Solved: If an elastic string is free at one end, the boundary condition to be satisfied
Chapter 10, Problem 9(choose chapter or problem)
If an elastic string is free at one end, the boundary condition to be satisfied there is that ux = 0. Find the displacement u(x, t) in an elastic string of length L, fixed at x = 0 and free at x = L, set in motion with no initial velocity from the initial position u(x, 0) = f(x), where f is a given function. Hint: Show that the fundamental solutions for this problem, satisfying all conditions except the nonhomogeneous initial condition, are un(x, t) = sin nx cos nat, where n = (2n 1)/2L, n = 1, 2, .... Compare this problem with of Section 10.6; pay particular attention to the extension of the initial data out of the original interval [0, L].
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