Wiener-Hopf Elastic Plate Solution - Revision history

Meylan at 22:16, 6 September 2009

2009-09-06T22:16:02Z

← Older revision		Revision as of 22:16, 6 September 2009
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			{{incomplete pages}}

	== Introduction ==		== Introduction ==

Adi Kurniawan at 05:39, 28 April 2009

2009-04-28T05:39:33Z

← Older revision		Revision as of 05:39, 28 April 2009
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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]].

	= Elastic ~~Plate~~ =		== Elastic plate ==

	We imagine two semi-infinite [[:category:Floating Elastic Plate\|Floating Elastic Plates]]		We imagine two semi-infinite [[:category:Floating Elastic Plate\|Floating Elastic Plates]]
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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( \alpha,z\right) </math>		We derive algebraic expressions for <math>\Phi^{\pm}\left( \alpha,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 Wiener-Hopf 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>

Adi Kurniawan: /* Introduction */

2009-04-28T05:38:35Z

Introduction

← Older revision		Revision as of 05:38, 28 April 2009
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	A simpler problem is the [[Wiener-Hopf Solution for a Semi-Infinite Dock]]		A simpler problem is the [[Wiener-Hopf Solution for a Semi-Infinite Dock]]


	The theory is described in [[:Category:Wiener-Hopf\|Wiener-Hopf]].		The theory is described in [[:Category:Wiener-Hopf\|Wiener-Hopf]].

Adi Kurniawan: /* Solution of the W-H Equation */

2008-12-24T06:02:25Z

Solution of the W-H Equation

← Older revision		Revision as of 06:02, 24 December 2008
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	</math></center>		</math></center>
	where <math>R_{1}\left( q^{\prime}\right) </math> is a residue of <math>\left[ f_{1}\left(		where <math>R_{1}\left( q^{\prime}\right) </math> is a residue of <math>\left[ f_{1}\left(
	\gamma\right) \right] ^{-1}<math> at </math>\gamma=q^{\prime}</math>		\gamma\right) \right] ^{-1}</math> at <math>\gamma=q^{\prime}</math>
	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	R_{1}\left( q^{\prime}\right) & =\left( \left. \frac{\mathrm{d}f_{1}\left(		R_{1}\left( q^{\prime}\right) & =\left( \left. \frac{\mathrm{d}f_{1}\left(

Adi Kurniawan: /* Solution of the W-H Equation */

2008-12-24T05:59:12Z

Solution of the W-H Equation

Adi Kurniawan: /* Derivation of the Wiener-Hopf equation */

2008-12-24T05:56:52Z

Derivation of the Wiener-Hopf equation

← Older revision		Revision as of 05:56, 24 December 2008
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	h\left( \alpha,z\right) =\int^{z}\frac{\sinh\gamma\left( z-t\right)		h\left( \alpha,z\right) =\int^{z}\frac{\sinh\gamma\left( z-t\right)
	}{\gamma}\left\{ \phi_{x}\left( 0,t\right) -\mathrm{i}\alpha		}{\gamma}\left\{ \phi_{x}\left( 0,t\right) -\mathrm{i}\alpha
	\phi\left( 0,t\right) \right\} dt.		\phi\left( 0,t\right) \right\} \mathrm{d}t.
	</math></center>		</math></center>
	Note that <math>\operatorname{Re}\gamma>0</math> when <math>\operatorname{Re}\alpha>0</math> and		Note that <math>\operatorname{Re}\gamma>0</math> when <math>\operatorname{Re}\alpha>0</math> and

Adi Kurniawan: /* Weierstrass's factor theorem */

2008-12-24T05:56:31Z

Weierstrass's factor theorem

← Older revision		Revision as of 05:56, 24 December 2008
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	</math></center>		</math></center>

	If <math>H\left( \alpha\right) </math> is even, then <math>dH\left( 0\right) /d\alpha=0</math>		If <math>H\left( \alpha\right) </math> is even, then <math>\mathrm{d}H\left( 0\right) /\mathrm{d}\alpha=0</math>
	and <math>-a_{n}</math> is a zero if <math>a_{n}</math> is a zero. Then we have the simpler		and <math>-a_{n}</math> is a zero if <math>a_{n}</math> is a zero. Then we have the simpler
	expression		expression

Adi Kurniawan: /* Weierstrass's factor theorem */

2008-12-24T05:55:53Z

Weierstrass's factor theorem

← Older revision		Revision as of 05:55, 24 December 2008
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	\alpha\right) </math>, i.e.,		\alpha\right) </math>, i.e.,
	<center><math>		<center><math>
	\frac{d\log H\left( \alpha\right) }{d\alpha} =\frac{1}{H\left(		\frac{\mathrm{d}\log H\left( \alpha\right) }{\mathrm{d}\alpha} =\frac{1}{H\left(
	\alpha\right) }\frac{dH\left( \alpha\right) }{d\alpha}		\alpha\right) }\frac{\mathrm{d}H\left( \alpha\right) }{\mathrm{d}\alpha}
	=\frac{d\log H\left( 0\right) }{d\alpha}+\sum_{n=0}^{\infty}\left[		=\frac{\mathrm{d}\log H\left( 0\right) }{\mathrm{d}\alpha}+\sum_{n=0}^{\infty}\left[
	\frac{1}{\alpha-a_{n}}+\frac{1}{a_{n}}\right] .		\frac{1}{\alpha-a_{n}}+\frac{1}{a_{n}}\right] .
	</math></center>		</math></center>
	Integrating both sides in <math>\left[ 0,\alpha\right] </math> we have		Integrating both sides in <math>\left[ 0,\alpha\right] </math> we have
	<center><math>		<center><math>
	\log H\left( \alpha\right) =\log H\left( 0\right) +\alpha\frac{d\log		\log H\left( \alpha\right) =\log H\left( 0\right) +\alpha\frac{\mathrm{d}\log
	H\left( 0\right) }{d\alpha}+\sum_{n=0}^{\infty}\left[ \log\left(		H\left( 0\right) }{\mathrm{d}\alpha}+\sum_{n=0}^{\infty}\left[ \log\left(
	1-\frac{\alpha}{a_{n}}\right) +\frac{\alpha}{a_{n}}\right] .		1-\frac{\alpha}{a_{n}}\right) +\frac{\alpha}{a_{n}}\right] .
	</math></center>		</math></center>
	Therefore, the expression for <math>H\left( \alpha\right) </math> is		Therefore, the expression for <math>H\left( \alpha\right) </math> is
	<center><math>		<center><math>
	H\left( \alpha\right) =H\left( 0\right) \exp\left[ \alpha\frac{d\log		H\left( \alpha\right) =H\left( 0\right) \exp\left[ \alpha\frac{\mathrm{d}\log
	H\left( 0\right) }{d\alpha}\right] \prod_{n=0}^{\infty}\left(		H\left( 0\right) }{\mathrm{d}\alpha}\right] \prod_{n=0}^{\infty}\left(
	1-\frac{\alpha}{a_{n}}\right) e^{\alpha/a_{n}}.		1-\frac{\alpha}{a_{n}}\right) e^{\alpha/a_{n}}.
	</math></center>		</math></center>

Adi Kurniawan: /* Elastic Plate */

2008-12-24T05:54:48Z

Elastic Plate

← Older revision		Revision as of 05:54, 24 December 2008
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	<center><math>		<center><math>
	\Phi^{-}\left( \alpha,z\right) =\int_{-\infty}^{0}\phi\left( x,z\right)		\Phi^{-}\left( \alpha,z\right) =\int_{-\infty}^{0}\phi\left( x,z\right)
	e^{\mathrm{i}\alpha x}dx		e^{\mathrm{i}\alpha x}\mathrm{d}x
	</math></center>		</math></center>
	and		and
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	\Phi^{+}\left( \alpha,z\right)		\Phi^{+}\left( \alpha,z\right)
	=\int_{0}^{\infty}\phi\left( x,z\right) e^{\mathrm{i}\alpha		=\int_{0}^{\infty}\phi\left( x,z\right) e^{\mathrm{i}\alpha
	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.

Meylan: Reverted edits by CpasmOnrel (CpasmOnrel); changed back to last version by Meylan

2008-07-11T22:53:30Z

Reverted edits by CpasmOnrel (CpasmOnrel); changed back to last version by Meylan

← Older revision		Revision as of 22:53, 11 July 2008
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	~~domgetc4t~~
	= Introduction =		= Introduction =

← Older revision		Revision as of 05:59, 24 December 2008
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	\phi_{z}\left( x,0\right) =\frac{1}{2\pi}\int_{-\infty-\mathrm{i}		\phi_{z}\left( x,0\right) =\frac{1}{2\pi}\int_{-\infty-\mathrm{i}
	\tau}^{\infty-\mathrm{i}\tau}\Phi_{z}^{-}e^{-\mathrm{i}\alpha		\tau}^{\infty-\mathrm{i}\tau}\Phi_{z}^{-}e^{-\mathrm{i}\alpha
	x}d\alpha=\mathrm{i}Q_{1}\left( -\lambda\right) e^{\mathrm{i}		x}\mathrm{d}\alpha=\mathrm{i}Q_{1}\left( -\lambda\right) e^{\mathrm{i}
	\lambda x}+\sum\limits_{q\in\mathcal{K}_{1}}\mathrm{i}Q_{1}\left(		\lambda x}+\sum\limits_{q\in\mathcal{K}_{1}}\mathrm{i}Q_{1}\left(
	q\right) e^{-\mathrm{i}qx} (4-51)		q\right) e^{-\mathrm{i}qx} (4-51)
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	\phi_{z}\left( x,0\right) =\frac{1}{2\pi}\int_{-\infty+\mathrm{i}		\phi_{z}\left( x,0\right) =\frac{1}{2\pi}\int_{-\infty+\mathrm{i}
	\tau}^{\infty+\mathrm{i}\tau}\Phi_{z}^{+}e^{-\mathrm{i}\alpha		\tau}^{\infty+\mathrm{i}\tau}\Phi_{z}^{+}e^{-\mathrm{i}\alpha
	x}d\alpha=-\sum\limits_{q\in\mathcal{K}_{2}}\mathrm{i}Q_{2}\left(		x}\mathrm{d}\alpha=-\sum\limits_{q\in\mathcal{K}_{2}}\mathrm{i}Q_{2}\left(
	q\right) e^{\mathrm{i}qx}.		q\right) e^{\mathrm{i}qx}.
	</math></center>		</math></center>
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	\gamma\right) \right] ^{-1}<math> at </math>\gamma=q^{\prime}</math>		\gamma\right) \right] ^{-1}<math> at </math>\gamma=q^{\prime}</math>
	<center><math>\begin{matrix}		<center><math>\begin{matrix}
	R_{1}\left( q^{\prime}\right) & =\left( \left. \frac{~~df_~~{1}\left(		R_{1}\left( q^{\prime}\right) & =\left( \left. \frac{\mathrm{d}f_{1}\left(
	\gamma\right) }{d\gamma}\right\| _{\gamma=q^{\prime}}\right) ^{-1}		\gamma\right) }{\mathrm{d}\gamma}\right\| _{\gamma=q^{\prime}}\right) ^{-1}
	\\		\\
	& =\left\{ 5D_{1}q^{\prime3}+\frac{b_{1}}{q^{\prime}}+\frac{H}{q^{\prime}		& =\left\{ 5D_{1}q^{\prime3}+\frac{b_{1}}{q^{\prime}}+\frac{H}{q^{\prime}