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Torsional vibration of a shaft is governed by the wave equation, where 0(x,t) is the angular displacement (angle of twist) along the shaft, z is the distance from the end of the shaft and t is time For a shaft of length 3T that is supported by frictionless bearings at each end, the boundary conditions are 0x(0, t) 0 (3m, t) = 0, t > 0. Suppose that the initial angular displacement and angular velocity are (,0) 4 cos(), 0t(x,0) = 3 +6 cos (2x), 0<z< 3T, respectively You may use the result that the eigenvalues of the boundary-value problem x -X 0, X' (0) = x'(37) = 0 are n 0,1,2,3,.. 32 with corresponding eigenfunctions nz Xn COS 3 Use the method of separation of variables to find the final solution 0(x, t). (t is possible to get half-marks for this question for a good but incomplete solution. However, half marks will still show as wrong with "How did I do?", so check your mark too.) e(r,t)