Outline. Objectives. Objectives. Objectives Progressive waves. Wave motion. Wave motion

Transcription

1 Chaper. Liew Sau Poh Wave moion Ouline. Progressive Waves. Wave Inensi.3 Principle of Superposiion.4 Sanding Waves.5 Elecromagneic Waves Objecives a) inerpre and use he progressive wave equaion, = a sin ( kx) or = a cos ( kx) b) skech and inerpre he displacemen-ime graph and he displacemen-disance graph c) use he formula = x/ d) derive and use he relaionship v = f e) define inensi and use he relaionship I A Objecives g) describe he variaion of inensi wih disance of a poin source in space h) sae he principle of superposiion i) use he principle of superposiion o explain he formaion of sanding waves j) derive and inerpre he sanding wave equaion k) disinguish beween progressive and sanding waves 3 4 Objecives l) sae ha elecromagneic waves are made up of elecrical vibraions E = E sin ( - kx) and magneic vibraions B = B sin ( - kx) m) sae he characerisics of elecromagneic waves n) compare elecromagneic waves wih mechanical waves o) sae he formula c = ( ) / and explain is significance p) sae he orders of he magniude of wavelenghs and frequencies for differen pes of elecromagneic waves.. Progressive waves 5 6 Wave moion A wave ravels along is medium, bu he individual paricles jus move up and down. 7 Wave moion All pes of raveling waves ranspor energ Sud of a single wave pulse shows ha i is begun wih a vibraion and ransmied hrough inernal forces in he medium. Coninuous waves sar wih vibraions oo. If he vibraion is SHM, hen he wave will be sinusoidal. 8

2 Wave characerisics Ampliude, A Wavelengh, Frequenc f and period Wave veloci, v = f Wave characerisics beween an wo idenical poins on adjacen waves Period : he ime required for wo idenical poins of adjacen waves o pass b a poin. 9 Wave characerisics Frequenc f: he inverse of he period. Ampliude A: he maximum displacemen Wave characerisics he moion of paricles in a wave can eiher be perpendicular o he wave direcion (ransverse) or parallel o i (longiudinal). Wave veloci, v = f Progressive waves is defined as he one in which he wave profile propagaes. he progressive waves have a definie speed called he speed of propagaion or wave speed. he direcion of he wave speed is alwas in he same direcion of he wave propagaion. Progressive waves here are wo pes of progressive wave, a. ransverse progressive waves b. Longiudinal progressive waves. 3 4 Progressive waves If he ension in he sring is, and is mass per uni lengh he wave speed is : Progressive waves he wave speed is : =. kg/m mg V sring V sring 5 6

3 Progressive waves raveling in he posiive x-direcion V f f f k k k(wave number) D( x, ) Asin( kx o) D( x, ) Asin( x o) Longiudinal and ransverse Waves 7 Longiudinal and ransverse Waves 8 Sound waves are longiudinal waves: he moion of paricles in a wave can eiher be perpendicular o he wave direcion (ransverse) or parallel o i (longiudinal). Longiudinal and ransverse Waves Earhquakes produce boh longiudinal and ransverse waves. Boh pes can ravel hrough solid maerial, bu onl longiudinal waves can propagae hrough a fluid in he ransverse direcion, a fluid has no resoring force. Surface waves are waves ha ravel along he boundar beween wo media. 9 Reflecion and ransmission of Waves A wave reaching he end of is medium, bu where he medium is sill free o move, will be refleced (b), and is reflecion will be uprigh. A wave hiing an obsacle will be refleced (a), and is reflecion will be invered. Reflecion and ransmission of Waves A wave encounering a denser medium will be parl refleced and parl ransmied; if he wave speed is less in he denser medium, he wavelengh will be shorer. v F M / L F Reflecion and ransmission of Waves wo- or hreedimensional waves can be represened b wave frons, which are curves of surfaces where all he waves have he same phase. Lines perpendicular o he wave frons are called ras; he poin in he direcion of propagaion of he wave. 3 4

4 Reflecion and ransmission of Waves he law of reflecion: he angle of incidence equals he angle of reflecion.. Wave Inensi 5 6 Wave Power and Inensi. Wave Inensi he power carried b a wave is given b I P Area P P 4 r aa v where a is a consan ha depends on he kind of wave. he inensi of a wave is he power per uni area a he wavefron. spherical wavefron 7 Definiion: he amoun of energ ransferred b waves ha passes hrough uni area per second of an plane surface normal o he direcion of propagaion of he waves. Uni: Js - m -, or W m - Area (sphere), A = 4 r Direcion of wave 8. Wave Inensi A poin source of disurbance generaes waves ha propagae ouwards in hree dimensions in a homogenous medium. I can radiae energ of P joules per second (P was). Since he source of disurbance is a poin source, he waves radiaed from he source ravel ouwards in all possible direcions, resuling in wavefrons being spherical in shape, wih he source as he cenre.. Wave Inensi Wavefron: a surface on which all paricles vibrae in phase wih one anoher. Spherical wavefron Direcion of he wave propagaion 9 3. Wave Inensi A a paricular insance, le he radius of one spherical wavefron be r. he surface area A of he wavefron is equal o he surface of a sphere given b A = 4 r he amoun of energ crossing he enire spherical surface of his wavefron ever second mus be P joules (he power radiaed b he source).. Wave Inensi herefore, he amoun of energ crossing uni area is P/ 4 r. B definiion, his amoun of energ is equal o he inensi I a disance r from he source. hus, he inensi, I = P/A = P/4 r. Hence, I P I /r 3 3

5 Sound and he Human Ear he human ear is an amazing insrumen. I can respond o sound inensiies ha differ b a facor of a rillion! he ear can do his because of is non-linear response o inensi. I urns ou o be beer o describe sound inensiies in erms of decibels (db): where I is in W/m and I = - W/m. An increase of log I I db corresponds o an inensi increase of a facor or. Pain hreshold ~ 3 db 33.3 Principle of Superposiion 33 Wave Wave Inensiies A whisper a m - V signal, 5km from 5 kw ransmier Sound, 4m from a loud rock band Sound, 5 m from a je aircraf 5 Inensi, W/m.6 x Microwaves, 34 inside a microwave oven 6.3 Superposiion he superposiion principle sas ha when wo waves pass hrough he same poin, he displacemen is he arihmeic sum of he individual displacemens Superposiion 35.3 Superposiion 36.3 Superposiion (a) exhibis desrucive inerference (b) exhibis consrucive inerference Superposiion hese figures show he sum of wo waves. In (a) he add consrucivel; in (b) he add desrucivel; and in (c) he add pariall desrucivel

6 In Phase 4 8 o ou of phase 4.4 Sanding Waves Beween in phase and 8 o ou of phase Sanding /Saionar waves he frequencies of he sanding waves on a paricular sring are called resonan frequencies. he are also referred o as he fundamenal and harmonics..4 Sanding /Saionar waves he wavelenghs and frequencies of sanding waves are: 45.4 Sanding /Saionar waves Sanding waves occur when boh ends of a sring are fixed. In ha case, onl waves which are moionless a he ends of he sring can persis. here are nodes, where he ampliude is alwas zero, and aninodes, where he ampliude varies from zero o he maximum value..4 Sanding /Saionar waves he characerisics of Saionar wave Nodes and aninodes are appear a paricular ime ha is deermine b he equaion of he saionar wave. 46 N A N A N A N 4 Node (N) is defined as a poin a which he displacemen is zero where he desrucive inerference occurred. Aninode (A) is defined as a poin a which he displacemen is maximum where he consrucive inerference occurred. he disance beween adjacen nodes or aninodes is? he disance beween a node and an adjacen aninode is? = (he disance beween adjacen nodes or aninodes) 47 48

7 .4 Sanding /Saionar waves he paern of he saionar wave is fixed hence he ampliude of each poins along he medium are differen. hus he nodes and aninodes appear a paricular disance and deermine b he equaion of he saionar wave..4 Sanding /Saionar waves he Equaion of Saionar wave B considering wave funcions(equaions) for wo progressive sinusoidal waves having he same ampliude, frequenc and wavelengh bu ravelling in opposie direcions in he same medium as shown below. a s i n ( k x ) a s i n ( k x ) where represens a wave ravelling in he +x direcion and represens one ravelling in he x direcion. B appling he principle of superposiion hence.4 Sanding /Saionar waves a s i n a s i n a s i n k x a s i n c o s k x c o s c o s k x k x s i n k x a s i n c o s k x c o s s i n k x he general equaion of saionar wave is given b A c o s k x s i n and A a 49.4 Sanding /Saionar waves Descripion of he equaion of saionar wave A c o s k x deermine he ampliude for an poin along he saionar wave. I is called he ampliude formula. Is value depend on he disance, x 5 where A : he ampliude of he saionar wave a : he ampliude of he progressive wave : he angular frequenc k : he wave number Sanding /Saionar waves Aninodes he poin wih maximum displacemen = A A c o s k x A c o s k x k x c o s k x,,,3,... k x m where m,,,3,... m x and k k m herefore x Aninodes are occur when 3 x,,,, Nodes he poin wih minimum displacemen = A c o s k x c o s k x k x kx kx x herefore x c o s 3 5,,,... n where n n and k k n 4 Nodes are occur when,3,5, 7,... x 3, 4 4 5, 4, s i n deermine he ime for aninodes and nodes will occur in he saionar wave. Aninodes he poin wih maximum displacemen = A herefore A s i n s i n n 4 A s i n 3 5,,,... n where n,3,5, 7,... n and 3 5,,, Aninodes occur when he ime are 55 Nodes he poin wih minimum displacemen = All he poin in he saionar wave a he equilibrium posiion ( = ) herefore A s i n s i n s i n,,,3 m m m,... where and m 3,,,,,,3,4,... Nodes occur when he ime are,... 56

8 .4 Sanding /Saionar waves Displacemen-disance graph for saionar wave A A A N A N A N A N A 4,, x Characerisics of Sanding Waves Nodes and aninodes remain saionar Nodes poins of leas ampliude Aninodes poins of maximum ampliude Resuling from inerference Waves of Equal ampliude Equal wavelengh Pass hru each oher In opposie direcions Ou of phase a nodes (regions of sable desrucive inerference) 58.5 Elecromagneic Waves 59 Inroducion: EM Waves Waves ha can ransmi energ wihou a medium or maerial o ravel hrough (ligh waves, hea waves, an waves in space, ec.) Inroducion: EM Waves Elecromagneic waves are ransverse waves. he consis of boh a changing elecric field and a changing magneic field. he fields are a righ angles o each oher and o he direcion of he wave. hp:// 6 6 Inroducion: EM Waves James Clerk Maxwell Developed he elecromagneic heor of ligh Developed he kineic heor of gases Explained he naure of color vision Explained he naure of 63 Died of cancer 64

9 Elecromagneic Waves In 865, James Clerk Maxwell provided a mahemaical heor ha showed a close relaionship beween all elecric and magneic phenomena exisence of elecromagneic waves ha propagae hrough space Einsein showed hese equaions are in agreemen wih he special heor of relaivi In emp space, q = and I = Maxwell prediced he exisence of elecromagneic waves he elecromagneic waves consis of oscillaing elecric and magneic fields he changing fields induce each oher which mainains he propagaion of he wave A changing elecric field induces a magneic field A changing magneic field induces an elecric field Elecromagneic Waves Elecromagneic waves are formed when an elecric field couples wih a magneic field. he magneic and elecric fields of an elecromagneic wave are perpendicular o each oher and o he direcion of he wave. James Clerk Maxwell and Heinrich Herz sudied how elecromagneic waves are formed and how fas he ravel. Elecromagneic Waves When ou lisen o he radio, wach V, or cook dinner in a microwave oven, ou are using elecromagneic waves. Radio waves, elevision waves, and microwaves are all pes of elecromagneic waves. he differ from each oher in wavelengh. Wavelengh is he disance beween wave cress Elecromagneic Waves Elecromagneic Radiaion: Elecromagneic waves are produced b he moion of elecricall charged paricles. hese waves are also called "elecromagneic radiaion" because he radiae from he elecricall charged paricles. he ravel hrough emp space as well as hrough air and oher subsances. Elecromagneic Waves Scieniss have observed ha elecromagneic radiaion has a dual "personali." Besides acing like waves, i acs like a sream of paricles (called "phoons") ha have no mass. he phoons wih he highes energ correspond o he shores wavelenghs Elecromagneic Waves When maer is heaed, i gives off ligh. For example, urning on an ordinar ligh bulb causes an elecric curren o flow hrough a meal filamen ha heas he filamen and produces ligh. he elecrical energ absorbed b he filamen excies he aoms' elecrons, causing hem o "wiggle". his absorbed energ released from he aoms in he form of ligh. 7 Elecromagneic Waves wrong, i sill provided insigh as o wh elecromagneic radiaion is given off. I is due o acceleraing elecrons, conribuion is ha he radius of he orbi of he elecrons mus be consrained o cerain values, and no jus an value. A such a place he elecron could orbi he nucleus and no give off an radiaion. 7

10 Plane em Waves Assume ha he vecors for he elecric and magneic fields in an em wave have a specific space-ime behavior ha is consisen wih Plane em Waves Assume an em wave ha ravels in he x direcion wih he elecric field in he direcion and he magneic field in he z direcion Plane em Waves, con he x-direcion is he direcion of propagaion Waves in which he elecric and magneic fields are resriced o being parallel o a pair of perpendicular axes are said o be linearl polarized waves Assume ha a an poin in space, he magniudes E and B of he fields depend upon x and onl Ras A ra is a line along which he wave ravels All he ras for he pe of linearl polarized waves ha have been discussed are parallel he collecion of waves is called a plane wave A surface connecing poins of equal phase on all waves, called he wave fron, is a geomeric plane Comparing EM vs Mechanical Waves EM Waves Can propagae hrough vacuum ransverse waves Originae from changing elecric / magneic fields Mechanical Waves Need a medium ransverse or longiudinal Originae from he oscillaion of he paricles of a Elecromagneic Vibraions Elecric vibraion, E = E sin (w kx) Magneic vibraion, B = B sin (w kx) E is perpendicular o B hp://phsicsclub.ne/phsleindex/waves.hml medium EM wave No charges, no currens EdA closed surface Changing magneic field creaes elecric field Changing elecric field creaes magneic field BdA closed surface Edl closed pah Bdl closed pah d d B d d E 79 E B E v B v E E sin( kx ) B B sin( kx ) z v v x E / B v k f f v k 8 v 3. m / s he speed of ligh!! 8

11 EM specrum Energ in EM wave f EM waves ranspor energ Energ densi: B E E / c B u E B c f c speed of ligh (m/s) f frequenc (Hz=/s) wavelengh (m) 8 Poning vecor (energ ranspored b EM wave per uni ime per uni area) S E B Average energ per uni ime per uni area S E rms B rms 8 Average inensi Energ ranspored b waves Displacemen D follows harmonic oscillaion: D D sin( ) Inensi (brighness for ligh) I is proporional o elecric field squared I D I I sin ( ) I Average over ime (one period of oscillaion) I: I I sin ( ) d sin xdx I I sin ( ) d ( cos x) dx I 83 Inensi of oscillaion I (energ per uni area/ per sec) is proporional o ampliude squared D 3D wave (from energ conservaion): D 4pr = D 4pr D /D =r /r Ampliude of he wave is inversel proporional o he disance o he source: D r 84 Properies of EM Waves -like, wih boh E and B saisfing a wave equaion Elecromagneic waves ravel a he speed of ligh o o his comes from he soluion of c Properies of em Waves he componens of he elecric and magneic fields of plane elecromagneic waves are perpendicular o each oher and perpendicular o he direcion of propagaion his can be summarized b saing ha elecromagneic waves are ransverse waves Properies of em Waves he magniudes of he fields in emp space are relaed b he expression c E B his also comes from he soluion of he parial differenials obained from 85 Derivaion of Speed: Some Deails space, he following parial derivaives can be found: E E B B o o and o o x x hese are in he form of a general wave equaion, wih v c o o 86 Elecromagneic waves obe he superposiion principle 87 Subsiuing he values for o and o gives c =.9979 x 8 m/s 88

12 E o B Raio Some Deails he simples soluion o he parial differenial equaions is a sinusoidal wave: E = E max cos (kx ) E E sin( kx B = B max cos (kx ) B B sin( kx he angular wave number is k = is he wavelengh he angular frequenc is = ) ) E o B Raio Deails, con he speed of he elecromagneic wave is? c k aking parial derivaions also gives Emax E c B k B max 89 9 em Wave Represenaion his is a picorial represenaion, a one insan, of a sinusoidal, linearl polarized plane wave moving in he x direcion E and B var Doppler Effec for Ligh Ligh exhibis a Doppler effec Remember, he Doppler effec is an apparen change in frequenc due o he moion of an observer or he source Since here is no medium required for ligh waves, onl he relaive speed, v, beween he source and he observer can be idenified sinusoidall wih x 9 9 Doppler Effec he equaion also depends on he laws of relaivi?? c v c v v is he relaive speed beween he source and he observer c is he speed of ligh ligh seen b he observer source 93 Doppler Effec For galaxies receding from he Earh v is enered as a negaive number wavelengh, acual wavelengh he ligh is shifed oward he red end of he specrum his is wha is observed in he red shif 94 Elecromagneic Waves When normal whie ligh, such as ha from he sun, is passed hrough a prism, he ligh separaes ino a coninuous specrum of colors: Coninuous (whie ligh) specra Bohr knew ha when pure elemens were excied b hea or elecrici, he gave off disinc colors raher han whie ligh. his phenomenon is mos commonl seen in modern-da neon lighs, ubes filled wih gaseous elemens (mos commonl neon). Elecromagneic Waves 95 96

13 Elecromagneic Waves Whie ligh specra: Elecromagneic Specrum he EM specrum is he enire range of wavelenghs (or frequencies) of EM waves, including he visible specrum he Elecromagneic Specrum Elecromagneic radiaion wih wavelenghs beween 4 nm and 7 nm is visible. In order of decreasing wavelengh (increasing frequenc), he colors are red (7 nm), orange, ellow, green, blue, viole (4 nm). ra X-ra UV Infrared wave Visible radio he Specrum of EM Waves Various pes of elecromagneic waves make up he em specrum here is no sharp division beween one kind of em wave and he nex All forms of he various pes of radiaion are produced b he same phenomenon acceleraing charges 4Å 7Å 99 he specrum of elecromagneic waves Various pes of elecromagneic waves, disinguished b frequenc or wavelengh, make up he EM specrum. Radio waves ( 4 m o ~. m ): Radio and elevision communicaion Microwaves (.3 m o -4 m): Radar ssems, microwave ovens Infrared waves ( -3 m o 7-7 m): Produced b ho objecs and molecules he EM Specrum he specrum of elecromagneic waves Visible (7 nm o 4 nm): Differen wavelenghs = differen colors Ulraviole (4-7 m o 6 - m) X-ras (-8 m o - m) Gamma ras ( - m o - 4 m) Emied b radioacive nuclei Noes on he EM Specrum Noe he overlap beween pes of waves Visible ligh is a small porion of he specrum pes are disinguished b frequenc or wavelengh 3 Radio Waves Wavelenghs of more han 4 m o abou. m Used in radio and elevision communicaion ssems Microwaves Wavelenghs from abou.3 m o -4 m Well suied for radar ssems Microwave ovens are an applicaion 4

14 Noes on he EM Specrum, Infrared waves Wavelenghs of abou -3 m o 7 x -7 m Produced b ho objecs and molecules Readil absorbed b mos maerials Visible ligh Par of he specrum deeced b he human ee Mos sensiive a abou 5.5 x -7 m (ellowgreen) Visible Ligh Specific Wavelenghs and Colors Noes on he EM Specrum Gamma ras Wavelenghs of abou - m o -4 m Emied b radioacive nuclei Highl peneraing and cause serious damage when absorbed b living issue Looking a objecs in differen porions of he specrum can produce differen informaion More Abou Visible Ligh Differen wavelenghs correspond o differen colors he range is from red ( ~7 x -7 m) o viole ( ~4 x - 7 m) Noes on he EM Specrum Ulraviole ligh Covers abou 4 x -7 m o 6 x - m Sun is an imporan source of uv ligh Mos uv ligh from he sun is absorbed in he sraosphere b ozone X-ras Wavelenghs of abou -8 m o - m Mos common source is acceleraion of high-energ elecrons sriking a meal arge Used as a diagnosic ool in medicine Wavelenghs and Informaion hese are images of he Crab Nebula he are (clockwise from upper lef) aken wih x-ras visible ligh radio waves infrared waves 6 8 Summar Summar Vibraing objecs are sources of waves, which ma be eiher a pulse or coninuous. Wavelengh: disance beween successive cress. Frequenc: number of cress ha pass a given poin per uni ime. Ampliude: maximum heigh of cres. Wave veloci, v = f

15 Elecromagneic Vibraions c Elecric vibraion, E = E sin (w kx) Magneic vibraion, B = B sin (w kx) E is perpendicular o B c o o hp://phsicsclub.ne/phsleindex/waves.hml C =.998 X 8 ms- = 8.85 X - Fm - (Free space permiivi) = 4 x -7 Hm - (Free space permiivi) Elecromagneic Waves Specrum Elecromagneic waves come in man wavelenghs and frequencies. Each one is useful in differen was. 3 Specrum of Elecromagneic Radiaion Region Wavelengh (Angsroms) Wavelengh (cenimeers) Frequenc (Hz) 4 Energ (ev) Radio > 9 > < 3 x 9 < -5 Microwave x 9-3 x Infrared x -5 3 x x 4 Visible x -5-4 x x x hp://science.hq.nasa.gov/kids/imagers/ems/index.hml 5 Ulraviole 4-4 x x 4-3 x X-Ras x 7-3 x 9 Gamma Ras 3-5 <. < -9 > 3 x 9 > 5 6