Difference between revisions of "Nonlinear Shallow Water Waves"
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<math> p = p_{atm} + \rho g(h(x,t) - z) </math> (pressure is hydrostatic) | <math> p = p_{atm} + \rho g(h(x,t) - z) </math> (pressure is hydrostatic) | ||
+ | |||
+ | Also, <math>\frac{D u}{D t} = \frac{-1}{\rho}\frac{\partial p}{\partial x}</math> | ||
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[[Category:789]] | [[Category:789]] |
Revision as of 20:53, 14 October 2008
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. Let [math]\displaystyle{ h(x,t) }[/math] be the local water depth or wave height. From previous lectures we have established the eqn. for Conservation of Mass.
[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 [math]\displaystyle{ \frac{D \rho}{D t} = 0 }[/math] i.e. [math]\displaystyle{ \nabla \cdot \vec{u} = 0 }[/math], the divergance of the velocity field is zero.
We can apply the Conservation of Momentum eqn. 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 velocity are negligible
and [math]\displaystyle{ \vec{u} = (u,v) }[/math], we have [math]\displaystyle{ \frac{D v}{D t} }[/math], thus, [math]\displaystyle{ 0 = \frac{-1}{\rho}\frac{\partial p}{\partial z} - g }[/math] or [math]\displaystyle{ \frac{\partial p}{\partial z} = -\rho g }[/math] and hence,
[math]\displaystyle{ p = p_{atm} + \rho g(h(x,t) - z) }[/math] (pressure is hydrostatic)
Also, [math]\displaystyle{ \frac{D u}{D t} = \frac{-1}{\rho}\frac{\partial p}{\partial x} }[/math]