The simplest example of a shock wave can be observed when a traffic light turns red on a road with lots of traffic, particularly if it's foggy out so that drivers can't see beyond the car in front of them. They won't see that traffic has stopped until they come to a sudden stop themselves. The edge of this pileup moves up the highway, against traffic, and marks a sharp jump in the velocity (and also the density) of cars.
In this animation we first see cars accelarating after a light turns green, giving a rarefaction wave, and then stopping at the next red light. The red curves map the trajectories of vehicles over time. The plot on top shows the density of cars.
This was created in MATLAB using traffic.m.
Some description of how this was computed, along with a brief introduction to finite volume methods: