I'm trying to animate an expanding circle as a texture on a sphere. I've tried using a wave texture, but that doesn't seem to be working. It has to start as a point (very small circle), then expand out a little way like a rock dropped in a pond (with no waves) and then disappear. The coordinates it starts on as well as its maximum size has to be random and repeat. It has to fade out at maximum size.
I'm stumped. Can this be done with anything but a custom written procedural?