Eigenfunction Matching for a Finite Rectangle using Symmetry - Revision history

Tertius: /* An infinite dimensional system of equations */

2012-03-19T01:18:48Z

An infinite dimensional system of equations

← Older revision		Revision as of 01:18, 19 March 2012
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	-k_{0}\chi_{0}\left( z\right) +\sum_{m=0}^{\infty} a^{s}_{m}k_{m}\chi_{m}\left(z\right) =		-k_{0}\chi_{0}\left( z\right) +\sum_{m=0}^{\infty} a^{s}_{m}k_{m}\chi_{m}\left(z\right) =
	\begin{cases}		\begin{cases}
	\quad \quad \quad \quad \quad 0,~~\quad~~ \quad \quad \quad \quad \quad -d\leq z\leq 0 \\		\quad \quad \quad \quad \quad 0,\quad \quad \quad \quad \quad -d\leq z\leq 0 \\
	-\sum_{m=1}^{\infty}b^{s}_{m}\kappa_{m}\tanh(\kappa_{m}L)\psi_{m}(z),\quad -h\leq z\leq -d.		-\sum_{m=1}^{\infty}b^{s}_{m}\kappa_{m}\tanh(\kappa_{m}L)\psi_{m}(z),\quad -h\leq z\leq -d.
	\end{cases}		\end{cases}

2012-03-16T06:37:03Z

An infinite dimensional system of equations

← Older revision		Revision as of 06:37, 16 March 2012
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	<center><math>		<center><math>
	D_{n} = \int\nolimits_{-h}^{-d}\chi_{n}(z) \mathrm{d} z = \int\nolimits_{-h}^{-d} \frac{\cos k_{n}(z+h)}{\cos k_{n}h} \mathrm{d} z=\frac{\sin k_{n}(h-d)}		D_{n} = \int\nolimits_{-h}^{-d}\chi_{n}(z) \mathrm{d} z = \int\nolimits_{-h}^{-d} \frac{\cos k_{n}(z+h)}{\cos k_{n}h} \mathrm{d} z=\frac{\sin k_{n}(h-d)}
	{k_{n}\cos k_{n}}.		{k_{n}\cos k_{n}h}.
	</math></center>		</math></center>

2012-03-16T06:36:19Z

An infinite dimensional system of equations

← Older revision		Revision as of 06:36, 16 March 2012
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	<center><math>		<center><math>
	D_{n} = \int\nolimits_{-h}^{-d}\chi_{n}(z) \mathrm{d} z = \int\nolimits_{-h}^{-d} \frac{\cos k_{n}(z+h)}{\cos k_{n}h} \mathrm{d} z=\frac{~~-(-1)^m k_{n}~~\sin k_{n}(h-d)}		D_{n} = \int\nolimits_{-h}^{-d}\chi_{n}(z) \mathrm{d} z = \int\nolimits_{-h}^{-d} \frac{\cos k_{n}(z+h)}{\cos k_{n}h} \mathrm{d} z=\frac{\sin k_{n}(h-d)}
	{~~\cos(~~k_{n}~~h)(~~\~~kappa_{m}^{2}-~~k_{n}~~^{2})~~}.		{k_{n}\cos k_{n}}.
	</math></center>		</math></center>

2012-03-16T06:33:51Z

An infinite dimensional system of equations

← Older revision		Revision as of 06:33, 16 March 2012
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	Solving the equations above will yield the coefficients of the water velocity potential in the dock covered region.		Solving the equations above will yield the coefficients of the water velocity potential in the dock covered region.


			<center><math>
			D_{n} = \int\nolimits_{-h}^{-d}\chi_{n}(z) \mathrm{d} z = \int\nolimits_{-h}^{-d} \frac{\cos k_{n}(z+h)}{\cos k_{n}h} \mathrm{d} z=\frac{-(-1)^m k_{n}\sin k_{n}(h-d)}
			{\cos(k_{n}h)(\kappa_{m}^{2}-k_{n}^{2})}.
			</math></center>

	== Numerical Solution ==		== Numerical Solution ==

2012-03-16T06:22:13Z

An infinite dimensional system of equations

← Older revision		Revision as of 06:22, 16 March 2012
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	\begin{cases}		\begin{cases}
	\quad \quad \quad \quad \quad 0,\quad \quad \quad \quad \quad \quad -d\leq z\leq 0 \\		\quad \quad \quad \quad \quad 0,\quad \quad \quad \quad \quad \quad -d\leq z\leq 0 \\
	-\sum_{m=1}^{\infty}b^{s}_{m}\kappa_{m}\tanh(\kappa_{m}L)\psi_{m}(z),\quad -h\leq z\leq -d		-\sum_{m=1}^{\infty}b^{s}_{m}\kappa_{m}\tanh(\kappa_{m}L)\psi_{m}(z),\quad -h\leq z\leq -d.
	\end{cases}		\end{cases}
	\end{align}		\end{align}
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	\begin{cases}		\begin{cases}
	\quad \quad \quad \quad \quad 0,\quad \quad \quad \quad \quad \quad \quad \quad -d\leq z\leq 0 \\		\quad \quad \quad \quad \quad 0,\quad \quad \quad \quad \quad \quad \quad \quad -d\leq z\leq 0 \\
	-\frac{1}{L}b^a_{0} -\sum_{m=1}^{\infty}b^{a}_{m}\kappa_{m}\coth(\kappa_{m}L)\psi_{m}(z),\quad -h\leq z\leq -d		-\frac{1}{L}b^a_{0} -\sum_{m=1}^{\infty}b^{a}_{m}\kappa_{m}\coth(\kappa_{m}L)\psi_{m}(z),\quad -h\leq z\leq -d.
	\end{cases}		\end{cases}
	\end{align}		\end{align}

2012-03-16T06:21:19Z

An infinite dimensional system of equations

← Older revision		Revision as of 06:21, 16 March 2012
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	-k_{0}\chi_{0}\left( z\right) +\sum_{m=0}^{\infty} a^{a}_{m}k_{m}\chi_{m}\left(z\right) =		-k_{0}\chi_{0}\left( z\right) +\sum_{m=0}^{\infty} a^{a}_{m}k_{m}\chi_{m}\left(z\right) =
	\begin{cases}		\begin{cases}
	\quad \quad \quad \quad \quad 0,\quad \quad \quad \quad \quad \quad -d\leq z\leq 0 \\		\quad \quad \quad \quad \quad 0,\quad \quad \quad \quad \quad \quad \quad \quad -d\leq z\leq 0 \\
	-\frac{1}{L}b^a_{0} -\sum_{m=1}^{\infty}b^{a}_{m}\kappa_{m}\coth(\kappa_{m}L)\psi_{m}(z),\quad -h\leq z\leq -d		-\frac{1}{L}b^a_{0} -\sum_{m=1}^{\infty}b^{a}_{m}\kappa_{m}\coth(\kappa_{m}L)\psi_{m}(z),\quad -h\leq z\leq -d
	\end{cases}		\end{cases}

2012-03-16T06:17:03Z

An infinite dimensional system of equations

← Older revision		Revision as of 06:17, 16 March 2012
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	<center><math>		<center><math>
	\begin{align}		\begin{align}
	-k_{0}\chi_{0}\left( z\right) +\sum_{m=0}^{\infty} a^{s}_{m}k_{m}\chi_{m}\left(z\right) =		-k_{0}\chi_{0}\left( z\right) +\sum_{m=0}^{\infty} a^{a}_{m}k_{m}\chi_{m}\left(z\right) =
	\begin{cases}		\begin{cases}
	\quad \quad \quad \quad \quad 0,\quad \quad \quad \quad \quad \quad -d\leq z\leq 0 \\		\quad \quad \quad \quad \quad 0,\quad \quad \quad \quad \quad \quad -d\leq z\leq 0 \\
	-\sum_{m=1}^{\infty}b^{s}_{m}\kappa_{m}\~~tanh~~(\kappa_{m}L)\psi_{m}(z),\quad -h\leq z\leq -d		-\frac{1}{L}b^a_{0} -\sum_{m=1}^{\infty}b^{a}_{m}\kappa_{m}\coth(\kappa_{m}L)\psi_{m}(z),\quad -h\leq z\leq -d
	\end{cases}		\end{cases}
	\end{align}		\end{align}

2012-03-16T06:09:03Z

An infinite dimensional system of equations

@@ Line 180: / Line 180: @@
-and for the second equation we multiply both sides by <math>\chi_{n}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
+The derivative of the '''antisymmetric''' potential <math>\partial_x\phi^{a}(x,z) \,</math> must be continuous across the
 <center><math>
   -k_{0}A_{0}\delta_{0n} + k_{m}a_{m}A_{m} = -\sum^{\infty}_{m=0}\kappa_{m}b_{m}B_{mn}, \quad n\in\mathbb{N}\cup\left\{0\right\}

2012-03-16T06:04:20Z

An infinite dimensional system of equations

← Older revision		Revision as of 06:04, 16 March 2012
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	The derivative of the '''symmetric''' potential <math>\phi^{s}(x,z) \,</math> must be continuous across the		The derivative of the '''symmetric''' potential <math>\partial_x\phi^{s}(x,z) \,</math> must be continuous across the
	transition from open water to the dock covered region. Therefore, the derivatives of the potentials at <math>x=-L</math> have to be equal.		transition from open water to the dock covered region. Therefore, the derivatives of the potentials at <math>x=-L</math> have to be equal.
	For this case we obtain:		For this case we obtain:

2012-03-16T05:57:42Z

An infinite dimensional system of equations

← Older revision		Revision as of 05:57, 16 March 2012
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	The ~~potential~~ must be continuous across the		The '''symmetric''' and '''antisymmetric''' potentials <math>\phi^{s,a}(x,z) \,</math> must be continuous across the
	transition from open water to the dock covered region. Therefore, the		transition from open water to the dock covered region. Therefore, the
	potentials at <math>x=-L</math> have to be equal.		potentials at <math>x=-L</math> have to be equal.
	We obtain ~~for the '''symmetric''' and '''antisymmetric''' cases~~:		For these cases we obtain:

	<center><math>		<center><math>