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Math 316 – Assignment 8 Due: Wednesday, March 23, 2011 1. Consider the wave equation utt = uxx for an infinite string (x in (−∞, ∞)), with initial conditions u(x, 0) = f (x), ut (x, 0) = g(x). a) Write out d’Alembert’s solution u(x, t). b) Sketch the solution for t = 1 and t = 2 and t = 4, and explain its behavior as t increases when g(x) = 0 and f (x) is given by the following graph: f (x) 6 1 @ @ @ -3 -2 -1 @ @ @ 1 2 3 - x 4 c) Consider now f (x) = 0 and 1 if − 6 < x < −2, 1 if 2 < x < 4, g(x) = 0 otherwise. Sketch the characteristic curves originating from x0 = −6, −2, 2, 4. Calculate the solutions u(x, 1) and u(x, 2) and sketch their profiles. Hint: Use the characteristic curves to distinguish several cases. 2. Find the solution to the following wave equation: utt = c2 uxx , 0 < x < 1, t > 0, u(0, t) = 0, u(1, t) = 0, u(x, 0) = 2 sin(2πx) + 3 sin(3πx), ut (x, 0) = sin(πx). 3. Find the solution to the following wave equation: utt = c2 uxx , 0 < x < L, t > 0, ux (0, t) = 0, ux (L, t) = 0, u(x, 0) = f (x), ut (x, 0) = g(x).