There are many types of waves, the two most common of which are longitudinal and transversal. They still follow the same formulaic models, but differ in the direction to which particles move.
Particles move parralel to the direction of the energy. For example, sound waves travel like this by having speaker drums move back and forth.
Particles move perpendicular to the direction of the energy. For example, ocean waves and light waves move like this. Motion is applied perpendicular to the motion of the energy.