This technique may be obvious to most people, still I thought I would show you how I alter my rims lately. Once the clay is centered and opened I usually make one pull or two straight up leaving the rim thick-ish but also even (not bulbous at the top). I cut at this point because to do so when the pot is almost finished the walls are way to weak and will distort.
The cut-off wire I use for this is super thin. I have tried with a thicker wire but got a lot of drag. I simply mark the pot at six points around the rim. It actually takes me about about 2 seconds only pressing the wire down three times. I am not aiming for perfection in the marking, just a rudimentary guide to shoot for when cutting. You could make your marks wherever you wish.
Photo below shows the rim with the guide marks.
Next I take my wire and while the wheel is slowly spinning beginning at one of the marks (I usually start in the up position) I cut, attempting to reach my low point at the next mark, then begin coming up again reaching the high point at the next mark, continuing around until I reach my first mark in the up position again.
This leaves me with a wavy but sharp edged and squared off rim. To smooth I wet the rim and holding two fingers opposite each other gently press down on the angled edges until the rim has a nice pleasing roundness.
At this point I continue throwing, being careful to stop my pulls slightly below the rim so as not to loose my wavy edge. And voila'. A nice wavy rim with some visual interest and movement.
One point to make. When you make your marks, I think you have to make them in even numbers, otherwise when you get back to the beginning you will be in the wrong position (I start up, but with only five marks I would be down when I get back to the beginning, while six marks puts me in the up position. Good Luck!