Eigenfunction Matching for a Semi-Infinite Dock - Revision history

Meylan: /* Governing Equations */

2017-04-25T00:43:36Z

Governing Equations

2010-03-16T20:29:11Z

Numerical Solution

← Older revision		Revision as of 20:29, 16 March 2010
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	To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to		To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
	be <math>M</math>. ~~In terms of~~ matrix, ~~we obtain~~		be <math>M</math>. This resulting system can be expressed in the block matrix form below,
	<center>		<center>
	<math>		<math>

2010-03-16T20:23:11Z

Numerical Solution

← Older revision		Revision as of 20:23, 16 March 2010
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	== Numerical Solution ==		== Numerical Solution ==

	To solve the system of equations (10) we set the upper limit of <math>l</math> to		To solve the system of equations (3) and (5), we set the upper limit of <math>l</math> to
	be <math>M</math>. In terms of matrix, we obtain		be <math>M</math>. In terms of matrix, we obtain
	<center>		<center>
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	A_0&0 \quad \cdots&0\\		A_0&0 \quad \cdots&0\\
	0&&\\		0&&\\
	\vdots&A_l~~\delta_{0l}~~&\vdots\\		\vdots&A_l&\vdots\\
	&&0\\		&&0\\
	0&\cdots \quad 0 &A_M		0&\cdots \quad 0 &A_M

2010-03-16T20:21:34Z

seems more intuitive, previous result seen in block matrix below anyway

2010-03-16T20:02:45Z

An infinite dimensional system of equations

← Older revision		Revision as of 20:02, 16 March 2010
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	</math>		</math>
	</center>		</center>
	for each <math>n</math>.		for each <math>m</math>.
	We solve these equations by multiplying both equations by		We solve these equations by multiplying both equations by
	<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:		<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:

2010-03-16T20:01:10Z

An infinite dimensional system of equations

← Older revision		Revision as of 20:01, 16 March 2010
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	for each <math>n</math>.		for each <math>n</math>.
	We solve these equations by multiplying both equations by		We solve these equations by multiplying both equations by
	<math>\phi_{l}(z)</math> and integrating from <math>-h</math> to <math>0</math> to obtain:		<math>\phi_{l}(z) \,</math> and integrating from <math>-h</math> to <math>0</math> to obtain:
	<center>		<center>
	<math>		<math>
	A_{0}\delta_{0l}+a_{l}A_{l}		A_{0}\delta_{0l}+a_{l}A_{l}
	=\sum_{n=0}^{\infty}b_{m}B_{ml}\,\,\,(3)		=\sum_{m=0}^{\infty}b_{m}B_{ml}\,\,\,(3)
	</math>		</math>
	</center>		</center>
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	</math>		</math>
	</center>		</center>
	If we ~~mutiple~~ equation (3) by <math>k_l</math> and subtract equation (4)		If we mutiply equation (3) by <math>k_l \,</math> and subtract equation (4)
	we obtain		we obtain
	<center>		<center>

2010-03-16T19:39:50Z

Expansion of the potential

← Older revision		Revision as of 19:39, 16 March 2010
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	We need to apply some boundary conditions at plus and minus infinity,		We need to apply some boundary conditions at plus and minus infinity,
	where are essentially the the solution cannot grow. This means that we		where are essentially the the solution cannot grow. This means that we
	only have the positive (or ~~negaitve~~) roots of the dispersion equation.		only have the positive (or negative) roots of the dispersion equation.
	However, it does not help us with the purely imaginary root. Here we		However, it does not help us with the purely imaginary root. Here we
	must use a different condition, essentially identifying one solution		must use a different condition, essentially identifying one solution

2009-08-30T10:30:12Z

2009-08-30T10:29:36Z

Additional code

← Older revision		Revision as of 10:29, 30 August 2009
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	[[Category:Eigenfunction Matching Method]]		[[Category:Eigenfunction Matching Method]]
	~~{{complete pages}}~~

2009-08-30T10:28:11Z

Additional code

← Older revision		Revision as of 10:28, 30 August 2009
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	[[Category:Eigenfunction Matching Method]]		[[Category:Eigenfunction Matching Method]]
	~~[[Category:Complete Pages]]~~		{{complete pages}}