Difference between revisions of "Template:Finite plate frequency domain"

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We consider the problem of small-amplitude waves which are incident on a semi-infinite floating elastic
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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>.  
We assume that the problem is invariant in the <math>y</math> direction, although we allow the waves to be
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These equations are derived in [[:Category:Floating Elastic Plate|Floating Elastic Plate]]
incident from an angle.  
+
The submergence of the plate is considered negligible.  
 +
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
 
each boundary, although other boundary conditions could easily be considered using
 
each boundary, although other boundary conditions could easily be considered using
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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>\begin{matrix}
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{{general dock type body equations}}
\Delta \phi = 0, \;\;\; -h < z \leq 0,
 
\end{matrix}</math></center>
 
 
<center><math>
 
<center><math>
\partial_z \phi = 0, \;\;\; z = - h,
+
\partial_x^2\left\{\beta(x) \partial_x^2\partial_z \phi\right\}
</math></center>
+
- \left( \gamma(x)\alpha - 1 \right) \partial_z \phi - \alpha\phi = 0, \;\;
<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,
 
  z = 0, \;\;\; -L \leq x \leq L,
 
</math></center>
 
</math></center>

Latest revision as of 00:44, 17 September 2009

We consider the problem of small-amplitude waves which are incident on finite floating elastic plate occupying water surface for [math]\displaystyle{ -L\lt x\lt L }[/math]. These equations are derived in Floating Elastic Plate The submergence of the plate is considered negligible. We assume that the problem is invariant in the [math]\displaystyle{ y }[/math] direction. 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 Plates in the non-dimensional form of 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).

[math]\displaystyle{ \Delta \phi = 0, \;\;\; -h \lt z \leq 0, }[/math]
[math]\displaystyle{ \partial_z \phi = 0, \;\;\; z = - h, }[/math]
[math]\displaystyle{ \partial_z\phi=\alpha\phi, \,\, z=0,\,x\lt -L,\,\,{\rm or}\,\,x\gt L }[/math]
[math]\displaystyle{ \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, \;\; z = 0, \;\;\; -L \leq x \leq L, }[/math]

where [math]\displaystyle{ \alpha = \omega^2 }[/math], [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \gamma }[/math] are the stiffness and mass constant for the plate respectively. The free edge conditions at the edge of the plate imply

[math]\displaystyle{ \partial_x^3 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L, }[/math]
[math]\displaystyle{ \partial_x^2 \partial_z\phi = 0, \;\; z = 0, \;\;\; x = \pm L, }[/math]