Cylindrical Eigenfunction Expansion - Revision history

Meylan at 09:34, 20 October 2009

2009-10-20T09:34:59Z

2009-02-24T07:03:42Z

Finite depth

← Older revision		Revision as of 07:03, 24 February 2009
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	== Finite depth ==		== Finite depth ==

	It is easily seen that this is the same as in the infinite-depth case except that we need to replace <math>\mathrm{e}^{ \alpha z}</math> by <math>\frac{\cosh(k (z+d))}{\cosh kd}</math> and <math>\alpha</math> by <math>k</math>. Therefore, we have		It is easily seen that this is the same as in the infinite-depth case except that we need to replace <math>\mathrm{e}^{ \alpha z}</math> by <math>\frac{\cosh(k (z+h))}{\cosh kh}</math> and <math>\alpha</math> by <math>k</math>. Therefore, we have

	<center><math>		<center><math>
	\phi^{\mathrm{In}}(r_l,\theta_l,z) = A \frac{g}{\omega} \, e^{\mathrm{i} k (O_x^l		\phi^{\mathrm{In}}(r_l,\theta_l,z) = A \frac{g}{\omega} \, e^{\mathrm{i} k (O_x^l
	\cos \chi + O_x^l \sin \chi)} \, \frac{\cosh(k (z+d))}{\cosh kd}		\cos \chi + O_x^l \sin \chi)} \, \frac{\cosh(k (z+h))}{\cosh kh}
	\sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 - \chi)}		\sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 - \chi)}
	J_\mu(k r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}.		J_\mu(k r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}.

2008-08-26T05:17:18Z

Outine of the theory

← Older revision		Revision as of 05:17, 26 August 2008
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	{{separation of variables in cylindrical coordinates in finite depth}}		{{separation of variables in cylindrical coordinates in finite depth}}

			{{separation of variables for a free surface}}

			{{separation of variables for the r and theta coordinates}}

	The potential <math>\phi</math> can thus be expressed in local cylindrical		The potential <math>\phi</math> can thus be expressed in local cylindrical

2008-08-25T09:06:40Z

The case of infinitely deep water

← Older revision		Revision as of 09:06, 25 August 2008
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	{{separation of variables in cylindrical coordinates in finite depth}}		{{separation of variables in cylindrical coordinates in finite depth}}

			The potential <math>\phi</math> can thus be expressed in local cylindrical
			coordinates as
			<center>
			<math>
			\phi (r,\theta,z) = \sum_{m = 0}^{\infty} \phi_m(z) \sum_{\nu = -
			\infty}^{\infty} \left[ D_{m\nu} I_\nu (k_m r) + E_{m\nu} K_\nu (k_m
			r) \right] \mathrm{e}^{\mathrm{i} \nu \theta},
			</math>
			</center>

	== The case of infinitely deep water ==		== The case of infinitely deep water ==

2008-08-25T08:44:24Z

← Older revision		Revision as of 08:44, 25 August 2008
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	{{sommerfeld radiation condition three dimensions}}		{{sommerfeld radiation condition three dimensions}}

	{{separation of ~~variable~~ in cylindrical coordinates in finite depth}}		{{separation of variables in cylindrical coordinates in finite depth}}

	== The case of infinitely deep water ==		== The case of infinitely deep water ==

2008-08-25T08:01:08Z

The case of water of finite depth

← Older revision		Revision as of 08:01, 25 August 2008
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	{{cylindrical equations}}		{{cylindrical equations}}
	{{sommerfeld radiation condition three dimensions}}		{{sommerfeld radiation condition three dimensions}}

	~~== The case of water of finite depth ==~~

	{{separation of variable in cylindrical coordinates in finite depth}}		{{separation of variable in cylindrical coordinates in finite depth}}

2008-08-25T07:55:56Z

Outine of the theory

← Older revision		Revision as of 07:55, 25 August 2008
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	{{cylindrical equations}}		{{cylindrical equations}}
			{{sommerfeld radiation condition three dimensions}}

	== The case of water of finite depth ==		== The case of water of finite depth ==

2008-08-25T07:50:28Z

Outine of the theory

@@ Line 11: / Line 11: @@
 = Outine of the theory =
+{{cylindrical equations}}
 == The case of water of finite depth ==

2008-08-25T07:48:57Z

The case of water of finite depth

2007-05-27T22:35:52Z

Introduction

← Older revision		Revision as of 22:35, 27 May 2007
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	There are any situations where we want to expand the three-dimensional linear water wave		There are any situations where we want to expand the three-dimensional linear water wave
	solution in cylindrical co-ordinates. For example, scattering from a		solution in cylindrical co-ordinates. For example, scattering from a
	[[Bottom Mounted Cylinder]] or scattering from a [[Circular Elastic Plate]]. In these cases it is easy to find		[[Bottom Mounted Cylinder]] or scattering from a [[Circular Floating Elastic Plate]]. In these cases it is easy to find
	the solution by an expansion in the cylindrical eigenfunctions. If the depth dependence can be		the solution by an expansion in the cylindrical eigenfunctions. If the depth dependence can be
	removed the solution reduces to a two dimensional problem (see [[Removing The Depth Dependence]]). While		removed the solution reduces to a two dimensional problem (see [[Removing The Depth Dependence]]). While