Week 8 Lecture 3: Problems 49, 50 Fourier analysis Courseware pp (don t look at French very confusing look in the Courseware instead)

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1 Week 8 Lecure 3: Problems 49, 5 Fourier lysis Coursewre pp 6-7 (do look Frech very cofusig look i he Coursewre ised) Fourier lysis ivolves ddig wves d heir hrmoics, so i would hve urlly followed fer he orml mode soluio o he wve equio secio. However I rerrged he lecures o ccommode problem solvig durig he uoril sessios. We will go bck o rvellig wves i he e lecure.

2 Superposiio of Wves (ledig o Fourier Alysis) impor chrcerisics Wves mii heir idividul iegriy whe overlppig (hey pss hrough oher wve uchged). I he overlp regio, he resul is lgebric sum of he vrious coribuios ech poi. he sum of y umber of hrmoic wves hvig he sme fudmel frequecy is lso hrmoic wve of h sme frequecy, eve if hey ll hve differe mpliudes d phses. he compoud wveform below is produced by ddig he 6 hrmoics bove fudmel Noe h he period (d herefore ω) of he compoud wveform is he sme s he fudmel!!! Compoud wveform

3 ur he rgume bckwrds You c lyze y periodic wveform (wih chrcerisic ω) io is frequecy compoes!!! Fourier Alysis Noe - do look Frech for his - errible Fourier s heorem (Fourier ) Ay periodic fucio c be creed by summig he hrmoics of wve, provided he hrmoic fudmel frequecy is he sme s h of he periodic fucio. (see emple of squre wve i he e slide) Fourier Alysis Ay periodic fucio c be represeed by series of hrmoiclly reled sies d cosies (hey c eve be phse shifed). Mhemiclly, if y () is periodic fucio wih frequecy ω, we c wrie i s: () cosω cosω cosω See Fourier syhesis Virul Physics websie lik - very cool b siω b siω b siω Where ω 3, ω 3, ω 4, re upper hrmoics of ω d d b re he mpliudes of ech hrmoic.

4 Syhesizig squre wve usig Fourier Alysis / Sme fudmel frequecy Need mpliude d phse shif for ech hrmoic See Cool Vibrio demos websie!! Off my web pge 4

5 hree higs o Noe: We kow wh ω, ω, ω re defied by ω of he fucio () we w o syhesize. We mus deermie he mpliude of he hrmoics,, 3, (lso ) d b, b, b 3, b Noe h oe of he si or cos erms hs phse shif erm i i. How do we ge phse shif i some of he hrmoics? By ddig si d cos erms ogeher! (see Fourier demo o he Virul Physics websie dd he f s cos d si) Deermiig coefficies d b for : for b : for : muliply boh sides of previous equio by cosω d iegre ech erm over he period muliply boh sides of previous equio by siω d iegre ech erm over he period iegre ech erm over he period Whe you do his, wh do you ge? 5

6 Ge buch of cos()cos() erms, si()si() erms d cos()si() erms, which simplify lo becuse: Iegrig over period cosω cosω d m siω siω d m cosω siω d m if m / if m if m / if m if m if m Becuse we re lef wih cos erm oly Becuse we re lef wih si erm oly Lef wih ohig!!! his will mke ll erms o he RHS of he equio (ecep oe) equl o zero! ry his for he cosω muliplicio d iegre over he firs erm (muliplyig cosω hrough ) looks like: () cosω d cosωd cosω cosωd cos cosω ωd b siω cosωd b siω cosωd / Solve his for d ge cosωd he, fid by muliplyig by cosω, d so o... () 6

7 If you do his up o you filly ge () cosω d do he sme for he siω muliplicio b () siω d do he sme bu jus iegre () d So he origil Fourier equio becomes: b siω () cosω combiio of hese produces phse shif for give hrmoic kes cre of y offse (oe i is / so h ll erms bove hve / i fro) where, d b re 7

8 Specil Cses Which Simplify he Alysis If he fucio f() or f() is odd: ie. f(-) -f() eg., 3, si(), -ve ve he he cos summio erm (he oe icludig ) goes o zero d oly he si epsio (b ) pplies. his correspods o he specil cse i Frech. If he fucio f() or f() is eve: ie. f(-) f() eg.,, cos(),, ve ve he he si erm (he b erm) goes o zero d oly he cos epsio pplies. If you co ideify fucio s eve or odd he you mus use he full Fourier epsio (such is he cse uoril problem 5) 8

9 Summry Pge Fourier Alysis Ay periodic fucio c be represeed by series of hrmoiclly reled si d cosie erms Fourier Equio: where mes ω of fucio is he sme s h of he fudmel of si/cosie erms b siω () cosω offse (shifs up d dow) () for y give vlue he combiio of hese produces phse shif cosω d b () siω d () eve fucio si erm odd fuco cos erm d if eve or odd he oe of he hrmoics re phse shifed wih respec 9 o he fudmel

10 E. he squre wve emple ( summry) y() () si cos b y ω ω geerl equio D D D D D π π π π Foud: π/ eve so - - ω ω ω 3 ω 4 ω 5 ω 6 ω 7 ω 8.57D D D uis

11 For he squre wve emple o he previous pge, wrie ou wh he Fourier equio looks like: Geerl equio y () cos π Pu he firs few coefficies for he squre wve io he equio y () () D π () 3 D π () 5 D π ( 7) D D π cos cos cos cos π 3π 5π 7π... y Ad sice () D he fudmel ω π D D D D cosω cos3ω cos5ω cos7ω... π 3π 5π 7π

12 Cos erms oly! You c lso ply wih he Virul Physics lik websie o ge squre wve!!

13 Squre wve demo o he Virul Physics websie You c lso simule swooh d oher wves d eve ply hem o see wh hey soud like!! Very cool!!! 3

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