Phys 2310 Fri. Nov. 7, 2014 Today s Topics. Begin Chapter 15: The Superposition of Waves Reading for Next Time
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1 Phys 3 Fr. Nov. 7, 4 Today s Topcs Bgn Chaptr 5: Th Suprposton of Wavs Radng for Nxt T
2 Radng ths Wk By Frday: Bgn Ch. 5 ( Addton of Wavs of th Sa Frquncy, Addton of Wavs of Dffrnt Frquncy, Rad Supplntary Matral: Anharonc Prodc Wavs
3 Howork ths Wk Chaptr 5: # (us xcl, 3, 6,, 33 Du Wd. Nov. 6 3
4 Chaptr 5: Addton wth Sa Frquncy Algbrac Mthod Rcall that lght wavs ar not affctd whn th bas cross. Th dsturbanc (-fld apltud just adds locally. Consdr two wavs: sn(!t " and sn(!t " whr "!(kx # s th spatal porton of th phas. Addng th two wavs (trg. dntts: (sn!t cos" cos!t sn" (sn!t cos" cos!t sn" and so whn th spatal and t-dpndnt portons ar sparatd: ( cos" cos" sn!t ( sn" sn" cos!t Now not that th trs n ( ar constant n t. So w dfn: cos" cos" cos" and sn" sn" sn" Squarng and addng to gt th ntnsty: cos("!" (not ntrfrnc tr Dvdng to gt th phas: tan" sn" sn" thus: cos" cos" cos" sn!t sn" cos!t Not th ntrfrnc tr vars wth OPL: $ (kx #! (kx # or: $ % & n(x! x (whr n(x! x s th OPL 4
5 5 Chaptr 5: Addton wth Sa Frquncy Suprposton of Many Wavs Now consdr that cas of any wavs: > ± ± N N N j N j N N t t cos sn tan and : cos( whr: cos( cos( ω ω Not that snc all hav th sa frquncy th rsult wll also hav th sa frquncy, only th apltud changs.
6 6 Chaptr 5: Addton wth Sa Frquncy Coplx Mthod and Phasor Addton Rcall that Phasor addton s lk vctor addton n polar coords. cos( cosns gvs : Slarly for Phasor addtonth law of cos( ] [ ( ( : and so for N ( ( that snc Rcall th coplx apltud. sknown as whr and thr ar N wavs : If ( ( * ( ω ω N j t N j t j j
7 Chaptr 5: Addton wth Sa Frquncy Coplx Mthod and Phasor Addton Consdr th Phasor addton of 5 wavs sn ωt sn( ωt 45 sn( ωt 5 sn( ωt o 8sn( ωt 8 o o o 7
8 Chaptr 5: Addton wth Sa Frquncy Standng Wavs Wnr showd that th fld at th surfac of a rror ust b zro, just lk th &M thory of a conductor. Rflctd wavs ar lk standng wavs wth a phas chang of π. In ordr to satsfy ths boundary condton consdr both wavs: I R I [sn(kx!t sn(kx!!t] And snc: sn" sn# sn (" #cos (" # w hav: (x,t I snkx cos!t (standng wav Not that on part oscllats n t but th apltud or "nvlop" dosn't, hnc th na. Ths rsults n nods and ant-nods. In addton, Wnr's xprnt showd that rflctd lght can ntrfr. Ths tchnqu can b usd to asur th wavlngth of lght fro th OPD. 8
9 Chaptr 5: Addton wth Dffrnt Frquncy Bats fro Two Wavs of Dffrnt Frquncy Consdr th addton of two wavs of dffrnt frquncy : If th apltuds ar th sa tr addton ylds : cos cos β cos ( β cos ( β w hav : cos [( k k x ( ω ω t] cos [( k k x ( ω ω t] Not that ths wav has a frquncy qual to th avrag of th two but s odulatd by a uch lowr frquncy wav ( ω ω : If w r-cast th dffrncs n frquncy and propagaton nubr : ω ( ω ω and ω ( ω ω and : k ( k k and k ( k k w hav : cos( k x ω tcos( kx ωt Not that th rrandanc bcos: 4 cos( k x ω t [cos( k x ω t cos( k cos ( k [ cos(k and x ω t x ω t] x ω t ] whch s: cos( k x ω t and snc: 9
10 Chaptr 5: Anharonc Prodc Wavs Fourr Srs Rcall that w statd that any prodc functon can b synthszd by as su of haronc wavs (sns and cosns. Ths s Fourr s Thor. Sns and cosns ar usd bcaus thy ar orthogonal,.., ndpndnt (s txt on lnar algbra.! f (x C C cos! " x # $!! # & C cos " % " / x # $ # & " % In th fgur at rght s how 6 haronc functons can gnrat th coplcatd rsult, and vc vrsa. Tradtonally w us both cosns and sns snc sns ar odd and can provd a phas shft too: If k! / " thn w can rforulat as: ' f (x A ( A coskx ( B sn kx (su ovr wavvctors ' whr th apltuds can b coputd by ntgratng both sds and notng orthogonalty: " sn akx cosbkx dx, cosakx cosbkx dx " # ab and sn akxsnbkx dx " # ab " whr # ab s f a * b and f a b (Kronckr dlta. Thus w fnd: " A f (x coskxdx and B f (x sn kxdx " " " So, gvn a functon w can coput th Fourr apltuds to odl t. Not th sytry btwn th functon and th apltuds of th Fourr srs. In addton a pur vn functon: f (x f (x wll contan only cosns and a purly odd functon:f (x f (x wll only hav sn trs. " Not that a coplx wavfor can thn b "odld" just fro th Fourr coffcnts (A, B. Ths gratly rducs a dgtal fl sz and s th bass of all coprsson algorths. All trad "fdlty" vs. fl sz. A falar xapl s MP3. Whn w odl th wavfor,.., tak a Fourr transfor, any ts pr scond (576 for arly vrsons of MP3, w nd up wth 576 sts (spctra of Fourr apltuds ach scond. Th frquncy rang w sapl (spctru dpnds on th applcaton. Lots of softwar s avalabl for rcordng and convrtng audo to MP3.
11 Chaptr 5: Anharonc Prodc Wavs Fourr Srs Knowng a functon, f(x, w can coput th coffcnts. Th nubr of trs dpnds on th prcson rqurd. Not how only 6 trs (n sn can rproduc th sawtooth functon rathr wll snc t s odd. Addng or trs s asy wth a coputr (vn xcl. A fw hundrd trs ay b ncssary to accuratly rproduc both th slops and th dscontnuts. Th nubr rqurd dpnds on th prcson rqurd. Not th dcrasng apltud of th hghr-ordr trs n th frquncy spctru (typcal.
12 Chaptr 5: Anharonc Prodc Wavs Fourr Srs xapl Lts follow along wth th book wth an xapl squar-wav. whn < x < λ / f ( x whn λ / < x < λ B B B B λ λ / sn kxdx λ so usng only sns ( A sn kxdx and thus : λ / [ cos kx] [cos kx] λ / or : π π ( cos π snc k π/ λ. Thus th Fourr coffcnts ar : π 4 / π, B, B 4 / 3π, B, B 4 / 5π and soon. Thus : 3 λ λ / 4 f ( x sn kx sn 3kx sn 5kx π 3 5 Not th dcrasng apltud of th hghr - ordr trs. S Fgur 7.and txt for a dscrpton of how as th pak wdth narrows hgr ordr haroncs ar ncssary to odl th wav.that s, rproducng sall faturs rqurs orfourr coffcnts (appltuds. 4 4, odd functon :
13 Chaptr 5: Non-prodc Wavs Fourr Intgrals If w accpt that w can odl any functon wth an nfnt srs of sns and cosns t ss rasonabl to gnralz Fourr srs to Fourr Intgrals. S fgur 7. for how In ths cas w hav th Fourr Transfor: f ( x A( kcos kxdk B( ksn kxdk wth : π A( k f ( xcos kxdx and B( k f ( xsn kxdx An ntrstng Fourr transfor par s th squar wav f ( x whn x < L / othrws. Its transfor s th snc functon : A( k sn( kl / L kl / 3
14 Chaptr 5: Pulss and Wav Packts Squar Puls Fourr transfor of a squar puls s calld th Snc functon Fnt Wavtran A gvn Fourr coponnt gos on forvr A fnt wavtran rqurs addtonal Fourr ods. Thnk of th wav bng odulatd by a squar puls. Thus a fnt wavtran (puls can b thought of as a wav packt. A th wavtran bcos long ts frquncy spctru shrnks and vc vrsa. Cohrnc Lngth Any ral -M wav s not absolutly onochoatc but has a natural wdth. Thr ust b a cohrnc lngth and a cohrnc t as a rsult. Naly: Δl c cδt c Cohrnc lngth and t ar a asur of spac and t ovr whch th wav has an approxatly constant wavlngth or frquncy. Not that wht lght (larg bandwdth can thn b undrstood as th suprposton of larg nubrs of onochroatc wavs (any Fourr ods 4
15 Chaptr 5: Non-prodc Wavs Dscrt Fourr Transfor Oftn our sgnal conssts of a st of dscrt ponts (.g. fro a dgtal dtctor Th Fourr ntgral s approxatd by a suaton Th rsultng transfor can thn b fltrd to rov unwantd frquncs (.g. ntrfrnc and thn transford back. Ths tchnqus can b furthr gnralzd to ut-dnsonal data, such as a -d ags S spatal fltrng xapls n th txtbook. Two xcllnt rfrncs: Th Fourr Transfor and Its Applcaton (Bracwll A Studnt s Gud to Fourr Transfors: wth Applcatons n Physcs and ngnrng (Jas 5
16 Radng ths Wk By Frday: Fnsh Radng Ch. 5 ( Addton of Wavs of th Sa Frquncy, Addton of Wavs of Dffrnt Frquncy Anharonc Prodc Wavs 6
17 Chaptr 5: Howork ths Wk Du Wd. Nov. 6 7
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