It does! It moves up and down. It doesn't move horizontally because,
as everybody knows by now, waves move energy, not material.
energy and not matter. The wave's energy causes the leaf to move up and down, but it does not carry the leaf with it towards the shore.
When a wave comes, the leaf floating on the water's surface will move with the motion of the wave. The leaf will rise and fall along with the water's movement but will remain on the surface due to buoyancy.
A wave is a transfer of energy through the water that causes objects on the surface to move up and down as the energy passes through them. However, the wave itself does not have the power to move objects horizontally, so it cannot push the leaf towards the shore. The leaf's movement is a result of the vertical motion caused by the wave passing underneath it.
The wave travels through the water without moving the water with it (the water moves but then as the wave passes the water moves back to where it was). The floating leaf stays with the water as the wave passes on its way to the shore.
Move up and down but stay in the same position
move move the leaf
A wave. The wave advances, but the individual water particles go back to their places. This is typical for all kinds of waves: energy is transferred from one part of the substance (water, in this case), to another.
The wave is not a current. The energy gets transmitted from one part of the water to another. A good comparison is a row of glass marbles - if another glass marble bumps into the first marble in the row, the LAST one will move away from the group. I strongly suggest you try it out. What happens here is that the energy gets transmitted from one marble to the next (in this case, it happens rather quickly). The situation in the water wave is similar. The energy gets transmitted, without the water itself moving (as it would in a current). The leaf is at the surface of the water wave. Deep in the water, the water wave is more of a compressive wave. At the surface, it is more of a transverse wave. At the surface, the water moves up and moves down, but it doesn't move longitudinally. The leaf follows the surface -- up and down.
When particles move up and down with a wave, this is known as vertical or transverse wave motion. In this type of wave, particles move perpendicular to the direction in which the wave is traveling. This motion creates crests and troughs in the wave pattern.
The particles of a transverse wave move perpendicular to the direction of the wave. As the wave passes through a medium, the particles move up and down or side to side in a direction that is perpendicular to the direction of the wave propagation.
compressional wave
transverse wave