Yup, Al's right. What you're experiencing is vibration in the wood. This can come from a variety of specific factors but is basically the wood repeatedly letting go and then recatching on the chisel. Same effect as driving on washboard surfaces.
Solutions are as Al said. Also, when turning a bowl relatively thin, turn a little on the rim, bring to the point where it's ready to sand, then turn the next little bit. This avoids making a cut way out on thin wood with only thin wood below it also. Kinda like stairstepping it.
Make sure you're using the bevel and cutting shavings, not using the tool as a scraper. If you get chatter (the other name for it), back off the cut till you're on smooth wood and redo it with a very light touch. Hard to eliminate the chatter starting at a point that is already chattering.
Worse comes to worst, pad you fingers with a soft cloth or leather glove and give a little support on the outside/inside of the bowl as you make the cut on the reverse surface.
Good luck,
dietrich