View source for Infinite Array Green Function

=Introduction=

We present here the solution to the [[:Category:Infinite Array|Infinite Array]] based 
on an infinite image system of [[Free-Surface Green Function|Free-Surface Green Functions]]

=Problem Formulation=

We begin by formulating the problem.
Cartesian coordinates <math>(x,y,z)</math> are chosen with <math>z</math> vertically upwards
such that <math>z=0</math> coincides with the mean free surface of the water.
An infinite array of identical bodies
are periodically spaced along
the <math>y</math>-axis with uniform separation <math>l</math>. The problem is to determine
the motion of the water and the bodies when plane waves are obliquely-incident
from <math>x=-\infty</math> upon the periodic array of bodies.

The bodies occupy <math>\Delta_m</math>, <math>-\infty < m < \infty</math>. Periodicity implies
that if <math>(x,y) \in \Delta_0</math>, then <math>(x,y+ml) \in \Delta_m</math>, 
<math>-\infty < m < \infty</math>.

We assume that we have the [[Standard Linear Wave Scattering Problem]].
The incident wave
potential given by
<center><math>
 \phi^{{\rm in}} = \frac{A}{k} 
 e^{ik (x\cos\theta+y\sin\theta)}\,e^{kz},
</math></center>
where <math>A</math> is the dimensionless amplitude and <math>\theta</math> is the direction of 
propagation of the wave (with <math>\theta = 0</math> corresponding to normal incidence.

=Transformation to an Integral Equation=

We now Floquet's theorem ([[Scott 1998]]) (also called ''the assumption of periodicity'' 
in the water wave context) which states the 
displacement from adjacent plates differ only by a phase factor.
If the potential under the ''central'' plate <math>\Delta_{0}</math> is given by <math>\phi( \mathbf{x}_{0},0)</math>,
<math>\mathbf{x}_{0}\in\Delta_{0}</math>, then by Floquet's theorem the potential
satisfies
<center><math>
 \phi(\mathbf{x}_{m},0) = \phi(\mathbf{x}_{0},0) e^{im\sigma l},
</math></center>
and the displacement of the plate <math>\Delta_{m}</math> satisfies
<center><math>
 w(\mathbf{x}_{m}) = w(\mathbf{x}_{0}) e^{im\sigma l},
</math></center>
where <math>\mathbf{x}_{m} \in \Delta_{m}</math>, <math>-\infty < m < \infty</math> and 
the phase difference is <math>\sigma = k\sin\theta</math> (see, for example, [[Linton 1998]]).

A standard approach to the solution of the equations of motion for
the water is the [[Green Function Solution Method]] in which
we transform the equations into a boundary integral
equation using the [[Free-Surface Green Function]]. In doing so we obtain
<center><math>
 \phi(\mathbf{x}) = \phi^{\rm in} (\mathbf{x},0)
 +\sum_{m=-\infty}^{\infty} \int_{\Delta_{m}} 
\left(G_{n_\xi}(\mathbf{x},\xi) \phi(\xi) - G(\mathbf{x},\xi) \phi_{n_\xi}(\xi) \right) d\xi
</math></center>
<math>G(\mathbf{x},\xi)</math> is 
the [[Free-Surface Green Function]] This
can be written alternatively as
<center><math> 
 \phi(\mathbf{x}) = \phi^{\rm in}(\mathbf{x})
 +\int_{\Delta_{0}} 
\sum_{m=-\infty}^{\infty} \left(G^{\mathbf{P}}_{n_\xi}(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l} \phi(\xi) 
- G^{\mathbf{P}} (\mathbf{x},\xi)e^{im\sigma l} \phi_{n_\xi}(\xi) \right) d\xi
</math></center>
where the kernel <math>G_{\mathbf{P}}</math> (referred to as the 
''periodic Green function'') is given by
<center><math>
 G^{\mathbf{P}} (\mathbf{x};\xi)
 = \sum_{m=-\infty}^{\infty} G(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l}.
</math></center>

=Accelerating the Convergence of the Periodic Green Function=

The spatial representation of the periodic Green function given by
equation is slowly convergent
and in the far field the terms decay in 
magnitude like <math>O(n^{-1/2})</math>. In this section we
show how to accelerate the convergence. We begin with the asymptotic
approximation of the Three-dimensional [[Free-Surface Green Function]] 
far from the source point,
<center><math>
 G(\mathbf{x},\xi) \sim -\frac{ik}{2}
 \,H_{0}( k |\mathbf{x}-\xi|), 
 |\mathbf{x}-\xi| \to \infty
</math></center>
[[Wehausen and Laitone 1960]] where <math>H_0 \equiv H_{0}^{(1)}</math> is the Hankel function 
of the first kind of order zero [[Abramowitz and Stegun 1964]]. In Linton [[Linton 1998]] 
various methods were described in which the convergence of the periodic 
Green functions was improved. One such method, which suits the particular 
problem being considered here, involves writing the periodic
Green function as
<center><math>
 G_{\mathbf{P}} (\mathbf{x};\xi) 
  =  \sum_{m=-\infty}^{\infty}
\left[
G\left(\mathbf{x};\xi)+(0,ml)\right) 
+ \frac{ik}{2} H_{0} 
\Big(k\sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big) e^{im\sigma l}
\right]
 -\sum_{m=-\infty}^{\infty}
 \frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big)
 e^{im\sigma l}
</math></center>
where <math>c</math> is a numerical smoothing parameter, introduced to avoid the 
singularity at <math>\mathbf{x} = \xi</math> in the Hankel 
function and
<center><math>
X = x-\xi,\quad \mathrm{and} \quad
Y_{m} = (y-\eta)-ml.
</math></center>
Furthermore we use the fact that second slowly convergent sum can be transformed to
<center><math> 
 -\sum_{m=-\infty}^{\infty}
 \frac{ik}{2}H_{0} \Big(k\sqrt{ (X+cl)^2 + Y_{m}^2 }\Big)
 e^{im\sigma l}
-\frac{i}{l} \sum_{m=-\infty}^{\infty}
 \frac{e^{ik \mu_{m} |X+c| }\,e^{i \sigma_{m} Y_0}}{\mu_{m}}
</math></center>
[[Linton 1998]] where
<math>\sigma_{m} = \sigma + 2 m \pi/l</math> and
<center><math>
\mu_m = \left[ 1-\left(\frac{\sigma_{m}}{k}
 \right)^{2} \right]^{\frac{1}{2}},
</math></center>
where the positive real or positive imaginary part of 
the square root is taken.
Combining these equations 
we obtain the accelerated version of the periodic Green function 
<center><math>
 G_{\mathbf{P}} (\mathbf{x};\xi) 
 = \sum_{m=-\infty}^{\infty}
 \left[ G\left(\mathbf{x};\xi+(0,ml)\right)  
 + \frac{ik}{2} H_{0} \Big(k \sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big)
 \right] e^{im\sigma l}
  -\frac{i}{l}\sum_{m=-\infty}^{\infty}
 \frac{e^{ik \mu_{m}|X+cl| }e^{i\sigma_{m}Y_0}}{\mu_{m}}.
</math></center>
The convergence of the two sums depends on the value
of <math>c</math>. For small <math>c</math> the first sum converges rapidly while the second converges
slowly. For large <math>c</math> the second sum converges rapidly while the first converges
slowly. 
The smoothing parameter <math>c</math> must be carefully chosen to balance these two
effects. Of course, the convergence also depends strongly on how close together the 
points <math>\mathbf{x}</math> and <math>\xi</math> are.

Note that some special combinations of wavelength <math>\lambda</math> and angle
of incidence <math>\theta</math> cause the periodic Green function to diverge
( [[Scott 1998]]).  This singularity is closely
related to the diffracted waves and will be explained shortly.

=The scattered waves  (modes)=

We begin with the accelerated periodic Green function, equation
setting <math>c=0</math> and considering the case when <math>X</math> is large
(positive or negative). We also note that for <math>m</math> sufficiently small
or large <math>i\mu_m</math> will be negative and the corresponding terms will
decay. Therefore
<center><math>
 G_{\mathbf{P}} (\mathbf{x};\xi)
 \sim - \frac{i}{l} \sum_{m=-M}^{N} \frac{e^{ik\mu_{m}|X|}\, e^{i\sigma_{m}Y_0}}
 {\mu_{m}}, X \to \pm \infty
</math></center>
where the integers <math>M</math> and <math>N</math>  satisfy the following
inequalities
<center><math> (M_N)
\left.
\begin{matrix}
\sigma_{-M-1}<-k<\sigma_{-M},\\
\sigma_{N}<k<\sigma_{N+1}.
\end{matrix}
\right\} 
</math></center>
These equations can be written as 
<center><math>
\frac{l}{2\pi}\left(\sigma+k-2\pi \right) < M < \frac{l}{2\pi}\left(
\sigma+k \right),
</math></center>
and
<center><math>
 \frac{l}{2\pi}\left( k - \sigma \right) > N > \frac{l}{2\pi}
 \left( k-\sigma - 2\pi \right)
</math></center>
[[Linton 1998]].
It is obvious that <math>G_{\mathbf{P}}</math> will diverge if <math>\sigma_m = \pm k</math>;
these values correspond to cut-off frequencies which are an expected
feature of periodic structures.

==The diffracted waves==

The diffracted waves are the plane waves which are observed as <math>x \to \pm
\infty</math>. Their amplitude and form are obtained by substituting the limit
of the periodic Green function  as  <math>x\to\pm\infty</math>
into the boundary integral equation for the potential.
This gives us
<center><math>
 \lim_{x\to\pm\infty}
 \phi^{s} ( \mathbf{x},0 )  = - \frac{i}{l} \sum_{m=-M}^{N} \int_{\Delta_0}
 \frac{e^{ik\mu_{m} |X| } e^{i\sigma_{m}Y_0}}{\mu_{m}}
 \left[ k\phi(\xi,0)
 - w(\xi) \right] 
 d\xi,
</math></center>
where <math>\phi^{s} = \phi-\phi^{\rm in}</math> is the scattered wave which
is composed of a finite number of plane waves. For <math>x \to -\infty</math> the scattered wave is given by
<center><math>
 \lim_{x\to-\infty}\phi^{s} 
 (\mathbf{x},0) = A_{m}^{-}\,e^{ik\mu_{m}x}e^{i\sigma_{m}y},
</math></center>
where the amplitudes <math>A_{m}^{-}</math> are 
<center><math>
 A_{m}^{-} = -\frac{i}{\mu_{m}l} \int_{\Delta_0}
 e^{ik\mu_{m}\xi } e^{-i\sigma_{m}\eta}
 \left[  k\phi\left( \xi\right)
 - w (\xi) \right] 
 d\xi.
</math></center>
Likewise as <math>x \to \infty</math> the scattered wave is given by
<center><math>
\lim_{x\to\infty}\phi^{s} (\mathbf{x},0) =
 A_{m}^{+} e^{-ik\mu_{m}x} e^{i\sigma_{m}y},
</math></center>
where <math>A_{m}^{+}</math> are
<center><math>
 A_{m}^{+} = -\frac{i}{\mu_{m}l}\int_{\Delta_0}
 e^{-ik\mu_{m}\xi }e^{-i\sigma_{m}\eta}
 \left[  k\phi (\xi,0)
 - w(\xi) \right]
 d\xi.
</math></center>
The diffracted waves propagate at various angles with respect to the 
normal direction of the array. The angles of diffraction, 
<math>\psi_{m}^{\pm}</math>, are given by 
<center><math>
 \psi_{m}^{\pm} = \tan^{-1}\left(  \frac{\sigma_{m}}{\pm k\mu_{m}}\right).
 (psi_m)
</math></center>
Notice that for <math>m=0</math> we have
<center><math>
 \psi_{0}^{\pm}=\pm\theta,
  (psi_0)
</math></center>
where <math>\theta</math> is the incident angle. This is exactly as expected since we
should always have a transmitted wave which travels in the same direction
as the incident wave and a reflected wave which travels in the negative
incident angle direction.

==The fundamental reflected and transmitted waves==

We need to be precise when we determine the wave
of order zero at <math>x\to\infty</math> because we have to include
the incident wave. There is always at least one set of propagating waves
corresponding to <math>m=0</math> which correspond
to simple reflection and transmission. 
The coefficient, <math>R</math>, for the fundamental reflected wave for the <math>m=0</math> mode 
is given by
<center><math>
R = A_{0}^{-}
 = -\frac{i}{\mu_{0}l}\int_{\Delta_0}e^{ik (\xi\cos\theta
 -\eta\sin\theta)}\left[ k\phi(\xi,0)
 - w(\xi)\right]
 d\xi.
</math></center>
The coefficient, <math>T</math>, for the fundamental transmitted wave for 
the <math>m=0</math> mode is given by
<center><math>
 T = 1 + A_{0}^{+}
 = 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} e^{-ik(\xi\cos\theta
 +\eta\sin\theta)}\left[ k\phi(\xi,0)
 - w(\xi) \right]
 d\xi.
</math></center>

==Conservation of energy==

The diffracted wave, taking into account the correction for <math>T</math>, must
satisfy the energy flux equation. This simply says that the energy of the
incoming wave must be equal to the energy of the outgoing waves. 
This gives us 
<center><math>
\cos\theta = \left( |R|^2+|T|^2 \right) \cos\theta
 + \sum_{m=-M,\,m  \neq 0}^{N}
 \left( |A_{m}^{-}|^2 \cos\psi_{m}^{-} +|A_{m}^{+}|^2 
 \cos\psi_{m}^{+} \right).
</math></center>
The energy balance equation can be used as an accuracy
check on the numerical results.


[[Category:Infinite Array]]