Boundary Values & Fourier Series · Hard problem

A string $0 \le x \le \pi$ is fixed at $x = 0$ and free at $x = \pi$: $u(0, t) = 0$, $u_x(\pi, t) = 0$, with $u_{tt} = u_{xx}$. It is released from rest with shape $u(x, 0) = \sin\dfrac{x}{2} + \sin\dfrac{3x}{2}$. At what smallest time $t > 0$ is the whole string momentarily flat ($u(x, t) = 0$ for every $x$)?