In a homework, we illustrated the exponential map on the Clifford torus. You can look at this with a computer. Paste the following into mathematica and move the slider to see the wave front move.

Manipulate[ParametricPlot[Mod[r{Cos[t],Sin[t]},1],{t,0,2Pi}],{r,0.1,5}]
 

Here is a more sophisticated picture, where the wave front density is colorized. We proved in this paper from 2024 that the wave front becomes dense in a precise way. The details on how the density behaves is highly nontrivial however.