Difference between revisions of "Interaction Theory for Cylinders"

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= Introduction =
+
{{complete pages}}
 +
 
 +
== Introduction ==
  
 
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>
Line 6: Line 8:
 
body is a cylinder.
 
body is a cylinder.
  
= Equations of Motion =  
+
== Equations of Motion ==
  
 
After we have [[Removing the Depth Dependence|Removed the Depth Dependence]]
 
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>  
 
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]]  
 
subject to [[Helmholtz's Equation]]  
 
<center><math>  
 
<center><math>  
\nabla^2 \phi -k^2\phi= 0,
+
\nabla^2 \phi + k^2\phi= 0,
 
</math></center>
 
</math></center>
 
where <math>k</math> is the positive real root of the [[Dispersion Relation for a Free Surface]]
 
where <math>k</math> is the positive real root of the [[Dispersion Relation for a Free Surface]]
 
<center><math>
 
<center><math>
k \tanh k d = \alpha\,.
+
k \tanh k h = \alpha,.
 
</math></center>
 
</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=
+
== Eigenfunction expansion of the potential ==
  
Each body is subject to an incident potential and moves in response to this
+
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
 
incident potential to produce a scattered potential. Each of these is
 
expanded using the [[Cylindrical Eigenfunction Expansion]]
 
expanded using the [[Cylindrical Eigenfunction Expansion]]
The scattered potential of a body
+
The scattered potential of cylinder
<math>\Delta_j</math> can be expressed as
+
<math>j</math> can be expressed as
<center><math> (basisrep_out_d)
+
<center><math>
 
\phi_j^\mathrm{S} (r_j,\theta_j,z) =  \sum_{\mu = -
 
\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},
 
\infty}^{\infty} A_{\mu}^j H^{(1)}_\mu (k r_j) \mathrm{e}^{\mathrm{i}\mu \theta_j},
Line 33: Line 38:
 
are polar coordinates centered at center of the <math>j</math>th cylinder.  
 
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
+
The incident potential upon cylinder <math>j</math> can be also be expanded in
 
regular cylindrical eigenfunctions,  
 
regular cylindrical eigenfunctions,  
<center><math> (basisrep_in_d)
+
<center><math>  
 
\phi_j^\mathrm{I} (r_j,\theta_j,z) =  
 
\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},
 
\sum_{\nu = - \infty}^{\infty} D_{\nu}^j J_\nu (k r_j) \mathrm{e}^{\mathrm{i}\nu \theta_j},
 
</math></center>
 
</math></center>
with discrete coefficients <math>D_{\nu}^j</math>. In these expansions, <math>J_\nu</math>
+
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  
 
and <math>H^{(1)}_\nu</math> denote Bessel and Hankel function  
respectively ([http://en.wikipedia.org/wiki/Bessel_function : Bessel functions])
+
respectively ([http://en.wikipedia.org/wiki/Bessel_function Bessel functions])
 
both of first kind and order <math>\nu</math>. For
 
both of first kind and order <math>\nu</math>. For
 
comparison with the [[Kagemoto and Yue Interaction Theory]]
 
comparison with the [[Kagemoto and Yue Interaction Theory]]
Line 53: Line 60:
 
Bessel functions of first and second kind, respectively, both of order <math>\nu</math>.
 
Bessel functions of first and second kind, respectively, both of order <math>\nu</math>.
  
=Derivation of the system of equations=
+
== Derivation of the system of equations ==
  
 
A system of equations for the unknown  
 
A system of equations for the unknown  
coefficients (in the expansion  (basisrep_out_d)) of the
+
coefficients of the
scattered wavefields of all bodies is developed. This system of
+
scattered wavefields of all cylinders is developed. This system of
 
equations is based on transforming the  
 
equations is based on transforming the  
scattered potential of <math>\Delta_j</math> into an incident potential upon
+
scattered potential of cylinder <math>j</math> into an incident potential upon cylinder
<math>\Delta_l</math> (<math>j \neq l</math>). Doing this for all bodies simultaneously,
+
<math>l</math> (<math>j \neq l</math>). Doing this for all cylinders simultaneously,
and relating the incident and scattered potential for each body, a system
+
and relating the incident and scattered potential for each cylinder, a system
 
of equations for the unknown coefficients is developed.  
 
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
+
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>
 
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
+
upon cylinder <math>l</math>, <math>j \neq l</math>. This can be accomplished by using
 
[[Graf's Addition Theorem]]
 
[[Graf's Addition Theorem]]
<center><math> (transf)
+
<center><math>
 
H^{(1)}_\tau(k r_j) \mathrm{e}^{\mathrm{i}\tau (\theta_j-\varphi_{jl})} =
 
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}) \,
 
\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,
 
J_\nu (k r_l) \mathrm{e}^{\mathrm{i}\nu (\pi - \theta_l + \varphi_{jl})}, \quad j \neq l,
 
</math></center>
 
</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>.
+
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 equation (transf), the scattered potential
+
Making use of the eigenfunction expansion as well as [[Graf's Addition Theorem]], the scattered potential
of <math>\Delta_j</math> (cf.~ (basisrep_out_d)) can be expressed in terms of the
+
of cylinder <math>j</math> can be expressed in terms of the
incident potential upon <math>\Delta_l</math> as
+
incident potential upon cylinder <math>l</math> as
 
<center><math>
 
<center><math>
 
\phi_j^{\mathrm{S}} (r_l,\theta_l,z)  
 
\phi_j^{\mathrm{S}} (r_l,\theta_l,z)  
Line 92: Line 99:
 
</math></center>
 
</math></center>
 
The ambient incident wavefield <math>\phi^{\mathrm{In}}</math> can also be
 
The ambient incident wavefield <math>\phi^{\mathrm{In}}</math> can also be
expanded in the eigenfunctions corresponding to the incident wavefield upon
+
expanded in the eigenfunctions corresponding to the incident wavefield upon cylinder
<math>\Delta_l</math>. Let <math>\tilde{D}_{\nu}^{l}</math> denote the coefficients of this
+
<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
 
ambient incident wavefield in the incoming eigenfunction expansion for
<math>\Delta_l</math> (cf. the example in [[Cylindrical Eigenfunction Expansion]]).
+
<math>\Delta_l</math> so that
 
 
 
<center><math>
 
<center><math>
\phi^{\mathrm{In}}(r_l,\theta_l,z) = A \frac{g}{\omega} \, e^{\mathrm{i} k (O_x^l
+
\tilde{D}_{\nu}^{l} = A \frac{g}{\omega} \, e^{\mathrm{i} k (x_l
\cos \chi + O_x^l \sin \chi)} \sum_{\mu = -\infty}^{\infty} \mathrm{e}^{\mathrm{i}\mu (\pi/2 -  \chi)}
+
\cos \chi + y_l \sin \chi)} \mathrm{e}^{\mathrm{i}\nu (\pi/2 -  \chi)}.
J_\mu(k r_l) \mathrm{e}^{\mathrm{i}\mu \theta_l}.
 
 
</math></center>
 
</math></center>
 +
  
 
The total
 
The total
incident wavefield upon body <math>\Delta_j</math> can now be expressed as  
+
incident wavefield upon cylinder <math>j</math> can now be expressed as  
 
<center><math>
 
<center><math>
 
\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +
 
\phi_l^{\mathrm{I}}(r_l,\theta_l,z) = \phi^{\mathrm{In}}(r_l,\theta_l,z) +
Line 110: Line 122:
 
(r_l,\theta_l,z)
 
(r_l,\theta_l,z)
 
</math></center>
 
</math></center>
 +
which can be written as
 
<center><math>
 
<center><math>
= \sum_{\nu = -\infty}^{\infty}
+
\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} +
 
  \Big[\tilde{D}_\nu^{l} +
 
\sum_{j=1,j \neq  l}^{n} \sum_{\tau =
 
\sum_{j=1,j \neq  l}^{n} \sum_{\tau =
Line 117: Line 132:
 
R_{jl})  \mathrm{e}^{\mathrm{i}(\tau - \nu) \varphi_{jl}} \Big] J_\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}.  
 
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>
 
</math></center>
  
= Final Equations =
+
== Final Equations ==
  
 
The scattered and incident potential can be related by the
 
The scattered and incident potential can be related by the
[[Diffraction Transfer Matrix]] for a [[Bottom Mounted Cylinder]] acting in the following way,
+
[[Diffraction Transfer Matrix]] for a [[Bottom Mounted Cylinder]] so that,
 
<center><math>
 
<center><math>
  D_{\mu}^l = \frac{J'_\mu(k a_j)}{H^{(1)}_\mu{}'(k a_j)} A_{\mu}^l.
+
  A_{\mu}^l= -\frac{J'_\nu(k a_l)}{H^{(1)}_\nu{}'(k a_l)} D_{\nu}^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_{\mu}^l= J'_\mu(k a_j)/H^{(1)}_\mu{}'(k a_j) .
 
 
</math></center>  
 
</math></center>  
  
The substitution of (inc_coeff) into  (diff_op) gives the
+
This gives the
 
required equations to determine the coefficients of the scattered
 
required equations to determine the coefficients of the scattered
 
wavefields of all bodies,  
 
wavefields of all bodies,  
 
<center><math>
 
<center><math>
A_{\mu}^l =
+
\frac{J'_\nu(k a_l)}{H^{(1)}_\nu{}'(k a_l)} \Big[
\sum_{\nu = -\infty}^{\infty} B_{\mu}^l
 
\Big[ \tilde{D}_{\nu}^{l} +
 
 
\sum_{j=1,j \neq  l}^{N} \sum_{\tau =
 
\sum_{j=1,j \neq  l}^{N} \sum_{\tau =
 
-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu}  (k
 
-\infty}^{\infty} A_{\tau}^j H^{(1)}_{\tau - \nu}  (k
R_{jl}) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{jl}} \Big],  
+
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></center>
 
<math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,n</math>.
 
<math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,n</math>.
 +
 +
[[Category:Interaction Theory]]

Latest revision as of 08:15, 19 October 2009


Introduction

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

After we have Removed the Depth Dependence the problem consists of [math]\displaystyle{ n }[/math] cylinders of radius [math]\displaystyle{ a_j }[/math] whose center is at [math]\displaystyle{ (x_j,y_j) }[/math] subject to Helmholtz's Equation

[math]\displaystyle{ \nabla^2 \phi + k^2\phi= 0, }[/math]

where [math]\displaystyle{ k }[/math] is the positive real root of the Dispersion Relation for a Free Surface

[math]\displaystyle{ k \tanh k h = \alpha,. }[/math]

where [math]\displaystyle{ \alpha }[/math] is the Infinite Depth wave number and [math]\displaystyle{ 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]\displaystyle{ j }[/math] can be expressed as

[math]\displaystyle{ \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]

with discrete coefficients [math]\displaystyle{ A_{\mu}^j }[/math], where [math]\displaystyle{ (r_j,\theta_j) }[/math] are polar coordinates centered at center of the [math]\displaystyle{ j }[/math]th cylinder.

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

[math]\displaystyle{ \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]

with discrete coefficients [math]\displaystyle{ D_{\nu}^j }[/math].

In these expansions, [math]\displaystyle{ J_\nu }[/math] and [math]\displaystyle{ H^{(1)}_\nu }[/math] denote Bessel and Hankel function respectively (Bessel functions) both of first kind and order [math]\displaystyle{ \nu }[/math]. For comparison with the Kagemoto and Yue Interaction Theory (which is written slightly differently), we remark that, for real [math]\displaystyle{ x }[/math],

[math]\displaystyle{ 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]

with [math]\displaystyle{ I_\nu }[/math] and [math]\displaystyle{ K_\nu }[/math] denoting the modified Bessel functions of first and second kind, respectively, both of order [math]\displaystyle{ \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]\displaystyle{ j }[/math] into an incident potential upon cylinder [math]\displaystyle{ l }[/math] ([math]\displaystyle{ 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]\displaystyle{ \phi_j^{\mathrm{S}} }[/math] of cylinder [math]\displaystyle{ j }[/math] needs to be represented in terms of the incident potential [math]\displaystyle{ \phi_l^{\mathrm{I}} }[/math] upon cylinder [math]\displaystyle{ l }[/math], [math]\displaystyle{ j \neq l }[/math]. This can be accomplished by using Graf's Addition Theorem

[math]\displaystyle{ 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]

where [math]\displaystyle{ (R_{jl},\varphi_{jl}) }[/math] are the polar coordinates of the mean centre position of cylinder [math]\displaystyle{ l }[/math] in the local coordinates of cylinder [math]\displaystyle{ j }[/math].

Making use of the eigenfunction expansion as well as Graf's Addition Theorem, the scattered potential of cylinder [math]\displaystyle{ j }[/math] can be expressed in terms of the incident potential upon cylinder [math]\displaystyle{ l }[/math] as

[math]\displaystyle{ \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]
[math]\displaystyle{ = \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]

The ambient incident wavefield [math]\displaystyle{ \phi^{\mathrm{In}} }[/math] can also be expanded in the eigenfunctions corresponding to the incident wavefield upon cylinder [math]\displaystyle{ l }[/math] (cf. the example in Cylindrical Eigenfunction Expansion)

[math]\displaystyle{ \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]

Let [math]\displaystyle{ \tilde{D}_{\nu}^{l} }[/math] denote the coefficients of this ambient incident wavefield in the incoming eigenfunction expansion for [math]\displaystyle{ \Delta_l }[/math] so that

[math]\displaystyle{ \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]


The total incident wavefield upon cylinder [math]\displaystyle{ 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=1,j \neq l}^{n} \, \phi_j^{\mathrm{S}} (r_l,\theta_l,z) }[/math]

which can be written as

[math]\displaystyle{ \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]

Therefore it follows that

[math]\displaystyle{ {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]

Final Equations

The scattered and incident potential can be related by the Diffraction Transfer Matrix for a Bottom Mounted Cylinder so that,

[math]\displaystyle{ A_{\mu}^l= -\frac{J'_\nu(k a_l)}{H^{(1)}_\nu{}'(k a_l)} D_{\nu}^l . }[/math]

This gives the required equations to determine the coefficients of the scattered wavefields of all bodies,

[math]\displaystyle{ \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]

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