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A string of length L is fixed at both ends. The speed of waves on the string is c. The string is initially displaced a distance h uniformly along its length, and is released from rest at t=0. (The string initially has a very large slope at x=0 and x=L; assume the slope is infinite.)
For w_n=ckn=cn*pi/L, the displacement of the string for t>=0 the displacement is
Sum(A_n*cos(wt) + B_n*sin(wt))
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The solution is ...
The attached file shows how to analyze a general solution of the wave equation using Fourier analysis.