Difference between revisions of "Template:Linear elastic plate on water time domain"

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but the dynamic condition needs to be modified to include the effect of the the plate
 
but the dynamic condition needs to be modified to include the effect of the the plate
 
<center><math> \rho g\zeta  + \rho \frac{\partial\Phi}{\partial t}
 
<center><math> \rho g\zeta  + \rho \frac{\partial\Phi}{\partial t}
= D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}
+
= \beta \frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}
 
, \ z=0; </math></center>
 
, \ z=0; </math></center>
 
We also have [[Laplace's Equation|Laplace's equation]]
 
We also have [[Laplace's Equation|Laplace's equation]]

Revision as of 01:17, 3 July 2009

We begin with the linear equations for a fluid. The kinematic condition is the same

[math]\displaystyle{ \frac{\partial\zeta}{\partial t} = \frac{\partial\Phi}{\partial z} , \ z=0; }[/math]

but the dynamic condition needs to be modified to include the effect of the the plate

[math]\displaystyle{ \rho g\zeta + \rho \frac{\partial\Phi}{\partial t} = \beta \frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} , \ z=0; }[/math]

We also have Laplace's equation

[math]\displaystyle{ \Delta \Phi = 0,\,\,-h\lt z\lt 0 }[/math]

and the usual non-flow condition at the bottom surface

[math]\displaystyle{ \partial_z \Phi = 0,\,\,z=-h, }[/math]

where [math]\displaystyle{ \zeta }[/math] is the surface displacement, [math]\displaystyle{ \Phi }[/math] is the velocity potential, and [math]\displaystyle{ \rho }[/math] is the fluid density.