Linear (Airy) wave theory gives the dispersion relation ω² = g k tanh(kh) with k = 2π/λ. The phase speed is c = ω/k: in deep water (h > λ/2) c = √(gλ/2π), in shallow water (h < λ/20) c = √(gh). Energy travels at the group speed cg = ½ c [1 + 2kh/sinh(2kh)] (half of c in deep water, equal to c in shallow). Two close wavelengths make a group that moves at c_g, visible when the second wave is on.

Water particles move in ellipses with horizontal and vertical amplitudes a cosh k(z+h)/sinh kh and a sinh k(z+h)/sinh kh: circles in deep water that shrink with depth, flat ellipses in shallow water. Energy density is ½ ρ g a². Simplifications: small amplitude (a k < 0.44, the Stokes breaking limit), no wind input, no currents, flat bottom; the display is vertically exaggerated and labelled. Educational model.