Template:Separation of variables for a submerged dock - Revision history

Mike smith: grammar

2009-08-08T23:50:02Z

grammar

← Older revision		Revision as of 23:50, 8 August 2009
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	The depth above the plate is <math>d</math> and below the plate is <math>h-d</math>. We introduce		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>\mu_n</math> ~~where~~		a new dispersion value <math>\mu_n</math>:
	<center>		<center>
	<math>		<math>
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	where <math>k_n^{d}</math> are the roots of the [[Dispersion Relation for a Free Surface ]] with depth <math>d</math>.		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 being the positive imaginary solution <math>k_0^{d}</math>,
	the second being zero, then ~~order~~ by increasing size. We then define a new function		the second being zero, then ordering by increasing size. We then define a new function
	<center>		<center>
	<math>		<math>

Mike smith: re-expressed mu, removed double up

2009-08-08T23:34:47Z

re-expressed mu, removed double up

← Older revision		Revision as of 23:34, 8 August 2009
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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 dispersion value <math>\~~mu_m~~</math> ~~which is either~~ <math>k_n^{d}</math>		a new dispersion value <math>\mu_n</math> where
	where <math>~~k_0~~^{d}</math> are the roots of the [[Dispersion Relation for a Free Surface ]] with depth <math>d</math>		<center>
	~~for <math>0 \leq n \leq N-M </math> or <math>n\pi/(h-d)</math> for <math>n</math> and integer~~.		<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
	~~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>
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	\chi_{n} =		\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>

Meylan at 11:11, 7 September 2008

2008-09-07T11:11:53Z

← Older revision		Revision as of 11:11, 7 September 2008
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	</center>		</center>
	depending on whether the root <math>\mu_n</math> is above or below.		depending on whether the root <math>\mu_n</math> is above or below.

	~~We define~~
	~~<center>~~
	~~<math>~~
	~~\int\nolimits_{-d}^{0}\phi_{n}^h(z)\chi_{m}^d(z) d z=B_{mn}~~
	~~</math>~~
	~~</center>~~
	~~where <math>B_{mn}</math> is either~~
	~~<center>~~
	~~<math>~~
	~~\int\nolimits_{-d}^{0}\phi_{n}^h(z)\phi_{m}^d(z) d z=B_{mn}~~
	~~</math>~~
	~~</center>~~
	or
	~~<center>~~
	~~<math>~~
	~~\int\nolimits_{-h}^{-d}\phi_{n}^h(z)\psi_{m}(z) d z=B_{mn}~~
	~~</math>~~
	~~</center>~~

Meylan at 11:07, 7 September 2008

2008-09-07T11:07:37Z

@@ Line 33: / Line 33: @@
 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
+We define
-of <math>k_n</math> and <math>\kappa_n</math> (with the proviso that we have at least one from each).
+<center>

Meylan at 11:03, 7 September 2008

2008-09-07T11:03:10Z

← Older revision		Revision as of 11:03, 7 September 2008
Line 1:		Line 1:
	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>
	~~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 \\		0,\,\,\,-h<z<-d \\
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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		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).

Meylan at 10:53, 7 September 2008

2008-09-07T10:53:12Z

← Older revision		Revision as of 10:53, 7 September 2008
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	</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).

Meylan at 10:52, 7 September 2008

2008-09-07T10:52:39Z

New page

The depth above the plate is <math>d</math> and below the plate is <math>h-d</math>. We now introduce
a new function
<center>
<math>
\chi_n =
\begin{cases}
\psi_{n}(z),\,\,\,-h<z<-d \\
0,\,\,\,-d<z< 0
\end{cases}
</math>
</center>
for <math>0 \leq n \leq M - 1 </math>
<center>
<math>
\chi_{n+M} =
\begin{cases}
0,\,\,\,-h<z<-d \\
\phi_{n}^{d}(z),\,\,\,-d<z< 0
\end{cases}
</math>
</center>
where
<center>
<math>
\phi_{m}^{d}\left( z\right) =\frac{\cos k_{m}^{d}(z+d)}{\cos k_{m}^{d}d},\quad m\geq0
</math>
</center>
where <math>^{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>
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).