Lect_32_post - Lecture 32 Standing waves Doppler effect...

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Lecture 32 Standing waves. Doppler effect.
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Reflected waves: fixed end A pulse travels through a rope towards the end that is tied to a hook in the wall (ie, fixed end ) The pulse is inverted (because of Newton s 3 rd law!) F on wall by string F on string by wall The force by the wall always pulls in the direction opposite to the pulse. Another way: Consider one wave going into the wall and another coming out of the wall. The superposition must give 0 at the wall. Virtual wave must be inverted:
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Reflected waves: free end A pulse travels through a rope towards the end that is tied to a ring that can slide up and down without friction along a vertical pole (ie, free end ) No force exerted on the free end, it just keeps going Fixed boundary condition Free boundary condition
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Standing waves A harmonic wave traveling along the + x direction is reflected at a fixed point. What is the result of its superposition with the reflected wave? No motion for these points ( nodes ) 2 k π λ = These points oscillate with the maximum possible amplitude ( antinodes ) Standing wave cos b ( ) cos a ( ) = 2sin a + b 2 " # $ $ % & ' ' sin a b 2 " # $ $ % & ' ' y 1 x , t ( ) = A cos kx ω t ( ) y 2 x , t ( ) = A cos kx + ω t ( ) y x , t ( ) = A cos kx ω t ( ) cos kx + ω t ( ) # $ % & ' ( y x , t ( ) = 2 A sin kx ( ) sin ω t ( ) If kx = 0, π ,2 π ,... x = 0, λ 2 , λ , 3 λ 2 ... y x , t ( ) = 0 If kx = π 2 , 3 π 2 ,... x = λ 4 , 3 λ 4 ... y x , t ( ) = ± 2 A sin ω t ( )
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+x -x Standing wave Node Antinode
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Standing waves and boundary conditions We obtained Nodes 0, , ,... 2 x λ λ = 3 Antinodes , ... 4 4 x λ λ = We need fixed ends to be nodes and free ends to be antinodes! Strong restriction on the waves that can survive with a given set of boundary conditions. y x , t ( ) = 2 A sin kx ( ) sin ω t ( )
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Normal modes Which standing waves can I have for a string of length L fixed at both ends? I need nodes at x = 0 and x = L Nodes 0, , ,... 2 x λ λ = , ,... for 1,2,... 2 2 L n n λ λ λ = = = 2 for 1,2,... n L n n λ = = Allowed standing waves ( normal modes ) between two fixed ends Mode n = n-th harmonic f n = v λ n = n v 2 L for n = 1,2,...
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2 free ends 1 fixed, 1 free 2 fixed ends 1 2 L λ = 2 L λ = 3 2 3 L λ = 4 2 L λ = Normal modes for fixed ends (lower row)
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