Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 4 / 14A / c / iii / Solution

Introduce $\tau =ct/L$. At the midpoint, the positive fronts arrive at $\tau =2m+1/2$ and the negative fronts at $\tau =2m+3/2$. HenceThe midpoint displacement is a square wave, initially zero, with period $2L/c$ and a phase delay $L/(2c)$. 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.