p09fl30 - Lecture 30: String I (11 Nov 09) review 9 Nov HW...

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Unformatted text preview: Lecture 30: String I (11 Nov 09) review 9 Nov HW HJ theory of oscillator A. Review continuum string 1. FW Secs. 25 and 38; G Sec. 13.2 2. The Lagrangian density for transverse ( u ) vibrations of a 1D string with mass density and spatially varying tension ( x ) is L = 1 2 ( x )( u t ) 2- 1 2 ( x )( u x ) 2 3. The Euler-Lagrange equations are obtained from t L ( u/t ) + x L ( u/x )- L u = 0 and for the string L are ( x ) 2 u t 2 = x ( x ) u x 4. Hamiltonian formulation G Sec. 13.4 (FW Sec. 45), using notation u = u/t : = L / u ; H = u-L here: = ( x ) u t ; H = Z dx [ 1 2 ( x )( u t ) 2 + 1 2 ( x )( u x ) 2 ] 5. The case ( x ) = and ( x ) = (constants) gives rise to the elementary form of the wave equation with constant speed c 2 = / : 1 c 2 2 u t 2 = 2 u x 2 1 6. Normal modes: separate time and space variables6....
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This note was uploaded on 12/14/2009 for the course PHYS 711 taught by Professor Bruch during the Fall '09 term at Wisconsin.

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p09fl30 - Lecture 30: String I (11 Nov 09) review 9 Nov HW...

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