View source for Interaction Theory for Cylinders

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== Introduction ==

We present an illustrative example of an interaction theory for the case of <math>n</math>
[[Bottom Mounted Cylinder|Bottom Mounted Cylinders]]. This theory was presented in [[Linton and Evans 1990]] and it
can be derived from the [[Kagemoto and Yue Interaction Theory]] by simply assuming that each
body is a cylinder.

== Equations of Motion = =

After we have [[Removing the Depth Dependence|Removed the Depth Dependence]]
the problem consists of <math>n</math> cylinders of radius <math>a_j</math> 
whose center is at <math>(x_j,y_j)</math>
subject to [[Helmholtz's Equation]] 
<center><math> 
\nabla^2 \phi + k^2\phi= 0,
</math></center>
where <math>k</math> is the positive real root of the [[Dispersion Relation for a Free Surface]]
<center><math>
k \tanh k h = \alpha,.
</math></center>
where <math>\alpha</math> is the [[Infinite Depth]] wave number and 
<math>h</math> is the water depth.

== Eigenfunction expansion of the potential ==

Each cylinder 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 cylinder 
<math>j</math> can be expressed as
<center><math>
\phi_j^\mathrm{S} (r_j,\theta_j,z) =  \sum_{\mu = -
\infty}^{\infty} A_{\mu}^j H^{(1)}_\mu (k r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j},
</math></center>
with discrete coefficients <math>A_{\mu}^j</math>, where <math>(r_j,\theta_j)</math>
are polar coordinates centered at center of the <math>j</math>th cylinder. 

The incident potential upon cylinder <math>j</math> can be also be expanded in
regular cylindrical eigenfunctions, 
<center><math> 
\phi_j^\mathrm{I} (r_j,\theta_j,z) = 
\sum_{\nu = - \infty}^{\infty} D_{\nu}^j J_\nu (k r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j},
</math></center>
with discrete coefficients <math>D_{\nu}^j</math>.

In these expansions, <math>J_\nu</math>
and <math>H^{(1)}_\nu</math> denote Bessel and Hankel function 
respectively ([http://en.wikipedia.org/wiki/Bessel_function Bessel functions])
both of first kind and order <math>\nu</math>. For
comparison with the [[Kagemoto and Yue Interaction Theory]]
(which is written slightly differently), we remark that, for real <math>x</math>,
<center><math> 
K_\nu (-\mathrm{i}x) = \frac{\pi \mathrm{i}^{\nu+1}}{2} H_\nu^{(1)}(x) \quad
\mathrm{and}  \quad
I_\nu (-\mathrm{i}x) = \mathrm{i}^{-\nu} J_\nu(x)
</math></center>
with <math>I_\nu</math> and <math>K_\nu</math> denoting the modified 
Bessel functions of first and second kind, respectively, both of order <math>\nu</math>.

== Derivation of the system of equations ==

A system of equations for the unknown 
coefficients of the
scattered wavefields of all cylinders is developed. This system of
equations is based on transforming the 
scattered potential of cylinder <math>j</math> into an incident potential upon cylinder 
<math>l</math> (<math>j \neq l</math>). Doing this for all cylinders simultaneously,
and relating the incident and scattered potential for each cylinder, a system
of equations for the unknown coefficients is developed. 

The scattered potential <math>\phi_j^{\mathrm{S}}</math> of cylinder <math>j</math> needs to be
represented in terms of the incident potential <math>\phi_l^{\mathrm{I}}</math>
upon cylinder <math>l</math>, <math>j \neq l</math>. This can be accomplished by using
[[Graf's Addition Theorem]]
<center><math>
H^{(1)}_\tau(k r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
\sum_{\nu = - \infty}^{\infty} H^{(1)}_{\tau + \nu} (k R_{jl}) \,
J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l,
</math></center>
where <math>(R_{jl},\varphi_{jl})</math> are the polar coordinates of the mean centre position of cylinder <math>l</math> in the local coordinates of cylinder <math>j</math>.

Making use of the eigenfunction expansion as well as [[Graf's Addition Theorem]], the scattered potential
of cylinder <math>j</math> can be expressed in terms of the
incident potential upon cylinder <math>l</math> as
<center><math>
\phi_j^{\mathrm{S}} (r_l,\theta_l,z) 
= \sum_{\tau = -
\infty}^{\infty} A_{\tau}^j \sum_{\nu = -\infty}^{\infty}
 H^{(1)}_{\tau-\nu} (k  R_{jl}) J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu
\theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}} 
</math></center>
<center><math>
=  \sum_{\nu =
-\infty}^{\infty}  \Big[ \sum_{\tau = - \infty}^{\infty} A_{\tau}^j
H^{(1)}_{\tau-\nu} (k R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu)
\varphi_{jl}} \Big] J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}.  
</math></center>
The ambient incident wavefield <math>\phi^{\mathrm{In}}</math> can also be
expanded in the eigenfunctions corresponding to the incident wavefield upon cylinder 
<math>l</math> (cf. the example in [[Cylindrical Eigenfunction Expansion]])
<center><math>
\phi^{\mathrm{In}}(r_l,\theta_l,z) = A \frac{g}{\omega} \, e^{\mathrm{i} k (x_l
\cos \chi + y_l \sin \chi)} \sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 -  \chi)}
J_\mu(k r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}.
</math></center>
Let <math>\tilde{D}_{\nu}^{l}</math> denote the coefficients of this
ambient incident wavefield in the incoming eigenfunction expansion for
<math>\Delta_l</math> so that
<center><math>
\tilde{D}_{\nu}^{l} = A \frac{g}{\omega} \, e^{\mathrm{i} k (x_l
\cos \chi + y_l \sin \chi)} \mathrm{e}^{\mathrm{i}\nu (\pi/2 -  \chi)}.
</math></center>


The total
incident wavefield upon cylinder <math>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>
which can be written as
<center><math>
\sum_{\nu = -\infty}^{\infty}
{D}_\nu^{l} J_\nu (kr_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}
 = \sum_{\nu = -\infty}^{\infty}
 \Big[\tilde{D}_\nu^{l} +
\sum_{j=1,j \neq  l}^{n} \sum_{\tau =
-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu}  (k
R_{jl})  \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] J_\nu (k
r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}. 
</math></center>
Therefore it follows that 
<center><math>
{D}_\nu^{l} 
 = \tilde{D}_\nu^{l} +
\sum_{j=1,j \neq  l}^{n} \sum_{\tau =
-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu}  (k
R_{jl})  \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}}. 
</math></center>

== Final Equations ==

The scattered and incident potential can be related by the
[[Diffraction Transfer Matrix]] for a [[Bottom Mounted Cylinder]] so that,
<center><math>
 A_{\mu}^l= -\frac{J'_\nu(k a_l)}{H^{(1)}_\nu{}'(k a_l)} D_{\nu}^l .
</math></center> 

This gives the
required equations to determine the coefficients of the scattered
wavefields of all bodies, 
<center><math>
\frac{J'_\nu(k a_l)}{H^{(1)}_\nu{}'(k a_l)} \Big[
\sum_{j=1,j \neq  l}^{N} \sum_{\tau =
-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu}  (k
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big]
 + A_{\nu}^l = -
\frac{J'_\nu(k a_l)}{H^{(1)}_\nu{}'(k a_l)}
\tilde{D}_{\nu}^{l},
</math></center>
<math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,n</math>.

[[Category:Interaction Theory]]