View source for Diffraction Transfer Matrix

==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,
<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> 

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. 

The [[Free-Surface Green Function]] for [[Finite Depth]]
in cylindrical polar coordinates
<center><math>
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
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]] is used. 
The elements of <math>{\mathbf B}_j</math> are therefore given by
<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>
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{\cos k_m(c+d)}{\cos kd} K_q
(k_m s) \mathrm{e}^{\mathrm{i}q \varphi}
</math></center>

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]].


===The diffraction transfer matrix of rotated bodies===

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

The additional angular dependence caused by the rotation of the
incident potential can be factored out of the normal derivative of the
incident potential such that
<center><math>
\frac{\partial \phi_{q\beta}^{\mathrm{I}}}{\partial n} =
\frac{\partial \phi_{q}^{\mathrm{I}}}{\partial n}
\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>
<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>
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>
({\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></center>

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>

[[Category:Interaction Theory]]
[[Category:Linear Water-Wave Theory]]