Which word means to move in a smooth, wave-like motion?

The term "undulate" specifically refers to a smooth, wave-like motion, often associated with the movement found in water waves, the flow of sand, or even certain biological motions, such as the way a snake slithers. This word captures the essence of a flowing, successive rise and fall, which is central to the concept of wave motion.

While "oscillate" suggests a back-and-forth motion that can resemble undulation, it is typically more rigid in context and often applies to repetitive movements along a fixed path rather than the smooth fluidity of undulation.

"Rotate," on the other hand, describes a circular motion around an axis, which does not carry the wave-like quality indicated in the question.

"Vibrate" refers to a rapid back-and-forth movement, often associated with sound waves or mechanical movements, which has more of a sharp, quick character than the smooth fluidity implied by undulation.

Therefore, "undulate" is the most fitting choice when defining a smooth, wave-like motion.