Question

# 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). In each of Problems 1 through 4, carry out the following steps. Let L = 10 and a = 1 in parts (b) through (d). a. Find the displacement u(x, t) for the given initial position f(x). Gb. Plot u(x, t) versus x for 0≤x≤ 10 and for several values oft between t = 0 and t = 20. G c. Plot u(x, t) versus t for 0 ≤ ≤ 20 and for several values of x. Gd. Construct an animation of the solution in time for at least one period. e. Describe the motion of the string in a few sentences. 0≤x≤L/2, 1. f(x) = (2x/L, 2(L-x)/L, (4x/L, 2. f(x) = 1, 4. f(x) = 3. f(x) = 8x(L- x)²/L³ (4(L-X)/L, 3L/4≤x≤L L/2 < x < L 0 0≤x≤L/2-1 1, L/2-1<x<L/2+1 (assume L> 2), 0, L/2+1 ≤ x ≤1 S4.x/L, 0≤x≤ L/4, L/4< x < 3L/4, Consider an elastic string of length L whose ends are held fixed. The string is set in motion from its equilibrium position with an initial velocity u(x, 0) = g(x). In each of Problems 5 through 8, carry out the following steps. Let L = 10 and a = 1 in parts (b) through (d). a. Find the displacement u(x, t) for the given g(x). G b. Plot u(x, t) versus x for 0 ≤ x ≤ 10 and for several values oft between t = 0 and 1 = 20. Gc. Plot u(x, f) versus t for 0 << 20 and for several values of x. Gd. Construct an animation of the solution in time for at least one period. e. Describe the motion of the string in a few sentences. 0≤x≤L/2, 5. g(x) = (2x/L, [2(L-x)/L, 6. g(x) = ¹, 4(L-x)/L, 7. g(x) = 8x(L-x)²/L³ L/2<x<L 0≤x≤ L/4, L/4 < x < 3L/4, 3L/4≤ x ≤L  Fig: 1