The Linearized Dispersion Relation of Navier-Stokes Equation in Shallow Water

A few number of variants of Navier-Stokes equations and their higher order generalizations are derived to describe the two way propagation of small amplitude, wavelength, gravity waves on the surface of water in a canal. These systems are also seen to model the propagation of long-crested waves on large lakes or the ocean and in other contexts. Depending on linearized terms and positing a solution of the form ei(kx-wt), the wave frequency is formulated in terms of wave number. Finally, the phase speed has been established with viscous term for long wavelength where the first three terms correspond to the expansion of the full linearized dispersion relation.