Support widening for the wave equation ($\mathrm{supp}u(\cdot ,t)\subseteq [-a-ct,a+ct]$)

For whole-line Cauchy data with support contained in $[-a,a]$, the displayed inclusion holds at $t\ge 0$. A short check uses D'Alembert's formula: if $x>a+ct$, both sampled endpoints and the whole interval of integration lie to the right of $a$, so all contributions vanish; the argument for $x<-a-ct$ is identical. The inclusion can be strict, and a nonzero response at the first permitted arrival is not guaranteed.