Basic water wave theory

Suppose we observe a calm lake from a profile view. We let x denote the horizontal direction and y denote the upward direction. For simplicity we set y = 0 to be the surface of the lake. At this point we will assume that the lake has a constant depth of h. If the lake is disturbed (by a steady wind, for example) and waves are formed what is the motion of the particle (x,y)? What is the motion of the point (x,0) on the surface of the water? How do nearby points interact?*

Given a particle that occupies the location (x,y) when the sea is at rest, we denote the position of this particle at time t by