WikiWaves - User contributions [en]

Kouzov 1963

2006-12-09T03:54:29Z

Shagman:

D.P. Kouzov, Diffraction of a Plane Hydro-elastic Wave on the Boundary of Two Elastic Plates (in Russian)
''Prikl. Mat. Mekh.'' '''27''', pp 541-546, 1963.

[[Category:Reference]]

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Oppenheim and Schafer 1975

2006-12-09T01:40:16Z

Shagman:

Oppenheim A.V. & R.W. Schafer. 1975. ''Digital Signal Processing''. Englewood Cliffs, New Jersey: Prentice-Hall.

[[Category:Reference]]

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Category:Eigenfunction Matching Method

2006-12-08T21:22:26Z

Shagman:

= Introduction =

A method for solving wave scattering problems in which the solution can be solve
in various regions using separation of variables. The solution in these regions
are then matched at various boundaries.

The method is described in [[Linton and McIver 2001]].

[[Eigenfunction Matching for a Semi-Infinite Dock]]

[[Category:Linear Water-Wave Theory]]
[[Category:Numerical Methods]]

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McIver and McIver 2006b

2006-12-08T20:50:19Z

Shagman:

[[Phil McIver|P. McIver]] & [[Maureen McIver| M. McIver]] 2006b Motion trapping structures in the three-dimensional water-wave problem. ''Journal of Engineering Mathematics'', to appear.

Concerns [[Trapped Modes]].

[[Category:Reference]]

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Category:Linear Water-Wave Theory

2006-12-08T18:33:20Z

Shagman:

These are articles which discuss linear water-waves.

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Removing the Depth Dependence

2006-12-08T15:21:21Z

Shagman:

We are considering the [[Frequency Domain Problem]] for linear wave waves.
If we have a problem in which the water depth is of constant depth <math>z=-d </math> (we are assuming
the free surface is at <math>z=0</math>) and all the scatters
are also constant with respect to the depth then we can remove the depth dependence by assuming
that the dependence on depth is given by

<math>
\Phi(x,y,z) = \cosh \big( k (z+d) \big) \phi(x,y)
</math>

where <math>k</math> is the positive root of the [[Dispersion Relation for a Free Surface]]
then the problem reduces to [[Helmholtz's Equation]]

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

in the region not occupied by the scatterers.

[[Category:Linear Water-Wave Theory]]

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Hazard and Lenoir 2002

2006-12-08T14:25:02Z

Shagman:

[[Christophe Hazard|C. Hazard]] and M. Lenoir, eds. R. Pike and P. Sabatier, Surface water waves, ''Scattering'' pp 618-636, Academic, 2002.

This is partly a review article and partly contains new research. It presents the theory of [[Scattering Frequencies]]
summarising the results in [[Hazard_Lenoir1993a | Hazard and Lenoir 1993]] except that the theory is
applied to a freely floating body. The paper also presents the [[Generalised Eigenfunction Expansion]]
for a rigid body in water of infinite depth.

[[Category:Reference]]

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Williams and Squire 2002

2006-12-08T07:54:52Z

Shagman:

T. D. Williams and V. A. Squire, Wave propagation across an oblique crack in an ice sheet,
''Intl. J. Offshore Polar Engng'',
''' 12''',
pp 157-162, 2002.

Contains a solution for a [[Floating Elastic Plate]] with a single crack.

[[Category:Reference]]

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Help:Contents

2006-12-07T23:23:17Z

Shagman:

[[Help:Formula MediaWiki and LaTeX]]

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Three-Dimensional Floating Elastic Plate

2006-12-07T21:43:29Z

Shagman:

= 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 and Porter 2004]]) 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 a 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]]

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Eigenfunction Matching for a Semi-Infinite Dock

2006-12-07T15:33:57Z

Shagman:

= Introduction =

This is the simplest problem in eigenfunction matching. It also is an easy
problem to understand the [[:Category:Wiener-Hopf|Wiener-Hopf]] and
[[:Category:Residue Calculus|Residue Calculus]]

=Governing Equations=

We begin with the [[Frequency Domain Problem]] for a dock which occupies
the region <math>x>0</math>.
The water is assumed to have
constant finite depth <math>H</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-H</math>. The
boundary value problem can therefore be expressed as
<center>
<math>
\Delta\phi=0, \,\, -H<z<0,
</math>
</center>
<center>
<math>
\phi_{z}=0, \,\, z=-H,
</math>
</center>
<center><math>
\phi_{z}=\alpha\phi, \,\, z=0,\,x<0,
</math></center>
<center>
<math>
\phi_{z}=0, \,\, z=0,\,x>0,
</math>
</center>
We
must also apply the [[Sommerfeld Radiation Condition]]
as <math>|x|\rightarrow\infty</math>. The subscript <math>z</math>
denotes the derivative in <math>z</math>-direction.

=Solution Method=

== Separation of variables==

We now separate variables and write the potential as
<center>
<math>
\phi(x,z)=\zeta(z)\rho(x)
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0
</math>
</center>
so that:
<center>
<math>
\zeta=\cos\mu(z+H)
</math>
</center>
where the separation constant <math>\mu^{2}</math> must
satisfy the [[Dispersion Relation for a Free Surface]]
<center>
<math>
k\tan\left( kH\right) =-\alpha,\quad x<0\,\,\,(1)
</math>
</center>
and
<center>
<math>
\kappa\tan(\kappa H)=0,\quad
x>0 \,\,\,(2)
</math>
</center>
Note that we have set <math>\mu=k</math> under the free
surface and <math>\mu=\kappa</math> under the plate. We denote the
positive imaginary solution of (1) by <math>k_{0}</math> and
the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math>. The solutions of
(2)
<math>\kappa_{m}=m\pi/H</math>, <math>m\geq 0</math>. We define
<center>
<math>
\phi_{m}\left( z\right) =\frac{\cos k_{m}(z+H)}{\cos k_{m}H},\quad m\geq0
</math>
</center>
as the vertical eigenfunction of the potential in the open
water region and
<center>
<math>
\psi_{m}\left( z\right) = \cos\kappa_{m}(z+H),\quad
m\geq 0
</math>
</center>
as the vertical eigenfunction of the potential in the dock
covered region. For later reference, we note that:
<center>
<math>
\int\nolimits_{-H}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}H\sin k_{m}H+k_{m}H}{k_{m}\cos
^{2}k_{m}H}\right)
</math>
</center>
and
<center>
<math>
\int\nolimits_{-H}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn}
</math>
</center>
where
<center><math>
B_{mn}=\frac{k_{n}\sin k_{n}H\cos\kappa_{m}H-\kappa_{m}\cos k_{n}H\sin
\kappa_{m}H}{\left( \cos k_{n}H\cos\kappa_{m}H\right) \left( k_{n}
^{2}-\kappa_{m}^{2}\right) }
</math></center>
and
<center>
<math>
\int\nolimits_{-H}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
C_{m}=\frac{1}{2}H,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = H
</math></center>

Therefore the potential can
be expanded as
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}a_{m}e^{k_{m}x}\phi_{m}(z), \;\;x<0
</math>
</center>
and
<center>
<math>
\phi(x,z)=\sum_{m=0}^{\infty}b_{m}
e^{-\kappa_{m}x}\psi_{m}(z), \;\;x>0
</math>
</center>
where <math>a_{m}</math> and <math>b_{m}</math>
are the coefficients of the potential in the open water and
the plate covered region respectively.

==Incident potential==

The incident potential is a wave of amplitude <math>A</math>
in displacement travelling in the positive <math>x</math>-direction.
The incident potential can therefore be written as
<center>
<math>
\phi^{\mathrm{I}} =e^{-k_{0}x}\phi_{0}\left(
z\right)
</math>
</center>

==An infinite dimensional system of equations==

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>x=0</math> have to be equal.
We obtain
<center>
<math>
\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{m} \phi_{m}\left( z\right)
=\sum_{m=0}^{\infty}b_{m}\psi_{m}(z)
</math>
</center>
and
<center>
<math>
-k_{0}\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{m}k_{m}\phi_{m}\left( z\right)
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}\psi
_{m}(z)
</math>
</center>
for each <math>n</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z)</math> and integrating from <math>-H</math> to <math>0</math> to obtain:
<center>
<math>
A_{0}\delta_{0l}+a_{l}A_{l}
=\sum_{n=0}^{\infty}b_{m}B_{ml}\,\,\,(8)
</math>
</center>
and
<center>
<math>
-k_{0}A_{0}\delta_{0l}+a_{l}k_{l}A_l
=-\sum_{m=0}^{\infty}b_{m}\kappa_{m}B_{ml} \,\,\,(9)
</math>
</center>
If we mutiple equation (8) by <math>-k_l</math> and add this to equation (9)
we obtain
<center>
<math>
-2k_{0}A_{0}\delta_{0l}
=-\sum_{m=0}^{\infty}b_{m}(k_l + \kappa_{m})B_{ml} \,\,\,(9)
</math>
</center>
Equation (10) gives the required equations to solve for the
coefficients of the water velocity potential in the dock covered region.

=Numerical Solution=

To solve the system of equations (10) we set the upper limit of <math>l</math> to
be <math>M</math>.

[[Category:Eigenfunction Matching Method]]

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Removing the Depth Dependence

2006-12-07T09:24:10Z

Shagman:

Two-Dimensional Floating Elastic Plate

2006-12-07T05:22:12Z

Shagman:

= 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 [http://en.wikipedia.org/wiki/Euler_Bernoulli_beam_equation 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
<center><math>
D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} = p
</math></center>
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).
<center><math>
\frac{\partial^2 \eta}{\partial x^2} = 0, \,\,\frac{\partial^3 \eta}{\partial x^3} = 0
</math></center>
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.
<center>
<math>p = \rho g \frac{\partial \phi}{\partial z} + \rho \frac{\partial \phi}{\partial t}</math>
</center>
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
<center>
<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>
</center>
<center><math>0=
\rho g \frac{\partial \phi}{\partial z} + i\omega \rho \phi, \, x\in F</math>
</center>
and the kinematic condition
<center>
<math>\frac{\partial\phi}{\partial z} = i\omega\eta</math>
</center>

The equation within the fluid is governed by [[Laplace's Equation]]
<center>
<math>\nabla^2\phi =0 </math>
</center>
and we have the no-flow condition through the bottom boundary
<center>
<math>\frac{\partial \phi}{\partial z} = 0, \, z=-h</math>
</center>
(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.

=Energy Balance=
An energy balence relation is derived in
[[Evans and Davies 1968]] which is simply a condition that the incident energy is equal to the sum of the radiated energy including both the energy
in the water and the energy in the plate.
If the properties of the first and last semi-infinite plates were identical, then
this would be the familiar requirement that
<center><math> |T_{\Lambda}(0)|^2 + |R_{1}(0)|^2 = |I|^2. </math></center>
However, when the first and last plates have different properties, then the
energy balance condition becomes the following
<center><math>
D|T_{\Lambda}(0)|^2 + |R_{1}(0)|^2 = |I|^2,
</math></center>
where <math>D</math> is found by applying Green's theorem to <math>\phi</math> and its conjugate,
[[Evans and Davies 1968]]
and is given by
<center><math> \begin{matrix} D & = & { \frac{\kappa_{\Lambda}(0)k_1(0)\cosh^2{(k_1(0)h)}}{\kappa_{1}(0)k_\Lambda(0)\cosh^2{(k_\Lambda(0)h)}} \times }\\ & & { \frac{\left(\frac{\beta_\Lambda}{\alpha}4k_\Lambda(0)^3(\kappa_{\Lambda}(0)^2 +k_y^2)\sinh^2{(k_\Lambda(0) h)} + \frac{1}{2}{\sinh{(2k_\Lambda(0)h)}}+k_\Lambda(0)h\right)} {\left(\frac{\beta_1}{\alpha}4k_1(0)^3(\kappa_{1}(0)^2 +k_y^2)\sinh^2{(k_1(0)h)} + \frac{1}{2}{\sinh{(2k_1(0)h)}}+k_1(0)h\right)}. } \end{matrix} </math></center>
The energy balance condition is useful to help check that the
solution is not incorrect (it does not of course guarantee the solution
is correct). The energy balance condition is surprisingly well
satisfied by our solutions, for example with <math>M=20</math> we can easily get ten decimal places.

= Solution Methods =

There are many different methods to solve the corresponding equations ranging from highly analytic such
as the [[:Category:Wiener-Hopf|Wiener-Hopf]] to very numerical based on [[Eigenfunction Matching Method]] which are
applicable and have advantages in different situations. We describe here some of the solutions
which have been developed grouped by problem

== Two Semi-Infinite Plates of Identical Properties ==

The simplest problem to consider is one where there are only two semi-infinite plates of identical properties separated by a crack. A related problem in acoustics was considered by [[Kouzov 1963]] who used an integral representation of the problem to solve it explicitly using the Riemann-Hilbert technique. Recently the crack problem has been considered by [[Squire and Dixon 2000]] and [[Williams and Squire 2002]] using a Green function method applicable to infinitely deep water and they obtained simple expressions for the reflection and transmission coefficients. [[Squire and Dixon 2001]] extended the single crack problem to a multiple crack problem in which the semi-infinite regions are separated by a region consisting of a finite number of plates of finite size with all plates having identical properties. [[Evans and Porter 2005]] further considered the multiple crack problem for finitely deep water and provided an explicit solution which is described in [[Two Semi-Infinite Elastic Plates of Identical Properties]]

== Two Semi-Infinite Plates of Different Properties ==

The next most simple problem is two semi-infinite plates of different properties. Often one of
the plates is taken to be open water which makes the problem simpler. In general, the solution method
developed for open water can be extended to two plates of different properties, the exception to
this is the [[:Category:Residue Calculus|Residue Calculus]] solution which applies only when one of the semi-infinite regions
is water.

===[[:Category:Wiener-Hopf|Wiener-Hopf]]===

The solution to the problem of two semi-infinite plates with different properties can be
solved by the [[:Category:Wiener-Hopf|Wiener-Hopf]] method. The first work on this problem was by [[Evans and Davies 1968]]
but they did not actually develop the method sufficiently to be able to calculate the solution.
The explicit solution was not found until the work of [[Chung and Fox 2002]] and the [[:Category:Wiener-Hopf|Wiener-Hopf]]
solution is described in [[Wiener-Hopf Elastic Plate Solution]]

===[[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]]===

The eigenfunction matching solution was developed by [[Fox and Squire 1994]]
for a single semi-infinite plate and extended
to plates of different properties by [[Barrett and Squire 1996]].
Essentially the solution is expanded on either side of the crack. The theory is
described in [[Eigenfunction Matching Method for Floating Elastic Plates]]

===[[:Category:Residue Calculus|Residue Calculus]]===

The solution using [[:Category:Residue Calculus|Residue Calculus]] was developed by [[Linton and Chung 200?]]

== Single Floating Plate ==

The problem of a single floating plate in two-dimensions was treated
by [[Newman 1994]], [[Meylan and Squire 1994]] and [[Hermans 2003]] using the
[[Free-Surface Green Function]], described in [[Green Function Methods for Floating Elastic Plates]]

== Multiple Floating Plates ==

The most general problem consists of multiple floating plate. The
methods which generalises to this are the [[Eigenfunction Matching Method for Floating Elastic Plates]]
([[Kohout et. al. 2006]]) and the [[Green Function Methods for Floating Elastic Plates]]
([[Hermans 2004]]).

[[Category:Floating Elastic Plate]]

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Fenton 1978

2006-12-07T05:10:46Z

Shagman:

J. D. Fenton, Wave forces on vertical bodies of revolution,
''J. Fluid Mech.'', '''85''', pp. 241-255, 1978.

Contains the [[Free-Surface Green Function]] for [[Finite Depth]]
in cylindrical polar coordinates.

[[Category:Reference]]

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Fernyhough and Evans 1995

2006-12-07T05:00:15Z

Shagman:

M Fernyhough and David V Evans,
Scattering by a Periodic Array of Rectangular Blocks,
''J. Fluid Mech.'', '''305''', p. 263-279, 1995.

[[Category:Reference]]

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Main Page

2006-12-07T02:40:59Z

Shagman:

==Welcome to '''Wikiwaves'''!==

'''Wikiwaves''' is a water waves [http://en.wikipedia.org/wiki/Wiki wiki] devoted to the collective creation of technical content for practicing scientists. Please [[Sign up instructions|sign up]], [[browse]] around the site, click on the edit links, and contribute something! The site is nominally centered around water waves, at the moment the focus in on linear water
wave theory, especially as applied to hydroelasticity but we welcome any content. If you are new to the [http://google.com/trends?q=wiki increasingly popular] wiki way, you may want to visit [http://en.wikipedia.org/ Wikipedia] to see a more general effort in action or
check out [[Wiki List]] for a list of wikis in a similar spirit to this one. This wiki was started 12 April 2006.

__NOTOC__

<hr>

{| cellspacing="3"
|- valign="top"
|width="55%" class="MainPageBG" style="border: 1px solid #ffffff; color: #000; background-color: #ffffff"|
<div style="padding: .4em .9em .9em">
===Getting Started===
*First [[Browse|browse]] around to get a feel for what is here.
*Then follow the [[Sign up instructions|sign up instructions]] to make yourself a profile page.
*After that, learn [[Simple wiki help|how to create and compose pages]] and contribute to the site!
*If you're having any problems, see the [[FAQ]] page.
*We like feedback. If you have suggestions, comments, or additional questions, add them to our [[requests]] page or contact [[About us|us]].
</div>

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<div style="padding: .4em .9em .9em">

===Featured Pages===
*[[Michael Meylan]]: A description of the research of Mike Meylan.
*[[:Category:Wave Scattering in the Marginal Ice Zone|Wave Scattering in the Marginal Ice Zone]]: A description of the geophysical problem in water wave scattering.
*[[:Category:Interaction Theory|Interaction Theory]]: Presents the theory of multiple body interactions.
</div>

|}

{| cellspacing="3"
|- valign="top"
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<div style="padding: .4em .9em .9em">

===Wikiwaves Announcements===
*This site is just beginning and all contributions are welcome.
*'''We have changed the standard format for references.''' Check out the [[FAQ]] for details.
*'''We are reorganising the site using categories.'''
</div>

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<div style="padding: .4em .9em .9em">

===Site Map===
*[[Browse]]
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*[[test | Test page]]
*[[FAQ]]
</div>
|}

<hr>

===About Us===

[[Image:NZIMA.jpg|thumb|right]]

This website was started by [[Michael Meylan]] and is being initially supported by a grant from
the [http://www.nzima.auckland.ac.nz/ New Zealand Institute of Mathematics].

== Useful Links ==

* [[FAQ]] (|Frequently asked questions) for the water-waves wiki
* [http://www.mediawiki.org/wiki/Help:FAQ MediaWiki FAQ]
* [http://meta.wikimedia.org/wiki/Help:Displaying_a_formula Using Latex in Wiki]
* [[Converting Latex to Wiki]]

Consult the [http://meta.wikipedia.org/wiki/MediaWiki_User%27s_Guide User's Guide] for information on using the wiki software.

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Converting Latex to Wiki

2006-12-07T01:50:14Z

Shagman:

This is a script to help convert TeX tags into the correct form for wiki. At the moment the script doesn't handle the <nowiki>\newcommand </nowiki> environment or figures and certain types of tables. [http://www.python.org/ Python] is required.

We are actively working on improving the program and we welcome your input. The code is ''relatively'' straight forward
to modify and you are welcome to add in your own lines to the code and we will use your updated version,
or if you have a request you can add these to the discussion page.

'''One thing to note''' is that the code looks for the line ''\begin{document}''
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you do not have line you will get a blank document.

'''Caveat Emptor''': This code IS NOT PERFECT. The script will do a lot of the leg work, but expect to make some minor modifications after the script has been run.

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The script is executed using the following syntax:

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The most important replacements made by the script are the following:
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!LaTeX !! replacement wikitext
|-
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|-
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|-
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|-
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|-
|\[..]section||''appropriate wiki heading level''
|}

==The Script==
For security, python scripts cannot be served off of the [[Main Page|wiki]]. Please change the extension for '.txt' to '.py' after downloading.

[[Media:Latex2wiki.txt|Latex2wiki.txt]]

==References==
This script is based ''heavily'' on [http://qwiki.caltech.edu/index.php/Converting_LaTex_To_Wiki Latex to Wiki on Qwiki] which in
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Goo and Yoshida 1990

2006-12-07T00:43:23Z

Shagman:

Goo, J.-S. & Yoshida, K. 1990 A numerical method for huge semisubmersible responses in waves.
SNAME Trans. 98, 365–387.

Outlines a method to apply [[Kagemoto and Yue Interaction Theory]] to bodies of
arbitrary geometry.

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Kagemoto and Yue Interaction Theory

2006-12-07T00:04:38Z

Shagman:

= Introduction =

This is an interaction theory which provides the exact solution (i.e. it is not based on a [[Wide Spacing Approximation]]).
The theory uses the [[Cylindrical Eigenfunction Expansion]] and [[Graf's Addition Theorem]] to represent the potential in local coordinates. The incident and scattered potential of each body are then related by the associated [[Diffraction Transfer Matrix]].

The basic idea is as follows: The scattered potential of each body is represented in the [[Cylindrical Eigenfunction Expansion]] associated with the local coordinates centred at the mean centre position of the body. Using [[Graf's Addition Theorem]], the scattered potential of all bodies (given in their local coordinates) can be mapped to an incident potential associated with the coordiates of all other bodies. Doing this, the incident potential of each body (which is given by the ambient incident potential plus the scattered potentials of all other bodies) is given in the [[Cylindrical Eigenfunction Expansion]] associated with its local coordinates. Using the [[Diffraction Transfer Matrix]], which relates the incident and scattered potential of each body in isolation, a system of equations for the coefficients of the scattered potentials of all bodies is obtained.

The theory is described in [[Kagemoto and Yue 1986]] and in
[[Peter and Meylan 2004]].

The derivation of the theory in [[Infinite Depth]] is also presented, see
[[Kagemoto and Yue Interaction Theory for Infinite Depth]].

[[Category:Interaction Theory]]

= Equations of Motion =

The problem consists of <math>n</math> bodies
<math>\Delta_j</math> with immersed body
surface <math>\Gamma_j</math>. Each body is subject to
the [[Standard Linear Wave Scattering Problem]] and the particluar
equations of motion for each body (e.g. rigid, or freely floating)
can be different for each body.
It is a [[Frequency Domain Problem]] with frequency <math>\omega</math>.
The solution is exact, up to the
restriction that the escribed cylinder of each body may not contain any
other body.
To simplify notation, <math>\mathbf{y} = (x,y,z)</math> always denotes a point
in the water, which is assumed to be of [[Finite Depth]] <math>d</math>,
while <math>\mathbf{x}</math> always denotes a point of the undisturbed water
surface assumed at <math>z=0</math>.

Writing <math>\alpha = \omega^2/g</math> where <math>g</math> is the acceleration due to
gravity, the potential <math>\phi</math> has to
satisfy the standard boundary-value problem
<center><math>
\nabla^2 \phi = 0, \; \mathbf{y} \in D
</math></center>
<center><math>
\frac{\partial \phi}{\partial z} = \alpha \phi, \;
{\mathbf{x}} \in \Gamma^\mathrm{f},
</math></center>
<center><math>
\frac{\partial \phi}{\partial z} = 0, \; \mathbf{y} \in D, \ z=-d,
</math></center>
where <math>D</math> is the
is the domain occupied by the water and
<math>\Gamma^\mathrm{f}</math> is the free water surface. At the immersed body
surface <math>\Gamma_j</math> of body <math>\Delta_j</math>, <math>j=1,\dots,N</math>, the water velocity potential has to
equal the normal velocity of the body <math>\mathbf{v}_j</math>,
<center><math>
\frac{\partial \phi}{\partial n} = \mathbf{v}_j, \; {\mathbf{y}}
\in \Gamma_j.
</math></center>
where the normal derivative is given by the particaluar equations of motion of the body.
Moreover, the [[Sommerfeld Radiation Condition]] is imposed.

=Eigenfunction expansion of the potential=

Each body is subject to an incident potential and moves in response to this
incident potential to produce a scattered potential. Each of these is
expanded using the [[Cylindrical Eigenfunction Expansion]]
The scattered potential of a body
<math>\Delta_j</math> can be expressed as
<center><math>
\phi_j^\mathrm{S} (r_j,\theta_j,z) =
\sum_{m=0}^{\infty} f_m(z) \sum_{\mu = -
\infty}^{\infty} A_{m \mu}^j K_\mu (k_m r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j},
</math></center>
with discrete coefficients <math>A_{m \mu}^j</math>, where <math>(r_j,\theta_j,z)</math>
are cylinderical polar coordinates centered at each body
<center><math>
f_m(z) = \frac{\cos k_m (z+d)}{\cos k_m d}.
</math></center>
and
where <math>k_n</math> are found from <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]
<center><math>
\alpha + k_m \tan k_m d = 0\,.
</math></center>
where <math>k_0</math> is the
imaginary root with positive imaginary part
and <math>k_m</math>, <math>m>0</math>, are given the positive real roots ordered
with increasing size.

The incident potential upon body <math>\Delta_j</math> can be also be expanded in
regular cylindrical eigenfunctions,
<center><math>
\phi_j^\mathrm{I} (r_j,\theta_j,z) = \sum_{n=0}^{\infty} f_n(z)
\sum_{\nu = - \infty}^{\infty} D_{n\nu}^j I_\nu (k_n r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j},
</math></center>
with discrete coefficients <math>D_{n\nu}^j</math>. In these expansions, <math>I_\nu</math>
and <math>K_\nu</math> denote the modified [http://en.wikipedia.org/wiki/Bessel_function : Bessel functions]
of the first and second kind, respectively, both of order <math>\nu</math>.
Note that the term for <math>m =0</math> or
<math>n=0</math>) corresponds to the propagating modes while the
terms for <math>m\geq 1</math> (<math>n\geq 1</math>) correspond to the evanescent modes.

=Derivation of the system of equations=

A system of equations for the unknown
coefficients of the
scattered wavefields of all bodies is developed. This system of
equations is based on transforming the
scattered potential of <math>\Delta_j</math> into an incident potential upon
<math>\Delta_l</math> (<math>j \neq l</math>). Doing this for all bodies simultaneously,
and relating the incident and scattered potential for each body, a system
of equations for the unknown coefficients is developed.
Making use of the periodicity of the geometry and of the ambient incident
wave, this system of equations can then be simplified.

The scattered potential <math>\phi_j^{\mathrm{S}}</math> of body <math>\Delta_j</math> needs to be
represented in terms of the incident potential <math>\phi_l^{\mathrm{I}}</math>
upon <math>\Delta_l</math>, <math>j \neq l</math>. This can be accomplished by using
[[Graf's Addition Theorem]]
<center><math>
K_\tau(k_m r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
\sum_{\nu = - \infty}^{\infty} K_{\tau + \nu} (k_m R_{jl}) \,
I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l,
</math></center>
which is valid provided that <math>r_l < R_{jl}</math>. Here, <math>(R_{jl},\varphi_{jl})</math> are the polar coordinates of the mean centre position of <math>\Delta_{l}</math> in the local coordinates of <math>\Delta_{j}</math>.

The limitation <math>r_l < R_{jl}</math> only requires that the escribed cylinder of each body
<math>\Delta_l</math> does not enclose any other origin <math>O_j</math> (<math>j \neq l</math>). However, the
expansion of the scattered and incident potential in cylindrical
eigenfunctions is only valid outside the escribed cylinder of each
body. Therefore the condition that the
escribed cylinder of each body <math>\Delta_l</math> does not enclose any other
origin <math>O_j</math> (<math>j \neq l</math>) is superseded by the more rigorous
restriction that the escribed cylinder of each body may not contain any
other body.

Making use of the eigenfunction expansion as well as [[Graf's Addition Theorem]], the scattered potential
of <math>\Delta_j</math> can be expressed in terms of the
incident potential upon <math>\Delta_l</math> as
<center><math>
\phi_j^{\mathrm{S}} (r_l,\theta_l,z)
= \sum_{m=0}^\infty f_m(z) \sum_{\tau = -
\infty}^{\infty} A_{m\tau}^j \sum_{\nu = -\infty}^{\infty}
(-1)^\nu K_{\tau-\nu} (k_m R_{jl}) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu
\theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}}
</math></center>
<center><math>
= \sum_{m=0}^\infty f_m(z) \sum_{\nu =
-\infty}^{\infty} \Big[ \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j
(-1)^\nu K_{\tau-\nu} (k_m R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu)
\varphi_{jl}} \Big] I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
</math></center>
The ambient incident wavefield <math>\phi^{\mathrm{In}}</math> can also be
expanded in the eigenfunctions corresponding to the incident wavefield upon
<math>\Delta_l</math>. Let <math>\tilde{D}_{n\nu}^{l}</math> denote the coefficients of this
ambient incident wavefield in the incoming eigenfunction expansion for
<math>\Delta_l</math> (cf. the example in [[Cylindrical Eigenfunction Expansion]]).
<center><math>
\phi^{\mathrm{In}}(r_l,\theta_l,z)= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
\tilde{D}_{n\nu}^{l} I_\nu (k_n
r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
</math></center>
The total
incident wavefield upon body <math>\Delta_j</math> can now be expressed as
<center><math>
\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +
\sum_{j=1,j \neq l}^{n} \, \phi_j^{\mathrm{S}}
(r_l,\theta_l,z)
</math></center>
This allows us to write
<center><math>
\sum_{n=0}^{\infty} f_n(z)
\sum_{\nu = - \infty}^{\infty} D_{n\nu}^j I_\nu (k_n r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}
</math></center>
<center><math>
= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
\Big[ \tilde{D}_{n\nu}^{l} +
\sum_{j=1,j \neq l}^{n} \sum_{\tau =
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_n
r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
</math></center>
It therefore follows tha
<center><math>
D_{n\nu}^j =
\tilde{D}_{n\nu}^{l} +
\sum_{j=1,j \neq l}^{n} \sum_{\tau =
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}}
</math></center>

= Final Equations =

The scattered and incident potential of each body <math>\Delta_l</math> can be related by the
[[Diffraction Transfer Matrix]] acting in the following way,
<center><math>
A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
\mu \nu}^l D_{n\nu}^l.
</math></center>

The substitution of this into the equation for relating
the coefficients <math>D_{n\nu}^l</math> and
<math>A_{m \mu}^l</math>gives the
required equations to determine the coefficients of the scattered
wavefields of all bodies,
<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\Big[ \tilde{D}_{n\nu}^{l} +
\sum_{j=1,j \neq l}^{N} \sum_{\tau =
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big],
</math></center>
<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</math>.

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Category:Administration

2006-12-06T22:23:53Z

Shagman:

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Michael Meylan

2006-10-25T09:40:15Z

Shagman:

Michael Meylan is a senior lecturer at the [http://www.auckland.ac.nz University of Auckland].
He completed his Ph.D. under [[Vernon Squire]]
in 1993 which was concerned with modelling ice floes using linear wave theory.
He has worked on various problem connected with linear water wave theory in the subsequent time.

Mike's [http://www.math.auckland.ac.nz/Directory/profile.php?upi=x0131 home page]

[[Image:Mikem.jpg|thumb|right|Photo taken in 1999]]

= Research =

== PhD Otago 1991 - 1993==
Mike's PhD thesis concerned a two-dimensional [[:Category:Floating Elastic Plate|Floating Elastic Plate]] which was solved
using a [[Green Function Solution Method]] ([[Meylan and Squire 1994]]). The motivation for the solution was to model ice floe
and at the time he was ignorant of the engineering applications (e.g. [[VLFS]]).
Mike independently derived the Green function which
was well known in water waves and goes back to [[John 1950]].
The derivation method was copied by [[Squire and Dixon 2000]]
(based on a close reading of his Phd thesis) for the case, not of a free surface,
but for a free surface covered by a plate
The results
of this research were published in the ''Journal of Geophysical Research'' and were largely
ignored by later researchers. His Phd thesis probably had a much greater influence, through
the researchers who followed at Otago and it is continuing to appear in journal citations.
The solution method using a Green function coupled with a Green function for the plate
(the later Green function does not extend to three dimensions because of the much
more complicated boundary conditions which exist). The solution method has been
superseded by more efficient methods, most notably the [[:Category:Wiener-Hopf|Wiener-Hopf]] method developed
by [[Tim Williams]] and the [[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]]
(which applied to multiple plates) developed by [[Kohout et. al. 2006]].

== Post-Doc in Otago 1994 - 1996 ==

[[Image:Mikem2006.jpg|thumb|right|Photo taken in 2006]]

Mike then extended the two-dimensional solution to a three-dimensional circular elastic plate
([[Meylan_Squire_1996a|Meylan and Squire 1996]]).
This solution again used a Green function method coupled with the eigenfunctions for a circular
plate (which can be computed in exact form, at least up to solving an equation involving
Bessel functions. The solution method has been superseded by [[Peter, Meylan, and Chung 2004]].
Mike also developed a method to solve for plates of arbitrary geometry, initially using
a variational method ([[Meylan_2001a|Meylan 2001]]) and later using the [[Finite Element Method]]
([[Meylan 2002]]).

== Post Doc Auckland 1996 - 1998 ==
Mike then worked on using the solution for a circular elastic plate to try and construct a model
for [[:Category:Wave Scattering in the Marginal Ice Zone|Wave Scattering in the Marginal Ice Zone]]
([[Meylan, Squire, and Fox 1997]]).
This model was developed independently of the model of [[Masson and LeBlond 1989]]
but shares many similarities with it.

Mike then began to work on a very abstract (and difficult problems) of an eigenfunction
expansion method for the non-selfadjoint operator which arises in the scattering model
of [[Meylan, Squire, and Fox 1997]]. This work is still
unpublished although a paper has been submitted. It is not a problem in water wave theory.

==Massey University 1999 - 2003 ==

Mike began working on the [[:Category:Time-Dependent Linear Water Waves|Time-Dependent Linear Water Wave]] problem.
He solved for the time-dependent motion of a [[:Category:Floating Elastic Plate|Floating Elastic Plate]]
assuming [[Shallow Depth]]. The solution was found using a [[Generalised Eigenfunction Expansion]]
and as a sum over [[Scattering Frequencies]] ([[Meylan 2002]]). This lead to a collaboration with
[[Christophe Hazard]] and to a solution of the problem of a [[:Category:Floating Elastic Plate|Floating Elastic Plate]]
on [[Finite Depth]] in the time domain.

[[Cynthia Wang]] worked with Mike as a masters (2000) and Phd student (2001-2003). Her master thesis concerned
wave scattering by a [[:Category:Floating Elastic Plate|Floating Elastic Plate]] on water of [[Variable Bottom Topography]]
([[Wang and Meylan 2002]]). Cynthia's PhD concerned a higher-order
coupled [[Boundary Element Method]] [[Finite Element Method]] for the three-dimensional
[[:Category:Floating Elastic Plate|Floating Elastic Plate]] ([[Wang and Meylan 2004]]) and applied this
method to the problem of an [[:Category:Infinite Array|Infinite Array]]
of [[:Category:Floating Elastic Plate|Floating Elastic Plates]]
([[Wang, Meylan, and Porter 2006]]).

Mike developed a method to solve for multiple floes using an extension of the method
of [[Meylan 2002]]. This was not published but was used to test the
multiple floe scattering method which was developed with [[Malte Peter]] using [[Kagemoto and Yue Interaction Theory]]
which was developed during his masters in 2002.
Specifically, in [[Peter and Meylan 2004]] the [[Kagemoto and Yue Interaction Theory]] was extended
to infinite depth and a coherent account of the theory for bodies of arbitrary geometry was given.
This work required the development of very sophisticated wave scattering code for bodies of
arbitrary geometry. As a direct result of this work a new expression for the [[Free-Surface Green Function]] was
developed and this was published separately ([[Peter and Meylan 2004b]]).

Mike also revisited the problem of a [[Circular Floating Elastic Plate]] and developed a method
based on the [[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]] ([[Peter, Meylan, and Chung 2004]]).
Rike Grotmaack worked with Mike for an honours project in 2002 on [[Wave Forcing of Small Bodies]]
([[Grotmaack and Meylan 2006]])

== Auckland 2003 - present ==

[[Malte Peter]] and Mike have continued to work together and have developed an alternative method
for the [[:Category:Infinite Array|Infinite Array]] based on [[Kagemoto and Yue Interaction Theory]]
([[Peter, Meylan, and Linton 2006]]). This method has been recently
extended to a [[Semi-Infinite Array]]. He has continuted to work of [[:Category:Wave Scattering in the Marginal Ice Zone|Wave Scattering in the Marginal Ice Zone]]
and has developed a model with [[Alison Kohout]]. This model is based on the multiple
[[:Category:Floating Elastic Plate|Floating Elastic Plate]]
solution using the [[:Category:Eigenfunction Matching Method|Eigenfunction Matching Method]]
([[Kohout et. al. 2006]]). He is presently
working on a theory based on [[Scattering Frequencies]] with [[Rodney Eatock Taylor]].

[[Category:People|Meylan, Michael]]

[[Category:People|Meylan, Michael]]

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