Chapter 18 Superposition and Standing Waves

Transcription

1 hpter 8 Superpositio d Stdig Wves 8. Superpositio d Iterereces y '(, y(, + y (, Overlppig wves lgericlly dd to produce resultt wve (or et wve). Overlppig wves do ot i y wy lter the trvel o ech other. Superpositio o Siusoidl Wves y(, ym si( k ω y (, ym si( k ω t + ) (, y + y y si( k ω t + ) cos y' m I two siusoidl wves o the sme mplitude d wvelegth trvel i the sme directio log stretched strig, they iterere to produce resultt siusoidl wve trvelig i tht directio. Phse Diereces d Resultig Itererece ypes Phse dierece mplitude o Resultt Wve ype o Itererece Degrees Rdis Wvelegths y m Fully costructive /. y m Itermedite 8.5 Fully destructive 4 4/.67 y m Itermedite 6. y m Fully costructive y m Itermedite Emple:

2 wo ideticl siusoidl wves, movig i the sme directio log stretched strig, iterere with ech other. dierece etwee them is o. he mplitude y m o ech wve is 9.8 mm, d the phse () Wht is the mplitude y m o the resultt wve due to the itererece o these two wves, d wht type o itererece occurs? () Wht phse dierece, i rdius d wvelegth, will give the resultt wve mplitude o 4.9 mm? y' ym si( k ω + ym si( k ωt + ) ym cossi( k ωt + ) o y' m ym cos 9.8 cos(5 ). 6mm itermedite o y ' m 4.9 ym cos 9.8 cos, 5.66rd.66,. 4 Itererece o Soud Wves: It is ote useul to epress pth dierece i terms o the phse gle etwee the two wves. r r or costructive itererece r or destructive itererece ( + ) Emple: wo ideticl spekers plced. m prt re drive y the sme oscilltor. listeer is origilly t poit O, locted 8. m rom the ceter o the lie coectig the two spekers. he listeer the moves to poit P, which is perpediculr distce.5 m rom O, d she eperieces the irst miimum i soud itesity. Wht is the requecy o the oscilltor? r r r v / 4/.6. (khz)

3 8. Stdig Wves y ym si( k ω ), y ym si( k + ω ) t t ' y + y y si( k)cos( y m ω y m si( k), whe k, re the positio o odes y m si( k) mimum, whe k ( + ), ( + ) re the positio o tiodes 8. Stdig Wves i Strigs Fied t Both Eds oudry coditio: si( k ), si( kl ) kl, L v v L Emple: strig, tied to siusoidl virtor t P d ruig over support t Q, is stretched y lock o mss m. he seprtio L etwee P d Q is. m, the lier desity o the strig is.6 g/m, d the requecy o the virtor is ied t Hz. he mplitude o the motio t P is smll eough or tht poit to e cosidered ode. ode lso eists t Q. () Wht mss m llows the virtor to set up the ourth hrmoic o the strig?

4 mg L. v,. 6, v.6 7 µ.6 4 m , m. 846kg Emple: hgig strig virtio with wter Oe ed o horizotl strig is ttched to virtig lde, d the other ed psses over pulley. sphere o mss. kg hgs o the ed o the strig. he strig is virtig i its secod hrmoic. cotier o wter is rised uder the sphere so tht the sphere is completely sumerged. ter this is doe, the strig virtes i its ith hrmoic. Wht is the rdius o the sphere? mg N, v, µ v v,, L.5 L / 9.6 / 4 ρ 4.6 g R R R. 77m Emple: middle strig o pio hs udmetl requecy o 6 Hz, d the ote hs udmetl requecy o 44 Hz. () lculte the requecy o the et two hrmoics o the strig. () I the strigs or d otes re ssumed to hve the sme mss per uit legth d the sme legth, determie the rtio o tesios i the two strig. (c) I rel pio, the ssumptio we mde i prt () is oly prtil true. he strig desities re equl, ut the strig is 64% s log s the strig. Wht is the rtio o their tesios? () strig: 6Hz, 6 54Hz, 6 786Hz () v L L µ, µ, µ 6 44,. 55 (c) L L.64,

5 8.4 Resoce 8.5 Stdig wves i ir colums Emple: Wid i ulvert sectio o drige culvert. m i legth mkes howlig oise whe the wid lows cross its ope eds. () Determie the requecies o the irst three hrmoics o the culvert i it is cylidricl i shpe d ope t oth eds. ke v4 m/s s the speed o soud i ir. L /( / ) v / L / v / L /( / ) v / () Wht re the three lowest turl requecies o the culvert i it is locked t oe ed? ( / 4) ( / 4) L / L / 8.6 Stdig Wves i Rods d 5

6 Memres 8.7 Bets: Itererece i ime itererece eect tht results rom the superpositio o two wves with slightly dieret requecies y cos( k ω, y cos( k ω y y + y cos( k ω + cos( k ω ) ' t ω + ω ω ω ω ω y' cos( k cos( cos( k ωt )cos( y ω ω cos( t t ω ω the et requecy ) ) 8.8 Nosiusoidl Wve Ptters Fourier s theorem y is wve o the strig stisy y ( ) y( L) y(, ( cos( k) + si( k) ) si( ω y cos ( k) + si( k) kl y si L y, <<L, < or >L Emple: wves:, otherwise, si. si L c e sythesized y hrmoic series o siusoidl 6

7 7 si d d si si si d ] [ si,, otherwise