Consider the wave equation a2 uxx = utt in an infinite one-dimensional medium subject to
Chapter 10, Problem 17(choose chapter or problem)
Consider the wave equation a2 uxx = utt in an infinite one-dimensional medium subject to the initial conditions u(x, 0) = 0, ut(x, 0) = g(x), < x < . (a) Using the form of the solution obtained in 13, show that(x) + (x) = 0,a(x) + a(x) = g(x).(b) Use the first equation of part (a) to show that (x) = (x). Then use the secondequation to show that 2a(x) = g(x) and therefore that(x) = 12a xx0g() d + (x0),where x0 is arbitrary. Finally, determine (x).(c) Show thatu(x, t) = 12a x+atxatg() d.
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