1. Principle of Linear Superposition and Interference Phenomena
Chapter 17

2. Combining waves
Consider two wave pulses moving toward each other with two upward pulses.
When the pulses merge, the wave
assumes a shape that is the sum of
the shapes of the individual pulses.
Wave pulses pass through each other and continue moving.

3. Combining waves
Consider two wave moving toward each other, but with one upward pulse and one downward pulse.
When the pulses merge, the wave shape is the sum of the shapes of the individual pulses. In this case, the pulses momentarily cancel as they pass each other.
Wave pulses pass through each other and continue moving.
Wave interference demonstration on YouTube

4. Principle of linear superposition
The disturbance created by two waves is the sum of the disturbances of the two waves.

5. Constructive wave interference
In-phase waves reinforce each other.
In phase Peaks line up with peaks, valleys line up with valleys.

6. Destructive wave interference
Out of phase waves cancel each other.
The waves cancel each other out.
Out of phase Peaks line up with valleys, valleys line up with peaks.

7. Destructive wave interference application
Noise canceling headphones use destructive interference to cancel out the noise.

8. Coherent wave sources
Coherent waves maintain the same phase pattern.

9. ΔL = |L1 - L2| Path length difference ΔL
Path length difference ΔL is the absolute value of the difference in the distances from each source to a point being considered.
ΔL = |L1 - L2| L1 L2
path 1
path 2

10. Path length difference
Find the path length difference ΔL for points 1, 2, 3, 4, 5, and 6.
Express the path length difference as a number of wavelengths.
For example: ΔL = 3½ wavelengths 1 2 3 4 5 6

11. Peaks reinforce peaks and valleys reinforce valleys.
Path length difference: constructive interference
Constructive interference occurs when the difference is path length is zero or an integer number of wavelengths. ΔL = 0, 1λ, 2λ, 3λ, etc.
Peaks reinforce peaks and valleys reinforce valleys.

12. Peaks and valleys cancel each other.
Path length difference: destructive interference
Destructive interference occurs when the difference is path length is a half integer number of wavelengths. ΔL = ½λ, 1½λ, 2½λ, 3½λ, etc.
Peaks and valleys cancel each other.

13. Constructive and destructive interference
Constructive interference
ΔL = 0 3λ
Destructive interference
ΔL = ½λ
3½λ 3λ

14. Example 1 What Does a Listener Hear?
Two in-phase loudspeakers A and B, are
separated by 3.2 m. A listener is stationed
at C, which is 2.4 m in front of speaker B.
Both speakers are playing identical 214 Hz
tones, and the speed of sound is 343 m/s.
Does the listener hear a loud sound, or no sound?
Critical thinking step:
Constructive interference would produce a loud sound.
Destructive interference would produce no sound (sound cancelation).
What is the path length difference for the waves from speakers A and B to the listener at C?

15. Example 1 What Does a Listener Hear?
Calculate the path length difference.
4 m
Calculate the wavelength.
Therefore ΔL=1 λ
Because the path length difference is equal to an integer number of wavelengths, the waves will constructively interference and the sound is loud.

16. Conceptual Example 2 Out-Of-Phase Speakers
To make speakers operate, two wires must be connected between each speaker and the amplifier.
The two speakers vibrate in phase when the connections are exactly the same way on each speaker.
The speakers are out of phase with each other when one speaker is connected backwards.

17. Diffraction
Diffraction is the bending of a wave when it interacts with the edges of an opening or an obstacle.

18. Diffraction
Smaller openings have more diffraction. Larger openings have less diffraction.
Maximum diffraction happens when the wavelength is the same size as the opening.

19. Diffraction Circular opening diffraction equation
short wavelength high frequencies
long wavelength low frequencies
width D of the speaker opening
Circular opening diffraction equation

20. Beat frequency
tuning forks
Two overlapping waves traveling in the same direction with slightly different frequencies create sound beats.
Beat frequency demo 1 on YouTube
Beat frequency demo 2 on YouTube

21. Beat frequency
Beat frequency is equal to the difference between the two frequencies.

22. Waves are reflected when they reach a boundary.
Reflected waves
Waves are reflected when they reach a boundary. 1 3 4 2
forward motion
backward motion
Reflected waves are upside down (inverted).
When the timing is right, the incoming wave and the reflected wave can form standing waves.

23. Transverse standing wavesf1
2 f1
3 f1
fundamental
2nd harmonic
3rd harmonic
1st harmonic
node
antinode
Harmonic frequencies are multiples of a fundamental frequency.
No motion at the nodes. Maximum motion at the antinodes.
Strings produce all harmonic frequencies.
Ends of the string can't move so they are always nodes .
Standing wave demo on YouTube

24. Transverse standing waves
An integer number (n) of half-wavelengths must exactly fit between the fixed ends of the string. (The integer n is the number of loops.)
½ λ
Fundamental vibration mode has just 1 half- wavelength.
Strings can produce all the harmonic frequencies.

25. How long is the string?
Guitar string vibrates at a fundamental frequency of Hz. The string tension is 226 N. The string linear density is kg/m3.
n = 1 for the fundamental frequency.
Side calculation for the wave speed on the stretched string.

26. Longitudinal standing waves
Longitudinal standing wave pattern on a spring.
node: minimum vibration
antinode:maximum vibration

27. Sound wave for a tube open at both ends
Air molecules have maximum motion at the tube's open ends so the open ends are antinodes. The distance from one antinode to the next antinode is ½ λ.
The natural modes of vibration occur when an integer number (n) of half-wavelengths exactly fit between the open ends of the tube.
All the harmonic frequencies can be produced.

28. Sound wave for a tube open only at one end
Air molecules have maximum motion at the tube's open end and no motion at the closed end so the open end is an antinode and the closed end is a node. The distance from one antinode to the next node is ¼ λ.
The natural modes of vibration occur when an odd number of quarter-wavelengths exactly fit between the open ends of the tube.
Only the odd harmonic frequencies can be produced.

29. Example - A tube that is 2 m long
open at both ends
open end and closed end

30. The End