Difference between revisions of "Diffraction Transfer Matrix"

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=Introduction=
+
{{complete pages}}
 +
 
 +
== Introduction ==
  
 
The diffraction transfer matrix relates the incident and scattered potential
 
The diffraction transfer matrix relates the incident and scattered potential
 
in [[Cylindrical Eigenfunction Expansion]]. The simplest problem is that
 
in [[Cylindrical Eigenfunction Expansion]]. The simplest problem is that
 
of a [[Bottom Mounted Cylinder]]. Here we present the theory for bodies of
 
of a [[Bottom Mounted Cylinder]]. Here we present the theory for bodies of
arbitrary geometery.
+
arbitrary geometry.
 +
While [[Kagemoto and Yue 1986]] presented theory 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 [[Goo and Yoshida 1990]] 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 [[Goo and Yoshida 1990]] used the representation of the free surface
 +
finite depth Green's function given by [[Black 1975]] and
 +
[[Fenton 1978]].  Their 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 [[John 1950]]. 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.
 +
The theory is extended to [[Infinite Depth]] in [[Diffraction Transfer Matrix for Infinite Depth]]
  
=Eigenfunction expansion of the potential=
+
== Eigenfunction expansion of the potential ==
  
 
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 expanded in the [[Cylindrical Eigenfunction Expansion]],
<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 = -
Line 17: Line 43:
 
with discrete coefficients <math>A_{m \mu}^j</math>, where
 
with discrete coefficients <math>A_{m \mu}^j</math>, where
 
<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>
 
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},
Line 28: Line 54:
 
and <math>K_\nu</math> denote the modified Bessel functions of the first and
 
and <math>K_\nu</math> denote the modified Bessel functions of the first and
 
second kind, respectively, both of order <math>\nu</math>.
 
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.
  
 
+
== Calculation of the diffraction transfer matrix for bodies of arbitrary geometry ==
=Calculation of the diffraction transfer matrix for bodies of arbitrary geometry=
 
  
 
The scattered and incident potential can therefore be related by a
 
The scattered and incident potential can therefore be related by a
 
diffraction transfer operator acting in the following way,
 
diffraction transfer operator 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}^j = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{m n
\mu \nu} D_{n\nu}^l.
+
\mu \nu}^j D_{n\nu}^j.
 
</math></center>  
 
</math></center>  
  
 
Before we can apply the interaction theory we require the diffraction
 
Before we can apply the interaction theory we require the diffraction
transfer matrices <math>\mathbf{B}_j</math> which relate the incident and the
+
transfer matrices <math>\mathbf{B}^j</math> which relate the incident and the
 
scattered potential for a body <math>\Delta_j</math> in isolation.  
 
scattered potential for a body <math>\Delta_j</math> in isolation.  
The elements of the diffraction transfer matrix, <math>({\mathbf B}_j)_{pq}</math>,
+
The elements of the diffraction transfer matrix, <math>({\mathbf B}^j)_{pq}</math>,
 
are the coefficients of the  
 
are the coefficients of the  
 
<math>p</math>th partial wave of the scattered potential due to a single
 
<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>.  
 
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
 
It should be noted that, instead of using the source strength distribution
Line 80: Line 81:
 
matrix from the solution of this integral equation.  
 
matrix from the solution of this integral equation.  
 
An outline of this method for water of finite
 
An outline of this method for water of finite
depth is given by [[kashiwagi00]]. We will present
+
depth is given by [[Kashiwagi 2000]]. We will present
 
here a derivation of the diffraction transfer matrices for the case
 
here a derivation of the diffraction transfer matrices for the case
 
infinite depth based on a solution
 
infinite depth based on a solution
Line 91: Line 92:
 
<center><math>
 
<center><math>
 
G(r,\theta,z;s,\varphi,c)=  \frac{1}{\pi} \sum_{m=0}^{\infty}
 
G(r,\theta,z;s,\varphi,c)=  \frac{1}{\pi} \sum_{m=0}^{\infty}
\frac{k_m^2+\alpha^2}{d(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+d) \cos
+
\frac{k_m^2+\alpha^2}{H(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+H) \cos
k_m(c+d) \sum_{\nu=-\infty}^{\infty} K_\nu(k_m r) I_\nu(k_m s) \mathrm{e}^{\mathrm{i}\nu
+
k_m(c+H) \sum_{\nu=-\infty}^{\infty} K_\nu(k_m r) I_\nu(k_m s) \mathrm{e}^{\mathrm{i}\nu
 
(\theta - \varphi)},
 
(\theta - \varphi)},
 
</math></center>
 
</math></center>
 
given by [[Black 1975]] and [[Fenton 1978]] is used.  
 
given by [[Black 1975]] and [[Fenton 1978]] is used.  
The elements of <math>{\mathbf B}_j</math> are therefore given by
+
The elements of <math>{\mathbf B}^j</math> are therefore given by
 
<center><math>
 
<center><math>
({\mathbf B}_j)_{pq} = \frac{1}{\pi}
+
({\mathbf B}^j)_{pq} = \frac{1}{\pi}
\frac{(k_m^2+\alpha^2)\cos^2 k_md}{d(k_m^2+\alpha^2)-\alpha}
+
\frac{(k_m^2+\alpha^2)\cos^2 k_mH}{H(k_m^2+\alpha^2)-\alpha}
\int\limits_{\Gamma_j} \cos k_m(c+d) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p
+
\int\limits_{\Gamma_j} \cos k_m(c+H) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p
 
\varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta}
 
\varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta}
 
</math></center>
 
</math></center>
Line 107: Line 108:
 
due to an incident potential of mode <math>q</math> of the form
 
due to an incident potential of mode <math>q</math> of the form
 
<center><math>
 
<center><math>
\phi_q^{\mathrm{I}}(s,\varphi,c) = \frac{\cos k_m(c+d)}{\cos kd} K_q
+
\phi_q^{\mathrm{I}}(s,\varphi,c) = \frac{\cos k_m(c+H)}{\cos k_m H} I_q
 
(k_m s) \mathrm{e}^{\mathrm{i}q \varphi}
 
(k_m s) \mathrm{e}^{\mathrm{i}q \varphi}
 
</math></center>
 
</math></center>
Line 114: Line 115:
 
the source strength distribution <math>\varsigma^j</math> so that the scattered
 
the source strength distribution <math>\varsigma^j</math> so that the scattered
 
potential can be written as  
 
potential can be written as  
<center><math> (int_eq_1)
+
<center><math>  
 
\phi_j^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma_j} G
 
\phi_j^\mathrm{S}(\mathbf{y}) = \int\limits_{\Gamma_j} G
 
(\mathbf{y},\mathbf{\zeta}) \, \varsigma^j (\mathbf{\zeta})
 
(\mathbf{y},\mathbf{\zeta}) \, \varsigma^j (\mathbf{\zeta})
Line 123: Line 124:
 
function <math>\varsigma^j</math> can be found by solving an
 
function <math>\varsigma^j</math> can be found by solving an
 
integral equation. The integral equation is described in
 
integral equation. The integral equation is described in
[[Weh_Lait]] and numerical methods for its solution are outlined in
+
[[Wehausen and Laitone 1960]] and in [[Green Function Solution Method]].
[[Sarp_Isa]].
+
 
 +
== Calculation of the coefficients <math>A_{m \mu}^j</math>, using a source strength distribution ==
  
 +
The idea is to represent the scattered potential for the body <math>\Delta_j</math> in terms
 +
of a source strength distribution <math>\varsigma^j</math>, as we already did in the previous section
 +
<center><math>
 +
\phi_j^\mathrm{S}(r_j,\theta_j,z) = \int\limits_{\Gamma_j} G
 +
(r_j,\theta_j,z;s,\varphi,c) \, \varsigma^j (s,\varphi,c)
 +
\mathrm{d}\sigma(s,\varphi,c), \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>. As we will use the [[Cylindrical Eigenfunction Expansion]],
 +
we express all the variables in a cylindrical coordinate system. The expression of the [[Free-Surface Green Function]]
 +
has already been expanded in cylindrical coordinates in the previous section, so we obtain the expression
 +
of the scattered potential as follow
 +
<center><math>
 +
\phi_j^\mathrm{S}(r_j,\theta_j,z) = \int\limits_{\Gamma_j} \Big[ \frac{1}{\pi} \sum_{m=0}^{\infty}
 +
\frac{k_m^2+\alpha^2}{H(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+H) \cos
 +
k_m(c+H) \sum_{\mu=-\infty}^{\infty} K_\mu(k_m r_j) I_\mu(k_m s) \mathrm{e}^{\mathrm{i}\mu
 +
(\theta_j - \varphi)} \Big] \varsigma^j (s,\varphi,c) \mathrm{d}\sigma(s,\varphi,c)
 +
</math></center>
 +
We can rearrange this expression in order to obtain a [[Cylindrical Eigenfunction Expansion]].
 +
<center><math>
 +
\phi_j^\mathrm{S}(r_j,\theta_j,z) = \sum_{m=0}^{\infty} \frac{\cos k_m(z+H)}{\cos k_m H}
 +
\sum_{\mu=-\infty}^{\infty} \Big[ \frac{1}{\pi} \frac{(k_m^2+\alpha^2) \cos k_m H}{H(k_m^2+\alpha^2)-\alpha}
 +
\int\limits_{\Gamma_j} \cos k_m(c+H) I_\mu(k_m s) \mathrm{e}^{-\mathrm{i}\mu \varphi} \varsigma^j (s,\varphi,c) \mathrm{d}\sigma(s,\varphi,c)
 +
\Big]
 +
K_\mu(k_m r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j}
 +
</math></center>
 +
An eigenfunction matching method permits us to identify the coefficients of the scattered potential expression
 +
<center><math>
 +
A_{m \mu}^j = \frac{1}{\pi} \frac{(k_m^2+\alpha^2) \cos k_m H}{H(k_m^2+\alpha^2)-\alpha}
 +
\int\limits_{\Gamma_j} \cos k_m(c+H) I_\mu(k_m s) \mathrm{e}^{-\mathrm{i}\mu \varphi} \varsigma^j (s,\varphi,c)
 +
\mathrm{d}\sigma(s,\varphi,c)
 +
</math></center>
  
=The diffraction transfer matrix of rotated bodies=
+
== The diffraction transfer matrix of rotated bodies ==
  
 
For a non-axisymmetric body, a rotation about the mean
 
For a non-axisymmetric body, a rotation about the mean
Line 162: Line 196:
 
</math></center>
 
</math></center>
 
<center><math>
 
<center><math>
({\mathbf B}_j)_{pq} = \frac{1}{\pi}
+
({\mathbf B}^j)_{pq} = \frac{1}{\pi}
 
\frac{(k_m^2+\alpha^2)\cos^2 k_md}{d(k_m^2+\alpha^2)-\alpha}
 
\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
 
\int\limits_{\Gamma_j} \cos k_m(c+d) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p
Line 170: Line 204:
 
body due to the standard incident modes.  
 
body due to the standard incident modes.  
  
The elements of the diffraction transfer matrix <math>\mathbf{B}_j</math> are
+
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
 
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
 
rotated by the angle <math>\beta</math>, the elements of the diffraction transfer
 
matrix of the rotated body are given by
 
matrix of the rotated body are given by
 
<center><math>
 
<center><math>
({\mathbf B}_j^\beta)_{pq} = \frac{1}{\pi}
+
({\mathbf B}^j_\beta)_{pq} = \frac{1}{\pi}
 
\frac{(k_m^2+\alpha^2)\cos^2 k_md}{d(k_m^2+\alpha^2)-\alpha}
 
\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
 
\int\limits_{\Gamma_j} \cos k_m(c+d) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p
Line 185: Line 219:
 
transfer matrix. The elements of the diffraction transfer matrix
 
transfer matrix. The elements of the diffraction transfer matrix
 
corresponding to the body rotated by the angle <math>\beta</math>,
 
corresponding to the body rotated by the angle <math>\beta</math>,
<math>\mathbf{B}_j^\beta</math>, are given by
+
<math>\mathbf{B}^j_\beta</math>, are given by
<center><math> (B_rot)
+
<center><math>
(\mathbf{B}_j^\beta)_{pq} = (\mathbf{B}_j)_{pq} \, \mathrm{e}^{\mathrm{i}(q-p) \beta}.
+
(\mathbf{B}^j_\beta)_{pq} = (\mathbf{B}^j)_{pq} \, \mathrm{e}^{\mathrm{i}(q-p) \beta}.
 
</math></center>
 
</math></center>
  
 
[[Category:Interaction Theory]]
 
[[Category:Interaction Theory]]
[[Category:Linear Water-Wave Theory]]
 

Latest revision as of 08:17, 19 October 2009


Introduction

The diffraction transfer matrix relates the incident and scattered potential in Cylindrical Eigenfunction Expansion. The simplest problem is that of a Bottom Mounted Cylinder. Here we present the theory for bodies of arbitrary geometry. While Kagemoto and Yue 1986 presented theory 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 Goo and Yoshida 1990 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 Goo and Yoshida 1990 used the representation of the free surface finite depth Green's function given by Black 1975 and Fenton 1978. Their 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 John 1950. 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. The theory is extended to Infinite Depth in Diffraction Transfer Matrix for Infinite Depth

Eigenfunction expansion of the potential

The scattered potential of a body [math]\displaystyle{ \Delta_j }[/math] can be expanded in the Cylindrical Eigenfunction Expansion,

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

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.

Calculation of the diffraction transfer matrix for bodies of arbitrary geometry

The scattered and incident potential can therefore be related by a diffraction transfer operator acting in the following way,

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

Before we can apply the interaction theory we require the diffraction transfer matrices [math]\displaystyle{ \mathbf{B}^j }[/math] which relate the incident and the scattered potential for a body [math]\displaystyle{ \Delta_j }[/math] in isolation. The elements of the diffraction transfer matrix, [math]\displaystyle{ ({\mathbf B}^j)_{pq} }[/math], are the coefficients of the [math]\displaystyle{ p }[/math]th partial wave of the scattered potential due to a single unit-amplitude incident wave of mode [math]\displaystyle{ q }[/math] upon [math]\displaystyle{ \Delta_j }[/math].

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 Kashiwagi 2000. 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.

The Free-Surface Green Function for Finite Depth in cylindrical polar coordinates

[math]\displaystyle{ G(r,\theta,z;s,\varphi,c)= \frac{1}{\pi} \sum_{m=0}^{\infty} \frac{k_m^2+\alpha^2}{H(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+H) \cos k_m(c+H) \sum_{\nu=-\infty}^{\infty} K_\nu(k_m r) I_\nu(k_m s) \mathrm{e}^{\mathrm{i}\nu (\theta - \varphi)}, }[/math]

given by Black 1975 and Fenton 1978 is used. The elements of [math]\displaystyle{ {\mathbf B}^j }[/math] are therefore given by

[math]\displaystyle{ ({\mathbf B}^j)_{pq} = \frac{1}{\pi} \frac{(k_m^2+\alpha^2)\cos^2 k_mH}{H(k_m^2+\alpha^2)-\alpha} \int\limits_{\Gamma_j} \cos k_m(c+H) I_p(\eta s) \mathrm{e}^{-\mathrm{i}p \varphi} \varsigma_q^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta} }[/math]

where [math]\displaystyle{ \varsigma_q^j(\mathbf{\zeta}) }[/math] is the source strength distribution due to an incident potential of mode [math]\displaystyle{ q }[/math] of the form

[math]\displaystyle{ \phi_q^{\mathrm{I}}(s,\varphi,c) = \frac{\cos k_m(c+H)}{\cos k_m H} I_q (k_m s) \mathrm{e}^{\mathrm{i}q \varphi} }[/math]

We assume that we have represented the scattered potential in terms of the source strength distribution [math]\displaystyle{ \varsigma^j }[/math] so that the scattered potential can be written as

[math]\displaystyle{ \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]

where [math]\displaystyle{ D }[/math] is the volume occupied by the water and [math]\displaystyle{ \Gamma_j }[/math] is the immersed surface of body [math]\displaystyle{ \Delta_j }[/math]. The source strength distribution function [math]\displaystyle{ \varsigma^j }[/math] can be found by solving an integral equation. The integral equation is described in Wehausen and Laitone 1960 and in Green Function Solution Method.

Calculation of the coefficients [math]\displaystyle{ A_{m \mu}^j }[/math], using a source strength distribution

The idea is to represent the scattered potential for the body [math]\displaystyle{ \Delta_j }[/math] in terms of a source strength distribution [math]\displaystyle{ \varsigma^j }[/math], as we already did in the previous section

[math]\displaystyle{ \phi_j^\mathrm{S}(r_j,\theta_j,z) = \int\limits_{\Gamma_j} G (r_j,\theta_j,z;s,\varphi,c) \, \varsigma^j (s,\varphi,c) \mathrm{d}\sigma(s,\varphi,c), \quad \mathbf{y} \in D, }[/math]

where [math]\displaystyle{ D }[/math] is the volume occupied by the water and [math]\displaystyle{ \Gamma_j }[/math] is the immersed surface of body [math]\displaystyle{ \Delta_j }[/math]. As we will use the Cylindrical Eigenfunction Expansion, we express all the variables in a cylindrical coordinate system. The expression of the Free-Surface Green Function has already been expanded in cylindrical coordinates in the previous section, so we obtain the expression of the scattered potential as follow

[math]\displaystyle{ \phi_j^\mathrm{S}(r_j,\theta_j,z) = \int\limits_{\Gamma_j} \Big[ \frac{1}{\pi} \sum_{m=0}^{\infty} \frac{k_m^2+\alpha^2}{H(k_m^2+\alpha^2)-\alpha}\, \cos k_m(z+H) \cos k_m(c+H) \sum_{\mu=-\infty}^{\infty} K_\mu(k_m r_j) I_\mu(k_m s) \mathrm{e}^{\mathrm{i}\mu (\theta_j - \varphi)} \Big] \varsigma^j (s,\varphi,c) \mathrm{d}\sigma(s,\varphi,c) }[/math]

We can rearrange this expression in order to obtain a Cylindrical Eigenfunction Expansion.

[math]\displaystyle{ \phi_j^\mathrm{S}(r_j,\theta_j,z) = \sum_{m=0}^{\infty} \frac{\cos k_m(z+H)}{\cos k_m H} \sum_{\mu=-\infty}^{\infty} \Big[ \frac{1}{\pi} \frac{(k_m^2+\alpha^2) \cos k_m H}{H(k_m^2+\alpha^2)-\alpha} \int\limits_{\Gamma_j} \cos k_m(c+H) I_\mu(k_m s) \mathrm{e}^{-\mathrm{i}\mu \varphi} \varsigma^j (s,\varphi,c) \mathrm{d}\sigma(s,\varphi,c) \Big] K_\mu(k_m r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j} }[/math]

An eigenfunction matching method permits us to identify the coefficients of the scattered potential expression

[math]\displaystyle{ A_{m \mu}^j = \frac{1}{\pi} \frac{(k_m^2+\alpha^2) \cos k_m H}{H(k_m^2+\alpha^2)-\alpha} \int\limits_{\Gamma_j} \cos k_m(c+H) I_\mu(k_m s) \mathrm{e}^{-\mathrm{i}\mu \varphi} \varsigma^j (s,\varphi,c) \mathrm{d}\sigma(s,\varphi,c) }[/math]

The diffraction transfer matrix of rotated bodies

For a non-axisymmetric body, a rotation about the mean centre position in the [math]\displaystyle{ (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]\displaystyle{ \beta }[/math] can be easily calculated from the diffraction transfer matrix of the non-rotated body. The rotation of the body influences the form of the elements of the diffraction transfer matrices in two ways. Firstly, the angular dependence in the integral over the immersed surface of the body is altered and, secondly, the source strength distribution function is different if the body is rotated. However, the source strength distribution function of the rotated body can be obtained by calculating the response of the non-rotated body due to rotated incident potentials. It will be shown that the additional angular dependence can be easily factored out of the elements of the diffraction transfer matrix.

The additional angular dependence caused by the rotation of the incident potential can be factored out of the normal derivative of the incident potential such that

[math]\displaystyle{ \frac{\partial \phi_{q\beta}^{\mathrm{I}}}{\partial n} = \frac{\partial \phi_{q}^{\mathrm{I}}}{\partial n} \mathrm{e}^{\mathrm{i}q \beta}, }[/math]

where [math]\displaystyle{ \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

[math]\displaystyle{ \varsigma_{q\beta}^j = \varsigma_q^j \, \mathrm{e}^{\mathrm{i}q \beta}. }[/math]
[math]\displaystyle{ ({\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]

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]\displaystyle{ \mathbf{B}^j }[/math] are given by equations (B_elem). Keeping in mind that the body is rotated by the angle [math]\displaystyle{ \beta }[/math], the elements of the diffraction transfer matrix of the rotated body are given by

[math]\displaystyle{ ({\mathbf B}^j_\beta)_{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+\beta)} \varsigma_{q\beta}^j(\mathbf{\zeta}) \mathrm{d}\sigma_\mathbf{\zeta} }[/math]

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]\displaystyle{ \beta }[/math], [math]\displaystyle{ \mathbf{B}^j_\beta }[/math], are given by

[math]\displaystyle{ (\mathbf{B}^j_\beta)_{pq} = (\mathbf{B}^j)_{pq} \, \mathrm{e}^{\mathrm{i}(q-p) \beta}. }[/math]