1. Principle of Superposition Interference Stationary Waves

2. What happens when two waves occupy the same space at the same time?

3. Principle of Superposition
When two waves occupy the same space at the same time, they add up algebraically.
Interference

4. Constructive Interference
Amplitudes increase

5. Destructive Interference
Amplitudes decrease
Sometimes, they completely cancel each other out

6. Interference Exercise 1
Two pulses are moving in opposite directions at 1.0 cms-1. The waves are 10 cm apart at t = 0. Sketch the shape of the string at 6.0, 7.0 and 8.0 seconds.

7. Interference Exercise 2
Sketch the shape of the string at t = 0.25s, t = 0.50s, t = 0.75 s and t= 1.00 s. (The figure is at t = 0.)

8. Interference Exercise 3
Sketch the shape of the string at t = 4.0s, t = 6.0s, and t= 10.0 s. (The figure is at t = 0.)

9. Interference and Path Difference
Consider two sources of waves at a distance from each other.
What conditions are necessary for complete destructive interference to occur at a point?

10. What conditions are necessary for complete destructive interference to occur at a point?
The same intensity or amplitude
The same type of wave
They have the same frequency or period

11. Their path difference must be N (λ/2) Where N = 1, 3,5,….
What conditions are necessary for complete destructive interference to occur at a point?
They must be in phase
Their path difference must be
N (λ/2)
Where N = 1, 3,5,….

12. Their path difference must be N (λ) Where N = 1, 2,3,….
What should be their path difference for constructive interference to occur at a point?
Their path difference must be
N (λ)
Where N = 1, 2,3,….

13. Example 1
Two loudspeakers A and B emit are driven by the same amplifier and emit in phase waves.

14. Example 1
For what frequencies does constructive interference occur at P?

15. Example 1
For what frequencies does destructive interference occur at P?

16. Example 2
Two identical sources of sound are situated 80 cm apart as shown.
They always vibrate in phase with each other.

17. Example 2
A microphone is placed 100 cm from S1 as shown. The speed of sound is 330 ms-1.
How many frequencies between 1.0 kHz to 4.0 kHz will produce no sound at M?

18. Standing Waves (a.k.a. Stationary Waves)

19. Formation of Standing Waves
Incident waves travel in one direction and are reflected back
Incident and reflected waves interfere
Points that constructively interfere form antinodes
Points that destructively interfere form nodes

20. Standing Waves (Which is a standing wave?)

21. Do standing waves have wave speed?
Yes
Wave speed is the speed of the incident and reflected waves.

22. Equations for Standing Waves
Fixed Ends
String Instruments
ƒ = n (v/2L)
Where n = 1,2,3,…

23. Equations for Standing Waves
One fixed end, One open end
Percussion instruments
Water pipe
ƒ = n (v/4L)
Where n = 1,3,5,…

24. Equations for Standing Waves
Two Open ends
Some wind instruments
ƒ = n (v/2L)
Where n = 1,2,3,…
Similar to two fixed ends

25. Example 1
One of the 63.5-cm long strings of a guitar is tuned to produce the note B3 (frequency 245 Hz) when vibrating in its fundamental mode. Find the speed of the transverse waves on the string.

26. Example 1
If the speed of sound in the air is 344 ms-1, find the frequency and wavelength of the sound wave produced in the air by the vibrating string.

27. Example 2 (A Tale of Two Pipes)
One day when the speed of sound is 345 ms-1, the fundamental frequency of a stopped organ pipe is 220 Hz. How long was this pipe?

28. Example 2 (A Tale of Two Pipes)
The second overtone of this pipe has the same wavelength as the third harmonic of an open pipe. How long is the open pipe?