1. Wave Interference
A material object will not share its space with another object, but more than one wave can exist at the same time in the same space.
Interference patterns occur when waves from different sources arrive at the same point—at the same time.

3. Tuning Forks Interference

4. Constructive Interference
In constructive interference, the crest of one wave overlaps the crest of another and their individual effects add together.
The result is a wave of increased amplitude.

5. Destructive Interference
In destructive interference, the crest of one wave overlaps the trough of another and their individual effects are reduced.
The high part of one wave fills in the low part of another.

6. Interference: Transverse and Longitudinal

7. “Beats”
When two tones of slightly different frequency are sounded together the sound is loud, then faint, then loud, then faint, and so on.
This periodic variation in the loudness of sound is called beats.

8. “Beats”

9. 10 Hz
12 Hz
Beat frequency:

10. Beat Frequency
When one fork vibrates 264 times per second, and the other fork vibrates 262 times per second, they are in step twice each second.
A beat frequency of 2 hertz is heard.

11. Example
One guitar string vibrates at a frequency of 256-Hz and a second guitar string is plucked. A beat frequency of 3-Hz is heard. What frequency is the second guitar string vibrating at? (There are two possibilities.)

12. Standing Waves

13. “Standing” Waves
By shaking a string (or Slinky) just right, you can cause the original and reflected waves to interfere and form a “standing” wave.
A standing wave is a wave that appears to stay in one place—it does not seem to move through the medium.

14. Parts of a Standing Wave
Certain parts of a standing wave remain stationary.
Nodes are the points of the wave with no displacement.
The positions on a standing wave with the largest displacement are known as antinodes.
Antinodes occur halfway between nodes.

15. Parts of a Standing Wave
Nodes

16. Parts of a Standing Wave
Antinodes
Nodes

17. Parts of a Standing Wave
Antinodes
Nodes

18. Shake the rope until you set up a standing wave of ½ wavelength.

19. Shake the rope until you set up a standing wave of ½ wavelength.
Shake with twice the frequency and produce a standing wave of 1 wavelength.

20. Shake the rope until you set up a standing wave of ½ wavelength.
Shake with twice the frequency and produce a standing wave of 1 wavelength.
Shake with three times the frequency and produce a standing wave of 1 ½ wavelengths.

21. Shake the rope until you set up a standing wave of ½ wavelength.
Shake with twice the frequency and produce a standing wave of 1 wavelength.
Shake with three times the frequency and produce a standing wave of 1 ½ wavelengths.
First Harmonic
L = 1/2 λ

22. Shake the rope until you set up a standing wave of ½ wavelength.
Shake with twice the frequency and produce a standing wave of 1 wavelength.
Shake with three times the frequency and produce a standing wave of 1 ½ wavelengths.
First Harmonic
L = 1/2 λ
Second Harmonic
L = 2/2 λ

23. Shake the rope until you set up a standing wave of ½ wavelength.
Shake with twice the frequency and produce a standing wave of 1 wavelength.
Shake with three times the frequency and produce a standing wave of 1 ½ wavelengths.
First Harmonic
L = 1/2 λ
Second Harmonic
L = 2/2 λ
Third Harmonic
L = 3/2 λ

24. n = 1, 2, 3, …
A standing wave forms only if half a wavelength or a multiple of half a wavelength fits exactly into the length (L) of the vibrating medium.

25. n = 1, 2, 3, …
Since there can only be certain wavelengths, there must only be certain frequencies that the string can produce.

26. n = 1, 2, 3, …
Since there can only be certain wavelengths, there must only be certain frequencies that the string can produce.

27. n = 1, 2, 3, …
Since there can only be certain wavelengths, there must only be certain frequencies that the string can produce.

28. n = 1, 2, 3, …
Since there can only be certain wavelengths, there must only be certain frequencies that the string can produce.

29. n = 1, 2, 3, …
Since there can only be certain wavelengths, there must only be certain frequencies that the string can produce.

30. n = 1, 2, 3, …
Since there can only be certain wavelengths, there must only be certain frequencies that the string can produce.

31. n = 1, 2, 3, …
Since there can only be certain wavelengths, there must only be certain frequencies that the string can produce.

32. n = 1, 2, 3, …
Since there can only be certain wavelengths, there must only be certain frequencies that the string can produce.
Harmonic Frequencies of Standing Waves on Strings
n = 1, 2, 3, …

33. n = 1, 2, 3, …
Since there can only be certain wavelengths, there must only be certain frequencies that the string can produce.
Harmonic Frequencies of Standing Waves on Strings
n = 1, 2, 3, …
A string has a set of natural frequencies that it vibrates at, and these are determined by the factors in this equation: the speed of the waves on the string and the length of the string.

34. Fundamental Frequency
The fundamental frequency is the lowest frequency (or longest wavelength) produced by the vibration of a string.
The fundamental frequency is often the loudest and lowest sound from the string, but there are also additional harmonics that are possible.
First harmonic, or fundamental frequency, is n = 1.
Second harmonic is n = 2.
Third harmonic is n = 3.
And so on.
The combination of harmonics on an instrument gives the instrument its unique sound or “timbre.”

38. Example: (#5 on Standing Waves Practice) The fundamental frequency of the biggest string on a bass guitar is 41.2 Hz. The length of the string is 76.0 cm. (a) Determine the speed of the wave disturbances on the string. (b) How much should you shorten the length of the string (by holding the string down with your finger) if you wanted to produce a pitch that is twice as high?

39. Guitar Strings
When a guitar string is plucked, a standing wave forms on the string.