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Transcript of Superposition and Interference Chapter 17. Expectations After this chapter, students will: ...
![Page 1: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/1.jpg)
Superposition and Interference
Chapter 17
![Page 2: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/2.jpg)
Expectations
After this chapter, students will: understand the principle of linear superposition apply the concept of interference to the interaction
of two or more waves use the concept of diffraction to understand what
happens when a wave passes through a limited opening
analyze situations that produce standing waves
![Page 3: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/3.jpg)
Superposition
We defined a wave as a traveling condition or disturbance in a material.
We can also think of a wave as a traveling instruction to each particle of matter in the material: Be displaced upward! Be displaced downward! Be compressed! Re rarefied!
What happens when two waves give instructions to the same bit of material?
![Page 4: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/4.jpg)
Superposition
What happens when two waves give instructions to the same bit of material?
The principle of linear superposition says that the two instructions add.
More formally: when two or more waves are present at the same place, the resultant disturbance is the sum of the individual disturbances due to the individual waves.
![Page 5: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/5.jpg)
Superposition
Here, two positive-going pulses meet at a point. At the moment both pulses are centered at the same point, the displacement there is the sum of the individual pulse heights.
This is an example of constructive interference.
![Page 6: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/6.jpg)
Superposition
Now, a positive-going pulse meets a negative-going one. The result, where they meet, is again the sum of the pulse heights, which in this case is zero.
This is an example of destructive interference.
![Page 7: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/7.jpg)
Superposition: Interference
Suppose two sources of waves, in phase, send waves to the same location. The interference depends on the difference in path lengths.
d1
d2
![Page 8: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/8.jpg)
Superposition: Interference
Constructive interference: path difference is an integer number of wavelengths.
Destructive interference: path difference is a half-integer number of wavelengths.
... 2, 1, ,0 21 mmdd
... 2, 1, ,0 2
121
mmdd
![Page 9: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/9.jpg)
Huygens’ Construction
Christiaan Huygens 1629 – 1695
Dutch natural philosopher
Advanced the wave theory of light; constructed practical telescopes and microscopes, pendulum and spring-regulated clocks
![Page 10: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/10.jpg)
Huygens’ Construction
Huygens said that each point on a wavefront can be thought of as a source of “wavelets:”
![Page 11: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/11.jpg)
Huygens’ Construction
The wavelets interfere. If the wavefront is infinite in extent, for every point, in every direction except straight ahead, there is another point whose wavelets interfere destructively.
![Page 12: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/12.jpg)
Huygens’ Construction
If the wavefront is limited, there are points near the edge which, for some directions, have no other point to serve as a “partner in destruction.” The wave spreads out at the edges: diffraction.
![Page 13: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/13.jpg)
Diffraction
We can calculate directions for destructive interference:
D
D/2
(m + ½)
Dm
D
m
sin :0
2
21
sin
![Page 14: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/14.jpg)
Diffraction
Note that this equation:
gives the first minimum in the diffraction pattern
(m = 0).
Applies to a “slit” opening: width (D) << height
D
sin
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Diffraction
For a circular opening, we calculate:
This time, D is the diameter of the opening.
Notice that , the “spread angle,” is small as long as the opening size D is much larger than .
D
22.1sin
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Interference: Different Frequencies
So far, we’ve been considering the interference of waves that have the same frequency.
What happens if two waves of different frequencies interfere at a location?
![Page 17: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/17.jpg)
Interference: Different Frequencies
placeholder: beat frequency plot
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Interference: Different Frequencies
The modulation or “beat” frequency in the superposition of the two waves is simply the difference between the interfering frequencies:
Note that the closer the two frequencies are to each other, the lower is the beat interference modulation frequency between them.
21 fffB
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Interference: Standing Waves
Consider a transverse pulse on a string, encountering a rigidly-fixed end of the string.
The pulse is inverted
and reflected from that
end.
![Page 20: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/20.jpg)
Interference: Standing WavesInstead of a pulse, consider a periodic wave that is so
reflected.
The “inverted” wave (phase-shifted by 180°, or half a cycle) interferes with the incident wave, producing a periodically-varying pattern in the string.
What if the length of the string is a multiple of half the wavelength?
![Page 21: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/21.jpg)
Interference: Standing Waves
![Page 22: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/22.jpg)
Interference: Standing Waves
Each half- section of the string becomes a “loop.”
nodes
antinodes
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Interference: Standing WavesIf the string’s length is an integer multiple of half a
wavelength:
But: So,
The natural or standing-wave frequencies of a string depend on the velocity and string length.
And velocity depends on tension and linear mass density. This tells us how to tune a stringed instrument.
... 3, 2, 1, 2
nnL
f
v L
nvf
f
nvL
2or ,
2
![Page 24: Superposition and Interference Chapter 17. Expectations After this chapter, students will: understand the principle of linear superposition apply.](https://reader035.fdocuments.in/reader035/viewer/2022062222/56649d9f5503460f94a89969/html5/thumbnails/24.jpg)
Standing Longitudinal Waves
We can calculate natural standing-wave frequencies for sound waves in tubes from the same sort of analysis.
The results:
... 7, 5, 3, 1, 4
... 4, 3, 2, 1, 2
nL
nvf
nL
nvf
n
n
tube open at both ends
tube closed at one end