Difference between revisions of "Kagemoto and Yue Interaction Theory"

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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 =  
 
= Equations of Motion =  
  
We assume
+
The problem consists of <math>n</math> bodies
the [[Frequency Domain Problem]] with frequency <math>\omega</math>.  
+
<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
 
To simplify notation, <math>\mathbf{y} = (x,y,z)</math> always denotes a point
in the water, which is assumed to be of [[Finite Depth]] <math>d</math>,
+
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
 
while <math>\mathbf{x}</math> always denotes a point of the undisturbed water
 
surface assumed at <math>z=0</math>.  
 
surface assumed at <math>z=0</math>.  
  
Writing <math>\alpha = \omega^2/g</math> where <math>g</math> is the acceleration due to
+
{{standard linear wave scattering equations}}
gravity, the potential <math>\phi</math> has to
+
 
satisfy the standard boundary-value problem
+
The [[Sommerfeld Radiation Condition]] is also imposed.
<center><math>
 
\nabla^2 \phi = 0, \;  \mathbf{y} \in D
 
</math></center>
 
<center><math> 
 
\frac{\partial \phi}{\partial z} = \alpha \phi, \;
 
{\mathbf{x}} \in \Gamma^\mathrm{f},
 
</math></center>
 
<center><math>
 
\frac{\partial \phi}{\partial z} = 0, \;  \mathbf{y} \in D, \ z=-d,
 
</math></center>
 
where <math>D</math> is the
 
is the domain occupied by the water and
 
<math>\Gamma^\mathrm{f}</math> is the free water surface. At the immersed body
 
surface <math>\Gamma_j</math> of <math>\Delta_j</math>, the water velocity potential has to
 
equal the normal velocity of the body <math>\mathbf{v}_j</math>,
 
<center><math>
 
\frac{\partial \phi}{\partial n} = \mathbf{v}_j, \;  {\mathbf{y}}
 
\in \Gamma_j.
 
</math></center>
 
Moreover, the Sommerfeld radiation condition is imposed  
 
<center><math>
 
\lim_{\tilde{r} \rightarrow \infty} \sqrt{\tilde{r}} \, \Big(
 
\frac{\partial}{\partial \tilde{r}} - \mathrm{i}k
 
\Big) (\phi - \phi^{\mathrm{In}}) = 0,
 
</math></center>
 
where <math>\tilde{r}^2=x^2+y^2</math>, <math>k</math> is the wavenumber and
 
<math>\phi^\mathrm{In}</math> is the ambient incident potential. 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_km)
 
\alpha + k_m \tan k_m d = 0.
 
</math></center>
 
For ease of notation, we write <math>k_0 = -\mathrm{i}k</math>. Note that <math>k_0</math> is a
 
(purely imaginary) root of  (eq_k_m).
 
  
 
=Eigenfunction expansion of the potential=
 
=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
 
The scattered potential of a body
<math>\Delta_j</math> can be expanded in singular cylindrical eigenfunctions,
+
<math>\Delta_j</math> can be expressed as
<center><math> (basisrep_out_d)
+
<center><math>  
 
\phi_j^\mathrm{S} (r_j,\theta_j,z) =  
 
\phi_j^\mathrm{S} (r_j,\theta_j,z) =  
 
\sum_{m=0}^{\infty} f_m(z) \sum_{\mu = -
 
\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},
 
\infty}^{\infty} A_{m \mu}^j K_\mu (k_m r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j},
 
</math></center>
 
</math></center>
with discrete coefficients <math>A_{m \mu}^j</math>, where
+
with discrete coefficients <math>A_{m \mu}^j</math>, where <math>(r_j,\theta_j,z)</math>
 +
are cylindrical polar coordinates centered at each body
 
<center><math>
 
<center><math>
f_m(z) = \frac{\cos k_m (z+d)}{\cos k_m d}.
+
f_m(z) = \frac{\cos k_m (z+h)}{\cos k_m h}.
 
</math></center>
 
</math></center>
 +
where <math>k_m</math> are found from <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]
 +
<center><math>
 +
\alpha + k_m \tan k_m h = 0\,.
 +
</math></center>
 +
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
 
The incident potential upon body <math>\Delta_j</math> can be also be expanded in
 
regular cylindrical eigenfunctions,  
 
regular cylindrical eigenfunctions,  
<center><math> (basisrep_in_d)
+
<center><math>  
 
\phi_j^\mathrm{I} (r_j,\theta_j,z) = \sum_{n=0}^{\infty} f_n(z)
 
\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},
 
\sum_{\nu = - \infty}^{\infty} D_{n\nu}^j I_\nu (k_n r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j},
 
</math></center>
 
</math></center>
 
with discrete coefficients <math>D_{n\nu}^j</math>. In these expansions, <math>I_\nu</math>
 
with discrete coefficients <math>D_{n\nu}^j</math>. In these expansions, <math>I_\nu</math>
and <math>K_\nu</math> denote the modified Bessel functions of the first and
+
and <math>K_\nu</math> denote the modified [http://en.wikipedia.org/wiki/Bessel_function : Bessel functions]
second kind, respectively, both of order <math>\nu</math>.
+
of the first and second kind, respectively, both of order <math>\nu</math>.
Note that in (basisrep_out_d) (and  (basisrep_in_d)) the term for <math>m =0<math> (
 
<math>n=0</math>) corresponds to the propagating modes while the
 
terms for <math>m\geq 1</math> (<math>n\geq 1</math>) correspond to the evanescent modes. For
 
future reference, we remark that, for real <math>x</math>,
 
<center><math> (H_K)
 
K_\nu (-\mathrm{i}x) = \frac{\pi i^{\nu+1}}{2} H_\nu^{(1)}(x) \quad
 
=and=  \quad
 
I_\nu (-\mathrm{i}x) = i^{-\nu} J_\nu(x)
 
</math></center>
 
with <math>H_\nu^{(1)}</math> and <math>J_\nu</math> denoting the Hankel function and the
 
Bessel function, respectively, both of first kind and order <math>\nu</math>.
 
 
 
= Representation of the ambient wavefield in the eigenfunction representation =
 
 
 
In Cartesian coordinates centred at the origin, the ambient wavefield is
 
given by
 
<center><math>
 
\phi^{\mathrm{In}}(x,y,z) = \frac{A g}{\omega}
 
 
 
f_0(z) \mathrm{e}^{\mathrm{i}k (x \cos \chi + y \sin \chi)},
 
</math></center>
 
where <math>A</math> is the amplitude (in displacement) and <math>\chi</math> is the
 
angle between the <math>x</math>-axis and the direction in which the wavefield
 
travels (also cf.~figure (fig:floes)).
 
This expression can be written in the eigenfunction expansion
 
centred at the origin as
 
<center><math>
 
\phi^{\mathrm{In}}(r,\theta,z) = \frac{A g}{\omega}
 
  
f_0(z)
+
Note that the term for <math>m =0</math> or
\sum_{\nu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\nu (\pi/2 - \theta + \chi)} J_\nu(k r)
+
<math>n=0</math> corresponds to the propagating modes while the  
</math></center>
+
terms for <math>m\geq 1</math> (<math>n\geq 1</math>) correspond to the evanescent modes.
\cite[p.~169]{linton01}. 
 
The local coordinates of each body are centred at their mean-centre
 
positions <math>O_l = (l R,0)</math>.
 
In order to represent the ambient wavefield, which is
 
incident upon all bodies, in the eigenfunction expansion of an
 
incoming wave in the local coordinates of the body, a phase factor has to be
 
defined,
 
<center><math> (phase_factor)
 
P_l = \mathrm{e}^{\mathrm{i}l R k \cos \chi},
 
</math></center>
 
which accounts for the position from the origin. Including this phase
 
factor and
 
making use of  (H_K), the ambient wavefield at the <math>l</math>th body is given by
 
<center><math>
 
\phi^{\mathrm{In}}(r_l,\theta_l,z) = \frac{A g}{\omega}  \, P_l \,
 
f_0(z) \sum_{\nu = -\infty}^{\infty}
 
\mathrm{e}^{\mathrm{i}\nu (\pi -  \chi)} I_\nu(k_0 r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
 
</math></center>
 
We can therefore define the coefficients of the ambient wavefield in
 
the eigenfunction expansion of an incident wave,
 
<center><math>
 
\tilde{D}^l_{n\nu} =
 
\begin{cases}
 
\frac{A g}{\omega}  P_l \mathrm{e}^{\mathrm{i}\nu (\pi - \chi)}, & n=0,\\
 
0, & n > 0.
 
\end{cases}
 
</math></center>
 
Note that the evanescent coefficients are all zero due to the
 
propagating nature of the ambient wave.
 
  
 
=Derivation of the system of equations=
 
=Derivation of the system of equations=
  
Following the ideas of general interaction theories
+
A system of equations for the unknown  
\cite[]{kagemoto86,JFM04}, a system of equations for the unknown  
+
coefficients of the
coefficients (in the expansion  (basisrep_out_d)) of the
 
 
scattered wavefields of all bodies is developed. This system of
 
scattered wavefields of all bodies is developed. This system of
 
equations is based on transforming the  
 
equations is based on transforming the  
Line 163: Line 90:
 
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:floes) we can see that this can be accomplished by using
 
 
[[Graf's Addition Theorem]]
 
[[Graf's Addition Theorem]]
<center><math> (transf)
 
K_\tau(k_m r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{j-l})} =
 
\sum_{\nu = - \infty}^{\infty} K_{\tau + \nu} (k_m |j-l|R) \,
 
I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{j-l})}, \quad j \neq l,
 
</math></center>
 
which is valid provided that <math>r_l < R</math>. The angles <math>\varphi_{n}</math>
 
account for the difference in direction depending if the <math>j</math>th body is
 
located to the left or to the right of the <math>l</math>th body and are
 
defined by
 
 
<center><math>
 
<center><math>
\varphi_n =  
+
K_\tau(k_m r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
\begin{cases}
+
\sum_{\nu = - \infty}^{\infty} K_{\tau + \nu} (k_m R_{jl}) \,
\pi, & n > 0,\\
+
I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l,
0, & n < 0.
 
\end{cases}
 
 
</math></center>
 
</math></center>
The limitation <math>r_l < R</math> only requires that the escribed cylinder of each body
+
which is valid provided that <math>r_l < R_{jl}</math>. Here, <math>(R_{jl},\varphi_{jl})</math>  are the polar coordinates of the mean centre position of <math>\Delta_{l}</math> in the local coordinates of <math>\Delta_{j}</math>.
 +
 
 +
The limitation <math>r_l < R_{jl}</math> only requires that the escribed cylinder of each body
 
<math>\Delta_l</math> does not enclose any other origin <math>O_j</math> (<math>j \neq l</math>). However, the
 
<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
Line 192: Line 109:
 
other body.  
 
other body.  
  
Making use of the eigenfunction expansion as well as equation (transf), the scattered potential
+
Making use of the eigenfunction expansion as well as [[Graf's Addition Theorem]], the scattered potential
of <math>\Delta_j</math> (cf.~ (basisrep_out_d)) can be expressed in terms of the
+
of <math>\Delta_j</math> can be expressed in terms of the
 
incident potential upon <math>\Delta_l</math> as
 
incident potential upon <math>\Delta_l</math> as
<center><math>\begin{matrix}
+
<center><math>
 
\phi_j^{\mathrm{S}} (r_l,\theta_l,z)  
 
\phi_j^{\mathrm{S}} (r_l,\theta_l,z)  
&= \sum_{m=0}^\infty f_m(z) \sum_{\tau = -
+
= \sum_{m=0}^\infty f_m(z) \sum_{\tau = -
 
\infty}^{\infty} A_{m\tau}^j \sum_{\nu = -\infty}^{\infty}
 
\infty}^{\infty} A_{m\tau}^j \sum_{\nu = -\infty}^{\infty}
(-1)^\nu K_{\tau-\nu} (k_m |j-l| R) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu
+
(-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_{j-l}} \\
+
\theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}}  
&= \sum_{m=0}^\infty f_m(z) \sum_{\nu =
+
</math></center>
 +
<center><math>
 +
= \sum_{m=0}^\infty f_m(z) \sum_{\nu =
 
-\infty}^{\infty}  \Big[ \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j
 
-\infty}^{\infty}  \Big[ \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j
(-1)^\nu K_{\tau-\nu} (k_m |j-l| R) \mathrm{e}^{\mathrm{i}(\tau - \nu)
+
(-1)^\nu K_{\tau-\nu} (k_m R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu)
\varphi_{j-l}} \Big] I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.   
+
\varphi_{jl}} \Big] I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.   
\end{matrix}</math></center>
+
</math></center>
 
The ambient incident wavefield <math>\phi^{\mathrm{In}}</math> can also be
 
The ambient incident wavefield <math>\phi^{\mathrm{In}}</math> can also be
 
expanded in the eigenfunctions corresponding to the incident wavefield upon
 
expanded in the eigenfunctions corresponding to the incident wavefield upon
 
<math>\Delta_l</math>. Let <math>\tilde{D}_{n\nu}^{l}</math> denote the coefficients of this
 
<math>\Delta_l</math>. Let <math>\tilde{D}_{n\nu}^{l}</math> denote the coefficients of this
 
ambient incident wavefield in the incoming eigenfunction expansion for
 
ambient incident wavefield in the incoming eigenfunction expansion for
<math>\Delta_l</math> (cf.~\S (sec:ambient)). The total
+
<math>\Delta_l</math> (cf. the example in [[Cylindrical Eigenfunction Expansion]]).  
incident wavefield upon body <math>\Delta_j</math> can now be expressed as
+
<center><math>
<center><math>\begin{matrix}
+
\phi^{\mathrm{In}}(r_l,\theta_l,z)= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
\phi_l^{\mathrm{I}}(r_l,\theta_l,z) &= \phi^{\mathrm{In}}(r_l,\theta_l,z) +
+
  \tilde{D}_{n\nu}^{l}  I_\nu (k_n
\sum_{j=-\infty,j \neq l}^{\infty} \, \phi_j^{\mathrm{S}}
 
(r_l,\theta_l,z)\\
 
&= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
 
\Big[ \tilde{D}_{n\nu}^{l} +
 
\sum_{j=-\infty,j \neq l}^{\infty} \sum_{\tau =
 
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
 
|j-l|R)  \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{j-l}} \Big] \times I_\nu (k_n
 
 
r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.  
 
r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.  
\end{matrix}</math></center>
 
The coefficients of the total incident potential upon <math>\Delta_l</math> are
 
therefore given by
 
<center><math> (inc_coeff)
 
D_{n\nu}^l = \tilde{D}_{n\nu}^{l} +
 
\sum_{j=-\infty,j \neq  l}^{\infty} \sum_{\tau =
 
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
 
|j-l| R)  \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{j-l}}.
 
</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>
 
A^l = B^l (D^l), \quad l\in {Z},
 
</math></center>
 
where <math>A^l</math> are the scattered modes due to the incident modes
 
<math>D^l</math>. Note that since it is assumed that all bodies are identical in
 
this setting, only one diffraction transfer operator, <math>B</math>, is required.
 
In the case of a countable number of modes (i.e.~when
 
the water depth is finite), <math>B</math> is an infinite dimensional matrix. When
 
the modes are functions of a continuous variable (i.e.~infinite
 
depth), <math>B</math> is the kernel of an integral operator.
 
The scattered and incident potential can therefore be related by a
 
diffraction transfer operator acting in the following way,
 
<center><math> (diff_op)
 
A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
 
\mu \nu} D_{n\nu}^l.
 
</math></center>
 
If the diffraction transfer operator is known (its calculation
 
is discussed later), the substitution of (inc_coeff) into  (diff_op) gives the
 
required equations to determine the coefficients of the scattered
 
wavefields of all bodies,
 
<center><math> (eq_op)
 
A_{m\mu}^l = \sum_{n=0}^{\infty}
 
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
 
\Big[ \tilde{D}_{n\nu}^{l} +
 
\sum_{j=-\infty,j \neq  l}^{\infty} \sum_{\tau =
 
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
 
|j-l| R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big],
 
</math></center>
 
<math>m \in {N}</math>, <math>l,\mu \in {Z}</math>.
 
 
=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>
 
To simplify notation, <math>\mathbf{y} = (x,y,z)</math> will always denote a point
 
in the water, which is assumed infinitely deep, while <math>\mathbf{x}</math> will
 
always denote a point of the undisturbed water surface assumed at <math>z=0</math>.
 
 
The problem consists of <math>N</math> vertically non-overlapping bodies, denoted
 
by <math>\Delta_j</math>, which are sufficiently far apart that there is no
 
intersection of the smallest cylinder which contains each body with
 
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.
 
 
The scattered potential of body <math>\Delta_j</math> can be expanded in
 
cylindrical eigenfunctions,
 
<center><math> (basisrep_out)
 
\phi_j^\mathrm{S} (r_j,\theta_j,z) = \mathrm{e}^{\alpha z} \sum_{\nu = -
 
\infty}^{\infty} A_{0 \nu}^j H_\nu^{(1)} (\alpha r_j) \mathrm{e}^{\mathrm{i}\nu
 
\theta_j}
 
+ \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 [[Abramowitz and Stegun 1964]].
 
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}
 
+ \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
 
<math>\Delta_l</math> (<math>j \neq l</math>). Doing this for all bodies simultaneously,
 
and relating the incident and scattered potential for each body, a system
 
of equations for the unknown coefficients will be developed.
 
 
The scattered potential <math>\phi_j^{\mathrm{S}}</math> of body <math>\Delta_j</math> needs to be
 
represented in terms of the incident potential <math>\phi_l^{\mathrm{I}}</math>
 
upon <math>\Delta_l</math>, <math>j \neq l</math>. From figure
 
(fig:floe_tri) we can see that this can be accomplished by using
 
Graf's addition theorem for Bessel functions given in
 
[[Abramowitz and Stegun 1964]],
 
<center><math> (transf)
 
\begin{matrix} (transf_h)
 
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>
which is valid provided that <math>r_l < R_{jl}</math>. This limitation
+
The total
only requires that the escribed cylinder of each body <math>\Delta_l</math> does
 
not enclose any other origin <math>O_j</math> (<math>j \neq l</math>). However, the
 
expansion of the scattered and incident potential in cylindrical
 
eigenfunctions is only valid outside the escribed cylinder of each
 
body. Therefore the condition that the
 
escribed cylinder of each body <math>\Delta_l</math> does not enclose any other
 
origin <math>O_j</math> (<math>j \neq l</math>) is superseded by the more rigorous
 
restriction that the escribed cylinder of each body may not contain any
 
other body. Making use of the equations  (transf)
 
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}^{\mathrm{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  
 
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{}_{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_{j=1,j
 
\neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
 
H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\mathrm{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_{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)
 
D_{0\mu}^l = D_{l0\mu}^{\mathrm{In}}
 
+ \sum_{j=1,j \neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
 
H_{\nu-\mu}^{(1)}
 
(\alpha R_{jl}) \mathrm{e}^{\mathbf{i}
 
(\nu - \mu) \vartheta_{jl}}
 
</math></center>
 
 
<center><math>
 
<center><math>
D_{\mu}^l(\eta) = D_{l\mu}^{\mathrm{In}}(\eta) +
+
\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +
\sum_{j=1,j \neq  l}^{N} \sum_{\nu =
+
\sum_{j=1,j \neq l}^{N} \, \phi_j^{\mathrm{S}}
-\infty}^{\infty} A_{\nu}^j (\eta) (-1)^\mu K_{\nu - \mu}  (\eta
+
(r_l,\theta_l,z)
R_{jl}) \mathrm{e}^{\mathrm{i}(\nu - \mu) \vartheta_{jl}}.
 
</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)
 
\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>
 
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)
 
\begin{matrix}
 
A_{0n}^l =& \sum_{\mu = -\infty}^{\infty} B_{ln\mu}^\mathrm{pp}
 
\Big[ D_{l0\mu}^{\mathrm{In}} + \sum_{j=1,j  
 
\neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
 
H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\mathbf{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_{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,\\
 
A_n^l (\eta) &= \sum_{\mu = -\infty}^{\infty}
 
B_{ln\mu}^\mathrm{dp} (\eta) \Big[
 
D_{l0\mu}^{\mathrm{In}} + \sum_{j=1,j
 
\neq l}^{N} \sum_{\nu = - \infty}^{\infty} A_{0\nu}^j
 
H_{\nu-\mu}^{(1)} (\alpha R_{jl}) \mathrm{e}^{\mathbf{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_{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{matrix}
 
 
</math></center>
 
</math></center>
<math>n \in \mathit{Z},\, l = 1, \ldots, N</math>. It has to be noted that all
+
This allows us to write
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
 
the integrals must be discretised. Implying a suitable truncation, the
 
four different diffraction transfer operators can be represented by
 
matrices which can be assembled in a big matrix <math>\mathbf{B}_l</math>,
 
 
<center><math>
 
<center><math>
\mathbf{B}_l = \left[
+
\sum_{n=0}^{\infty} f_n(z)
\begin{matrix}{cc} \mathbf{B}_l^{\mathrm{pp}} & \mathbf{B}_l^{\mathrm{pd}}\\
+
\sum_{\nu = - \infty}^{\infty} D_{n\nu}^l I_\nu (k_n r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}
\mathbf{B}_l^{\mathrm{dp}} & \mathbf{B}_l^{\mathrm{dd}}
 
\end{matrix} \right],
 
 
</math></center>
 
</math></center>
the infinite depth diffraction transfer matrix.
 
Truncating the coefficients accordingly, defining <math>{\mathbf 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>{\mathbf T}_{jl}</math> given by
 
<center><math> (T_elem_deep)
 
({\mathbf T}_{jl})_{pq} = H_{p-q}^{(1)}(\alpha R_{jl}) \, \mathrm{e}^{\mathrm{i}(p-q)
 
\vartheta_{jl}}
 
</math></center>
 
for the propagating modes, and
 
 
<center><math>
 
<center><math>
({\mathbf T}_{jl})_{pq} = (-1)^{q} \, K_{p-q} (\eta R_{jl}) \, \mathrm{e}^{\mathbf{i}
+
  = \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
(p-q) \vartheta_{jl}}
+
\Big[ \tilde{D}_{n\nu}^{l} +
</math></center>
+
\sum_{j=1,j \neq l}^{N} \sum_{\tau =
for the decaying modes, a linear system of equations
+
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
for the unknown coefficients follows from equations (eq_op),
+
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_n
<center><math> (eq_Binf)
+
r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.  
{\mathbf a}_l =  
 
{\mathbf {B}}_l \Big(
 
{\mathbf d}_l^{\mathrm{In}} +
 
\sum_{j=1,j \neq l}^{N} trans {\mathbf T}_{jl} \,
 
{\mathbf a}_j \Big), \quad  l=1, \ldots, N,
 
</math></center>
 
where the left superscript <math>\mathrm{t}</math> indicates transposition.
 
The matrix <math>{\mathbf \hat{B}}_l</math> denotes the infinite depth diffraction
 
transfer matrix <math>{\mathbf 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.
 
 
 
==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>({\mathbf 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 Function]]
 
in cylindrical eigenfunctions,
 
<center><math> (green_inf)\begin{matrix}
 
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)} \\
 
  +& \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,
 
\end{matrix}
 
</math></center>
 
<math>r > s</math>, given by [[Peter and Meylan 2004]].
 
 
 
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({\mathbf{\zeta}})  
 
\mathrm{d}\sigma_{\mathbf{\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)
 
({\mathbf 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}
 
 
</math></center>
 
</math></center>
and
+
It therefore follows that
 
<center><math>
 
<center><math>
({\mathbf B}_j)_{pq} = \frac{1}{\pi^2} \frac{\eta^2}{\eta^2 +
+
D_{n\nu}^l  =  
\alpha^2} \int\limits_{\Gamma_j} \psi(c,\eta) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p
+
  \tilde{D}_{n\nu}^{l} +
\varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta}
+
\sum_{j=1,j \neq  l}^{N} \sum_{\tau =
</math></center>
+
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n
for the propagating and the decaying modes respectively, where
+
R_{jl})  \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}}  
<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_modesinf)
 
\phi_q^{\mathrm{I}}(s,\varphi,c) = \mathrm{e}^{\alpha c} H_q^{(1)} (\alpha
 
s) \mathrm{e}^{\mathrm{i}q \varphi}
 
</math></center>
 
for the propagating modes, and
 
<center><math>
 
\phi_q^{\mathrm{I}}(s,\varphi,c) = \psi(c,\eta) K_q (\eta s) \mathrm{e}^{\mathrm{i}q \varphi}
 
 
</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
+
= Final Equations =
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
+
The scattered and incident potential of each body <math>\Delta_l</math> can be related by the
incident potential can be factored out of the normal derivative of the
+
[[Diffraction Transfer Matrix]] acting in the following way,
incident potential such that
 
 
<center><math>
 
<center><math>
\frac{\partial \phi_{q\beta}^{\mathrm{I}}}{\partial n} =
+
A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
\frac{\partial \phi_{q}^{\mathrm{I}}}{\partial n}
+
\mu \nu}^l D_{n\nu}^l.
\mathrm{e}^{\mathrm{i}q \beta},
 
</math></center>
 
where <math>\phi_{q\beta}^{\mathrm{I}}</math> is the rotated incident potential.
 
Since the integral equation for the determination of the source
 
strength distribution function is linear, the source strength
 
distribution function due to the rotated incident potential is thus just
 
given by
 
<center><math>
 
\varsigma_{q\beta}^j = \varsigma_q^j \, \mathrm{e}^{\mathrm{i}q \beta}.
 
</math></center>
 
This is also the source strength distribution function of the rotated
 
body due to the standard incident modes.
 
 
 
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_elemrot)
 
(\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},
 
</math></center>
 
and
 
<center><math>
 
(\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},
 
</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.
 
 
 
==Representation of the ambient wavefield in the eigenfunction representation==
 
In Cartesian coordinates centred at the origin, the ambient wavefield is
 
given by
 
<center><math>
 
\phi^{\mathrm{In}}(x,y,z) = A \frac{g}{\omega} \, \mathrm{e}^{\mathrm{i}\alpha (x
 
\cos \chi + y \sin \chi)+ \alpha z},
 
</math></center>
 
where <math>A</math> is the amplitude (in displacement) and <math>\chi</math> is the
 
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>
 
\phi^{\mathrm{In}}(x,y,z) = A \frac{g}{\omega} \mathrm{e}^{\alpha z}
 
\sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 - \theta + \chi)}
 
J_\mu(\alpha r)
 
</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>
 
\phi^{\mathrm{In}}(r_l,\theta_l,z) = A \frac{g}{\omega} \, e^{\mathrm{i}\alpha (O_x^l
 
\cos \chi + O_x^l \sin \chi)} \, \mathrm{e}^{\alpha z}
 
\sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 -  \chi)}
 
J_\mu(\alpha r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}.
 
</math></center>
 
 
 
==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>
 
\left[ \begin{matrix}
 
{\mathbf a}_1\\ {\mathbf a}_2\\ \\ \vdots \\ \\ {\mathbf a}_N
 
\end{matrix} \right]
 
= \left[ \begin{matrix}
 
{{\mathbf B}}_1 {\mathbf d}_1^\mathrm{In}\\ {{\mathbf B}}_2 {\mathbf
 
d}_2^\mathrm{In}\\ \\ \vdots \\ \\ {{\mathbf B}}_N {\mathbf d}_N^\mathrm{In}
 
\end{matrix} \right]
 
+
 
\left[ \begin{matrix}
 
\mathbf{0} & {{\mathbf B}}_1 {\mathbf T}_{21} & {{\mathbf B}}_1
 
{\mathbf T}_{31} & \dots & {{\mathbf B}}_1 {\mathbf T}_{N1}\\
 
{{\mathbf B}}_2  {\mathbf T}_{12} & \mathbf{0} & {{\mathbf B}}_2
 
{\mathbf T}_{32} & \dots & {{\mathbf B}}_2  {\mathbf T}_{N2}\\
 
& & \mathbf{0} & &\\
 
\vdots & & & \ddots & \vdots\\
 
& & & & \\
 
{{\mathbf B}}_N  {\mathbf T}_{1N} & & \dots & 
 
& \mathbf{0}
 
\end{matrix} \right]
 
\left[ \begin{matrix}
 
{\mathbf a}_1\\ {\mathbf a}_2\\ \\ \vdots \\ \\ {\mathbf a}_N
 
\end{matrix} \right],
 
</math></center>
 
where <math>\mathbf{0}</math> denotes the zero-matrix which is of the same
 
dimension as <math>{{\mathbf 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.
 
 
 
=Finite Depth Interaction Theory=
 
 
 
We will compare the performance of the infinite depth interaction theory
 
with the equivalent theory for finite
 
depth. As we have stated previously, the finite depth theory was
 
developed by [[Kagemoto and Yue 1986]] and extended to bodies of arbitrary
 
geometry by [[Goo and Yoshida 1990]]. 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>
 
\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}
 
+ \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 for a Free Surface]]
 
<center><math>
 
\alpha = k \mathrm{tanh} k d,\,
 
</math></center>
 
and the values of <math>k_m</math>, <math>m>0</math>, are given as positive real roots of
 
[[Dispersion Relation for a Free Surface]]
 
<center><math>
 
\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>
 
\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}
 
+ \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>
 
{\mathbf a}_l = {\mathbf B}_l \Big( {\mathbf d}_l^\mathrm{In} +
 
\sum_{j=1,j \neq l}^{N}  {\mathbf T}_{jl} \,
 
{\mathbf a}_j \Big),  l=1, \ldots, N,
 
</math></center>
 
where <math>{\mathbf a}_l</math> is the coefficient vector of the scattered
 
wave, <math>{\mathbf d}_l^\mathrm{In}</math> is the coefficient vector of the
 
ambient incident wave, <math>{\mathbf B}_l</math> is the diffraction transfer
 
matrix of <math>\Delta_l</math> and <math>{\mathbf T}_{jl}</math> is the coordinate transformation
 
matrix analogous to  (T_elem_deep).
 
  
The calculation of the diffraction transfer matrices is
+
The substitution of this into the equation for relating
also similar to the infinite depth case. [[Free-Surface Green Function]] for [[Finite Depth]]
+
the coefficients <math>D_{n\nu}^l</math> and
in cylindrical polar coordinates
+
<math>A_{m \mu}^l</math>gives the
 +
required equations to determine the coefficients of the scattered
 +
wavefields of all bodies,
 
<center><math>
 
<center><math>
G(r,\theta,z;s,\varphi,c)= \frac{\mathrm{i}}{2} \,
+
A_{m\mu}^l = \sum_{n=0}^{\infty}
\frac{\alpha^2-k^2}{d(\alpha^2-k^2)-\alpha}\, \cosh (z+d) \cosh k(c+d)
+
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\sum_{\nu=-\infty}^{\infty} H_\nu^{(1)}(k r) J_\nu(k s) \mathrm{e}^{\mathrm{i}\nu
+
\Big[ \tilde{D}_{n\nu}^{l} +
(\theta - \varphi)}
+
\sum_{j=1,j \neq  l}^{N} \sum_{\tau =
</math></center>
+
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
<center><math>
+
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big],
+ \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 [[Black 1975]] and [[Fenton 1978]], needs to be used instead
 
of the infinite depth Green's function  (green_inf).
 
The elements of <math>{\mathbf B}_j</math> are therefore given by
 
<center><math>
 
({\mathbf B}_j)_{pq} = \frac{\mathrm{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}
 
</math></center>
 
and
 
<center><math>
 
({\mathbf 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}
 
</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>
 
\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}
 
</math></center>
 
for the propagating modes, and
 
<center><math>
 
\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}
 
 
</math></center>
 
</math></center>
for the decaying modes.
+
<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</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].