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Some examples of things that move back and forth include a swing, a pendulum, a rocking chair, and a seesaw.
The particles in a solid move back and forth in place.
"Oscillate" means to move back and forth in a regular rhythm.
Transverse waves cause matter to move back and forth perpendicular to the direction of the wave. Examples include light waves and water waves.
Transverse waves cause particles to move perpendicular to the direction of the wave. This leads to a back-and-forth motion of particles. Examples of transverse waves include electromagnetic waves such as light and water waves.
Those are called oscillatory waves. These waves exhibit a periodic motion where energy is transferred in a back-and-forth manner. Examples include water waves and sound waves.