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5hw5 - Physics 315 Oscillations and Waves Homework 5 Due in...

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Physics 315: Oscillations and Waves Homework 5: Due in class on Wednesday, Oct. 7th 1. Consider a uniform string of length l , tension T , and mass per unit length ρ which is stretched between two immovable walls. Show that the total energy of the string, which is the sum of its kinetic and potential energies, is E = 1 2 integraldisplay l 0 bracketleftBigg ρ parenleftbigg ∂y ∂t parenrightbigg 2 + T parenleftbigg ∂y ∂x parenrightbigg 2 bracketrightBigg dx, where y ( x,t ) is the string’s (relatively small) transverse displacement. Now, the general motion of the string can be represented as a linear superposition of the normal modes: y ( x,t ) = summationdisplay n =1 , A n sin parenleftBig x l parenrightBig cos parenleftbigg vt l φ n parenrightbigg , where v = radicalbig T/ρ . Demonstrate that E = summationdisplay n =1 , E n , where E n = 1 4 2 n A 2 n is the energy of the n th normal mode. Here, m = ρl is the mass of the string, and ω n = nπv/l the angular frequency of the n th normal mode.
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