 Solution

Fixed endpoints select the spatial normal modes $\sin (n\pi x/L)$, with angular frequencies ${\omega }_n=n\pi c/L$. Zero initial velocity selects cosine time dependence. Matching the two nonzero initial amplitudes givesEach term separately satisfies the wave equation and fixed-end boundary conditions. At $t=0$ the prescribed profile is recovered, and differentiating in time gives zero initial velocity.