1. Wave Interactions

2. Wave Interference
Two different material objects can never occupy the same space at the same time
Think about a concert, can you hear different sounds at the same time?

3. Wave Interference
Because mechanical waves are the displacement of matter, two waves can occupy the same space at the same time
Superposition

4. Wave Interference
Displacement in the same direction produces a constructive interference.
When 2 waves are added together, the resultant wave is larger.

5. Wave Interference
Displacement in the opposite direction produces a destructive interference.
When 2 waves are added together, the resultant wave is smaller.

6. Wave Interference
Law of Superposition is limited to transverse and longitudinal waves

7. Reflection
At a free boundary (the end is free to move up and down), the reflected pulse is identical to the incident pulse

8. Reflection
At a fixed boundary(not able to move up and down), the reflected pulse is inverted to the incident pulse

9. Standing Wave Are fixed at both ends
Waves reflected from fixed ends, create waves traveling in both directions
According to the Law of Superposition, the waves combine

10. Standing Waves on a String
There is a node at each end because the string is fixed at the ends.
The diagram shows three possible standing wave patterns.
Standing waves are produced by interference as waves travel in opposite directions after plucking or bowing the string.
The lowest frequency (one loop) is called the fundamental frequency (f1).
To show how standing waves are produced, go to the following web site:
Choose “Standing Waves (Explanation by superposition).”
When showing this, pause and point out that the nodes are produced by waves that always cancel.
This demonstration shows a six loop standing wave.

11. Standing Wave Depending on the frequency a pattern is established
String will have regions of zero displacement
String will have regions of maximum displacement
Only certain frequencies produce standing waves

12. Standing Wave Nodes- Point at which two waves cancel
Antinodes- Point at which two waves vibrate with the greatest amplitude.

13. Standing Waves on a String
To the left is a snapshot of a single loop standing wave on a string of length, L.
What is the wavelength for this wave?
Answer:  = 2L
What is the frequency?
Answer:

14. Fundamental Harmonic Wavelength (λ)1 = 2 L
Frequency (f) = v/ λ1 f1= v/ 2L

15. 2nd Harmonic Wavelength (λ)2 = L
Frequency (f) = v/ λ2 f2= 2v/ 2L f2=2f1

16. 3nd Harmonic Wavelength (λ)3 = 2L/3
Frequency (f) = v/ λ3 f3= 3v/2L f3=3f1

17. 4th Harmonic Wavelength (λ)4 = L/2
Frequency (f) = v/ λ4 f4= 4v/2L f4=4f1

18. Now that students understand why f1 = v/(2L), help them see why the wavelengths for the next three harmonics are as shown. It is helpful to look at just one segment of the wave instead of the four that are shown for each mode.
Point out the “harmonic” terminology for each mode. These are sometimes called overtones instead of harmonics. The second harmonic is called the 1st overtone, and so on.
Ask students to come up with a general equation for f in terms of v and L, using n where n represent the number of loops. Show them f1 = v/(2L), f2 = v/L, f3 = (3v)/(2L), and so on, and see if they see the pattern to write fn = ????
The answer is on next slide.

19. Harmonics n is the number of loops or harmonic number.
v is the speed of the wave on the string.
Depends on tension and density of the string
L is the length of the vibrating portion of the string.
How could you change the frequency (pitch) of a string?
Shorter strings (decreased L) have higher pitches.
Higher-tension strings (increased v) have higher pitchers.
More dense strings (decreased v) have lower pitches.
Point out to students that v in the equation is the speed of the waves as they move back and forth on the string. It is not the speed of the sound. The speed of waves on the string ranges from 100’s to 1000’s of m/s depending on the string tension and density. The vibrating string then produces sound waves that travel at 346 m/s (at 25°C) through the air.