Wiener-Hopf Solution for a Semi-Infinite Dock - Revision history

Meylan at 22:14, 6 September 2009

2009-09-06T22:14:44Z

← Older revision		Revision as of 22:14, 6 September 2009
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	{{~~incompete page~~}}		{{incomplete pages}}

	== Introduction ==		== Introduction ==

Meylan at 22:14, 6 September 2009

2009-09-06T22:14:20Z

← Older revision		Revision as of 22:14, 6 September 2009
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	{{incompete ~~pages~~}}		{{incompete page}}

	== Introduction ==		== Introduction ==

Meylan at 22:14, 6 September 2009

2009-09-06T22:14:00Z

← Older revision		Revision as of 22:14, 6 September 2009
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			{{incompete pages}}

	== Introduction ==		== Introduction ==

Adi Kurniawan: /* Governing Equations */

2008-12-24T06:03:39Z

Governing Equations

← Older revision		Revision as of 06:03, 24 December 2008
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	<center><math>		<center><math>
	\Phi^{-}\left( s,z\right) =\int_{-\infty}^{0}\phi\left( x,z\right)		\Phi^{-}\left( s,z\right) =\int_{-\infty}^{0}\phi\left( x,z\right)
	e^{\mathrm{i}s x}dx		e^{\mathrm{i}s x}\mathrm{d}x
	</math></center>		</math></center>
	and		and
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	\Phi^{+}\left( s,z\right)		\Phi^{+}\left( s,z\right)
	=\int_{0}^{\infty}\phi\left( x,z\right) e^{\mathrm{i}s		=\int_{0}^{\infty}\phi\left( x,z\right) e^{\mathrm{i}s
	x}dx.		x}\mathrm{d}x.
	</math></center>		</math></center>
	Notice that the superscript `<math>+</math>' and `<math>-</math>' correspond to the integral domain.		Notice that the superscript `<math>+</math>' and `<math>-</math>' correspond to the integral domain.

Adi Kurniawan at 04:12, 23 September 2008

2008-09-23T04:12:31Z

← Older revision		Revision as of 04:12, 23 September 2008
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	= Introduction =		== Introduction ==

	We present here the [[:Category:Wiener-Hopf\|Wiener-Hopf]] solution to the problem of a		We present here the [[:Category:Wiener-Hopf\|Wiener-Hopf]] solution to the problem of a
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	The theory is described in [[:Category:Wiener-Hopf\|Wiener-Hopf]].		The theory is described in [[:Category:Wiener-Hopf\|Wiener-Hopf]].

			==Governing Equations==

	=Governing Equations=

	We begin with the [[Frequency Domain Problem]] for a dock which occupies		We begin with the [[Frequency Domain Problem]] for a dock which occupies
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	transforms are regular and are able to calculate the inverse transform.		transforms are regular and are able to calculate the inverse transform.

	=Weierstrass's factor theorem =		==Weierstrass's factor theorem ==

	As mentioned above, we will require splitting a ratio of two functions of a		As mentioned above, we will require splitting a ratio of two functions of a
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	</math></center>		</math></center>

	=Derivation of the Wiener-Hopf equation=		==Derivation of the Wiener-Hopf equation==

	We derive algebraic expressions for <math>\Phi^{\pm}\left( s,z\right) </math>		We derive algebraic expressions for <math>\Phi^{\pm}\left( s,z\right) </math>
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	</math></center>		</math></center>

	= [[Dispersion Relation for a Floating Elastic Plate]] =		== [[Dispersion Relation for a Floating Elastic Plate]] ==

	Functions <math>f_{1}</math> and <math>f_{2}</math> are		Functions <math>f_{1}</math> and <math>f_{2}</math> are
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	and we order them with increasing size.		and we order them with increasing size.

	= Solution of the W-H Equation=		== Solution of the W-H Equation==

	Using the Mittag-Leffler theorem ([[Carrier, Krook and Pearson 1966]] section 2.9), functions <math>\Phi_{z}^{\pm}</math> can be expressed by a series of fractional functions that contribute to the solutions. Thus, we have series expansions of <math>\Phi_{z}^{-}</math> and <math>\Phi_{z}^{+}</math>		Using the Mittag-Leffler theorem ([[Carrier, Krook and Pearson 1966]] section 2.9), functions <math>\Phi_{z}^{\pm}</math> can be expressed by a series of fractional functions that contribute to the solutions. Thus, we have series expansions of <math>\Phi_{z}^{-}</math> and <math>\Phi_{z}^{+}</math>

Meylan: /* Derivation of the Wiener-Hopf equation */

2008-04-28T03:48:26Z

Derivation of the Wiener-Hopf equation

← Older revision		Revision as of 03:48, 28 April 2008
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	</math></center>		</math></center>
	<center><math>		<center><math>
	f_{2}\left( s \right) = 1		f_{2}\left( s \right) = \frac{\alpha}{s\tanh s h}
	</math></center>		</math></center>
	<center><math>		<center><math>

Meylan: /* Derivation of the Wiener-Hopf equation */

2008-04-28T03:47:17Z

Derivation of the Wiener-Hopf equation

← Older revision		Revision as of 03:47, 28 April 2008
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	0,z\right) -\phi_{x}\left( 0,z\right) \right\} </math>,		0,z\right) -\phi_{x}\left( 0,z\right) \right\} </math>,
	<center><math>		<center><math>
	g\left( s,z\right) =\frac{h_{z}\left( s,-H\right) }{s		g\left( s,z\right) =\frac{h_{z}\left( s,-h\right) }{s
	}\left( \tanh s H\cosh s\left( z+H\right) -\sinh s\left(		}\left( \tanh s h\cosh s\left( z+h\right) -\sinh s\left(
	z+H\right) \right)		z+h\right) \right)
	</math></center>		</math></center>
	<center><math>		<center><math>
	+h\left( s,z\right) \left( 1-\frac{\cosh s\left( z+H\right)		+h\left( s,z\right) \left( 1-\frac{\cosh s\left( z+h\right)
	}{\cosh s H}\right) ,		}{\cosh s h}\right) ,
	</math></center>		</math></center>
	<center><math>		<center><math>
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	<center><math>		<center><math>
	\Phi_{z}^{\pm}\left( s,0\right) =\Phi^{\pm}\left( s,0\right)		\Phi_{z}^{\pm}\left( s,0\right) =\Phi^{\pm}\left( s,0\right)
	~~\gamma~~\tanh~~\gamma H~~\pm g_{z}\left( s,0\right)		s \tanh sh\pm g_{z}\left( s,0\right)
	</math></center>		</math></center>
	where <math>\Phi_{z}^{\pm}\left( s,0\right) </math> denotes the <math>z</math>-derivative. We		where <math>\Phi_{z}^{\pm}\left( s,0\right) </math> denotes the <math>z</math>-derivative. We
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	We therefore have		We therefore have
	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	f_{1}\left( ~~\gamma~~\right) \Phi_{z}^{-}\left( s,0\right) +C_{1}\left(		f_{1}\left( s\right) \Phi_{z}^{-}\left( s,0\right) +C_{1}\left(
	s\right) & =0 \\		s\right) & =0 \\
	f_{2}\left( ~~\gamma~~\right) \Phi_{z}^{+}\left( s,0\right) +C_{2}\left(		f_{2}\left( s\right) \Phi_{z}^{+}\left( s,0\right) +C_{2}\left(
	s\right) & =0		s\right) & =0
	\end{matrix}</math></center>		\end{matrix}</math></center>
	where		where
	<center><math>		<center><math>
	f_{1}\left( ~~\gamma~~\right) = 1 -\frac{\alpha}{s\tanh~~\gamma~~ h}		f_{1}\left( s \right) = 1 -\frac{\alpha}{s\tanh s h}
	</math></center>		</math></center>
	<center><math>		<center><math>
	f_{2}\left( ~~\gamma~~\right) = 1 ~~-\frac{\alpha}{s\tanh\gamma h}~~		f_{2}\left( s \right) = 1
	</math></center>		</math></center>
	<center><math>		<center><math>
	C_{1}\left( s\right) =-\frac{\~~rho\omega^{2}~~g_{z}\left(		C_{1}\left( s\right) =-\frac{\alpha g_{z}\left(
	s,0\right) }{~~\gamma~~\tanh~~\gamma H~~}~~+P_{1}\left( s\right)~~		s,0\right) }{s\tanh s h}
	,\;C_{2}\left( s\right) =\frac{\~~rho\omega^{2}~~g_{z}\left(		,\;C_{2}\left( s\right) =\frac{\alpha g_{z}\left(
	s,0\right) }{~~\gamma~~\tanh~~\gamma H~~}~~-P_{2}\left( s\right)~~ .		s,0\right) }{s\tanh s h}.
	</math></center>		</math></center>

Meylan: /* Derivation of the Wiener-Hopf equation */

2008-04-28T03:11:58Z

Derivation of the Wiener-Hopf equation

← Older revision		Revision as of 03:11, 28 April 2008
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	</math></center>		</math></center>
	<center><math>		<center><math>
	\Phi_{z} ~~=0,~~		\Phi_{z}^{+} =0,
	~~</math></center>~~
	~~where~~
	~~<center><math>~~
	~~P_{j}\left( s\right) =D_{j}\left[ c_{3}^{j}-\mathrm{i}c_{2}~~
	~~^{j}s-\left( s+2k^{2}\right) \left( c_{1}^{j}-\mathrm{i}~~
	~~c_{0}^{j}s\right) \right] ,\;j=1,2,~~
	~~</math></center>~~
	~~<center><math>~~
	~~c_{\mathrm{i}}^{1}=\left. \left( \frac{\partial}{\partial x}\right)~~
	^{~~i}\phi_{z}\right\| _{x=0-,z=0},\;c_{\mathrm{i}}~~
	~~^{2}=\left. \left( \frac{\partial}{\partial x}\right) ^{i~~
	~~}\phi_{z}\right\| _{x=0~~+~~,z=0~~}~~,\;i~~=0,~~1,2,3.~~
	</math></center>		</math></center>
	We therefore have		We therefore have
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	where		where
	<center><math>		<center><math>
	f_{j}\left( \gamma\right) =D_{j}\gamma~~^{4~~}~~-m_~~{j}\~~omega^{2}+~~\~~rho~~		f_{1}\left( \gamma\right) = 1 -\frac{\alpha}{s\tanh\gamma h}
	g-\frac{\~~rho\omega^{2}~~}{~~\gamma~~\tanh\gamma H}~~,\;j=1,2,~~		</math></center>
			<center><math>
			f_{2}\left( \gamma\right) = 1 -\frac{\alpha}{s\tanh\gamma h}
	</math></center>		</math></center>
	<center><math>		<center><math>

Meylan: /* Derivation of the Wiener-Hopf equation */

2008-04-28T03:09:28Z

Derivation of the Wiener-Hopf equation

← Older revision		Revision as of 03:09, 28 April 2008
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	apply the integral transform to the free-surface conditions in <math>x<0</math> and <math>x>0</math>,		apply the integral transform to the free-surface conditions in <math>x<0</math> and <math>x>0</math>,
	<center><math>		<center><math>
	~~\left\{ D_{1}\gamma^{4}-m_{1}\omega^{2}+\rho g\right\}~~ \Phi_{z}^{-}~~\left(~~		\Phi_{z}^{-} -\alpha \Phi^{-}\left( s,0\right) =0,
	~~s,0\right)~~ -\~~rho\omega^{2}~~\Phi^{-}\left( s,0~~\right) +P_{1}\left(~~
	s\right) =0,
	</math></center>		</math></center>
	<center><math>		<center><math>
	~~\left\{ D_{2}\gamma^{4}-m_{2}\omega^{2}+\rho g\right\}~~ \Phi_{z}~~^{+}\left(~~		\Phi_{z} =0,
	~~s,0\right) -\rho\omega^{2}\Phi^{+}\left( s,0\right) -P_{2}\left(~~
	~~s\right)~~ =0,
	</math></center>		</math></center>
	where		where

Meylan: /* Derivation of the Wiener-Hopf equation */

2008-04-28T03:07:13Z

Derivation of the Wiener-Hopf equation

← Older revision		Revision as of 03:07, 28 April 2008
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	s,z\right)		s,z\right)
	</math></center>		</math></center>
	where ~~<math>\gamma=\sqrt{s^{2}+k^{2}}</math> and~~ <math>g\left( s,z\right) </math> is a		where <math>g\left( s,z\right) </math> is a
	function determined by <math>\left\{ \mathrm{i}s\phi\left(		function determined by <math>\left\{ \mathrm{i}s\phi\left(
	0,z\right) -\phi_{x}\left( 0,z\right) \right\} </math>,		0,z\right) -\phi_{x}\left( 0,z\right) \right\} </math>,
	<center><math>		<center><math>
	g\left( s,z\right) =\frac{h_{z}\left( s,-H\right) }{~~\gamma~~		g\left( s,z\right) =\frac{h_{z}\left( s,-H\right) }{s
	}\left( \tanh~~\gamma~~ H\cosh~~\gamma~~\left( z+H\right) -\sinh~~\gamma~~\left(		}\left( \tanh s H\cosh s\left( z+H\right) -\sinh s\left(
	z+H\right) \right)		z+H\right) \right)
	</math></center>		</math></center>
	<center><math>		<center><math>
	+h\left( s,z\right) \left( 1-\frac{\cosh~~\gamma~~\left( z+H\right)		+h\left( s,z\right) \left( 1-\frac{\cosh s\left( z+H\right)
	}{\cosh~~\gamma~~ H}\right) ,		}{\cosh s H}\right) ,
	</math></center>		</math></center>
	<center><math>		<center><math>
	h\left( s,z\right) =\int^{z}\frac{\sinh~~\gamma~~\left( z-t\right)		h\left( s,z\right) =\int^{z}\frac{\sinh s\left( z-t\right)
	}{~~\gamma~~}\left\{ \phi_{x}\left( 0,t\right) -\mathrm{i}s		}{s}\left\{ \phi_{x}\left( 0,t\right) -\mathrm{i}s
	\phi\left( 0,t\right) \right\} dt.		\phi\left( 0,t\right) \right\} dt.
	</math></center>		</math></center>
	~~Note that <math>\operatorname{Re}\gamma>0</math> when <math>\operatorname{Re}s>0</math> and~~		We have, by differentiating both sides with respect to <math>z</math> at <math>z=0</math>
	~~<math>\operatorname{Re}\gamma<0</math> when <math>\operatorname{Re}s<0</math>.~~ We have, by
	differentiating both sides with respect to <math>z</math> at <math>z=0</math>
	<center><math>		<center><math>
	\Phi_{z}^{\pm}\left( s,0\right) =\Phi^{\pm}\left( s,0\right)		\Phi_{z}^{\pm}\left( s,0\right) =\Phi^{\pm}\left( s,0\right)