= Solution
Expand the initial <velocity> in the <Fourier sine basis>, $V(x)=\sum_{n\ge1}V_n\sin(n\pi x/L)$, where $V_n=(2/L)\int_0^LV(x)\sin(n\pi x/L)\,dx$. Zero initial <displacement> removes the cosine-in-time coefficients. Thus
$$
\boxed{y(x,t)=\sum_{n\ge1}\frac{V_n}{\omega_n}\sin\frac{n\pi x}{L}\sin(\omega_nt).}
$$
The <trigonometric identity> $2\sin a\sin b=\cos(a-b)-\cos(a+b)$ rewrites each summand as
$$
\frac{V_n}{2\omega_n}\left[\cos\left(\frac{n\pi x}{L}-\omega_nt\right)-\cos\left(\frac{n\pi x}{L}+\omega_nt\right)\right].
$$
It is therefore a superposition of <travelling waves> of <speeds> $+c$ and $-c$, whose interference enforces the fixed endpoints. For general square-integrable initial <velocity>, the series is interpreted in the standard finite-energy sense.
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