Difference between revisions of "Template:Separation of variables for a submerged dock"
Mike smith (talk | contribs) (re-expressed mu, removed double up) |
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| − | The depth above the plate is <math>d</math> and below the plate is <math>h-d</math>. We | + | The depth above the plate is <math>d</math> and below the plate is <math>h-d</math>. We introduce |
| − | a new dispersion value <math>\ | + | a new dispersion value <math>\mu_n</math> where |
| − | where <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 the positive imaginary solution <math>k_0^{d}</math> and | We also order the roots with the first the positive imaginary solution <math>k_0^{d}</math> and | ||
the second being zero, then order by increasing size. We then define a new function | the second being zero, then order by increasing size. We then define a new function | ||
| − | |||
<center> | <center> | ||
<math> | <math> | ||
\chi_n = | \chi_n = | ||
\begin{cases} | \begin{cases} | ||
| − | \ | + | 0,\,\,\, \qquad-d<z< 0 \\ |
| − | + | \psi_{n}(z),\,\,\,-h<z<-d | |
\end{cases} | \end{cases} | ||
</math> | </math> | ||
| Line 20: | Line 28: | ||
\chi_{n} = | \chi_{n} = | ||
\begin{cases} | \begin{cases} | ||
| − | + | \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> | ||
Revision as of 23:34, 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] where
[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 the positive imaginary solution [math]\displaystyle{ k_0^{d} }[/math] and the second being zero, then order 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.
