 Solution

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

c t = L /2

, the interval of integration has half-width

L /2

. Evaluating the cumulative integral in part a gives the even function

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

It has value

L^{2} / (8 c)

at zero and maxima

L^{2} / (4 c)

x = \pm L /2

, then falls to zero at the two wavefronts.

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