Consider an elastic string of length L. The end x = 0 is held fixed, while the end x = L

Chapter 10, Problem 10

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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 ux ( 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) = 0 0 x L/2 1 1, L/2 1 < x < L/2 + 1 (assume L > 2), 0, L/2 + 1 x 1 a. Find the displacement u( x, t). G b. With L = 10 and a = 1, plot u versus x for 0 x 10 and for several values of t. Pay particular attention to values of t between 3 and 7. Observe how the initial disturbance is reflected at each end of the string. G c. With L = 10 and a = 1, plot u versus t for several values of x. G d. Construct an animation of the solution in time for at least one period. e. Describe the motion of the string in a few sentences. G