On a property of solutions of the wave equation

We consider solutions of the wave equation in $S'(R_{n+1})$ ($n$ is odd) that become zero in the doubly connected region $\vert q^0\vert>\vert\widetilde q\vert+a$. We show that if a condition of sufficient decrease at infinity is imposed on the solution the solution also vanishes in the region $\vert q^0\vert\leqslant\vert\widetilde q\vert-a$.