Consider an elastic string of length L whose ends are held fixed. The string is set in motion with no initial velocity from an initial position u(x, 0) = f(x). Let L10 and a 1 in parts (b) and (c). (A computer algebra system is recommended.) 2x L f(x)= 2(L-X), osxs늘 <XSL L (a) Find the displacement u(x, t) for the given initial position f(x). (Use a to represent an arbitrary constant.) u(x,t) = Σ 1)3. 1 (b) Plot u(x, t) versus x for 0 ≤ x ≤ 10 and for several values of t between t = 0 and t = 10. u(x, 0) u(x, 2.5) 1.0F 0.85 0.6 468 10 A b P V 1.0 0.8 0.6 0.4 0.4 0.2 0.2 0.0 x 0.0 0 2 68 10 0 2 4 6 8 10 u(x, 7.5) 0.0 x -0.2 2 4 6 8 10 u(x, 10) 0.0 -0.2 2 -0.4 -0.4 -0.6 x -0.8 -1.01 -0.6 -0.8 -1.0 x u(x, 0) 0.5 u(x, 2.5) 0.5F u(x, 7.5) u(x, 10) 0.4 0.4 0.0 x 0.0 x 2 4 6 8 10 0.3 0.3 -0.1 -0.1 2 4 6 8 10 0.2 0.2 -0.2 -0.2 0.1 0.1 -0.3 -0.3 -0.4 -0.4 0.0 x 0.0 x 0 2 4 8 10 0 2 4 6 8 10 -0.51 -0.5 u(x, 0) u(x, 2.5) 0.0 -0.2 2 4 6 8 10 u(x, 7.5) 1.0r x 0.8 0.6 0.4 0.2 0.0 u(x, 10) 1.0 0.8 0.6 PVAL 0.0 x -0.2 -0.4 2 4 6 8 10 -0.6 -0.8 -1.0 -0.4 -0.6 -0.8 -1.0 -0.1 x 0 2 468 10 u(x, 7.5) 0.5F 0.4 0.4 0.2 0.0 0 2 4 6 8 10 u(x, 10) 0.5 P v b b u(x, 0) 0.0 x u(x, 2.5) 0.0 -0.1 246 8 10 2 4 6 8 10 -0.2 -0.3 -0.4 -0.5 -0.2 -0.3 -0.4 -0.5 0.3 0.2 0.1 0.0 0 2 4 6 8 10 x 0.4 0.3 0.2 0.1 0.0 0 2 468 10 x x (c) Plot u(x, t) versus t for 0 ≤t≤ 50 and for several values of x. u(1, t) u(3, t) MA MA -0.4 u(1, t) 0.4 0.2 20 30 -0.2 u(3, t) 0.4 0.2 MM 0.0 -0.2 -0.4 u(1, t) 1.0 0.5 0.0 -0.5 -1.0t u(1, t) 1.0 0.5 0.0 -0.5 -1.0b 10 20 30 40 20 40 ما 0.0 -0.2 -0.4 AA u(3, t) 1.0 r 0.5 10/ 20 30 40 u(5, t) 0.4 0.2 0.0 1 10 20 400 -0.2 -0.4 u(5, t) 0.4 0.2 MA 0.0 -0.2 -0.4 u(5, t) 1.0 0.5 10 20 30 40 50 MA MA 0.0 -0.5 -1.0t u(3, t) 1.0 0.5 10 20 30 40 -0.5 -1.0t (5, 1) 1.0 0.5 0.0 0.0 40 50 10 20 30 40 50 20 40 -0.5 -1.0 MA MAA -0.5 -1.0b (d) Describe the motion of the string in a few sentences. This answer has not been graded yet.