Difference between revisions of "Template:Separation of variables for a submerged dock"

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The depth above the plate is <math>d</math> and below the plate is <math>h-d</math>. We now introduce
+
The depth above the plate is <math>d</math> and below the plate is <math>h-d</math>. We introduce
a new function
+
a new dispersion value <math>\mu_n</math>:
 +
<center>
 +
<math>
 +
\mu_n =
 +
\begin{cases}
 +
k_n^{d},\qquad \qquad\mbox{for}\,\, 0 \leq n \leq N-M\\
 +
n\pi/(h-d),\,\,\mbox{otherwise}
 +
\end{cases}
 +
</math>
 +
</center>
 +
 
 +
where <math>k_n^{d}</math> are the roots of the [[Dispersion Relation for a Free Surface ]] with depth <math>d</math>.
 +
We also order the roots with the first being the positive imaginary solution <math>k_0^{d}</math>,
 +
the second being zero, then ordering by increasing size. We then define a new function
 
<center>
 
<center>
 
<math>
 
<math>
 
\chi_n =  
 
\chi_n =  
 
\begin{cases}
 
\begin{cases}
\psi_{n}(z),\,\,\,-h<z<-d \\
+
0,\,\,\, \qquad-d<z< 0 \\
0,\,\,\,-d<z< 0
+
\psi_{n}(z),\,\,\,-h<z<-d
 
\end{cases}
 
\end{cases}
 
</math>
 
</math>
 
</center>
 
</center>
for <math>0 \leq n \leq M - 1 </math>
+
or
 
<center>
 
<center>
 
<math>
 
<math>
\chi_{n+M} =  
+
\chi_{n} =  
 
\begin{cases}
 
\begin{cases}
0,\,\,\,-h<z<-d \\
+
\phi_{n}^{d}(z),\,\,\,-d<z< 0 \\
\phi_{n}^{d}(z),\,\,\,-d<z< 0
+
0,\,\,\qquad-h<z<-d
 
\end{cases}
 
\end{cases}
 
</math>
 
</math>
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</math>
 
</math>
 
</center>
 
</center>
where <math>k^{d}</math> is the roots of the [[Free Surface Dispersion Equation]] with depth <math>d</math>
+
depending on whether the root <math>\mu_n</math> is above or below.
for <math>0 \leq n \leq N-M </math>
 
and we choose the values of <math>N</math> so that we have the <math>N+1</math> smallest values
 
of <math>k_n</math> and <math>\kappa_n</math> (with the proviso that we have at least one from each).
 

Latest revision as of 23:50, 8 August 2009

The depth above the plate is [math]\displaystyle{ d }[/math] and below the plate is [math]\displaystyle{ h-d }[/math]. We introduce a new dispersion value [math]\displaystyle{ \mu_n }[/math]:

[math]\displaystyle{ \mu_n = \begin{cases} k_n^{d},\qquad \qquad\mbox{for}\,\, 0 \leq n \leq N-M\\ n\pi/(h-d),\,\,\mbox{otherwise} \end{cases} }[/math]

where [math]\displaystyle{ k_n^{d} }[/math] are the roots of the Dispersion Relation for a Free Surface with depth [math]\displaystyle{ d }[/math]. We also order the roots with the first being the positive imaginary solution [math]\displaystyle{ k_0^{d} }[/math], the second being zero, then ordering by increasing size. We then define a new function

[math]\displaystyle{ \chi_n = \begin{cases} 0,\,\,\, \qquad-d\lt z\lt 0 \\ \psi_{n}(z),\,\,\,-h\lt z\lt -d \end{cases} }[/math]

or

[math]\displaystyle{ \chi_{n} = \begin{cases} \phi_{n}^{d}(z),\,\,\,-d\lt z\lt 0 \\ 0,\,\,\qquad-h\lt z\lt -d \end{cases} }[/math]

where

[math]\displaystyle{ \phi_{m}^{d}\left( z\right) =\frac{\cos k_{m}^{d}(z+d)}{\cos k_{m}^{d}d},\quad m\geq0 }[/math]

depending on whether the root [math]\displaystyle{ \mu_n }[/math] is above or below.