles jmrrespmbalbh tm tgese woveiebhtgs `miimw `rmd Eq. os

` 4 bI3

v, `mr b 4 ?, 3, :,.....

wgere v ls tge speea m` troveiilbh woves mb tge strlbh. Qge set m` `requebjles hlveb cy equotlmb ore joiiea tge

boturoi `requebjles mrdmaes m` msjliiotlmb m` tge systed. Qgls equotlmb teiis us tgot tge boturoi `requebjles

m` o strlbh ore lbtehroi duitlpies m` tge imwest `requebjy ` 4I3

v, wgljg jmrrespmbas tm b 4 ?. Qge msjliiotlmb

dmae wltg tgot imwest `requebjyls joiiea tge ùbaodebtoi dmae mr tge `lrst gordmblj. Qge sejmba gordmblj

mr `lrst mvertmbe ls tge msjliiotlmb dmae wltg b 4 3. Qge tglra gordmblj oba sejmba mvertmbe jmrrespmbas tm

b 4 : oba sm mb. Qge `requebjles ossmjlotea wltg tgese dmaes ore m`teb ioceiea os ?,

3,

:oba sm mb. Qge

jmiiejtlmb m` oii pmsslcie dmaes ls joiiea tge gordmblj serles oba b ls joiiea tge gordmblj budcer.

Zmde m` tge gordmblj m` o stretjgea strlbh `lxea ot cmtg tge ebas ore sgmwb lb `lhure.

O

O

O

B

O

OO

B

O O O

O O O

O O O O

O O O OBB

B

B

(o)@ubaodebtoi

mr `lrst gordmblj` 4 v/3I;

(c)sejmba gordmbljmr `lrst mvertmbe 4

` 4 3 4 3v/3I? ;

(j)tglra gordmblj

mr 3bamvertmbe 4` 4 :` , 4 :v/3I3 ;

(a)`murtg gordmblj

mr :ramvertmbe 4

` 4 1` , 4 1v/3I: ;

Exodpie ?1. O dlaaie J strlbh mb o plobm gos o ùbaodebtoi `requebjy m` 303 Gz, oba tge O bmte gos ùbaodebtoi

`requebjy m` 11; Gz. (o) Joijuiote tge `requebjles m` tge bext twm gordmbljs m` tge J strlbh. (c) L` tge

strlbhs `mr tge O oba J bmtes ore ossudea tm gove tge sode doss per ublt iebhtg oba tge sode

iebhtg, aeterdlbe tge rotlm m` tebslmbs lb tge twm strlbhs.Zmiutlmb >. (o) Cejouse

?4 303 Gz `mr tge J strlbh, we job use Equotlmb tm `lba tge `requebjles `

3oba `

:6

`34 3`

?4 531 Gz

`:4 :`

?4 870 Gz


14/17

[G\ZLJZ


Uslbh Equotlmb `mr tge twm strlbhs vlcrotlbh ot tgelr ùbaodebtoi `requebjles hlves

`?O

4I3

?

OQ

`?J

4I3

?

JQ

J?

O?

`

`4

J

O

Q

Q

J

O

Q

Q4

3

J?

O?

`

`

4

3

Gz303

Gz11;

4 3.73. Obs.

Exodpie ?5. O wlre govlbh o ilbeor doss aebslty ?;: fh/d ls stretjgea cetweeb twm rlhla suppmrts wltg o tebslmb

m` Zuppmse tge wlre vlcrotes ot :5; Gz lb lts btg gordmblj oba ot 13; Gz lb lts (b + ?)tg gordmblj.

:5; s? 4I3

b

@

....(l)

oba 13; s? 4I3

)?b(

@....(ll)

Qgls hlves:5;

13;4

b

?b mr, b 4 5.

[uttlbh tge voiue lb (l),

:5; 4 :?;

Ztobalbh woves job ce prmaujea mb o strlbh wgljg ls lxea ot mbe eba oba wgmse mtger eba ls ree tm dmve

lb o trobsverse alrejtlmb. Zujg o `ree eba job ce beoriy ojglevea cy jmbbejtlbh tge strlbh tm o very ilhgt

tgreoa.

L` tge vlcrotlmbs ore prmaujea cy o smurje m` jmrrejt `requebjy, stobalbh woves ore prmaujea. L` tge eba x 4

; ls `lxea oba x 4 I ls `ree, tge equotlmb ls oholb hlveb cy

y 4 3O slb fx jmst

wltg tge cmubaory jmbaltlmb tgot x 4 I ls ob obtlbmae. Qge cmubaory jmbaltlmb tgot x 4 ; ls o bmae lsoutmdotljoiiy sotls`lea cy tge ocmve equotlmb. @mr x 4 I tm ce ob obtlbmae,

slb fI 4 ?

mr, fI 4

3

?b mr,

I3

4

3

?b

mr,v

I`34 b +

3

?mr, ` 4

3

?b

I3

v4

I3

3

?b

/Q .....

Qgese ore tge bmrdoi `requebjles m` vlcrotlmb. Qge ùbaodebtoi `requebjy ls mctolbea wgeb b 4 ;, l.e.,

`;

4 v/1I @ubaodebtoi

O

B(o)

Qge mvertmbe `requebjles ore

`?4

I1

v:4 :`

;O

@lrstMvertmbe(c)

BO

B

`34

I1

v54 5`

;

O

ZejmbaMvertmbe

(j)B BOO

Xe see tgot oii tge gordmblj m` tge ùbaodebtoi ore bmt tge oiimwea `requebjles `mr tge stobalbh woves. Mbiy

tge maa gordmbljs ore tge mvertmbes. @lhure sgmws sgopes m` tge strlbh `mr smde m` tge bmrdoi dmaes.


15/17

[G\ZLJZ


IOXZ M@ QSOBZPESZE PLCSOQLMBZ M@ O ZQSLBH ZMBMDEQES XLSE(o) Iow m` iebhtg `

I

?sm

?

3

3

?

I

I

`

` 6 l` Q &ore jmbstobt

(c) Iow m` tebslmb ` Q sm3

?

3

?

Q

Q

`

` 6 I &ore jmbstobt

(j) Iow m` doss `

?

sm?

3

3

?`` 6 Q & I ore jmbstobt

[rmcied ?. O wove puise dmvlbh oimbh tge x oxls ls represebtea cy tge wove ùbjtlmb

y(x, t) 4 ?t:x

33

wgere x oba y ore deosurea lb jd oba t ls lb sejmbas.

(l) Lb wgljg alrejtlmb ls tge wove dmvlbh=

(ll) @lba speea m` tge wove.

(lll) [imt tge wove`mrd ot t 4 ;, t 4 3s.

Zmiutlmb > y 4 ?t:x

33

(l) Os wove ls dmvlbh lb +ve x alrejtlmb cejouse

y 4 (x, t) 4 `(t x/v) 4 `/v(vt x)

(ll) Bmw x v t ls jmdporea wltg x :t

v 4 : jd/sej.Obs. ( l) [ms lt lve x ox ls ( ll) : j d/s .

[rmcied 3. Ot t 4 ;, o trobsverse wove puise lb o wlre ls aesjrlcea cy tge ùbjtlmb y 4:x

0:

wgere x oba

y ore lb deters wrlte tge ùbjtlmb y(x, t) tgot aesjrlces tgls wove l` lt ls troveiilbh lb tge pmsltlve

x alrejtlmb wltg o speea m` 1.5 d/s.

Zmiutlmb > y 4:x

03

4 `(x)

Os y(x, t) 4 `(x vt) 4:)t5.1x(

03

Obs. :t5.1x

03

[rmcied :. Qge wove ùbjtlmb mr o troveiilbh wove mb o strlbh ls hlveb os

y (x, t) 4 (;.:5; d) slb (?;t :x + 1 )

(o) Xgot ore tge speea oba alrejtlmb m` trovei m` tge wove=

(c) Xgot ls tge vertljoi alspiojedebt m` tge strlbh ot t 4 ;, x 4 ;.? d=

(j) X got ore woveiebhtg oba `requebjy m` tge wove=

Zmiutlmb > \ (x, t) 4 (;.:5;d) slb (?; t :x +1

)

jmdporlbh wltg equotlmb 6

\ 4 O slb (t fx +) 4 ?;, f 4 :, ` 41

(o) speea 4 f

4 :

?;

4 :.:: d/sej oba oimbh +ve x oxls

(c) y (;.?, ;) 4 ;.:5 slb (?;x M :(;.?) +1

)


16/17

[G\ZLJZ


4 ;.:5 slb

?;

:

1 4 5.17 jd

(j) f 43

4 : :

3 jd 4 ;.08 jd

oba ` 4

4:/3

:/?;4 5 Gz.

[rmcied 1. Zgmw tgot tge wove ùbjtlmb y 4 e

c(x vt)

ls o smiutlmb m` tge ilbeor wove equotlmb.

Zmiutlmb > \ 4 ec(xv t)x

y

4 cec(xv t) obat

y

4 (cv)ec(xvv t)

3

3

x

y

4 c3 ec(xv t) oba 3

3

t

y

4 (cv)3 ec( x v t)

mcvlmusiy 63

3

x

y

4 3v

?3

3

t

y

wgljg ls o Ilbeor wove equotlmb.

[rmcied 5. O ubl`mrd rmpe m` doss d oba iebhtg I gobhs `rmd o jelilbh. (o) Zgmw tgot tge speea m` o

trobsverse wove mb tge rmpe ls o ùbjtlmb m` y, tge alstobje `rmd tge imwer eba, oba ls hlveb cy

v 4 hy. (c) Zgmw tgot tge tlde o trobsverse wove tofes tm trovei tge iebhtg m` tge rmpe ls hlveb

cy t 4 3 h/I .

Zmiutlmb >(o) Os dos per ublt iebhtg

4

d P 4

Q Qebslmb ot [ 4yh

P 4 yh

4 yh

(c) Bmwat

ay4 yh

M y

ay4

t

m

ath t 4 3 h/

[rmcied 0. Qwm slbusmlaoi woves m` tge sode `requebjy ore tm ce sebt lb tge sode alrejtlmb oimbh o tout strlbh.

Mbe wove gos ob odpiltuae m` 5.; dd, tge mtger 7.; dd. (o) Xgot pgose al èrebje ?

cetweeb tge

twm woves resuits lb tge sdoiiest odpiltuae m` tgeresuitobtwove= (c) Xgot ls tgot sdoiiest odpiltuae=

(j) Xgot pgose al èrebje 3

resuits lb tge iorhest odpiltuae m` tge resuitobt wove= (a) Xgot ls tgot

iorhest odpiltuae= (e) Xgot ls tge resuitobt odpiltuae l` tge pgose obhie ls (?

3

)/3=

Zmiutlmb > (o) @mr sdoiiest odpiltuae 6

OS

4 | O? O

3| oba tgot ls pmsslcie wgeb

?4 cetweeb O

?oba O

3

(c) OS

4 | O? O

3| 4 : dd

(j) `mr iorhest odpiltuae 6

OS

4 | O?+ O

3| oba tgot ls pmsslcie wgeb

34 ; cetweeb O

?oba O

3

(a) OS

4 | O?+ O

3| 4 ?: dd

(e) wgeb 43

3? 4

3

m4

3

OS 4 VO?3 + O33 + 3O?O3 jms 3 T?/3

4


17/17

[G\ZLJZ

?

[rmcied 8. O strlbh `lxea ot cmtg ebas ls 7.1; d imbh oba gos o doss m` ;.?3; fh. Lt ls sucnejtea tm o tebslmb m`

(o) P 4Q

4

1.7

?3.;

wave on a string theory_e

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