1. Consider the motion of a semi-infinite string with Dirichlet boundary condition at x = 0, Utt = 4uxx (0 0) (

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1 Consider The Motion Of A Semi Infinite String With Dirichlet Boundary Condition At X 0 Utt 4uxx 0 X T 0 1 (286.97 KiB) Viewed 39 times

1. Consider the motion of a semi-infinite string with Dirichlet boundary condition at x = 0, Utt = 4uxx (0<x<∞, t > 0) (t > 0) (0 < x <∞) (0 ≤ x <∞) where u(0, t) = 0 u(x,0) = f(x) ut(x,0) = 0 f(x) = 0 x - 10 12 - x 0 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 = 1, t = 5, t = 5.5, and t = 6. = (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 the end of the string. It is possible — and instructive — to do this all by hand, but it is easier to use Desmos; and Desmos will also produce nice movies for you showing how the wave evolves. See Assignment 8 for help on how to graph the solution using Desmos. ) Utt = 4UTr ut (0, t) = = 0 u(x,0) = f(x) ut (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.