Answer Happy

1. Consider the motion of a semi-infinite string with Dirichlet boundary condition at x = 0, (0<x<∞, t > 0) (t > 0) (0 < x <∞) (0 ≤ x <∞) where Utt = 4uxx u(0, t) = 0 u(x, 0) = f(x) ut(x,0) = 0 f(x) = 0 - 10 X - 12- 0 X for x 10 for 10 ≤ x ≤ 11 for 11 < x < 12 for x > 12. Use the method of images to solve the problem, and sketch the graph of the solution u(x, t) for t t = 5, t = 5.5, and t = 6. 19 (You will first define h(x) to be the odd extension of f(x) to the entire real line, and then use D'Alembert's formula u(x, t) = ½ [h(x + at) + h(x − at)] to find the solution of the wave equation with initial data given by h(x). Then graph u for x ≥ 0. If you do this correctly, the graphs at t = = 5, t = 5.5, and t 6 will show what happens when the left-moving wave arising from the initial disturbance reaches =