math212,homework15december4january,solution

math212,homework15december4january,solution - Math 212 ,...

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Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 14 December 2007 – 4 January 2008 , Solution Exercise: Analyse the vibrations of an elastic solid cylinder occupying the region 0 r 1, 0 z 1, in cylindrical coordinates if its top and bottom are held fixed, its circular surface is free, and the initial velocity u t is zero. that is, find the general solution of v tt = c 2 ( v rr + r - 1 v r + r - 2 v θθ + v zz ) , v ( r, θ, 0 , t ) = v ( r, θ, 1 , t ) = v r (1 , θ, z, t ) = v t ( r, θ, z, 0) = 0 . Solution: First we look for separated solutions, v = R ( r )Θ( θ ) Z ( z ) T ( t ), of the PDE, which satisfies the given boundary and initial conditions. This leads to T 00 /T = c 2 [( R 00 /R ) + ( R 0 / ( rR )) + (Θ 00 / ( r 2 Θ)) + ( Z 00 /Z )), and Z (0) = Z (1) = R 0 (1) = T 0 (0) = 0. Due to the periodicity w.r.t. θ , the function Θ satisfies the periodic boundary conditions Θ(0) = Θ(2 π ), and Θ 0 (0) = Θ 0 (2 π ). Hence Z 00 = - βZ , Θ 00 = -
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This note was uploaded on 02/07/2011 for the course MATH 212 taught by Professor Friedmannbrock during the Fall '08 term at American University of Beirut.

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math212,homework15december4january,solution - Math 212 ,...

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