Show that the initial-value problem for the one-dimensional
Question and Solution
Show that the initial-value problem for the one-dimensional wave equation 8 < : utt.x; t / D c2uxx.x; t / u.x; 0/ D p.x/
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Show that the initial-value problem for the one-dimensional wave equation 8 < : utt.x; t / D c2uxx.x; t / u.x; 0/ D p.x/ ut.x; 0/ D q.x/ has the solution u.x; t / D 1 2 p.x?ct/Cp.xCct/ C 1 2c Z xCct x?ct q.s/ ds: (Note that we have used subscripts x and t instead of 1 and 2 to denote the partial derivatives here. This is common usage in dealing with partial differential equations.)
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