View source for Kagemoto and Yue Interaction Theory for Infinite Depth

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==Introduction==

[[Kagemoto and Yue Interaction Theory]] applies in [[Finite Depth]] water.
The theory was extended by [[Peter and Meylan 2004]] to [[Infinite Depth]] water
and we present this theory here. 

==Eigenfunction expansion of the potential==

The scattered potential of body <math>\Delta_j</math> can be expanded using
the [[Cylindrical Eigenfunction Expansion]] for [[Infinite Depth]],
<center><math> 
\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> 
([http://en.wikipedia.org/wiki/Bessel_function : Bessel functions]). 
The incident potential upon body <math>\Delta_j</math> can be expanded in
cylindrical eigenfunctions, 
<center><math> 
\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==

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>. This can be accomplished by using
[[Graf's Addition Theorem]] to obtain, 
<center><math> 
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,
</math></center>
<center><math> 
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,
</math></center>
which is valid provided that <math>r_l < R_{jl}</math>. This limitation
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>
\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}}
</math></center>
<center><math>
 \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
</math></center>
<center><math>
= \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}
</math></center>
<center><math>
 + \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. 
</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> (cf. the example in [[Cylindrical Eigenfunction Expansion]]). Let <math>D_{l0\mu}^{\mathrm{In}}</math> denote the coefficients of this
ambient incident wavefield corresponding to the propagating modes and
<math>D_{l\mu}^{\mathrm{In}} (\cdot)</math>  denote the coefficients functions
corresponding to the decaying modes (which are identically zero) of
the incoming eigenfunction expansion for <math>\Delta_l</math>. The total
incident wavefield upon body <math>\Delta_j</math> can now be expressed as 
<center><math>
\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></center>
<center><math>
= \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}
</math></center>
<center><math>
+ \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.
</math></center>
The coefficients of the total incident potential upon <math>\Delta_l</math> are
therefore given by
<center><math>
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>
D_{\mu}^l(\eta) = 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}}.
</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. The [[Diffraction Transfer Matrix for Infinite Depth]]
(strictly an operator)
<math>B_l</math> relates the coefficients of the incident and scattered
partial waves, such that
<center><math> 
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>.
Fro the [[Finite Depth]] case, <math>B_l</math> is an infinite dimensional matrix. For 
[[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 the [[Diffraction Transfer Matrix for Infinite Depth]] in the
following ways,
<center><math> 
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,
</math></center> 
<center><math>
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.
</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 Matrix for Infinite Depth]] are substituted we
obtain the
required equations to determine the coefficients and coefficient
functions of the scattered wavefields of all bodies,
<center><math> 
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]
</math></center>
<center><math> 
 + \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,
</math></center>
<center><math> 
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]
</math></center>
<center><math> 
 + \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,
</math></center>
<math>n \in \mathbb{Z},\, l = 1, \ldots, N</math>. It has to be noted that all
equations are coupled so that it is necessary to solve for all
scattered coefficients and coefficient functions simultaneously. 

For numerical calculations, the infinite sums have to be truncated and
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>
\mathbf{B}_l = \left[ 
\begin{matrix} \mathbf{B}_l^{\mathrm{pp}} & \mathbf{B}_l^{\mathrm{pd}}\\
\mathbf{B}_l^{\mathrm{dp}} & \mathbf{B}_l^{\mathrm{dd}}
\end{matrix} \right],
</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}_{lj}</math> given by
<center><math> 
({\mathbf T}_{lj})_{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>
({\mathbf T}_{lj})_{pq} = (-1)^{q} \, K_{p-q} (\eta R_{jl}) \, \mathrm{e}^{\mathbf{i}
(p-q) \vartheta_{jl}}
</math></center>
for the decaying modes, a linear system of equations
for the unknown coefficients follows
<center><math> 
{\mathbf a}_l = 
{\mathbf {B}}_l \Big( 
{\mathbf d}_l^{\mathrm{In}} +
\sum_{j=1,j \neq l}^{N} {\mathbf T}_{lj} \,
 {\mathbf a}_j \Big), \quad  l=1, \ldots, N.
</math></center>
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.






[[Category:Interaction Theory]]