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 incompressible, viscous effects are negligible and that the typical wave lengths are much larger than the water depth. For "shallow" water we assume no variations in the y-dirn. Let the domain of interest be denoted by [math]\displaystyle{ \Omega }[/math] with z-axis vertical and x-axis horizontal, [math]\displaystyle{ \vec{x} = (x,z) }[/math]. Let [math]\displaystyle{ h_{o} }[/math] be the at rest water depth.

[math]\displaystyle{ \frac{D \rho}{D t} (\vec{x} ,t) + \rho(\vec{x} ,t)\nabla \cdot \vec{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]