= Solution
{c}
Let $w$ be the difference of two sufficiently regular solutions of the <wave equation> with the prescribed data. By <linearity>, $w$ has zero <initial displacement>, zero <initial velocity>, and homogeneous <Dirichlet boundary conditions>. Define its <wave energy> by
$$
E(t)=\frac12\int_0^\pi(w_t^2+w_x^2)\,dx\geq0.
$$
For $\alpha=0$, <differentiation under the integral sign> and <integration by parts> yield
$$
E'(t)=\int_0^\pi w_t(w_{tt}-w_{xx})\,dx+[w_tw_x]_0^\pi=0.
$$
The <boundary term> vanishes because $w=0$ at both fixed endpoints implies $w_t=0$ there. The zero <initial data> give $E(0)=0$, so $w_t=w_x=0$ for all times. Thus $w$ is spatially constant, and its zero <boundary values> give $\boxed{w=0}$. This proves <uniqueness> for the undamped <wave equation>.
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