 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-4/14a/c/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

Introduce

τ = c t / L

. At the midpoint, the positive fronts arrive at

τ = 2 m + 1/2

and the negative fronts at

τ = 2 m + 3/2

. Hence

y (L /2, t) = {a, 0, 2 m + \frac{1}{2} < c t / L < 2 m + \frac{3}{2} for some m \geq 0, otherwise, away from jump times.

The midpoint displacement is a square wave, initially zero, with period

2 L / c

and a phase delay

L / (2 c)

. At a front, choosing

H (0) = 1/2

gives value

a /2

. The dotted vertical segments below denote jumps; they are not continuous changes through intermediate displacements.

 New to topics? Read the docs here!