Linear water-wave theory: dispersion, orbit shapes with depth and the transition from deep to shallow water.
The science
For small-amplitude waves on water of depth h the surface is eta = a cos(kx - w t) and the frequency obeys the dispersion relation w^2 = g k tanh(k h), with k = 2 pi / lambda. The phase speed is c = w / k and the speed of the energy (group velocity) is cg = c/2 (1 + 2kh / sinh 2kh).
Each water particle moves in a closed ellipse with semi-axes a cosh(k(h - z)) / sinh(kh) (horizontal) and a sinh(k(h - z)) / sinh(kh) (vertical), where z is the depth below the surface. In deep water (h greater than lambda/2) the orbits are circles that decay as exp(-kz); near the bed the vertical motion vanishes. Long waves in shallow water travel at sqrt(g h), the tsunami regime.
Simplifications: linear theory (fails when H/lambda exceeds about 1/7 or H/h exceeds 0.78 where waves break), no currents, flat bed, no viscosity, orbits and heights are drawn magnified for visibility. Checks: Newton inversion of the dispersion relation and the two analytic limits.