A mechanical wave is the transfer of energy through a medium (any material through a wave can travel). Please note that it is energy that is moving and not the material through which the energy travels. For example, the diagram shows how one wave of energy moves though water. Each colored dot represent one water molecule. I chose different colors for the dots to make it easier to compare them.
1. Dot of one color are all the same distance from the shore, on the right side of the diagram.
2. At different times, the relation of the dots to each other varies. For example, at 1 sec, the red and purple dots are in line with each other, but the green dot is higher.
3. At 4 seconds, the green and purple dots appear to be in line with each other and the red dot is higher.
4. 
At 6 seconds, the green and red dots are in line with each other and the purple dot is higher.
So what causes the dots to rise and fall? The dots rise on the crest of an energy wave that is traveling along the surface of the water. Again, it is important to note that the wave energy causes the surface of the water to rise and fall, but the molecules making up the water ARE NOT carried forward with the wave. Instead, as the water molecules move up and down they transfer energy to their neighboring water molecules which then rise and fall transferring energy to their neighbors and so on. Thus the energy causes displacement of the medium (the water moves up and down) as it moves.
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