Difference between revisions of "Category:Interaction Theory"

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Interaction theory is based on calculating a solution for a number of individual scatterers
 
Interaction theory is based on calculating a solution for a number of individual scatterers
without simply discretising the total problem. THe theory is generally applied in
+
without simply discretising the total problem. The theory is generally applied in
 
three dimensions.
 
three dimensions.
 
Essentially the [[Cylindrical Eigenfunction Expansion]]
 
Essentially the [[Cylindrical Eigenfunction Expansion]]
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a solution without any approximation. This solution method is valid, provided only that
 
a solution without any approximation. This solution method is valid, provided only that
 
an escribed circle can be drawn around each body.  
 
an escribed circle can be drawn around each body.  
 
= Illustrative Example =
 
 
 
We present an illustrative example of an interaction theory for the case of <math>n</math>
 
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
+
[[Linton and Evans 1990]] presented an [[Interaction Theory for Cylinders]]
can be derived from the [[Kagemoto and Yue Interaction Theory]] by simply assuming that each
+
which was [[Kagemoto and Yue Interaction Theory]] simplified by assuming that each
body is a cylinder.
+
body is a [[Bottom Mounted 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>
 
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 d = \alpha\,.
 
</math></center>
 
 
 
=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>\Delta_j</math> can be expressed as
 
<center><math> (basisrep_out_d)
 
\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 body <math>\Delta_j</math> can be also be expanded in
 
regular cylindrical eigenfunctions,
 
<center><math> (basisrep_in_d)
 
\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 (in the expansion  (basisrep_out_d)) of the
 
scattered wavefields of all bodies is developed. This system of
 
equations is based on transforming the
 
scattered potential of <math>\Delta_j</math> into an incident potential upon
 
<math>\Delta_l</math> (<math>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.
 
 
 
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]]
 
<center><math> (transf)
 
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 <math>\Delta_l</math> in the local coordinates of <math>\Delta_j</math>.
 
 
 
Making use of the eigenfunction expansion as well as equation (transf), the scattered potential
 
of <math>\Delta_j</math> (cf.~ (basisrep_out_d)) can be expressed in terms of the
 
incident potential upon <math>\Delta_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
 
<math>\Delta_l</math>. 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> (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 (O_x^l
 
\cos \chi + O_x^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>
 
 
 
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>
 
= \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>
 
 
 
= Final Equations =
 
 
 
The scattered and incident potential can be related by the
 
[[Diffraction Transfer Matrix]] for a [[Bottom Mounted Cylinder]] acting in the following way,
 
<center><math> (diff_op)
 
A_{\mu}^l = J'_\mu(k a_j)/H^{(1)}_\mu{}'(k a_j)D_{\mu}^l.
 
</math></center>
 
Therefore, the diffraction transfer matrix of the <math>l</math>th cylinder (having radius <math>a_l</math>) is diagonal and defined as
 
<center><math> (diff_op)
 
(B^l)_{\mu\mu} = J'_\mu(k a_j)/H^{(1)}_\mu{}'(k a_j).
 
</math></center>
 
 
 
The substitution of (inc_coeff) into  (diff_op) gives the
 
required equations to determine the coefficients of the scattered
 
wavefields of all bodies,
 
<center><math> (eq_op)
 
A_{\mu}^l =
 
\sum_{\nu = -\infty}^{\infty} B_{\mu}^l
 
\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],
 
</math></center>
 
<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,n</math>.
 
  
  
 
[[Category:Linear Water-Wave Theory]]
 
[[Category:Linear Water-Wave Theory]]

Latest revision as of 08:16, 19 October 2009


Interaction theory is based on calculating a solution for a number of individual scatterers without simply discretising the total problem. The theory is generally applied in three dimensions. Essentially the Cylindrical Eigenfunction Expansion surrounding each body is used coupled with some way of mapping these. Various approximations were developed until the the Kagemoto and Yue Interaction Theory which contained a solution without any approximation. This solution method is valid, provided only that an escribed circle can be drawn around each body. We present an illustrative example of an interaction theory for the case of [math]\displaystyle{ n }[/math] Linton and Evans 1990 presented an Interaction Theory for Cylinders which was Kagemoto and Yue Interaction Theory simplified by assuming that each body is a Bottom Mounted Cylinder.