= Solution
For the <wave equation>, introduce characteristic coordinates $\xi=x-ct$, $\eta=x+ct$ when $c\ne0$. The equation becomes $u_{\xi\eta}=0$, so $u=F(x-ct)+G(x+ct)$. The <initial conditions> imply $F+G=\phi$ and $-cF'+cG'=0$. Constants can be absorbed into $F,G$, giving the <D'Alembert formula>
$$
\boxed{u(x,t)=\frac12\bigl[\phi(x-ct)+\phi(x+ct)\bigr].}
$$
If $\phi$ vanishes outside $[-\alpha,\alpha]$, then $u(x,t)=0$ whenever $|x|>\alpha+|c|t$. Thus \b[Property P fails: the wave equation has <finite propagation speed> $|c|$]. No arbitrarily short time can transmit the disturbance arbitrarily far. If $c=0$, the zero initial velocity instead gives $u(x,t)=\phi(x)$, and the conclusion still holds.
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