Solution
= Solution
{c}
At $ct=L/2$, the <interval> of integration has half-width $L/2$. Evaluating the cumulative <integral> in part <5/a> gives the <even function>
$$
\boxed{y\!\left(x,\frac L{2c}\right)=\begin{cases}
\dfrac{x^2+L^2/4}{2c},&|x|\leq L/2,\\[4pt]
\dfrac{L^2-(|x|-L/2)^2}{4c},&L/2\leq|x|\leq3L/2,\\[4pt]
0,&|x|\geq3L/2.
\end{cases}}
$$
It has value $L^2/(8c)$ at zero and maxima $L^2/(4c)$ at $x=\pm L/2$, then falls to zero at the two wavefronts.