Template:Standard linear wave scattering equations: Difference between revisions

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The equations are the following
The equations are the following
<center><math>
<center><math>
\nabla^{2}\phi=0, \, -h<z<0,\,\,\,\mathbf{x}\notin \Omega
\Delta\phi=0, \, -h<z<0,\,\,\,\mathbf{x} \in \Omega
</math></center>
</math></center>
<center><math>
<center><math>
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</math></center>
</math></center>
<center><math>
<center><math>
  \frac{\partial\phi}{\partial z} = \alpha \phi,\,z=0,\,\,\mathbf{x}\notin\Omega,
  \partial_n \phi = \alpha \phi,\,z=0,\,\,\mathbf{x} \in F,
</math></center>
</math></center>
<center><math>
<center><math>
\frac{\partial\phi}{\partial z} = \mathcal{L}\phi, \, z\in\partial\Omega,
\partial_n\phi = \mathcal{L}\phi, \, z\in\partial\Omega,
</math></center>
</math></center>
where <math>\alpha</math> is the wavenumber in [[Infinite Depth]] which is given by  
where <math>\alpha</math> is the wavenumber in [[Infinite Depth]] which is given by  
<math>\alpha=\omega^2/g</math> where <math>g</math> is gravity. <math>\mathcal{L}</math> is a linear
<math>\alpha=\omega^2/g</math> where <math>g</math> is gravity. <math>\mathcal{L}</math> is a linear
operator which relates the normal and potential on the body surface through the physics
operator which relates the normal and potential on the body surface through the physics
of the body.
of the body.

Revision as of 01:42, 18 August 2009

The equations are the following

[math]\displaystyle{ \Delta\phi=0, \, -h\lt z\lt 0,\,\,\,\mathbf{x} \in \Omega }[/math]
[math]\displaystyle{ \frac{\partial\phi}{\partial z}=0, \, z=-h, }[/math]
[math]\displaystyle{ \partial_n \phi = \alpha \phi,\,z=0,\,\,\mathbf{x} \in F, }[/math]
[math]\displaystyle{ \partial_n\phi = \mathcal{L}\phi, \, z\in\partial\Omega, }[/math]

where [math]\displaystyle{ \alpha }[/math] is the wavenumber in Infinite Depth which is given by [math]\displaystyle{ \alpha=\omega^2/g }[/math] where [math]\displaystyle{ g }[/math] is gravity. [math]\displaystyle{ \mathcal{L} }[/math] is a linear operator which relates the normal and potential on the body surface through the physics of the body.