Solved: Consider the wave equation a2uxx = utt in an infinite one-dimensional medium

Chapter 10, Problem 16

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Consider the wave equation a2uxx = 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. Show that f ( x at) = 2, 1 + at < x < 1 + at, 0, otherwise. Also determine f ( x + at). d. Sketch the solution found in part b at t = 0, t = 1/2a, t = 1/a, and t = 2/a, obtaining the results shown in Figure 10.7.7. Observe that an initial displacement produces two waves moving in opposite directions away from the original location; each wave consists of one-half of the initial displacement.