Kagemoto and Yue Interaction Theory: Difference between revisions

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= Introduction =
This is an interaction theory which provides the exact solution (i.e. it is not based on a [[Wide Spacing Approximation]]).
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 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]]. [[Interaction Theory for Cylinders]] presents a simplified version for cylinders.  


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 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 coordinates 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
The theory is described in [[Kagemoto and Yue 1986]] and in
[[Peter and Meylan 2004]].
[[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:Linear Water-Wave Theory]]
[[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>h</math>,
while <math>\mathbf{x}</math> always denotes a point of the undisturbed water
surface assumed at <math>z=0</math>.


{{standard linear wave scattering equations}}


The [[Sommerfeld Radiation Condition]] is also imposed.


=Eigenfunction expansion of the potential=


We extend the finite depth interaction theory of [[kagemoto86]] to
Each body is subject to an incident potential and moves in response to this
water of infinite depth and bodies of arbitrary geometry. The sum
incident potential to produce a scattered potential. Each of these is
over the discrete roots of the dispersion equation in the finite depth
expanded using the [[Cylindrical Eigenfunction Expansion]]
theory becomes
The scattered potential of a body
an integral in the infinite depth theory. This means that the infinite
<math>\Delta_j</math> can be expressed as
dimensional diffraction
<center><math>
transfer matrix
\phi_j^\mathrm{S} (r_j,\theta_j,z) =
in the finite depth theory must be replaced by an integral
\sum_{m=0}^{\infty} f_m(z) \sum_{\mu = -
operator. In the numerical solution of the equations, this
\infty}^{\infty} A_{m \mu}^j K_\mu (k_m r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j},
integral operator is approximated by a sum and a linear system
</math></center>
of equations is obtained. We also show how the calculations
with discrete coefficients <math>A_{m \mu}^j</math>, where <math>(r_j,\theta_j,z)</math>
of the diffraction transfer matrix for bodies of arbitrary
are cylindrical polar coordinates centered at each body
geometry developed by [[goo90]] can be extended to
<center><math>
infinite depth, and how the diffraction transfer matrix for rotated bodies can
f_m(z) = \frac{\cos k_m (z+h)}{\cos k_m h}.
be easily calculated. This interaction theory is applied to the wave forcing
</math></center>
of multiple ice floes and a method to solve
where <math>k_m</math> are found from <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]
the full diffraction problem in this case is presented. Convergence
<center><math>
studies comparing the interaction method with the full diffraction
\alpha + k_m \tan k_m h = 0\,.
calculations and the finite and infinite depth interaction methods are
</math></center>
carried out.
where <math>k_0</math> is the
imaginary root with negative 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,  
==Introduction==
<center><math>
The scattering of water waves by floating or submerged
\phi_j^\mathrm{I} (r_j,\theta_j,z) = \sum_{n=0}^{\infty} f_n(z)
bodies is of wide practical importance.
\sum_{\nu = - \infty}^{\infty} D_{n\nu}^j I_\nu (k_n r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j},
Although the problem is non-linear, if the
wave amplitude is sufficiently small, the
problem can be linearised. The linear problem is still the basis of
the engineering design of most off-shore structures and is the standard
model of geophysical phenomena such as the wave forcing of ice floes.
While analytic solutions have been found for simplified problems
(especially for simple geometries or in two dimensions) the full
three-dimensional linear diffraction problem can only be solved by
numerical methods involving the discretisation of the body's surface.
The resulting linear system of equations has a dimension equal to the
number of unknowns used in the discretisation of the body.
 
If more than one body is present, all bodies scatter the
incoming waves. Therefore, the scattered wave from one body is
incident upon all the others and, given that they are not too far apart,
this notably changes the total incident wave upon them.
Therefore, the diffraction calculation must be conducted
for all bodies simultaneously. Since each body must be discretised this
can lead to a very large number of unknowns. However, the scattered
wavefield can be represented in an eigenfunction basis with a
comparatively small number of unknowns. If we can express the problem in this
basis, using what is known as an {\em Interaction Theory},
the number of unknowns can be much reduced, especially if there is a large
number of bodies.
 
The first interaction theory that was not based on an approximation was
the interaction theory
developed by [[kagemoto86]]. Kagemoto and Yue found an exact
algebraic method to solve the linear wave problem for vertically
non-overlapping bodies in water of finite depth.  The only restriction
of their theory was that the smallest escribed circle for each body must not
overlap any other body. The interaction of the bodies was
accounted for by taking the scattered wave of each body to be the
incident wave upon all other bodies (in addition to the ambient
incident wave). Furthermore, since the cylindrical eigenfunction expansions
are local, these were mapped from one body to another using
Graf's addition theorem for Bessel functions.
Doing this for all bodies,
\citeauthor{kagemoto86} were able to solve for the
coefficients of the scattered wavefields of all bodies simultaneously.
The only difficulty with this method was that the
solutions of the single diffraction problems had to be available in
the cylindrical eigenfunction expansion of an outgoing
wave. \citeauthor{kagemoto86} therefore only solved for axisymmetric
bodies because the single diffraction solution for axisymmetric
bodies was
available in the required representation. 
 
The extension of the Kagemoto and Yue scattering theory to bodies of
arbitrary geometry was performed by [[goo90]] who found a general
way to solve the single diffraction problem in the required
cylindrical eigenfunction representation. They used a
representation of the finite depth free surface Green's function in
the eigenfunction expansion of cylindrical outgoing waves
centred at an arbitrary point of the water surface (above the
body's mean centre position in this case). This Green's function was
presented by
[[black75]] and further investigated by [[fenton78]]
who corrected some statements about the Green's function which Black had
made. This Green's function is
based on the cylindrical
eigenfunction expansion of the finite depth free surface Green's
function given by [[john2]]. The results of
\citeauthor{goo90} were recently used by [[chakrabarti00]] to
solve for arrays of cylinders which can be divided into modules.
 
The development of the Kagemoto and Yue interaction theory was
motivated by problems
in off-shore engineering. However, the theory can also be applied
to the geophysical problem of wave scattering by ice floes. At the
interface of the open and frozen ocean an interfacial region known
as the Marginal Ice Zone (MIZ) forms. The MIZ largely controls
the interaction of the open and frozen ocean, especially the interaction
through wave processors. This is because
the MIZ consists of vast fields
of ice floes whose size is comparable to the dominant wavelength, which
means that it strongly scatters incoming waves. A method of
solving for the wave response of a single ice floe of arbitrary
geometry in water of infinite depth was presented by
[[JGR02]]. The ice floe was modelled as a floating, flexible
thin plate and its motion was expanded in the free plate modes of
vibration. Converting the problem for the water into an integral
equation and substituting the free modes, a system of equations for
the coefficients in the modal expansion was obtained. However,
to understand wave propagation and scattering in the MIZ we need to
understand the way in which large numbers of interacting
ice floes scatter waves. For this reason, we require an interaction
theory. While the Kagemoto and Yue interaction theory could be used,
their theory requires that the water depth is finite.
While the water depth in the Marginal Ice Zone varies,
it is generally located far from shore above the deep ocean. This means
that the finite depth must be chosen large in order to be able to
apply their theory.  Furthermore, when ocean waves propagate beneath
an ice floe the wavelength is increased so that it becomes more
difficult to make the
water depth sufficiently deep that it may be approximated as infinite. For this
reason, in this paper we develop  the equivalent interaction theory to
Kagemoto and Yue's in infinite depth. Also, because of
the complicated geometry of an ice floe, this interaction theory is
for bodies of arbitrary geometry. 
 
In the first part of this paper Kagemoto and Yue's interaction theory
is extended to water of infinite depth. We represent the incident and
scattered potentials in the cylindrical eigenfunction expansions
and we use an analogous infinite depth Green's function
to the one used by \citeauthor{goo90} \cite[given by][]{malte03}. We
show how the infinite
depth diffraction transfer matrices can be obtained with the use of
this Green's function and we illustrate how the rotation of a body
about its mean centre position in the plane can be accounted for without
recalculating the diffraction transfer matrix.
 
In the second part of the paper, using \citeauthor{JGR02}'s single
floe result,
the full diffraction calculation for the motion and scattering from many
interacting ice floes is calculated and presented. For two square
interacting ice floes the convergence of the method obtained from the
developed interaction theory is compared to the result of the full
diffraction calculation. The solutions of more than two interacting
ice floes and of other shapes in different arrangements are presented as well.
We also compare the convergence of the finite depth and infinite
depth methods in
deep water.
 
\section{The extension of Kagemoto and Yue's interaction
theory to bodies of arbitrary shape in water of infinite depth}
 
[[kagemoto86]] developed an interaction theory for
vertically non-overlapping axisymmetric structures in water of finite
depth. While their theory was valid for bodies of
arbitrary geometry, they did not develop all the necessary
details to apply the theory to arbitrary bodies.
The only requirements to apply this scattering theory is
that the bodies are vertically non-overlapping and
that the smallest cylinder which completely contains each body does not
intersect with any other body.
In this section we will extend their theory to bodies of
arbitrary geometry in water of infinite depth. The extension of
\citeauthor{kagemoto86}'s finite depth interaction theory to bodies of
arbitrary geometry was accomplished by [[goo90]].
 
 
The interaction theory begins by representing the scattered potential
of each body in the cylindrical eigenfunction expansion. Furthermore,
the incoming potential is also represented in the cylindrical
eigenfunction expansion. The operator which maps the incoming and
outgoing representation is called the diffraction transfer matrix and
is different for each body.
Since these representations are local to each body, a mapping of
the eigenfunction representations between different bodies
is required. This operator is called the coordinate transformation
matrix.
 
The cylindrical eigenfunction expansions will be introduced before we
derive a system of
equations for the coefficients of the scattered wavefields. Analogously to
[[kagemoto86]], we represent the scattered wavefield of
each body as an incoming wave upon all other bodies. The addition of
the ambient incident wave yields the complete incident potential and
with the use of diffraction transfer matrices which relate the
coefficients of the incident potential to those of the scattered
wavefield a system of equations for the unknown coefficients of the
scattered wavefields of all bodies is derived.
 
 
===Eigenfunction expansion of the potential===
The equations of motion for the water are derived from the linearised
inviscid theory. Under the assumption of irrotational motion the
velocity vector field of the water can be written as the gradient
field of a scalar velocity potential <math>\Phi</math>. Assuming that the motion
is time-harmonic with the radian frequency <math>\omega</math> the
velocity potential can be expressed as the real part of a complex
quantity,
<center><math> (time)
\Phi(\mathbf{y},t) = \Re \{\phi (\mathbf{y}) \mathrm{e}^{-\mathrm{i}\omega t} \}.
</math></center>
</math></center>
To simplify notation, <math>\mathbf{y} = (x,y,z)</math> will always denote a point
with discrete coefficients <math>D_{n\nu}^j</math>. In these expansions, <math>I_\nu</math>
in the water, which is assumed infinitely deep, while <math>\mathbf{x}</math> will
and <math>K_\nu</math> denote the modified [http://en.wikipedia.org/wiki/Bessel_function : Bessel functions]
always denote a point of the undisturbed water surface assumed at <math>z=0</math>.
of the first and second kind, respectively, both of order <math>\nu</math>.


The problem consists of <math>N</math> vertically non-overlapping bodies, denoted
Note that the term for <math>m =0</math> or
by <math>\Delta_j</math>, which are sufficiently far apart that there is no
<math>n=0</math> corresponds to the propagating modes while the  
intersection of the smallest cylinder which contains each body with
terms for <math>m\geq 1</math> (<math>n\geq 1</math>) correspond to the evanescent modes.
any other body. Each body is subject to an incident wavefield which is
incoming, responds to this wavefield and produces a scattered wave field which
is outgoing. Both the incident and scattered potential corresponding
to these wavefields can be represented in the cylindrical
eigenfunction expansion valid outside of the escribed cylinder of the
body. Let <math>(r_j,\theta_j,z)</math> be the local cylindrical coordinates of
the <math>j</math>th body, <math>\Delta_j</math>, <math>j \in \{1, \ldots , N\}</math>, and </math>\alpha =
\omega^2/g<math> where </math>g<math> is the acceleration due to gravity. Figure
(fig:floe_tri) shows these coordinate systems for two bodies.


\begin{figure}
=Derivation of the system of equations=
\begin{center}
\includegraphics[height=5.5cm]{floe_tri}
\caption{Plan view of the relation between two bodies.} (fig:floe_tri)
\end{center}
\end{figure}


The scattered potential of body <math>\Delta_j</math> can be expanded in
A system of equations for the unknown  
cylindrical eigenfunctions,
coefficients of the
<center><math> (basisrep_out)
scattered wavefields of all bodies is developed. This system of
\phi_j^\mathrm{S} (r_j,\theta_j,z) &= \mathrm{e}^{\alpha z} \sum_{\nu = -
equations is based on transforming the  
\infty}^{\infty} A_{0 \nu}^j H_\nu^{(1)} (\alpha r_j) \mathrm{e}^{\mathrm{i}\nu
\theta_j}\\
&\quad + \int\limits_0^{\infty} \big( \cos \eta z + \frac{\alpha}{\eta}
\sin \eta z \big) \sum_{\nu = -
\infty}^{\infty} A_{\nu}^j (\eta) K_\nu (\eta r_j) \mathrm{e}^{\mathrm{i}\nu
\theta_j} \mathrm{d}\eta,
</math></center>
where the coefficients <math>A_{0 \nu}^j</math> for the propagating modes are
discrete and the coefficients <math>A_{\nu}^j (\cdot)</math> for the decaying
modes are functions. <math>H_\nu^{(1)}</math> and <math>K_\nu</math> are the Hankel function
of the first kind and the modified Bessel function of the second kind
respectively, both of order <math>\nu</math> as defined in [[abr_ste]].
The incident potential upon body <math>\Delta_j</math> can be expanded in
cylindrical eigenfunctions,
<center><math> (basisrep_in)
\phi_j^\mathrm{I} (r_j,\theta_j,z) &= \mathrm{e}^{\alpha z} \sum_{\mu = -
\infty}^{\infty} D_{0 \mu}^j J_\mu (\alpha r_j) \mathrm{e}^{\mathrm{i}\mu
\theta_j}\\
& \quad + \int\limits_0^{\infty} \big( \cos \eta z + \frac{\alpha}{\eta}
\sin \eta z \big) \sum_{\mu = -
\infty}^{\infty} D_{\mu}^j (\eta) I_\mu (\eta r_j) \mathrm{e}^{\mathrm{i}\mu
\theta_j} \mathrm{d}\eta,
</math></center>
where the coefficients <math>D_{0 \mu}^j</math> for the propagating modes are
discrete and the coefficients <math>D_{\mu}^j (\cdot)</math> for the decaying
modes are functions. <math>J_\mu</math> and <math>I_\mu</math> are the Bessel function and
the modified Bessel function respectively, both of the first kind and
order <math>\mu</math>. To simplify the notation, from now on <math>\psi(z,\eta)</math> will
denote the vertical eigenfunctions corresponding to the decaying modes,
<center><math>
\psi(z,\eta) = \cos \eta z + \alpha / \eta \, \sin \eta z.
</math></center>
 
===The interaction in water of infinite depth===
Following the ideas of [[kagemoto86]], a system of equations for the unknown
coefficients and coefficient functions of the scattered wavefields
will be developed. This system of equations is based on transforming the
scattered potential of <math>\Delta_j</math> into an incident potential upon
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,
<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
and relating the incident and scattered potential for each body, a system
of equations for the unknown coefficients will be developed.  
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
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>
represented in terms of the incident potential <math>\phi_l^{\mathrm{I}}</math>
upon <math>\Delta_l</math>, <math>j \neq l</math>. From figure
upon <math>\Delta_l</math>, <math>j \neq l</math>. This can be accomplished by using
(fig:floe_tri) we can see that this can be accomplished by using
[[Graf's Addition Theorem]]
Graf's addition theorem for Bessel functions given in
<center><math>
\citet[eq. 9.1.79]{abr_ste},
K_\tau(k_m r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
<center><math> (transf)
\sum_{\nu = - \infty}^{\infty} K_{\tau + \nu} (k_m R_{jl}) \,
<center><math>\begin{matrix} (transf_h)
I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l,
H_\nu^{(1)}(\alpha r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} &=
\sum_{\mu = - \infty}^{\infty} H^{(1)}_{\nu + \mu} (\alpha R_{jl}) \,
J_\mu (\alpha r_l) \mathrm{e}^{\mathrm{i}\mu (\pi - \theta_l + \vartheta_{jl})},
\quad j \neq l,\\
(transf_k)
K_\nu(\eta r_j) \mathrm{e}^{\mathrm{i}\nu (\theta_j - \vartheta_{jl})} &= \sum_{\mu = -
\infty}^{\infty} K_{\nu + \mu} (\eta R_{jl}) \, I_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu
(\pi - \theta_l + \vartheta_{jl})}, \quad j \neq l,
\end{matrix}</math></center>
</math></center>
</math></center>
which is valid provided that <math>r_l < R_{jl}</math>. This limitation
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>.
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
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
expansion of the scattered and incident potential in cylindrical
eigenfunctions is only valid outside the escribed cylinder of each
eigenfunctions is only valid outside the escribed cylinder of each
Line 313: Line 107:
origin <math>O_j</math> (<math>j \neq l</math>) is superseded by the more rigorous
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
restriction that the escribed cylinder of each body may not contain any
other body. Making use of the equations  (transf)
other body.  
the scattered potential of <math>\Delta_j</math> can be expressed in terms of the
incident potential upon <math>\Delta_l</math>,
<center><math>\begin{matrix}
\phi_j^{\mathrm{S}} (r_l,\theta_l,z) &=
\mathrm{e}^{\alpha z} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
\sum_{\mu = -\infty}^{\infty} H_{\nu - \mu}^{(1)} (\alpha R_{jl})
J_\mu (\alpha r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l} \mathrm{e}^{\mathrm{i}(\nu-\mu)
\vartheta_{jl}}\\
& \quad + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\nu = -
\infty}^{\infty} A_{\nu}^j (\eta) \sum_{\mu = -\infty}^{\infty}
(-1)^\mu K_{\nu-\mu} (\eta R_{jl}) I_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu
\theta_l}  \mathrm{e}^{\mathrm{i}(\nu-\mu) \vartheta_{jl}} \mathrm{d}\eta\\
&= \mathrm{e}^{\alpha z} \sum_{\mu = -\infty}^{\infty} \Big[ \sum_{\nu = -
\infty}^{\infty} A_{0\nu}^j H_{\nu - \mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\i
(\nu-\mu) \vartheta_{jl}} \Big] J_\mu (\alpha r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}\\
& + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\mu = -\infty}^{\infty}
\Big[ \sum_{\nu = - \infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu-\mu}
(\eta R_{jl}) \mathrm{e}^{\mathrm{i}(\nu-\mu)
\vartheta_{jl}} \Big] I_\mu (\eta r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l} \mathrm{d}\eta.
\end{matrix}</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>. A detailed illustration of how to accomplish this will be given
later. Let <math>D_{l0\mu}^{\mathrm{In}}</math> denote the coefficients of this
ambient incident wavefield corresponding to the propagating modes and
<math>D_{l\mu}^{\mathrm{In}} (\cdot)</math>  denote the coefficients functions
corresponding to the decaying modes (which are identically zero) of
the incoming eigenfunction expansion for <math>\Delta_l</math>. The total
incident wavefield upon body <math>\Delta_j</math> can now be expressed as
<center><math>\begin{matrix}
&\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +
\sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \, \phi_j^{\mathrm{S}}
(r_l,\theta_l,z)\\
&= \mathrm{e}^{\alpha z} \sum_{\mu = -\infty}^{\infty} \Big[
D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j
\neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\i
(\nu - \mu) \vartheta_{jl}} \Big] J_\mu (\alpha r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}\\
& + \int\limits_0^{\infty} \psi (z,\eta) \sum_{\mu =
-\infty}^{\infty} \Big[  D_{l\mu}^{\mathrm{In}}(\eta) +
\sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq  l}}^{N} \sum_{\nu =
-\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu}  (\eta
R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}} \Big] I_\mu (\eta r_l)
\mathrm{e}^{\mathrm{i}\mu \theta_l} \mathrm{d}\eta.
\end{matrix}</math></center>
The coefficients of the total incident potential upon <math>\Delta_l</math> are
therefore given by
<center><math> (inc_coeff)
<center><math>\begin{matrix}
D_{0\mu}^l &= D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j
\neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\i
(\nu - \mu) \vartheta_{jl}},\\
D_{\mu}^l(\eta) &= D_{l\mu}^{\mathrm{In}}(\eta) +
\sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq  l}}^{N} \sum_{\nu =
-\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu}  (\eta
R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}}.
\end{matrix}</math></center>
</math></center>
 
In general, it is possible to relate the total incident and scattered
partial waves for any body through the diffraction characteristics of
that body in isolation. There exist diffraction transfer operators
<math>B_l</math> that relate the coefficients of the incident and scattered
partial waves, such that
<center><math> (eq_B)
A_l = B_l (D_l), \quad l=1, \ldots, N,
</math></center>
where <math>A_l</math> are the scattered modes due to the incident modes <math>D_l</math>.
In the case of a countable number of modes, (i.e. when
the depth is finite), <math>B_l</math> is an infinite dimensional matrix. When
the modes are functions of a continuous variable (i.e. infinite
depth), <math>B_l</math> is the kernel of an integral operator.
For the propagating and the decaying modes respectively, the scattered
potential can be related by diffraction transfer operators acting in the
following ways,
<center><math> (diff_op)
<center><math>\begin{matrix}
A_{0\nu}^l &= \sum_{\mu = -\infty}^{\infty} B_{l\nu\mu}^\mathrm{pp} D_{0\mu}^l
+ \int\limits_{0}^{\infty} \sum_{\mu = -\infty}^{\infty}
B_{l\nu\mu}^\mathrm{pd} (\xi) D_{\mu}^l (\xi) \mathrm{d}\xi,\\
A_\nu^l (\eta) &= \sum_{\mu = -\infty}^{\infty}
B_{l\nu\mu}^\mathrm{dp} (\eta) D_{0\mu}^l + \int\limits_{0}^{\infty}
\sum_{\mu = -\infty}^{\infty} B_{l\nu\mu}^\mathrm{dd} (\eta;\xi)
D_{\mu}^l (\xi) \mathrm{d}\xi.
\end{matrix}</math></center>
</math></center>
The superscripts <math>\mathrm{p}</math> and <math>\mathrm{d}</math> are used to distinguish
between propagating and decaying modes, the first superscript denotes the kind
of scattered mode, the second one the kind of incident mode.
If the diffraction transfer operators are known (their calculation
will be discussed later), the substitution of
equations  (inc_coeff) into equations  (diff_op) give the
required equations to determine the coefficients and coefficient
functions of the scattered wavefields of all bodies,
<center><math> (eq_op)
<center><math>\begin{matrix}
&\begin{aligned}
&A_{0n}^l = \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{pp}
\Big[ D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j
\neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\i
(\nu - \mu) \vartheta_{jl}} \Big]\\
& \ + \int\limits_{0}^{\infty} \sum_{\mu = -\infty}^{\infty}
B_{ln\mu}^\mathrm{pd} (\xi) \Big[D_{l\mu}^{\mathrm{In}}(\eta) +
\sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq  l}}^{N} \sum_{\nu =
-\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu}  (\eta
R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}} \Big] \mathrm{d}\xi,
\end{aligned}\\
&\begin{aligned}
&A_n^l (\eta) = \sum_{\mu = -\infty}^{\infty}
B_{ln\mu}^\mathrm{dp} (\eta) \Big[
D_{l0\mu}^{\mathrm{In}} + \sum_{\genfrac{}{}{0pt}{}{j=1}{j
\neq l}}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\i
(\nu - \mu) \vartheta_{jl}}\Big]\\
& \ + \int\limits_{0}^{\infty}
\sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{dd} (\eta;\xi)
\Big[ D_{l\mu}^{\mathrm{In}}(\eta) +
\sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq  l}}^{N} \sum_{\nu =
-\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu}  (\eta
R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}}\Big] \mathrm{d}\xi,
\end{aligned}
\end{matrix}</math></center>
</math></center>
<math>n \in \mathds{Z},\, l = 1, \ldots, N</math>. It has to be noted that all
equations are coupled so that it is necessary to solve for all
scattered coefficients and coefficient functions simultaneously.  


For numerical calculations, the infinite sums have to be truncated and
Making use of the eigenfunction expansion as well as [[Graf's Addition Theorem]], the scattered potential
the integrals must be discretised. Implying a suitable truncation, the
of <math>\Delta_j</math> can be expressed in terms of the
four different diffraction transfer operators can be represented by
incident potential upon <math>\Delta_l</math> as
matrices which can be assembled in a big matrix <math>\mathbf{B}_l</math>,
<center><math>
<center><math>
\mathbf{B}_l = \left[
\phi_j^{\mathrm{S}} (r_l,\theta_l,z)  
\begin{matrix}{cc} \mathbf{B}_l^{\mathrm{pp}} & \mathbf{B}_l^{\mathrm{pd}}\\
= \sum_{m=0}^\infty f_m(z) \sum_{\tau = -
\mathbf{B}_l^{\mathrm{dp}} & \mathbf{B}_l^{\mathrm{dd}}
\infty}^{\infty} A_{m\tau}^j \sum_{\nu = -\infty}^{\infty}
\end{matrix} \right],
(-1)^\nu K_{\tau-\nu} (k_m R_{jl}) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu
</math></center>
\theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}}  
the infinite depth diffraction transfer matrix.
Truncating the coefficients accordingly, defining <math>{\bf a}^l</math> to be the
vector of the coefficients of the scattered potential of body
<math>\Delta_l</math>, <math>\mathbf{d}_l^{\mathrm{In}}</math> to be the vector of
coefficients of the ambient wavefield, and making use of a coordinate
transformation matrix <math>{\bf T}_{jl}</math> given by
<center><math> (T_elem_deep)
<center><math>\begin{matrix}
({\bf T}_{jl})_{pq} &= H_{p-q}^{(1)}(\alpha R_{jl}) \, \mathrm{e}^{\mathrm{i}(p-q)
\vartheta_{jl}}\\
=for the propagating modes, and=
({\bf T}_{jl})_{pq} &= (-1)^{q} \, K_{p-q} (\eta R_{jl}) \, \mathrm{e}^{\i
(p-q) \vartheta_{jl}}
\end{matrix}</math></center>
</math></center>
for the decaying modes, a linear system of equations
for the unknown coefficients follows from equations  (eq_op),
<center><math> (eq_B_inf)
{\bf a}_l = {\bf \hat{B}}_l \Big( {\bf d}_l^{\mathrm{In}} +
\sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \trans {\bf T}_{jl} \,
{\bf a}_j \Big), \quad  l=1, \ldots, N,
</math></center>
where the left superscript <math>\mathrm{t}</math> indicates transposition.
The matrix <math>{\bf \hat{B}}_l</math> denotes the infinite depth diffraction
transfer matrix <math>{\bf B}_l</math> in which the elements associated with
decaying scattered modes have been multiplied with the appropriate
integration weights depending on the discretisation of the continuous variable.
 
 
 
\subsection{Calculation of the diffraction transfer matrix for bodies
of arbitrary geometry}
 
Before we can apply the interaction theory we require the diffraction
transfer matrices <math>\mathbf{B}_j</math> which relate the incident and the
scattered potential for a body <math>\Delta_j</math> in isolation.
The elements of the diffraction transfer matrix, <math>({\bf B}_j)_{pq}</math>,
are the coefficients of the
<math>p</math>th partial wave of the scattered potential due to a single
unit-amplitude incident wave of mode <math>q</math> upon <math>\Delta_j</math>.
 
While \citeauthor{kagemoto86}'s interaction theory was valid for
bodies of arbitrary shape, they did not explain how to actually obtain the
diffraction transfer matrices for bodies which did not have an axisymmetric
geometry. This step was performed by [[goo90]] who came up with an
explicit method to calculate the diffraction transfer matrices for bodies of
arbitrary geometry in the case of finite depth. Utilising a Green's
function they used the standard
method of transforming the single diffraction boundary-value problem
to an integral equation for the source strength distribution function
over the immersed surface of the body.
However, the representation of the scattered potential which
is obtained using this method is not automatically given in the
cylindrical eigenfunction
expansion. To obtain such cylindrical eigenfunction expansions of the
potential [[goo90]] used the representation of the free surface
finite depth Green's function given by [[black75]] and
[[fenton78]].  \citeauthor{black75} and
\citeauthor{fenton78}'s representation of the Green's function was based
on applying Graf's addition theorem to the eigenfunction
representation of the free surface finite depth Green's function given
by [[john2]]. Their representation allowed the scattered potential to be
represented in the eigenfunction expansion with the cylindrical
coordinate system fixed at the point of the water surface above the
mean centre position of the body.
 
It should be noted that, instead of using the source strength distribution
function, it is also possible to consider an integral equation for the
total potential and calculate the elements of the diffraction transfer
matrix from the solution of this integral equation.
An outline of this method for water of finite
depth is given by [[kashiwagi00]]. We will present
here a derivation of the diffraction transfer matrices for the case
infinite depth based on a solution
for the source strength distribution function. However,
an equivalent derivation would be possible based on the solution
for the total velocity potential.
 
To calculate the diffraction transfer matrix in infinite depth, we
require the representation of the infinite depth free surface Green's
function in cylindrical eigenfunctions,
<center><math> (green_inf)
G(r,\theta,z;s,\varphi,c) &= \frac{\mathrm{i}\alpha}{2} \,  \mathrm{e}^{\alpha (z+c)}
\sum_{\nu=-\infty}^{\infty} H_\nu^{(1)}(\alpha r) J_\nu(\alpha s) \mathrm{e}^{\mathrm{i}\nu
(\theta - \varphi)}\\ &\quad + \frac{1}{\pi^2} \int\limits_0^{\infty}
\psi(z,\eta) \frac{\eta^2}{\eta^2+\alpha^2} \psi(c,\eta)
\sum_{\nu=-\infty}^{\infty} K_\nu(\eta r) I_\nu(\eta s) \mathrm{e}^{\mathrm{i}\nu
(\theta - \varphi)} \mathrm{d}\eta,
</math></center>
<math>r > s</math>, given by [[malte03]].
 
We assume that we have represented the scattered potential in terms of
the source strength distribution <math>\varsigma^j</math> so that the scattered
potential can be written as
<center><math> (int_eq_1)
\phi_j^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma_j} G
(\mathbf{y},\mathbf{\zeta}) \, \varsigma^j (\mathbf{\zeta})
\mathrm{d}\sigma_\mathbf{\zeta}, \quad \mathbf{y} \in D,
</math></center>
where <math>D</math> is the volume occupied by the water and <math>\Gamma_j</math> is the
immersed surface of body <math>\Delta_j</math>. The source strength distribution
function <math>\varsigma^j</math> can be found by solving an
integral equation. The integral equation is described in
[[Weh_Lait]] and numerical methods for its solution are outlined in
[[Sarp_Isa]].
Substituting the eigenfunction expansion of the Green's function
(green_inf) into  (int_eq_1), the scattered potential can
be written as
<center><math>\begin{matrix}
&\phi_j^\mathrm{S}(r_j,\theta_j,z) = \mathrm{e}^{\alpha z} \sum_{\nu = -
\infty}^{\infty} \bigg[ \frac{\mathrm{i}\alpha}{2}  
\int\limits_{\Gamma_j} \mathrm{e}^{\alpha c} J_\nu(\alpha s) \mathrm{e}^{-\mathrm{i}\nu
\varphi} \varsigma^j(\mathbf{\zeta})
\mathrm{d}\sigma_\mathbf{\zeta} \bigg] H_\nu^{(1)} (\alpha r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j}\\
& \, + \int\limits_{0}^{\infty} \psi(z,\eta) \sum_{\nu = -
\infty}^{\infty}  \bigg[ \frac{1}{\pi^2} \frac{\eta^2
}{\eta^2 + \alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_\nu(\eta s)
\mathrm{e}^{-\mathrm{i}\nu \varphi} \varsigma^j({\bf{\zeta}})
\mathrm{d}\sigma_{\bf{\zeta}} \bigg] K_\nu (\eta r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j} \mathrm{d}\eta,
\end{matrix}</math></center>
where
<math>\mathbf{\zeta}=(s,\varphi,c)</math> and <math>r>s</math>.
This restriction implies that the eigenfunction expansion is only valid
outside the escribed cylinder of the body.
 
The columns of the diffraction transfer matrix are the coefficients of
the eigenfunction expansion of the scattered wavefield due to the
different incident modes of unit-amplitude. The elements of the
diffraction transfer matrix of a body of arbitrary shape are therefore given by
<center><math> (B_elem)
<center><math>\begin{matrix}
({\bf B}_j)_{pq} &= \frac{\mathrm{i}\alpha}{2} \int\limits_{\Gamma_j}
\mathrm{e}^{\alpha c} J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta})
\mathrm{d}\sigma_\mathbf{\zeta}\\
=and= 
({\bf B}_j)_{pq} &= \frac{1}{\pi^2} \frac{\eta^2}{\eta^2 +
\alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p
\varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta}
\end{matrix}</math></center>
</math></center>
for the propagating and the decaying modes respectively, where
<math>\varsigma_q^j(\mathbf{\zeta})</math> is the source strength distribution
due to an incident potential of mode <math>q</math> of the form
<center><math> (test_modes_inf)
<center><math>\begin{matrix}
\phi_q^{\mathrm{I}}(s,\varphi,c) &=  \mathrm{e}^{\alpha c} H_q^{(1)} (\alpha
s) \mathrm{e}^{\mathrm{i}q \varphi}\\
=for the propagating modes, and= 
\phi_q^{\mathrm{I}}(s,\varphi,c) &= \psi(c,\eta) K_q (\eta s) \mathrm{e}^{\mathrm{i}q \varphi}
\end{matrix}</math></center>
</math></center>
</math></center>
for the decaying modes.
===The diffraction transfer matrix of rotated bodies===
For a non-axisymmetric body, a rotation about the mean
centre position in the <math>(x,y)</math>-plane will result in a
different diffraction transfer matrix. We will show how the
diffraction transfer matrix of a body rotated by an angle <math>\beta</math> can
be easily calculated from the diffraction transfer matrix of the
non-rotated body. The rotation of the body influences the form of the
elements of the diffraction transfer matrices in two ways. Firstly, the
angular dependence in the integral over the immersed surface of the
body is altered and, secondly, the source strength distribution
function is different if the body is rotated. However, the source
strength distribution function of the rotated body can be obtained by
calculating the response of the non-rotated body due to rotated
incident potentials. It will be shown that the additional angular
dependence can be easily factored out of the elements of the
diffraction transfer matrix.
The additional angular dependence caused by the rotation of the
incident potential can be factored out of the normal derivative of the
incident potential such that
<center><math>
<center><math>
\frac{\partial \phi_{q\beta}^{\mathrm{I}}}{\partial n} =
= \sum_{m=0}^\infty f_m(z) \sum_{\nu =
\frac{\partial \phi_{q}^{\mathrm{I}}}{\partial n}
-\infty}^{\infty}  \Big[ \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j
\mathrm{e}^{\mathrm{i}q \beta},
(-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>
</math></center>
where <math>\phi_{q\beta}^{\mathrm{I}}</math> is the rotated incident potential.
The ambient incident wavefield <math>\phi^{\mathrm{In}}</math> can also be
Since the integral equation for the determination of the source
expanded in the eigenfunctions corresponding to the incident wavefield upon
strength distribution function is linear, the source strength
<math>\Delta_l</math>. Let <math>\tilde{D}_{n\nu}^{l}</math> denote the coefficients of this
distribution function due to the rotated incident potential is thus just
ambient incident wavefield in the incoming eigenfunction expansion for
given by
<math>\Delta_l</math> (cf. the example in [[Cylindrical Eigenfunction Expansion]]).
<center><math>
<center><math>
\varsigma_{q\beta}^j = \varsigma_q^j \, \mathrm{e}^{\mathrm{i}q \beta}.
\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>
</math></center>
This is also the source strength distribution function of the rotated
The total
body due to the standard incident modes.
incident wavefield upon body <math>\Delta_j</math> can now be expressed as
 
The elements of the diffraction transfer matrix <math>\mathbf{B}_j</math> are
given by equations  (B_elem). Keeping in mind that the body is
rotated by the angle <math>\beta</math>, the elements of the diffraction transfer
matrix of the rotated body are given by
<center><math> (B_elem_rot)
<center><math>\begin{matrix}
(\mathbf{B}_j^\beta)_{pq} &= \frac{\mathrm{i}\alpha}{2} \int\limits_{\Gamma_j}
\mathrm{e}^{\alpha c} J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p (\varphi+\beta)}
\varsigma_{q\beta}^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta},\\
=and= 
(\mathbf{B}_j^\beta)_{pq} &= \frac{1}{\pi^2} \frac{\eta^2}{\eta^2 +
\alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p
(\varphi+\beta)} \varsigma_{q\beta}^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta},
\end{matrix}</math></center>
</math></center>
for the propagating and decaying modes respectively.
 
Thus the additional angular dependence caused by the rotation of
the body can be factored out of the elements of the diffraction
transfer matrix. The elements of the diffraction transfer matrix
corresponding to the body rotated by the angle <math>\beta</math>,
<math>\mathbf{B}_j^\beta</math>, are given by
<center><math> (B_rot)
(\mathbf{B}_j^\beta)_{pq} = (\mathbf{B}_j)_{pq} \, \mathrm{e}^{\mathrm{i}(q-p) \beta}.
</math></center>
As before, <math>(\mathbf{B})_{pq}</math> is understood to be the element of
<math>\mathbf{B}</math> which corresponds to the coefficient of the <math>p</math>th scattered
mode due to a unit-amplitude incident wave of mode <math>q</math>. Equation
(B_rot) applies to propagating and decaying modes likewise.
 
 
\subsection{Representation of the ambient wavefield in the eigenfunction
representation}
In Cartesian coordinates centred at the origin, the ambient wavefield is
given by
<center><math>
<center><math>
\phi^{\mathrm{In}}(x,y,z) = A \frac{g}{\omega} \, \mathrm{e}^{\mathrm{i}\alpha (x
\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +
\cos \chi + y \sin \chi)+ \alpha z},
\sum_{j=1,j \neq l}^{N} \, \phi_j^{\mathrm{S}}
(r_l,\theta_l,z)
</math></center>
</math></center>
where <math>A</math> is the amplitude (in displacement) and <math>\chi</math> is the
This allows us to write
angle between the <math>x</math>-axis and the direction in which the wavefield travels.
The interaction theory requires that the ambient wavefield, which is
incident upon
all bodies, is represented in the eigenfunction expansion of an
incoming wave in the local coordinates of the body. The ambient wave
can be represented in an eigenfunction expansion centred at the origin
as
<center><math>
<center><math>
\phi^{\mathrm{In}}(x,y,z) = A \frac{g}{\omega} \mathrm{e}^{\alpha z}
\sum_{n=0}^{\infty} f_n(z)
\sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 - \theta + \chi)}
\sum_{\nu = - \infty}^{\infty} D_{n\nu}^l I_\nu (k_n r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}
J_\mu(\alpha r)
</math></center>
</math></center>
\cite[p. 169]{linton01}.
Since the local coordinates of the bodies are centred at their mean
centre positions <math>O_l = (O_x^l,O_y^l)</math>, a phase factor has to be defined
which accounts for the position from the origin. Including this phase
factor the ambient wavefield at the <math>l</math>th body is given
by
<center><math>
<center><math>
\phi^{\mathrm{In}}(r_l,\theta_l,z) = A \frac{g}{\omega} \, e^{\mathrm{i}\alpha (O_x^l
= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
\cos \chi + O_x^l \sin \chi)} \, \mathrm{e}^{\alpha z}
\Big[  \tilde{D}_{n\nu}^{l} +
\sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 - \chi)}
\sum_{j=1,j \neq  l}^{N} \sum_{\tau =
J_\mu(\alpha r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}.
-\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>
</math></center>
 
It therefore follows that
===Solving the resulting system of equations===
After the coefficient vector of the ambient incident wavefield, the
diffraction transfer matrices and the coordinate
transformation matrices have been calculated, the system of
equations  (eq_B_inf),
has to be solved. This system can be represented by the following
matrix equation,
<center><math>
<center><math>
\left[ \begin{matrix}{c}
D_{n\nu}^l  =  
{\bf a}_1\\ {\bf a}_2\\ \\ \vdots \\ \\ {\bf a}_N
  \tilde{D}_{n\nu}^{l} +
\end{matrix} \right]
\sum_{j=1,j \neq  l}^{N} \sum_{\tau =
= \left[ \begin{matrix}{c}
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
\hat{{\bf B}}_1 {\bf d}_1^\mathrm{In}\\ \hat{{\bf B}}_2 {\bf
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}}  
d}_2^\mathrm{In}\\ \\ \vdots \\ \\ \hat{{\bf B}}_N {\bf d}_N^\mathrm{In}
\end{matrix} \right]+
\left[ \begin{matrix}{ccccc}
\mathbf{0} & \hat{{\bf B}}_1 \trans {\bf T}_{21} & \hat{{\bf B}}_1
\trans {\bf T}_{31} & \dots & \hat{{\bf B}}_1 \trans {\bf T}_{N1}\\
\hat{{\bf B}}_2 \trans {\bf T}_{12} & \mathbf{0} & \hat{{\bf B}}_2
\trans {\bf T}_{32} & \dots & \hat{{\bf B}}_2 \trans {\bf T}_{N2}\\
& & \mathbf{0} & &\\
\vdots & & & \ddots & \vdots\\
& & & & \\
\hat{{\bf B}}_N \trans {\bf T}_{1N} & & \dots &  
& \mathbf{0}
\end{matrix} \right]
\left[ \begin{matrix}{c}
{\bf a}_1\\ {\bf a}_2\\ \\ \vdots \\ \\ {\bf a}_N
\end{matrix} \right],
</math></center>
</math></center>
where <math>\mathbf{0}</math> denotes the zero-matrix which is of the same
dimension as <math>\hat{{\bf B}}_j</math>, say <math>n</math>. This matrix equation can be
easily transformed into a classical <math>(N \, n)</math>-dimensional linear system of
equations.


= Final Equations =


==Finite Depth Interaction Theory==
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,
We will compare the performance of the infinite depth interaction theory
<center><math>
with the equivalent theory for finite
A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
depth. As we have stated previously, the finite depth theory was
\mu \nu}^l D_{n\nu}^l.
developed by [[kagemoto86]] and extended to bodies of arbitrary
geometry by [[goo90]]. We will briefly present this theory in
our notation and the comparisons will be made in a later section.
 
In water of constant finite depth <math>d</math>, the scattered potential of a body
<math>\Delta_j</math> can be expanded in cylindrical eigenfunctions,
<center><math> (basisrep_out_d)
\phi_j^\mathrm{S} (r_j,\theta_j,z) &= \frac{\cosh k(z+d)}{\cosh kd}
\sum_{\nu = - \infty}^{\infty} A_{0 \nu}^j H_\nu^{(1)} (k r_j) \mathrm{e}^{\mathrm{i}\nu
\theta_j}\\
&\quad + \sum_{m=1}^{\infty} \frac{\cos k_m (z+d)}{\cos kd} \sum_{\nu = -
\infty}^{\infty} A_{m \nu}^j K_\nu (k_m r_j) \mathrm{e}^{\mathrm{i}\nu
\theta_j},
</math></center>
with discrete coefficients <math>A_{m \nu}^j</math>. The positive wavenumber <math>k</math>
is related to <math>\alpha</math> by the dispersion relation
<center><math> (eq_k)
\alpha = k \tanh k d,
</math></center>
and the values of <math>k_m</math>, <math>m>0</math>, are given as positive real roots of
the dispersion relation
<center><math> (eq_k_m)
\alpha + k_m \tan k_m d = 0.
</math></center>  
</math></center>  
The incident potential upon body <math>\Delta_j</math> can be also be expanded in
cylindrical eigenfunctions,
<center><math> (basisrep_in_d)
\phi_j^\mathrm{I} (r_j,\theta_j,z) &= \frac{\cosh k(z+d)}{\cosh kd}
\sum_{\mu = - \infty}^{\infty} D_{0 \mu}^j J_\mu (k r_j) \mathrm{e}^{\mathrm{i}\mu
\theta_j}\\
& \quad + \sum_{m=1}^{\infty} \frac{\cos k_m (z+d)}{\cos kd} \sum_{\mu = -
\infty}^{\infty} D_{m\mu}^j I_\mu (k_m r_j) \mathrm{e}^{\mathrm{i}\mu
\theta_j},
</math></center>
with discrete coefficients <math>D_{m\mu}^j</math>. A system of equations for the
coefficients of the scattered wavefields for the bodies are derived
in an analogous way to the infinite depth case. The derivation is
simpler because all the coefficients are discrete and the
diffraction transfer operator can be represented by an
infinite dimensional matrix.
Truncating the infinite dimensional matrix as well as the
coefficient vectors appropriately, the resulting system of
equations is given by 
<center><math>
{\bf a}_l = {\bf B}_l \Big( {\bf d}_l^\mathrm{In} +
\sum_{\genfrac{}{}{0pt}{}{j=1}{j \neq l}}^{N} \trans {\bf T}_{jl} \,
{\bf a}_j \Big), \quad  l=1, \ldots, N,
</math></center>
where <math>{\bf a}_l</math> is the coefficient vector of the scattered
wave, <math>{\bf d}_l^\mathrm{In}</math> is the coefficient vector of the
ambient incident wave, <math>{\bf B}_l</math> is the diffraction transfer
matrix of <math>\Delta_l</math> and <math>{\bf T}_{jl}</math> is the coordinate transformation
matrix analogous to  (T_elem_deep).
The calculation of the diffraction transfer matrices is
also similar to the infinite depth case. The finite depth
Green's function
<center><math> (green_d)
&G(r,\theta,z;s,\varphi,c)\\ &= \frac{\i}{2} \,
\frac{\alpha^2-k^2}{d(\alpha^2-k^2)-\alpha}\, \cosh k(z+d) \cosh k(c+d)
\sum_{\nu=-\infty}^{\infty} H_\nu^{(1)}(k r) J_\nu(k s) \mathrm{e}^{\mathrm{i}\nu
(\theta - \varphi)}\\ 
& \quad + \frac{1}{\pi} \sum_{m=1}^{\infty}
\frac{k_m^2+\alpha^2}{d(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+d) \cos
k_m(c+d) \sum_{\nu=-\infty}^{\infty} K_\nu(k_m r) I_\nu(k_m s) \mathrm{e}^{\mathrm{i}\nu
(\theta - \varphi)},
</math></center>
given by [[black75]] and [[fenton78]], needs to be used instead
of the infinite depth Green's function  (green_inf).
The elements of <math>{\bf B}_j</math> are therefore given by
<center><math> (B_elem_d)
<center><math>\begin{matrix}
({\bf B}_j)_{pq} &= \frac{\i}{2} \,
\frac{(\alpha^2-k^2)\cosh^2 kd}{d(\alpha^2-k^2)-\alpha} \int\limits_{\Gamma_j}
\cosh k(c+d) J_p(\alpha s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta})
\mathrm{d}\sigma_\mathbf{\zeta}\\
=and= 
({\bf B}_j)_{pq} &= \frac{1}{\pi}
\frac{(k_m^2+\alpha^2)\cos^2 k_md}{d(k_m^2+\alpha^2)-\alpha}
\int\limits_{\Gamma_j} \cos k_m(c+d) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p
\varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta}
\end{matrix}</math></center>
</math></center>
for the propagating and the decaying modes respectively, where
<math>\varsigma_q^j(\mathbf{\zeta})</math> is the source strength distribution
due to an incident potential of mode <math>q</math> of the form
<center><math> (test_modes_d)
<center><math>\begin{matrix}
\phi_q^{\mathrm{I}}(s,\varphi,c) &=  \frac{\cosh k_m(c+d)}{\cosh kd}
H_q^{(1)} (k s) \mathrm{e}^{\mathrm{i}q \varphi}\\
=for the propagating modes, and= 
\phi_q^{\mathrm{I}}(s,\varphi,c) &= \frac{\cos k_m(c+d)}{\cos kd} K_q
(k_m s) \mathrm{e}^{\mathrm{i}q \varphi}
\end{matrix}</math></center>
</math></center>
for the decaying modes.


 
The substitution of this into the equation for relating
 
the coefficients <math>D_{n\nu}^l</math> and
==Numerical Results==
<math>A_{m \mu}^l</math>gives the
 
required equations to determine the coefficients of the scattered
In this section we will present some calculations using the interaction
wavefields of all bodies,  
theory in finite and infinite depth and the full
diffraction method in finite and infinite depth.
These will be based on calculations for ice floes. We begin with some
convergence tests which aim to compare the various methods. It needs
to be noted that this comparison is only of numerical nature since the
interactions methods as well as the full diffraction calculations
are exact in an analytical sense. However, numerical calculations
require truncations which affect the different methods in different
ways. Especially the dependence on these truncations will be investigated.
 
===Convergence Test===
We will present some convergence tests that aim to compare the
performance of the interaction theory with the full diffraction
calculations and to compare the
performance of the finite and infinite depth interaction methods in deep water.
The comparisons will be conducted for the case of two square ice floes
in three different arrangements.
In the full diffraction calculation the ice floes
are discretised in <math>24 \times 24 = 576</math> elements. For the full diffraction
calculation the resulting linear system of equations to be solved is
therefore 1152. As will be seen, once the diffraction
transfer matrix has been calculated (and saved), the dimension of the
linear system of equations to be solved in the interaction method is
considerably smaller. It is given by twice the dimension of the
diffraction transfer matrix. The most challenging situation for the
interaction theory is when the bodies are close together. For this
reason we choose the distance such that the escribed circles
of the two ice floes just overlap. It must be recalled that the
interaction theory is valid as long as the escribed cylinder of a body
does not intersect with any other body.
 
Both ice floes have non-dimensionalised
stiffness <math>\beta = 0.02</math>, mass <math>\gamma = 0.02</math> and Poisson's ratio
is chosen as <math>\nu=0.3333</math>. The wavelength of
the ambient incident wave is <math>\lambda = 2</math>. Each ice floe has
side length 2. The ambient
wavefield is of unit amplitude and propagates in the <math>x</math>-direction.
Three different arrangements are chosen to compare the results of the
finite depth interaction method in deep water and the infinite depth
interaction method with the corresponding full diffraction
calculations. In the first arrangement the second ice floe is located
behind the first, in the second arrangement it is located
beside, and the third arrangement it is both
beside and behind. The exact positions of the ice floes
are given in table (tab:pos).
 
\begin{table}
\begin{center}
\begin{tabular}{@{}ccc@{}}
arrangement & <math>O_1</math> & <math>O_2</math>\<center><math>3pt]
1 & <math>(-1.4,0)</math> & <math>(1.4,0)</math>\\
2 & <math>(0,-1.4)</math> & <math>(0,1.4)</math>\\
3 & <math>(-1.4,-0.6)</math> & <math>(1.4,0.6)</math>
\end{tabular}
\caption{Positions of the ice floes in the different arrangements.} (tab:pos)
\end{center}
\end{table}
 
Figure (fig:tsf) shows the
solutions corresponding to the three arrangements in the case of water
of infinite depth. To illustrate the effect on the water in the
vicinity of the ice floes, the water displacement is also shown.
It is interesting
to note that the ice floe in front is barely influenced by the
floe behind while the motion of the floe behind is quite
different from its motion in the absence of the floe in front.
 
\begin{figure}
\begin{center}
 
\includegraphics[width=8.2cm]{int_n11_x0_y28_v5_b02_chi2}\<center><math>0.4cm]
\includegraphics[width=8.2cm]{int_n11_x0_y28_v5_b02_chi0}\<center><math>0.4cm]
\includegraphics[width=8.2cm]{int_n11_x12_y28_v5_b02_chi2}
 
\end{center}
\caption{Surface displacement of the ice floes
and the water in their vicinity,\newline arrangements 1, 2 and 3.} (fig:tsf)
\end{figure}
 
To compare the results, a measure of
the error from the full diffraction calculation is used. We calculate
the full diffraction solution with a sufficient number of points
so that we may use it to approximate the exact solution.
<center><math>
E_2 = \left( \, \int\limits_{\Delta}
\big| w_{i}(\mathbf{x})-w_{f}(\mathbf{x}) \big|^2 \mathrm{d}\mathbf{x} \,
\right)^{1/2},
</math></center>
where <math>w_{i}</math> and <math>w_{f}</math> are the solutions of the
interaction method and the corresponding full diffraction calculation
respectively. It would also be possible to compare other errors, the
maximum difference of the solutions for example, but the results are
very similar.
 
It is worth noting that the finite depth interaction method
only converges up to a certain depth if used with the
eigenfunction expansion of the finite depth Green's function  (green_d).
This is because of the factor
<math>\alpha^2-k^2</math> in the term of propagating modes of the Green's
function. The Green's function can
be rewritten by making use of the dispersion relation  (eq_k)
\cite[as suggested by][p. 26, for example]{linton01}
and the depth restriction of the finite depth interaction method for
bodies of arbitrary geometry can be circumvented.
 
The truncation parameters for the interaction methods will
now be considered for both finite and infinite depth.
The number of propagating modes and angular decaying
components are free parameters in both methods. In
finite depth, the number of decaying roots of the dispersion relation
needs to be chosen while in infinite depth the discretisation of
a continuous variable must be selected.
In the infinite depth case we are free to choose the number of
points as well as the points themselves. In water of finite depth, the depth
can also be considered a free parameter as long as it is chosen large
enough to account for deep water.
 
Truncating the infinite sums in the eigenfunction expansion of the
outgoing water velocity potential for infinite depth with
truncation parameters <math>T_H</math> and <math>T_K</math> and discretising the integration
by defining a set of nodes, </math>0\leq\eta_1 < \ldots < \eta_m < \ldots <
\eta_{_{T_R}}<math>, with weights </math>h_m<math>, the potential for infinite depth
can be approximated by
<center><math>
\phi (r,\theta,z) &=  \mathrm{e}^{\alpha z} \sum_{\nu = -
T_H}^{T_H} A_{0\nu} H_\nu^{(1)} (\alpha r) \mathrm{e}^{\mathrm{i}\nu
\theta}\\
&\quad + \sum_{m=1}^{T_R} h_m \, \psi (z,\eta_m) \sum_{\nu = -
T_K}^{T_K} A_{\nu} (\eta_m) K_\nu (\eta_m r) \mathrm{e}^{\mathrm{i}\nu \theta}.
</math></center>
In the following, the integration weights are chosen to be </math>h_m =
1/2\,(\eta_{m+1}-\eta_{m-1})<math>, </math>m=2, \ldots, T_R-1<math> and </math>h_1 =
\eta_2-\eta_1<math> as well as </math>h_{_{T_R}} =
\eta_{_{T_R}}-\eta_{_{T_R-1}}<math>, which corresponds to the mid-point
quadrature rule.
Different quadrature rules such as Gaussian quadrature
could be considered. Although in general this would lead to better
results, the mid-point rule allows a clever
choice of the discretisation points so that the convergence with
Gaussian quadrature is no better.
In finite depth, the analogous truncation leads to
<center><math>
<center><math>
\phi (r,\theta,z) &= \frac{\cosh k (z+d)}{\cosh kd} \sum_{\nu = -
A_{m\mu}^l = \sum_{n=0}^{\infty}
T_H}^{T_H} A_{0\nu} H_\nu^{(1)} (k r) \mathrm{e}^{\mathrm{i}\nu \theta}\\
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
& \quad + \sum_{m = 1}^{T_R} \frac{\cos k_m(z+d)}{\cos k_m d}
\Big[ \tilde{D}_{n\nu}^{l} +
\sum_{\nu = - T_K}^{T_K} A_{m\nu} K_\nu (k_m r) \mathrm{e}^{\mathrm{i}\nu \theta}.
\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></center>
In both cases, the dimension of the diffraction transfer matrix,
<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</math>.
<math>\mathbf{B}</math>, is given by <math>2 \, T_H+1+T_R \, (2 \, T_K+1)</math>.
 
Since the choice of the number of propagating
modes and angular decaying components affects the finite and
infinite depth methods in similar ways, the dependence on these
parameters will not be further presented. Thorough convergence tests
have shown that in the settings investigated here, it is sufficient to
choose <math>T_H</math> to be 11 and <math>T_K</math> to be 5. Further increasing these
parameter values does not result in smaller errors (as compared
to the full diffraction calculation with 576 elements per floe).
We will now compare the convergence of the infinite depth and
the finite depth methods if <math>T_H</math> and <math>T_K</math> are
fixed (with the previously mentioned values) and <math>T_R</math> is varied. To be able to
compare the results, the discretisation of the continuous variable
will always be the same for fixed <math>T_R</math> and these are
shown in table (tab:discr).
It should be noted that if only one node is used the integration
weight is chosen to be 1.
 
\begin{table}
\begin{center}
\begin{tabular}{@{}cl@{}}
<math>T_R</math> & discretisation of <math>\eta</math>\<center><math>3pt]
1 & \{ 2.1 \}\\
2 & \{ 1.2, 2.7 \}\\
3 & \{ 0.8, 1.8, 3.0 \}\\
4 & \{ 0.4, 1.4, 2.2, 3.2 \}\\
5 & \{ 0.2, 1.0, 1.8, 3.0, 4.6 \}
\end{tabular}
\caption{The different discretisations used in the convergence tests.} (tab:discr)
\end{center}
\end{table}
Figures (fig:behind), (fig:beside) and (fig:shifted) show the convergence for arrangement 1, arrangement 2 and arrangement 3, respectively, for
the infinite depth method and the finite depth method with depth 2
(plot (a)) and depth 4 (plot (b)).
Since the ice floes are located beside each other
in arrangement 2 the average errors are the same for both floes.
As can be seen from figures (fig:behind), (fig:beside) and
(fig:shifted) the convergence of the infinite depth method
is similar to that of the finite depth method. Used with depth 2, the
convergence of the finite depth method is generally better than that
of the infinite depth method while used with depth 4, the infinite depth
method achieves the better results. Tests with other depths show that
the performance of the finite depth method decreases with increasing
water depth as expected. In general, since the wavelength is 2, a depth
of <math>d=2</math> should approximate infinite depth and hence there is no
advantage to using the infinite depth theory. However, as mentioned
previously, for certain situations such as ice floes it is not necessarily
true that <math>d=2</math> will approximate infinite depth. 
 
\begin{figure}
\begin{center}
\begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}} 
\includegraphics[height=0.38\columnwidth]{behind_d2}&&
\includegraphics[height=0.38\columnwidth]{behind_d4}
\end{tabular}
\caption{Development of the errors as <math>T_R</math> is increased in
arrangement 1.} (fig:behind)
\end{center}
\end{figure}
 
\begin{figure}
\begin{center}
\begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}} 
\includegraphics[height=0.38\columnwidth]{beside_d2}&&
\includegraphics[height=0.38\columnwidth]{beside_d4}
\end{tabular}
\caption{Development of the errors as <math>T_R</math> is increased in
arrangement 2.} (fig:beside)
\end{center}
\end{figure}
 
\begin{figure}
\begin{center}
\begin{tabular}{p{0.46\columnwidth}p{0.02\columnwidth}p{0.46\columnwidth}} 
\includegraphics[height=0.38\columnwidth]{shifted_d2}&&
\includegraphics[height=0.38\columnwidth]{shifted_d4}
\end{tabular}
\caption{Development of the errors as <math>T_R</math> is increased in
arrangement 3.} (fig:shifted)
\end{center}
\end{figure}
 
===Multiple ice floe results===
We will now present results for multiple ice floes of different
geometries and in different arrangements on water of infinite depth.
We choose the floe arrangements arbitrarily, since there are
no known special ice floe arrangements, such as those that give
rise to resonances in the infinite limit.
In all plots, the wavelength <math>\lambda</math> has been chosen to
be <math>2</math>, the stiffness <math>\beta</math> and the mass <math>\gamma</math> of the ice
floes to be 0.02 and Poisson's ratio <math>\nu</math> is <math>0.3333</math>. The ambient
wavefield of amplitude 1 propagates in
the positive direction of the <math>x</math>-axis, thus it travels from left to
right in the plots. 
 
Figure (fig:int_arb) shows the
displacements of multiple interacting ice floes of different shapes and
in different arrangements. Since square elements have been used to
represent the floes, non-rectangular geometries are approximated.
All ice floes have an area of 4 and the escribing circles do not
intersect with any of the other ice floes.
The plots show the displacement of the ice floes at time <math>t=0</math>.
 
\begin{figure}
\begin{center}
\begin{tabular}{p{0.46\columnwidth}p{0.03\columnwidth}p{0.46\columnwidth}}
\includegraphics[width=0.45\columnwidth]{mult_n15_tr_tr_tr_tr_in} &&
\includegraphics[width=0.45\columnwidth]{mult_n15_tr_tr_tr_tr_out}\<center><math>0.2cm]
\includegraphics[width=0.45\columnwidth]{mult_n15_rh_rh_rh} &&
\includegraphics[width=0.45\columnwidth]{mult_n15_sq_sq_sq_sq_sq}\\
\end{tabular}
\end{center}
\caption{Surface displacement of interacting ice floes of different geometries.} (fig:int_arb)
\end{figure}
 
 
 
 
==Summary==
The finite depth interaction theory developed by
[[kagemoto86]] has been extended to water of infinite
depth. Furthermore, using the eigenfunction
expansion of the infinite depth free surface Green's function we have
been able to calculate the diffraction transfer matrices for bodies of
arbitrary geometry. We also showed how the diffraction transfer
matrices can be calculated efficiently for different orientations of
the body.
 
The convergence of the infinite depth interaction method is similar to
that of the finite depth method. Generally, it can be said that the
greater the water depth in the finite depth method the poorer its
performance. Since bodies in the water can change the water depth
which is required to allow the water to be approximated as infinitely
deep (ice floes for example) it is recommendable to use the infinite
depth method if the water depth may be considered
infinite. Furthermore, the infinite depth method requires the infinite
depth single diffraction solutions which are easier to
compute than the finite depth solutions.
It is also possible that the
convergence of the infinite depth method may be further improved
by a novel to optimisation of the discretisation of the continuous variable.</math>

Latest revision as of 10:24, 2 May 2010

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. Interaction Theory for Cylinders presents a simplified version for cylinders.

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 coordinates 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.

Equations of Motion

The problem consists of [math]\displaystyle{ n }[/math] bodies [math]\displaystyle{ \Delta_j }[/math] with immersed body surface [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \mathbf{y} = (x,y,z) }[/math] always denotes a point in the water, which is assumed to be of Finite Depth [math]\displaystyle{ h }[/math], while [math]\displaystyle{ \mathbf{x} }[/math] always denotes a point of the undisturbed water surface assumed at [math]\displaystyle{ z=0 }[/math].

The equations are the following

[math]\displaystyle{ \begin{align} \Delta\phi &=0, &-h\lt z\lt 0,\,\,\mathbf{x} \in \Omega \\ \partial_z\phi &= 0, &z=-h, \\ \partial_z \phi &= \alpha \phi, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]


(note that the last expression can be obtained from combining the expressions:

[math]\displaystyle{ \begin{align} \partial_z \phi &= -\mathrm{i} \omega \zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \\ \mathrm{i} \omega \phi &= g\zeta, &z=0,\,\,\mathbf{x} \in \partial \Omega_{\mathrm{F}}, \end{align} }[/math]

where [math]\displaystyle{ \alpha = \omega^2/g \, }[/math])

[math]\displaystyle{ \partial_n\phi = \mathcal{L}\phi, \quad \mathbf{x}\in\partial\Omega_B, }[/math]

where [math]\displaystyle{ \mathcal{L} }[/math] is a linear operator which relates the normal and potential on the body surface through the physics of the body.

The Sommerfeld Radiation Condition is also 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]\displaystyle{ \Delta_j }[/math] can be expressed as

[math]\displaystyle{ \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]

with discrete coefficients [math]\displaystyle{ A_{m \mu}^j }[/math], where [math]\displaystyle{ (r_j,\theta_j,z) }[/math] are cylindrical polar coordinates centered at each body

[math]\displaystyle{ f_m(z) = \frac{\cos k_m (z+h)}{\cos k_m h}. }[/math]

where [math]\displaystyle{ k_m }[/math] are found from [math]\displaystyle{ \alpha }[/math] by the Dispersion Relation for a Free Surface

[math]\displaystyle{ \alpha + k_m \tan k_m h = 0\,. }[/math]

where [math]\displaystyle{ k_0 }[/math] is the imaginary root with negative imaginary part and [math]\displaystyle{ k_m }[/math], [math]\displaystyle{ m\gt 0 }[/math], are given the positive real roots ordered with increasing size.

The incident potential upon body [math]\displaystyle{ \Delta_j }[/math] can be also be expanded in regular cylindrical eigenfunctions,

[math]\displaystyle{ \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]

with discrete coefficients [math]\displaystyle{ D_{n\nu}^j }[/math]. In these expansions, [math]\displaystyle{ I_\nu }[/math] and [math]\displaystyle{ K_\nu }[/math] denote the modified : Bessel functions of the first and second kind, respectively, both of order [math]\displaystyle{ \nu }[/math].

Note that the term for [math]\displaystyle{ m =0 }[/math] or [math]\displaystyle{ n=0 }[/math] corresponds to the propagating modes while the terms for [math]\displaystyle{ m\geq 1 }[/math] ([math]\displaystyle{ 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]\displaystyle{ \Delta_j }[/math] into an incident potential upon [math]\displaystyle{ \Delta_l }[/math] ([math]\displaystyle{ 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]\displaystyle{ \phi_j^{\mathrm{S}} }[/math] of body [math]\displaystyle{ \Delta_j }[/math] needs to be represented in terms of the incident potential [math]\displaystyle{ \phi_l^{\mathrm{I}} }[/math] upon [math]\displaystyle{ \Delta_l }[/math], [math]\displaystyle{ j \neq l }[/math]. This can be accomplished by using Graf's Addition Theorem

[math]\displaystyle{ 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]

which is valid provided that [math]\displaystyle{ r_l \lt R_{jl} }[/math]. Here, [math]\displaystyle{ (R_{jl},\varphi_{jl}) }[/math] are the polar coordinates of the mean centre position of [math]\displaystyle{ \Delta_{l} }[/math] in the local coordinates of [math]\displaystyle{ \Delta_{j} }[/math].

The limitation [math]\displaystyle{ r_l \lt R_{jl} }[/math] only requires that the escribed cylinder of each body [math]\displaystyle{ \Delta_l }[/math] does not enclose any other origin [math]\displaystyle{ O_j }[/math] ([math]\displaystyle{ 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]\displaystyle{ \Delta_l }[/math] does not enclose any other origin [math]\displaystyle{ O_j }[/math] ([math]\displaystyle{ 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]\displaystyle{ \Delta_j }[/math] can be expressed in terms of the incident potential upon [math]\displaystyle{ \Delta_l }[/math] as

[math]\displaystyle{ \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]
[math]\displaystyle{ = \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]

The ambient incident wavefield [math]\displaystyle{ \phi^{\mathrm{In}} }[/math] can also be expanded in the eigenfunctions corresponding to the incident wavefield upon [math]\displaystyle{ \Delta_l }[/math]. Let [math]\displaystyle{ \tilde{D}_{n\nu}^{l} }[/math] denote the coefficients of this ambient incident wavefield in the incoming eigenfunction expansion for [math]\displaystyle{ \Delta_l }[/math] (cf. the example in Cylindrical Eigenfunction Expansion).

[math]\displaystyle{ \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]

The total incident wavefield upon body [math]\displaystyle{ \Delta_j }[/math] can now be expressed as

[math]\displaystyle{ \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]

This allows us to write

[math]\displaystyle{ \sum_{n=0}^{\infty} f_n(z) \sum_{\nu = - \infty}^{\infty} D_{n\nu}^l I_\nu (k_n r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l} }[/math]
[math]\displaystyle{ = \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]

It therefore follows that

[math]\displaystyle{ D_{n\nu}^l = \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]

Final Equations

The scattered and incident potential of each body [math]\displaystyle{ \Delta_l }[/math] can be related by the Diffraction Transfer Matrix acting in the following way,

[math]\displaystyle{ A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n \mu \nu}^l D_{n\nu}^l. }[/math]

The substitution of this into the equation for relating the coefficients [math]\displaystyle{ D_{n\nu}^l }[/math] and [math]\displaystyle{ A_{m \mu}^l }[/math]gives the required equations to determine the coefficients of the scattered wavefields of all bodies,

[math]\displaystyle{ 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]

[math]\displaystyle{ m \in \mathbb{N} }[/math], [math]\displaystyle{ \mu \in \mathbb{Z} }[/math], [math]\displaystyle{ l=1,\dots,N }[/math].

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