Physics 207: Lecture 29, Pg 1 Lecture 29 Goals: Chapter 20 Chapter 20  Work with a few important characteristics of sound waves. (e.g., Doppler effect) Chapter 21 Chapter 21  Recognize standing waves are the superposition of two traveling waves of same frequency  Study the basic properties of standing waves  Model interference occurs in one and two dimensions  Understand beats as the superposition of two waves of unequal frequency. Assignment Assignment  HW13, Due Friday, May 7 h  Thursday, Finish up, begin review for final

Physics 207: Lecture 29, Pg 2 Doppler effect, moving sources/receivers

Physics 207: Lecture 29, Pg 3 Doppler effect, moving sources/receivers l If the source of sound is moving  Toward the observer  seems smaller  Away from observer  seems larger l If the observer is moving  Toward the source  seems smaller  Away from source  seems larger Doppler Example Audio Doppler Example Visual

Physics 207: Lecture 29, Pg 4 Doppler Example A speaker sits on a small moving cart and emits a short 1 Watt sine wave pulse at 340 Hz (the speed of sound in air is ~340 m/s, so = 1m ). The cart is 30 meters away from the wall and moving towards it at 20 m/s. l The sound reflects perfectly from the wall. To an observer on the cart, what is the Doppler shifted frequency of the directly reflected sound? l Considering only the position of the cart, what is the intensity of the reflected sound? (In principle on would have to look at the energy per unit time in the moving frame.) t0t0 A 30 m

Physics 207: Lecture 29, Pg 5 Doppler Example l The sound reflects perfectly from the wall. To an observer on the cart, what is the Doppler shifted frequency of the directly reflected sound? At the wall: f wall = 340 / (1-20/340) = 361 Hz Wall becomes “source” for the subsequent part At the speaker f ’ = f wall (1+ 20/340) = 382 Hz t0t0 30 m t1t1

Physics 207: Lecture 29, Pg 6 Example Sound Intensity l Considering only the position of the cart, what is the intensity of the reflected sound to this observer? (In principle one would have to look at the energy per unit time in the moving frame.) v cart  t + v sound  t = 2 x 30 m = 60 m  t = 60 / (340+20) = 0.17 s  d sound = 340 * 0.17 m = 58 m I = 1 / (4  58 2 ) = 2.4 x W/m 2 or 74 dBs t0t0 30 m t1t1

Physics 207: Lecture 29, Pg 7 Doppler effect, moving sources/receivers l Three key pieces of information  Time of echo  Intensity of echo  Frequency of echo Plus prior knowledge of object being studied l With modern technology (analog and digital) this can be done in real time.

Physics 207: Lecture 29, Pg 8 Ch. 21: Wave Superposition l Q: What happens when two waves “collide” ? l A: They ADD together!  We say the waves are “superimposed”.

Physics 207: Lecture 29, Pg 9 Interference of Waves l 2D Surface Waves on Water In phase sources separated by a distance d d

Physics 207: Lecture 29, Pg 10 Principle of superposition l The superposition of 2 or more waves is called interference Constructive interference: These two waves are in phase. Their crests are aligned. Their superposition produces a wave with amplitude 2a Destructive interference: These two waves are out of phase. The crests of one are aligned with the troughs of the other. Their superposition produces a wave with zero amplitude

Physics 207: Lecture 29, Pg 11 Interference: space and time l Is this a point of constructive or destructive interference? What do we need to do to make the sound from these two speakers interfere constructively?

Physics 207: Lecture 29, Pg 12 Interference of Sound Sound waves interfere, just like transverse waves do. The resulting wave (displacement, pressure) is the sum of the two (or more) waves you started with.

Physics 207: Lecture 29, Pg 13 Interference of Sound Sound waves interfere, just like transverse waves do. The resulting wave (displacement, pressure) is the sum of the two (or more) waves you started with.

Physics 207: Lecture 29, Pg 14 Example Interference A speaker sits on a pedestal 2 m tall and emits a sine wave at 340 Hz (the speed of sound in air is 340 m/s, so = 1m ). Only the direct sound wave and that which reflects off the ground at a position half-way between the speaker and the person (also 2 m tall) makes it to the persons ear. l How close to the speaker can the person stand (A to D) so they hear a maximum sound intensity assuming there is no “phase change” at the ground (this is a bad assumption)? The distances AD and BCD have equal transit times so the sound waves will be in phase. The only need is for AB = t1t1 t0t0 t0t0 A B A D C d h

Physics 207: Lecture 29, Pg 15 Example Interference l The geometry dictates everything else. AB = AD = BC+CD = BC + (h 2 + (d/2) 2 ) ½ = d AC = AB+BC =  +BC = (h 2 + d/2 2 ) ½ Eliminating BC gives +d = 2 (h 2 + d 2 /4) ½  + 2 d + d 2 = 4 h 2 + d d = 4 h 2 /  d = 2 h 2 /  – ½ = 7.5 m t1t1 t0t0 t0t0 A B A D C Because the ground is more dense than air there will be a phase change of  and so we really should set AB to  /2 or 0.5 m.

Physics 207: Lecture 29, Pg 16 Exercise Superposition l Two continuous harmonic waves with the same frequency and amplitude but, at a certain time, have a phase difference of 170° are superimposed. Which of the following best represents the resultant wave at this moment? (A) (E) (D) (C) (B) Original wave (the other has a different phase)

Physics 207: Lecture 29, Pg 17 Wave motion at interfaces Reflection of a Wave, Fixed End l When the pulse reaches the support, the pulse moves back along the string in the opposite direction l This is the reflection of the pulse l The pulse is inverted

Physics 207: Lecture 29, Pg 18 Reflection of a Wave, Fixed End Animation

Physics 207: Lecture 29, Pg 19 Reflection of a Wave, Free End Animation

Physics 207: Lecture 29, Pg 20 Transmission of a Wave, Case 1 l When the boundary is intermediate between the last two extremes ( The right hand rope is massive or massless.) then part of the energy in the incident pulse is reflected and part is transmitted l Some energy passes through the boundary Here  rhs >  lhs Animation

Physics 207: Lecture 29, Pg 21 Transmission of a Wave, Case 2 l Now assume a heavier string is attached to a light string l Part of the pulse is reflected and part is transmitted l The reflected part is not inverted Animation

Physics 207: Lecture 29, Pg 22 Standing waves l Two waves traveling in opposite direction interfere with each other. If the conditions are right, same k & , their interference generates a standing wave: D Right (x,t)= a sin(kx-  t) D Left (x,t)= a sin(kx+  t) A standing wave does not propagate in space, it “stands” in place. A standing wave has nodes and antinodes D(x,t)= D L (x,t) + D R (x,t) D(x,t)= 2a sin(kx) cos(  t) The outer curve is the amplitude function A(x) = 2a sin(kx) when  t = 2  n n = 0,1,2,… k = wave number = 2π/λ Nodes Anti-nodes

Physics 207: Lecture 29, Pg 23 Standing waves on a string l Longest wavelength allowed is one half of a wave Fundamental: /2 = L  = 2 L Recall v = f Overtones m > 1

Physics 207: Lecture 29, Pg 24 Vibrating Strings- Superposition Principle l Violin, viola, cello, string bass l Guitars l Ukuleles l Mandolins l Banjos D(x,0) Antinode D(0,t)

Physics 207: Lecture 29, Pg 25 Standing waves in a pipe Open end: Must be a displacement antinode (pressure minimum) Closed end: Must be a displacement node (pressure maximum) Blue curves are displacement oscillations. Red curves, pressure. Fundamental:  /2 /2 /4

Physics 207: Lecture 29, Pg 26 Standing waves in a pipe

Physics 207: Lecture 29, Pg 27 Combining Waves Fourier Synthesis

Physics 207: Lecture 29, Pg 28 Organ Pipe Example A 0.9 m organ pipe (open at both ends) is measured to have it’s first harmonic (i.e., its fundamental) at a frequency of 382 Hz. What is the speed of sound (refers to energy transfer) in this pipe? L=0.9 m f = 382 Hz and f = v with = 2 L / m (m = 1) v = 382 x 2(0.9) m  v = 687 m/s

Physics 207: Lecture 29, Pg 29 Standing Waves l What happens to the fundamental frequency of a pipe, if the air (v =300 m/s) is replaced by helium (v = 900 m/s)? Recall: f = v (A) Increases(B) Same(C) Decreases

Physics 207: Lecture 29, Pg 30 Lecture 29 Assignment Assignment  HW13, Due Friday, May 7 th