1. Study the basic properties of standing waves
Lecture 29
Goals:
Chapter 20
Work with a few important characteristics of sound waves. (e.g., Doppler effect)
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
HW12, Due Friday, May 8th
Thursday, Finish up, begin review for final, evaluations 1

2. Doppler effect, moving sources/receivers

3. Doppler effect, moving sources/receivers
If the source of sound is moving
Toward the observer 
 seems smaller
Away from observer 
 seems larger
If the observer is moving
Toward the source 
 seems smaller
Away from source 
 seems larger
Doppler Example Audio
Doppler Example Visual

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 l = 1m ). The cart is 30 meters away from the wall and moving towards it at 20 m/s.
The sound reflects perfectly from the wall. To an observer on the cart, what is the Doppler shifted frequency of the directly reflected sound?
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.) t0
30 m A

5. Doppler Example
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: fwall = 340 / (1-20/340) = 361 Hz
Wall becomes “source” for the subsequent part
At the speaker f ’ = fwall (1+ 20/340) = 382 Hz t0
30 m t1

6. Example Interference
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.)
vcart Dt + vsound Dt = 2 x 30 m = 60 m
Dt = 60 / (340+20) = 0.17 s  dsound = 340 * 0.17 m = 58 m
I = 1 / (4p 582) = 2.4 x 10-5 W/m2 or 74 dBs t0
30 m t1

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

8. Superposition
Q: What happens when two waves “collide” ?
A: They ADD together!
We say the waves are “superimposed”.

9. Interference of Waves 2D Surface Waves on Water
In phase sources separated by a distance d d

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

11. Interference: space and time
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?

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.

13. Example Interference
A speaker sits on a pedestal 2 m tall and emits a sine wave at 343 Hz (the speed of sound in air is 343 m/s, so l = 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.
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)? t1 t0 A B D C d h
The distances AD and BCD have equal transit times so the sound waves will be in phase. The only need is for AB = l

14. Example Interference The geometry dictates everything else.
AB = l AD = BC+CD = BC + (h2 + (d/2)2)½ = d
AC = AB+BC = l +BC = (h2 + d/22)½
Eliminating BC gives l+d = 2 (h2 + d2/4)½
l + 2ld + d2 = 4 h2 + d2
1 + 2d = 4 h2 / l  d = 2 h2 / l – ½
= 7.5 m t1 t0
7.5 t0 D A A
4.25 B
3.25 C
Because the ground is more dense than air there will be a phase change of p and so we really should set AB to l/2 or 0.5 m.

15. Exercise Superposition
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?
Original wave
(the other has a different phase)
(A)
(B)
(D)
(C)
(E)

16. Wave motion at interfaces Reflection of a Wave, Fixed End
When the pulse reaches the support, the pulse moves back along the string in the opposite direction
This is the reflection of the pulse
The pulse is inverted

17. Reflection of a Wave, Fixed End
Animation

18. Reflection of a Wave, Free End
Animation

19. Transmission of a Wave, Case 1
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
Some energy passes
through the boundary
Here mrhs > mlhs
Animation

20. Transmission of a Wave, Case 2
Now assume a heavier string is attached to a light string
Part of the pulse is reflected and part is transmitted
The reflected part is not inverted
Animation

21. Standing waves
Two waves traveling in opposite direction interfere with each other.
If the conditions are right, same k & w, their interference generates a standing wave:
DRight(x,t)= a sin(kx-wt) DLeft(x,t)= a sin(kx+wt)
A standing wave does not propagate in space, it “stands” in place.
A standing wave has nodes and antinodes
Anti-nodes
D(x,t)= DL(x,t) + DR(x,t)
D(x,t)= 2a sin(kx) cos(wt)
The outer curve is the amplitude function
A(x) = 2a sin(kx)
when wt = 2pn n = 0,1,2,…
k = wave number = 2π/λ
Nodes

22. Standing waves on a string
Longest wavelength allowed is one half of a wave
Fundamental: l/2 = L  l = 2 L
Recall v = f l
Overtones m > 1

23. Vibrating Strings- Superposition Principle
Violin, viola, cello, string bass
Guitars
Ukuleles
Mandolins
Banjos
D(x,0)
Antinode D(0,t)

24. 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: l/ l/ l/4

25. Standing waves in a pipe

26. Combining Waves
Fourier Synthesis

27. Lecture 29
Assignment
HW12, Due Friday, May 8th 1