Solved: Consider the wave equation a2 uxx = utt in an infinite one-dimensional medium

Chapter 10, Problem 16

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Consider the wave equation a2 uxx = utt in an infinite one-dimensional medium subject to the initial conditions u(x, 0) = f(x), ut(x, 0) = 0, < x < . (a) Using the form of the solution obtained in 13, show that and must satisfy (x) + (x) = f(x), (x) + (x) = 0. (b) Solve the equations of part (a) for and , and thereby show that u(x, t) = 1 2 f(x at) + f(x + at) . This form of the solution was obtained by DAlembert in 1746. Hint: Note that the equation (x) = (x) is solved by choosing (x) = (x) + c. (c) Let f(x) = 2, 1 < x < 1, 0, otherwise

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