WikiWaves - User contributions [en]

Three-Dimensional Floating Elastic Plate

2006-09-11T08:31:47Z

Mdmgreen: /* Equations of Motion for the Water */

= Equations of Motion =

For a classical thin plate, the equation of motion is given by
<center><math>
D\nabla ^4 w - \rho _i h \frac{\partial^2 w}{\partial t^2} = p \qquad (1)
</math></center>
Equation (1) is subject to the free edge boundary
conditions for a thin plate
<center><math>
\left[ \nabla^2 - (1-\nu)
\left(\frac{\partial^2}{\partial s^2} + \kappa(s)
\frac{\partial}{\partial n} \right) \right] w = 0, \qquad(1.1)
</math></center>
<center><math>
\left[ \frac{\partial}{\partial n} \nabla^2 +(1-\nu)
\frac{\partial}{\partial s}
\left( \frac{\partial}{\partial n} \frac{\partial}{\partial s}
-\kappa(s) \frac{\partial}{\partial s} \right) \right] w = 0, \qquad(1.2)
</math></center>
where <math>\nu</math> is Poisson's ratio and
<center><math>
\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}
= \frac{\partial^2}{\partial n^2} + \frac{\partial^2}{\partial s^2}
+ \kappa(s) \frac{\partial}{\partial n}.
</math></center>
Here, <math>\kappa(s)</math> is the curvature of the boundary, <math>\partial \Delta</math>,
as a function of arclength <math>s</math> along <math>\partial \Delta</math>;
<math>\partial/\partial s</math> and <math>\partial/\partial n</math> represent derivatives
tangential and normal to the boundary <math>\partial \Delta</math>, respectively.
[[Porter paper?]] where <math>n</math> and <math>s</math> denote the normal and tangential
directions respectively.

The pressure, <math>p</math>, is given by the linearized Bernoulli's equation at the
water surface,
<center><math>
p=-\rho \frac{\partial \phi }{\partial t}-\rho gw.\,\,\, (2)
</math></center>
where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density
of the water, and <math>g</math> is the acceleration due to gravity.

We now introduce non-dimensional variables. We non-dimensionalise the length
variables with respect to <math>a</math> where the surface area of the floe is <math>4a^{2}.</math>
We non-dimensionalise the time variables with respect to <math>\sqrt{g/a}</math>.
In the non-dimensional variables equations (1) and (2)
become
<center><math>
\beta \nabla^{4}{w}+\gamma \frac{\partial^2 w}{\partial t^2}=\frac{\partial {\Phi}}{\partial {t}}-{w}, \qquad(3)% (n-d_ice)
</math></center>
where
<center><math>
\beta =\frac{D}{g\rho a^4}\;\;{\mathrm and}\;\; \gamma =\frac{\rho_i h}{\rho a}.
</math></center>

We assume the [[Frequency Domain Problem]] with frequency <math>\omega</math>.
This leads to the following equation
<center><math>
\beta \nabla ^{4}w+\alpha \gamma w=-i\omega\phi -w. \qquad(4)%(plate2)
</math></center>

=Equations of Motion for the Water=

We require the equation of motion for the water to solve equation ({plate2}).
We begin [[Standard Linear Wave Scattering Problem]] equations with the boundary condition
under the plate modified as appropriate.
<center><math>
\left.
\begin{matrix}
\nabla ^{2}\phi =0, & -\infty <z<0, \\
{\frac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\
{\frac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\;
\mathbf{x}\in \Delta , \\
{\frac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{
x}\notin \Delta ,
\end{matrix}
\right\} \qquad(5)
</math></center>
The vector <math>\mathbf{x=(}x,y)</math> is a
point on the water surface and <math>\Delta </math> is the region of the water surface
occupied by the plate.

The boundary value problem (5) is subject to an incident wave which
is imposed through the
[[Sommerfeld Radiation Condition]]
<center><math>
\lim_{\left| \mathbf{x}\right| \rightarrow \infty }\sqrt{|\mathbf{x}|}\left(
\frac{\partial }{\partial |\mathbf{x}|}-i\alpha \right) (\phi -\phi ^{
\mathrm{In}})=0, \qquad(6)
</math></center>
where the incident potential <math>\phi ^{\mathrm{In}}</math> is
<center><math>
\phi ^{\mathrm{In}}(x,y,z)=\frac{A}{{\omega }}e^{i\alpha (x\cos \theta
+y\sin \theta )}e^{\alpha z}, \qquad (7)(input)
</math></center>
where <math>A</math> is the non-dimensional wave amplitude.

= Solution of the Equations of Motion =

There are a number of methods to solve this problem. We will describe a
method which generalises the [[Linear Wave Scattering for Floating Rigid Body]] to a plate which
has an infinite number of degrees of freedom. Many other methods of solution
have been presented, most of which consider some kind of regular plate shape
(such as a circle or square).
The standard solution method to the linear wave problem is to transform the
boundary value problem into an integral equation using a Green function
Performing such a transformation, the boundary
value problem (5) and (6) become
<center><math>
\phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x}
;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x}
)\right) dS_{\mathbf{y}}. \qquad(8)(water)
</math></center>
where <math>G_{\alpha }</math> is the [[Free-Surface Green Function]]

=Solving for the Elastic Plate Motion=

To determine the ice floe motion we must solve equations ((plate2)) and (
(water)) simultaneously. We do this by expanding the floe motion in the
free modes of vibration of a thin plate. The major difficulty with this
method is that the free modes of vibration can be determined analytically
only for very restrictive geometries, e.g. a circular thin plate. Even the
free modes of vibration of a square plate with free edges must be determined
numerically. This is the reason why the solution of [[Meylan and Squire 1996]] was
only for a circular floe.

Since the operator <math>\nabla ^{4},</math> subject to the free edge boundary
conditions, is self adjoint a thin plate must possess a set of modes <math>w_{i}</math>
which satisfy the free boundary conditions and the following eigenvalue
equation
<center><math>
\nabla ^{4}w_{i}=\lambda _{i}w_{i}.
</math></center>
The modes which correspond to different eigenvalues <math>\lambda _{i}</math> are
orthogonal and the eigenvalues are positive and real. While the plate will
always have repeated eigenvalues, orthogonal modes can still be found and
the modes can be normalized. We therefore assume that the modes are
orthonormal, i.e.
<center><math>
\iint_{\Delta }w_{i}\left( \mathbf{Q}\right) w_{j}\left( \mathbf{Q}\right)
dS_{\mathbf{Q}}=\delta _{ij}
</math></center>
where <math>\delta _{ij}</math> is the Kronecker delta. The eigenvalues <math>\lambda _{i}</math>
have the property that <math>\lambda _{i}\rightarrow \infty </math> as <math>i\rightarrow
\infty </math> and we order the modes by increasing eigenvalue. These modes can be
used to expand any function over the wetted surface of the ice floe <math>\Delta </math>
.

We expand the displacement of the floe in a finite number of modes <math>N,</math> i.e.
<center><math>
w\left( \mathbf{x}\right) =\sum_{i=1}^{N}c_{i}w_{i}\left( \mathbf{x}\right) .
(expansion)
</math></center>
From the linearity of ((water)) the potential can be written in the
following form
<center><math>
\phi =\phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i} (expansionphi)
</math></center>
where <math>\phi _{0}</math> and <math>\phi _{i}</math> satisfy the integral equations
<center><math>
\phi _{0}(\mathbf{x})=\phi ^{\mathrm{In}}(\mathbf{x})+\iint_{\Delta }\alpha
G_{\alpha }(\mathbf{x};\mathbf{y})\phi (\mathbf{y})dS_{\mathbf{y}}
(phi0)
</math></center>
and
<center><math>
\phi _{i}(\mathbf{x})=\iint_{\Delta }G_{\alpha }(\mathbf{x};\mathbf{y}
)\left( \alpha \phi _{i}(\mathbf{x})+i\sqrt{\alpha }w_{i}(\mathbf{y})\right)
dS_{\mathbf{y}}. (phii)
</math></center>
The potential <math>\phi _{0}</math> represents the potential due the incoming wave
assuming that the displacement of the ice floe is zero. The potentials <math>\phi
_{i}</math> represent the potential which is generated by the plate vibrating with
the <math>i</math>th mode in the absence of any input wave forcing.

We substitute equations ((expansion)) and ((expansionphi)) into
equation ((plate2)) to obtain
<center><math>
\beta \sum_{i=1}^{N}\lambda _{i}c_{i}w_{i}-\alpha \gamma
\sum_{i=1}^{N}c_{i}w_{i}=i\sqrt{\alpha }\left( \phi
_{0}+\sum_{i=1}^{N}c_{i}\phi _{i}\right) -\sum_{i=1}^{N}c_{i}w_{i}.
(expanded)
</math></center>
To solve equation ((expanded)) we multiply by <math>w_{j}</math> and integrate over
the plate (i.e. we take the inner product with respect to <math>w_{j})</math> taking
into account the orthogonality of the modes <math>w_{i}</math>, and obtain
<center><math>
\beta \lambda _{j}c_{j}+\left( 1-\alpha \gamma \right) c_{j}=\iint_{\Delta }i
\sqrt{\alpha }\left( \phi _{0}\left( \mathbf{Q}\right)
+\sum_{i=1}^{N}c_{i}\phi _{i}\left( \mathbf{Q}\right) \right) w_{j}\left(
\mathbf{Q}\right) dS_{\mathbf{Q}} (final)
</math></center>
which is a matrix equation in <math>c_{i}.</math>

We cannot solve equation ((final)) without determining the modes of
vibration of the thin plate <math>w_{i}</math> (along with the associated eigenvalues <math>
\lambda _{i})</math> and solving the integral equations ((phi0)) and (\ref
{phii}). We use the finite element method to determine the modes of
vibration [[Zienkiewicz]] and the integral equations ((phi0)) and (
(phii)) are solved by a constant panel method [[Sarp_Isa]]. The same
set of nodes is used for the finite element method and to define the panels
for the integral equation.

[[Category:Floating Elastic Plate]]

Three-Dimensional Floating Elastic Plate

2006-09-11T08:31:01Z

Mdmgreen: /* Solution of the Equations of Motion */

= Equations of Motion =

For a classical thin plate, the equation of motion is given by
<center><math>
D\nabla ^4 w - \rho _i h \frac{\partial^2 w}{\partial t^2} = p \qquad (1)
</math></center>
Equation (1) is subject to the free edge boundary
conditions for a thin plate
<center><math>
\left[ \nabla^2 - (1-\nu)
\left(\frac{\partial^2}{\partial s^2} + \kappa(s)
\frac{\partial}{\partial n} \right) \right] w = 0, \qquad(1.1)
</math></center>
<center><math>
\left[ \frac{\partial}{\partial n} \nabla^2 +(1-\nu)
\frac{\partial}{\partial s}
\left( \frac{\partial}{\partial n} \frac{\partial}{\partial s}
-\kappa(s) \frac{\partial}{\partial s} \right) \right] w = 0, \qquad(1.2)
</math></center>
where <math>\nu</math> is Poisson's ratio and
<center><math>
\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}
= \frac{\partial^2}{\partial n^2} + \frac{\partial^2}{\partial s^2}
+ \kappa(s) \frac{\partial}{\partial n}.
</math></center>
Here, <math>\kappa(s)</math> is the curvature of the boundary, <math>\partial \Delta</math>,
as a function of arclength <math>s</math> along <math>\partial \Delta</math>;
<math>\partial/\partial s</math> and <math>\partial/\partial n</math> represent derivatives
tangential and normal to the boundary <math>\partial \Delta</math>, respectively.
[[Porter paper?]] where <math>n</math> and <math>s</math> denote the normal and tangential
directions respectively.

The pressure, <math>p</math>, is given by the linearized Bernoulli's equation at the
water surface,
<center><math>
p=-\rho \frac{\partial \phi }{\partial t}-\rho gw.\,\,\, (2)
</math></center>
where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density
of the water, and <math>g</math> is the acceleration due to gravity.

We now introduce non-dimensional variables. We non-dimensionalise the length
variables with respect to <math>a</math> where the surface area of the floe is <math>4a^{2}.</math>
We non-dimensionalise the time variables with respect to <math>\sqrt{g/a}</math>.
In the non-dimensional variables equations (1) and (2)
become
<center><math>
\beta \nabla^{4}{w}+\gamma \frac{\partial^2 w}{\partial t^2}=\frac{\partial {\Phi}}{\partial {t}}-{w}, \qquad(3)% (n-d_ice)
</math></center>
where
<center><math>
\beta =\frac{D}{g\rho a^4}\;\;{\mathrm and}\;\; \gamma =\frac{\rho_i h}{\rho a}.
</math></center>

We assume the [[Frequency Domain Problem]] with frequency <math>\omega</math>.
This leads to the following equation
<center><math>
\beta \nabla ^{4}w+\alpha \gamma w=-i\omega\phi -w. \qquad(4)%(plate2)
</math></center>

=Equations of Motion for the Water=

We require the equation of motion for the water to solve equation ({plate2}).
We begin [[Standard Linear Wave Scattering Problem]] equations with the boundary condition
under the plate modified as appropriate.
<center><math>
\left.
\begin{matrix}
\nabla ^{2}\phi =0, & -\infty <z<0, \\
{\frac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\
{\frac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\;
\mathbf{x}\in \Delta , \\
{\frac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{
x}\notin \Delta ,
\end{matrix}
\right\} \qquad(5)%(bvp_nond)
</math></center>
The vector <math>\mathbf{x=(}x,y)</math> is a
point on the water surface and <math>\Delta </math> is the region of the water surface
occupied by the plate.

The boundary value problem ((bvp)) is subject to an incident wave which
is imposed through the
[[Sommerfeld Radiation Condition]]
<center><math>
\lim_{\left| \mathbf{x}\right| \rightarrow \infty }\sqrt{|\mathbf{x}|}\left(
\frac{\partial }{\partial |\mathbf{x}|}-i\alpha \right) (\phi -\phi ^{
\mathrm{In}})=0, \qquad(6) %(sommerfield)
</math></center>
where the incident potential <math>\phi ^{\mathrm{In}}</math> is
<center><math>
\phi ^{\mathrm{In}}(x,y,z)=\frac{A}{{\omega }}e^{i\alpha (x\cos \theta
+y\sin \theta )}e^{\alpha z}, \qquad (7)(input)
</math></center>
where <math>A</math> is the non-dimensional wave amplitude.

= Solution of the Equations of Motion =

There are a number of methods to solve this problem. We will describe a
method which generalises the [[Linear Wave Scattering for Floating Rigid Body]] to a plate which
has an infinite number of degrees of freedom. Many other methods of solution
have been presented, most of which consider some kind of regular plate shape
(such as a circle or square).
The standard solution method to the linear wave problem is to transform the
boundary value problem into an integral equation using a Green function
Performing such a transformation, the boundary
value problem (5) and (6) become
<center><math>
\phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x}
;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x}
)\right) dS_{\mathbf{y}}. \qquad(8)(water)
</math></center>
where <math>G_{\alpha }</math> is the [[Free-Surface Green Function]]

=Solving for the Elastic Plate Motion=

To determine the ice floe motion we must solve equations ((plate2)) and (
(water)) simultaneously. We do this by expanding the floe motion in the
free modes of vibration of a thin plate. The major difficulty with this
method is that the free modes of vibration can be determined analytically
only for very restrictive geometries, e.g. a circular thin plate. Even the
free modes of vibration of a square plate with free edges must be determined
numerically. This is the reason why the solution of [[Meylan and Squire 1996]] was
only for a circular floe.

Since the operator <math>\nabla ^{4},</math> subject to the free edge boundary
conditions, is self adjoint a thin plate must possess a set of modes <math>w_{i}</math>
which satisfy the free boundary conditions and the following eigenvalue
equation
<center><math>
\nabla ^{4}w_{i}=\lambda _{i}w_{i}.
</math></center>
The modes which correspond to different eigenvalues <math>\lambda _{i}</math> are
orthogonal and the eigenvalues are positive and real. While the plate will
always have repeated eigenvalues, orthogonal modes can still be found and
the modes can be normalized. We therefore assume that the modes are
orthonormal, i.e.
<center><math>
\iint_{\Delta }w_{i}\left( \mathbf{Q}\right) w_{j}\left( \mathbf{Q}\right)
dS_{\mathbf{Q}}=\delta _{ij}
</math></center>
where <math>\delta _{ij}</math> is the Kronecker delta. The eigenvalues <math>\lambda _{i}</math>
have the property that <math>\lambda _{i}\rightarrow \infty </math> as <math>i\rightarrow
\infty </math> and we order the modes by increasing eigenvalue. These modes can be
used to expand any function over the wetted surface of the ice floe <math>\Delta </math>
.

We expand the displacement of the floe in a finite number of modes <math>N,</math> i.e.
<center><math>
w\left( \mathbf{x}\right) =\sum_{i=1}^{N}c_{i}w_{i}\left( \mathbf{x}\right) .
(expansion)
</math></center>
From the linearity of ((water)) the potential can be written in the
following form
<center><math>
\phi =\phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i} (expansionphi)
</math></center>
where <math>\phi _{0}</math> and <math>\phi _{i}</math> satisfy the integral equations
<center><math>
\phi _{0}(\mathbf{x})=\phi ^{\mathrm{In}}(\mathbf{x})+\iint_{\Delta }\alpha
G_{\alpha }(\mathbf{x};\mathbf{y})\phi (\mathbf{y})dS_{\mathbf{y}}
(phi0)
</math></center>
and
<center><math>
\phi _{i}(\mathbf{x})=\iint_{\Delta }G_{\alpha }(\mathbf{x};\mathbf{y}
)\left( \alpha \phi _{i}(\mathbf{x})+i\sqrt{\alpha }w_{i}(\mathbf{y})\right)
dS_{\mathbf{y}}. (phii)
</math></center>
The potential <math>\phi _{0}</math> represents the potential due the incoming wave
assuming that the displacement of the ice floe is zero. The potentials <math>\phi
_{i}</math> represent the potential which is generated by the plate vibrating with
the <math>i</math>th mode in the absence of any input wave forcing.

We substitute equations ((expansion)) and ((expansionphi)) into
equation ((plate2)) to obtain
<center><math>
\beta \sum_{i=1}^{N}\lambda _{i}c_{i}w_{i}-\alpha \gamma
\sum_{i=1}^{N}c_{i}w_{i}=i\sqrt{\alpha }\left( \phi
_{0}+\sum_{i=1}^{N}c_{i}\phi _{i}\right) -\sum_{i=1}^{N}c_{i}w_{i}.
(expanded)
</math></center>
To solve equation ((expanded)) we multiply by <math>w_{j}</math> and integrate over
the plate (i.e. we take the inner product with respect to <math>w_{j})</math> taking
into account the orthogonality of the modes <math>w_{i}</math>, and obtain
<center><math>
\beta \lambda _{j}c_{j}+\left( 1-\alpha \gamma \right) c_{j}=\iint_{\Delta }i
\sqrt{\alpha }\left( \phi _{0}\left( \mathbf{Q}\right)
+\sum_{i=1}^{N}c_{i}\phi _{i}\left( \mathbf{Q}\right) \right) w_{j}\left(
\mathbf{Q}\right) dS_{\mathbf{Q}} (final)
</math></center>
which is a matrix equation in <math>c_{i}.</math>

We cannot solve equation ((final)) without determining the modes of
vibration of the thin plate <math>w_{i}</math> (along with the associated eigenvalues <math>
\lambda _{i})</math> and solving the integral equations ((phi0)) and (\ref
{phii}). We use the finite element method to determine the modes of
vibration [[Zienkiewicz]] and the integral equations ((phi0)) and (
(phii)) are solved by a constant panel method [[Sarp_Isa]]. The same
set of nodes is used for the finite element method and to define the panels
for the integral equation.

[[Category:Floating Elastic Plate]]

Three-Dimensional Floating Elastic Plate

2006-09-11T08:30:07Z

Mdmgreen: /* Equations of Motion for the Water */

= Equations of Motion =

For a classical thin plate, the equation of motion is given by
<center><math>
D\nabla ^4 w - \rho _i h \frac{\partial^2 w}{\partial t^2} = p \qquad (1)
</math></center>
Equation (1) is subject to the free edge boundary
conditions for a thin plate
<center><math>
\left[ \nabla^2 - (1-\nu)
\left(\frac{\partial^2}{\partial s^2} + \kappa(s)
\frac{\partial}{\partial n} \right) \right] w = 0, \qquad(1.1)
</math></center>
<center><math>
\left[ \frac{\partial}{\partial n} \nabla^2 +(1-\nu)
\frac{\partial}{\partial s}
\left( \frac{\partial}{\partial n} \frac{\partial}{\partial s}
-\kappa(s) \frac{\partial}{\partial s} \right) \right] w = 0, \qquad(1.2)
</math></center>
where <math>\nu</math> is Poisson's ratio and
<center><math>
\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}
= \frac{\partial^2}{\partial n^2} + \frac{\partial^2}{\partial s^2}
+ \kappa(s) \frac{\partial}{\partial n}.
</math></center>
Here, <math>\kappa(s)</math> is the curvature of the boundary, <math>\partial \Delta</math>,
as a function of arclength <math>s</math> along <math>\partial \Delta</math>;
<math>\partial/\partial s</math> and <math>\partial/\partial n</math> represent derivatives
tangential and normal to the boundary <math>\partial \Delta</math>, respectively.
[[Porter paper?]] where <math>n</math> and <math>s</math> denote the normal and tangential
directions respectively.

The pressure, <math>p</math>, is given by the linearized Bernoulli's equation at the
water surface,
<center><math>
p=-\rho \frac{\partial \phi }{\partial t}-\rho gw.\,\,\, (2)
</math></center>
where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density
of the water, and <math>g</math> is the acceleration due to gravity.

We now introduce non-dimensional variables. We non-dimensionalise the length
variables with respect to <math>a</math> where the surface area of the floe is <math>4a^{2}.</math>
We non-dimensionalise the time variables with respect to <math>\sqrt{g/a}</math>.
In the non-dimensional variables equations (1) and (2)
become
<center><math>
\beta \nabla^{4}{w}+\gamma \frac{\partial^2 w}{\partial t^2}=\frac{\partial {\Phi}}{\partial {t}}-{w}, \qquad(3)% (n-d_ice)
</math></center>
where
<center><math>
\beta =\frac{D}{g\rho a^4}\;\;{\mathrm and}\;\; \gamma =\frac{\rho_i h}{\rho a}.
</math></center>

We assume the [[Frequency Domain Problem]] with frequency <math>\omega</math>.
This leads to the following equation
<center><math>
\beta \nabla ^{4}w+\alpha \gamma w=-i\omega\phi -w. \qquad(4)%(plate2)
</math></center>

=Equations of Motion for the Water=

We require the equation of motion for the water to solve equation ({plate2}).
We begin [[Standard Linear Wave Scattering Problem]] equations with the boundary condition
under the plate modified as appropriate.
<center><math>
\left.
\begin{matrix}
\nabla ^{2}\phi =0, & -\infty <z<0, \\
{\frac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\
{\frac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\;
\mathbf{x}\in \Delta , \\
{\frac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{
x}\notin \Delta ,
\end{matrix}
\right\} \qquad(5)%(bvp_nond)
</math></center>
The vector <math>\mathbf{x=(}x,y)</math> is a
point on the water surface and <math>\Delta </math> is the region of the water surface
occupied by the plate.

The boundary value problem ((bvp)) is subject to an incident wave which
is imposed through the
[[Sommerfeld Radiation Condition]]
<center><math>
\lim_{\left| \mathbf{x}\right| \rightarrow \infty }\sqrt{|\mathbf{x}|}\left(
\frac{\partial }{\partial |\mathbf{x}|}-i\alpha \right) (\phi -\phi ^{
\mathrm{In}})=0, \qquad(6) %(sommerfield)
</math></center>
where the incident potential <math>\phi ^{\mathrm{In}}</math> is
<center><math>
\phi ^{\mathrm{In}}(x,y,z)=\frac{A}{{\omega }}e^{i\alpha (x\cos \theta
+y\sin \theta )}e^{\alpha z}, \qquad (7)(input)
</math></center>
where <math>A</math> is the non-dimensional wave amplitude.

= Solution of the Equations of Motion =

There are a number of methods to solve this problem. We will describe a
method which generalises the [[Linear Wave Scattering for Floating Rigid Body]] to a plate which
has an infinite number of degrees of freedom. Many other methods of solution
have been presented, most of which consider some kind of regular plate shape
(such as a circle or square).
The standard solution method to the linear wave problem is to transform the
boundary value problem into an integral equation using a Green function
Performing such a transformation, the boundary
value problem ((bvp)) and ((summerfield)) become
<center><math>
\phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x}
;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x}
)\right) dS_{\mathbf{y}}. (water)
</math></center>
where <math>G_{\alpha }</math> is the [[Free-Surface Green Function]]

=Solving for the Elastic Plate Motion=

To determine the ice floe motion we must solve equations ((plate2)) and (
(water)) simultaneously. We do this by expanding the floe motion in the
free modes of vibration of a thin plate. The major difficulty with this
method is that the free modes of vibration can be determined analytically
only for very restrictive geometries, e.g. a circular thin plate. Even the
free modes of vibration of a square plate with free edges must be determined
numerically. This is the reason why the solution of [[Meylan and Squire 1996]] was
only for a circular floe.

Since the operator <math>\nabla ^{4},</math> subject to the free edge boundary
conditions, is self adjoint a thin plate must possess a set of modes <math>w_{i}</math>
which satisfy the free boundary conditions and the following eigenvalue
equation
<center><math>
\nabla ^{4}w_{i}=\lambda _{i}w_{i}.
</math></center>
The modes which correspond to different eigenvalues <math>\lambda _{i}</math> are
orthogonal and the eigenvalues are positive and real. While the plate will
always have repeated eigenvalues, orthogonal modes can still be found and
the modes can be normalized. We therefore assume that the modes are
orthonormal, i.e.
<center><math>
\iint_{\Delta }w_{i}\left( \mathbf{Q}\right) w_{j}\left( \mathbf{Q}\right)
dS_{\mathbf{Q}}=\delta _{ij}
</math></center>
where <math>\delta _{ij}</math> is the Kronecker delta. The eigenvalues <math>\lambda _{i}</math>
have the property that <math>\lambda _{i}\rightarrow \infty </math> as <math>i\rightarrow
\infty </math> and we order the modes by increasing eigenvalue. These modes can be
used to expand any function over the wetted surface of the ice floe <math>\Delta </math>
.

We expand the displacement of the floe in a finite number of modes <math>N,</math> i.e.
<center><math>
w\left( \mathbf{x}\right) =\sum_{i=1}^{N}c_{i}w_{i}\left( \mathbf{x}\right) .
(expansion)
</math></center>
From the linearity of ((water)) the potential can be written in the
following form
<center><math>
\phi =\phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i} (expansionphi)
</math></center>
where <math>\phi _{0}</math> and <math>\phi _{i}</math> satisfy the integral equations
<center><math>
\phi _{0}(\mathbf{x})=\phi ^{\mathrm{In}}(\mathbf{x})+\iint_{\Delta }\alpha
G_{\alpha }(\mathbf{x};\mathbf{y})\phi (\mathbf{y})dS_{\mathbf{y}}
(phi0)
</math></center>
and
<center><math>
\phi _{i}(\mathbf{x})=\iint_{\Delta }G_{\alpha }(\mathbf{x};\mathbf{y}
)\left( \alpha \phi _{i}(\mathbf{x})+i\sqrt{\alpha }w_{i}(\mathbf{y})\right)
dS_{\mathbf{y}}. (phii)
</math></center>
The potential <math>\phi _{0}</math> represents the potential due the incoming wave
assuming that the displacement of the ice floe is zero. The potentials <math>\phi
_{i}</math> represent the potential which is generated by the plate vibrating with
the <math>i</math>th mode in the absence of any input wave forcing.

We substitute equations ((expansion)) and ((expansionphi)) into
equation ((plate2)) to obtain
<center><math>
\beta \sum_{i=1}^{N}\lambda _{i}c_{i}w_{i}-\alpha \gamma
\sum_{i=1}^{N}c_{i}w_{i}=i\sqrt{\alpha }\left( \phi
_{0}+\sum_{i=1}^{N}c_{i}\phi _{i}\right) -\sum_{i=1}^{N}c_{i}w_{i}.
(expanded)
</math></center>
To solve equation ((expanded)) we multiply by <math>w_{j}</math> and integrate over
the plate (i.e. we take the inner product with respect to <math>w_{j})</math> taking
into account the orthogonality of the modes <math>w_{i}</math>, and obtain
<center><math>
\beta \lambda _{j}c_{j}+\left( 1-\alpha \gamma \right) c_{j}=\iint_{\Delta }i
\sqrt{\alpha }\left( \phi _{0}\left( \mathbf{Q}\right)
+\sum_{i=1}^{N}c_{i}\phi _{i}\left( \mathbf{Q}\right) \right) w_{j}\left(
\mathbf{Q}\right) dS_{\mathbf{Q}} (final)
</math></center>
which is a matrix equation in <math>c_{i}.</math>

We cannot solve equation ((final)) without determining the modes of
vibration of the thin plate <math>w_{i}</math> (along with the associated eigenvalues <math>
\lambda _{i})</math> and solving the integral equations ((phi0)) and (\ref
{phii}). We use the finite element method to determine the modes of
vibration [[Zienkiewicz]] and the integral equations ((phi0)) and (
(phii)) are solved by a constant panel method [[Sarp_Isa]]. The same
set of nodes is used for the finite element method and to define the panels
for the integral equation.

[[Category:Floating Elastic Plate]]

Three-Dimensional Floating Elastic Plate

2006-09-11T08:27:31Z

Mdmgreen: /* Equations of Motion */

= Equations of Motion =

For a classical thin plate, the equation of motion is given by
<center><math>
D\nabla ^4 w - \rho _i h \frac{\partial^2 w}{\partial t^2} = p \qquad (1)
</math></center>
Equation (1) is subject to the free edge boundary
conditions for a thin plate
<center><math>
\left[ \nabla^2 - (1-\nu)
\left(\frac{\partial^2}{\partial s^2} + \kappa(s)
\frac{\partial}{\partial n} \right) \right] w = 0, \qquad(1.1)
</math></center>
<center><math>
\left[ \frac{\partial}{\partial n} \nabla^2 +(1-\nu)
\frac{\partial}{\partial s}
\left( \frac{\partial}{\partial n} \frac{\partial}{\partial s}
-\kappa(s) \frac{\partial}{\partial s} \right) \right] w = 0, \qquad(1.2)
</math></center>
where <math>\nu</math> is Poisson's ratio and
<center><math>
\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}
= \frac{\partial^2}{\partial n^2} + \frac{\partial^2}{\partial s^2}
+ \kappa(s) \frac{\partial}{\partial n}.
</math></center>
Here, <math>\kappa(s)</math> is the curvature of the boundary, <math>\partial \Delta</math>,
as a function of arclength <math>s</math> along <math>\partial \Delta</math>;
<math>\partial/\partial s</math> and <math>\partial/\partial n</math> represent derivatives
tangential and normal to the boundary <math>\partial \Delta</math>, respectively.
[[Porter paper?]] where <math>n</math> and <math>s</math> denote the normal and tangential
directions respectively.

The pressure, <math>p</math>, is given by the linearized Bernoulli's equation at the
water surface,
<center><math>
p=-\rho \frac{\partial \phi }{\partial t}-\rho gw.\,\,\, (2)
</math></center>
where <math>\Phi </math> is the velocity potential of the water, <math>\rho </math> is the density
of the water, and <math>g</math> is the acceleration due to gravity.

We now introduce non-dimensional variables. We non-dimensionalise the length
variables with respect to <math>a</math> where the surface area of the floe is <math>4a^{2}.</math>
We non-dimensionalise the time variables with respect to <math>\sqrt{g/a}</math>.
In the non-dimensional variables equations (1) and (2)
become
<center><math>
\beta \nabla^{4}{w}+\gamma \frac{\partial^2 w}{\partial t^2}=\frac{\partial {\Phi}}{\partial {t}}-{w}, \qquad(3)% (n-d_ice)
</math></center>
where
<center><math>
\beta =\frac{D}{g\rho a^4}\;\;{\mathrm and}\;\; \gamma =\frac{\rho_i h}{\rho a}.
</math></center>

We assume the [[Frequency Domain Problem]] with frequency <math>\omega</math>.
This leads to the following equation
<center><math>
\beta \nabla ^{4}w+\alpha \gamma w=-i\omega\phi -w. \qquad(4)%(plate2)
</math></center>

=Equations of Motion for the Water=

We require the equation of motion for the water to solve equation ({plate2}).
We begin [[Standard Linear Wave Scattering Problem]] equations with the boundary condition
under the plate modified as appropriate.
<center><math>
\left.
\begin{matrix}
\nabla ^{2}\phi =0, & -\infty <z<0, \\
{\frac{\partial \phi }{\partial z}=0}, & z\rightarrow -\infty , \\
{\frac{\partial \phi }{\partial z}=}-i\sqrt{\alpha }w, & z\;=\;0,\;\;
\mathbf{x}\in \Delta , \\
{\frac{\partial \phi }{\partial z}-}\alpha \phi {=}p, & z\;=\;0,\;\;\mathbf{
x}\notin \Delta ,
\end{matrix}
\right\} (bvp_nond)
</math></center>
The vector <math>\mathbf{x=(}x,y)</math> is a
point on the water surface and <math>\Delta </math> is the region of the water surface
occupied by the plate.

The boundary value problem ((bvp)) is subject to an incident wave which
is imposed through the
[[Sommerfeld Radiation Condition]]
<center><math>
\lim_{\left| \mathbf{x}\right| \rightarrow \infty }\sqrt{|\mathbf{x}|}\left(
\frac{\partial }{\partial |\mathbf{x}|}-i\alpha \right) (\phi -\phi ^{
\mathrm{In}})=0, (summerfield)
</math></center>
where the incident potential <math>\phi ^{\mathrm{In}}</math> is
<center><math>
\phi ^{\mathrm{In}}(x,y,z)=\frac{A}{{\omega }}e^{i\alpha (x\cos \theta
+y\sin \theta )}e^{\alpha z}, (input)
</math></center>
where <math>A</math> is the non-dimensional wave amplitude.

= Solution of the Equations of Motion =

There are a number of methods to solve this problem. We will describe a
method which generalises the [[Linear Wave Scattering for Floating Rigid Body]] to a plate which
has an infinite number of degrees of freedom. Many other methods of solution
have been presented, most of which consider some kind of regular plate shape
(such as a circle or square).
The standard solution method to the linear wave problem is to transform the
boundary value problem into an integral equation using a Green function
Performing such a transformation, the boundary
value problem ((bvp)) and ((summerfield)) become
<center><math>
\phi (\mathbf{x})=\phi ^{i}(\mathbf{x})+\iint_{\Delta }G_{\alpha }(\mathbf{x}
;\mathbf{y})\left( \alpha \phi (\mathbf{x})+i\sqrt{\alpha }w(\mathbf{x}
)\right) dS_{\mathbf{y}}. (water)
</math></center>
where <math>G_{\alpha }</math> is the [[Free-Surface Green Function]]

=Solving for the Elastic Plate Motion=

To determine the ice floe motion we must solve equations ((plate2)) and (
(water)) simultaneously. We do this by expanding the floe motion in the
free modes of vibration of a thin plate. The major difficulty with this
method is that the free modes of vibration can be determined analytically
only for very restrictive geometries, e.g. a circular thin plate. Even the
free modes of vibration of a square plate with free edges must be determined
numerically. This is the reason why the solution of [[Meylan and Squire 1996]] was
only for a circular floe.

Since the operator <math>\nabla ^{4},</math> subject to the free edge boundary
conditions, is self adjoint a thin plate must possess a set of modes <math>w_{i}</math>
which satisfy the free boundary conditions and the following eigenvalue
equation
<center><math>
\nabla ^{4}w_{i}=\lambda _{i}w_{i}.
</math></center>
The modes which correspond to different eigenvalues <math>\lambda _{i}</math> are
orthogonal and the eigenvalues are positive and real. While the plate will
always have repeated eigenvalues, orthogonal modes can still be found and
the modes can be normalized. We therefore assume that the modes are
orthonormal, i.e.
<center><math>
\iint_{\Delta }w_{i}\left( \mathbf{Q}\right) w_{j}\left( \mathbf{Q}\right)
dS_{\mathbf{Q}}=\delta _{ij}
</math></center>
where <math>\delta _{ij}</math> is the Kronecker delta. The eigenvalues <math>\lambda _{i}</math>
have the property that <math>\lambda _{i}\rightarrow \infty </math> as <math>i\rightarrow
\infty </math> and we order the modes by increasing eigenvalue. These modes can be
used to expand any function over the wetted surface of the ice floe <math>\Delta </math>
.

We expand the displacement of the floe in a finite number of modes <math>N,</math> i.e.
<center><math>
w\left( \mathbf{x}\right) =\sum_{i=1}^{N}c_{i}w_{i}\left( \mathbf{x}\right) .
(expansion)
</math></center>
From the linearity of ((water)) the potential can be written in the
following form
<center><math>
\phi =\phi _{0}+\sum_{i=1}^{N}c_{i}\phi _{i} (expansionphi)
</math></center>
where <math>\phi _{0}</math> and <math>\phi _{i}</math> satisfy the integral equations
<center><math>
\phi _{0}(\mathbf{x})=\phi ^{\mathrm{In}}(\mathbf{x})+\iint_{\Delta }\alpha
G_{\alpha }(\mathbf{x};\mathbf{y})\phi (\mathbf{y})dS_{\mathbf{y}}
(phi0)
</math></center>
and
<center><math>
\phi _{i}(\mathbf{x})=\iint_{\Delta }G_{\alpha }(\mathbf{x};\mathbf{y}
)\left( \alpha \phi _{i}(\mathbf{x})+i\sqrt{\alpha }w_{i}(\mathbf{y})\right)
dS_{\mathbf{y}}. (phii)
</math></center>
The potential <math>\phi _{0}</math> represents the potential due the incoming wave
assuming that the displacement of the ice floe is zero. The potentials <math>\phi
_{i}</math> represent the potential which is generated by the plate vibrating with
the <math>i</math>th mode in the absence of any input wave forcing.

We substitute equations ((expansion)) and ((expansionphi)) into
equation ((plate2)) to obtain
<center><math>
\beta \sum_{i=1}^{N}\lambda _{i}c_{i}w_{i}-\alpha \gamma
\sum_{i=1}^{N}c_{i}w_{i}=i\sqrt{\alpha }\left( \phi
_{0}+\sum_{i=1}^{N}c_{i}\phi _{i}\right) -\sum_{i=1}^{N}c_{i}w_{i}.
(expanded)
</math></center>
To solve equation ((expanded)) we multiply by <math>w_{j}</math> and integrate over
the plate (i.e. we take the inner product with respect to <math>w_{j})</math> taking
into account the orthogonality of the modes <math>w_{i}</math>, and obtain
<center><math>
\beta \lambda _{j}c_{j}+\left( 1-\alpha \gamma \right) c_{j}=\iint_{\Delta }i
\sqrt{\alpha }\left( \phi _{0}\left( \mathbf{Q}\right)
+\sum_{i=1}^{N}c_{i}\phi _{i}\left( \mathbf{Q}\right) \right) w_{j}\left(
\mathbf{Q}\right) dS_{\mathbf{Q}} (final)
</math></center>
which is a matrix equation in <math>c_{i}.</math>

We cannot solve equation ((final)) without determining the modes of
vibration of the thin plate <math>w_{i}</math> (along with the associated eigenvalues <math>
\lambda _{i})</math> and solving the integral equations ((phi0)) and (\ref
{phii}). We use the finite element method to determine the modes of
vibration [[Zienkiewicz]] and the integral equations ((phi0)) and (
(phii)) are solved by a constant panel method [[Sarp_Isa]]. The same
set of nodes is used for the finite element method and to define the panels
for the integral equation.

[[Category:Floating Elastic Plate]]

Floating Elastic Plate

2006-05-30T08:48:29Z

Mdmgreen: /* Equations of Motion */

= Introduction =

The floating elastic plate is one of the best studied problems in hydroelasticity. It can be used to model a range of
physical structures such as a floating break water, an ice floe or a [[VLFS]]). The equations of motion were formulated
more than 100 years ago and a discussion of the problem appears in [[Stoker_1957a|Stoker 1957]]. The problem can
be divided into the two and three dimensional formulations which are closely related. The plate is assumed to be isotropic while the water motion is irrotational and inviscid.

= Two Dimensional Problem =

== Equations of Motion ==

When considering a two dimensional problem, the <math>y</math> variable is dropped and the plate is regarded as a beam. There are various beam theories that can be used to describe the motion of the beam. The simplest theory is the Bernoulli-Euler beam which is commonly used in the two dimensional hydroelastic analysis. Other beam theories include the Timoshenko beam theory and Reddy-Bickford beam theory where shear deformation of higher order is considered.

For a Bernoulli-Euler beam on the surface of the water, the equation of motion is given
by the following

<math>D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p</math>

where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the beam,
<math>h</math> is the thickness of the beam (assumed constant), <math> p</math> is the pressure
and <math>\eta</math> is the beam vertical displacement.

The edges of the plate satisfy the natural boundary condition (i.e. free-edge boundary conditions).

<math>\frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0</math>

at the edges of the plate.

The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero
pressure at the surface), i.e.

<math>p = \rho g \frac{\partial \phi}{\partial z} + \rho \frac{\partial \phi}{\partial t}</math>

where <math>\rho</math> is the water density and <math>g</math> is gravity, and <math>\phi</math>
is the velocity potential. The velocity potential is governed by Laplace's equation through out
the fluid domain subject to the free surface condition and the condition of no flow through the
bottom surface. If we denote the beam-covered (or possible multiple beams covered) region of the fluid by <math>P</math> and the free surface by <math>F</math> the equations of motion for the
[[Frequency Domain Problem]] with frequency <math>\omega</math> for water of
[[Finite Depth]] are the following. At the surface
we have the dynamic condition

<math>D\frac{\partial^4 \eta}{\partial x^4} +\left(\rho g- \omega^2 \rho_i h \right)\eta =
i\omega \rho \phi, \, z=0, \, x\in P</math>

<math>0=
\rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F</math>

and the kinematic condition

<math>\frac{\partial\phi}{\partial z} = i\omega\eta</math>

The equation within the fluid is governed by [[Laplace's Equation]]

<math>\nabla^2\phi =0 </math>

and we have the no-flow condition through the bottom boundary

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>

(so we have a fluid of constant depth with the bottom surface at <math>z=-h</math> and the
free surface or plate covered surface are at <math>z=0</math>).
<math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> D </math>
the thickness and flexural rigidity of the plate.

== Solution Method ==

There are many different methods to solve the corresponding equations ranging from highly analytic such
as the [[Wiener-Hopf]] to very numerical based on [[Eigenfunction Matching Method]] which are
applicable and have advantages in different situations.

= Three Dimensional Problem =

== Equations of Motion ==

For a classical thin plate, the equation of motion is given by

<math>D\nabla ^4 w + \rho _i h w = p</math>

Floating Elastic Plate

2006-05-30T08:43:49Z

Mdmgreen: /* Three Dimensional Problem */

2006-05-23T05:47:17Z

Mdmgreen: Kamigoto oil storage facility

Kamigoto oil storage facility

VLFS

2006-05-23T05:46:14Z

Mdmgreen:

File:Container.jpg

2006-05-23T05:44:57Z

Mdmgreen:

Floating Elastic Plate

2006-05-23T05:39:33Z

Mdmgreen: /* Equations of Motion */

= Introduction =

The floating elastic plate is one of the best studied problems in hydroelasticity. It can be used to model a range of
physical structures such as a floating break water, an ice floe or a [[VLFS]]). The equations of motion were formulated
more than 100 years ago and a discussion of the problem appears in [[Stoker_1957a|Stoker 1957]]. The problem can
be divided into the two and three dimensional formulations which are closely related.

= Two Dimensional Problem =

= Equations of Motion =

For a thin plate floating on the surface of the water, the motion of the plate is derived from the Classical Thin Plate Theory which was developed by Kirchhoff. The equation of motion for a thin plate (also known as Kirchhoff's equation) is given
by the following

<math>D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p</math>

where <math>D</math> is the flexural rigidity, <math>\rho_i</math> is the density of the plate,
<math>h</math> is the thickness of the plate (assumed constant), <math> p</math> is the pressure
and <math>\eta</math> is the plate displacement.

The pressure is given by the linearised Bernoulli equation at the wetted surface (assuming zero
pressure at the surface), i.e.

<math>p = \rho g \frac{\partial \phi}{\partial z} + \rho \frac{\partial \phi}{\partial t}</math>

where <math>\rho</math> is the water density and <math>g</math> is gravity, and <math>\phi</math>
is the velocity potential. The velocity potential is governed by Laplace's equation through out
the fluid domain subject to the free surface condition and the condition of no flow through the
bottom surface. If we denote the region of the fluid surface covered in the plate (or possible
multiple plates) by <math>P</math> and the free surface by <math>F</math> the equations of motion for the
[[Frequency Domain Problem]] with frequency <math>\omega</math> for water of
[[Finite Depth]] are the following. At the surface
we have the dynamic condition

<math>D\frac{\partial^4 \eta}{\partial x^4} +\left(\rho g- \omega^2 \rho_i h \right)\eta =
i\omega \rho \phi, \, z=0, \, x\in P</math>

<math>0=
\rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F</math>

and the kinematic condition

<math>\frac{\partial\phi}{\partial z} = i\omega\eta</math>

the equation within the fluid is [[Laplace's Equation]]

<math>\nabla^2\phi =0 </math>

and we have the no-flow condition through the bottom boundary

<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>

(so we have a fluid of constant depth with the bottom surface at <math>z=-h</math> and the
free surface or plate covered surface are at <math>z=0</math>).
<math> g </math> is the acceleration due to gravity, <math> \rho_i </math> and <math> \rho </math>
are the densities of the plate and the water respectively, <math> h </math> and <math> D </math>
the thickness and flexural rigidity of the plate.

Finally we need to include
some boundary conditions at the edge of the plate. The most common boundary conditions
in pratical applications are that the edges are free, this means that we have the additional
conditions that

<math>\frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0</math>

at the edges of the plate.

= Solution Method =

There are many different methods to solve the corresponding equations ranging from highly analytic such
as the [[Wiener-Hopf]] to very numerical based on [[Eigenfunction Matching Method]] which are
applicable and have advantages in different situations.