I'd like to imitate a ripple, imagining the effect off pushing the central ring down as a result of impact and seeing it oscillate with heavy damping, something like this. . My assumption is that I can just do an offset for the other rings, with the time difference (C) being proportional to the radius of each ring and it will look decent. How can I do this effect? I had a go with the 'Elastic' interpolation mode, but I wasn't able to get enough control over motion.
If you want to make it so theres a ripple when you push from the middle ring, try using dynamic paint in the physics tab. Make a separate object as a brush, and your circles as the canvas, and change surface type to waves. Also, give it a bit of subdivision so it looks nicer.