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notes14b - Superposition Interference and Standing Waves In...

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Superposition, Interference, and Standing Waves In the first part of Chap. 14, we considered the motion of a single wave in space and time What if there are two waves present simultaneously – in the same place and time Let the first wave have λ 1 and T 1 , while the second wave has λ 2 and T 2 The two waves (or more) can be added to give a resultant wave this is the Principle of Linear Superposition Consider the simplest example: λ 1 = λ 2
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Since both waves travel in the same medium, the wave speeds are the same, then T 1 =T 2 We make the additional condition, that the waves have the same phase – i.e. they start at the same time Constructive Interference The waves have A 1 =1 and A 2 =2. Here the sum of the amplitudes A sum =A 1 +A 2 = 3 (y=y 1 +y 2 ) A 1 A 2 Sum
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If the waves ( λ 1 = λ 2 and T 1 =T 2 ) are exactly out of phase, i.e. one starts a half cycle later than the other Destructive Interference If A 1 =A 2 , we have complete cancellation: A sum =0 A 1 A 2 sum These are special cases. Waves may have different wavelengths, periods, and amplitudes and may have some fractional phase difference. y=y 1 +y 1 =0
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Here are a few more examples: exactly out of phase ( π ), but different amplitudes Same amplitudes, but out of phase by ( π /2)
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Example Problem Speakers A and B are vibrating in phase. They are directed facing each other, are 7.80 m apart, and are each playing a 73.0-Hz tone. The speed of sound is 343 m/s. On a line between the speakers there are three points where constructive interference occurs. What are the distances of these three points from speaker A? Solution: Given: f A = f B =73.0 Hz, L=7.80 m, v=343 m/s
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m 70 . 4 Hz 73.0 m/s 343 v v = = = = λ λ T m 9 . 3 2 70 . 4 0 2 8 . 7 : 0 2 2 2 2 2 = = = = = + = x n n L x L x x L λ λ λ x is the distance to the first constructive interference point The next point (node) is half a wave-length away.
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