Interaction Theory for Cylinders - Revision history

Meylan: /* = Equations of Motion = */

2009-10-19T08:15:43Z

= Equations of Motion =

← Older revision		Revision as of 08:15, 19 October 2009
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	body is a cylinder.		body is a cylinder.

	== Equations of Motion = =		== Equations of Motion ==

	After we have [[Removing the Depth Dependence\|Removed the Depth Dependence]]		After we have [[Removing the Depth Dependence\|Removed the Depth Dependence]]

Meylan at 08:15, 19 October 2009

2009-10-19T08:15:29Z

← Older revision		Revision as of 08:15, 19 October 2009
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	= Introduction =		{{complete pages}}

			== Introduction ==

	We present an illustrative example of an interaction theory for the case of <math>n</math>		We present an illustrative example of an interaction theory for the case of <math>n</math>
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	body is a cylinder.		body is a cylinder.

	= Equations of Motion =		== Equations of Motion = =

	After we have [[Removing the Depth Dependence\|Removed the Depth Dependence]]		After we have [[Removing the Depth Dependence\|Removed the Depth Dependence]]
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	where <math>k</math> is the positive real root of the [[Dispersion Relation for a Free Surface]]		where <math>k</math> is the positive real root of the [[Dispersion Relation for a Free Surface]]
	<center><math>		<center><math>
	k \tanh k h = k_\~~infty~~,.		k \tanh k h = \alpha,.
	</math></center>		</math></center>
	where <math>k_\~~infty~~</math> is the [[Infinite Depth]] wave number and		where <math>\alpha</math> is the [[Infinite Depth]] wave number and
	<math>h</math> is the water depth.		<math>h</math> is the water depth.

	=Eigenfunction expansion of the potential=		== Eigenfunction expansion of the potential ==

	Each cylinder is subject to an incident potential and moves in response to this		Each cylinder is subject to an incident potential and moves in response to this
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	Bessel functions of first and second kind, respectively, both of order <math>\nu</math>.		Bessel functions of first and second kind, respectively, both of order <math>\nu</math>.

	=Derivation of the system of equations=		== Derivation of the system of equations ==

	A system of equations for the unknown		A system of equations for the unknown
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	</math></center>		</math></center>

	= Final Equations =		== Final Equations ==

	The scattered and incident potential can be related by the		The scattered and incident potential can be related by the

Adi Kurniawan: /* Equations of Motion */

2008-12-24T05:21:44Z

Equations of Motion

← Older revision		Revision as of 05:21, 24 December 2008
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	where <math>k</math> is the positive real root of the [[Dispersion Relation for a Free Surface]]		where <math>k</math> is the positive real root of the [[Dispersion Relation for a Free Surface]]
	<center><math>		<center><math>
	k \tanh k d = k_\infty,.		k \tanh k h = k_\infty,.
	</math></center>		</math></center>
	where <math>k_\infty</math> is the [[Infinite Depth]] wave number and		where <math>k_\infty</math> is the [[Infinite Depth]] wave number and
	<math>d</math> is the water depth.		<math>h</math> is the water depth.

	=Eigenfunction expansion of the potential=		=Eigenfunction expansion of the potential=

Meylan: /* Final Equations */

2007-05-14T09:49:48Z

Final Equations

← Older revision		Revision as of 09:49, 14 May 2007
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	-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu} (k		-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu} (k
	R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big]		R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big]
	- A_{\nu}^l =		+ A_{\nu}^l = -
	\frac{J'_\nu(k a_l)}{H^{(1)}_\nu{}'(k a_l)}		\frac{J'_\nu(k a_l)}{H^{(1)}_\nu{}'(k a_l)}
	\tilde{D}_{\nu}^{l},		\tilde{D}_{\nu}^{l},

Meylan: /* Final Equations */

2007-05-14T09:48:00Z

Final Equations

← Older revision		Revision as of 09:48, 14 May 2007
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	[[Diffraction Transfer Matrix]] for a [[Bottom Mounted Cylinder]] so that,		[[Diffraction Transfer Matrix]] for a [[Bottom Mounted Cylinder]] so that,
	<center><math>		<center><math>
	A_{\mu}^l= -\frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)} D_{\mu}^l .		A_{\mu}^l= -\frac{J'_\nu(k a_l)}{H^{(1)}_\nu{}'(k a_l)} D_{\nu}^l .
	</math></center>		</math></center>

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	wavefields of all bodies,		wavefields of all bodies,
	<center><math>		<center><math>
	\frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)} \Big[		\frac{J'_\nu(k a_l)}{H^{(1)}_\nu{}'(k a_l)} \Big[
	\sum_{j=1,j \neq l}^{N} \sum_{\tau =		\sum_{j=1,j \neq l}^{N} \sum_{\tau =
	-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \mu} (k		-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu} (k
	R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\mu) \varphi_{jl}} \Big]		R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big]
	- A_{\mu}^l =		- A_{\nu}^l =
			\frac{J'_\nu(k a_l)}{H^{(1)}_\nu{}'(k a_l)}
	\frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)}		\tilde{D}_{\nu}^{l},
	\tilde{D}_{\mu}^{l},
	</math></center>		</math></center>
	<math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,n</math>.		<math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,n</math>.

	[[Category:Interaction Theory]]		[[Category:Interaction Theory]]

Meylan: /* Final Equations */

2007-05-14T09:43:15Z

Final Equations

← Older revision		Revision as of 09:43, 14 May 2007
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	wavefields of all bodies,		wavefields of all bodies,
	<center><math>		<center><math>
	~~A_{\mu}^l = -~~\frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)}		\frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)} \Big[
	\Big[ ~~\tilde{D}_{\mu}^{l} +~~
	\sum_{j=1,j \neq l}^{N} \sum_{\tau =		\sum_{j=1,j \neq l}^{N} \sum_{\tau =
	-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \mu} (k		-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \mu} (k
	R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\mu) \varphi_{jl}} \Big],		R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\mu) \varphi_{jl}} \Big]
			- A_{\mu}^l =

			\frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)}
			\tilde{D}_{\mu}^{l},
	</math></center>		</math></center>
	<math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,n</math>.		<math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,n</math>.

	[[Category:Interaction Theory]]		[[Category:Interaction Theory]]

Meylan: /* Final Equations */

2007-05-14T09:39:57Z

Final Equations

← Older revision		Revision as of 09:39, 14 May 2007
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	<center><math>		<center><math>
	A_{\mu}^l = -\frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)}		A_{\mu}^l = -\frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)}
	~~\sum_{\nu = -\infty}^{\infty}~~		\Big[ \tilde{D}_{\mu}^{l} +
	\Big[ \tilde{D}_{\nu}^{l} +
	\sum_{j=1,j \neq l}^{N} \sum_{\tau =		\sum_{j=1,j \neq l}^{N} \sum_{\tau =
	-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu} (k		-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \mu} (k
	R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big],		R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\mu) \varphi_{jl}} \Big],
	</math></center>		</math></center>
	<math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,n</math>.		<math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,n</math>.

	[[Category:Interaction Theory]]		[[Category:Interaction Theory]]

Meylan: /* Final Equations */

2007-05-01T07:16:58Z

Final Equations

← Older revision		Revision as of 07:16, 1 May 2007
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	wavefields of all bodies,		wavefields of all bodies,
	<center><math>		<center><math>
	A_{\mu}^l = \frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)}		A_{\mu}^l = -\frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)}
	\sum_{\nu = -\infty}^{\infty}		\sum_{\nu = -\infty}^{\infty}
	\Big[ \tilde{D}_{\nu}^{l} -		\Big[ \tilde{D}_{\nu}^{l} +
	\sum_{j=1,j \neq l}^{N} \sum_{\tau =		\sum_{j=1,j \neq l}^{N} \sum_{\tau =
	-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu} (k		-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu} (k

Meylan: /* Final Equations */

2007-05-01T07:16:21Z

Final Equations

← Older revision		Revision as of 07:16, 1 May 2007
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	[[Diffraction Transfer Matrix]] for a [[Bottom Mounted Cylinder]] so that,		[[Diffraction Transfer Matrix]] for a [[Bottom Mounted Cylinder]] so that,
	<center><math>		<center><math>
	A_{\mu}^l= \frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)} D_{\mu}^l .		A_{\mu}^l= -\frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)} D_{\mu}^l .
	</math></center>		</math></center>

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	A_{\mu}^l = \frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)}		A_{\mu}^l = \frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)}
	\sum_{\nu = -\infty}^{\infty}		\sum_{\nu = -\infty}^{\infty}
	\Big[ \tilde{D}_{\nu}^{l} +		\Big[ \tilde{D}_{\nu}^{l} -
	\sum_{j=1,j \neq l}^{N} \sum_{\tau =		\sum_{j=1,j \neq l}^{N} \sum_{\tau =
	-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu} (k		-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu} (k

Meylan: /* Final Equations */

2007-05-01T07:11:25Z

Final Equations

← Older revision		Revision as of 07:11, 1 May 2007
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	wavefields of all bodies,		wavefields of all bodies,
	<center><math>		<center><math>
	A_{\mu}^l = \frac{H^{(1)}_\mu{}'~~(k a_l)}{J'_\mu~~(k a_l)}		A_{\mu}^l = \frac{J'_\mu(k a_l)}{H^{(1)}_\mu{}'(k a_l)}
	\sum_{\nu = -\infty}^{\infty}		\sum_{\nu = -\infty}^{\infty}
	\Big[ \tilde{D}_{\nu}^{l} +		\Big[ \tilde{D}_{\nu}^{l} +