Let $u$ solve the 3-d wave equation: $u_{tt}-\Delta u =0$ such that $u=g$ and $u_t=h$ for $t=0$ and where $g$ and $h$ are both assumed to be compactly supported and smooth. I have shown that there exists $C>0$ such that $|u(x,t)|<C\cdot t^{-1}$ for all $x\in \mathbb{R}^3$ and $t>0$.
Now I have to show that is optimal in the sense that it is not true in general that $|u(x,t)|< C /t^a$ for $a>1$. How do I do this?
Use an explicit example. The function $g$ is not helpful; its contribution dissipates as $t^{-2}$ in three dimensions as it is stretched over a sphere of radius $t$ (this should also be familiar from physics: light intensity, etc). I would take $g=0$ and $h=1$ in the unit ball (tapering off in some way to make $h$ smooth and compactly supported).
Looking at the integral formula for solution (case $n=3$), you should see that the relevant average of $h$ decays as $t^{-2}$, and it has a factor of $t^{3-2}$, so the example works. (Since the kernel is positive, the solution can be estimated from below by disregarding the part of $h$ outside the unit ball.)