Template:Separation of variables for a submerged dock: Difference between revisions
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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 now introduce | ||
a new dispersion value <math>\mu_m</math> which is either <math>k_n^{d}</math> | |||
where <math>k_0^{d}</math> are the roots of the [[Dispersion Relation for a Free Surface ]] with depth <math>d</math> | |||
for <math>0 \leq n \leq N-M </math> or <math>n\pi/(h-d)</math> for <math>n</math> and integer. | |||
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 | |||
a new function | a new function | ||
<center> | <center> | ||
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</math> | </math> | ||
</center> | </center> | ||
or | |||
<center> | <center> | ||
<math> | <math> | ||
\chi_{n | \chi_{n} = | ||
\begin{cases} | \begin{cases} | ||
0,\,\,\,-h<z<-d \\ | 0,\,\,\,-h<z<-d \\ | ||
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</math> | </math> | ||
</center> | </center> | ||
depending on whether the root <math>\mu_n</math> is above or below. | |||
and we choose the values of <math>N</math> so that we have the <math>N+1</math> smallest values | 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). | of <math>k_n</math> and <math>\kappa_n</math> (with the proviso that we have at least one from each). | ||
Revision as of 11:03, 7 September 2008
The depth above the plate is [math]\displaystyle{ d }[/math] and below the plate is [math]\displaystyle{ h-d }[/math]. We now introduce a new dispersion value [math]\displaystyle{ \mu_m }[/math] which is either [math]\displaystyle{ k_n^{d} }[/math] where [math]\displaystyle{ k_0^{d} }[/math] are the roots of the Dispersion Relation for a Free Surface with depth [math]\displaystyle{ d }[/math] for [math]\displaystyle{ 0 \leq n \leq N-M }[/math] or [math]\displaystyle{ n\pi/(h-d) }[/math] for [math]\displaystyle{ n }[/math] and integer. 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 a new function
[math]\displaystyle{ \chi_n = \begin{cases} \psi_{n}(z),\,\,\,-h\lt z\lt -d \\ 0,\,\,\,-d\lt z\lt 0 \end{cases} }[/math]
or
[math]\displaystyle{ \chi_{n} = \begin{cases} 0,\,\,\,-h\lt z\lt -d \\ \phi_{n}^{d}(z),\,\,\,-d\lt z\lt 0 \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.
and we choose the values of [math]\displaystyle{ N }[/math] so that we have the [math]\displaystyle{ N+1 }[/math] smallest values of [math]\displaystyle{ k_n }[/math] and [math]\displaystyle{ \kappa_n }[/math] (with the proviso that we have at least one from each).
