Doppler effect, moving sources/receivers

Transcription

1 Goals: Lecture 29 Chapter 20 Work with a ew iportant characteristics o sound waves. (e.g., Doppler eect) Chapter 21 Recognize standing waves are the superposition o two traveling waves o sae requency Study the basic properties o standing waves Model intererence occurs in one and two diensions Understand beats as the superposition o two waves o unequal requency. Assignent HW12, Due Friday, May 8 th Thursday, Finish up, begin review or inal, evaluations Physics 207: Lecture 29, Pg 1 Doppler eect, oving sources/receivers Physics 207: Lecture 29, Pg 2 Page 1

2 Doppler eect, oving sources/receivers I the source o sound is oving Toward the observer λ sees saller Away ro observer λ sees larger I the observer is oving Toward the source λ sees saller observer observer observer source v 1 s v source v 1 + s v v + v o 1 source Away ro source λ sees larger observer v v o 1 source Doppler Exaple Audio Doppler Exaple Visual Physics 207: Lecture 29, Pg 3 Doppler Exaple A speaker sits on a sall oving cart and eits a short 1 Watt sine wave pulse at 340 Hz (the speed o sound in air is 340 /s, so λ 1 ). The cart is 30 eters away ro the wall and oving towards it at 20 /s. The sound relects perectly ro the wall. To an observer on the cart, what is the Doppler shited requency o the directly relected sound? Considering only the position o the cart, what is the intensity o the relected sound? (In principle on would have to look at the energy per unit tie in the oving rae.) t 0 A 30 Physics 207: Lecture 29, Pg 4 Page 2

3 Doppler Exaple The sound relects perectly ro the wall. To an observer on the cart, what is the Doppler shited requency o the directly relected sound? At the wall: wall 340 / (1-20/340) 361 Hz observer source v 1 s v Wall becoes source or the subsequent part At the speaker wall (1+ 20/340) 382 Hz observer v + v o 1 source t 0 t 1 30 Physics 207: Lecture 29, Pg 5 Exaple Intererence Considering only the position o the cart, what is the intensity o the relected sound to this observer? (In principle one would have to look at the energy per unit tie in the oving rae.) v cart t + v sound t 2 x t 60 / (340+20) 0.17 s d sound 340 * I 1 / (4π 58 2 ) 2.4 x 10-5 W/ 2 or 74 dbs t 0 t 1 30 Physics 207: Lecture 29, Pg 6 Page 3

4 Doppler eect, oving sources/receivers Three key pieces o inoration Tie o echo Intensity o echo Frequency o echo Plus prior knowledge o object being studied With odern technology (analog and digital) this can be done in real tie. Physics 207: Lecture 29, Pg 7 Superposition Q: What happens when two waves collide? A: They ADD together! We say the waves are superiposed. Physics 207: Lecture 29, Pg 8 Page 4

5 Intererence o Waves 2D Surace Waves on Water In phase sources separated by a distance d d Physics 207: Lecture 29, Pg 9 Principle o superposition The superposition o 2 or ore waves is called intererence Destructive intererence: Constructive intererence: These two waves are out o These two waves are in phase. phase. Their crests are aligned. The crests o one are aligned with the troughs o the other. Their superposition produces a wave with aplitude 2a Their superposition produces a wave with zero aplitude Physics 207: Lecture 29, Pg 10 Page 5

6 Is this a point o constructive or destructive intererence? Intererence: space and tie What do we need to do to ake the sound ro these two speakers interere constructively? Physics 207: Lecture 29, Pg 11 Intererence o Sound Sound waves interere, just like transverse waves do. The resulting wave (displaceent, pressure) is the su o the two (or ore) waves you started with. A D ( r2, t) cos[ 2π ( r2 / λ t / T ) + φ 2 ] r r r r A D( r1, t) cos[ 2π ( r1 / λ t / T ) + φ1] r1 Maxiu constructive intererence φ 2π r + φ1 φ2 2π λ λ λ φ r + ( φ1 φ2) λ 2π 2π Maxiu destructive intererence φ 2π r + φ1 φ2 2π ( + λ 0,1,2,... 1) 2 r Physics 207: Lecture 29, Pg 12 Page 6

7 Exaple Intererence A speaker sits on a pedestal 2 tall and eits a sine wave at 343 Hz (the speed o sound in air is 343 /s, so λ 1 ). Only the direct sound wave and that which relects o the ground at a position hal-way between the speaker and the person (also 2 tall) akes it to the persons ear. How close to the speaker can the person stand (A to D) so they hear a axiu sound intensity assuing there is no phase change at the ground (this is a bad assuption)? t 0 A B t 0 d C t 1 D h The distances AD and BCD have equal transit ties so the sound waves will be in phase. The only need is or AB λ Physics 207: Lecture 29, Pg 13 Exaple Intererence The geoetry dictates everything else. AB λ AD BC+CD BC + (h 2 + (d/2) 2 ) ½ d AC AB+BC λ +BC (h 2 + d/2 2 ) ½ Eliinating BC gives λ+d 2 (h 2 + d 2 /4) ½ λ + 2λd + d 2 4 h 2 + d 2 t 0 A B t d 4 h 2 / λ d 2 h 2 / λ ½ 7.5 t C 4.25 D Because the ground is ore dense than air there will be a phase change o π and so we really should set AB to λ/2 or 0.5. Physics 207: Lecture 29, Pg 14 Page 7

8 Exercise Superposition Two continuous haronic waves with the sae requency and aplitude but, at a certain tie, have a phase dierence o 170 are superiposed. Which o the ollo wing best represents the resultant wave at this oent? Original wave (the other has a dierent phase) (A) (B) (D) (C) (E) Physics 207: Lecture 29, Pg 15 Wave otion at interaces Relection o a Wave, Fixed End When the pulse reaches the support, the pulse oves back along the string in the opposite direction This is the relection o the pulse The pulse is inverted Physics 207: Lecture 29, Pg 16 Page 8

9 Aniation Relection o a Wave, Fixed End Physics 207: Lecture 29, Pg 17 Relection o a Wave, Free End Aniation Physics 207: Lecture 29, Pg 18 Page 9

10 Transission o a Wave, Case 1 When the boundary is interediate between the last two extrees ( The right hand rope is assive or assless.) then part o the energy in the incident pulse is relected and part is transitted Soe energy passes through the boundary Here µ rhs > µ lhs Aniation Physics 207: Lecture 29, Pg 19 Transission o a Wave, Case 2 Now assue a heavier string is attached to a light string Part o the pulse is relected and part is transitted The relected part is not inverted Aniation Physics 207: Lecture 29, Pg 20 Page 10

11 Standing waves Two waves traveling in opposite direction interere with each other. I the conditions are right, sae k & ω, their intererence generates a standing wave: D Right (x,t) a sin(kx-ωt) D Let (x,t) a sin(kx+ωt) A standing wave does not propagate in space, it stands in place. A standing wave has nodes and antinodes Anti-nodes D(x,t) D L (x,t) + D R (x,t) D(x,t) 2a sin(kx) cos(ωt) The outer curve is the aplitude unction A(x) 2a sin(kx) when ωt 2πn n 0,1,2, k wave nuber 2 / Nodes Physics 207: Lecture 29, Pg 21 Standing waves on a string Longest wavelength allowed is one hal o a wave Fundaental: λ/2 L λ 2 L λ 2 L 1,2,3,... Recall v λ v v 2L Overtones > 1 Physics 207: Lecture 29, Pg 22 Page 11

12 Vibrating Strings- Superposition Principle Violin, viola, cello, string bass Guitars Ukuleles Mandolins Banjos D(x,0) Antinode D(0,t) Physics 207: Lecture 29, Pg 23 Standing waves in a pipe Open end: Must be a displaceent antinode (pressure iniu) Closed end: Must be a displaceent node (pressure axiu) Blue curves are displaceent oscillations. Red curves, pressure. Fundaental: λ/2 λ/2 λ/4 Physics 207: Lecture 29, Pg 24 Page 12

13 λ 2 L v 2 L 1, 2,3,... Standing waves in a pipe λ 2 L v 2 L 1, 2,3,... λ 4 L v 4 L 1,3,5,... Physics 207: Lecture 29, Pg 25 Cobining Waves Fourier Synthesis Physics 207: Lecture 29, Pg 26 Page 13

14 Lecture 29 Assignent HW12, Due Friday, May 8 th Physics 207: Lecture 29, Pg 27 Page 14