1. Physics 2 Class 16 Wave superposition, interference, and reflection

2. Adding waves: superposition
When two waves are incident on the same place at the same time, their amplitudes can usually just be added.
In the next few slides we will look at some special cases: (both with two waves at the same frequency and with the same amplitude)
same frequency and amplitude
superposition at a point in space
same direction, but different phase
opposite directions

3. Adding waves
In most simple systems, the amplitude of two waves that cross or overlap can just be added together.

4. Adding two equal amplitude waves at a point

5. iClicker check 16.1
For which phase difference  is the superposition amplitude a maximum?
=0
=/2
= 

6. iClicker check 16.2
For which phase difference  is the superposition amplitude a minimum?
=0
=/2
= 

7. Adding two waves that have the same frequency and direction, but different phase
What happens when the phase difference is 0?
What happens when the phase difference is /2?
What happens when the phase difference is ?

8. Constructive interference when =0. Destructive interference when =.
The two wave in the middle of the page travel at different speeds and therefore move in and out of phase with one another as time progresses.
Link to animation of two sine waves adding in and out of phase (Kettering)

9. What happens when a wave is incident on an immovable object?
It reflects, travelling in the opposite direction and upside down.
At the point of contact with the wall, the amplitude of the sum of the waves must be zero. The only way to add two waves together and get zero is for them to have opposite signs at all times.
(Now watch patiently while your instructor plays with the demonstration.)

10. Same frequency, opposite directions
What happens when kx is 0?
What happens when kx is /2?
What happens when kx is ?
The animation is near the bottom of the page and shows standing waves. Standing waves manifest thenselves in sound reflections from walls, producing dead spots in a room.
Link to animation of two sine waves traveling in opposite directions

11. Standing waves
Interfering waves traveling in opposite directions can produce fixed points called nodes.
y1 = ym sin(kx – wt)
y2 = ym sin(kx + wt) v=w/k
yT = y1 + y2 = 2ym cos(wt) sin(kx)
yT=0 when kx = 0, p, 2p...
yT(t)=maximum when kx = p/2, 3p/2, 5p/2 ...

12. Standing waves - ends fixed
Amplitude will resonate when an integer number of half-wavelengths fit in the opening.
Example: violin
Fundamental mode (1st harmonic, n = 1)
Watch while your instructor once again plays with the string. We are sort of like cats this way. Show us a string and we must play with it.
2nd harmonic, n = 2
l=2L/n
5th harmonic, n = 5

13. Both Ends Fixed
Once again, watch patiently while your instructor plays with the string.

14. Standing waves - one end free
Free end will be an anti-node at resonance.
Demo: spring (slinky) with one end free.
l=4L/n, n odd
Fundamental mode (1st harmonic), n = 1
3rd harmonic, n = 3
9th harmonic, n = 9

15. One End Open (e.g. Organ Pipe)

16. Standing waves - both ends free
Example: wind instrument
l=2L/n
5th harmonic, n = 5
2nd harmonic, n = 2
Fundamental mode (1st harmonic), n = 1

17. Both Ends Open (e.g. Organ Pipe)