Difference between revisions of "Template:Separation of variables for a submerged dock"
| Line 26: | Line 26: | ||
</math> | </math> | ||
</center> | </center> | ||
| − | where <math>^{d}</math> is the roots of the [[Free Surface Dispersion Equation]] with depth <math>d</math> | + | where <math>k^{d}</math> is the roots of the [[Free Surface Dispersion Equation]] with depth <math>d</math> |
for <math>0 \leq n \leq N-M </math> | 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 | 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 10:53, 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 function
[math]\displaystyle{ \chi_n = \begin{cases} \psi_{n}(z),\,\,\,-h\lt z\lt -d \\ 0,\,\,\,-d\lt z\lt 0 \end{cases} }[/math]
for [math]\displaystyle{ 0 \leq n \leq M - 1 }[/math]
[math]\displaystyle{ \chi_{n+M} = \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]
where [math]\displaystyle{ k^{d} }[/math] is the roots of the Free Surface Dispersion Equation with depth [math]\displaystyle{ d }[/math] for [math]\displaystyle{ 0 \leq n \leq N-M }[/math] 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).
