N coupled oscillators

Transcription

1 Waves

2 Waves. N coupled oscillaors owards he coiuous limi. Sreched srig ad he wave equaio 3. he d lember soluio 4. Siusoidal waves, wave characerisics ad oaio

3 N coupled oscillaors Cosider fleible elasic srig o which are aached N paricles of mass m, each a disace l apar. he srig is fied a each ed. Small rasverse displacemes are applied rasverse oscillaios p p 0 3 p p ll agles small, i.e. i i No sigifica horizoal force, sice l p N N si a / i ad cos i i p cos p cos 0. Bu vericall: F p m p si p p l si p p l p p p 0 p 0 p p 0 wih 0 / ml 3

4 N coupled oscillaors: special cases Firs cosider he special cases N= ad N= N= / ml N= r q ad q q q q 0 0 q

5 N coupled oscillaors: geeral case We have p 0 p 0 p p 0 Cosider rial soluio p p cos For simplici we re o allowig for addiioal phase offse, e.g. cos p. Equivale o imposig all masses sar a res p p Subsiuig i rial soluio gives N equaios: 0 p 0 p p p p p p,,..., N For p=0 ad p=n+ we ow p =0 Sice ω is same for all masses we see RHS ca deped o p & so or ca LHS 5

6 N coupled oscillaors: geeral case Loo for form for p ha p p. Leaves idepede of p p. Saisfies p =0 for p=0 ad N+ r p C si p so p p Csi p si p C si p cos p p cos which saisfies. p ad requirig N,,3,... saisfies. p p C si N p p p 0 0 cos N 0 cos N si 0 N 6

7 N coupled oscillaors: he soluio p p 0 3 p p p l p C si cos N p N N Displaceme for mass p whe oscillaig i mode ad agular frequec: si 0 N oe we have smuggled bac i he phase offse φ 0 / ml lhough he value of ca go beod N, his jus geeraes duplicae soluios, i.e. here are N ormal modes i oal. 7

8 N coupled oscillaors: modes for N=5 Loo a each mode for N=5, wih sapsho ae a =0 = = =5 =3 =4 Noe how he displaceme of ever paricle falls o a sie curve! 8

9 N coupled oscillaors: N ver large Le s cosider srig-mass ssem of fied legh L ad mass M, i.e. L N l M Nm ad defie liear desi m/ l We ll focus o case N ver large, which is sarig o approimae o real srig is small Loo firs a mode umbers which are low i value compared wih N ml si N m / l N l L so ormal frequecies are ieger muliples of a lowes frequec ω L 9

10 N coupled oscillaors: N ver large is small We have L, wha abou displacemes? p Geeral resul p C si cos. Now separaio bewee N paricles becomes smaller & we approach a coiuous variable., C si cos L i.e. srig ges closer ad closer o lig o a sie curve pl 0

11 N coupled oscillaors: N ver large =N Loo firs a frequec of highes mode Now cosider displaceme of successive masses N 0 si N p p 0 pn si N p N si N si p si p So we have somehig lie his Realisig his, we see eq of moio for a give mass is from which we ca recover he resul N 0 m l

12 Ssem of sprigs ad N masses: logiudial oscillaios u p m m m m p p p p p p p p p u u u u u u u u mu m / 0 wih Le u p be displaceme from equilibrium posiio of mass p Equaios of moio have same form as for masses o srig same soluios!

13 Sreched srig Cosider a segme of srig of liear desi ρ sreched uder esio, small F F si si cos cos sice δθ F small F 0 ad so a 3

14 Sreched srig a i.e. Noe he parial double derivaive, raher ha d d which implies. depeds o boh ad. his ad much of wha follows will cocer parial differeial equaios. Now, a so sec ad sice small sec ad sec herefore ad so 4

15 Sreched Srig ad he wave equaio We have performed a similar aalsis o he N oscillaig masses o a srig. - he we had N coordiaes, p, p=...n. - Now we have,, which are coiuous variables Have obaied Now ρ/ has dimesios of /speed ad ideed we will see ha hese parameers do defie he veloci of ravellig waves o he srig. his is he wave equaio c c 5

16 Jea-Bapise le Rod d lember Lived i Paris Mahemaicia ad phsicis lso a music heoris ad co-edior wih Didero of a famous ecclopaedia 6

17 d lember soluio of wave equaio v v u u v u v v u u v v u u c v c u v v u u v u v v u u v v u u c v c u c v v u u v v u u c c v c u is a fucio of ad. Defie ew variables so ha is ow a fucio of u ad v Chai rule o ge firs derivaives ad he secod derivaives 7

18 d lember soluio of wave equaio c c v c u is a fucio of ad. Defie ew variables so ha is ow a fucio of u ad v 0 v u, v g u f v u, c g c f Here f ad g are a fucios of -c ad +c. he are deermied b iiial codiios. So geeral soluio of wave equaio is 8

19 Ierpreaio of d lember soluio Les focus o f-c par of soluio c, c g c f is geeral soluio of ime = :, c f, c f ime = : ] [, c c f c c c f c f i.e. displaceme a ime ad posiio = displaceme a ime a disace c - o he lef of soluio describes wave ravellig o righ wih speed c 9

20 Ierpreaio of d lember soluio, f c Focus o =0 ad cosider siuaios a =0 ad =Δ 0,0 0, c Wave moves o righ wih speed c 0

21 Ierpreaio of d lember soluio, g c Focus o =0 ad cosider siuaios a =0 ad =Δ 0,0 0, c Wave moves o lef wih speed c

22 d lember s soluio wih boudar codiios, c g c f fucios f ad g are deermied b iiial codiios Suppose a ime =0, he wave has iiial displaceme U ad veloci V,0 g f U ' ' d d d d,0 cg cf c g c c f c V b V c g f d b V c U g d b V c U f d Iegraig gives ad combiig wih ields

23 d lember s soluio wih boudar codiios, c g c f fucios f ad g are deermied b iiial codiios Suppose a ime =0, he wave has iiial displaceme U ad veloci V b V c U g d b V c U f d c c c b c b V c c U c U V V c c U c U d d d, 3

24 a d lember s soluio wih boudar codiios Eample: recagular wave of legh a released from res, U c U c 0 a / c U a a a a / c 3a / c a a a a 4

25 Siusoidal waves ver commo fucioal depedece for f ad g..., f c g c...is siusoidal. I his case i is usual o wrie:, cos Bcos wih ad ω ad ad B cosas or si-ω... ec choice does maer, uless we are comparig oe wave wih aoher ad he relaive phases become impora speed of wave frequec c / / / where ω is agular frequec f / wavelegh where is he wave-umber or wave-vecor if also used o idicae direcio of wave 5

26 Noaio choices Siusoidal soluio, cos wriig here, for compacess, ol he forward-goig soluio Usig he relaioships bewee,ω, λ & c his ca be epressed i ma forms, cos[ c] lso oe ha someimes i is coveie o wrie, cos Chages ohig for cosie, riviall so, & pracicall o eve for sie fucio, as overall sig ca be absorbed i cosa & sill describes forward-goig wave. ver freque approach is o use comple oaio we alread made use of his whe aalsig ormal modes, ad ou will have see i i circui aalsis, Re ep[ i ] or, Im ep[ i ] if is impora o pic ou sie fucio. Noe ha ofe he Re or Im is implici, ad i ges omied i discussio. 6

27 Phase differeces Ofe impora o specif phase shifs. Ol meaigful o do so whe we are comparig oe wave o aoher., cos, cos wave wave I his eample wave leads wave b π/, i.e. φ=-π/ Ca be epressed wih comple oaio, Re ep[ i ] Nicer sill o subsume phase io ampliude, Re ep[ i ] wih π/ ep i =0 ω 7

28 Waves. Sadig waves. rasverse waves i aure: elecromageic radiaio 3. Polarisaio 4. Dispersio 5. Iformaio rasfer ad wave paces 6. Group veloci 8

29 Sadig waves Cosider a srig wih waves of equal ampliude movig i opposie direcios or, if ou prefer, si si si cos, si cos i.e. has facorised io space ad ime-depede pars. his meas ever poi o srig is movig wih a cerai ime-depedece cosω, bu he ampliude of he moio is a fucio of he disace from he ed of he srig eample a srig o wo which wo waveleghs are ecied =δ =0 Saioar pois are he odes occur ever λ/. Bewee hese are he aiodes. λ λ 9

30 Sadig waves Boudar codiio ha each ed of a fied srig mus be a ode..., si cos wih 0, L, 0...meas ha ol cerai discree frequecies he modes are available. hese modes are muliples of he basic mode, which is he fudameal. Waveleghs mus fi, hece wavelegh of mode L Recallig c / =0 =δ =0 mode 4 =L gives agular frequec of mode L λ λ We alread obaied his resul whe discussig lump srig wih N large! 30

31 Sadig waves violi srig E srig of a violi is o be ued o a frequec of 640 Hz. Is legh ad mass from bridge o ed are 33 cm ad 0.5 g respecivel. Wha esio is required? Fudameal give b f L Lf bove parameers give 68 N 3

32 rasverse waves i aure: EM radiaio he mos impora eample of waves i aure is elecromageic radiaio, i.e. ligh ec. his will be properl covered i EM lecures. Here is jus a aser. Mawell s equaios i free space for elecric field E, ad mageic iducace B E 0. B 0 B E E B James Cler Mawell ε 0 = permiivi of free space = F/m μ 0 = permeabili of free space = 4π 0-7 Hm - Vecor calculus idei plus 3: Maig use of 4 ad : B B B B B E B 3

33 rasverse waves i aure: EM radiaio Mawell s equaios i free space ield: B B 0 0 equivale epressio is obaiable for E which is he wave equaio wih c ms he speed of ligh! Le s he cosider a wave ravellig i z-direcio usig comple oaio: B B0 ep[ i z] ha we ae real B i B j B ep[ i z] par is implici z From Mawell B 0, ha is 0 ad so z B B B z 0 B field has o compoe i direcio of propagaio - a oscillaio is i rasverse plae. Same argume follows for E field wave is rasverse! Mawell s equaios also impl ha E ad B vecors are rasverse o each oher o show here eercise for he sude! 33 B z

34 rasverse waves i aure: EM radiaio EM waves i vacuum: boh E ad B vecors oscillae rasverse o he direcio of propagaio ad, i phase, rasverse o each oher B-field E-field 34

35 rasverse vs logiudial waves For coupled oscillaors we cosidered boh rasverse ad logiudial eciaios. he same is rue here ca cerail have logiudial waves Some ssems suppor ol rasverse waves, some ol logiudial, some boh rasverse ol: sreched srig, EM waves i vacuum... Logiudial ol: soud waves i air his because air has o elasic resisace o chage i shape, ol o chage i desi Boh: sreched sprig, crsal... rasverse waves have a impora aribue o available o logiudial waves: POLRISION 35

36 Polarisaio rasverse vibraios ca be i oe of wo direcios or boh orhogoal o he direcio of wave propagaio. We al of wo differe direcios of polarisaio. I ca eve be ha wave velociies are differe for he wo polarisaio saes, due o e.g. he differe ieraomic spacigs i a crsal. Some possibilies for polarisaio of E vecor i EM wave ravellig i z-direcio: plae of oscillaio liear E liear E E E 0 0 E cos z liear E E E 0 E cos z 0 E circular roaes wih ime E E E0 / cos z E / cos z 0 E E E E 0 0 cos z si z 36

37 Dispersio For our sreched srig we foud ha he wave veloci is, c / i.e. depeds ol o properies of srig ad has o depedece o frequec or wavelegh of wave. Bu his is a idealised ssem! For mos ssems he veloci of a wave does have a depedece o ω ad λ DISPERSION Oe well ow eample is ligh i a prism. Ligh i a medium m wih refracive ide Has a veloci c m, where c m c /. Bu he refracive ide, ad hece wave veloci, varies wih wavelegh. Hece ligh is be a differe agles b prism accordig o wavelegh. 37

38 Dispersio lump srig revisied he sreched srig has a idealised mass / ui legh. Bu earlier we aalsed ormal modes of he lump srig. We foud: si 0 N wih 0 / ml ad L / ; also we have / / L Recall ormal modes for N=5: = = =5 =3 =4 L Loo a behaviour of ω vs for =...N, recallig ha wave speed=ω/ 38

39 Dispersio curve for lump srig For a lump srig wih N=00 masses oher properies arbirar calculae ω ad for each ormal mode his is o liear! Veloci of wave correspodig o each mode depeds o ω or. his is dispersio. ω/ω 0 Sauraes owards cu-off agular frequec of ω 0 icreasig Noe also ha here is a cu-off frequec a maimum frequec above which i is o possible o ecie ssem/rasmi waves his is a proper ofe foud i a dispersive ssem. 39

40 Iformaio rasfer & wave paces o rasmi iformaio i is ecessar o modulae a wave. Cosider he simples case of urig a wave o ad he off: For a cerai rage of -ω his sigal has displaceme =si-ω, ouside his rage he displaceme =0. his is o a sigle wave, for which =si-ω would appl for all -ω! I is i fac a wave pace. N, D cos wave pace ca be formed b summig ogeher a possibl ifiie umber of waves of differe frequecies his is a Fourier series d ear opic 40

41 Modulaio pure sie wave carries o iformaio o ecode iformaio for radio rasmissio eed o modulae he wave. Geeral priciple as follows: Sigal, picall characerised b low frequec variaio e.g. voice: a few 00 Hz -Hz Carrier wave High frequec e.g. ~ MHz Carrier sigal is modulaed Modulaed sigal, which is rasmied, received ad he de-modulaed Various opios eis for he modulaio sraeg 4

42 Modulaio sraegies Pulse modulaio Simpl ur sie wave off ad o, e.g. morse code mpliude modulaio Modulae ampliude, e.g. Offse + sigal si [π f carrier ] Frequec modulaio Ecode iformaio i modulaio of frequec also phase modulaio 4

43 Wave paces a o eample Sum ogeher wo waves which differ b δω ad δ i agular frequec ad wave-umber, respecivel: o give cos si si si No eacl a pace, more a ifiie series of sausages would eed a ifiie umber of ipu waves o mae a discree wave pace 43

44 Group veloci he veloci of he wave pace is ow as he group veloci. I almos all cases his is he veloci a which iformaio is rasmied. I a dispersive medium he group veloci is o he same as he veloci of he idividual waves, which is ow as he phase veloci & i a dispersive medium he phase veloci, ω/, varies wih frequec & wavelegh Cosider our o eample: cos si Describes evelope so evelope moves wih veloci Group veloci d v g d while phase veloci v p ad ideed 44

45 Caroo of wave pace /7 ravellig wave pace. he mars he op of he wave pace which moves wih he group veloci. he idicaes a compoe wave cres which eers he pace, moves hrough i, ad leaves, wih he phase veloci. 45

46 Caroo of wave pace /7 ravellig wave pace. he mars he op of he wave pace which moves wih he group veloci. he idicaes a compoe wave cres which eers he pace, moves hrough i, ad leaves, wih he phase veloci. 46

47 Caroo of wave pace 3/7 ravellig wave pace. he mars he op of he wave pace which moves wih he group veloci. he idicaes a compoe wave cres which eers he pace, moves hrough i, ad leaves, wih he phase veloci. 47

48 Caroo of wave pace 4/7 ravellig wave pace. he mars he op of he wave pace which moves wih he group veloci. he idicaes a compoe wave cres which eers he pace, moves hrough i, ad leaves, wih he phase veloci. 48

49 Caroo of wave pace 5/7 ravellig wave pace. he mars he op of he wave pace which moves wih he group veloci. he idicaes a compoe wave cres which eers he pace, moves hrough i, ad leaves, wih he phase veloci. 49

50 Caroo of wave pace 6/7 ravellig wave pace. he mars he op of he wave pace which moves wih he group veloci. he idicaes a compoe wave cres which eers he pace, moves hrough i, ad leaves, wih he phase veloci. 50

51 Caroo of wave pace 7/7 ravellig wave pace. he mars he op of he wave pace which moves wih he group veloci. he idicaes a compoe wave cres which eers he pace, moves hrough i, ad leaves, wih he phase veloci. 5

52 Differe epressios for he group veloci We have alread saed d v g d bu v p so v g v p dv d p also, sice / v g v p dv p d or if cosiderig ligh, & a medium wih refracive ide, we have v p c / c d v g d v g c / Observe ha! 5

53 Dispersio ad he spreadig of he wave pace oher cosequece of dispersio is ha a wave-pace will o reai is shape perfecl, bu will spread ou. Ca have cosequeces for sigal deecio 53

54 Group ad phase velociies for lump srig V g /v p Veloci Calculae phase ad group veloci for he lump srig wih N=00 ω/ω 0 v p v g Dispersio curve Raio of v g o v p ω/ω 0 Phase ad group veloci ~ he same a firs, bu v g 0 as ω ω 0 cu-off 54

55 Waves i deep waer Waves i waer wih λ > cm below which surface esio effecs are impora, bu sill small compared o waer deph, have a dispersio relaio g i.e. drive b gravi, hece ofe called gravi waves - ocea swell v d d g g v p So group veloci is half ha of phase veloci 55

56 Esimaig how far awa a sea sorm is g Dispersio relaio meas ha v p g / ad so waves of loger waveleghs ravel more quicl. Sorm Measure ierval bewee successive wave cress a log wa L disa L Period of wave: ime of arrival of wave cress: Elimiae λ: / g v p Hece / d L / g 0 d p 0 L / v 0 L g / L / e.g. if L=000 m ad τ=0 s he -dτ/d= s / hour L / g 0 So if measure period ad rae of decrease i period ca obai L! 56

57 Waves 3. Eerg sored i a mechaical wave. Wave equaio revisied - separaio of variables 3. Wave o srig wih fied eds 4. Waves a boudaries 5. Impedace 6. Oher eamples: - Lossless rasmissio lie - Logiudial waves i a bar - cousic waves 57

58 Eerg sored i a mechaical wave vibraig srig mus carr eerg bu how much? Les calculae he ieic eerg desi i.e. KE / ui legh ad poeial eerg desi +d ds Cosider small segme of srig of liear desi ρ bewee ad +d displaced i he direcio. s usual, assume displaceme is small + d Kieic eerg, K d u d KE desi dk d 58

59 Eerg sored i a mechaical wave + d +d ds Poeial eerg, U, is wor doe b deformaio d d s U... d d / d s U d d d desi PE U 59

60 Eerg sored i a mechaical wave d d desi KE K d d desi PE U ' z f ' z vf ] ' [ d d z f v K ] ' [ z f U We have show, z f v f ad we ow soluios ae form, sa herefore ad so / v hese are equal sice - oe maifesaio of he Virial heorem. 60

61 Eerg sored i a mechaical wave Le s eplicil calculae KE ad PE for a siusoidal wave si Evaluae hese quaiies over a ieger umber of waveleghs d ] cos[ d cos K d ] cos[ d cos U v / / Now ad so hese epressios are equal hus oal eerg / ui legh Moreover, we ca evaluae eerg flow/ui ime = power o geerae wave P = eerg / wavelegh disace ravelled / ime v P 6

62 Wave equaio revisied solvig b separaio of variables c We have alread solved he wave equaio usig he d lember approach Les aac i agai, ow looig for soluios which have he separaed form, X i.e. ha facorise io fucios ha are separae fucios of ad d d d d X c X c X X 6

63 Wave equaio revisied solvig b separaio of variables Separaed soluio o wave equaio e.g., X gives X X c Ca ol be saisfied if boh sides equal a cosa: X X c C he separaio cosa Les cosider firs he case whe C is egaive. We wrie C=- X X c hese are SHM equaios, so we ca wrie X cos Bsi so e.g. i case B=0 ad D=0 we have, Dcos c Esi c F cos si wih,b,d ad E cosas defied b iiial codiios wih ω=c ad F=E... a form we have see before, whe maipulaig d lember soluio 63

64 Wave equaio revisied solvig b separaio of variables Wave equaio wih separaed variables: X X c C C egaive alread cosidered. Here are some oher possibiliies:, e C is posiive = Be De c Ee c C = 0, B D E Sill ohers eis e.g. C is comple. Which of soluios is releva depeds o he phsical siuaio. Here we are usuall ieresed i C is egaive, sice he we ge oscillaios! Bu eve he ma possibiliies eiss =differe values of, ad we will ofe ge a liear superposiio of hese. 64

65 Wave o srig wih fied eds Cosider a srig wih fied eds, iiiall a res, give a iiial displaceme ad released. Describe is subseque moio. 0 L Kow from separaio of variables ha a soluio o wave equaio is, cos Bsi C cos c Dsi c ad we also have four boudar codiios. hree of hem are as follows:. Srig iiiall a res, i.e. / 0 for all D 0. 0,= L,=0 L where a ieger. his is he eigevalue eq. ad discreises. Each value of correspods o a ormal mode. I is clear ha for his coiuous ssem ca go o ifii... 65

66 Wave o srig wih fied eds herefore, b he priciple of superposiio, he soluio is:, F c si cos L L i.e. sum of all possible soluios wih coefficies F give b iiial displaceme, which is he boudar codiio we have o e ivoed,0 F si L h where h is paer of iiial displaceme Simples case is whe h is jus a ormal mode, e.g. 5 h si F5 ad F 0 whe 5 L 5 5c, si cos L L 66

67 Wave o srig wih fied eds Cosider a more complicaed siuaio whe h is o a sigle ormal mode h si si L L he i is clear F =, F =0.5 ad F =0 whe or, so c c, si cos si cos L L L L I coras o case wih jus a sigle ormal mode he subseque moio is o equal o iiial displaceme varig ampliude. Sice he shorer waves are faser he shape of he wave varies durig oscillaio. Eve if he iiial displaceme loos lie his i.e. pluced srig, i ca be epressed as sum of ormal modes! Fourier series ear 0 L 67

68 Wave o srig wih fied eds: eergies of ormal modes L c L F cos si, Normal mode for our problem, wih give boudar codiios: Calculae ieic eerg, K, ad poeial eerg, U, for each mode K L d 0 U L d 0 L c L c F L L c L c F K L 0 si 4 d si si L c L F L L c L F U L 0 cos 4 d cos cos Sice we have / c 4 4 LF E L c LF U K E 68

69 Wave o srig wih fied eds: eergies of ssem Now le s calculae eerg of ssem, wih arbirar iiial displaceme, F si L cos c L his is simple eesio of eercise for idividual ormal modes, Bu wih addiioal erms E E m cross - erms bu cross-erms all iclude facors of he pe L 0 si si L m L d wih m which are zero! E E i.e. oal eerg is sum of eergies of ormal modes 69

70 Waves a boudaries Wha happes for a wave propagaig o a srig whe i ecouers a boudar across which he srig characerisics chage, e.g. chage of liear desi ρ,? ρ ρ =0 Sice he esio,, is cosa across boudar, i follows phase velociies chage c, /, We mus allow for he possibili of hree waves: Icide wave Refleced wave rasmied wave 70

71 Waves a boudaries ρ ρ =0 Le s wrie dow he hree waves Icide Refleced rasmied si si ' ''si we have swiched o wriig siω- raher ha si-ω, as his oaio is usual for hese problems. Bu his does o affec ahig. Observe he + sig i argume for lef-goig wave Noe: We assume same agular frequec, ω, o boh srigs. his is required b boudar codiios see laer. bu wavelegh, ad hece wave umber, medium-depede he relaive values of, ad are o be deermied 7

72 Boudar codiios wo boudar codiios: ρ ρ =0 Here: Srig is coiuous Verical compoe of force o lef of boudar mus be balaced b verical compoe o righ, si 'si, ''si 0, 0, 0, 0, e wave o lef of boudar wave o righ of boudar 7

73 pplig boudar codiios pplig 0, 0, 0, 0, wih, si 'si, ''si Srig coiuous Balaced verical esio si 'si ' '' ''si cos 'cos ' '' ''cos Solve o obai ampliude reflecio ad rasmissio coefficies r ' '' 73

74 pplig boudar codiios We have r ' ad '' Les cosider some specific cases: = < > ρ he r=0, = o reflecio ad full rasmissio he is egaive ad ca wrie refleced wave ' si ' si he is posiive i.e. here is a phase chage a a rare-dese boudar sice c / /, hece r ' Full reflecio wih phase chage ad o wave i secod srig 74

75 Power flow a a boudar We have ad ' r '' P ad earlier we showed ha ' R r 4 '' R 4 R R r So raios of refleced o icide power, R r, ad rasmied o icide power, R, are give b ad so, as epeced 75

76 Reflecio from a mass a he boudar Cosider siuaio where a fiie poi mass, M, is fied a he boudar bewee wo semi-ifiie srigs of desi ρ ad ρ M ρ ρ =0 Solve as before, bu i his case oe of he boudar codiios chages Sum of forces a boudar ac o mass ad geeraes rasverse acceleraio 0, 0, M 0, M 0, Oher boudar codiio ssem coiuous, so 0, 0, uchaged 76

77 Reflecio from a mass a he boudar M ρ ρ =0 Here, wih d derivaives ivolved, i s coveie o use epoeial oaio, Re ep[ i ] 'ep[ i ], Re ''ep[ i ] Ssem coiuous ' '', as before New codiio ii' i'' M ' M'' i ' i M / '' 77

78 Reflecio from a mass a he boudar ' '' We have ad ' ' / ' M i i ep ' i R M i M i r ep '' i M i From hese ca show Here θ φ is phase shif of refleced rasmied wave w.r.. icide wave. cos cos, ] 'ep[ ] ep[ Re, R i i cos, ] ''ep[ Re, i Here R ad are real umbers 78

79 Reflecio from a mass a he boudar ep ' i R M i M i r ep '' i M i / 4 4 M M R M M a a / 4 4 M M a For compleeess we have: ad so oce more oe, i.e. eerg coserved R r 79

80 Reflecio from a mass a he boudar Cosider special case where secod srig has zero desi: ρ =0. he =0 ad i as if we are jus ermiaig he firs srig wih he mass. R / 4 M 4 M oal reflecio a a M M a M 0 if M is small π if M is large 80

81 Impedace Familiar wih cocep of elecrical impedace from circui heor = a measure of opposiio o a ime varig elecric curre V Z I differe compoes have differe impedaces, some frequec depede Z R R Z C Z L i / C il resisor capacior iducor For srucures alog which elecromageic waves propagae, i.e. rasmissio lies, eve free space, al of characerisic impedace Bu cocep of impedace ad characerisic impedace ca be used i oher wave-carrig ssems. Here is a admiedl wooll defiiio: Impedace is raio of push variable i.e. volage or pressure o a flow variable i.e. curre or paricle veloci 8

82 Impedace & waves o srig he characerisic impedace Z is defied as he applied drivig force acig i he -direcio divided b he veloci of he srig i he -direcio For rasverse waves o a srig: v F Z so wih si, c Z lso, ca ae reflecio ad rasmissio coefficies from mass a a boudar problem ad wrie hese i erms of impedaces ' M M Z Z Z Z Z Z M i M i r '' Z M Z Z Z M i Here we have he characerisic impedaces which are resisive, ad a impedace for he mass iself of which is iducive /,, Z M i Z M 8

83 Oher impora eamples rasmissio lies Logiudial elasic waves - Oscillaios i a solid bar - cousic waves i gas 83

84 Lossless rasmissio lies Cosider a ssem made of iducors ad capaciors, ad he le i become coiuous so ha we ow spea of iducace / ui legh L & capaciace / ui legh C e.g. coaial cable. Le i have zero resisace lossless. V V V I I I Self-iducace of δ = L δ volage chage I V L' V I L' Capaciace of δ = C δ volage chage V Q / C' V I C' 84

85 Lossless rasmissio lies V V V I I I We have show V I L' V C' I hese are he elegraph equaios ad I I L' C' ad V V L' C' Wave equaio! So waves of form e.g. V V0 si I I si 0 ca ravel dow lie wih c / C' L'

86 Characerisic impedace of rasmissio lie V V V I I I We have V I L' ad V V0 si I I si 0 V0 cos L' I0 cos Characerisic impedace Z V I 0 0 L' C' L' L' L' C' +ve for forward ravellig wave 86

87 Reflecio a a ermiaed lie Cosider how wave reflecs for rasmissio lie of characerisic impedace Z 0 ermiaed a =0 b a impedace of Z Z 0 icide, V 0 = refleced, V 0 = =0 Z We have: V, ep i[ ] 'ep i[ ] Z0I, ep i[ ] 'ep i[ ] Now a =0 he raio V/I mus = he ermiaig impedace! Hece, Z Z ' ' 0 so reflecio coefficie r ' Z 0 he r -: full reflecio wih phase shif Z Z Z Z Z =Z 0 he r 0 : o reflecio mached impedace; all power rasmied o ermiaig load Z he r : full reflecio 0 0 V 0, I0, 87 Z

88 Logiudial waves i a solid bar Cosider a solid bar, iiiall i equilibrium, i which a disurbace perurbs he posiio ad hicess of a slice of maerial uder equilibrium uder disurbace from wave F F F direcio of wave We deoe b F he magiude of he ew sress force srechig he maerial ad δf he ecess force acceleraig he segme o he righ 88

89 Logiudial waves i a solid bar Force per ui area o slice is give b Hooe s law ad Youg s modulus of he maerial, Y F Y ifiiel hi slice F Y So ecess force is give b F Y Mass of slice wih desi ρ is give b ad acceleraio is Newo II gives Y Y So veloci of waves is v Y Seel has Y= 0 Nm - v 5ms ad ρ= 8000 g m -3 89

90 cousic waves Soud waves are logiudial waves associaed wih compressio of medium. uder equilibrium p p uder disurbace from soud wave p p p p p direcio of soud wave Cosider a slice of gas, iiiall a equilibrium, i a ube of cross-secioal area. Slice is bewee ad +δ. disurbace moves he slice o +ψ ad chages is widh o δ+δψ. Pressure has chaged from p o p+ψ p o LHS of slice, ad o p+ψ p +δψ p o RHS. 90

91 cousic waves Slice has had is volume chaged b a fracioal amou.δψ/.δ, ad his happes as a resul of a pressure chage ψ p. he relaioship is deermied b he elasici of he gas, he bul modulus κ / / p ifiiel hi slice p From his we see Mass of slice p Force o slice i -dir cceleraio of slice p which will be useful i geig wave eq. hese relaios ad Newo II ield: 9

92 cousic waves We have obaied resul, a wave equaio describig moio of a slice of gas a posiio ψ. Oe migh worr ha a slice of gas is a raher iagible eperimeal observable. Isead oe ca phrase problem i erms of he pressure variaios, ψ p, which are cerail measurable. p Sice he ψ p also saisfies wave equaio, i.e. p p Z si, K v K K Z / v Eiher wa, phase veloci of waves is Characerisic impedace ca be defied b so for forward-ravellig wave 9

93 cousic waves speed of soud We have show ha v /, so we ca calculae v if we ow κ ad ρ. o calculae κ i is coveie o use his form of defiiio: p V V We also eed o specif wha else happes o ssem as p chages. Isohermal compressio No emperaure chages jusifiable if he pressure chages are slow eough o allow ube o echage hea freel wih surroudigs. P R R Ideal gas PV R so p v p / V V V diabaic compressio Pressure chages occur so rapidl ha hea cao be echaged from dese o less dese regios. Good approimaio o reali. diabaic chages i a ideal gas pv cosa where C p / C i.e. he raio of specific heas a cosa p ad cosa V V P V p V so p v p / R / M i.e. veloci idepede of pressure for ideal gas pical value for air a room emperaure ~350 ms - 93