Consider an elastic string of length L. The end x = 0 is held fixed, while the end x = L is free; thus the boundary conditions are u(0,t) = 0 and u_x(L, t) = 0. The string is set in motion with no initial velocity from the initial position u(x, 0) = f{x), where f(x) = {1, 0, L/2 - 1 < x L/2 + 1 (L > 2), otherwise. Find the displacement u(x, t).
