PHY226 Topic 4 Fourier series (Lectures 5-8) Online Problems

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1 PHY6 PHY6 Topic Fourier series (ectures 5-8. Itroductio to Fourier series pge. Why re they useful? pge 3.3 Towrds fidig the Fourier coefficiets pge. Averge vlue of fuctio pge 5.5 Orthogolity pge 5.6 Fourier coefficiets derivtio pge 6.7 Summry of results pge 7.8 Emples pge 8.9 Eve d Odd fuctios pge. Hlf rge series pge 3. Comple series pge 5. Prsevl s Theorem pge 7.3 Appedi: Proof of Orthogolity pge 8 Olie Problems for topic pge 9 Alstir Buckley Sept Itroductio - Pge of

2 Topic. Fourier Series PHY6 Refereces Jord & Smith Ch.6, Bos Ch.7, Kreyszig Ch. Some fu jv pplet demostrtios re vilble o the web. Try puttig Fourier series pplet ito Google d lookig t the sites from jhu, Flstd d Mths Olie Gllery.. Itroductio to Fourier Series Cosider legth of strig fied betwee rigid supports. The full behviour of this system c be foud by solvig wve equtio prtil differetil equtio. We will do this lter i the course. For ow we will just recll the bsic properties of wves of strigs which we lredy kow: There is fudmetl mode of vibrtio. Cll the frequecy of this mode f d the time period T. The there re vrious hrmoics. These hve frequecy f, 3f, f, 5f,, f, I prctice, whe pio or guitr or other strig is hit or plucked, it does ot vibrte purely i oe mode the displcemet of the strig is ot purely siusoidl, the soud emitted is ot ll of oe frequecy. I prctice, oe ormlly hers lrge mout of the fudmetl plus smller mouts of vrious hrmoics. The proportios i which the differet frequecies re preset vries hece guitr souds differet from violi or pio, d violi souds differet if it is bowed from if it is plucked! (See pges & Remember tht if the fudmetl frequecy hs frequecy f, its period T /f. A hrmoic wve of frequecy f will the hve period T/, but obviously lso repets with period T. So if we dd together siusoidl wves of frequecy f, f, 3f, f, the result is (osiusoidl wveform which is periodic with the sme period T s the fudmetl frequecy, f /T. [E.g. ply with ] Sometimes we use the gulr frequecy ω where the th hrmoic hs ω f /Τ. The vrious hrmoics re the of the form Asiω t. siωt.5 siωt.5 si3ωt Alstir Buckley Sept Itroductio - Pge of

3 PHY6 y (t siωt+.5 siωt+.5 si3ωt For ll the fuctios bove, the verge vlue over period is zero. If we dd costt term, the wveform remis periodic but its verge vlue is o loger zero: y (t + siωt+.5 siωt+.5si3ωt Wht is relly useful is tht this works i reverse: Ay periodic fuctio with period T c be epressed s the sum of costt term plus hrmoic (sie d cosie curves of gulr frequecy ω, ω, 3ω,... where ω /T. i.e. we c write Ft ( + ( cosωt+ bsi ωt + ( cos ωt+ bsi ωt + K + cos ωt+ bsi ωt where ω /T. We will lter prove this result mthemticlly, d lter i the semester will see tht it c be deduced from the geerl solutio of the wve equtio. For ow you my be ble to persude yourself of its plusibility by plyig with the vrious websites for emple, the demostrtios of how squre or trigulr wveforms c be mde from sums of hrmoic wves. The more terms i the sum, the closer the pproimtio to the desired wveform. Hece i geerl, ifiite umber of terms re eeded.. Why is this useful? I lecture we solved the forced hrmoic oscilltor equtio d dt d( t ( t + γ + ω ( t F cosωt. dt Such equtio could describe, for emple, the respose of electricl CR circuit to siusoidl drivig voltge. But wht would hppe if we pplied squre wve drivig voltge?? Usig Fourier theory, we would just eed to epress the squre wveform s sum of siusoidl terms. The the respose would be the sum of the solutios for ech term (which would ll hve similr form, but ivolve differet multiples of ω thus lso hve differet mplitudes. Throughout physics there re my similr situtios. Fourier series mes tht if we c solve problem for siusoidl fuctio the we c solve it for y periodic fuctio! Alstir Buckley Sept Itroductio - Pge 3 of

4 Ad periodic fuctios pper everywhere! Emples of periodicity i time: pulsr, tri of electricl pulses, the temperture vritio over hours or the verge dily temperture over yer (pproimtely. Emples of periodicity i spce: crystl lttice, rry of mgetic domis, etc. Other Forms If we wt to work i terms of t ot ω, the formul becomes t t f ( t + cos + b si. T T Or similrly for fuctio f( which is periodic i spce with repetitio legth, we hve f ( + cos + b si. (Ay vlue of T or c be used, lthough to keep the lgebr stright forwrd, most questios will set T s or eve s metres. PHY6.3 Towrds Fidig the Fourier Coefficiets To mke thigs esy let s sy tht the ptter repets itself every metres, so. The Fourier series c the be epressed more simply i the form f ( + cos + b si. Now we wt to fid epressios for the coefficiets d b. To do this we eed two other bits of preprtory mthemtics Alstir Buckley Sept Itroductio - Pge of

5 PHY6. Averge Vlue of Fuctio Cosider fuctio y f(. The verge vlue of the fuctio betwee d b is defied to be b f ( d. b Geometriclly this mes tht the re uder the curve f( betwee d b is equl to the re of rectgle of width (b- d height equl to this verge vlue. Note tht while verge vlues c be foud by evlutig the bove itegrl, sometimes they c be idetified more quickly from symmetry cosidertios, sketch grph d commo sese! Two prticulrly importt results re: The verge vlue of sie or cosie fuctio over period is zero: si si cos d d d. The verge vlue of cos or si over period is ½: si d cos d. Actully both these results c be geerlized. It is esily show tht: si d cos d d si d cos d for Hece si d cos d d si d cos d ( Note: We hve writte ll the itegrls over [, ] but y itervl of width c be used, e.g. [, ], [3., 5.], etc..5 Orthogolity (Proofs i the Appedi Sies d cosies hve importt property clled orthogolity : The product of two differet sie or cosie fuctios, itegrted over period, gives zero: si cosmd for ll, m si si m d cos cos m d for ll m Agi we c itegrte over y period. Equipped with these results we c ow fid the Fourier coefficiets Alstir Buckley Sept Itroductio - Pge 5 of

6 .6 Fourier Coefficiets Derivtio PHY6 Erlier we sid y fuctio f( with period c be writte f ( + cos + b si. Tke this equtio d itegrte both sides over period (y period: + + f ( d d cos d b si d Clerly o the RHS the oly o-zero term is the term: f ( d d ( hece f ( d i.e. / is the verge vlue of the fuctio f(. Now tke the origil equtio gi, multiply both sides by cos, the itegrte over period: + + f ( cos d cos d coscos d b si cos d O the RHS, this time oly the term survives s it is the oly term where (see Orthogolity. f ( cos d cos cos d cos d hece f ( cos d. The method for fidig the coefficiets should thus be cler. To fid geerl epressio for we c tke the equtio, multiply both sides by cosm, the itegrte over period: + + f ( cosmd cosmd coscosmd b si cosmd O the RHS, oly the m term survives the itegrtio: f ( cosmd m cos md m hece m f ( cosmd. I similr wy, multiplyig both sides by sim, the itegrtig over period we get: b m f ( si m d Alstir Buckley Sept Itroductio - Pge 6 of

7 .7 Summry of Results PHY6 A fuctio f( with period c be epressed s f ( + cos + b si where f ( d, f ( cos d, b f ( si d. The more geerl epressio from pge c be writte s:- A fuctio f( with period c be epressed s f ( + cos + b si where f ( d, f ( cos d, b f ( si d. Note The formul for c be obtied from the formul for just by settig. The itegrls bove re writte over [, ] d [, ] but y coveiet itervl of width oe period my be used, d this will be depedet o the ture of the fuctio (see emples d the olie problems. 3 The equtios c be esily dpted to work with other vribles or periodicities. For emple, for fuctio periodic i time with period T just replce by t d by T. A few books use the ltertive form d Ft d t d fid vlues of d d θ. ( + cos( ω + θ Alstir Buckley Sept Itroductio - Pge 7 of

8 .8 Emples Emple. Fid Fourier series for the squre wve show. We hve < < f ( The period is. < < PHY6 3 Thik: After rechig your swer, sk yourself: is this result sesible? - Does the term / look like pproprite vlue for the verge vlue of the fuctio over period? - Would we epect this fuctio to be mde mily of sies or of cosies? (See lter for symmetry. - I wht proportios would we epect to fid the fudmetl d the vrious hrmoics? (You c lso try checkig your swer by buildig the series t or Alstir Buckley Sept Itroductio - Pge 8 of

9 PHY6 Emple. Fid Fourier series of the fuctio show: Agi the period is. But this time it is esiest to work with the rge [-, ]. If we wted we could use the rge [,] d get the sme swer, but it would be fiddlier. Betwee - d, f( is stright lie with grdiet d Y-itercept of. So we c write f( + < <. ets fid ( ( f d d ( ets fid f ( cos d ( + cos d cos d + cos d We must itegrte cos d by prts: udv uv vdu u dv cos d du d v cos d si So cos d si si si + cos d (see p.3 verge vlue Goig bck to, f ( cos d (see p.3 verge vlue ( + cos d + cos d Alstir Buckley Sept Itroductio - Pge 9 of

10 PHY6 Now let s fid the b coefficiets. b f ( si d ( + si d si d + si d We must itegrte si d by prts: udv uv vdu u dv si d du d v si d cos So si d cos cos d cos + si (see p.3 verge vlue Goig bck to b, b b f ( si d cos + si si d + si d cos + si cos + si + ( cos( + si( Remember tht cos( cos( d si( si( So b cos si cos + Wht will b be for differet vlues of? 3 cos ( cos ( cos3 ( cos ( Hece f ( + + si si + si 3 si ( + si Notes Where fuctio hs discotiuities, the Fourier Series coverges to the midpoit of the jump (e.g. i emple t,, etc the series hs vlue ½. I geerl the lowest frequecy terms provide the mi shpe, the higher hrmoics dd the detil. Whe fuctios hve discotiuities, more higher hrmoics re eeded. Hece i both the bove emples the terms drop off quite slowly. I geerl, for smoother fuctios the terms drop off fster. Alstir Buckley Sept Itroductio - Pge of

11 .9 Eve d Odd Fuctios PHY6 For eve fuctio, f e (- f e ( i.e. the grph y f( hs reflectio symmetry i the y-is. For odd fuctio, f o (- - f o ( i.e. the grph y f( hs 8º rottiol symmetry bout the origi. Ay sum of eve fuctios is lso eve fuctio. Hece cos is lwys eve fuctio. Therefore the Fourier series of eve fuctio cotis oly cosie terms. Similrly, the Fourier series of odd fuctio cotis oly sie terms. It is eceptiolly useful to remember this! E.g. if you re sked to fid the Fourier series of fuctio which is eve, you c immeditely stte tht b for ll, meig tht there will be o sie terms. You should lso remember the followig fcts (esily verified lgebriclly or by sketchig grphs: The product of eve fuctio d eve fuctio is eve The product of odd fuctio d odd fuctio is eve The product of eve fuctio d odd fuctio is odd Alstir Buckley Sept Itroductio - Pge of

12 PHY6 Alstir Buckley Sept Itroductio - Pge of Emple.3 Fid Fourier series of the fuctio show: The period is. As discussed erlier we c itegrte over y full period e.g. or / / The fuctio is eve d c be writte f( for 3. Therefore there will be o sie terms (b for ll d I feel like itegrtig betwee d. The series will hve form + cos ( f where d f ( d d f cos (. So ] [ ( / 3 / / 3 / d d f (si 6 (si si cos cos ( / 3 / / 3 / d d f (si 3 (si (si 6 (si Epressio for is ot very pretty d esy to mke mistkes with. Write out tble to help with ssigmet of coefficiets. 3 (si 3 (si (si 6 (si 3 3 (si 9 (si 3 (si (si (si 5 (si 5 6 (si 8 (si (si (si 7 8 (si (si 8 So... cos 5 6 cos 3 cos ( + + f

13 . Hlf-Rge Series Sometimes we wt to fid Fourier series represettio of fuctio which is vlid just over some restricted itervl. We could do this i the orml wy d the stte tht the fuctio is oly vlid over specific itervl. However, the fct tht we c do this llows us to use clever trick tht reduces the compleity of problem. We will study this by cosiderig the followig emple: PHY6 Emple Cosider guitr strig of legth tht is beig plucked. (Note o pplictio: If strig ws relesed from this positio, fidig this Fourier series would be crucil step i determiig the displcemet of the strig t ll subsequet times see lter i course. We could, s before, pply the Fourier series to preted ifiite series of plucked strigs d the sy tht the epressio ws oly vlid betwee d. However this series would coti both sie d cosie terms s there is either eve or odd symmetry, d so would tke ges to solve. There is much more clever wy to proceed. Note tht we re oly told the form of the fuctio o the itervl [, ]. All tht mtters is tht the series correspods to the give fuctio i the give itervl. Wht hppes outside the give itervl is irrelevt. The wy to tckle such problem is to cosider rtificil fuctio which coicides with the give fuctio over the give itervl but eteds it d is periodic. Clerly we could do this i ifiite umber of differet wys, however i the previous sectio, we observed tht the Fourier series of odd d eve fuctios re prticulrly simple. It is therefore sesible to choose odd or eve rtificil fuctio! If the origil fuctio is defied o the rge [, ] the there re lwys odd d eve rtificil fuctios with period. I this cse these look like Odd etesio Eve etesio These fuctios re clled the odd etesio d eve etesio respectively. Their correspodig Fourier series re clled the hlf-rge sie series d hlf-rge cosie series. Alstir Buckley Sept Itroductio - Pge 3 of

14 PHY6 Theory We sw erlier tht for fuctio with period the Fourier series is:- f ( + cos + b b f ( si d si, where f ( cos d, I this cse we hve fuctio of period so the formule become f ( + cos + b si, where b f ( si d Rememberig lso tht b f ( cos d, f e ( d f e ( d, we get the followig results: b b Hlf-rge cosie series: f ( + cos, where f ( cos d. Hlf-rge sie series: f ( b si, where b f ( si d. Note : The resultig series is oly vlid over the specified itervl! Emple. Fid Fourier series which represets the displcemet y(, betwee d, of the plucked strig show. et us choose to fid the hlf-rge sie series. We hve d y( ( d < < < < So m d m d m bm dsi Y( d si d ( si + / / Usig itegrtio by prts, it c be show tht the result is: 8d m b m si for m odd b for m eve 8d So for < < we hve Y( si si si si Work out the full solutio for yourself. This questio is swered i the olie problems. Alstir Buckley Sept Itroductio - Pge of

15 Further Results. Comple Series. For the wves o strigs we eed rel stdig wves. But i some other res of physics, especilly solid stte physics, it is more coveiet to cosider comple or ruig wves. Remember tht: ik ik ik ik i ik ik cos k ( e + e ; si k ( e e ( e e i The comple form of the Fourier series c be derived by ssumig solutio of the form f i ( c e d the by evlutig the coefficiets s i sectio 3, tkig the epressio d multiplyig both sides by e -im d itegrtig over period: im i im f ( e d c e e d c e i( m For mthe itegrl vishes. For m the itegrl gives. Hece c f ( e i d d PHY6 Comple Fourier Series for fuctio of period : i i ( c e where c f ( e d f A fuctio f( with period c be epressed s:- i i / f ( c e where c f ( e d Alstir Buckley Sept Itroductio - Pge 5 of

16 PHY6 et s hve look t emple of comple Fourier series. Emple.5 Fid the comple Fourier series for f( i the rge - < < if the repet period is. Alstir Buckley Sept Itroductio - Pge 6 of

17 PHY6. Prsevl s Theorem Cosider gi the Fourier series f ( + cos + b si. Squre both sides the itegrte over period: ( [ f ] d + cos + b si d The RHS will give both squred terms d cross term. Whe we itegrte, ll the cross terms will vish. All the squres of the cosies d sies itegrte to give (hlf the period. Hece [ f ( ] d + [ + b ] The eergy i vibrtig strig or electricl sigl is proportiol to itegrl like [ f ] d (. Hece Prsevl s theorem tells us tht the totl eergy i vibrtig system is equl to the sum of the eergies i the idividul modes. From PHY Alstir Buckley Sept Itroductio - Pge 7 of

18 .3 Appedi: Orthogolity PHY6 At fudmetl mthemticl level, the reso the Fourier series works the reso y periodic fuctio c be epressed s sum of sie d cosie fuctios is tht sies d cosies re orthogol. I geerl, set of fuctios u (, u (,, u (, is sid to be orthogol o the itervl [, b] if b m u ( um ( d (where c is costt. c m Here we will prove tht fuctio si, cosm, etc re orthogol o the itervl [, ].. si cosmd Hece si cosmd for m si( + m si( m d cos( cos( + m + m + m m [Usig si( + b si( b sicosb]. si si md cos( m cos( + m d [Usig cos( b cos( + b sisib] Hece si si m d for m. 3. cos cosmd Hece cos cos m d for m si( si( m m m m cos( + m + cos( m d si( si( + m + m + m m [Usigcos( b + cos( + b coscosb] Alstir Buckley Sept Itroductio - Pge 8 of

19 PHY6 For m the itegrls becomes:. si cos d si cos d. si d ( cos d si 3. cos d ( + cos d + si For m the first two itegrls become d d the third becomes d Note. Similr results c be proved for fuctio of periodicity.. The results ( re esy to remember: A itegrls over sies d cosies over full period give zero, uless the itegrd is squre i which cse the itegrl is lwys equl to hlf the rge of the itegrl. Olie Problems (Topic, questios - f ( for -/ to /. Give squre wve fuctio f ( tht repets f ( for / to 3 / every, show tht y itegrl rge is cceptble so log s m mi. Fid the comple epoetil Fourier series for the fuctio specified s f ( e for < < 3. Fid the Fourier series for the fuctio the repet period is. if t < / f ( t cos(3 t if t < / where if / t <. Fid the Fourier series for the swtooth fuctio f ( for < f ( f ( for < 5. Sketch the fuctio Fourier series. f ( t for t - to f ( t for two periods d fid its f ( t 5 for t to Alstir Buckley Sept Itroductio - Pge 9 of

20 PHY6 6. The fuctio f(t with period τ is defied i the itervl [-τ/, τ/] by if τ / < t f ( t Fid the Fourier series. if < < τ / 7. I m thikig of the fuctio y cosθ. Sketch it, tell me wht the period is, d fid its Fourier series. 8. A guitrist pulls strig s show below. Drw the odd d eve etesios of the plot d deduce the fuctios f( for both the << rges 9. Fid the hlf rge cosie series for questio 8 bove.. Fid the Fourier series tht represets the displcemet y( betwee d of the strig below.. Show tht the hlf rge cosie series c be used to clculte the Fourier series betwee << for the swtooth fuctio i questio.. A fuctio f( is defied oly o the rge << d o this rge f(k where k is positive costt. Sketch the fuctio d lso show tht the hlf k rge Fourier sie series of the fuctio is f ( si odd Alstir Buckley Sept Itroductio - Pge of