Solution
= Solution
The terms $u_{tt}$ and $c^2u_{xx}$ must have the same dimensions. Likewise $au$ must match $u_{tt}$. Thus
$$
\boxed{[c]=\mathrm{length}/\mathrm{time},\qquad[a]=\mathrm{time}^{-2}.}
$$
Only $c^2$ occurs in the equation, so its characteristic speed is $|c|$ when $c\ne0$.