Eigenfunction Matching for a Semi-Infinite Rectangle - Revision history

Tertius: /* Numerical Solution */

2012-03-16T01:28:57Z

Numerical Solution

← Older revision		Revision as of 01:28, 16 March 2012
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	To solve the system of equations, we set the upper limit of <math>m</math> to be <math>N</math>. This resulting system can be expressed in the block matrix form below,		To solve the system of equations, we set the upper limit of <math>m</math> to be <math>N</math>. This resulting system can be expressed in the block matrix form below,


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Tertius: /* Numerical Solution */

2012-03-16T01:28:34Z

Numerical Solution

← Older revision		Revision as of 01:28, 16 March 2012
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	To solve the system of equations, we set the upper limit of <math>m</math> to be <math>N</math>. This resulting system can be expressed in the block matrix form below,		To solve the system of equations, we set the upper limit of <math>m</math> to be <math>N</math>. This resulting system can be expressed in the block matrix form below,


	<center>		<center>

Tertius: /* Numerical Solution */

2012-03-16T01:28:16Z

Numerical Solution

← Older revision		Revision as of 01:28, 16 March 2012
Line 191:		Line 191:

	To solve the system of equations, we set the upper limit of <math>m</math> to be <math>N</math>. This resulting system can be expressed in the block matrix form below,		To solve the system of equations, we set the upper limit of <math>m</math> to be <math>N</math>. This resulting system can be expressed in the block matrix form below,

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

	We then simply need to solve the linear system of equations.		We then simply need to solve the linear system of equations.

Tertius: /* Governing Equations */

2012-03-15T06:05:29Z

Governing Equations

← Older revision		Revision as of 06:05, 15 March 2012
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	Where <math>\alpha = \omega^2/g</math>. We must also apply the [[Sommerfeld Radiation Condition]]		Where <math>\alpha = \omega^2/g</math>.

			We must also apply the [[Sommerfeld Radiation Condition]]
	as <math>\|x\|\rightarrow\infty</math>. This essentially implies		as <math>\|x\|\rightarrow\infty</math>. This essentially implies
	that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave		that the only wave at infinity is propagating away and at negative infinity there is a unit incident wave

Tertius: /* Introduction */

2012-03-15T03:22:27Z

Introduction

← Older revision		Revision as of 03:22, 15 March 2012
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	== Introduction ==		== Introduction ==

	The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d + ...</math> freeboard with depth of submergence being <math>d</math> through which no flow is possible.		The problem consists of a region to the left with a free water surface and a region to the right with a rigid fixed semi infinite rectangular block with height <math>d + ...</math> freeboard with depth of submergence being <math>d</math> through which no flow is possible.
	We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We can extend this solution to a finite dock using symmetry		We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We can extend this solution to a finite dock using symmetry
	[[Eigenfunction Matching for a Finite Rectangle using Symmetry]]		[[Eigenfunction Matching for a Finite Rectangle using Symmetry]]

Tertius: /* Introduction */

2012-03-15T02:43:44Z

Introduction

← Older revision		Revision as of 02:43, 15 March 2012
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	== Introduction ==		== Introduction ==

	The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d + </math> freeboard with depth of submergence being <math>d</math> through which no flow is possible.		The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d + ...</math> freeboard with depth of submergence being <math>d</math> through which no flow is possible.
	We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We can extend this solution to a finite dock using symmetry		We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We can extend this solution to a finite dock using symmetry
	[[Eigenfunction Matching for a Finite Rectangle using Symmetry]]		[[Eigenfunction Matching for a Finite Rectangle using Symmetry]]

Tertius: /* Introduction */

2012-03-15T02:42:04Z

Introduction

← Older revision		Revision as of 02:42, 15 March 2012
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	== Introduction ==		== Introduction ==

	The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d +</math> freeboard with depth of submergence being <math>d</math> through which no flow is possible.		The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d + </math> freeboard with depth of submergence being <math>d</math> through which no flow is possible.
	We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We can extend this solution to a finite dock using symmetry		We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We can extend this solution to a finite dock using symmetry
	[[Eigenfunction Matching for a Finite Rectangle using Symmetry]]		[[Eigenfunction Matching for a Finite Rectangle using Symmetry]]

Tertius: /* Introduction */

2012-03-15T02:28:34Z

Introduction

← Older revision		Revision as of 02:28, 15 March 2012
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	== Introduction ==		== Introduction ==

	The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d</math> ~~fully submerged~~ through which no flow is possible.		The problem consists of a region to the left with a free water surface and a region to the right with a rigid semi infinite rectangular block with height <math>d +</math> freeboard with depth of submergence being <math>d</math> through which no flow is possible.
	We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We can extend this solution to a finite dock using symmetry		We begin with the simple problem when the waves are normally incident (so that the problem is truly two-dimensional). We can extend this solution to a finite dock using symmetry
	[[Eigenfunction Matching for a Finite Rectangle using Symmetry]]		[[Eigenfunction Matching for a Finite Rectangle using Symmetry]]

Tertius: /* Governing Equations */

2012-03-15T01:53:42Z

Governing Equations

← Older revision		Revision as of 01:53, 15 March 2012
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	\Delta\phi &= 0, \,\,\, -h\leq z\leq 0, \,\, x < 0, \\		\Delta\phi &= 0, \,\,\, -h\leq z\leq 0, \,\, x < 0, \\

	\Delta\phi &= 0, \,\,\, -h\leq z\leq -d, \,\, x < 0. \\		\Delta\phi &= 0, \,\,\, -h\leq z\leq -d, \,\, x > 0. \\

	\partial_z\phi &= \alpha\phi, \,\,\, z=0, \,\, x<0, \\		\partial_z\phi &= \alpha\phi, \,\,\, z=0, \,\, x<0, \\

Tertius: /* Governing Equations */

2012-03-15T01:53:14Z

Governing Equations

← Older revision		Revision as of 01:53, 15 March 2012
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	<center><math>		<center><math>
	\begin{align}		\begin{align}
	\Delta\phi &= 0, \,\,\, -h\leq z\leq 0, \,\, x~~\leq~~ 0, \\		\Delta\phi &= 0, \,\,\, -h\leq z\leq 0, \,\, x < 0, \\

	\Delta\phi &= 0, \,\,\, -h\leq z\leq -d, \,\, x~~\geq~~ 0. \\		\Delta\phi &= 0, \,\,\, -h\leq z\leq -d, \,\, x < 0. \\

	\partial_z\phi &= \alpha\phi, \,\,\, z=0, \,\, x<0, \\		\partial_z\phi &= \alpha\phi, \,\,\, z=0, \,\, x<0, \\