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= u(0, t) = 0 u(x,0) = f(x) u₁(x,0) = 0 f(x) 0 10 12- x 0 Use the method of images to solve the problem, and sketch the graph of the solution u(x, t) for t = 1, t = 5, t = 5.5, and t = 6. for x ≤ 10 for 10 ≤ x ≤ 11 for 11 ≤ x ≤ 12 for x ≥ 12. Utt = 4uxx ut (0, t) = 0 u(x,0) = f(x) u₁(x,0) = 0 2. Repeat problem 1, but this time for the semi-infinite string with Neumann boundary conditions at x = 0, (0 < x <∞, t > 0) (t > 0) (0 ≤ x <∞) (0 ≤ x <∞) with the same function f(x) as in problem 1. Again use the method of images to solve the problem (this time you will need to use the even extension of f(x) to the entire real line), and sketch the graph of the solution u(x, t) for t = 1, t = 5, t = 5.5, and t = 6.