Huygens' principle says that points on a wave can be thought of as the sources of new waves. This makes a lot of sense to me in the context of a moving wave and diffraction. However, in the context of a wave pulse, like a single stone being dropped into a pond, it feels as if Huygens' principle would predict that the circular wave pulse would reflect back in on itself.

When I drop a stone in water it makes one ring coming out, Huygens' principle would expect that that ring would send waves back towards the center where the stone hit. What's going on here?