Topic 4 Fourier Series. Today

Transcription

1 Topic 4 Fourier Series Toy

2 Wves with repetig uctios Sigl geertor Clssicl guitr Pio Ech istrumet is plyig sigle ote mile C 6Hz)

3 st hrmoic hrmoic 3 r hrmoic 4 th hrmoic 6Hz 5Hz 783Hz 44Hz A sigle ote will coti ieret rctios o ech hrmoic y B siω t y B siω t y 3 3 B3 siω t y 4 4 B4 siω t The summe output repets with the perio o the st hrmoic Summe hrmoics y totl B siωt B siωt... B siωt

4 Fourier clime tht y repetig ptter coul e represete y summe series o ie sie terms ω t) ωt siωt ω rie/ourier/ourier.html

5 Fourier clime tht y repetig ptter coul e represete y summe series o ie sie terms ω t) ωt siωt ω rie/ourier/ourier.html

6 si ) T t T t t si ) t t t ω ω T the T ω ω ω ω, The ptter will repet with perio o thest hrmoic requecy ω ω so sice Fourier si where T is the perio o the repetig uctio

7 si ) Or i the ptter repets with perio where is istce we c write Here perio is 4

8 F) ) si F) si F) For these cses is tke to e to simpliy epressios F) si F) F) 3 si 3

9 Nee to i, ) si some ckgrou work require.

10 Bckgrou I - Itegrtig over si

11 Itegrtig over or ll si or ll

12 Itegrtig over or ll si or ll

13 Bckgrou II - useul itegrls A B A B) A B) si 4 m m { m) } { m) }

14 si or ll or ll m si si m or ll m si m or ll m Rememer: o eve o si or ll

15 Fiig coeiciets o the Fourier Series Rememer how the Fourier series c e writte like this or perio ) si For simplicity let s mke so we c write ) si

16 Fiig coeiciets o the Fourier Series ) ) si Tke this equtio itegrte oth sies over perio Clerly o the RHS the oly o-zero term is the term ) ) Repet perio si )

17 ) Fiig coeiciets o the Fourier Series si This time multiply oth sies y ) itegrte over perio ) si

18 Fiig coeiciets o the Fourier Series ) si O RHS, oly the term survives s it is oly term where Orthogolity) ) Hece we i )

19 Fiig coeiciets o the Fourier Series To i ll coeiciets multiply oth sies o the Fourier series y m), the itegrte over perio: )m m m si m O the RHS, oly the m term survives the itegrtio m m m )m m ) m

20 Coeiciets o the Fourier Series I similr wy, multiplyig oth sies o the Fourier series y sim), the itegrtig over perio we get: m )si m

21 Coeiciets o the Fourier Series The Fourier series c e writte with perio s ) si The Fourier series coeiciets c e ou y:- ) ) )si

22 Coeiciets o the Fourier Series Coeiciets o the Fourier Series si ) ) ) ) si The Fourier series c e writte with perio s The Fourier series coeiciets c e ou y:-

23 Fiig the coeiciets o Fourier Series Step. Write ow the uctio ) i terms o. Wht is the perio? Step. Use equtio to i? ) 3 Step 3. Use equtio to i? ) Step 4. Use equtio to i? ) si

24 Emple 4. - pge Use equtio to i?. The uctio )? Wht s the perio? ) < < < < ) [ ] 3. Use equtio to i? ) ) ) Perio is si

25 4. Use equtio to i? si ) si ) si )si ) )

26 Step 5. Write out vlues o or,, 3, 4, 5,. ) ) ) ) ) ) ) ) ) ) ) ) ) si

27 Step 5. Write out vlues o or,, 3, 4, 5,.... si 5 5 si 3 3 si ) si )

28 So wht oes the Fourier series look like i we oly use irst ew terms? 3

29 ecture 7 ecture 7 Fourier Series Fourier Series More emples o Fourier series Descriig pulses with Fourier series si ) ) ) )si

30 Emple 4. - pge Use equtio to i?. The uctio )? Wht s the perio? ) < < < < ) [ ] 3. Use equtio to i? ) ) ) Perio is si

31 4. Use equtio to i? si ) si ) si )si ) )

32 Step 5. Write out vlues o or,, 3, 4, 5,. ) ) ) ) ) ) ) ) ) ) ) ) ) si

33 Step 5. Write out vlues o or,, 3, 4, 5,.... si 5 5 si 3 3 si ) si )

34 Fourier Series - QUIZ ). Wht is whe 3? ). Wht is whe 5? ) ) 3. Wht is whe? 4. Wht is whe 7? ) 5. Wht is whe 5? ) ) 6. Wht is whe? 7. Wht is whe 4? ) ) 3 5 ) ) ) ) ) 4 ) )

35 Fourier Series - QUIZ 8. Wht is? I 4 ) I 4 [ ]

36 Fourier Series - QUIZ 9. Wht is I 5) ) I 5) [ ]

37 Fiig coeiciets o the Fourier Series Fi Fourier series to represet this repet ptter. Steps to clculte coeiciets o Fourier series. Write ow the uctio ) i terms o. Wht is perio? ) < < < Perio is

38 Fiig coeiciets o the Fourier Series Fiig coeiciets o the Fourier Series < < < ) Steps to clculte coeiciets o Fourier series. Use equtio to i? ) )

39 Fiig coeiciets o the Fourier Series ) < < < First 5 terms to 5). Use equtio to i? 3. Use equtio to i? ) )si et sie - i coeiciets Right sie - i coeiciets

40 < < < ) vu uv uv v v si Itegrte y prts ) ) u u

41 vu uv uv v v si si si Itegrte y prts si si u u

42 si ) ) ) ) )

43 < < < ) vu uv uv v v si si Itegrte y prts )si u u si si )si

44 vu uv uv v v si si Itegrte y prts u u si si

45 si )

46 ) 3 5 si si si3 si4 si

47 ) 3 5 si si si 3 si 4 si Check our Fourier series usig Fourier_checker.ls

48 Fourier Series o eve o repetig uctios ) ) ) )

49 Oly sie terms require to eie o uctio Oly ie terms require to eie eve uctio Oly eve uctio c hve oset.

50 Fourier Series pplie to pulses Or spce ) ser light pulse or < or or > Iitil isplcemet o guitr strig Electroic wveuctio o molecule

51 ecomes ut oly look etwee This pproch is ie ut it les to lot o work i the itegrtio stge.

52 O uctio oly sie terms) Eve uctio oly ie terms) Wht is perio o the repetig ptter ow?

53 Hl-rge sie series where We sw erlier tht or uctio with perio the Fourier series is:- ) si where ) )si I the hl rge cse we hve uctio o perio which is o so cotis oly sie terms

54 Hl-rge sie series where I the hl rge cse we hve uctio o perio which is o so cotis oly sie terms si ) si ) )si ) )si where The series is vli oly etwee

55 Hl-rge sie series ) si )si ) ODD si ODD ODD ODD EVEN EVEN EVEN )si

56 Hl-rge ie series For uctio with perio the Fourier series is:- ) si where ) )si We hve uctio o perio ut this time it is eve so cotis oly ie terms

57 Hl-rge ie series We hve uctio o perio ut this time it is eve so cotis oly ie terms ) ) ) ) ) ) EVEN EVEN EVEN EVEN )

58 The Fourier series or pulse such s c e writte s either hl rge sie or ie series. However the series is oly vli etwee Hl rge sie series ) si where )si Hl rge ie series ) where )

59 Fi Hl Rge Sie Series which represets the isplcemet ), etwee 6, o the pulse show to the right ) or < 6 The pulse is eie s with legth 6 So )si si Itegrte y prts uv uv vu

60 )si si Itegrte y prts uv uv vu So so set u si v 6 6 v si u si

61 6 6 si 6 )si si si si

62 Fi Hl Rge Sie Series which represets the isplcemet ), etwee 6, o the pulse show to the right Hl rge sie series ) si where )si ) si si si si si

63 si 5 3 si 3 si 4 3 si 6 6 si ) Check our Fourier series usig Fourier_checker.ls

64 Fourier Series pplie to pulses Why is this useul? I Qutum you hve see tht there re speciic solutios to the wve equtio withi potetil well suject to the give oury coitios. Ψ ) Ψ ) B si

65 ecture 7 Summry Prctice questios olie t Norml series Eve o uctios Pulses

66 ecture 8 Summry Prctice questios olie t Comple Fourier series Prsevl s theorum Revisio & Prctice

67 ecture 8 ecture 8 Fourier Fourier si ) ) )si )

68 Comple Fourier Series I my res o physics, especilly Qutum mechics, it is more coveiet to ier wves writte i their comple orm e i The comple orm o the Fourier series c e erive y ssumig i solutio o the orm ) c e the multiplyig oth sies y itegrtig over perio e i ) si si i e i e i im e )

69 Comple Fourier Series The comple orm o the Fourier series c e erive y ssumig im i im ) e c i solutio o the orm ) c e the multiplyig oth sies y itegrtig over perio: im e e e c e i m)

70 Comple Fourier Series Comple Fourier Series m i im i im e c e e c e ) ) For itegrl vishes. For m itegrl. So e c i ) m m i m e m i ) ) si )

71 For perio o the comple Fourier series is i c e i where c ) e ) The more geerl epressio with perio is i where i ) c e c ) e

72 Emple 4.5 Fi the comple Fourier series or ) i the rge - < < i the repet perio is 4. c i i ) e perio is 4, so we write c e 4 Itegrtio y prts u v uv v u u u v e i v e i i

73 e c i ) perio is 4, so we write e c i 4 u v uv u v 4 e i e i i i 4 4 i i e i e i i i e e i e v u i i e i v u ) ) i i i i i i i i e e e e i e e i e e i

74 i i i i i i i Sice the C ) ) i i e e e e C i i e e ) We wt to i ctul vlues or C so it woul e helpul to covert epressio or C ito sie ie terms usig the str epressios:- si i i i e e ) i i i i i si so C ) sice ) c e i so i i ) ) e

75 Prsevl s Theorem pplie to Fourier Series The eergy i virtig strig or electricl sigl is proportiol to the squre o the mplitue o the wve

76 Prsevl s Theorem pplie to Fourier Series Cosier gi the str Fourier series with perio tke or simplicity s. ) si Squre oth sies the itegrte over perio: [ ) ] si

77 [ ) ] si The RHS will give oth squre terms cross term. Whe we itegrte, ll the cross terms will vish. All the squres o the ies sies itegrte to give hl the perio). [ ) ] [ ] Prsevl s theorem sys the totl eergy i virtig system is equl to the sum o the eergies i the iiviul moes.

78 Prctice revisio Try olie questios & 7