If an elastic string is free at one end, the boundary condition to be satisfied there is

Chapter 10, Problem 9

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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(n x) 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].