The phrase "move back and forth" describes a motion that goes back and forth repeatedly. Synonyms for this phrase include oscillate, swing, sway, rock, wobble, undulate, and fluctuate. These words can convey different types of movement, such as a gentle swaying motion or a violent, uncontrolled movement. Other options include wiggle, jostle, jiggle, and shimmy. These words can add a playful or informal tone to the description of movement. Finally, verbs such as shuttle, traverse, and survey can describe a similar movement back and forth, but with a destination or purpose in mind rather than simply a repetitive motion.