Chapter 14: Wave Motion

Transcription

1 Chaper 14: Wave Moion Tpes of mechanical waves Mechanical waves are disurbances ha ravel hrough some maerial or subsance called medium for he waves. ravel hrough he medium b displacing paricles in he medium ravel in he perpendicular o or along he movemen of he paricles or in a combinaion of boh ransverse waves: waves in a sring ec. longiudinal waves: sound waves ec. waves in waer ec.

2 Tpes of mechanical waves (con d) Longiudinal and ransverse waves sound wave longiudinal wave C compression R rarefacion air compressed air rarefied

3 Tpes of mechanical waves (con d) Longiudinal-ransverse waves

4 0 Tpes of mechanical waves (con d) Periodic waves When paricles of he medium in a wave undergo periodic moion as he wave propagaes, he wave is called periodic. λ wavelengh 0 A ampliude T/4 T period

5 Mahemaical descripion of a wave Wave funcion The wave funcion describes he displacemen of paricles in a wave as a funcion of ime and heir posiions: (, ) ; is displacemen a, A sinusoidal wave is described b he wave funcion: (, ) angular frequenc ω π f f λ v wavelengh Acos[ ω ( Acos[ ω ( Acosπ f / v)] / v ( / v )] Acosπ ( / λ ) / T ) sinusoidal wave moving in + direcion veloci of wave, NOT of paricles of he medium period f 1/T (, ) Acos[ ω( + v / )] sinusoidal wave moving in - direcion v->-v phase veloci

6 Mahemaical descripion of a wave (con d) Wave funcion (con d) (, ) Acosπ ( / λ / T λ wavelengh ) ( + λ, ) (, + T ) 0 0 T/4 T period

7 Mahemaical descripion of a wave (con d) Wave number and phase veloci wave number: k π / λ (, ) Acos( k ω) phase The speed of wave is he speed wih which we have o move along a poin of a given phase. So for a fied phase, k ω cons. d / d ω / k v phase veloci (, ) Acos( k ω) Acos[ k( v)]

8 Mahemaical descripion of a wave (con d) Paricle veloci and acceleraion in a sinusoidal wave ) cos( ), ( k A ω ), ( ) cos( ) /, ( ), ( ) sin( ) /, ( ), ( k A a k A v ω ω ω ω ω acceleraion u in ebook veloci Also ), ( ) cos( ) /, ( k k A k ω ) /, ( ) /, ( ) / ( ) /, ( v k ω wave equaion

9 Mahemaical descripion of a wave (con d) General soluion o he wave equaion ), ( ), ( ), ( v k ω wave equaion ) ( ), ( v f ± ) cos( k ω Soluions: such as The mos general form of he soluion: ) ( ) ( ), ( v g v f + +

10 Speed of a ransverse wave Wave speed on a sring F 1 F F 1 F F 1 F F F + Newon s nd law Consider a small segmen of sring whose lengh in he equilibrium posiion is. The mass of he segmen is m µ. The componen of he force (ension) a boh ends have equal in magniude and opposie in direcion because his is a ransverse wave. F1 / F ( / ), F / F ( / ) + The oal componen of he forces is: F F1 + F F[( / ) + µ ( / ) ( / ) ] mass acceleraion

11 Speed of a ransverse wave (con d) Wave speed on a sring (con d) F F F F 1 F The oal componen of he forces is: ) / ( ] ) / ( ) / [( 1 F F F F + + µ ) / )( / ( ]/ ) / ( ) / [( F + µ 0 ) / )( / ( / F µ F 1 wave eq. ) ) /( ( / ineria force resoring F v µ

12 F 1 Energ in wave moion Toal energ of a shor sring segmen of mass F a F F F 1 F ω vk, v F / µ dm µd A poin a, he force does work on he sring segmen righ of poin a. F 1 Power is he rae of work done : P ma P(, ) F1 0 (, )( (, ) / ) F( (, ) / )( (, ) / ) (, ) ( ( / ) / ) Acos( k kasin( k ωasin( k P(, ) de / d FkωA µ vω A µ Fω A sin sin sin ( k ω) ω) ( k ω) work done ω) ω) ( k ω)

13 Energ in wave moion (con d) Maimum power of a sinusoidal wave on a sring: P ma µ Fω A Average power of a sinusoidal wave on a sring The average of sin ( k ω) over a period: 1 π 0 π sin θ dθ 1 The average power: P ave ( 1/ ) µ Fω A

14 Wave inensi Wave inensi for a hree dimensional wave from a poin source: I P W/m 4πr in unis of power/uni area r 1 4π r πr I 1 I1 4 r I I 1 r r 1

15 Wave inerference, boundar condiion, and superposiion The principle of superposiion When wo waves overlap, he acual displacemen of an poin a an ime is obained b adding he displacemen he poin would have if onl he firs wave were presen and he displacemen i would have if onl he second wave were presen: (, ) (, ) + (, ) 1

16 Wave inerference, boundar condiion, and superposiion (con d) Inerference Consrucive inerference (posiive-posiive or negaive-negaive) Desrucive inerference (posiive-negaive)

17 Wave inerference, boundar condiion, and superposiion (con d) Reflecion inciden wave refleced wave Free end (, ) Acos( k ω ) + Bcos( k+ ω) For < B B ( (, ) / ) 0 B + A A B B Verical componen of he force a he boundar is zero.

18 Wave inerference, boundar condiion, and superposiion (con d) Reflecion (con d) Fied end (, ) Acos( k ω ) + Bcos( k+ ω) For < B A B (, ) 0 B B A Displacemen a he boundar is zero.

19 Wave inerference, boundar condiion, and superposiion (con d) Reflecion (con d) A high/low densi

20 Wave inerference, boundar condiion, and superposiion (con d) Reflecion (con d) A low/high densi

21 Sanding waves on a sring Superposiion of wo waves moving in he same direcion Superposiion of wo waves moving in he opposie direcion

22 Sanding waves on a sring (con d) Superposiion of wo waves moving in he opposie direcion creaes a sanding wave when wo waves have he same speed and wavelengh. inciden refleced (, ) 1(, ) + (, ) Acos( k ω) Acos( k ω) A(sink)(sinω ) Nnode, ANaninode sin k 0 when k nπ or ( n nπ / k nλ / 0,1,,..)

23 Normal modes of a sring There are infinie numbers of modes of sanding waves fundamenal λ 1 / L λ n ( n 1,,3,...) firs overone λ λ n L / n second overone 3 3 λ / f n n v L f 1 1 L F µ λ 4 hird overone fied end L fied end

24 Sound Sound waves Sound is a longiudinal wave in a medium The simples sound waves are sinusoidal waves which have definie frequenc, ampliude and wavelengh. The audible range of frequenc is beween 0 and 0,000 Hz.

25 Sound waves (con d) Sound wave (sinusoidal wave) undisurbed cl. of air (, ) ( +, ) 1 S disurbed cl. of air Sinusoidal sound wave funcion: (, ) Acos( k ω) Change of volume: V S( ) 1 Pressure: S[ ( +, ) (, )] dv / V (, ) / ( QV Sd) pressure bulk modulus B p(, ) /( dv / V ) + p(, ) B( (, )/ ) BkAsin( k ω)

26 Pressure ampliude and ear Pressure ampliude for a sinusoidal sound wave Pressure: p(, ) BkAsin( k ω) Ear Pressure ampliude: p ma BkA

27 Percepion of sound waves Fourier s heorem and frequenc specrum Fourier s heorem: An periodic funcion of period T can be wrien as ( ) [ A sin(πf ) + B cos(πf where n n n fundamenal freq. 1 1/ T, fn nf1 ( n f Implicaion of Fourier s heorem: n 1,,3,...) n )]

28 Percepion of sound waves Timbre or one color or one quali Frequenc specrum noise music piano piano

29 Speed of sound waves (ref. onl) The speed of sound waves in a fluid in a pipe movable pison pa pa longiudinal momenum carried b he fluid in moion fluid in original volume of he fluid in equilibrium moion veloci of wave (ρva)v Av veloci of fluid v v ( p + p) A v v fluid in moion v v pa change in volume of he fluid in moion bulk modulus B: -pressure change/frac. vol. change change in pressure in he fluid in moion fluid a res boundar moves a speed of wave Av p ( Av ) /( Av) v p B v

30 Speed of sound waves (ref. onl) (con d) The speed of sound waves in a fluid in a pipe (con d) longiudinal impulse change in momenum pa B speed of a longiudinal wave in a fluid v v A ρvav v The speed of sound waves in a solid bar/rod B ρ Y v, Y ρ Young s modulus

31 Speed of sound waves (con d) The speed of sound waves in gases bulk modulus of a gas In ebook speed of a longiudinal wave in a fluid v R T M B RT M γp γ p 0 0 γp ρ γ 0 raio of hea capaciies equilibrium pressure of gas gas consan J/(mol K) emperaure in Kelvin molar mass - P in ebook (background pressure). - ρ densi

32 Decibel scale Sound level (Decibel scale) As he sensiivi of he ear covers a broad range of inensiies, i is bes o use logarihmic scale: Definiion of sound inensi: ( uni decibel or db) I β (10 db)log, I I Sound inensi in db Inensi (W/m ) Miliar je plane a 30 m Threshold of pain 10 1 Whisper Hearing hres. (100Hz) W/m

33 Sanding sound waves Sound wave in a pipe wih wo open ends

34 Sanding sound waves Sanding sound wave in a pipe wih wo open ends

35 Sanding sound waves Sound wave in a pipe wih one closed and one open end

36 Sanding sound waves Sanding wave in a pipe wih wo closed ends Displacemen

37 Normal modes Normal modes in a pipe wih wo open ends nd normal mode L L n λ n or n ( n λ n 1,,3,...) f n v n ( n L 1,,3,...)

38 Normal modes Normal modes in a pipe wih an open and a closed end (sopped pipe) L 4L n λ n or n ( n 4 λ n 1,3,5,...) f n v n ( n 4L 1,3,5,...)

39 Resonance Resonance When we appl a periodicall varing force o a ssem ha can oscillae, he ssem is forced o oscillae wih a frequenc equal o he frequenc of he applied force (driving frequenc): forced oscillaion. When he applied frequenc is close o a characerisic frequenc of he ssem, a phenomenon called resonance occurs. Resonance also occurs when a periodicall varing force is applied o a ssem wih normal modes. When he frequenc of he applied force is close o one of normal modes of he ssem, resonance occurs.

40 Inerference of waves Two sound waves inerfere each oher consrucive desrucive d1 d d1 d nλ ( n n 0,1,, / ) λ ( consrucive) ( desrucive)

41 Beas Two inerfering sound waves can make bea Two waves wih differen frequenc creae a bea because of inerference beween hem. The bea frequenc is he difference of he wo frequencies.

42 Beas (con d) Two inerfering sound waves can make bea (con d) f. Suppose he wo waves have frequencies a and For simplici, consider wo sinusoidal waves of equal inensi: a ( ) Asin πf Then he resuling combined wave will be: a ; b ( ) Asin πf 1 1 ( ) + b( ) Asin[ (π )( fa fb) ]cos[ (π )( fa fb) ] 1 1 ( Qsin a sin b sin ( a b)cos ( a + b)) a + As human ears does no disinguish negaive and posiive ampliude, he hear wo ma. or min. inensi per ccle, so (1/) f a -f b f a -f b is he bea frequenc f bea. f b b

43 Doppler effec Moving lisener Source a res Lisener moving righ Source a res Lisener moving lef

44 Doppler effec (con d) Moving lisener (con d) The wavelengh of he sound wave does no change wheher he lisener is moving or no. The ime ha wo subsequen wave cress pass he lisener changes when he lisener is moving, which effecivel changes he veloci of sound. freq. lisener hears freq. source generaes veloci of sound a source veloci of lisener f L f s v vl f L v ± v λ L v v ± / v f - for a lisener moving awa from + for a lisener moving owards he source. L s

45 Moving source Doppler effec (con d) When he source moves

46 Doppler effec (con d) Moving source (con d) The wave veloci relaive o he wave medium does no change even when he source is moving. The wavelengh, however, changes when he source is moving. This is because, when he source generaes he ne cres, he he disance beween he previous and ne cres i.e. he wavelengh changed b he speed of he source. The source a res When he source is moving λ v f s s λ v vs v v ± ± f f f s s + for a receding source - for a approaching source s s

47 Doppler effec (con d) Moving source and lisener f L v ± v λ L v v ± ± v v L s f s - for a lisener moving awa from + for a lisener moving owards he source. The signs of v L and v S are measured in he direcion from he lisener L o he source S. Effec of change of source speed + for a receding source - for a approaching source v v s v < vs

48 Eample 1 Doppler effec (con d) A police siren emis a sinusoidal wave wih frequenc f s 300 Hz. The speed of sound is 340 m/s. a) Find he wavelengh of he waves if he siren is a res in he air, b) if he siren is moving a 30 m/s, find he wavelenghs of he waves ahead of and behind he source. a) λ s v / f 340 m/s /300 Hz 1.13 m. b) In fron of he siren: λ ( v v s ) / fs (340 m/s - 30 m/s)/300 Hz 1.03 m Behind he siren: λ ( v + v s ) / f (340 m/s + 30 m/s)/300 Hz 1.3 m s

49 Eample Doppler effec (con d) If a lisener l is a res and he siren in Eample 1 is moving awa from L a 30 m/s, wha frequenc does he lisener hear? v 340 m/s f L fs (300 Hz) 76 Hz. v + v 340 m/s + 30 m/s Eample 3 s If he siren is a res and he lisener is moving oward he lef a 30 m/s, wha frequenc does he lisener hear? f L v v v L fs 340 m/s -30 m/s) 340 m/s (300 Hz) 74 Hz.

50 Eample 4 Doppler effec (con d) If he siren is moving awa from he lisener wih a speed of 45 m/s relaive o he air and he lisener is moving oward he siren wih a speed of 15 m/s relaive o he air, wha frequenc does he lisener hear? v + vl 340 m/s + 15 m/s f L fs (300 Hz) 77 Hz. v + v 340 m/s + 45 m/s Eample 5 s The police car wih is 300-MHz siren is moving oward a warehouse a 30 m/s, inending o crash hrough he door. Wha frequenc does he driver of he police car hear refleced from he warehouse? Freq. reaching he warehouse Freq. heard b he driver f f W L v v v fs s v + v v L fw 340 m/s 340 m/s 30 m/s (300 Hz) 39 Hz. 340 m/s + 30 m/s (39 Hz) 358 Hz. 340 m/s

51 Problem 1 Eercises A ransverse wave on a rope is given b: (, ) (0.750cm)cosπ[(0.400cm 1 ) + (50 s (a) Find he ampliude, period, frequenc, wavelengh, and speed of propagaion. (b) Skech he shape of he rope a he following values of : s, and s. (c) Is he wave raveling in he + or direcion? (d) The mass per uni lengh of he rope is kg/m. Find he ension. (e) Find he average power of his wave. Soluion (, ) Acos π ( / λ + / T ) (a) A0.75 cm, λ/ cm, f15 Hz, T1/f s and vλf6.5 m/s. (b) Homework (c) To sa wih a wave fron as increases, decreases. Therefore he wave is moving in direcion. (d) v ( F / µ ), he ension is F µ v (0.050 kg / m)(6.5m / s) 19.6 N. (e) P av (1/ ) µ Fω A 54. W. 1 ) ]

52 Eercises Problem A riangular wave pulse on a au sring ravels in he posiive + direcion wih speed v. The ension in he sring is F and he linear mass densi of he sring is µ. A 0 he shape of he pulse I given b (,0) 0 h( L + ) / L h( L ) / L 0 for < L for L < < 0 for 0 < < L for > L (a) Draw he pulse a 0. (b) Deermine he wave funcion (,) a all imes. (c) Find he insananeous power in he wave. Show ha he power is zero ecep for L < (-v) < L and ha in his inerval he power is consan. Find he value of his consan. Soluion (a) -L h L

53 Eercises Problem (con d) Soluion (b) The wave moves in he + direcion wih speed v, so in he eperession for (,0) replace wih v: 0 for < L (, ) h( L + v) / L h( L + v) / L for L < < 0 for 0 < < L 0 for > L (c) F(0)0 0 for < L P (, ) F F( h / L)( hv / L) F( h / L)( hv / L) Fv( h / L) Fv( h / L) for L < < 0 for 0 < < L F(0)(0) 0 for > L Thus he insananeous power is zero ecep for L < (-v) < L where I has he consan value Fv(h/L).

54 Problem 3 Eercises The sound from a rumpe radiaes uniforml in all direcions in air. A a disance of 5.00 m from he rumpe he sound inensi level is 5.0 db. A wha disance is he sound inensi level 30.0 db? Soluion The disance is proporional o he reciprocal of he square roo of he inensi and hence o 10 raised o half of he sound inensi levels divided b 10: I β /10 P / 4πd, β 10 log( I / I0) I I010, I d I β /10 /10 ( /10 /10)/ 1 1 β 10 β1 β / I 10 ( d / d1) d1 d (5.0 (5.00m) ) / 6.9 m.

55 Problem 4 Eercises An organ pipe has wo successive harmonics wih frequencies 1,37 and 1,764 Hz. (a) Is his an open or sopped pipe? (b) Wha wo harmonics are hese? (c) Wha is he lengh of he pipe? Soluion (a) For an open pipe, he difference beween successive frequencies is he fundamenal, in his case 39 Hz, and all frequencies are ineger muliples of his frequenc. If his is no he case, he pipe canno be an open pipe. For a sopped pipe, he difference beween he successive frequencies is wice he fundamenal, and each frequenc is an odd ineger muliple of he fundamenal. In his case, f Hz, and 137 Hz 7f 1, 1764 Hz 9f 1. So his is a sopped pipe. (b) n7 for 1,37 Hz, n9 for 1,764 Hz. (c) f v /(4 ), so L v /( 4 f1) (344m / s) /(784 Hz) m. 1 L

56 Problem 5 Eercises Two idenical loudspeakers are locaed a poins A and B,.00 m apar. The loudspeakers are driven b he same amplifier and produce sound waves wih a frequenc of 784 Hz. Take he speed of sound in air o be 344 m/s. A small microphone is moved ou from Poin B along a line perpendicular o he line connecing A and B. (a) A wha disances from B will here be desrucive inerference? (b) A wha disances from B will here be consrucive inerference? (c) If he frequenc is made low enough, here will be no posiions along he line BC a which desrucive inerference occurs. How low mus he frequenc be for his o be he case? A B.00 m C

57 Problem 5 Eercises Soluion (a) If he separaion of he speakers is denoed b h, he condiion for desrucive inerference is + h βλ, where β is an odd muliple of one-half. Adding o boh sides, squaring, canceling he erm from boh sides and solving for gives: [ h /(βλ) βλ / ]. Using λ v / f and h from he given daa ields: 9.01m for β 1/,.71m for β 3/, 1.7m for β 5/, 0.53m for β 7 /, 0.06m for β 9 /. (b) Repeaing he above argumen for inegral values for β, consrucive inerference occurs a 4.34 m, 1.84 m, 0.86 m, 0.6 m. (c) If h λ /, here will be desrucive inerference a speaker B. If h < λ /, he pah difference can never be as large as λ /. The minimum frequenc is hen v/(h)(344 m/s)/(4.0 m)86 Hz.

58 Problem 6 Eercises A.00 MHz sound wave ravels hrough a pregnan woman s abdomen and is refleced from feal hear wall of her unborn bab. The hear wall is moving oward he sound receiver as he hear beas. The refleced sound is hen mied wih he ransmied sound, and 5 beas per second are deeced. The speed of sound in bod issue is 1,500 m/s. Calculae he speed of he feal hear wall a he insance his measuremen is made. Soluion Le f 0.00 MHz be he frequenc of he generaed wave. The frequenc wih which he hear wall receives his wave is f H [(v+v H )/v]f 0, and his is also he frequenc wih which he hear wall re-emis he wave. The deeced frequenc of his refleced wave is f [v/(v-v H )]f H, wih he minus sign indicaing ha he hear wall, acing now as a source of waves, is moving oward he receiver. Now combining f [(v+v H )/(v-v H )]f 0, and he bea frequenc is: f f f [( v + v ) /( v v )] f v f /( v v ). Solving for v H, bea ' 0 H H 0 H 0 H v H v[ f bea /( 6 f0 + fbea )] (1500m / s){85hz /[( Hz) + 85Hz)} m / s.