 Solution

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

x = L /2

, the integration interval first moves through positive arguments, then crosses zero, and finally contains all of

[- L, L]

. The result is

y (\frac{L}{2}, t) = ⎩ ⎨ ⎧ L t /2, \frac{L ^{2} + ( c t - L /2 ) ^{2}}{4 c}, \frac{L ^{2}}{2 c}, 0 \leq c t \leq L /2, L /2 \leq c t \leq 3 L /2, c t \geq 3 L /2.

The final plateau is the integral of the entire initial velocity profile divided by

2 c

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