Difference between revisions of "Template:Boundary value problem for a fixed body"

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The [[Standard Linear Wave Scattering Problem]]
 
The [[Standard Linear Wave Scattering Problem]]
 
in [[Finite Depth]] for a fixed body is  
 
in [[Finite Depth]] for a fixed body is  
<center><math>
+
{{standard linear wave scattering equations without body condition}}
\nabla^{2}\phi=0, \, -h<z<0,\,\,\,\mathbf{x}\notin \Omega
+
{{frequency domain equations for a rigid body}}
</math></center>
 
<center><math>
 
\frac{\partial\phi}{\partial z}=0, \, z=-h,
 
</math></center>
 
<center><math>
 
\frac{\partial\phi}{\partial z} = \alpha \phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
 
</math></center>
 
<center><math>
 
\frac{\partial\phi}{\partial n} = 0, \, \mathbf{x}\in\partial\Omega.
 
</math></center>
 
where <math>n</math> is the outward (from the fluid) normal.
 

Latest revision as of 01:42, 21 August 2009

The Standard Linear Wave Scattering Problem in Finite Depth for a fixed body is

[math]\displaystyle{ \begin{align} \Delta\phi &=0, &-h\lt z\lt 0,\,\,\mathbf{x} \in \Omega \\ \partial_z\phi &= 0, &z=-h, \\ \partial_z \phi &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]


(note that the last expression can be obtained from combining the expressions:

[math]\displaystyle{ \begin{align} \partial_z \phi &= -\mathrm{i} \omega \zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \\ \mathrm{i} \omega \phi &= g\zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

where [math]\displaystyle{ \alpha = \omega^2/g \, }[/math]) The body boundary condition for a rigid body is just

[math]\displaystyle{ \partial_{n}\phi=0,\ \ \mathbf{x}\in\partial\Omega_{\mathrm{B}}, }[/math]