math212,homework15december4january,solution

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

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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 /T = c 2 [( R /R ) + ( R / ( rR )) + (Θ / ( r 2 Θ)) + ( Z /Z )), and Z (0) = Z (1) = R (1) = T (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) = Θ (2 π ). Hence Z = - βZ , Θ = - α Θ, R + r - 1 R - αr - 2 R + γR = 0, and T = - c 2 ( γ + β ) T , for some numbers α, β, γ . The problem for Θ has solutions α = - n 2 , ( n = 1 , 1 , 2 , . . . ), Θ = a cos + b sin , ( a, b IR), and the problem for Z has solutions β = m 2 π 2 , ( m = 1 , 2 , . . . ), Z = sin mπz . The problem for R is a singular Sturm-Liouville problem which has to be accomplished by the
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