In the case of mechanical waves , a material medium is disturbed. Originally physicists thought that waves can only form inside a material medium. After Einstein it became clear that light waves propagate inside the vacuum and are examples of electromagnetic waves . Waves can also be classified as transverse or longitudinal. Transverse waves cause the medium particles to move in a direction perpendicular to that of the wave propagation.
Longitudinal waves cause the medium particles to move parallel to the direction of wave propagation.
There are also waves where the motion of the particles is neither transverse nor longitudinal. For example, in ocean waves, the water particles move in a circular fashion around their equlibrium position. As you can see, waves are a common phenomenon that manifests itself in a very wide variety of different systems. To quantitatively describe wave motion, we need to know the displacement of each of the medium particles from their equilibrium positions at every instant in time. Such a function is referred to as the wavefunction .