Transverse Wave Motion

Transcription

1 Trasverse Wave Moio

2 Defiiio of Waves wave is a disurbae ha moves hrough a medium wihou givig he medium, as a whole, a permae displaeme. The geeral ame for hese waves is progressive wave. If he disurbae akes plae perpediular o he direio of propagaio of he wave, he wave is alled rasverse. If he disurbae is alog he direio of propagaio of he wave, i is alled logiudial.

3 Charaerisis of Waves a poi, he disurbae is a fuio of ime ad a a isa, he disurbae is a fuio of he posiio of he poi. I a soud wave, he disurbae is pressure-variaio i a medium. I he rasmissio of ligh i a medium or vauum, he disurbae is he variaio of he sreghs of he eleri ad magei fields. I a progressive wave moio, i is he disurbae ha moves ad o he pariles of he medium.

4 Progressive Waves To demosrae wave moio, ake he loose ed of a log rope whih is fied a he oher ed quikl up ad dow Cress ad roughs of he waves move dow he rope If he rope is ifii log suh waves are alled progressive waves

5 Sadig Waves If he rope is fied a boh eds, he progressive waves ravelig o i are refleed ad ombied o form sadig waves The firs four harmois of he sadig waves allowed bewee he wo fied eds of a srig

6 Trasverse vs Logiudial Waves Trasverse wave: he displaemes or osillaios i he medium are rasverse o he direio of propagaio e.g. eleromagei (EM waves, waves o srigs Logiudial wave: he osillaios are parallel o he direio of wave propagaio e.g. soud waves

7 Plae Waves Take a plae perpediular o he direio of wave propagaio ad all osillaors lig wihi ha plae have a ommo phase Over suh a plae, all parameers desribig he wave moio remai osa The ress ad roughs are plaes of maimum ampliude of osillaio, whih are rad ou of phase Cres = a plae of maimum posiive ampliude Trough = a plae of maimum egaive ampliude

8 The Wave Equaio The wave equaio of small eleme of srig of liear desi ad osa esio T where T is he phase or wave veloi. ( +d, +d T +d T (,

9 a si small ver si si( igored be a hus ad small ver T d d T d T d T d T d ds d ds d ds d d d s d d d ds d d d d d

10 Waves i Oe Dimesio Suppose a wave moves alog he -ais wih osa veloi ad wihou a hage of shape (i.e. wih o dispersio ad he disurbae akes plae parallel o he -ais, he (, = f ( ( defies a oe-dimesioal wave alog he posiive direio of he -ais (forward wave

11 Waves i Oe Dimesio wave whih is he same i all respe bu movig i he opposie direio (i.e. alog he direio of dereasig is give b Eq. ( wih he sig of v haged: (, = f ( + ( This is kow as bakward wave.

12 (, ( (, ( ( f f f f f

13 Waves i Oe Dimesio Eqs. ( ad ( saisf he seod-order parial differeial equaio: (3 Eq. (3 is kow as he o-dispersive wave equaio.

14 Displaeme Lous of osillaor displaemes a -a

15 Soluio of Wave equaio soluio o he wave equaio a si( a si (, where is he osillaio freque ad The wave is movig i he posiive direio.

16 The Wave Equaio posiio = 0, wave equaio a si osillaor o is righ a some posiio will be se i moio a some laer ime. a si( a si ( Have a phase lag wih respe o he osillaor a = 0. The wavelegh is he separaio i spae bewee a wo osillaors wih a phase differee rad.

17 The Wave Equaio The period of osillaio observer a a poi would be passed b waveleghs per seod. If he wave is movig o he lef he sig is haged. Wave movig o righ Wave movig o lef a si( a si( a a si si ( (

18 Equivale Wave Epressios a si ( a si ( a si ( a si( k where k Cosie fuios are equall valid. For boh sie ad osie is alled wave umber. ae i k

19 The Wave Equaio si(, os( si(, os( si( k k k a k k ka k a k a k a

20 Three Veloiies i Wave Moio. Parile veloi Simple harmoi veloi of he osillaor abou is equilibrium posiio. Wave or phase veloi The veloi wih whih plaes of equal phase, ress or roughs, progress hrough he medium 3. Group veloi umber of waves of differe frequeies, waveleghs ad veloiies ma be superposed o form a group. Moio of suh a pulse would be desribed b is group veloi

21 Parile Veloi arrows show he direio ad magiude of he parile veloi

22 Lous of osillaor displaemes i a oiuous medium as a wave passes over hem ravellig i he posiive -direio The wavelegh is defied as he disae bewee a wo osillaors havig a phase differee of rad

23 Wave or Phase veloi The wave or phase veloi is I is he rae a whih disurbae moves aross he osillaors. The osillaor or parile veloi is a simple harmoi veloi a si( k a os( k

24 Wave or Phase Veloi Wave or Phase Veloi = he rae a whih disurbae moves aross he osillaors Wave or Phase Veloi = Osillaor or Parile Veloi is a simple harmoi veloi Osillaor or Parile Veloi =

25 Charaerisi Impedae of a Srig (he srig as a fored osillaor medium hrough whih waves propagae will prese a impedae o hose waves If he medium is lossless, ad possesses o resisive or dissipaio mehaism, for a srig he impedae is deermied b ieria ad elasii The presee of a loss mehaism will irodue a omple erm io he impedae

26 Charaerisi Impedae of a Srig (he srig as a fored osillaor The rasverse impedae is defie as: rasverse rasverse fore veloi F v Charaerisi impedae of he srig: T sie T

27 Charaerisi Impedae of a Srig (he srig as a fored osillaor The srig as a fored osillaor wih a verial fore F 0 e i drivig i a oe ed For small : T T T e F i a si 0

28 Charaerisi Impedae of a Srig (he srig as a fored osillaor displaeme of he progressive waves ma be represeed epoeiall b: ampliude ma be omple he ed of he srig, where = 0 ( k i e 0 ( 0 0 k i i e ikt T e F T i F ikt F 0 0 ( 0 k i e T i F

29 Charaerisi Impedae of a Srig (he srig as a fored osillaor rasverse veloi: v F 0 T e i ( k veloi ampliude: v F / 0 rasverse impedae: T sie T Charaerisi Impedae of he srig Sie he veloi is deermied b he ieria ad he elasii, he impedae is also govered b hese properies

30 Refleio ad Trasmissio Suppose a srig osiss of wo seios smoohl joied a a poi = 0 wih a esio T Waves o a srig of impedae = refleed ad rasmied a he boudar = 0 where he srig hages o impedae = = =

31 Refleio ad Trasmissio Iide wave: Refleed wave: Trasmied wave: fid he refleio ad rasmissio ampliude oeffiies i.e. he relaive values of B ad wih respe o ( k i i e ( k i r e B ( k i e

32 ( k i i e ( k i r e B ( k i e fid he refleio ad rasmissio ampliude oeffiies i.e. he relaive values of B ad wih respe o

33 Boudar odiio No. a he impedae disoiui a = 0 Refleio ad Trasmissio. geomerial odiio ha he displaeme is he same immediael o he lef ad righ of = 0 for all ime, so ha here is o disoiui of displaeme r i ( ( ( k i k i k i e e B e 0 ( Eq B

34 Boudar odiio No. a he impedae disoiui a = 0 Refleio ad Trasmissio. damial odiio ha here is a oiui of he rasverse fore T(/ a = 0, ad herefore a oiuous slope r i T T a = 0 for all T k TB k T k T B T T

35 Refleio ad Trasmissio These oeffiies are idepede of ad T T ( Eq ( B Refleio oeffiie of ampliude: B Trasmissio oeffiie of ampliude: Solvig Eqs. ( ad (

36 ( ( ( ( k i k i r i k i k i r i e B ik e ik e B e r i T T ( ( k i k i e ik e 0 0, B ik ik r i 0 0, ik

37 r i T T 0 0, ik B ik ik T B T T k B k k ( B T T

38 Refleio ad Trasmissio B If =, B / = iide wave is ompleel refleed wih a phase hage of (odiios ha eessar for sadig waves o eis If = 0 ( =0 is a free ed of he srig B / =, / = he flik a he ed of a whip or free ed srig

39 If =, B / = iide wave is ompleel refleed wih a phase hage of (odiios ha eessar for sadig waves o eis If = 0 ( =0 is a free ed of he srig B / =, / = he flik a he ed of a whip or free ed srig

40 Refleio ad Trasmissio of Eerg Wha happes o he eerg i a wave whe i mees a boudar bewee wo media of differe impedae values? (he wave fuio of rasferrig eerg hroughou a medium Cosider eah ui legh, mass, of he srig as a simple harmoi osillaor of maimum ampliude Toal eerg: E = wave freque The rae a whih eerg is beig arried alog he srig: (eerg veloi

41 Refleio ad Trasmissio of Eerg The rae a whih eerg leaves he boudar, via he refleed ad rasmied waves: he rae of eerg arrivig a he boudar = 0 is he eerg arrivig wih he iide wave: eerg is oserved, ad all eerg arrivig a he boudar i he iide wave leaves he boudar i he refleed ad rasmied waves ( 4 ( B B

42 Refleed ad Trasmied Iesi Coeffiies If = o eerg is refleed ad he impedaes are said o be mahed Eerg Iide Eerg Refleed B B 4 Eerg Iide Eerg d Trasmie

43 Mahig of Impedaes Wh Impora? Log disae ables arrig eerg mus be aurael mahed a all jois o avoid wasage from eerg refleio Eample: The power rasfer from a geeraor is a maimum whe he load mahes he geeraor impedae loudspeaker is mahed o he impedae of he power oupu of a amplifier b hoosig he orre urs raio o he ouplig rasformer

44 Mahig of Impedaes Iserio of a ouplig eleme bewee wo mismahed impedaes Goal: o elimiae eerg refleio ad mah he impedaes Remark: whe a smooh joi eiss bewee wo srigs of differe impedaes, eerg will be refleed a he boudar Require o mah he impedaes = ad 3 = 3 3 b he smooh iserio of a srig of legh l ad impedae = Our problem is o fid he values of l ad

45 Mahig of Impedaes The impedaes ad 3 of wo srigs are mahed b he iserio of a legh l of a srig of impedae

46 Mahig of Impedaes we seek o make he raio Trasmie Iide d Eerg Eerg 3 3 Boudar odiios: ad T(/ are oiuous aross he juios = 0 ad = l

47 Mahig of Impedaes Bewee ad he oiui of gives: Coiui of T(/ gives Dividig he above equaio b ad remember = 0 ( ( ( ( k i k i k i k i e B e e B e 0 (a B B B ik ik T B ik ik T T/ k T / B B

48 Mahig of Impedaes = l Coiui of gives: e ik l B e ik l 3 Coiui of T(/ gives: ik l ik l e B e 3 3 From he four boudar equaios, solve for he raio 3 / 3 4 r r os k l r r si k l Refer o he H.J. Pai, The Phsis of Vibraios ad Waves,6 h Ediio, pg -3 for deail derivaio

49 Mahig of Impedaes Eerg Iide Eerg d Trasmie r l k r r l k r r si os 4 have we si ad 0 os 4, / hoose we if l k l k l whe 4 r r r r r

50 Sadig Waves o a Srig of Fied Legh

51 Sadig Waves srig of fied legh l wih boh eds rigidl lamped Cosider wave wih a ampliude a ravelig i he posiive -direio ad a ampliude b ravelig i he egaive -direio The displaeme o he srig a a poi is give b: ae i ( k i ( k be wih he boudar odiio ha = 0 a = 0 ad = l

52

53 Sadig Waves Boudar odiio: = 0 a = 0 0 ( a b e a = b i wave i eiher direio meeig he ifiie impedae a eiher ed is ompleel refleed wih a phase hage i ampliude ae i ik ik i e e i ae si k epressio of whih saisfies he sadig wave ime depede form of he wave equaio: k 0

54 Sadig Waves Boudar odiio: = 0 a = l Limiig he value of allowed frequeies o: l l f l l kl 0 si si l kl i ae e e ae be ae i ikl ikl i kl i kl i si ( (

55 Sadig Waves ormal frequeies or modes of vibraio: si si l Suh allowed frequeies defie he legh of he srig as a ea umber of half waveleghs l The firs four harmois, =,, 3, 4 of he sadig waves allowed bewee he wo fied eds of a srig (Fudameal mode

56

57 si k Sadig Waves For >, here will be a umber of posiios alog he srig where he displaeme is alwas zero alled odes or odal poi These pois our where k r 0 si here are ( posiios equall spaed alog he srig i he h harmoi where he displaeme is alwas zero Sadig waves arise whe a sigle mode is eied ad he iide ad refleed waves are superposed If he ampliudes of hese progressive waves are equal ad opposie (resulig from omplee refleio, odal pois will eis l r si l 0 ( r 0,,, 3,...,

58 Sadig Waves he omplee epressio for he displaeme of he h harmoi is give b: a ( i os i si si we a epress his i he form: os B si si / where he ampliude of he h mode is give b B a

59 Sadig Wave Raio If a progressive wave ssem is pariall refleed from a boudar, le he ampliude refleio oeffiie B / = r, for r < The maimum ampliude a reiforeme is ( + B, he miimum ampliude ( B The raio of he maimum o miimum ampliudes is alled sadig wave raio (SWR SWR B B r r Refleio oeffiie: r B SWR SWR

60 Eerg i Eah Normal Mode of a Vibraig Srig vibraig srig possesses boh kiei ad poeial eerg Kiei eerg of a eleme of legh d ad liear desi d Toal kiei eerg: E ( kiei 0 d

61 Eerg i Eah Normal Mode of a Vibraig Srig Poeial eerg = he work doe b hee esio T i eedig a eleme of legh d o a ew legh ds whe he srig is vibraig E ( poeial T ( ds d T d T egle higher powers of / d (... (...

62 Eerg i Eah Normal Mode of a Vibraig Srig For sadig waves: os B si si si B os si os B si os E ( kiei si B os si d l 0 E ( poeial T l 0 os B si os d

63 Eerg i Eah Normal Mode of a Vibraig Srig where m is he mass of he srig = he square of he maimum displaeme of he mode T ( ( poeial ( kiei 4 4 B m B l E ( B

64 a a ad 4 si si a a ad 4 si os si 0 / 4 ( / ( si 0 l d l l os 0 / 4 ( / ( si 0 l d l l d B E l 0 si os si ( kiei d B T E l 0 os si os poeial (

65 os si ( kiei l B E 4 4 kiei ( B m B l E a ime : si os poeial ( l B T E a ime : 4 4 poeial ( m l E T ( ( ( poeial ( kiei 4 4 B m B l E E

66 Wave Groups ad Group Veloi Waves o our as a miure of a umber or group of ompoe frequeies e.g. whie ligh is omposed of visible wavelegh sperum of 400 m o 700 m The behavior of suh a group leads o he group veloi dispersio auses he spaial separaio of a whie ligh io ompoes of differe wavelegh (differe olour

67 Superposiio of wo waves of almos equal frequeies group osiss of wo ompoes of equal ampliude a bu frequeies ad whih differ b a small amou. Their displaemes: Superposiio of ampliude ad phase: os( os( k a k a ( ( os ( ( os k k k k a a wave ssem wih a freque ( + / whih is ver lose o he freque of eiher ompoe bu wih a maimum ampliude of a, modulaed i spae ad ime b a ver slowl varig evelope of freque ( / ad wave umber (k k /

68 Superposiio of wo waves of almos equal frequeies

69 Superposiio of wo waves of almos equal frequeies The veloi of he ew wave is ( /( k k If he phase veloiies / k / k, gives ( k k k k k k so ha he ompoe frequeies ad heir superposiio, or group will ravel wih he same veloi, he profile of heir ombiaio i Figure 5. remaiig osa

70 Superposiio of wo waves of almos equal frequeies For he wo freque ompoes have differe phase veloiies so ha /k /k Group veloi k k k The superposiio of he wo waves will o loger remai osa ad he group profile will hage wih ime Dispersive medium = medium i whih he phase veloi is freque depede (i.e. /k o osa

71 Superposiio of wo waves of almos equal frequeies If a group oai a umber of ompoes of frequeies whih are earl equal he origial, epressio for he group veloi is wrie: k d dk v g Sie = kv (v is he phase veloi group veloi: v g d d ( kv dk dk v k dv dk v g v dv d

72 o-dispersive medium where /k is osa, so ha v g = v, for isae free spae behaviour owards ligh waves ormal dispersio relaio, v g < v aomalous dispersio relaio, v g > v

73 Sadig Waves as Normal Modes of Vibraig Srig

74 Charaerisi of a Normal Mode all he masses move i SHM a he same freque ormal modes are ompleel idepede of eah oher geeral moio of he ssem is a superposiio of he ormal modes Sadig Waves as Normal Modes ll of hese properies of ormal modes are shared b sadig waves o a vibraig srig all he pariles of he srig perform SHM wih he same freque he sadig waves are he ormal modes of he vibraig srig

75 Superposiio of Normal Modes he epressio for he -h ormal mode of a vibraig srig of legh L he moio of he srig will be a superposiio of ormal modes give b:

76 (, si k os Displaeme zero (odes our whe sie erm = 0 si k 0 k ( 0,,,...

77 Eample: superposiio of he 3rd ormal mode wih a relaive ampliude of.0 ad he 3h ormal mode wih a relaive ampliude of 0.5 3rd harmoi 3 (, 0 of a srig a = 0 (a 3h harmoi 3 (, 0 of a srig a = 0 (b ( The superposiio of he wo harmois o give he resula shape of he srig a = 0

78 To eie he wo ormal modes i his wa, we would somehow have o osrai he shape of he srig as i ( ad he release i a ime = 0 I is impraial o do his ad i praie we pluk a srig o ause i o vibrae Eample he srig is displaed a disae d a oe quarer of is legh Iiiall, he srig has a riagular shape ad his shape learl does o mah a of he shapes of he ormal modes

79 i is possible o reprodue his riagular shape b addig ogeher he ormal modes of he srig wih appropriae ampliudes The firs hree eied ormal modes of he srig: (, 0, (, 0 ad 3 (, 0 L (,0 si (,0 si (,0 si 3 3 L 3 L The superposiio of he firs hree ormal modes gives a good reproduio of he iiial riagular shape of he srig eep for he sharp orer Eve usig jus he firs hree ormal modes we ge a surprisigl good fi o he riagular shape B addig more ormal modes, we would ahieve eve beer agreeme, espeiall wih respe o he sharp orer

80 mpliudes of Normal Modes Whe we pluk a srig we eie ma of is ormal modes ad he subseque moio of he srig is give b he superposiio of hese ormal modes aordig o equaio The iiial shape of he srig f (, i.e. a = 0 is give b

81 mpliudes of Normal Modes shape f ( of he srig wih fied ed pois [f (0 = f (L = 0] a be wrie as a superposiio of hese sie fuios wih appropriae values for he oeffiies,,..., i.e. i he form Fourier series The epasio of he above equaio is kow as a Fourier series ad he ampliudes,,... as Fourier oeffiies or Fourier ampliude

82 mpliudes of Normal Modes where m ad are iegers Bu: Fourier ampliude

83 Eample srig of legh L is displaed a is mid-poi b a disae d ad released a = 0, as show i figure below. Fid he firs hree ormal modes ha are eied ad heir ampliudes i erms of he iiial displaeme d.

84 Soluio: Le he shape of he srig a ime = 0 b he fuio = f ( Ispeio of figure shows ha: To ope wih he kik i f ( a = L/, we spli he iegral i he Fourier ampliude equaio ( i slide-9 io wo pars, so ha

85 Soluio (oiued..: Subsiuig for f ( over he appropriae rages of, he righ-had side of his equaio beomes: Useful formula for he idefiie iegrals The fial resul is

86 Soluio (oiued..: = 0 for eve values of : we ol eie hose modes ha have odd values of, sie modes wih eve have a ode a he mid-poi of he srig ad so will o be eied he ampliudes of hese ormal modes: frequeies give b: