Chapter 15: Superposition and Interference of Waves

Transcription

1 Chaper 5: Superposiion and Inerference of Waves Real waves are rarely purely sinusoidal (harmonic, bu hey can be represened by superposiions of harmonic waves In his chaper we explore wha happens when harmonic waves combine o form more complicaed waves

2 The sum (or difference of wo soluions of he wave equaion is again a soluion of he wave equaion > superposiion principle Lineariy of he wave equaion b v a v x b x a The soluions o he wave equaion are linear True for small displacemens where he resoring force remains linear ( If a(x, and b(x, are soluions o he wave equaion, hen so is, (, (, ( x b x a x Wha does his mean? v x ( ( b a v x b a Check: a v x a b v x b is rue if and

3 Inerference The superposiion principle allows us o add waves of differen ampliudes, wavelenghs, and frequencies, each of hem moving in differen direcions. > The resul: Inerference Mahemaically: he sums are algebraic sums Examples: (Red curves: algebraic sum of he blue and green curves

4 Simple example for desrucive inerference: ( x, ( x, Imporan rigonomeric relaionships for general cases of inerference: sin( a cos( a b b sin( kx sin( kx ω ω π sin( kx Simple example for consrucive inerference: Example: ( x, ( x, sin( kx sin( kx ω ω ω sin( acos( b cos( asin( b cos( acos( b sin( asin( b ( x, ( x, sin( kx sin( kx sin( kx ω? ω ω ϕ [sin( kx ω sin( kx ω ϕ] a a b

5 > if sin( if kx π ϕ ω ϕ sin( cos( cos( sin( sin( ϕ ω ϕ ω ϕ ω kx kx kx use: sin( cos( sin( cos( ( ] sin( cos( cos( sin( [sin( ] sin( [sin( ϕ ω ω ϕ ϕ ω ϕ ω ω ϕ ω ω kx kx kx kx kx kx kx plug in: Coherence: Phase is sable in ime.

6 Sanding Waves Through Inerference Adding wo opposiely raveling waves o ge a sanding wave lef( x, sin( ω kx x, sin( ω kx righ ( ( x, righ ( x, lef( x, ( x, sin( ω kx sin( ω kx [ ] sin( ωcos( kx sin( kxcos( ω sin( ωcos( kx sin( kxcos( ω sin( ωcos( kx ( x, sin( ωcos( kx noe ha ω and k are same for raveling waves and resuling sanding wave

7 Superposiion of raveling waves

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9 Inerference of inciden and refleced waves To ge node a wall, refleced wave mus have opposie sign as inciden wave (bu same magniude a wall

10 Beas Beas occur when wo waves of nearly he same frequency are superposed. Example: Two combs wih differen ooh spacing Where he wo combs superpose, a new paern emerges (here called Moire paern. A similar hing happens when wo waves superpose.

11 Two sine waves wih nearly same wavelenghs superpose o a wave ha has an envelope ha is again a sine wave bu wih longer wavelengh. > beas > How o calculae he wavelengh of he beas?

12 To simplify analysis, fix posiion x and look a wave as funcion of ime. (Could equivalenly fix ime and look a wave as a funcion of posiion Les define wo new frequencies: (x, sin( ω kx sin( ω fix posiion x δω (x, sin( ω kx sin( ω δω ω ω and Ω ω ( ω ω ω Ω (x, δω δω δω sin Ω sin( Ω cos cos( sin Ω (x, ( x, ( x, Ω ( ω sin( ω sin( ω ω kx kx and sin( ω sin( ω δω ω ω ω δω δω δω sin Ω sin( cos cos( sin Ω Ω δω (x, (x, sin( Ω cos δω ( x, ( x, sin( Ωcos( average frequency: Ω (fas bea frequency: ω bea δω (slow

13 f bea ωbea π ω ω π Noe: since he ear is only sensiive o he ampliude, i canno ell he difference beween cos( δω and cos( δω he perceived beaing frequency is f pulse fbea f f ( f f pulse frequency

14 Demo: Bea frequency ω ω Can hear average frequency And pulse frequency f f Ω π pulse fbea f f (fas Wha happens o he pulse frequency when he wo frequencies ge close o each oher? Wha happens o he pulse frequency when he wo frequencies move apar?

15 Spaial Inerference Phenomena (fix ime by aking snapsho and look a x-dependence

16 In phase: Consrucive inerference n x nλ, ±, ±, Phase difference: n π Ou of phase: Desrucive Inerference x ( n λ n, ±, ±, Phase difference: ( n π

17 LL -L minima when L(n/λ, maxima when Lnλ, n, ±, ±, Recall: λ π cos( x λ cos( x λ λ π cos( x cos( x λ π λ cos( x sin( x 4 4

18 Locaions of maxima and minima: L dsin θ for maxima: n λ L n λ sin θ d for minima: n λ L n λ sinθ d where n, ±, ±, ± 3

19 Demo: Inerference from wo speakers S S Raise your hand when an acousic minimum passes you.

20 Collision of pulses

21 Wha happens here?? Noe ha he cener poin a he do remains moionless. Connecion o reflecion of pulse on fixed end

22 Reflecion from a fixed end Think of his as a real pulse moving o he righ and an imaginary pulse moving o he lef. Before reflecion: only real pulse visible ( imaginary one is somewhere beyond wall Afer reflecion: previously imaginary pulse is real now and moves o lef Wha happens during he reflecion? superposiion of lef and righ moving pulses (such ha rope end remains fixed, see previous slide

23 Reflecion from a verically free end Wha happens here?

24 Reflecion due o change in µ (noe ension is consan! v? Ligher o heavier medium

25 heavier o ligher medium

26 Power is conserved (power inpower ou µω Av µω Av µω Av v v v and v v i i i r r r i r ω ω ω i r

27 Fourier decomposiion of waves

28 f( A sin(nω φ n n ' n g(x A sin(nkx θ n This is analogous o decomposing a vecor ino orhogonal direcion componens (x, y, and, excep here we are decomposing a funcion ino orhogonal frequency componens n n

29 8 ( cos(nω π n oddn (.8 cos( ω cos(3ω cos(5ω

30 Real Waves

31 f f f ( bea Ch 5 Useful Equaions Inerference due o wo sources L d sinθ if wo sources are in phase ( φ : nλ for maxima: L nλ sinθ d n λ for minima: L n λ sinθ d where n, ±, ±, ± 3 If wo sources are ou of phase ( φ π n λ for maxima: L n λ sinθ d nλ for minima: L nλ sinθ d µω i Av i i µω r Av r r µω Av v v v and v v ω ω ω i r i r