Abstract: Two preferred embodiments of an apparatus are disclosed for imparting combined synchronous reciprocating and oscillatory motions to a cutting tool of the type used with an engine lathe to cut a continuous circumferential serpentine-patterned groove in the surface of a workpiece held and rotated about an axis of rotation by the lathe. The tool is axially journaled and mounted on a slide block permitting oscillatory and reciprocating motions of the tool relative to the rotating workpiece. Direct inflexible drive mechanisms impart the reciprocating and oscillatory motions, where the motions are synchronous with each other and are timed so as to define an integral number of waves around the circumference of the workpiece. The first embodiment employs a cam, cam followers, and direct mechanical or hydraulic linkages therefrom utilizing a minimum of parts to impart the motions. The second embodiment employs two separate servo motors to impart the motions.