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Wave scattering by a [[Floating Elastic Plate]] on water of Variable Bottom Topography was treated in
Wave scattering by a [[Floating Elastic Plate]] on water of Variable Bottom Topography was treated in
[[Wang and Meylan 2002]].
[[Wang and Meylan 2002]] and is described in [[Floating Elastic Plate on Variable Bottom Topography]]


\title{The Linear Wave Response of a Floating Thin Plate on Water of
Variable Depth}
\author{Cynthia D. Wang and Michael H. Meylan \\
Institute of Information and Mathematical Sciences,\\
Massey University, New Zealand}
We present a solution for the linear wave forcing of a floating
two-dimensional thin plate on water of variable depth. The solution method
is based on reducing the problem to a finite domain which contains both the
region of variable water depth and the floating thin plate. In this finite
region the outward normal derivative of the potential on the boundary is
expressed as a function of the potential. This is accomplished by using
integral operators for the radiating boundaries and the boundary under the
plate. Laplace's equation in the finite domain is solved using the boundary
element method and the integral equations are solved by numerical
integration. The same discretisation is used for the boundary element method
and to integrate the integral equations. The results show that there is a
significant region where the solution for a plate with a variable depth
differs from the simpler solutions for either variable depth but no plate or
a plate with constant depth. Furthermore, the presence of the plate
increases the frequency of influence of the variable depth.
=Introduction=
The linear wave forcing of a floating thin plate can be used to model a wide
range of physical systems; for example very large floating structures \cite
{Kashiwagihydro98}, sea ice floes [[Squire_Review]] and breakwaters \cite
{Stoker}. For this reason it is the one of the best studied hydroelastic
problems and several solution methods have been developed. These methods
have focused on providing a fast solution and for this reason have
exclusively solved for water of constant depth. Furthermore, since all the
models tests have been conducted in water of constant depth, only the
constant depth solution is required to compare theory and experiment.
However, because of the large size of floating hydroelastic structures, it
is unlikely that the water depth will be constant under the entire
structure. For this reason the effect of a variation in the water depth
under a floating thin plate is investigated in this paper.
As mentioned, the linear wave forcing of a floating thin plate has been
extensively studied and standard solution methods have now been developed.
These methods are based on expanding the plate motion in basis functions
(often thin plate or beam modes) and on solving the equations of motion for
the water using a Green function or by a further expansion in modes \cite
{Kashiwagihydro98}. A solution of the water equations by either a Green
function or an expansion in modes requires the water depth to be constant.
Therefore these standard methods are unsuitable to be extended to the case
of variable water depth.
The linear wave scattering by variable depth (or bottom topography) in the
absence of a floating plate has been considered by many authors. Two
approaches have been developed. The first is analytical and the solution is
derived in an almost closed form ([[Porter95]], [[Staziker96]] and
[[Porter00]]). However this approach is unsuitable to be generalised to
the case when a thin plate is also floating on the water surface because of
the complicated free surface boundary condition which the floating plate
imposes. The second approach is numerical, an example of which is the method
developed by [[Liu82]], in which the boundary element method in a finite
region is coupled to a separation of variables solution in the semi-infinite
outer domains. This method is well suited to the inclusion of the plate as
will be shown. For both the analytic and numerical approach the region of
variable depth must be bounded.
In this paper, a solution method for the linear wave forcing of a two
dimensional floating plate on water of variable depth will be derived from
[[Liu82]], [[Hazard]] and [[jgrfloe1d]]. The method is based on
dividing the water domain into two semi-infinite domains and a finite
domain. The finite domain contains both the plate and the region of variable
water depth. Laplace's equation in the finite domain is solved by the
boundary element method. Laplace's equation in the semi-infinite domains is
solved by separation of variables. The solution in the semi-infinite domains
gives an integral equation relating the normal derivative of the potential
and the potential on the boundary of the finite and semi-infinite domains.
The thin plate equation is expressed as an integral equation relating the
normal derivative of the potential and the potential under the surface of
the plate. The boundary element equations and the integral equations are
solved simultaneously using the same discretisation.
=Problem Formulation=
We consider a thin plate, floating on the water surface above a sea bed of
variable depth, which is subject to an incoming wave. The plate is
approximated as infinitely long in the <math>y</math> directions which reduces the
problem to two dimensions, <math>x</math> and <math>z</math>. The <math>x</math>-axis is horizontal and the <math>
z <math>-axis points vertically up with the free water surface at </math>z=0.</math> The
incoming wave is assumed to be travelling in the positive <math>x</math>-direction with
a single radian frequency <math>\omega </math>. The theory which will be developed
could be extended to oblique incident waves using the standard method \cite
{OhkusuISOPE}. However, to keep the treatment straightforward, this will not
be done. We assume that the wave amplitude is sufficiently small that the
problem can be approximated as linear. From the linearity and the single
frequency wave assumption it follows that all quantities can be written as
the real part of a complex quantity whose time dependence is <math>e^{-i\omega t}</math>
.
The linear boundary value problem for the water is the following,
<center><math>
\left.
\begin{matrix}{c}
\nabla ^{2}\phi =0, \\
-\rho g\,w+i\omega \rho \phi =p,\qquad z=0, \\
\phi _{n}=0,\qquad z=d\left( x\right) ,
\end{matrix}
\right\}  (1)
</math></center>
where <math>\rho </math> is the density of the water, <math>p</math> is the pressure on water
surface, <math>w</math> is the displacement of the water surface, <math>g</math> is the
gravitational acceleration, <math>d(x)</math> is the water depth and <math>\phi _{n}</math> is the
outward normal derivative of the potential. We assume that the water depth
is constant outside a finite region, <math>-l<x<l</math>, but allow the depth to be
different at either end. Therefore
<center><math>
d\left( x\right) =\left\{
\begin{matrix}{c}
-H_{1},\;\;x<-l, \\
d\left( x\right) ,\;\;-l<x<l, \\
-H_{2},\;\;x>l,
\end{matrix}
\right.
</math></center>
where <math>H_{1}</math> and <math>H_{2}</math> are the water depths in the left and right hand
domains of constant depth.
A thin plate, of negligible draft, floats on the surface of the water and
occupies the region <math>-L\leq x\leq L\ </math>as is shown in Figure \ref{fig_region}
. Without loss of generality, we assume that <math>l</math> is sufficiently large that <math>
L<l.</math> For any point on the water surface not under the plate the pressure is
the constant atmospheric pressure whose time-dependent part is zero. Under
the plate, the pressure and the displacement are related by the
Bernoulli-Euler equation
<center><math>
D\frac{\partial ^{4}w}{\partial x^{4}}-\rho ^{\prime }a\omega
^{2}\,w=p,\qquad -L\leq x\leq L= and = z=0,  (2)
</math></center>
where <math>\rho ^{\prime }</math> is the density of the plate, <math>a</math> is the plate
thickness, and <math>D</math> is the bending rigidity of the plate. We assume that the
plate edges are free so the bending moment and shear must vanish at both
ends of the plate, i.e.
<center><math>
\frac{\partial ^{2}w}{\partial x^{2}}=\frac{\partial ^{3}w}{\partial x^{3}}
=0,\qquad =at= \ x=-L= and = x=L.
</math></center>
The kinematic boundary condition at the surface allows us to express the
displacement as a function of the outward normal derivative of the
potential,
<center><math>
w=\frac{i\phi _{n}}{\omega },\qquad z=0.  (3)
</math></center>
Substituting equations (2) and (3)\ into
equation (1), we obtain the following boundary value problem for
the potential only,
<center><math>
\left.
\begin{matrix}{c}
\nabla ^{2}\phi =0, \\
-\rho \left( g\phi _{n}-\omega ^{2}\,\phi \right) =\left\{
\begin{matrix}{c}
0,\;x\notin \left[ -L,L\right] ,\;z=0, \\
D\,\frac{\partial ^{4}\phi _{n}}{\partial x^{4}}-\rho ^{\prime }a\,\omega
^{2}\phi _{n},\;x\in \left[ -L,L\right] ,\;z=0,
\end{matrix}
\right. \\
\phi _{n}=0,\qquad z=d\left( x\right) ,
\end{matrix}
\right\}  (4)
</math></center>
together with the free plate edge conditions
<center><math>
\frac{\partial ^{2}\phi _{n}}{\partial x^{2}}=\frac{\partial ^{3}\phi _{n}}{
\partial x^{3}}=0,\qquad =at = x=\pm L=, \ \ = z=0.
(5)
</math></center>
==Radiation Boundary Conditions==
Equation (4) is subject to radiation conditions as <math>
x\rightarrow \pm \infty .</math> We assume that there is a wave incident from the
left which gives rise to a reflected and transmitted wave. Therefore the
following boundary conditions for the potential apply as <math>x\rightarrow \pm
\infty </math>
<center><math>
\lim_{x\rightarrow -\infty }\phi \left( x,z\right) =\cosh \left(
k_{t}^{\left( 1\right) }\left( z+H_{1}\right) \right) e^{ik_{t}^{\left(
1\right) }x}+R\,\cosh \left( k_{t}^{\left( 1\right) }\left( z+H_{1}\right)
\right) e^{-k_{t}^{\left( 1\right) }x},  (6)
</math></center>
and
<center><math>
\lim_{x\rightarrow \infty }\phi \left( x,z\right) =T\cosh \left(
k_{t}^{\left( 2\right) }\left( z+H_{2}\right) \right) e^{k_{t}^{\left(
2\right) }x},  (7)
</math></center>
where <math>R</math> and <math>T</math> are the reflection and transmission coefficients
respectively and <math>k_{t}^{\left( j\right) }\,\left( j=1,2\right) </math> are the
positive real solutions of the following equation
<center><math>
gk_{t}^{\left( j\right) }\tanh \left( k_{t}^{\left( j\right) }\,H_{j}\right)
=\omega ^{2}.
</math></center>
==(8)Non-dimensionalisation==
Non-dimensional variables are now introduced. We non-dimensionalise the
space variables with respect to the water depth on the left hand side, <math>
H_{1},<math> and the time variables with respect to</math>\;\sqrt{g/H_{1}}</math>. The
non-dimensional variables, denoted by an overbar, are
<center><math>
\bar{x}=\frac{x}{H_{1}},\;\bar{z}=\frac{z}{H_{1}},\;\bar{t}=t\sqrt{\frac{g
}{H_{1}}},\;=and= \;\bar{\phi}=\frac{1}{H_{1}\sqrt{H_{1}g}}\,\phi .
</math></center>
Applying this non-dimensionalisation to equation (4)
we obtain
<center><math>
\left.
\begin{matrix}{c}
\nabla ^{2}\bar{\phi}=0, \\
\left( \bar{\phi}_{n}-\nu \bar{\phi}\right) =\left\{
\begin{matrix}{c}
0,\qquad \bar{x}\notin \left[ -L,L\right] ,\;\bar{z}=0, \\
-\beta \,\frac{\partial ^{4}\bar{\phi}_{n}}{\partial \bar{x}^{4}}+\gamma
\nu \,\bar{\phi}_{n},\qquad \bar{x}\in \left[ -L,L\right] ,\;\bar{z}=0,
\end{matrix}
\right. \\
\bar{\phi}_{n}=0,\qquad \bar{z}=\bar{d}\left( \bar{x}\right) ,
\end{matrix}
\right\}  (9)
</math></center>
where
<center><math>
\beta =\frac{D}{\rho gH_{1}^{4}},\;\gamma =\frac{\rho ^{\prime }a}{\rho
H_{1}},\;=and= \;\nu =\frac{\omega ^{2}H_{1}}{g}.  (10)
</math></center>
We will refer to <math>\beta </math> as the stiffness, <math>\gamma </math> as the mass and <math>\nu </math>
as the wavenumber. The non-dimensional water depth is
<center><math>
\bar{d}\left( \bar{x}\right) =\left\{
\begin{matrix}{c}
-1,\;\;\bar{x}<-\bar{l}, \\
\bar{d}\left( \bar{x}\right) ,\;\;-\bar{l}<\bar{x}<\bar{l}, \\
-H,\;\;\bar{x}>\bar{l},
\end{matrix}
\right.
</math></center>
where <math>H=H_{2}/H_{1}.</math> Equation (9) is also subject to the
non-dimensional free edge conditions of the plate (5) and the
radiation conditions (6) and (7). With the
understanding that all variables have been non-dimensionalised, from now
onwards we omit the overbar.
=Reduction to a Finite Domain=
We solve equation (9) by reducing the problem to a finite
domain which contains both the region of variable depth and the floating
thin plate. This finite domain <math>\Omega =\left\{ -l\leq x\leq l,\;d\left(
x\right) \leq z\leq 0\right\} </math> is shown in Figure \ref{fig_region}. We will
solve Laplace's equation in <math>\Omega </math> using the boundary element method. To
accomplish this we need to express the normal derivative of the potential on
the boundary of <math>\Omega \;(\partial \Omega )</math> as a function of the potential
on the boundary.
==Green's Function Solution for the Floating Thin Plate==
We begin with the boundary condition under the plate which is the following
<center><math>
\beta \frac{\partial ^{4}\phi _{n}}{\partial x^{4}}-\left( \gamma \nu
-1\right) \phi _{n}=\nu \phi ,\;-L\leq x\leq L= and = z=0,  (11)
</math></center>
together with the free edge boundary conditions (5).
Following [[jgrfloe1d]] we can transform equations (5) and
(11) to an integral equation using the Green function, <math>g,</math> which
satisfies
<center><math>
\left.
\begin{matrix}{c}
\beta \frac{\partial ^{4}}{\partial x^{4}}g\left( x,\xi \right) -\left(
\gamma \nu -1\right) g\left( x,\xi \right) =\nu \delta \left( x-\xi \right) ,
\\
\frac{\partial ^{2}}{\partial x^{2}}g\left( x,\xi \right) =\frac{\partial
^{3}}{\partial x^{3}}g\left( x,\xi \right) =0,\qquad =at = x=-L,\;x=L.
\end{matrix}
\right\}
</math></center>
This gives us the following expression for <math>\phi _{n}</math> as a function of the
potential under the plate <math>\phi ,</math>
<center><math>
\phi _{n}\left( x\right) =\int_{-L}^{L}g\left( x,\xi \right) \;\phi \left(
\xi \right) \;d\xi .  (12)
</math></center>
We will write this in operator notation as <math>\phi _{n}=\mathbf{g}\phi </math> where
<math>\mathbf{g}</math> denotes the integral operator with kernel <math>g\left( x,\xi
\right) .</math>
==Solution in the Semi-infinite Domains==
We now solve Laplace's equation in the semi-infinite domains <math>\Omega
^{-}=\left\{ x<-l,\;-1\leq z\leq 0\right\} <math> and </math>\Omega ^{+}=\left\{
x>l,\;-H\leq z\leq 0\right\} </math> which are shown in Figure (\ref{fig_region}).
Since the water depth is constant in these regions we can solve Laplace's
equation by separation of variables. The potential in the region <math>\Omega
^{-} </math> satisfies the following equation
<center><math>
\left.
\begin{matrix}{c}
\nabla ^{2}\phi =0,\;\;\mathbf{x}\in \Omega ^{-}, \\
\phi _{n}-\nu \phi =0,\;\;z=0, \\
\phi _{n}=0,\;\;z=-1, \\
\phi =\tilde{\phi}\left( z\right) ,\;\;x=-\,l, \\
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) =\cosh \left(
k_{t}^{\left( 1\right) }\left( z+1\right) \right) e^{ik_{t}^{\left( 1\right)
}x} \\
+R\,\cosh \left( k_{t}^{\left( 1\right) }\left( z+1\right) \right)
e^{-ik_{t}^{\left( 1\right) }x},
\end{matrix}
\right\}  (13)
</math></center>
where <math>\mathbf{x}=\left( x,z\right) \ </math>and <math>\tilde{\phi}\left( z\right) </math> is
an arbitrary continuous function<math>.</math>\ Our aim is to find the outward normal
derivative of the potential on <math>x=-l</math> as a function of <math>\tilde{\phi}\left(
z\right) </math>.
We solve equation (13) by separation of variables \cite{Liu82,
Hazard} and obtain the following expression for the potential in the region <math>
\Omega ^{-},</math>
<center><math>\begin{matrix}
\phi \left( x,z\right) &=&\cosh \left( k_{t}^{\left( 1\right) }\left(
z+1\right) \right) \,e^{ik_{t}^{\left( 1\right) }x}+R\cosh \left(
k_{t}^{\left( 1\right) }\left( z+1\right) \right) \,e^{-ik_{t}^{\left(
1\right) }x}  \notag \\
&&+\sum_{m=1}^{\infty }\left\langle \tilde{\phi}\left( z\right) ,\tau
_{m}^{\left( 1\right) }\left( z\right) \right\rangle \tau _{m}^{\left(
1\right) }\left( z\right) e^{k_{m}^{\left( 1\right) }\left( x+l\right) }.
(14)
\end{matrix}</math></center>
The functions <math>\tau _{m}^{\left( 1\right) }\left( z\right) </math> (<math>m\geq 1)</math> are
the orthonormal modes given by
<center><math>
\tau _{m}^{\left( 1\right) }\left( z\right) =\left( \frac{1}{2}+\frac{\sin
\left( 2k_{m}^{\left( 1\right) }\,\right) }{4k_{m}^{\left( 1\right) }}
\right) ^{-\frac{1}{2}}\cos \left( k_{m}^{\left( 1\right) }\left(
z+1\right) \right) ,\;\;m\geq 1.
</math></center>
The evanescent eigenvalues <math>k_{m}^{\left( 1\right) }</math> are the positive real
solutions of the dispersion equation
<center><math>
-k_{m}^{\left( 1\right) }\,\tan \left( k_{m}^{\left( 1\right) }H_{j}\right)
=\nu ,\;\;m\geq 1,  (15)
</math></center>
ordered by increasing size. The inner product in equation (13) is
the natural inner product for the region <math>-1\leq z\leq 0</math> given by
<center><math>
\left\langle \tilde{\phi}\left( z\right) ,\tau _{m}^{\left( j\right) }\left(
z\right) \right\rangle =\int_{-1}^{0}\tilde{\phi}\left( z\right) ,\tau
_{m}^{\left( j\right) }\left( z\right) dx.  (16)
</math></center>
The reflection coefficient is determined by taking an inner product of
equation (14) with respect to <math>\cosh \left( k_{t}^{\left( 1\right)
}\left( z+1\right) \right) .<math> This gives us the following expression for </math>R</math>
,
<center><math>
R=\frac{\left\langle \tilde{\phi}\left( z\right) ,\cosh \left(
k_{t}^{\left( 1\right) }\left( z+1\right) \right) \right\rangle }{\frac{1}{2
}+\sinh \left( 2k_{t}^{\left( 1\right) }\,\right) /4k_{0}^{\left( 1\right) }}
e^{-ik_{t}^{\left( 1\right) }l}-e^{-2ik_{t}^{\left( 1\right) }l}.
(17)
</math></center>
The normal derivative of the potential on the boundary of <math>\Omega ^{-}</math> and <math>
\Omega <math> </math>\left( x=-l\right) </math> is calculated using equation (14) and
we obtain,
<center><math>
\left. \phi _{n}\right| _{x=-l}=\mathbf{Q}_{1}\tilde{\phi}\left( z\right)
-2ik_{t}^{\left( 1\right) }\cosh \left( k_{t}^{\left( 1\right) }\left(
z+1\right) \right) \,e^{-ik_{t}^{\left( 1\right) }l},
</math></center>
where the outward normal is with respect to the <math>\Omega </math> domain. The
integral operator <math>\mathbf{Q}_{1}</math> is given by
<center><math>\begin{matrix}
\mathbf{Q}_{1}\tilde{\phi}\left( z\right) &=&\sum_{m=1}^{\infty
}k_{m}^{\left( 1\right) }\left\langle \tilde{\phi}\left( z\right) ,\tau
_{m}^{\left( 1\right) }\left( z\right) \right\rangle \,\tau _{m}^{\left(
1\right) }\left( z\right)  (18) \\
&&+ik_{t}\frac{\left\langle \tilde{\phi}\left( z\right) ,\cosh \left(
k_{t}^{\left( 1\right) }\left( z+1\right) \right) \right\rangle \cosh \left(
k_{t}^{\left( 1\right) }\left( z+1\right) \right) }{\frac{1}{2}+\sinh
\left( 2k_{t}^{\left( 1\right) }\,\right) /4k_{t}^{\left( 1\right) }}.
\notag
\end{matrix}</math></center>
We can combine the two terms of equation (18) and express <math>\mathbf{Q}
_{1}</math> as
<center><math>
\mathbf{Q}_{1}\tilde{\phi}\left( z\right) =\sum_{m=0}^{\infty }k_{m}^{\left(
1\right) }\left\langle \tilde{\phi}\left( z\right) ,\tau _{m}^{\left(
1\right) }\left( z\right) \right\rangle \,\tau _{m}^{\left( 1\right) }\left(
z\right)  (19)
</math></center>
where <math>k_{0}^{\left( 1\right) }=ik_{t}^{\left( 1\right) }</math> and
<center><math>
\tau _{0}^{\left( 1\right) }\left( z\right) =\left( \frac{1}{2}+\frac{\sin
\left( 2k_{0}^{\left( 1\right) }\right) }{4k_{0}^{\left( 1\right) }}\right)
^{-\frac{1}{2}}\cos \left( k_{0}^{\left( 1\right) }\left( z+1\right)
\right) .
</math></center>
As well as providing a more compact notation, equation (19)
will be useful in the numerical calculation of <math>\mathbf{Q}_{1}.</math>
Similarly, we now consider the potential in the region <math>\Omega ^{+}</math> which
satisfies
<center><math>
\left.
\begin{matrix}{c}
\nabla ^{2}\phi =0,\;\;\mathbf{x}\in \Omega ^{+}, \\
\phi _{n}-\nu \phi =0,\;\;z=0, \\
\phi _{n}=0=,= \;\;z=-H, \\
\phi =\tilde{\phi}\left( z\right) ,\;\;x=\,l, \\
\lim\limits_{x\rightarrow -\infty }\phi \left( x,z\right) =T\cosh \left(
k_{t}^{\left( 2\right) }\left( z+H_{2}\right) \right) e^{ik_{t}^{\left(
2\right) }x}.
\end{matrix}
\right\}  (20)
</math></center>
Solving equation (20) by separation of variables as before we obtain
<center><math>
\left. \phi _{n}\right| _{x=l}=\mathbf{Q}_{2}\tilde{\phi}\left( z\right) ,
</math></center>
where the outward normal is with respect to the <math>\Omega </math> domain. The
integral operator <math>\mathbf{Q}_{2}</math> is given by
<center><math>
\mathbf{Q}_{2}\tilde{\phi}\left( z\right) =\sum_{m=0}^{\infty }k_{m}^{\left(
2\right) }\left\langle \tilde{\phi}\left( z\right) ,\tau _{m}^{\left(
2\right) }\left( z\right) \right\rangle \tau _{m}^{\left( 2\right) }\left(
z\right) .  (21)
</math></center>
The orthonormal modes <math>\tau _{m}^{\left( 2\right) }</math> are given by
<center><math>
\tau _{m}^{\left( 2\right) }\left( z\right) =\left( \frac{H}{2}+\frac{\sin
\left( 2k_{m}^{\left( 2\right) }\,H\right) }{4k_{m}^{\left( 2\right) }}
\right) ^{-\frac{1}{2}}\cos \left( k_{m}^{\left( 2\right) }\left(
z+H\right) \right) ,
</math></center>
The eigenvalues <math>k_{m}^{\left( 2\right) }</math> are the positive real solutions <math>
\left( m\geq 1\right) <math> and positive imaginary solutions </math>\left( m=0\right) </math>
of the dispersion equation
<center><math>
-k_{m}^{\left( 2\right) }\,\tan \left( k_{m}^{\left( 2\right) }H\right) =\nu
.
</math></center>
The inner product is the same as that given by equation (16)
except that the integration is from <math>z=-H</math> to <math>z=0.</math> The transmission
coefficient, <math>T,</math> is given by
<center><math>
T=\frac{\left\langle \tilde{\phi}\left( z\right) ,\cosh \left(
k_{t}^{\left( 2\right) }\left( z+1\right) \right) \right\rangle }{\frac{1}{2
}+\sinh \left( 2k_{t}^{\left( 2\right) }H\right) /4k_{t}^{\left( 2\right) }}
e^{-ik_{t}^{\left( 2\right) }l}.  (22)
</math></center>
==The equation for the finite domain==
We now consider the finite domain <math>\Omega .</math> In this domain, Laplace's
equation is subject to the boundary conditions given by equations (\ref
{integral_plate}), (18) and (21) as well as the free surface and
sea floor boundary conditions. This gives the following equation for the
potential in <math>\Omega ,</math>
<center><math>
\left.
\begin{matrix}{c}
\nabla ^{2}\phi =0,\;\;\;\mathbf{x}\in \Omega , \\
\phi _{n}=\mathbf{Q}_{1}\phi -2ik_{t}^{\left( 1\right) }\cosh \left(
k_{t}^{\left( 1\right) }\left( z+1\right) \right) \,e^{-ik_{t}^{\left(
1\right) }l},\;\;\;\mathbf{x}\in \partial \Omega _{1}, \\
\phi _{n}=\mathbf{Q}_{2}\phi ,\;\;\;\mathbf{x}\in \partial \Omega _{2}, \\
\phi _{n}=\nu \phi ,\;\;\;\mathbf{x}\in \partial \Omega _{3}\cup \partial
\Omega _{5}, \\
\phi _{n}=\mathbf{g}\phi ,\;\;\;\mathbf{x}\in \partial \Omega _{4}, \\
\phi _{n}=0,\;\;\;\mathbf{x}\in \partial \Omega _{6}.
\end{matrix}
\right\}  (23)
</math></center>
This boundary value problem is shown in Figure \ref{fig_sigma}. The boundary
of <math>\Omega </math> (<math>\partial \Omega )</math> has been divided into the six boundary
regions <math>\partial \Omega _{i}</math> shown in the figure. They are respectively,
the boundary of <math>\Omega ^{-}</math> and <math>\Omega </math> (<math>\partial \Omega _{1}),</math> the
boundary of <math>\Omega ^{+}</math> and <math>\Omega </math> (<math>\partial \Omega _{2}),</math> the free
water surface to the left (<math>\partial \Omega _{3}</math>) and right of the plate (<math>
\partial \Omega _{5}<math>), the plate (</math>\partial \Omega _{4}</math>), and the sea
floor (<math>\partial \Omega _{6}</math>). Equation (23) is the
boundary value problem which we will solve numerically.
=Numerical Solution Method=
We have reduced the problem to Laplace's equation in a finite domain subject
to certain boundary conditions (23). These boundary
conditions give the outward normal derivative of the potential as a function
of the potential but this is not always a point-wise condition; on some
boundaries it is given by an integral equation. We must solve both Laplace's
equation and the integral equations numerically. We will solve Laplace's
equation by the boundary element method and the integral equations by
numerical integration. However, the same discretisation of the boundary will
be used for both numerical solutions.
We begin by applying the boundary element method to equation (\ref
{finitedomain}). This gives us the following equation relating the potential
and its outward normal derivative on the boundary <math>\partial \Omega </math>
<center><math>
\frac{1}{2}\phi \left( \mathbf{x}\right) =\int_{\partial \Omega }\left(
G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi \left( \mathbf{x}
^{\prime }\right) -G\left( \mathbf{x},\mathbf{x}^{\prime }\right) \phi
_{n}\left( \mathbf{x}^{\prime }\right) \right) d\mathbf{x}^{\prime },\;\;
\mathbf{x}\in \partial \Omega .  (24)
</math></center>
In equation (24) <math>G\left( \mathbf{x},\mathbf{x}^{\prime
}\right) </math> is the free space Green function given by
<center><math>
G\left( \mathbf{x},\mathbf{x}^{\prime }\right) =\frac{1}{2\pi }\ln \,\left|
\mathbf{x}-\mathbf{x}^{\prime }\right| ,  (25)
</math></center>
and <math>G_{n}\left( \mathbf{x},\mathbf{x}^{\prime }\right) </math> is the outward
normal derivative of <math>G</math> (with respect to the <math>\mathbf{x}^{\prime }</math>
coordinate).
We solve equation (24) by a modified constant panel method
which reduces it to the following matrix equation
<center><math>
\frac{1}{2}\vec{\phi}=\mathbf{G}_{n}\vec{\phi}-\mathbf{G}\vec{\phi}_{n}.
(26)
</math></center>
In equation (26) <math>\vec{\phi}\mathcal{\ }</math>and <math>\vec{\phi}
_{n}</math> are vectors which approximate the potential and its normal derivative
around the boundary <math>\partial \Omega </math>, and <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math>
are matrices corresponding to the Green function and the outward normal
derivative of the Green function respectively. The method used to calculate
the elements of the matrices <math>\mathbf{G}</math> and <math>\mathbf{G}_{n}</math> will be
discussed in section \ref{Green}.
The outward normal derivative of the potential, <math>\vec{\phi}_{n},</math> and the
potential, <math>\vec{\phi},</math> are related by the conditions on the boundary <math>
\partial \Omega </math> in equation (23). This can be expressed as
<center><math>
\vec{\phi}_{n}=\mathbf{A}\,\vec{\phi}-\vec{f},  (27)
</math></center>
where <math>\mathbf{A}</math> is the block diagonal matrix given by
<center><math>
\mathbf{A}\mathbb{=}\left[
\begin{matrix}{cccccc}
\mathbf{Q}_{1} &  &  &  &  &  \\
& \mathbf{Q}_{2} &  &  &  &  \\
&  & \nu \,\mathbf{I} &  &  &  \\
&  &  & \mathbf{g} &  &  \\
&  &  &  & \nu \,\mathbf{I} &  \\
&  &  &  &  & 0
\end{matrix}
\right] ,  (28)
</math></center>
<math>\mathbf{Q}_{1}</math>, <math>\mathbf{Q}_{2}</math>, and <math>\mathbf{g}</math> are matrix
approximations of the integral operators of the same name and <math>\vec{f}</math> is
the vector
<center><math>
\vec{f}=\left[
\begin{matrix}{c}
2ik_{t}^{\left( 1\right) }\cosh \left( k_{t}^{\left( 1\right) }\left(
z+1\right) \right) \,e^{ik_{t}^{\left( 1\right) }l} \\
0 \\
\vdots \\
0
\end{matrix}
\right] .  (29)
</math></center>
\ The methods used to construct the matrices <math>\mathbf{Q}_{1},\mathbf{Q}_{2}</math>
, and <math>\mathbf{g}</math> will be described in sections 31 and \ref
{numericalg} respectively.
Substituting equation (27) into equation (\ref
{panelEqn_boundary}) we obtain the following matrix equation for the
potential
<center><math>
\left( \frac{1}{2}-\mathbf{G}_{n}+\mathbf{GA}\right) \vec{\phi}=\mathbf{G}
\vec{f}
</math></center>
which can be solved straightforwardly. The reflection and transmission
coefficients are calculated from <math>\vec{\phi}</math> using equations (\ref
{reflection}) and (22) respectively.
==Numerical Calculation of <math>\mathbf{G==</math> and <math>\mathbf{G}_{n}</math>\label
{Green}}
The boundary element equation (24) is solved numerically by
a modified constant panel method. In this method, the boundary is divided
into panels over which the potential, <math>\phi ,</math> or its outward normal
derivative, <math>\phi _{n},</math> are assumed to be constant. The free-space Green's
function, <math>G,</math> and its normal derivative, <math>G_{n}</math> are more rapidly varying
and have a singularity at <math>\mathbf{x}=\mathbf{x}^{\prime }</math>. For this
reason, over each panel, while <math>\phi </math> and <math>\phi _{n}</math> are assumed constant,
<math>G</math> and <math>G_{n}</math> are integrated exactly. For example, we use the following
approximation to calculate the integral of <math>G</math> and <math>\phi </math> over a single
panel
<center><math>
\int_{\mathbf{x}_{i}-h/2}^{\mathbf{x}_{i}+h/2}G\left( \mathbf{x},\mathbf{x}
^{\prime }\right) \phi \left( \mathbf{x}^{\prime }\right) d\mathbf{x}
^{\prime }\approx \phi \left( \mathbf{x}_{i}\right) \int_{\mathbf{x}
_{i}-h/2}^{\mathbf{x}_{i}+h/2}G\left( \mathbf{x},\mathbf{x}^{\prime }\right)
d\mathbf{x}^{\prime },  (30)
</math></center>
where <math>\mathbf{x}_{i}</math> is the midpoint of the panel and <math>h</math> is the panel
length. The integral on the right hand side of equation (30),
because of the simple structure of <math>G</math>, can be calculated exactly.
==Numerical Calculation of <math>\mathbf{Q==_{1}</math> and <math>\mathbf{Q}_{2}
(31)</math>}
We will discuss the numerical approximation of the operator <math>\mathbf{Q}_{1}</math>
. The operator <math>\mathbf{Q}_{2}</math> is approximated in a similar fashion. We
begin with equation (19) truncated to a finite number (<math>N</math>) of
evanescent modes,
<center><math>
\mathbf{Q}_{1}\phi =\sum_{m=0}^{N}k_{m}^{\left( 1\right) }\left\langle \phi
\left( z\right) ,\tau _{m}^{\left( 1\right) }\left( z\right) \right\rangle
\tau _{m}^{\left( 1\right) }\left( z\right) .
</math></center>
We calculate the inner product
<center><math>
\left\langle \phi \left( z\right) ,\tau _{m}^{\left( 1\right) }\left(
z\right) \right\rangle =\int\nolimits_{-1}^{0}\phi \left( z\right) \,\tau
_{m}^{\left( 1\right) }\left( z\right) \,dz
</math></center>
with the same panels as we used to approximate the integrals of the Green
function and its normal derivative in subsection \ref{Green}. Similarly, we
assume that <math>\phi </math> is constant over each panel and integrate <math>\tau
_{m}^{\left( 1\right) }\left( z\right) </math> exactly. This gives us the
following matrix factorisation of <math>\mathbf{Q}_{1},</math>
<center><math>
\mathbf{Q}_{1}\,\vec{\phi}=\mathbf{S}\,\mathbf{R}\,\vec{\phi}.
</math></center>
The components of the matrices <math>\mathbf{S}</math> and <math>\mathbf{R}</math> are
<center><math>\begin{matrix}
s_{im} &=&\tau _{m}^{\left( 1\right) }\left( z_{i}\right) , \\
r_{mj} &=&k_{m}^{\left( 1\right) }\int_{z_{j}-h/2}^{z_{j}+h/2}\tau
_{m}^{\left( 1\right) }\left( s\right) ds
\end{matrix}</math></center>
where <math>z_{j}</math> is the value of the <math>z</math> coordinate in the centre of the panel
and <math>h</math> is the panel length. The integral operator <math>\mathbf{Q}_{2}</math> is
approximated by a matrix in exactly the same manner.
==Numerical Calculation of <math>\mathbf{g(32)==</math>}
The method used to approximate <math>\mathbf{g}</math> by a matrix is similar to the
methods used for <math>\mathbf{Q}_{1}</math> and <math>\mathbf{Q}_{2}</math> and follows \cite
{jgrfloe1d} with some modification. The Green function for the plate can be
expressed as
<center><math>
g\left( x,\xi \right) =\left\{
\begin{matrix}{c}
A_{1}e^{i\alpha x}+B_{1}e^{-i\alpha x}+C_{1}e^{\alpha x}+D_{1}e^{-\alpha
x},\qquad x<\xi , \\
A_{2}e^{i\alpha x}+B_{2}e^{-i\alpha x}+C_{2}e^{\alpha x}+D_{2}e^{-\alpha
x},\qquad x>\xi ,
\end{matrix}
\right.
</math></center>
where the coefficients are determined by solving a linear system \cite
{jgrfloe1d}. Again the same panels are used to approximate the integral
operator by a matrix as were used for the boundary integral equation (\ref
{integral_eqn}). Over each panel we assume that the potential is constant
and integrate the Green function <math>g</math> exactly.
=Results=
We will now present some results, concentrating on comparing the reflection
coefficient for constant and variable depth profiles. This will allow us to
determine when the variable depth profile has a significant effect. To
reduce the number of figures we restrict ourselves to four values of the
stiffness <math>\beta </math> and two variable depth profiles.
==Profiles for the variable depth==
We will consider two different profiles for the variable depth. The first
will be the profile which was used by [[Staziker96]]. This corresponds to
a rise from a uniform depth to a maximum height of half the uniform depth at
<math>x=0</math>. The formula for this profile is the following
<center><math>
d\left( x\right) =\left\{
\begin{matrix}{c}
-1,\;\;x<-l, \\
-\left( \frac{1}{2}\left( \frac{x+l}{l}\right) ^{2}-\frac{x+l}{l}
+1\right) ,\;\;-l<x<l, \\
-1,\;\;x>l.
\end{matrix}
\right.  (33)
</math></center>
Following [[Staziker96]] we will refer to this variable depth profile as
the ''hump. ''
In the second profile the depth rises linearly. The depth in the right hand
region is half the depth in the left hand region. The formula for this
profile is
<center><math>
d\left( x\right) =\left\{
\begin{matrix}{c}
-1,\;\;x<-l, \\
\frac{x+l}{4l}-1,\;\;-l<x<l, \\
-\frac{1}{2},\;\;x>l.
\end{matrix}
\right.  (34)
</math></center>
We will refer to this variable depth profile as the ''simple slope'' .
==Convergence study==
We now present a convergence study. Since we have two parameters, the panel
size used to discretise the boundary and the number of evanescent modes (<math>N</math>
) used to approximate the integral equations <math>\mathbf{Q}_{1}</math> and <math>\mathbf{Q}
_{2},</math> we must present two convergence studies considering each parameter
separately. We begin by considering the panel size used to discretise the
boundary. We expect that the panel size should be proportional to the
wavelength and therefore inversely proportional to the wavenumber <math>
k_{t}^{\left( 1\right) }</math> (assuming that the water depth at either end is of
a similar size). We therefore use the following formula for the panel size
<center><math>
=panel size = =\frac{1}{\kappa \,k_{t}^{\left( 1\right) }},
</math></center>
where <math>\kappa </math> is a constant of proportionality which will be determined
from the convergence study. Table 35 shows the absolute value
of the reflection coefficient for a plate of length <math>L=2.5,</math> stiffness <math>
\beta =1,<math> and mass </math>\gamma =0<math> for </math>\nu =1,</math> 2, and 3 for a constant depth
and for the hump with <math>l=2.5.</math> The number of evanescent modes, <math>N,</math> was
fixed to be 5. Three values of <math>\kappa </math> were considered, <math>\kappa =10,</math> 20,
and 40. The results in Table 35 show that good convergence is
achieved when <math>\kappa =20.</math>
Table 37 shows a similar convergence study for the number of
evanescent modes. We have considered <math>0,</math> 5 and <math>10</math> evanescent modes (<math>N</math>)
and fixed <math>\kappa </math> to be <math>\kappa =20</math>. All other parameters as the same as
those in Table 35. The results in Table 37 show
that good convergence is achieved when the number of modes is 5. For all
subsequent calculations the panel size will be determined by setting <math>\kappa
=20<math> and </math>N=5.</math>
==Comparison with existing results==
Before presenting our results for a plate on water of variable depth we will
make comparisons with two results from the literature. This is to establish
that our method gives the correct solution for the simpler cases of either
variable depth but no plate, or a plate floating on constant depth. We begin
by comparing our results with [[Staziker96]] who solved for wave
scattering by variable depth only. One problem which they solved was to
determine the absolute value of the reflection coefficient for a hump depth
profile with fixed frequency <math>\nu =1</math> and variable hump length <math>l.</math> The
solution to this problem by our method is shown in Figure (\ref{staziker1.ps}
). This figure is identical to Figure 2 in [[Staziker96]] (p. 290) which
establishes that our method gives the correct solution for a variable depth
in the absence of the plate.
The second comparison is with [[jgrfloe1d]] in which the problem of a
thin plate on water of constant depth was solved (to model an ice floe). One
problem which they solved was the absolute value of the reflection
coefficient as a function of plate length for fixed <math>\nu .</math> The dimensional
parameters which they used were, density <math>\rho ^{\prime }=922.5\,</math>kgm<math>^{-3},</math>
thickness <math>h=1</math>m, and bending rigidity <math>D=</math>5.4945<math>\times 10^{8}</math>kgm<math>^{2}</math>s<math>
^{-2}.<math> The water density was 1025kgm</math>^{-3}</math> and the incoming wave was
chosen to have wavelength <math>100</math>m. The solution to this problem by our method
is shown in Figure (\ref{fig_mike_refl}). This figure is identical to Figure
3 in [[jgrfloe1d]] (p. 895) which establishes that our method gives the
correct result for a plate on water of constant depth.
==Reflection==
We will consider the absolute value of the reflection coefficient as a
function of wavenumber <math>\nu </math> for various values of the parameters. Figure (
\ref{plot4hump}) shows the absolute value of the reflection coefficient as a
function of <math>\nu </math> with <math>\beta =0.01,</math> <math>0.1,</math> 1 and 10 and <math>\gamma =0</math> for
the hump depth profile. Both the length of the hump and the length of the
plate were fixed to be <math>l=L=2.5</math>. The solution for the plate and hump (solid
line), plate with constant depth (dashed line) and hump only (dotted line)
are shown. The two simpler solutions are drawn so that the full solution may
be compared to these simpler cases. Figure (\ref{plot4hump}) shows the
existence of two asymptotic regimes. When <math>\nu </math> is small (low frequency or
large wavelength)\ the reflection is dominated by the hump and the plate is
transparent. For large <math>\nu </math> the reflection is dominated by the plate and
the hump is transparent. As the value of the stiffness <math>\beta </math> is increased
the hump dominated region becomes smaller and the plate dominated region
becomes larger. This not unexpected because increasing the value of <math>\beta </math>
increases the influence of the plate. It is apparent that, especially for
smaller values of stiffness, there is a large region where the hump and
plate solution is significantly different from the plate only solution, even
though the hump only reflection is practically zero. This is because the
wavelength under the plate is larger than the open water wavelength so the
depth variation is felt more strongly when the plate is present.
Figure (\ref{plot4plane}) is equivalent to Figure (\ref{plot4hump}) except
that the depth profile is the simple slope and the results are also very
similar<math>.</math> Figure (\ref{plot4hump_gamma}) is equivalent to Figure (\ref
{plot4hump}) except that the value of stiffness is fixed (<math>\beta =0.1)</math> and
the value of <math>\gamma </math> is varied. This figure shows that for realistic
(small) values of <math>\gamma </math> this parameter is not significant. This explains
why <math>\gamma </math> is often neglected (e.g. [[OhkusuISOPE]]) and why we have
chosen <math>\gamma =0</math> for Figures (\ref{plot4hump}) and (\ref{plot4plane}).
Figure (\ref{plot4ahump}) is equivalent to figure (\ref{plot4hump}) except
that the hump has been moved by <math>L</math> to the left so that the minimum depth is
directly underneath the left (incoming) end of the plate. Comparing figure (
\ref{plot4ahump}) with figure (\ref{plot4ahump}) we see that moving the hump
has made a significant change to the reflection coefficient for low
frequencies, especially as the stiffness <math>\beta </math> is increased.
==Displacements==
Finally we investigate the displacement of the plate for some of the regimes
we have considered. We present the displacement for the variable and
constant depth profiles so that we may compare the effect of the variable
depth. We divide the displacement by <math>i\omega /k_{t}^{\left( 1\right) }\sinh
\left( k_{t}^{\left( 1\right) }\right) </math> so that the incoming wave now has
unit amplitude in displacement at the water surface. Figure (\ref
{deflbeta4hump_nu05}) shows the displacement of the plate for <math>\nu =0.5</math> and
<math>\beta =0.01,</math> <math>0.1,</math> 1 and 10 and <math>\gamma =0.</math> The plate length is <math>L=2.5</math>.
The solid line is the real part of the displacement and the dotted line is
the imaginary part of the displacement. The thicker line is the solution
with the hump depth profile (<math>l=2.5)</math> and the thinner line is the solution
with a constant depth. It is apparent from this figure, that the bending of
the plate is increased by the presence of the hump, but that this effect is
not very strong. Figures (\ref{deflbeta4hump_nu1}) and (\ref
{deflbeta4hump_nu2}) are identical to Figure (\ref{deflbeta4hump_nu05})
except that <math>v=1</math> and <math>\nu =2</math> respectively. These figures also show only a
slight increase in the bending of the plate due to the hump. It appears
that, while the variable depth does have a significant effect on the
reflection coefficient, the effect on the plate displacement is not
necessarily as strong.
=Summary=
We have presented a solution for the linear wave forcing of a floating thin
plate on water of variable depth. The solution method was based on reducing
the problem to a finite domain which contained both the region of variable
water depth and the floating thin plate. In this finite region the outward
normal derivative of the potential around the boundary was expressed as a
function of the potential. This was accomplished by using integral operators
for the boundary under the plate and the radiating boundaries. The integral
operator for the plate was calculated using a Green function as described in
[[jgrfloe1d]]. The integral operators for the radiation boundary
conditions were calculated by solving Laplace's equation in the
semi-infinite outer domains using separation of variables as described in
[[Liu82, Hazard]]. Laplace's equation in the finite domain was solved
using the boundary element method.
The results showed that, for certain parameter regimes, there was
significant difference between the absolute value of the reflection
coefficient for the variable depth and constant depth profiles. Furthermore,
the region of influence of the variable depth was increased by the presence
of the plate due to the increased wavelength under the plate. Finally, there
was a slight increase in the bending of the plate due to the presence of the
variable depth profile.
\bibliographystyle{IEEE}
\bibliography{mike,others}
\pagebreak
=Tables=
\begin{table}[h] \centering
(35)
\begin{tabular}{lll}
<math>\nu =1</math> &  &  \\ \hline
\multicolumn{1}{|c}{<math>\kappa </math>} & \multicolumn{1}{|c}{plate only} &
\multicolumn{1}{|c|}{plate and hump} \\ \hline
\multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.2983746166} &
\multicolumn{1}{|c|}{0.2491046427} \\ \hline
\multicolumn{1}{|c}{20} & \multicolumn{1}{|c}{0.2957175612} &
\multicolumn{1}{|c|}{0.2470511349} \\ \hline
\multicolumn{1}{|c}{40} & \multicolumn{1}{|c}{0.2948480461} &
\multicolumn{1}{|c|}{0.2465182527} \\ \hline
<math>\nu =2</math> &  &  \\ \hline
\multicolumn{1}{|c}{<math>\kappa </math>} & \multicolumn{1}{|c}{plate only} &
\multicolumn{1}{|c|}{plate and hump} \\ \hline
\multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.3457203891} &
\multicolumn{1}{|c|}{0.1934227806} \\ \hline
\multicolumn{1}{|c}{20} & \multicolumn{1}{|c}{0.3461627544} &
\multicolumn{1}{|c|}{0.1947144005} \\ \hline
\multicolumn{1}{|c}{40} & \multicolumn{1}{|c}{0.3463681703} &
\multicolumn{1}{|c|}{0.1952205755} \\ \hline
<math>\nu =3</math> &  &  \\ \hline
\multicolumn{1}{|c}{<math>\kappa </math>} & \multicolumn{1}{|c}{plate only} &
\multicolumn{1}{|c|}{plate and hump} \\ \hline
\multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.0277944414} &
\multicolumn{1}{|c|}{0.2605833203} \\ \hline
\multicolumn{1}{|c}{20} & \multicolumn{1}{|c}{0.0249319083} &
\multicolumn{1}{|c|}{0.2568361963} \\ \hline
\multicolumn{1}{|c}{40} & \multicolumn{1}{|c}{0.0236686063} &
\multicolumn{1}{|c|}{0.2550926942} \\ \hline
\end{tabular}
\caption{<math>\left|R\right|</math> for <math>\kappa</math> = 10, 20, and 40 and
<math>\nu </math> = 1,2, and 3 for the plate only and the plate and hump.
<math>\beta=1</math>, <math>\gamma =0</math> and 5 evanscent modes are
used. (36)}
\end{table}
\pagebreak
\begin{table}[t] \centering
(37)
\begin{tabular}{ccc}
<math>\nu =1</math> &  &  \\ \hline
\multicolumn{1}{|c}{<math>N</math>} & \multicolumn{1}{|c}{plate only} &
\multicolumn{1}{|c|}{plate and hump} \\ \hline
\multicolumn{1}{|c}{0} & \multicolumn{1}{|c}{0.2934754434} &
\multicolumn{1}{|c|}{0.2448842047} \\ \hline
\multicolumn{1}{|c}{5} & \multicolumn{1}{|c}{0.2957175612} &
\multicolumn{1}{|c|}{0.2470511349} \\ \hline
\multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.2957697025} &
\multicolumn{1}{|c|}{0.2470814786} \\ \hline
<math>\nu =2</math> &  &  \\ \hline
\multicolumn{1}{|c}{<math>N</math>} & \multicolumn{1}{|c}{plate only} &
\multicolumn{1}{|c|}{plate and hump} \\ \hline
\multicolumn{1}{|c}{0} & \multicolumn{1}{|c}{0.3392954189} &
\multicolumn{1}{|c|}{0.2021691049} \\ \hline
\multicolumn{1}{|c}{5} & \multicolumn{1}{|c}{0.3461627544} &
\multicolumn{1}{|c|}{0.1947144005} \\ \hline
\multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.3459741914} &
\multicolumn{1}{|c|}{0.1944525347} \\ \hline
<math>\nu =3</math> &  &  \\ \hline
\multicolumn{1}{|c}{<math>N</math>} & \multicolumn{1}{|c}{plate only} &
\multicolumn{1}{|c|}{plate and hump} \\ \hline
\multicolumn{1}{|c}{0} & \multicolumn{1}{|c}{0.0262065602} &
\multicolumn{1}{|c|}{0.2296877448} \\ \hline
\multicolumn{1}{|c}{5} & \multicolumn{1}{|c}{0.0249319083} &
\multicolumn{1}{|c|}{0.2568361963} \\ \hline
\multicolumn{1}{|c}{10} & \multicolumn{1}{|c}{0.0259483771} &
\multicolumn{1}{|c|}{0.2578144528} \\ \hline
\end{tabular}
\caption{<math>\left|R\right|</math> for 0, 5, and 10 evanescent modes (<math>N</math>)
and <math>\nu </math> = 1,2, and 3 for the plate only and the plate and hump.
<math>\beta=1</math>, <math>\gamma =0</math> and
<math>\kappa = 20</math>.(38)}
\end{table}




Revision as of 08:52, 15 August 2006

A problem in which the scattering comes from a variation in the bottom topography.

Wave scattering by a Floating Elastic Plate on water of Variable Bottom Topography was treated in Wang and Meylan 2002 and is described in Floating Elastic Plate on Variable Bottom Topography

Retrieved from "https://www.wikiwaves.org/index.php?title=Variable_Bottom_Topography&oldid=3355"