= Solution
Here the <D'Alembert formula with initial velocity> becomes
$$
u(x,t)=\frac12\int_{x-t}^{x+t}g(\xi)\,d\xi.
$$
When $t>1$ and
$$
1-t\leq x\leq t-1,
$$
the integration interval contains the whole support $[-1,1]$. Since $g$ is <odd>,
$$
u(x,t)=\frac12\int_{-1}^{1}\xi\,d\xi=0.
$$
Thus the requested region containing the line $x=0$ is
$$
\boxed{t>1,\qquad |x|\leq t-1.}
$$
This is the <central zero region for odd compactly supported initial velocity>. The initial disturbance splits into left-moving and right-moving waves; after they pass, their opposite signed impulses cancel and leave an expanding undisturbed region around the origin.
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