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
+
The derivation of the theory in [[Infinite Depth]] is also presented, see
[[Kagemoto and Yue Interaction Theory for Infinite Depth]]
+
[[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 for a Free Surface]]
 
<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 the [[Cylindrical Eigenfunction Expansion]],
+
<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> or
+
 
<math>n=0</math>) corresponds to the propagating modes while the  
+
Note that the term for <math>m =0</math> or
 +
<math>n=0</math> corresponds to the propagating modes while the  
 
terms for <math>m\geq 1</math> (<math>n\geq 1</math>) correspond to the evanescent modes.
 
terms for <math>m\geq 1</math> (<math>n\geq 1</math>) correspond to the evanescent modes.
  
Line 95: Line 78:
  
 
A system of equations for the unknown  
 
A system of equations for the unknown  
coefficients (in the expansion  (basisrep_out_d)) of the
+
coefficients 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 109: Line 92:
 
upon <math>\Delta_l</math>, <math>j \neq l</math>. This can be accomplished by using
 
upon <math>\Delta_l</math>, <math>j \neq l</math>. This can be accomplished by using
 
[[Graf's Addition Theorem]]
 
[[Graf's Addition Theorem]]
<center><math> (transf)
+
<center><math>
 
K_\tau(k_m r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
 
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}) \,
 
\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,
 
I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l,
 
</math></center>
 
</math></center>
which is valid provided that <math>r_l < R_{jl}</math>. Here, <math>R_{jl}</math> and <math>\varphi_{jl}</math> are the distance and the angle, respectively, between bodies <math>\Delta_{j}</math> and <math>\Delta_{l}</math>.  
+
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
 
The limitation <math>r_l < R_{jl}</math> only requires that the escribed cylinder of each body
Line 126: 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>
 
<center><math>
Line 146: Line 129:
 
<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. the example in [[Cylindrical Eigenfunction Expansion]]). The total
+
<math>\Delta_l</math> (cf. the example in [[Cylindrical Eigenfunction Expansion]]).  
 +
<center><math>
 +
\phi^{\mathrm{In}}(r_l,\theta_l,z)= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
 +
\tilde{D}_{n\nu}^{l}  I_\nu (k_n
 +
r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.
 +
</math></center>
 +
The total
 
incident wavefield upon body <math>\Delta_j</math> can now be expressed as  
 
incident wavefield upon body <math>\Delta_j</math> can now be expressed as  
 
<center><math>
 
<center><math>
 
\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +
 
\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +
\sum_{j=-\infty,j \neq l}^{\infty} \, \phi_j^{\mathrm{S}}
+
\sum_{j=1,j \neq l}^{N} \, \phi_j^{\mathrm{S}}
 
(r_l,\theta_l,z)
 
(r_l,\theta_l,z)
 
</math></center>
 
</math></center>
 +
This allows us to write
 
<center><math>
 
<center><math>
= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
+
\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></center>
 +
<center><math>
 +
= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
 
  \Big[  \tilde{D}_{n\nu}^{l} +
 
  \Big[  \tilde{D}_{n\nu}^{l} +
\sum_{j=-\infty,j \neq  l}^{\infty} \sum_{\tau =
+
\sum_{j=1,j \neq  l}^{N} \sum_{\tau =
 
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
 
-\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_{jl})  \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_n
 
r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.  
 
r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.  
 +
</math></center>
 +
It therefore follows that
 +
<center><math>
 +
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></center>
 
</math></center>
  
 
= Final Equations =
 
= Final Equations =
  
The scattered and incident potential can be related by the
+
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,
 
[[Diffraction Transfer Matrix]] acting in the following way,
<center><math> (diff_op)
+
<center><math>
 
A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
 
A_{m \mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
\mu \nu} D_{n\nu}^l.
+
\mu \nu}^l D_{n\nu}^l.
 
</math></center>  
 
</math></center>  
  
The substitution of (inc_coeff) into (diff_op) gives the
+
The substitution of this into the equation for relating
 +
the coefficients <math>D_{n\nu}^l</math> and
 +
<math>A_{m \mu}^l</math>gives the
 
required equations to determine the coefficients of the scattered
 
required equations to determine the coefficients of the scattered
 
wavefields of all bodies,  
 
wavefields of all bodies,  
<center><math> (eq_op)
+
<center><math>
 
A_{m\mu}^l = \sum_{n=0}^{\infty}
 
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
+
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
 
\Big[ \tilde{D}_{n\nu}^{l} +
 
\Big[ \tilde{D}_{n\nu}^{l} +
\sum_{j=-\infty,j \neq  l}^{\infty} \sum_{\tau =
+
\sum_{j=1,j \neq  l}^{N} \sum_{\tau =
 
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
 
-\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],  
+
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big],  
 
</math></center>
 
</math></center>
<math>m \in {N}</math>, <math>l,\mu \in {Z}</math>.
+
<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].