Template:Standard linear wave scattering equations without body condition: Difference between revisions

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<center><math>
<center><math>
\Delta\phi=0, \, -h<z<0,\,\,\,\mathbf{x} \in \Omega
\begin{align}
\Delta\phi &=0, &-h<z<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></center>
</math></center>
(note that the last expression can be obtained from combining the expressions:
<center><math>
<center><math>
\partial_n\phi = 0, \, z=-h,
\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></center>
</math></center>
<center><math>
 
\partial_n \phi  =\mathrm{i} \omega \zeta\,z=0,\,\,\mathbf{x} \in F,
where <math>\alpha = \omega^2/g \,</math>)
</math></center>
<center><math>
\mathrm{i} \omega \phi  = -\zeta,\,z=0,\,\,\mathbf{x} \in F,
</math></center>
(note that The latter two can be combined as
<center><math>
\partial_n \phi  = \alpha \phi,\,z=0,\,\,\mathbf{x} \in F,
</math></center>
where <math>\alpha = \omega^2 </math>)

Latest revision as of 10:40, 6 November 2010

[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])

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