The "shock wave" effect is because the exhaust gases are traveling at supersonic speed.

At first the flame expands - that gives you the point at the top. Then lower down, it cools from the outside in. That gives you the point at the bottom.

I think it has some thing to do with to do with the way the exhaust nozzles flare out and the tighten again.

I am not a rocket scientist so it could be anything.