= Solution
Introduce $\tau=ct/L$. At the midpoint, the positive fronts arrive at $\tau=2m+1/2$ and the negative fronts at $\tau=2m+3/2$. Hence
$$
\boxed{y(L/2,t)=\begin{cases}a,&2m+\tfrac12<ct/L<2m+\tfrac32\text{ for some }m\geq0,\\0,&\text{otherwise, away from jump times.}\end{cases}}
$$
\b[The 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/ib/paper-4-midpoint.png]
{title=Midpoint displacement under repeated reflections of an applied step}
Back to article page
