Boundary Values & Fourier Series · Superchallenge problem

A string $0 \le x \le \pi$ with fixed ends obeys $u_{tt} = u_{xx}$. It starts flat, $u(x,0) = 0$, and is given the uniform velocity $u_t(x,0) = 1$. Find the largest displacement ever reached at the point $x = \dfrac{\pi}{4}$, that is, $\displaystyle\max_{t\ge0}u\!\left(\dfrac\pi4, t\right)$.