WikiWaves - User contributions [en]

Peter and Meylan 2007

2010-09-23T11:38:31Z

Mpeter: Created page with 'Peter, M. A. & Meylan, M. H. 2007 Water-wave scattering by a semi-infinite periodic array of arbitrary bodies. J. Fluid Mech. 575, 473–494.'

Peter, M. A. & Meylan, M. H. 2007 Water-wave scattering by a semi-infinite periodic array of arbitrary bodies. J. Fluid Mech. 575, 473–494.

Rayleigh-Bloch Waves

2010-09-23T11:37:17Z

Mpeter:

Rayleigh-Bloch waves are waves which travel along an
[[:Category:Infinite Array|Infinite Array]].
Generally, they propagate along the array with a dominant wavenumber which is greater than that for freely-propagating waves and they decay exponentially away from the array. They have other names (guided waves, surface waves) in acoustics and electromagnetics. Only in very exceptional cases are they excited by plane incident waves ([[Porter and Evans 2005]]). Rayleigh-Bloch waves (if they can be supported by the structure) will be excited in a [[Semi-Infinite Array]] provided that <math> \ k < \pi/R \ </math> (where <math>k</math> is the wavenumber of the incident wave and <math>R</math> is the body spacing of the array).

[[Image:RB_wiki.jpg|thumb|right|Rayleigh-Bloch wave travelling down an infinite array of bottom-mounted cylinders]]

Rayleigh-Bloch waves are observed for a very general class of arrays, where the medium is governed by the two-dimensional [[Helmholtz's Equation]] (cf. [[Linton and McIver 2002]]). In the water-wave context this means that the structures must have a depth dependence which can be removed (cf. [[Removing The Depth Dependence]]), so that the problem reduces to two dimensions. In the more general water-wave context, no Rayleigh-Bloch waves have been found (see [[Peter and Meylan 2007]] for a discussion). It seems likely that such waves will exist only for very special geometries and frequencies, but this remains only a conjecture at the present time. There are many off-shore structures which do support Rayleigh-Bloch waves, e.g. [[Bottom Mounted Cylinder|Bottom-Mounted Cylinders]].
Some more information on Rayleigh--Bloch waves can be found in [[Porter and Evans 1999]] and [[Linton and McIver 2002]].

The Rayleigh-Bloch wavenumber <math>\beta</math> is the value for which the operator
<math>-\nabla^2</math>, subject to the periodicity conditions

<math>
\phi|_{x=0} = \mathrm{e}^{\mathrm{i} R \beta }\phi|_{x=R}, \quad \partial_x
\phi|_{x=0} = \mathrm{e}^{\mathrm{i} R \beta } \partial_x \phi|_{x=R}
</math>

and appropriate boundary conditions at the body surfaces, has an
eigenvalue in the interval <math> \ (k,\pi/R] \ </math>. This corresponds to a wave
travelling down the array with phase factor

<math>
\ Q_j = \mathrm{e}^{\mathrm{i} j R \beta}.
</math>

An example is given in the plot.

[[Category:Infinite Array]]

Eigenfunction Matching for a Circular Dock

2008-06-03T22:38:17Z

Mpeter: /* An infinite dimensional system of equations */

=Introduction=

We show here a solution for a dock on [[Finite Depth]] water, which is circular. This is the three-dimensional
analog of the [[Eigenfunction Matching for a Semi-Infinite Dock]].

=Governing Equations=

We begin with the [[Frequency Domain Problem]].
We will use a cylindrical coordinate system, <math>(r,\theta,z)</math>,
assumed to have its origin at the centre of the circular
plate which has radius <math>a</math>. The water is assumed to have
constant finite depth <math>h</math> and the <math>z</math>-direction points vertically
upward with the water surface at <math>z=0</math> and the sea floor at <math>z=-h</math>. The
boundary value problem can therefore be expressed as
<center>
<math>
\Delta\phi=0, \,\, -H<z<0,
</math>
</center>
<center>
<math>
\phi_{z}=0, \,\, z=-h,
</math>
</center>
<center><math>
\phi_{z}=\alpha\phi, \,\, z=0,\,r>a,
</math></center>
<center>
<math>
\phi_{z}=0, \,\, z=0,\,r<a
</math>
</center>
We must also apply the [[Sommerfeld Radiation Condition]]
as <math>r\rightarrow\infty</math>. The subscript <math>z</math>
denotes the derivative in <math>z</math>-direction.

=Solution Method=

== Separation of variables==

We now separate variables, noting that since the problem has
circular symmetry we can write the potential as
<center>
<math>
\phi(r,\theta,z)=\zeta(z)\sum_{n=-\infty}^{\infty}\rho_{n}(r)e^{i n \theta}
</math>
</center>
Applying Laplace's equation we obtain
<center>
<math>
\zeta_{zz}+\mu^{2}\zeta=0
</math>
</center>
so that:
<center>
<math>
\zeta=\cos\mu(z+H)
</math>
</center>
where the separation constant <math>\mu^{2}</math> must
satisfy the [[Dispersion Relation for a Free Surface]]
<center>
<math>
k\tan\left( kh\right) =-\alpha,\quad r>a
</math>
</center>
and
<center>
<math>
\kappa\tan(\kappa h)=0,\quad
r<a
</math>
</center>
Note that we have set <math>\mu=k</math> under the free
surface and <math>\mu=\kappa</math> under the dock. We denote the
positive imaginary solution of the [[Dispersion Relation for a Free Surface]]
by <math>k_{0}</math> and
the positive real solutions by <math>k_{m}</math>, <math>m\geq1</math>. The solutions of
second equation will be denoted by
<math>\kappa_{m} = m\pi/h</math>, <math>m\geq 0</math>.

We define
<center>
<math>
\phi_{m}\left( z\right) =\frac{\cos k_{m}(z+h)}{\cos k_{m}h},\quad m\geq0
</math>
</center>
as the vertical eigenfunction of the potential in the open
water region and
<center>
<math>
\psi_{m}\left( z\right) = \cos\kappa_{m}(z+h),\quad
m\geq 0
</math>
</center>
as the vertical eigenfunction of the potential in the dock
covered region. For later reference, we note that:
<center>
<math>
\int\nolimits_{-h}^{0}\phi_{m}(z)\phi_{n}(z) d z=A_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
A_{m}=\frac{1}{2}\left( \frac{\cos k_{m}h\sin k_{m}h+k_{m}h}{k_{m}\cos
^{2}k_{m}h}\right)
</math>
</center>
and
<center>
<math>
\int\nolimits_{-h}^{0}\phi_{n}(z)\psi_{m}(z) d z=B_{mn}
</math>
</center>
where
<center><math>
B_{mn}=\frac{k_{n}\sin k_{n}h\cos\kappa_{m}h-\kappa_{m}\cos k_{n}h\sin
\kappa_{m}h}{\left( \cos k_{n}h\cos\kappa_{m}h\right) \left( k_{n}
^{2}-\kappa_{m}^{2}\right) }
</math></center>
and
<center>
<math>
\int\nolimits_{-h}^{0}\psi_{m}(z)\psi_{n}(z) d z=C_{m}\delta_{mn}
</math>
</center>
where
<center>
<math>
C_{m}=\frac{1}{2}h,\quad,m\neq 0 \quad \mathrm{and} \quad C_0 = h
</math></center>

We now solve for the function <math>\rho_{n}(r)</math>.
Using Laplace's equation in polar coordinates we obtain
<center>
<math>
\frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}r^{2}}+\frac{1}{r}
\frac{\mathrm{d}\rho_{n}}{\mathrm{d}r}-\left(
\frac{n^{2}}{r^{2}}+\mu^{2}\right) \rho_{n}=0
</math>
</center>
where <math>\mu</math> is <math>k_{m}</math> or
<math>\kappa_{m},</math> depending on whether <math>r</math> is
greater or less than <math>a</math>. We can convert this equation to the
standard form by substituting <math>y=\mu r</math> (provided that
<math>\mu\neq 0</math>to obtain
<center>
<math>
y^{2}\frac{\mathrm{d}^{2}\rho_{n}}{\mathrm{d}y^{2}}+y\frac{\mathrm{d}\rho_{n}
}{\rm{d}y}-(n^{2}+y^{2})\rho_{n}=0
</math>
</center>
The solution of this equation is a linear combination of the
modified Bessel functions of order <math>n</math>, <math>I_{n}(y)</math> and
<math>K_{n}(y)</math> ([[Abramowitz and Stegun 1964]]). Since the solution must be bounded
we know that under the plate the solution will be a linear combination of
<math>I_{n}(y)</math> while outside the plate the solution will be a
linear combination of <math>K_{n}(y)</math>.
The case <math>\kappa_0 =0</math> is a special case and the solution under
the dock is <math>(r/a)^{|n|}</math>.
Therefore the potential can
be expanded as
<center>
<math>
\phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}\sum_{m=0}^{\infty}a_{mn}K_{n}
(k_{m}r)e^{i n\theta}\phi_{m}(z), \;\;r>a
</math>
</center>
and
<center>
<math>
\phi(r,\theta,z)=\sum_{n=-\infty}^{\infty}b_{0n}(r/a)^{|n|} e^{i n\theta}\psi_{0}(z)+
\sum_{n=-\infty}^{\infty}\sum_{m=1}^{\infty}b_{mn}
I_{n}(\kappa_{m}r)e^{i n\theta}\psi_{m}(z), \;\;r<a
</math>
</center>
where <math>a_{mn}</math> and <math>b_{mn}</math>
are the coefficients of the potential in the open water and
the plate covered region respectively.

==Incident potential==

The incident potential is a wave of amplitude <math>A</math>
in displacement travelling in the positive <math>x</math>-direction.
The incident potential can therefore be written as
<center>
<math>
\phi^{\mathrm{I}} =\frac{A}{i\sqrt{\alpha}}e^{k_{0}x}\phi_{0}\left(
z\right)
=\sum\limits_{n=-\infty}^{\infty}e_{n}I_{n}(k_{0}r)\phi_{0}\left(z\right)
e^{i n \theta}
</math>
</center>
where <math>e_{n}=A/\left(i\sqrt{\alpha}\right)</math>
(we retain the dependence on <math>n</math> for situations
where the incident potential might take another form).

==An infinite dimensional system of equations==

The potential and its derivative must be continuous across the
transition from open water to the plate covered region. Therefore, the
potentials and their derivatives at <math>r=a</math> have to be equal
for each angle and we obtain
<center>
<math>
e_{n}I_{n}(k_{0}a)\phi_{0}\left( z\right) + \sum_{m=0}^{\infty}
a_{mn} K_{n}(k_{m}a)\phi_{m}\left( z\right)
= b_{0n} \psi_{0}(z) +\sum_{m=1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)\psi_{m}(z)
</math>
</center>
and
<center>
<math>
e_{n}k_{0}I_{n}^{\prime}(k_{0}a)\phi_{0}\left( z\right) +\sum
_{m=0}^{\infty} a_{mn}k_{m}K_{n}^{\prime}(k_{m}a)\phi_{m}\left( z\right)
=b_{0n} \frac{|n|}{a} \psi_{0}(z) +\sum_{m=1}^{\infty}
b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)\psi
_{m}(z)
</math>
</center>
for each <math>n</math>.
We solve these equations by multiplying both equations by
<math>\phi_{l}(z)</math> and integrating from <math>-H</math> to <math>0</math> to obtain:
<center>
<math>
e_{n}I_{n}(k_{0}a)A_{0}\delta_{0l}+a_{ln}K_{n}(k_{l}a)A_{l}
=b_{0n}B_{0l} + \sum_{m=1}^{\infty}b_{mn}I_{n}(\kappa_{m}a)B_{ml}
</math>
</center>
and
<center>
<math>
e_{n}k_{0}I_{n}^{\prime}(k_{0}a)A_{0}\delta_{0l}+a_{ln}k_{l}K_{n}^{\prime
}(k_{l}a)A_{l}
= b_{0n}B_{0l}\frac{|n|}{a} +
\sum_{m=1}^{\infty}b_{mn}\kappa_{m}I_{n}^{\prime}(\kappa_{m}a)B_{ml}
</math>
</center>

= Numerical Solution =

To solve the system of equations we set the upper limit of <math>l</math> to
be <math>M</math>.

= Matlab Code =

A program to calculate the coefficients for circular dock problems can be found here
[http://www.math.auckland.ac.nz/~meylan/code/eigenfunction_matching/circle_dock_matching_one_n.m circle_dock_matching_one_n.m]
Note that this problem solves only for a single n.

== Additional code ==

This program requires
[http://www.math.auckland.ac.nz/~meylan/code/dispersion/dispersion_free_surface.m dispersion_free_surface.m]
to run

[[Category:Problems with Cylindrical Symmetry]]
[[Category:Eigenfunction Matching Method]]
[[Category:Pages with Matlab Code]]

Interaction Theory for Infinite Arrays

2006-07-18T15:00:17Z

Mpeter: /* System of equations */

=Introduction =

We want to use the [[Kagemoto and Yue Interaction Theory]] to derive a system of equations for the infinite array.

= System of equations =

We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely

<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\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>.

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by <math>R</math>, we have <math>R_{jl} = |j-l| R</math> and

<center><math>
\varphi_{n} =
\begin{cases}
\pi, & n>0,\\
0, & n<0.
\end{cases}
</math></center>

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients <math>A_{m\mu}^l</math> can be written as <math>A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu}</math>, where the phase factor <math>P_l</math> is given by

<center><math>
\ P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},
</math></center>

where <math>\chi</math> is the angle which the direction of the ambient waves makes with the <math>x</math>-axis. The same can be done for the coefficients of the ambient wave, i.e. <math>\tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu}</math>.

Therefore, the system simplifies to
<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq 0}^{\infty} P_{j} K_{\tau - \nu} (k_n
|j|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big].
</math></center>

Introducing the constants

<center><math>
\sigma^n_\nu = \sum_{j=-\infty,j \neq 0}^{\infty} P_{j} K_\nu(k_n|j|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR),
</math></center>

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big],
</math></center>

<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>.
Note that this system of equations is for the body centred at the origin only. The scattered waves of all other bodies can be obtained from its solution by the simple formula <math>A_{m\mu}^l = P_l A_{m\mu}</math>.

[[Category:Infinite Array]]

Interaction Theory for Infinite Arrays

2006-07-18T14:59:59Z

Mpeter: /* System of equations */

=Introduction =

We want to use the [[Kagemoto and Yue Interaction Theory]] to derive a system of equations for the infinite array.

= System of equations =

We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely

<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\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>.

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by <math>R</math>, we have <math>R_{jl} = |j-l| R</math> and

<center><math>
\varphi_{n} =
\begin{cases}
\pi, & n>0,\\
0, & n<0.
\end{cases}
</math></center>

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients <math>A_{m\mu}^l</math> can be written as <math>A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu}</math>, where the phase factor <math>P_l</math> is given by

\
<center><math>
P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},
</math></center>

where <math>\chi</math> is the angle which the direction of the ambient waves makes with the <math>x</math>-axis. The same can be done for the coefficients of the ambient wave, i.e. <math>\tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu}</math>.

Therefore, the system simplifies to
<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq 0}^{\infty} P_{j} K_{\tau - \nu} (k_n
|j|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big].
</math></center>

Introducing the constants

<center><math>
\sigma^n_\nu = \sum_{j=-\infty,j \neq 0}^{\infty} P_{j} K_\nu(k_n|j|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR),
</math></center>

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big],
</math></center>

<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>.
Note that this system of equations is for the body centred at the origin only. The scattered waves of all other bodies can be obtained from its solution by the simple formula <math>A_{m\mu}^l = P_l A_{m\mu}</math>.

[[Category:Infinite Array]]

Interaction Theory for Infinite Arrays

2006-07-18T14:57:32Z

Mpeter: /* System of equations */

=Introduction =

We want to use the [[Kagemoto and Yue Interaction Theory]] to derive a system of equations for the infinite array.

= System of equations =

We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely

<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\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>.

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by <math>R</math>, we have <math>R_{jl} = |j-l| R</math> and

<center><math>
\varphi_{n} =
\begin{cases}
\pi, & n>0,\\
0, & n<0.
\end{cases}
</math></center>

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients <math>A_{m\mu}^l</math> can be written as <math>A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu}</math>, where the phase factor <math>P_l</math> is given by

<center><math>
P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},
</math></center>

where <math>\chi</math> is the angle which the direction of the ambient waves makes with the <math>x</math>-axis. The same can be done for the coefficients of the ambient wave, i.e. <math>\tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu}</math>.

Therefore, the system simplifies to
<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq 0}^{\infty} P_{j} K_{\tau - \nu} (k_n
|j|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big].
</math></center>

Introducing the constants

<center><math>
\sigma^n_\nu = \sum_{j=-\infty,j \neq 0}^{\infty} P_{j} K_\nu(k_n|j|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR),
</math></center>

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big],
</math></center>

<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>.
Note that this system of equations is for the body centred at the origin only. The scattered waves of all other bodies can be obtained from its solution by the simple formula <math>A_{m\mu}^l = P_l A_{m\mu}</math>.

[[Category:Infinite Array]]

Interaction Theory for Infinite Arrays

2006-07-18T14:56:05Z

Mpeter: /* System of equations */

=Introduction =

We want to use the [[Kagemoto and Yue Interaction Theory]] to derive a system of equations for the infinite array.

= System of equations =

We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely

<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\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>.

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by <math>R</math>, we have <math>R_{jl} = |j-l| R</math> and

<center><math>
\varphi_{n} =
\begin{cases}
\pi, & n>0,\\
0, & n<0.
\end{cases}
</math></center>

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients <math>A_{m\mu}^l</math> can be written as <math>A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu}</math>, where the phase factor <math>P_l</math> is given by

<center><math>
P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},
</math></center>

where <math>\chi</math> is the angle which the direction of the ambient waves makes with the <math>x</math>-axis. The same can be done for the coefficients of the ambient wave, i.e. <math>\tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu}</math>.

Therefore, the system simplifies to
<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq 0}^{\infty} P_{j-l} K_{\tau - \nu} (k_n
|j-l|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big].
</math></center>

Introducing the constants

<center><math>
\sigma^n_\nu = \sum_{j=-\infty,j \neq 0}^{\infty} P_{j-l} K_\nu(k_n|j-l|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j-l}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR),
</math></center>

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big],
</math></center>

<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>.
Note that this system of equations is for the body centred at the origin only. The scattered waves of all other bodies can be obtained from its solution by the simple formula <math>A_{m\mu}^l = P_l A_{m\mu}</math>.

[[Category:Infinite Array]]

Interaction Theory for Infinite Arrays

2006-07-18T14:55:02Z

Mpeter: /* System of equations */

=Introduction =

We want to use the [[Kagemoto and Yue Interaction Theory]] to derive a system of equations for the infinite array.

= System of equations =

We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely

<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\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>.

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by <math>R</math>, we have <math>R_{jl} = |j-l| R</math> and

<center><math>
\varphi_{n} =
\begin{cases}
\pi, & n>0,\\
0, & n<0.
\end{cases}
</math></center>

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients <math>A_{m\mu}^l</math> can be written as <math>A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu}</math>, where the phase factor <math>P_l</math> is given by

<center><math>
P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},
</math></center>

where <math>\chi</math> is the angle which the direction of the ambient waves makes with the <math>x</math>-axis. The same can be done for the coefficients of the ambient wave, i.e. <math>\tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu}</math>.

Therefore, the system simplifies to
<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq l}^{\infty} P_{j-l} K_{\tau - \nu} (k_n
|j-l|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big].
</math></center>

Introducing the constants

<center><math>
\sigma^n_\nu = \sum_{j=-\infty,j \neq l}^{\infty} P_{j-l} K_\nu(k_n|j-l|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j-l}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR),
</math></center>

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big],
</math></center>

<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>.
Note that this system of equations is for the body centred at the origin only. The scattered waves of all other bodies can be obtained from its solution by the simple formula <math>A_{m\mu}^l = P_l A_{m\mu}</math>.

[[Category:Infinite Array]]

Interaction Theory for Infinite Arrays

2006-07-18T14:54:16Z

Mpeter: /* System of equations */

=Introduction =

We want to use the [[Kagemoto and Yue Interaction Theory]] to derive a system of equations for the infinite array.

= System of equations =

We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely

<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\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>.

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by <math>R</math>, we have <math>R_{jl} = |j-l| R</math> and

<center><math>
\varphi_{n} =
\begin{cases}
\pi, & n>0,\\
0, & n<0.
\end{cases}
</math></center>

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients <math>A_{m\mu}^l</math> can be written as <math>A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu}</math>, where the phase factor <math>P_l</math> is given by

<center><math>
P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},
</math></center>

where <math>\chi</math> is the angle which the direction of the ambient waves makes with the <math>x</math>-axis. The same can be done for the coefficients of the ambient wave, i.e. <math>\tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu}</math>.

Therefore, the system simplifies to
<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq l}^{\infty} P_{j-l} K_{\tau - \nu} (k_n
|j-l|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big].
</math></center>

Introducing the constants

<center><math>
\sigma^n_\nu = \sum_{j=-\infty,j \neq l}^{\infty} P_{j-l} K_\nu(k_n|j-l|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j-l}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR),
</math></center>

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big],
</math></center>

<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>.
Note that this system of equations is for the body centred at the origin. The scattered waves of all other bodies can be obtained by the simple formula <math>A_{m\mu}^l = P_l A_{m\mu}</math>.

[[Category:Infinite Array]]

Interaction Theory for Infinite Arrays

2006-07-18T14:52:41Z

Mpeter: /* System of equations */

=Introduction =

We want to use the [[Kagemoto and Yue Interaction Theory]] to derive a system of equations for the infinite array.

= System of equations =

We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely

<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\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>.

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by <math>R</math>, we have <math>R_{jl} = |j-l| R</math> and

<center><math>
\varphi_{n} =
\begin{cases}
\pi, & n>0,\\
0, & n<0.
\end{cases}
</math></center>

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients <math>A_{m\mu}^l</math> can be written as <math>A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu}</math>, where the phase factor <math>P_l</math> is given by

<center><math>
P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},
</math></center>

where <math>\chi</math> is the angle which the direction of the ambient waves makes with the <math>x</math>-axis. The same can be done for the coefficients of the ambient wave, i.e. <math>\tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu}</math>.

Therefore, the system simplifies to
<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq l}^{\infty} P_{j-l} K_{\tau - \nu} (k_n
|j-l|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big],
</math></center>
<math>m \in \mathbb{N}</math>, <math>\mu,l \in \mathbb{Z}</math>.

Introducing the constants

<center><math>
\sigma^n_\nu = \sum_{j=-\infty,j \neq l}^{\infty} P_{j-l} K_\nu(k_n|j-l|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j-l}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR),
</math></center>

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big],
</math></center>

<math>m \in \mathbb{N}</math>, <math>\mu,l \in \mathbb{Z}</math>.
Note that this system of equations is for the body centred at the origin. The scattered waves of all other bodies can be obtained by the simple formula <math>A_{m\mu}^l = P_l A_{m\mu}</math>.

[[Category:Infinite Array]]

Interaction Theory for Infinite Arrays

2006-07-18T14:50:14Z

Mpeter: /* System of equations */

=Introduction =

We want to use the [[Kagemoto and Yue Interaction Theory]] to derive a system of equations for the infinite array.

= System of equations =

We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely
<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\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>.

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by <math>R</math>, we have <math>R_{jl} = |j-l| R</math> and

<center><math>
\varphi_{n} =
\begin{cases}
\pi, & n>0,\\
0, & n<0.
\end{cases}
</math></center>

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients <math>A_{m\mu}^l</math> can be written as <math>A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu}</math>, where the phase factor <math>P_l</math> is given by

<center><math>
P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},
</math></center>

where <math>\chi</math> is the angle which the direction of the ambient waves makes with the <math>x</math>-axis. The same can be done for the coefficients of the ambient wave, i.e. <math>\tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu}</math>.

Therefore, the system simplifies to
<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq l}^{\infty} P_{j-l} K_{\tau - \nu} (k_n
|j-l|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big],
</math></center>
<math>m \in \mathbb{N}</math>, <math>\mu,l \in \mathbb{Z}</math>.

Introducing the constants

<center><math>
\sigma^n_\nu = \sum_{j=-\infty,j \neq l}^{\infty} P_{j-l} K_\nu(k_n|j-l|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j-l}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR),
</math></center>

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} \Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big].
</math></center>

[[Category:Infinite Array]]

Interaction Theory for Infinite Arrays

2006-07-18T14:49:41Z

Mpeter: /* System of equations */

=Introduction =

We want to use the [[Kagemoto and Yue Interaction Theory]] to derive a system of equations for the infinite array.

= System of equations =

We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely
<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\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>.

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by <math>R</math>, we have <math>R_{jl} = |j-l| R</math> and

<center><math>
\varphi_{n} =
\begin{cases}
\pi, & n>0,\\
0, & n<0.
\end{cases}
</math></center>

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients <math>A_{m\mu}^l</math> can be written as <math>A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu}</math>, where the phase factor <math>P_l</math> is given by

<center><math>
P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},
</math></center>

where <math>\chi</math> is the angle which the direction of the ambient waves makes with the <math>x</math>-axis. The same can be done for the coefficients of the ambient wave, i.e. <math>\tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu}</math>.

Therefore, the system simplifies to
<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq l}^{\infty} P_{j-l} K_{\tau - \nu} (k_n
|j-l|R) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big],
</math></center>
<math>m \in \mathbb{N}</math>, <math>\mu,l \in \mathbb{Z}</math>.

Introducing the constants

<center><math>
\sigma^n_\nu = \sum_{j=-\infty,j \neq l}^{\infty} P_{j-l} K_\nu(k_n|j-l|R) \mathrm{e}^{\mathrm{i}\nu \varphi_{j-l}} = \sum_{j=1}^{\infty} (P_{-j} + (-1)^\nu P_j) K_\nu(k_njR),
</math></center>

which can be evaluated separately since they do not contain any unknowns, the problem reduces to

<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu} Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sigma^n_{\tau-\nu} \Big].
</math></center>

[[Category:Infinite Array]]

Interaction Theory for Infinite Arrays

2006-07-18T14:45:48Z

Mpeter: /* System of equations */

=Introduction =

We want to use the [[Kagemoto and Yue Interaction Theory]] to derive a system of equations for the infinite array.

= System of equations =

We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely
<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\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>.

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by <math>R</math>, we have <math>R_{jl} = |j-l| R</math> and

<center><math>
\varphi_{n} =
\begin{cases}
\pi, & n>0,\\
0, & n<0.
\end{cases}
</math></center>

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients <math>A_{m\mu}^l</math> can be written as <math>A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu}</math>, where the phase factor <math>P_l</math> is given by

<center><math>
P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},
</math></center>

where <math>\chi</math> is the angle which the direction of the ambient waves makes with the <math>x</math>-axis. The same can be done for the coefficients of the ambient wave, i.e. <math>\tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu}</math>.

Therefore, the system simplifies to
<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=-\infty,j \neq l}^{\infty} P_{j-l} K_{\tau - \nu} (k_n
R |j-l|) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big],
</math></center>
<math>m \in \mathbb{N}</math>, <math>\mu,l \in \mathbb{Z}</math>.

[[Category:Infinite Array]]

Interaction Theory for Infinite Arrays

2006-07-18T14:44:49Z

Mpeter: /* System of equations */

=Introduction =

We want to use the [[Kagemoto and Yue Interaction Theory]] to derive a system of equations for the infinite array.

= System of equations =

We start with the final system of equations of the [[Kagemoto and Yue Interaction Theory]], namely
<center><math>
A_{m\mu}^l = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}^l
\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>.

For the infinite array, some simplifications of this system can be made. First of all, the bodies are aligned in an evenly spaced array. Denoting the spacing by <math>R</math>, we have <math>R_{jl} = |j-l| R</math> and

<center><math>
\varphi_{n} =
\begin{cases}
\pi, & n>0,\\
0, & n<0.
\end{cases}
</math></center>

Moreover, owing to the periodicity of the array as well as the ambient wave, the coefficients <math>A_{m\mu}^l</math> can be written as <math>A_{m\mu}^l = P_l A_{m\mu}^0 = P_l A_{m\mu}</math>, where the phase factor <math>P_l</math> is given by

<center><math>
P_l = \mathrm{e}^{\mathrm{i}Rk\cos \chi},
</math></center>

where <math>\chi</math> is the angle which the direction of the ambient waves makes with the <math>x</math>-axis. The same can be done for the coefficients of the ambient wave, i.e. <math>\tilde{D}_{n\nu}^{l} = P_l \tilde{D}_{n\nu}</math>.

Therefore, the system simplifies to
<center><math>
A_{m\mu} = \sum_{n=0}^{\infty}
\sum_{\nu = -\infty}^{\infty} B_{mn\mu\nu}
\Big[ \tilde{D}_{n\nu} + (-1)^\nu
\sum_{\tau = -\infty}^{\infty} A_{n\tau} \sum_{j=1,j \neq l}^{\infty} P_{j-l} K_{\tau - \nu} (k_n
R |j-l|) \mathrm{e}^{\mathrm{i}(\tau -\nu) \varphi_{j-l}} \Big],
</math></center>
<math>m \in \mathbb{N}</math>, <math>\mu \in \mathbb{Z}</math>, <math>l=1,\dots,N</math>.

[[Category:Infinite Array]]

Interaction Theory for Infinite Arrays

2006-07-18T14:42:03Z

Mpeter: /* System of equations */

Interaction Theory for Infinite Arrays

2006-07-18T14:41:23Z

Mpeter: /* System of equations */

Interaction Theory for Infinite Arrays

2006-07-18T14:39:11Z

Mpeter: /* System of equations */

Interaction Theory for Infinite Arrays

2006-07-18T14:36:57Z

Mpeter:

Interaction Theory for Infinite Arrays

2006-07-18T14:36:33Z

Mpeter: /* System of equations */

Interaction Theory for Infinite Arrays

2006-07-18T14:36:10Z

Mpeter: /* System of equations */

2006-06-16T11:55:58Z

Mpeter: /* Final Equations */

= Introduction =

This is an interaction theory which provides the exact solution (i.e. it is not based on a [[Wide Spacing Approximation]]).
The theory uses the [[Cylindrical Eigenfunction Expansion]] and [[Graf's Addition Theorem]] to represent the potential in local coordinates. The incident and scattered potential of each body are then related by the associated [[Diffraction Transfer Matrix]].

The basic idea is as follows: The scattered potential of each body is represented in the [[Cylindrical Eigenfunction Expansion]] associated with the local coordinates centred at the mean centre position of the body. Using [[Graf's Addition Theorem]], the scattered potential of all bodies (given in their local coordinates) can be mapped to an incident potential associated with the coordiates of all other bodies. Doing this, the incident potential of each body (which is given by the ambient incident potential plus the scattered potentials of all other bodies) is given in the [[Cylindrical Eigenfunction Expansion]] associated with its local coordinates. Using the [[Diffraction Transfer Matrix]], which relates the incident and scattered potential of each body in isolation, a system of equations for the coefficients of the scattered potentials of all bodies is obtained.

The theory is described in [[Kagemoto and Yue 1986]] and in
[[Peter and Meylan 2004]].

The derivation of the theory in [[Infinite Depth]] is also presented, see
[[Kagemoto and Yue Interaction Theory for Infinite Depth]].

[[Category:Linear Water-Wave Theory]]

= Equations of Motion =

We assume
the [[Frequency Domain Problem]] with frequency <math>\omega</math>.
To simplify notation, <math>\mathbf{y} = (x,y,z)</math> always denotes a point
in the water, which is assumed to be of [[Finite Depth]] <math>d</math>,
while <math>\mathbf{x}</math> always denotes a point of the undisturbed water
surface assumed at <math>z=0</math>.

Writing <math>\alpha = \omega^2/g</math> where <math>g</math> is the acceleration due to
gravity, the potential <math>\phi</math> has to
satisfy the standard boundary-value problem
<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 <math>\Delta_j</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>
Moreover, the [[Sommerfeld Radiation Condition]] is imposed
<center><math>
\lim_{\tilde{r} \rightarrow \infty} \sqrt{\tilde{r}} \, \Big(
\frac{\partial}{\partial \tilde{r}} - \mathrm{i}k
\Big) (\phi - \phi^{\mathrm{In}}) = 0,
</math></center>
where <math>\tilde{r}^2=x^2+y^2</math>, <math>k</math> is the wavenumber and
<math>\phi^\mathrm{In}</math> is the ambient incident potential. The
positive wavenumber <math>k</math>
is related to <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]
<center><math> (eq_k)
\alpha = k \tanh k d,
</math></center>
and the values of <math>k_m</math>, <math>m>0</math>, are given as positive real roots of
the dispersion relation
<center><math> (eq_km)
\alpha + k_m \tan k_m d = 0.
</math></center>
For ease of notation, we write <math>k_0 = -\mathrm{i}k</math>. Note that <math>k_0</math> is a
(purely imaginary) root of (eq_k_m).

=Eigenfunction expansion of the potential=

The scattered potential of a body
<math>\Delta_j</math> can be expanded in the [[Cylindrical Eigenfunction Expansion]],
<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>
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 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>.

Kagemoto and Yue Interaction Theory

2006-06-16T11:55:40Z

Mpeter: /* Final Equations */

= Introduction =

This is an interaction theory which provides the exact solution (i.e. it is not based on a [[Wide Spacing Approximation]]).
The theory uses the [[Cylindrical Eigenfunction Expansion]] and [[Graf's Addition Theorem]] to represent the potential in local coordinates. The incident and scattered potential of each body are then related by the associated [[Diffraction Transfer Matrix]].

The basic idea is as follows: The scattered potential of each body is represented in the [[Cylindrical Eigenfunction Expansion]] associated with the local coordinates centred at the mean centre position of the body. Using [[Graf's Addition Theorem]], the scattered potential of all bodies (given in their local coordinates) can be mapped to an incident potential associated with the coordiates of all other bodies. Doing this, the incident potential of each body (which is given by the ambient incident potential plus the scattered potentials of all other bodies) is given in the [[Cylindrical Eigenfunction Expansion]] associated with its local coordinates. Using the [[Diffraction Transfer Matrix]], which relates the incident and scattered potential of each body in isolation, a system of equations for the coefficients of the scattered potentials of all bodies is obtained.

The theory is described in [[Kagemoto and Yue 1986]] and in
[[Peter and Meylan 2004]].

The derivation of the theory in [[Infinite Depth]] is also presented, see
[[Kagemoto and Yue Interaction Theory for Infinite Depth]].

[[Category:Linear Water-Wave Theory]]

= Equations of Motion =

We assume
the [[Frequency Domain Problem]] with frequency <math>\omega</math>.
To simplify notation, <math>\mathbf{y} = (x,y,z)</math> always denotes a point
in the water, which is assumed to be of [[Finite Depth]] <math>d</math>,
while <math>\mathbf{x}</math> always denotes a point of the undisturbed water
surface assumed at <math>z=0</math>.

Writing <math>\alpha = \omega^2/g</math> where <math>g</math> is the acceleration due to
gravity, the potential <math>\phi</math> has to
satisfy the standard boundary-value problem
<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 <math>\Delta_j</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>
Moreover, the [[Sommerfeld Radiation Condition]] is imposed
<center><math>
\lim_{\tilde{r} \rightarrow \infty} \sqrt{\tilde{r}} \, \Big(
\frac{\partial}{\partial \tilde{r}} - \mathrm{i}k
\Big) (\phi - \phi^{\mathrm{In}}) = 0,
</math></center>
where <math>\tilde{r}^2=x^2+y^2</math>, <math>k</math> is the wavenumber and
<math>\phi^\mathrm{In}</math> is the ambient incident potential. The
positive wavenumber <math>k</math>
is related to <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]
<center><math> (eq_k)
\alpha = k \tanh k d,
</math></center>
and the values of <math>k_m</math>, <math>m>0</math>, are given as positive real roots of
the dispersion relation
<center><math> (eq_km)
\alpha + k_m \tan k_m d = 0.
</math></center>
For ease of notation, we write <math>k_0 = -\mathrm{i}k</math>. Note that <math>k_0</math> is a
(purely imaginary) root of (eq_k_m).

=Eigenfunction expansion of the potential=

The scattered potential of a body
<math>\Delta_j</math> can be expanded in the [[Cylindrical Eigenfunction Expansion]],
<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>
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 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 {N}</math>, <math>\mu \in {Z}</math>, <math>l=1,\dots,N</math>.

Kagemoto and Yue Interaction Theory

2006-06-16T11:54:50Z

Mpeter: /* Derivation of the system of equations */

= Introduction =

This is an interaction theory which provides the exact solution (i.e. it is not based on a [[Wide Spacing Approximation]]).
The theory uses the [[Cylindrical Eigenfunction Expansion]] and [[Graf's Addition Theorem]] to represent the potential in local coordinates. The incident and scattered potential of each body are then related by the associated [[Diffraction Transfer Matrix]].

The basic idea is as follows: The scattered potential of each body is represented in the [[Cylindrical Eigenfunction Expansion]] associated with the local coordinates centred at the mean centre position of the body. Using [[Graf's Addition Theorem]], the scattered potential of all bodies (given in their local coordinates) can be mapped to an incident potential associated with the coordiates of all other bodies. Doing this, the incident potential of each body (which is given by the ambient incident potential plus the scattered potentials of all other bodies) is given in the [[Cylindrical Eigenfunction Expansion]] associated with its local coordinates. Using the [[Diffraction Transfer Matrix]], which relates the incident and scattered potential of each body in isolation, a system of equations for the coefficients of the scattered potentials of all bodies is obtained.

The theory is described in [[Kagemoto and Yue 1986]] and in
[[Peter and Meylan 2004]].

The derivation of the theory in [[Infinite Depth]] is also presented, see
[[Kagemoto and Yue Interaction Theory for Infinite Depth]].

[[Category:Linear Water-Wave Theory]]

= Equations of Motion =

We assume
the [[Frequency Domain Problem]] with frequency <math>\omega</math>.
To simplify notation, <math>\mathbf{y} = (x,y,z)</math> always denotes a point
in the water, which is assumed to be of [[Finite Depth]] <math>d</math>,
while <math>\mathbf{x}</math> always denotes a point of the undisturbed water
surface assumed at <math>z=0</math>.

Writing <math>\alpha = \omega^2/g</math> where <math>g</math> is the acceleration due to
gravity, the potential <math>\phi</math> has to
satisfy the standard boundary-value problem
<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 <math>\Delta_j</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>
Moreover, the [[Sommerfeld Radiation Condition]] is imposed
<center><math>
\lim_{\tilde{r} \rightarrow \infty} \sqrt{\tilde{r}} \, \Big(
\frac{\partial}{\partial \tilde{r}} - \mathrm{i}k
\Big) (\phi - \phi^{\mathrm{In}}) = 0,
</math></center>
where <math>\tilde{r}^2=x^2+y^2</math>, <math>k</math> is the wavenumber and
<math>\phi^\mathrm{In}</math> is the ambient incident potential. The
positive wavenumber <math>k</math>
is related to <math>\alpha</math> by the [[Dispersion Relation for a Free Surface]]
<center><math> (eq_k)
\alpha = k \tanh k d,
</math></center>
and the values of <math>k_m</math>, <math>m>0</math>, are given as positive real roots of
the dispersion relation
<center><math> (eq_km)
\alpha + k_m \tan k_m d = 0.
</math></center>
For ease of notation, we write <math>k_0 = -\mathrm{i}k</math>. Note that <math>k_0</math> is a
(purely imaginary) root of (eq_k_m).

=Eigenfunction expansion of the potential=

The scattered potential of a body
<math>\Delta_j</math> can be expanded in the [[Cylindrical Eigenfunction Expansion]],
<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>
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 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=-\infty,j \neq l}^{\infty} \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 {N}</math>, <math>l,\mu \in {Z}</math>.