Template:Finite plate frequency domain - Revision history

Meylan at 00:44, 17 September 2009

2009-09-17T00:44:09Z

← Older revision		Revision as of 00:44, 17 September 2009
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	in the non-dimensional form of [[Tayler_1986a\|Tayler 1986]] ([[Dispersion Relation for a Floating Elastic Plate]]).		in the non-dimensional form of [[Tayler_1986a\|Tayler 1986]] ([[Dispersion Relation for a Floating Elastic Plate]]).
	We also assume that the waves are normally incident (incidence at an angle will be discussed later).		We also assume that the waves are normally incident (incidence at an angle will be discussed later).
	~~<center><math>~~		{{general dock type body equations}}
	~~\Delta \phi = 0, \;\;\; -h < z \leq 0,~~
	~~</math></center>~~
	~~<center><math>~~
	~~\partial_z \phi = 0, \;\;\; z = - h,~~
	~~</math></center>~~
	~~<center><math>~~
	~~\partial_z\phi=\alpha\phi, \,\, z=0,\,x<-L,\,\,~~{~~\rm or~~}~~\,\,x>L~~
	~~</math></center>~~
	<center><math>		<center><math>
	\partial_x^2\left\{\beta(x) \partial_x^2\partial_z \phi\right\}		\partial_x^2\left\{\beta(x) \partial_x^2\partial_z \phi\right\}

Meylan at 00:14, 2 September 2009

2009-09-02T00:14:54Z

← Older revision		Revision as of 00:14, 2 September 2009
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	</math></center>		</math></center>
	<center><math>		<center><math>
	\partial_x^2\beta(x) \partial_x^2\partial_z \phi		\partial_x^2\left\{\beta(x) \partial_x^2\partial_z \phi\right\}
	- \left( \gamma(x)\alpha - 1 \right) \partial_z \phi - \alpha\phi = 0, \;\;		- \left( \gamma(x)\alpha - 1 \right) \partial_z \phi - \alpha\phi = 0, \;\;
	z = 0, \;\;\; -L \leq x \leq L,		z = 0, \;\;\; -L \leq x \leq L,

Meylan at 00:12, 2 September 2009

2009-09-02T00:12:50Z

← Older revision		Revision as of 00:12, 2 September 2009
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	</math></center>		</math></center>
	<center><math>		<center><math>
	\beta \partial_x^4\partial_z \phi		\partial_x^2\beta(x) \partial_x^2\partial_z \phi
	- \left( \gamma\alpha - 1 \right) \partial_z \phi - \alpha\phi = 0, \;\;		- \left( \gamma(x)\alpha - 1 \right) \partial_z \phi - \alpha\phi = 0, \;\;
	z = 0, \;\;\; -L \leq x \leq L,		z = 0, \;\;\; -L \leq x \leq L,
	</math></center>		</math></center>

Mike smith: orientation correction

2009-03-31T05:54:50Z

orientation correction

← Older revision		Revision as of 05:54, 31 March 2009
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	<center><math>		<center><math>
	\beta \partial_x^4\partial_z \phi		\beta \partial_x^4\partial_z \phi
	- \left( \gamma\alpha - 1 \right) \partial_z \phi + \alpha\phi = 0, \;\;		- \left( \gamma\alpha - 1 \right) \partial_z \phi - \alpha\phi = 0, \;\;
	z = 0, \;\;\; -L \leq x \leq L,		z = 0, \;\;\; -L \leq x \leq L,
	</math></center>		</math></center>

Meylan at 02:27, 23 March 2009

2009-03-23T02:27:29Z

← Older revision		Revision as of 02:27, 23 March 2009
Line 1:		Line 1:
	We consider the problem of small-amplitude waves which are incident on finite floating elastic		We consider the problem of small-amplitude waves which are incident on finite floating elastic
	plate occupying water surface for <math>-L<x<L</math>. The submergence of the plate is considered negligible.		plate occupying water surface for <math>-L<x<L</math>.
			These equations are derived in [[:Category:Floating Elastic Plate\|Floating Elastic Plate]]
			The submergence of the plate is considered negligible.
	We assume that the problem is invariant in the <math>y</math> direction.		We assume that the problem is invariant in the <math>y</math> direction.
	We also assume that the plate edges are free to move at		We also assume that the plate edges are free to move at

Meylan at 22:53, 5 November 2008

2008-11-05T22:53:09Z

← Older revision		Revision as of 22:53, 5 November 2008
Line 8:		Line 8:
	in the non-dimensional form of [[Tayler_1986a\|Tayler 1986]] ([[Dispersion Relation for a Floating Elastic Plate]]).		in the non-dimensional form of [[Tayler_1986a\|Tayler 1986]] ([[Dispersion Relation for a Floating Elastic Plate]]).
	We also assume that the waves are normally incident (incidence at an angle will be discussed later).		We also assume that the waves are normally incident (incidence at an angle will be discussed later).
	<center><math>~~\begin{matrix}~~		<center><math>
	\Delta \phi = 0, \;\;\; -h < z \leq 0,		\Delta \phi = 0, \;\;\; -h < z \leq 0,
	~~\end{matrix}~~</math></center>		</math></center>
	<center><math>		<center><math>
	\partial_z \phi = 0, \;\;\; z = - h,		\partial_z \phi = 0, \;\;\; z = - h,

Meylan at 07:41, 4 November 2008

2008-11-04T07:41:45Z

← Older revision		Revision as of 07:41, 4 November 2008
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	We consider the problem of small-amplitude waves which are incident on ~~a semi-infinite~~ floating elastic		We consider the problem of small-amplitude waves which are incident on finite floating elastic
	plate occupying water surface for <math>x>0</math>. The submergence of the plate is considered negligible.		plate occupying water surface for <math>-L<x<L</math>. The submergence of the plate is considered negligible.
	We assume that the problem is invariant in the <math>y</math> direction~~, although we allow the waves to be~~		We assume that the problem is invariant in the <math>y</math> direction.
	~~incident from an angle~~.
	We also assume that the plate edges are free to move at		We also assume that the plate edges are free to move at
	each boundary, although other boundary conditions could easily be considered using		each boundary, although other boundary conditions could easily be considered using

Meylan at 07:40, 4 November 2008

2008-11-04T07:40:03Z

New page

We consider the problem of small-amplitude waves which are incident on a semi-infinite floating elastic
plate occupying water surface for <math>x>0</math>. The submergence of the plate is considered negligible.
We assume that the problem is invariant in the <math>y</math> direction, although we allow the waves to be
incident from an angle.
We also assume that the plate edges are free to move at
each boundary, although other boundary conditions could easily be considered using
the methods of solution presented here. We begin with the [[Frequency Domain Problem]] for a semi-infinite
[[Floating Elastic Plate|Floating Elastic Plates]]
in the non-dimensional form of [[Tayler_1986a|Tayler 1986]] ([[Dispersion Relation for a Floating Elastic Plate]]).
We also assume that the waves are normally incident (incidence at an angle will be discussed later).
<center><math>\begin{matrix}
\Delta \phi = 0, \;\;\; -h < z \leq 0,
\end{matrix}</math></center>
<center><math>
\partial_z \phi = 0, \;\;\; z = - h,
</math></center>
<center><math>
\partial_z\phi=\alpha\phi, \,\, z=0,\,x<-L,\,\,{\rm or}\,\,x>L
</math></center>
<center><math>
\beta \partial_x^4\partial_z \phi
- \left( \gamma\alpha - 1 \right) \partial_z \phi + \alpha\phi = 0, \;\;
z = 0, \;\;\; -L \leq x \leq L,
</math></center>
where <math>\alpha = \omega^2</math>, <math>\beta</math> and <math>\gamma</math>
are the stiffness and mass constant for the plate respectively. The free edge conditions
at the edge of the plate imply
<center><math>
\partial_x^3 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L,
</math></center>
<center><math>
\partial_x^2 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L,
</math></center>