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Unformatted text preview: 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 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 bracketleftBigg parenleftbigg y t parenrightbigg 2 + T parenleftbigg y x parenrightbigg 2 bracketrightBigg dx, where y ( x, t ) is the strings (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 n x l parenrightBig cos parenleftbigg n v t l n parenrightbigg , where v = radicalbig T/ . Demonstrate that E = summationdisplay n =1 , E n , where E n = 1 4 m 2 n A 2 n is the energy of the n th normal mode. Here, m = l is the mass of the string, and n = n v/l...
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