Nonlinear Shallow Water Waves

Introduction

We want to consider waves occurring at the interface of the two fluids water and air. We assume that water is

[math]\displaystyle{ \frac{D \rho}{D t} (\vec{x} ,t) + \rho(\vec{x} ,t)\nabla \cdot u(\vec{x} ,t) = 0, x \in \Omega }[/math]

Since water is incompressible i.e. [math]\displaystyle{ \frac{D \rho}{D t} = 0 }[/math] and then [math]\displaystyle{ \nabla \cdot \vec{u} = 0 }[/math] i.e. the divergance of the velocity field is zero.

Conservation of momentum reads as follows

[math]\displaystyle{ \frac{D \vec{u}}{D t} (\vec{x} ,t) = \frac{-1}{\rho} \nabla p + g(0,-1) }[/math]

Assuming that changes in the vertical vel. are negligible and [math]\displaystyle{ \vec{u} }[/math] we have [math]\displaystyle{ \frac{D v}{D t} }[/math], thus, [math]\displaystyle{ 0 = \frac{-1}{\rho} }[/math]