 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-v/5/a/solution

For

c > 0

and

L > 0

y (x, t) = \frac{f ( x - c t ) + f ( x + c t )}{2} + \frac{1}{2 c} \int_{x - c t}^{x + c t} g (s) d s .

For the specified initial data, define the cumulative initial-velocity integral

A (s) = ⎩ ⎨ ⎧ 0, (L^{2} - s^{2}) /2, (L^{2} + s^{2}) /2, L^{2}, s \leq - L, - L \leq s \leq 0, 0 \leq s \leq L, s \geq L .

Then

y (x, t) = [A (x + c t) - A (x - c t)] / (2 c)

\pm L

, so the resulting solution is interpreted piecewise classically, or as a weak wave-equation solution across the corresponding characteristic curves.

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