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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.

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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.

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

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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

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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

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Path Difference = n If a wave is shifted 1 or 2, etc., then total constructive inteference will occur. wave 1 wave 2 1 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.

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

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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? 1.1 3. 3 2.2 4. 4

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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

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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.

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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? 1.1 2.2

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. S1S1. center line If two identical sources S 1 and S 2 are 180° out of phase, as shown here, then if point P is moved to a location 2 further from S 1 than from S 2, there will be __________ at P.. S2S2. PATH 2PATH 1. P A. total constructive interference B. total destructive interference C. something in between

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. S1S1. center line. S2S2. PATH 2PATH 1. P A. total constructive interference B. total destructive interference C. something in between If two identical sources S 1 and S 2 are 180° out of phase, as shown here, then if point P is moved to a location further from S 1 than from S 2, there will be __________ at P.

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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.

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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.

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Finding the maxima

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Path Difference for Maxima therefore

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

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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)

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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) 1.increases 2.decreases 3.remains the same

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S2S2 SOURCE 2 S1S1 SOURCE 1 Water wave patterns spreading out from two identical point sources S 1 and S 2 (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)

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S2S2 SOURCE 2 S1S1 SOURCE 1 (Click to continue stepwise animation) 5. (continued)

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S2S2 SOURCE 2 S1S1 SOURCE 1 (Click to continue stepwise animation) 5. (continued)

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SOURCE 2 S1S1 SOURCE 1 S2S2 (Click to continue stepwise animation) 5. (continued)

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SOURCE 2 S1S1 SOURCE 1 S2S2 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 5. (continued)

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SOURCE 2 S1S1 SOURCE 1 S2S2 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 5. (continued)

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SOURCE 2 S1S1 SOURCE 1 S2S2 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.] 5. (continued) P

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.. S2S2. A E S1S1 B C Which one of the points A, B, C, D, and E is on a second line of constructive interference (n = 2) from the center? . D Two identical, in-phase sources of water waves

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