Category:Interaction Theory: Difference between revisions

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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. Essentially the [[Cylindrical Eigenfunction Expansion]]
without simply discretising the total problem. THe theory is generally applied in
surrounding each body is used coupled with some way of mapping these.
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.
 
= Illustrative Example =
 
We present an illustrative example of an interaction theory for the case of <math>n</math>
[[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 =
 
The problem consists of <math>n</math> cylinders of radius
<math>a_j</math> after we have [[Removing the Depth Dependence|Removed the Depth Dependence]]
subject to [[Helmholtz's equation]] with
surface <math>\Gamma_j</math>.
<center><math>
\nabla^2 \phi = 0, \;  \mathbf{y} \in D
</math></center>
<center><math> 
\frac{\partial \phi}{\partial z} = \alpha \phi, \;
{\mathbf{x}} \in \Gamma^\mathrm{f},
</math></center>
<center><math>
\frac{\partial \phi}{\partial z} = 0, \;  \mathbf{y} \in D, \ z=-d,
</math></center>
where <math>D</math> is the
is the domain occupied by the water and
<math>\Gamma^\mathrm{f}</math> is the free water surface. At the immersed body
surface <math>\Gamma_j</math> of body <math>\Delta_j</math>, <math>j=1,\dots,N</math>, the water velocity potential has to
equal the normal velocity of the body <math>\mathbf{v}_j</math>,
<center><math>
\frac{\partial \phi}{\partial n} = \mathbf{v}_j, \;  {\mathbf{y}}
\in \Gamma_j.
</math></center>
where the normal derivative is given by the particaluar equations of motion of the body.
Moreover, the [[Sommerfeld Radiation Condition]] is imposed.
 
=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_{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></center>
with discrete coefficients <math>A_{m \mu}^j</math>, where
<center><math>
f_m(z) = \frac{\cos k_m (z+d)}{\cos k_m d}.
</math></center>
and
where <math>k_n</math> are found from <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]
<center><math> (eq_km)
\alpha + k_m \tan k_m d = 0\,.
</math></center>
where <math>k_0</math> is the
imaginary root with positive imaginary part
and <math>k_m</math>, <math>m>0</math>, are given the positive real roots ordered
with increasing size.
 
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_{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></center>
with discrete coefficients <math>D_{n\nu}^j</math>. In these expansions, <math>I_\nu</math>
and <math>K_\nu</math> denote the modified [http://en.wikipedia.org/wiki/Bessel_function : Bessel functions]
of the first and 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
terms for <math>m\geq 1</math> (<math>n\geq 1</math>) correspond to the evanescent modes.
 
=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.
Making use of the periodicity of the geometry and of the ambient incident
wave, this system of equations can then be simplified.
 
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)
K_\tau(k_m r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
\sum_{\nu = - \infty}^{\infty} K_{\tau + \nu} (k_m R_{jl}) \,
I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l,
</math></center>
which is valid provided that <math>r_l < R_{jl}</math>. Here, <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>.
 
The limitation <math>r_l < R_{jl}</math> 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 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_{m=0}^\infty f_m(z) \sum_{\tau = -
\infty}^{\infty} A_{m\tau}^j \sum_{\nu = -\infty}^{\infty}
(-1)^\nu K_{\tau-\nu} (k_m  R_{jl}) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu
\theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}}
</math></center>
<center><math>
= \sum_{m=0}^\infty f_m(z) \sum_{\nu =
-\infty}^{\infty}  \Big[ \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j
(-1)^\nu K_{\tau-\nu} (k_m R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu)
\varphi_{jl}} \Big] I_\nu (k_m 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}_{n\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]]). 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=-\infty,j \neq l}^{\infty} \, \phi_j^{\mathrm{S}}
(r_l,\theta_l,z)
</math></center>
<center><math>
= \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty}
\Big[  \tilde{D}_{n\nu}^{l} +
\sum_{j=1,j \neq  l}^{N} \sum_{\tau =
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
R_{jl})  \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_n
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]] 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>
 
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_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu}^{l} +
\sum_{j=1,j \neq  l}^{N} \sum_{\tau =
-\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu}  (k_n
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>.


The theory was fully developed in [[Kagemoto and Yue Interaction Theory]] which contained
a solution without any approximation.


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

Revision as of 10:42, 18 June 2006

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.

Illustrative Example

We present an illustrative example of an interaction theory for the case of [math]\displaystyle{ n }[/math] 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

The problem consists of [math]\displaystyle{ n }[/math] cylinders of radius [math]\displaystyle{ a_j }[/math] after we have Removed the Depth Dependence subject to Helmholtz's equation with surface [math]\displaystyle{ \Gamma_j }[/math].

[math]\displaystyle{ \nabla^2 \phi = 0, \; \mathbf{y} \in D }[/math]
[math]\displaystyle{ \frac{\partial \phi}{\partial z} = \alpha \phi, \; {\mathbf{x}} \in \Gamma^\mathrm{f}, }[/math]
[math]\displaystyle{ \frac{\partial \phi}{\partial z} = 0, \; \mathbf{y} \in D, \ z=-d, }[/math]

where [math]\displaystyle{ D }[/math] is the is the domain occupied by the water and [math]\displaystyle{ \Gamma^\mathrm{f} }[/math] is the free water surface. At the immersed body surface [math]\displaystyle{ \Gamma_j }[/math] of body [math]\displaystyle{ \Delta_j }[/math], [math]\displaystyle{ j=1,\dots,N }[/math], the water velocity potential has to equal the normal velocity of the body [math]\displaystyle{ \mathbf{v}_j }[/math],

[math]\displaystyle{ \frac{\partial \phi}{\partial n} = \mathbf{v}_j, \; {\mathbf{y}} \in \Gamma_j. }[/math]

where the normal derivative is given by the particaluar equations of motion of the body. Moreover, the Sommerfeld Radiation Condition is imposed.

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]\displaystyle{ \Delta_j }[/math] can be expressed as

[math]\displaystyle{ (basisrep_out_d) \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+d)}{\cos k_m d}. }[/math]

and where [math]\displaystyle{ k_n }[/math] are found from [math]\displaystyle{ \alpha }[/math] by the Dispersion Relation for a Free Surface

[math]\displaystyle{ (eq_km) \alpha + k_m \tan k_m d = 0\,. }[/math]

where [math]\displaystyle{ k_0 }[/math] is the imaginary root with positive imaginary part and [math]\displaystyle{ k_m }[/math], [math]\displaystyle{ m\gt 0 }[/math], are given the positive real roots ordered with increasing size.

The incident potential upon body [math]\displaystyle{ \Delta_j }[/math] can be also be expanded in regular cylindrical eigenfunctions,

[math]\displaystyle{ (basisrep_in_d) \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 in (basisrep_out_d) (and (basisrep_in_d)) 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.

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]\displaystyle{ \Delta_j }[/math] into an incident potential upon [math]\displaystyle{ \Delta_l }[/math] ([math]\displaystyle{ 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. Making use of the periodicity of the geometry and of the ambient incident wave, this system of equations can then be simplified.

The scattered potential [math]\displaystyle{ \phi_j^{\mathrm{S}} }[/math] of body [math]\displaystyle{ \Delta_j }[/math] needs to be represented in terms of the incident potential [math]\displaystyle{ \phi_l^{\mathrm{I}} }[/math] upon [math]\displaystyle{ \Delta_l }[/math], [math]\displaystyle{ j \neq l }[/math]. This can be accomplished by using Graf's Addition Theorem

[math]\displaystyle{ (transf) K_\tau(k_m r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} = \sum_{\nu = - \infty}^{\infty} K_{\tau + \nu} (k_m R_{jl}) \, I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l, }[/math]

which is valid provided that [math]\displaystyle{ r_l \lt R_{jl} }[/math]. Here, [math]\displaystyle{ (R_{jl},\varphi_{jl}) }[/math] are the polar coordinates of the mean centre position of [math]\displaystyle{ \Delta_{l} }[/math] in the local coordinates of [math]\displaystyle{ \Delta_{j} }[/math].

The limitation [math]\displaystyle{ r_l \lt R_{jl} }[/math] only requires that the escribed cylinder of each body [math]\displaystyle{ \Delta_l }[/math] does not enclose any other origin [math]\displaystyle{ O_j }[/math] ([math]\displaystyle{ 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]\displaystyle{ \Delta_l }[/math] does not enclose any other origin [math]\displaystyle{ O_j }[/math] ([math]\displaystyle{ 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 eigenfunction expansion as well as equation (transf), the scattered potential of [math]\displaystyle{ \Delta_j }[/math] (cf.~ (basisrep_out_d)) can be expressed in terms of the incident potential upon [math]\displaystyle{ \Delta_l }[/math] as

[math]\displaystyle{ \phi_j^{\mathrm{S}} (r_l,\theta_l,z) = \sum_{m=0}^\infty f_m(z) \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j \sum_{\nu = -\infty}^{\infty} (-1)^\nu K_{\tau-\nu} (k_m R_{jl}) I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l} \mathrm{e}^{\mathrm{i}(\tau-\nu) \varphi_{jl}} }[/math]
[math]\displaystyle{ = \sum_{m=0}^\infty f_m(z) \sum_{\nu = -\infty}^{\infty} \Big[ \sum_{\tau = - \infty}^{\infty} A_{m\tau}^j (-1)^\nu K_{\tau-\nu} (k_m R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_m r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}. }[/math]

The ambient incident wavefield [math]\displaystyle{ \phi^{\mathrm{In}} }[/math] can also be expanded in the eigenfunctions corresponding to the incident wavefield upon [math]\displaystyle{ \Delta_l }[/math]. Let [math]\displaystyle{ \tilde{D}_{n\nu}^{l} }[/math] denote the coefficients of this ambient incident wavefield in the incoming eigenfunction expansion for [math]\displaystyle{ \Delta_l }[/math] (cf. the example in Cylindrical Eigenfunction Expansion). The total incident wavefield upon body [math]\displaystyle{ \Delta_j }[/math] can now be expressed as

[math]\displaystyle{ \phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) + \sum_{j=-\infty,j \neq l}^{\infty} \, \phi_j^{\mathrm{S}} (r_l,\theta_l,z) }[/math]
[math]\displaystyle{ = \sum_{n=0}^\infty f_n(z) \sum_{\nu = -\infty}^{\infty} \Big[ \tilde{D}_{n\nu}^{l} + \sum_{j=1,j \neq l}^{N} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n R_{jl}) \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] I_\nu (k_n r_l) \mathrm{e}^{\mathrm{i}\nu \theta_l}. }[/math]

Final Equations

The scattered and incident potential can be related by the Diffraction Transfer Matrix acting in the following way,

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

The substitution of (inc_coeff) into (diff_op) gives the required equations to determine the coefficients of the scattered wavefields of all bodies,

[math]\displaystyle{ (eq_op) A_{m\mu}^l = \sum_{n=0}^{\infty} \sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu}^{l} + \sum_{j=1,j \neq l}^{N} \sum_{\tau = -\infty}^{\infty} A_{n\tau}^j (-1)^\nu K_{\tau - \nu} (k_n R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big], }[/math]

[math]\displaystyle{ m \in \mathbb{N} }[/math], [math]\displaystyle{ \mu \in \mathbb{Z} }[/math], [math]\displaystyle{ l=1,\dots,N }[/math].

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