= Solution
Use unit speed and write the <Cauchy data> as $u(0)=f$, $u_t(0)=g$. For finite-energy data define the <wave energy estimate> quantity
$$
E(t)=\frac12\int_{\mathbb R^n}\bigl(u_t^2+|\nabla_xu|^2\bigr)\,dx.
$$
Multiply $u_{tt}-\Delta_xu=0$ by $u_t$ and use <integration by parts>. With compact support or sufficient decay the boundary flux is zero, so
$$
E'(t)=\int u_t(u_{tt}-\Delta_xu)\,dx=0,
\qquad
\boxed{\|u_t(t)\|_2^2+\|\nabla_xu(t)\|_2^2=\|g\|_2^2+\|\nabla f\|_2^2.}
$$
For general finite-energy solutions, cutoff or approximation arguments justify this identity; equivalently the local estimate below, applied in both time directions and with radii tending to infinity, gives the same equality. Arbitrary <smooth> data need not have finite global energy; then the global bound with an infinite right side is uninformative, while the local estimate remains useful.
If $f\in H^1$ and $g\in L^2$, the <fundamental theorem of calculus> and the <wave energy estimate> further give
$$
\|u(t)\|_2\leq\|f\|_2+|t|\sqrt{\|g\|_2^2+\|\nabla f\|_2^2}.
$$
Together these yield an a priori bound for $\|u(t)\|_{H^1}+\|u_t(t)\|_2$ on each bounded time interval. No existence assumption is proved by the estimate itself; it controls any sufficiently regular solution.
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