2. 2-D waves in water
A bobber moves up and down in simple harmonic motion and produces water waves.
Bright rings are wave crests; dark rings are wave troughs.
These waves are produced by a point source.
Top view of a sine wave.

3. Two point sources How can we explain this pattern?
Where the pattern is brightest, a maximum occurs.
Where the pattern is darkest, a minimum occurs.

4. Interference
Interference is the superposition (i.e. addition) of waves.
Wave 1
Wave 2

5. Total Constructive Interference
The wave crests of one wave coincide with the wave crests of the other wave. The result is a wave crest that has twice the amplitude.
wave 1
wave 2
wave 1 + wave 2

6. Total Destructive Interference
The wave crests of one wave coincide with the wave troughs of the other wave. The result is a wave of zero amplitude.
wave 1
wave 2
wave 1 + wave 2

7. Path Difference = n
If a wave is shifted 1 or 2, etc., then total constructive inteference will occur.
wave 1 1
wave 2
Wave 1 TRAVELS FARTHER than wave 2 by an amount 1. The same result would occur if it traveled farther by an amount 2 3 etc. The difference in the distance the waves travel from their sources is called path difference. When the path difference at a point = n, total constructive interference occurs.

8. In Phase
If a wave is shifted 1 or 2, etc., then total constructive inteference will occur.
wave 1 1
wave 2
1 wavelength is 360 for a sine function. Because total constructive interference occurs, we say the waves are in phase.

9. Poll
wave 1
wave 2
What is the path difference between wave 1 and wave 2? That is, how much farther does wave 1 travel than wave 2?
1 3. 3 
2  

10. Path Difference = (2n-1)/2
If a wave is shifted  or 3, etc., then total destructive inteference will occur.
wave 1

wave 2
 corresponds to a phase difference of

11. Out of Phase
If a wave is shifted  or 3, etc., then total destructive inteference will occur.
wave 1

wave 2
 corresponds to a phase difference of 180. When the phase difference is 180, the waves are out of phase, and total destructive interference occurs.

12. Poll
wave 1
wave 2
What is the path difference between wave 1 and wave 2? That is, how much farther does wave 1 travel than wave 2?
1  
2  

13. If two identical sources S1 and S2 are 180° out of phase, as shown here, then if point P is moved to a location 2l further from S1 than from S2, there will be __________ at P.
A. total constructive
interference
B. total destructive
interference
C. something
in between . P . . .
PATH 1
PATH 2 . S1 S2
center
line

14. If two identical sources S1 and S2 are 180° out of phase, as shown here, then if point P is moved to a location l further from S1 than from S2, there will be __________ at P.
A. total constructive
interference
B. total destructive
interference
C. something
in between . P . . .
PATH 1
PATH 2 . S1 S2
center
line

15. Two point sources How can we explain this pattern?
Interference of two waves. The maxima correspond to total constructive interference. The minima correspond to total destructive interference.

16. Two point sources
The path difference from the sources at a maximum is n.
The path difference from the sources at a minimum is (2n-1)/2.

17. Finding the maxima

18. Path Difference for Maxima
therefore

19. Example
Label each line of maxima with the integer n corresponding to a path difference of 0, 1, 2, etc.

20. Example
If the distance d between the sources is increased, what happens to the angle  to the first maxima? (i.e. the “spread” of the maxima)

21. Poll
If you decrease the wavelengths of the waves produced by the sources, the angle of the first maxima (i.e. the spread in the maxima)
increases
decreases
remains the same

22. (Click to continue stepwise animation)
SOURCE 1 S2
SOURCE 2
Water wave patterns spreading out from two identical point sources S1 and S2 (the crests are in white) can be superimposed by sliding them towards each other on the track until they overlap.
(Click to continue stepwise animation)

23. (Click to continue stepwise animation)
5. (continued) S1
SOURCE 1 S2
SOURCE 2
(Click to continue stepwise animation)

24. (Click to continue stepwise animation)
5. (continued) S1
SOURCE 1 S2
SOURCE 2
(Click to continue stepwise animation)

25. (Click to continue stepwise animation)
5. (continued) S1
SOURCE 1
SOURCE 2 S2
(Click to continue stepwise animation)

26. A. crest B. trough C. point of zero displacement
5. (continued) S1
SOURCE 1
SOURCE 2 S2
a. Along the red lines, where there is a crest from one wave, there will be a _______________ from the other wave.
A. crest B. trough C. point of zero displacement

27. A. become more spread out B. become less spread out
5. (continued) S1
SOURCE 1
SOURCE 2 S2
b. If we continue to slide the sources closer together, the pattern of red lines will _______________.
A. become more spread out
B. become less spread out
C. remain unchanged

28. A. crest B. trough C. point of zero displacement
5. (continued)
SOURCE 1
SOURCE 2 P  S1 S2
c. The pattern is now not shown, but the red lines show the directions in which there is constructive interference. Thus, at a particular instant, there is a _________ arriving at point P from each source.
A. crest B. trough C. point of zero displacement
D. [A and B are both possible correct answers.]
E. [A, B, and C are all possible correct answers.]

30. Which one of the points A, B, C, D, and E is on a second line of constructive interference (n = 2) from the center? .  E  B . A  C  D  . .   S1 S2
Two identical, in-phase
sources of water waves