Lectures 5-8: Fourier Series

Transcription

1 cturs 5-8: Fourir Sris PHY6 Rfrcs Jord & Smith Ch.6, Bos Ch.7, Kryszig Ch. Som fu jv pplt dmostrtios r vilbl o th wb. Try puttig Fourir sris pplt ito Googl d lookig t th sits from jhu, Flstd d Mths Oli Gllry.. Itroductio to Fourir Sris Cosidr lgth of strig fid btw rigid supports. Th full bhviour of this systm c b foud by solvig wv qutio prtil diffrtil qutio. W will do this ltr i th cours. For ow w will just rcll th bsic proprtis of wvs of strigs which w lrdy kow: Thr is fudmtl mod of vibrtio. Cll th frqucy of this mod f d th tim priod T. Th thr r vrious hrmoics. Ths hv frqucy f, 3f, 4f, 5f,, f, I prctic, wh pio or guitr or othr strig is hit or pluckd, it dos ot vibrt purly i o mod th displcmt of th strig is ot purly siusoidl, th soud mittd is ot ll of o frqucy. I prctic, o ormlly hrs lrg mout of th fudmtl plus smllr mouts of vrious hrmoics. Th proportios i which th diffrt frqucis r prst vris hc guitr souds diffrt from violi or pio, d violi souds diffrt if it is bowd from if it is pluckd! (S pgs &) Rmmbr tht if th fudmtl frqucy hs frqucy f, its priod T /f. A hrmoic wv of frqucy f will th hv priod T/, but obviously lso rpts with priod T. So if w dd togthr siusoidl wvs of frqucy f, f, 3f, 4f, th rsult is (o-siusoidl) wvform which is priodic with th sm priod T s th fudmtl frqucy, f /T. [E.g. ply with ] Somtims w us th gulr frqucy ω whr th th hrmoic hs ω f /Τ. Th vrious hrmoics r th of th form Asiω t. Illustrtio: siωt.5 siωt.5 si3ωt y (t) siωt+.5 siωt+.5 si3ωt For ll th fuctios bov, th vrg vlu ovr priod is zro. If w dd costt trm, th wvform rmis priodic but its vrg vlu is o logr zro: y (t) + siωt+.5 siωt+.5 si3ωt Phil ightfoot 8/9 ctur 5 - Pg of

2 PHY6 Wht is rlly usful is tht this works i rvrs: Ay priodic fuctio with priod T c b prssd s th sum of costt trm plus hrmoic (si d cosi) curvs of gulr frqucy ω, ω, 3ω,... whr ω /T. i.. w c writ Ft ( ) + ( cosωt+ bsi ωt) + ( cos ωt+ bsi ωt) + K + cos ωt+ bsi ωt whr ω /T. W will ltr prov this rsult mthmticlly, d ltr i th smstr will s tht it c b dducd from th grl solutio of th wv qutio. For ow you my b bl to prsud yourslf of its plusibility by plyig with th vrious wbsits for mpl, th dmostrtios of how squr or trigulr wvforms c b md from sums of hrmoic wvs. Th mor trms i th sum, th closr th pproimtio to th dsird wvform. Hc i grl, ifiit umbr of trms r dd. Why is this usful? d d( t) I lctur 4 w solvd th forcd hrmoic oscilltor qutio ( t) + γ + ω ( t) F cosωt. dt dt Such qutio could dscrib, for mpl, th rspos of lctricl CR circuit to siusoidl drivig voltg. But wht would hpp if w pplid squr wv drivig voltg?? Usig Fourir thory, w would just d to prss th squr wvform s sum of siusoidl trms. Th th rspos would b th sum of th solutios for ch trm (which would ll hv similr form, but ivolv diffrt multipls of ω thus lso hv diffrt mplituds). Throughout physics thr r my similr situtios. Fourir sris ms tht if w c solv problm for siusoidl fuctio th w c solv it for y priodic fuctio! Ad priodic fuctios ppr vrywhr! Empls of priodicity i tim: pulsr, tri of lctricl pulss, th tmprtur vritio ovr 4 hours or th vrg dily tmprtur ovr yr (pproimtly). Empls of priodicity i spc: crystl lttic, rry of mgtic domis, tc. Othr Forms If w wt to work i trms of t ot ω, th formul bcoms t t f ( t) + cos + b si. T T Or similrly for fuctio f() which is priodic i spc with rptitio lgth, w hv + cos + b si. (Ay vlu of T or c b usd, lthough to kp th lgbr stright forwrd, most qustios will st T s or v s mtrs.). Towrds Fidig th Fourir Cofficits To mk thigs sy lt s sy tht th pttr rpts itslf vry mtrs, so. Th Fourir sris c th b prssd mor simply i th form + cos + b si. Now w wt to fid prssios for th cofficits d b. To do this w d two othr bits of prprtory mthmtics Phil ightfoot 8/9 ctur 5 - Pg of

3 () Avrg Vlu of Fuctio Cosidr fuctio y f(). Th vrg vlu of th fuctio b btw d b is dfid to b d. b Gomtriclly this ms tht th r udr th curv f() btw d b is qul to th r of rctgl of width (b-) d hight qul to this vrg vlu. Not tht whil vrg vlus c b foud by vlutig th bov itgrl, somtims thy c b idtifid mor quickly from symmtry cosidrtios, sktch grph d commo ss! Two prticulrly importt rsults r: Th vrg vlu of si or cosi fuctio ovr priod is zro: si si cos d d d. Th vrg vlu of cos or si ovr priod is ½: si d cos d. PHY6 Actully both ths rsults c b grlizd. It is sily show tht: si cos d d d si cos d d for Hc si d cos d d si d cos d ( ) Not: W hv writt ll th itgrls ovr [, ] but y itrvl of width c b usd,.g. [, ], [3., 5.], tc. (b) Orthogolity (Proofs i th Appdi) Sis d cosis hv importt proprty clld orthogolity : Th product of two diffrt si or cosi fuctios, itgrtd ovr priod, givs zro: si cosmd for ll, m Agi w c itgrt ovr y priod. si si m d cos cos m d for ll m Equippd with ths rsults w c ow fid th Fourir cofficits 3. Fourir Cofficits Drivtio Erlir w sid y fuctio f() with priod c b writt + cos + b si. Tk this qutio d itgrt both sids ovr priod (y priod): + + d d cos d b si d Clrly o th RHS th oly o-zro trm is th trm: d d ( ) hc w fid d. i.. / is th vrg vlu of th fuctio f(). Phil ightfoot 8/9 ctur 5 - Pg 3 of

4 PHY6 Now tk th origil qutio gi, multiply both sids by cos, th itgrt ovr priod: + + cos d cos d coscos d b si cos d O th RHS, this tim oly th trm survivs s it is th oly trm whr (s Orthogolity.) cos d cos cos d cos d hc cos d. Th mthod for fidig th cofficits should thus b clr. To fid grl prssio for w c tk th qutio, multiply both sids by cosm, th itgrt ovr priod: + + cosmd cosmd coscosmd b si cosmd O th RHS, oly th m trm survivs th itgrtio: cosmd m cos md m hc m cosmd. I similr wy, multiplyig both sids by sim, th itgrtig ovr priod w gt: b m si m d 4. Summry of Rsults A fuctio f() with priod c b prssd s + cos + b si whr d, cos d, b si d. Th mor grl prssio from pg c b writt s:- A fuctio f() with priod c b prssd s + cos + b si whr d, cos d, b si d. Not ) Th formul for c b obtid from th formul for just by sttig. ) Th itgrls bov r writt ovr [, ] d [, ] but y covit itrvl of width o priod my b usd, d this will b dpdt o th tur of th fuctio (s mpls d Phil s Problms). 3) Th qutios c b sily dptd to work with othr vribls or priodicitis. For mpl, for fuctio priodic i tim with priod T just rplc by t d by T. d 4) A fw books us th ltrtiv form Ft () + dcos( ω t+ θ ) d fid vlus of d d θ. 5. Empls Empl Fid Fourir sris for th squr wv show. W hv < < Th priod is. < < 3 Usig our formul for th cofficits w hv: ( ) + f d [ ] d d Phil ightfoot 8/9 ctur 5 - Pg 4 of

5 cos d () cos + () cos d d si cos d PHY6 So ll th cofficits r zro for. b si d () si d + () si d cos si d cos cos cos cos b So wh odd ; cos - so b odd So wh v ; cos so b v Th stdrd Fourir sris prssio is + cos + b si So th rsultig sris is: + si + si 3 + si ) odd si Thik: Aftr rchig your swr, sk yourslf: is this rsult ssibl? - Dos th trm / look lik pproprit vlu for th vrg vlu of th fuctio ovr priod? - Would w pct this fuctio to b md mily of sis or of cosis? (S ltr for symmtry). - I wht proportios would w pct to fid th fudmtl d th vrious hrmoics? (You c lso try chckig your swr by buildig th sris t or ) Empl Fid Fourir sris of th fuctio show: Agi th priod is. But this tim it is sist to work with th rg [-, ]. N.B. If w wtd w could us th rg [,] d gt th sm swr, but it would b mor fiddly. Btw - d, f() is stright li with grdit d Y-itrcpt of. So w c writ f() + < <. ( ) ( ) f d d ( ) cos d ( + )cos d cos d + cos d W must itgrt cos d by prts: udv uv vdu so st u d cos d dv Phil ightfoot 8/9 ctur 5 - Pg 5 of

6 PHY6 So du d d v cos d si. So cos d si si si + cos d (s p.3 vrg vlu) Goig bck to, cos d Now lt s fid th b cofficits. ( + )cos d + cos d (s p.3 vrg vlu) b si d ( + )si d si d + si d W must itgrt si d by prts: udv uv vdu so st u d si d dv So du d d v si d cos. So si d cos cos d cos + si Goig bck to b b si d si d + si d cos + si + ( ) b cos + si cos + si cos( ) + si( ) Rmmbr tht cos( ) cos( ) d si( ) si( ) So b cos si cos +. Wht will b b for diffrt vlus of? 3 4 cos ( ) cos () cos3 ( ) cos 4 () Hc + + si si + si 3 si ( ) + si Nots ) Whr fuctio hs discotiuitis, th Fourir Sris covrgs to th midpoit of th jump (.g. i mpl t,, tc th sris hs vlu ½). ) I grl th lowst frqucy trms provid th mi shp, th highr hrmoics dd th dtil. Wh fuctios hv discotiuitis, mor highr hrmoics r dd. Hc i both th bov mpls th trms drop off quit slowly. I grl, for smoothr fuctios th trms drop off fstr. Phil ightfoot 8/9 ctur 5 - Pg 6 of

7 6. Ev d Odd Fuctios PHY6 For v fuctio, f (-) f () i.. th grph y f() hs rflctiol symmtry i th y-is. For odd fuctio, f o (-) - f o () i.. th grph y f() hs 8º rottiol symmtry bout th origi. Ay sum of v fuctios is lso v fuctio. Hc cos is lwys v fuctio. Thrfor th Fourir sris of v fuctio cotis oly cosi trms. Similrly, th Fourir sris of odd fuctio cotis oly si trms. It is cptiolly usful to rmmbr this! E.g. if you r skd to fid th Fourir sris of fuctio which is v, you c immditly stt tht b for ll, mig tht thr will b o si trms. You should lso rmmbr th followig fcts (sily vrifid lgbriclly or by sktchig grphs): Th product of v fuctio d v fuctio is v Th product of odd fuctio d odd fuctio is v Th product of v fuctio d odd fuctio is odd Empl 3 Fid Fourir sris of th fuctio show: Th priod is. As discussd rlir w c itgrt ovr y full priod.g. / or / Th fuctio is v d c b writt f() for 3. Thrfor thr will b o si trms 4 4 (b for ll ) d I fl lik itgrtig btw d. Th sris will hv form + cos whr d d cos d. So 3 / 4 3 / d [ ] 4 / 4 d / 4 cos d 3 / 4 / 4 cos d si 3 / 4 / 4 6 (si ) (si ) (si ) (si ) (si ) (si ) 4 4 Eprssio for is ot vry prtty d sy to mk mistks with. Writ out tbl to hlp with ssigmt of cofficits. Phil ightfoot 8/9 ctur 5 - Pg 7 of

8 (si ) (si ) (si ) (si ) PHY (si ) (si ) (si ) (si ) (si ) (si ) (si ) (si ) (si ) (si ) (si ) (si ) So cos + cos cos Hlf-Rg Sris Somtims w wt to fid Fourir sris rprsttio of fuctio which is vlid just ovr som rstrictd itrvl. W could do this i th orml wy d th stt tht th fuctio is oly vlid ovr spcific itrvl. Howvr, th fct tht w c do this llows us to us clvr trick tht rducs th complity of problm. W will study this by cosidrig th followig mpl: Empl 4 Cosidr guitr strig of lgth which is big pluckd. (Not o pplictio: If strig ws rlsd from this positio, fidig this Fourir sris would b crucil stp i dtrmiig th displcmt of th strig t ll subsqut tims s ltr i cours.) W could, s bfor, pply th Fourir sris to prtd ifiit sris of pluckd strigs d th sy tht th prssio ws oly vlid btw d. Howvr this sris would coti both si d cosi trms s thr is ithr v or odd symmtry, d so would tk gs to solv. Thr is much mor clvr wy to procd. Not tht w r oly told th form of th fuctio o th itrvl [, ]. All tht mttrs is tht th sris corrspods to th giv fuctio i th giv itrvl. Wht hpps outsid th giv itrvl is irrlvt. Th wy to tckl such problm is to cosidr rtificil fuctio which coicids with th giv fuctio ovr th giv itrvl, but tds it d is priodic. Clrly w could do this i ifiit umbr of diffrt wys, howvr i th prvious sctio, w obsrvd tht th Fourir sris of odd d v fuctios r prticulrly simpl. It is thrfor ssibl to choos odd or v rtificil fuctio! If th origil fuctio is dfid o th rg [, ] th thr r lwys odd d v rtificil fuctios with priod. I this cs ths look lik: Ths fuctios r clld th odd tsio d v tsio rspctivly. Thir corrspodig Fourir sris r clld th hlf-rg si sris d hlf-rg cosi sris. Thory W sw rlir tht for fuctio with priod th Fourir sris is:- + cos + b si, whr cos d, b si d I this cs w hv fuctio of priod so th formul bcom + cos + b si, whr cos d, b si d Phil ightfoot 8/9 ctur 5 - Pg 8 of

9 b b Rmmbrig lso tht f ( ) d f ( ) d, w gt th followig rsults: b Hlf-rg cosi sris: + cos, whr cos d. Hlf-rg si sris: b si, whr b si d. Not : Th rsultig sris is oly vlid ovr th spcifid itrvl! Empl 5 Fid Fourir sris which rprsts th displcmt y(), btw d, of th pluckd strig show. PHY6 t us choos to fid th hlf-rg si sris. d < < W hv y( ) ( ) d < < / m d m d m So bm dsi Y( ) d si d ( )si + / Usig itgrtio by prts, it c b show tht th rsult is: b 8d m si m for m odd for m v So for < < w hv 8d Y( ) si si si si Work out th full solutio for yourslf. This qustio is swrd i Phil s problms. 8. Furthr Rsults ) Compl Sris. For th wvs o strigs w d rl stdig wvs. But i som othr rs of physics, spcilly solid stt physics, it is mor covit to cosidr compl or ruig wvs. Rmmbr tht: ik ik ik ik i ik ik cos k ( + ); si k ( ) ( ) i Th compl form of th Fourir sris c b drivd by ssumig solutio of th form f ) i ( c d th by vlutig th cofficits s i sctio 3, tkig th prssio d multiplyig both sids by -im d itgrtig ovr priod: im i im d c d c i( m) i For mth itgrl vishs. For m th itgrl givs. Hc c d Compl Fourir Sris for fuctio of priod : Th mor grl prssio c b writt s:- A fuctio f() with priod c b prssd s:- d i i ( c whr c d f ) i i / c whr c d Phil ightfoot 8/9 ctur 5 - Pg 9 of

10 t s hv look t mpl of compl Fourir sris. PHY6 Empl 6 Fid th compl Fourir sris for f() i th rg - < < if th rpt priod is 4. c i / i / d d th priod is 4. So w c writ c d. 4 Itgrtio by prts c 4 i i i + d u dv uv v du with u d dv i / d so du d d i 4 i 4 i i i i v i i i i + C i i + i i i + i i i i i i ( + ) + ( ) Sic i i i i i i i i i th C ( + ) + ( ) It is kow tht sic i i + i cos + isi d i cos isi th i i i i i si so w sy C cos si cos i cos ( ) d ( ) i i C cos So ( ) d sic i i c th ( ) i b) Prsvl s Thorm Cosidr gi th Fourir sris + cos + b si. Squr both sids th itgrt ovr priod: [ ] d cos b si d + + Th RHS will giv both squrd trms d cross trm. Wh w itgrt, ll th cross trms will vish. All th squrs of th cosis d sis itgrt to giv (hlf th priod). Hc 4 [ ] d + [ + b ] Th rgy i vibrtig strig or lctricl sigl is proportiol to itgrl lik [ f ) ] d (. Hc Prsvl s thorm tlls us tht th totl rgy i vibrtig systm is qul to th sum of th rgis i th idividul mods. Tk from PHY Phil ightfoot 8/9 ctur 5 - Pg of

11 Appdi: Orthogolity PHY6 At fudmtl mthmticl lvl, th rso th Fourir sris works th rso y priodic fuctio c b prssd s sum of si d cosi fuctios is tht sis d cosis r orthogol. I grl, st of fuctios u (), u (),, u (), is sid to b orthogol o th itrvl [, b] if b m u ( ) um ( ) d (whr c is costt). c m Hr w will prov tht fuctio si, cosm, tc r orthogol o th itrvl [, ].. si cosmd si( + m) si( m) d [Usig si( + b) si( b) si cosb] cos( ) cos( ) + m + m + m m Hc si cosmd for m.. si si md cos( m) cos( + m) d [Usig cos( b) cos( + b) si si b] si( ) si( ) m m m m Hc si si md for m. 3. cos cosmd cos( + m) + cos( m) d [Usig cos( b) + cos( + b) cos cosb] si( ) si( ) + m + m + m m Hc cos cos m d for m. For m th itgrls bcoms:. si cos si cos d d 4. si d ( cos) d si 3. cos d ( + cos) d + si For m th first two itgrls bcom d d th third bcoms d Not. Similr rsults c b provd for fuctio of priodicity.. Th rsults ( ) r sy to rmmbr: A itgrls ovr sis d cosis ovr full priod giv zro, ulss th itgrd is squr i which cs th itgrl is lwys qul to hlf th rg of th itgrl. Phil ightfoot 8/9 ctur 5 - Pg of