Difference between revisions of "Infinite Array Green Function"

From WikiWaves
Jump to navigationJump to search
m (Reverted edits by DronvArroc (DronvArroc); changed back to last version by Meylan)
 
(16 intermediate revisions by 4 users not shown)
Line 75: Line 75:
  
 
The spatial representation of the periodic Green function given by
 
The spatial representation of the periodic Green function given by
equation ((G_periodic)) is slowly convergent
+
equation is slowly convergent
 
and in the far field the terms decay in  
 
and in the far field the terms decay in  
 
magnitude like <math>O(n^{-1/2})</math>. In this section we
 
magnitude like <math>O(n^{-1/2})</math>. In this section we
Line 134: Line 134:
 
  \left[ G\left(\mathbf{x};\xi+(0,ml)\right)   
 
  \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)
 
  + \frac{ik}{2} H_{0} \Big(k \sqrt{\left( X+cl\right)^2 + Y_{m}^2}\Big)
  \right] \exp^{im\sigma l}
+
  \right] e^{im\sigma l}
 
   -\frac{i}{l}\sum_{m=-\infty}^{\infty}
 
   -\frac{i}{l}\sum_{m=-\infty}^{\infty}
  \frac{\exp^{ik \mu_{m}|X+cl| }\exp^{i\sigma_{m}Y_0}}{\mu_{m}}.
+
  \frac{e^{ik \mu_{m}|X+cl| }e^{i\sigma_{m}Y_0}}{\mu_{m}}.
 
</math></center>
 
</math></center>
 
The convergence of the two sums depends on the value
 
The convergence of the two sums depends on the value
Line 192: Line 192:
 
The diffracted waves are the plane waves which are observed as <math>x \to \pm
 
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
 
\infty</math>. Their amplitude and form are obtained by substituting the limit
of the periodic Green function ((new_G_p_trunc)) as  <math>x\to\pm\infty</math>
+
of the periodic Green function as  <math>x\to\pm\infty</math>
into the boundary integral equation for the potential ((bem_eq_2)).
+
into the boundary integral equation for the potential.
 
This gives us
 
This gives us
 
<center><math>
 
<center><math>
 
  \lim_{x\to\pm\infty}
 
  \lim_{x\to\pm\infty}
 
  \phi^{s} ( \mathbf{x},0 )  = - \frac{i}{l} \sum_{m=-M}^{N} \int_{\Delta_0}
 
  \phi^{s} ( \mathbf{x},0 )  = - \frac{i}{l} \sum_{m=-M}^{N} \int_{\Delta_0}
  \frac{\exp^{ik\mu_{m} |X| } \exp^{i\sigma_{m}Y_0}}{\mu_{m}}
+
  \frac{e^{ik\mu_{m} |X| } e^{i\sigma_{m}Y_0}}{\mu_{m}}
 
  \left[ k\phi(\xi,0)
 
  \left[ k\phi(\xi,0)
 
  - w(\xi) \right]  
 
  - w(\xi) \right]  
Line 207: Line 207:
 
<center><math>
 
<center><math>
 
  \lim_{x\to-\infty}\phi^{s}  
 
  \lim_{x\to-\infty}\phi^{s}  
  (\mathbf{x},0) = A_{m}^{-}\,\exp^{ik\mu_{m}x}\exp^{i\sigma_{m}y},
+
  (\mathbf{x},0) = A_{m}^{-}\,e^{ik\mu_{m}x}e^{i\sigma_{m}y},
 
</math></center>
 
</math></center>
where the amplitudes <math>A_{m}^{-}</math> are identified from ((phi_s))
+
where the amplitudes <math>A_{m}^{-}</math> are  
as
 
 
<center><math>
 
<center><math>
 
  A_{m}^{-} = -\frac{i}{\mu_{m}l} \int_{\Delta_0}
 
  A_{m}^{-} = -\frac{i}{\mu_{m}l} \int_{\Delta_0}
  \exp^{ik\mu_{m}\xi } \exp^{-i\sigma_{m}\eta}
+
  e^{ik\mu_{m}\xi } e^{-i\sigma_{m}\eta}
 
  \left[  k\phi\left( \xi\right)
 
  \left[  k\phi\left( \xi\right)
 
  - w (\xi) \right]  
 
  - w (\xi) \right]  
Line 221: Line 220:
 
<center><math>
 
<center><math>
 
\lim_{x\to\infty}\phi^{s} (\mathbf{x},0) =
 
\lim_{x\to\infty}\phi^{s} (\mathbf{x},0) =
  A_{m}^{+} \exp^{-ik\mu_{m}x} \exp^{i\sigma_{m}y},
+
  A_{m}^{+} e^{-ik\mu_{m}x} e^{i\sigma_{m}y},
 
</math></center>
 
</math></center>
 
where <math>A_{m}^{+}</math> are
 
where <math>A_{m}^{+}</math> are
 
<center><math>
 
<center><math>
 
  A_{m}^{+} = -\frac{i}{\mu_{m}l}\int_{\Delta_0}
 
  A_{m}^{+} = -\frac{i}{\mu_{m}l}\int_{\Delta_0}
  \exp^{-ik\mu_{m}\xi }\exp^{-i\sigma_{m}\eta}
+
  e^{-ik\mu_{m}\xi }e^{-i\sigma_{m}\eta}
 
  \left[  k\phi (\xi,0)
 
  \left[  k\phi (\xi,0)
 
  - w(\xi) \right]
 
  - w(\xi) \right]
Line 259: Line 258:
 
<center><math>
 
<center><math>
 
R = A_{0}^{-}
 
R = A_{0}^{-}
  = -\frac{i}{\mu_{0}l}\int_{\Delta_0}\exp^{ik (\xi\cos\theta
+
  = -\frac{i}{\mu_{0}l}\int_{\Delta_0}e^{ik (\xi\cos\theta
 
  -\eta\sin\theta)}\left[ k\phi(\xi,0)
 
  -\eta\sin\theta)}\left[ k\phi(\xi,0)
 
  - w(\xi)\right]
 
  - w(\xi)\right]
Line 268: Line 267:
 
<center><math>
 
<center><math>
 
  T = 1 + A_{0}^{+}
 
  T = 1 + A_{0}^{+}
  = 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} \exp^{-ik(\xi\cos\theta
+
  = 1 - \frac{i}{\mu_{0}l} \int_{\Delta_0} e^{-ik(\xi\cos\theta
 
  +\eta\sin\theta)}\left[ k\phi(\xi,0)
 
  +\eta\sin\theta)}\left[ k\phi(\xi,0)
 
  - w(\xi) \right]
 
  - w(\xi) \right]
Line 285: Line 284:
 
  \left( |A_{m}^{-}|^2 \cos\psi_{m}^{-} +|A_{m}^{+}|^2  
 
  \left( |A_{m}^{-}|^2 \cos\psi_{m}^{-} +|A_{m}^{+}|^2  
 
  \cos\psi_{m}^{+} \right).
 
  \cos\psi_{m}^{+} \right).
  (NRGbal)
 
 
</math></center>
 
</math></center>
The energy balance equation ((NRGbal)) can be used as an accuracy
+
The energy balance equation can be used as an accuracy
 
check on the numerical results.
 
check on the numerical results.
 
 
 
=Concluding Remarks=
 
 
Motivated by the problem of modelling wave propagation in the
 
marginal ice zone and by the general problem of scattering by
 
arrays of bodies we have presented a solution to the problem
 
of wave scattering by an infinite array of floating elastic
 
plates. The model is based on the linear theory and assumes that
 
the floe submergence is negligible. For this reason it is applicable
 
to low to moderate wave heights and to floes whose size is large compared
 
to the floe thickness.
 
The solution method is similar to that used to solve
 
for a single plate except that the periodic Green function must
 
be used. The periodic Green function is obtained by summing the free-surface
 
Green function for all <math>y</math>. However the periodic Green function is slowly
 
convergent and therefore a method, which is commonly used in Optics, is
 
devised to accelerate the convergence. The acceleration method involves
 
expressing the infinite sum of the Green function in its far-field form.
 
From the far-field representation of the periodic Green Function, we
 
calculate the diffracted wave
 
far from the array and the cut-off frequencies. We have checked our
 
numerical calculations for energy balance and against the
 
limiting case when the plates are rigid and joined where the
 
solution reduces to that of a rigid dock. We have also presented
 
solutions for a range of elastic plate geometries.
 
 
The solution method could be extended to water of finite depth using
 
a similar periodic Green function, but in this case based on the free
 
surface Green function for finite depth water. The same problems of slow
 
convergence would arise and a similar method for convergence of the
 
series would be required. We believe that a method similar to the one
 
presented here could be used to accelerate the convergence of the periodic
 
Green function for water of finite depth. Another extension would be to consider
 
a doubly periodic lattice in which a doubly periodic Green function
 
would be required. This should also be able to be computed quickly by
 
methods similar to the ones presented here.
 
 
The most important extension of this work concerns using it to construct
 
a scattering model for the marginal ice zone. Such a model would be based
 
on the solutions presented here but would be a far from trivial extension
 
of it. For a model to be realistic the effects which have been induced by
 
the assumption of periodicity would have to be removed. One method to do
 
this would be to consider random spacings of the ice floes and to average the
 
result over these. This should lead to a scattered wave field over
 
all directions rather than discrete scattering angles as well as a reflected
 
and transmitted wave. These solutions would then have to be combined, again with
 
some kind of averaging and wide spacing approximation, to compute the
 
scattering by multiple rows of ice floes.
 
 
 
 
\bibliography{/home/groups/seaice/bibdata/mike,/home/groups/seaice/bibdata/others,/home/meylan/Papers/Periodic_Cynthia/ALMOSTALL,/home/meylan/Papers/Periodic_Cynthia/Cynthia}
 
 
 
 
\pagebreak
 
\begin{figure}
 
[p]
 
\begin{center}
 
\includegraphics[scale = 0.7]
 
{array}
 
\caption{A schematic diagram of the periodic array of floating elastic
 
plates.}
 
(fig_array)
 
\end{center}
 
\end{figure}
 
 
\begin{figure}[ptb]
 
\begin{center}
 
\includegraphics[scale = 0.7
 
 
 
]
 
{floes4jfs}
 
\caption{Diagram of the four plate geometries for which we will
 
calculate solutions. }
 
(floes4jfs)
 
\end{center}
 
\end{figure}
 
 
\begin{figure}
 
[p]
 
\begin{center}
 
\includegraphics[scale = 0.7]{stiff_compare}
 
\caption{The reflection coefficient <math>R</math> (solid line)
 
and the transmission coefficient <math>T</math> (dashed line) as a function
 
of <math>k</math> for a two-dimensional dock of length 4 for the incident
 
angles shown. The crosses are the same problem solved using
 
the three-dimensional array code with the dock boundary
 
condition and using plates of geometry 1 with <math>l=4</math>.}
 
(fig_RnT4stiff)
 
\end{center}
 
\end{figure}
 
 
\begin{figure}
 
[p]
 
\begin{center}
 
\includegraphics[scale = 0.7
 
 
 
]
 
{diffracted_square}
 
\caption{The diffracted waves <math>A_m^\pm</math> for a periodic array
 
of geometry one plates with <math>k=\pi/2</math>, <math>l=6</math>, <math>\beta=0.1,</math> <math>\gamma=0</math>,
 
and <math>l=6</math>. The solid line
 
is <math>A_0^\pm</math>, the chained line is <math>A_{-2}^\pm</math> and <math>A_1^\pm</math> and the
 
dashed line is <math>A_{-1}^\pm</math> and <math>A_2^\pm</math>.}
 
(fig_square_vartheta_beta0p1)
 
\end{center}
 
\end{figure}
 
 
\begin{figure}
 
[p]
 
\begin{center}
 
\includegraphics[scale = 0.7
 
 
 
]
 
{diffracted_triangle}
 
\caption{As for figure (fig_square_vartheta_beta0p1) except that
 
the pate has geometry 2. }
 
(fig_triangle_vartheta_beta0p1)
 
\end{center}
 
\end{figure}
 
 
\begin{figure}
 
[p]
 
\begin{center}
 
\includegraphics[scale = 0.7
 
 
 
]
 
{disp_P_square}
 
\caption{The real part of the displacement <math>w</math> for five plates of geometry one
 
which are part of a periodic
 
array, <math>l= 6</math>, <math>\theta=\pi/6</math>, <math>\beta=0.1</math>, <math>\gamma=0</math> and
 
(a) <math>k=\pi/2</math> and (b) <math>k=\pi/4=8</math>.}
 
(fig_disp_p_square)
 
\end{center}
 
\end{figure}
 
 
\begin{figure}
 
[p]
 
\begin{center}
 
\includegraphics[scale = 0.7
 
 
 
]
 
{disp_P_triangle}
 
\caption{As for figure (fig_disp_p_square) except that the
 
plate is of geometry two.}
 
(fig_disp_p_triangle)
 
\end{center}
 
\end{figure}
 
 
\begin{figure}
 
[p]
 
\begin{center}
 
\includegraphics[scale = 0.7
 
 
 
]
 
{disp_P_parallelogram}
 
\caption{As for figure (fig_disp_p_square) except that the
 
plate is of geometry three.}
 
(fig_disp_p_parallelogram)
 
\end{center}
 
\end{figure}
 
 
\begin{figure}
 
[p]
 
\begin{center}
 
\includegraphics[scale = 0.7
 
 
 
]
 
{disp_P_trapezoid}
 
\caption{As for figure (fig_disp_p_square) except that the
 
plate is of geometry four.}
 
(fig_disp_p_trapezoid)
 
\end{center}
 
\end{figure}
 
 
\begin{figure}
 
[p]
 
\begin{center}
 
\includegraphics[scale = 0.7
 
 
 
]
 
{energy_vs_k}
 
\caption{The total reflected energy <math>E_R</math> divided by
 
<math>\cos\theta</math> as a function of <math>k</math> for floes of geometry 1 (a)
 
and 2 (b). <math>l= 6</math>, <math>\beta=0.1</math>, <math>\gamma=0</math>, and
 
<math>\theta = 0</math> (solid line), <math>\pi/6</math> (dashed line), and <math>\pi/3</math>
 
(chained line).}
 
(energy_vs_k)
 
\end{center}
 
\end{figure}
 
 
 
  
  
 
[[Category:Infinite Array]]
 
[[Category:Infinite Array]]

Latest revision as of 09:11, 9 January 2009

Introduction

We present here the solution to the Infinite Array based on an infinite image system of Free-Surface Green Functions

Problem Formulation

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

The bodies occupy [math]\displaystyle{ \Delta_m }[/math], [math]\displaystyle{ -\infty \lt m \lt \infty }[/math]. Periodicity implies that if [math]\displaystyle{ (x,y) \in \Delta_0 }[/math], then [math]\displaystyle{ (x,y+ml) \in \Delta_m }[/math], [math]\displaystyle{ -\infty \lt m \lt \infty }[/math].

We assume that we have the Standard Linear Wave Scattering Problem. The incident wave potential given by

[math]\displaystyle{ \phi^{{\rm in}} = \frac{A}{k} e^{ik (x\cos\theta+y\sin\theta)}\,e^{kz}, }[/math]

where [math]\displaystyle{ A }[/math] is the dimensionless amplitude and [math]\displaystyle{ \theta }[/math] is the direction of propagation of the wave (with [math]\displaystyle{ \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]\displaystyle{ \Delta_{0} }[/math] is given by [math]\displaystyle{ \phi( \mathbf{x}_{0},0) }[/math], [math]\displaystyle{ \mathbf{x}_{0}\in\Delta_{0} }[/math], then by Floquet's theorem the potential satisfies

[math]\displaystyle{ \phi(\mathbf{x}_{m},0) = \phi(\mathbf{x}_{0},0) e^{im\sigma l}, }[/math]

and the displacement of the plate [math]\displaystyle{ \Delta_{m} }[/math] satisfies

[math]\displaystyle{ w(\mathbf{x}_{m}) = w(\mathbf{x}_{0}) e^{im\sigma l}, }[/math]

where [math]\displaystyle{ \mathbf{x}_{m} \in \Delta_{m} }[/math], [math]\displaystyle{ -\infty \lt m \lt \infty }[/math] and the phase difference is [math]\displaystyle{ \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

[math]\displaystyle{ \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]

[math]\displaystyle{ G(\mathbf{x},\xi) }[/math] is the Free-Surface Green Function This can be written alternatively as

[math]\displaystyle{ \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]

where the kernel [math]\displaystyle{ G_{\mathbf{P}} }[/math] (referred to as the periodic Green function) is given by

[math]\displaystyle{ G^{\mathbf{P}} (\mathbf{x};\xi) = \sum_{m=-\infty}^{\infty} G(\mathbf{x},\xi+(0,ml,0))e^{im\sigma l}. }[/math]

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]\displaystyle{ 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,

[math]\displaystyle{ G(\mathbf{x},\xi) \sim -\frac{ik}{2} \,H_{0}( k |\mathbf{x}-\xi|), |\mathbf{x}-\xi| \to \infty }[/math]

Wehausen and Laitone 1960 where [math]\displaystyle{ 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

[math]\displaystyle{ 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]

where [math]\displaystyle{ c }[/math] is a numerical smoothing parameter, introduced to avoid the singularity at [math]\displaystyle{ \mathbf{x} = \xi }[/math] in the Hankel function and

[math]\displaystyle{ X = x-\xi,\quad \mathrm{and} \quad Y_{m} = (y-\eta)-ml. }[/math]

Furthermore we use the fact that second slowly convergent sum can be transformed to

[math]\displaystyle{ -\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]

Linton 1998 where [math]\displaystyle{ \sigma_{m} = \sigma + 2 m \pi/l }[/math] and

[math]\displaystyle{ \mu_m = \left[ 1-\left(\frac{\sigma_{m}}{k} \right)^{2} \right]^{\frac{1}{2}}, }[/math]

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

[math]\displaystyle{ 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]

The convergence of the two sums depends on the value of [math]\displaystyle{ c }[/math]. For small [math]\displaystyle{ c }[/math] the first sum converges rapidly while the second converges slowly. For large [math]\displaystyle{ c }[/math] the second sum converges rapidly while the first converges slowly. The smoothing parameter [math]\displaystyle{ 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]\displaystyle{ \mathbf{x} }[/math] and [math]\displaystyle{ \xi }[/math] are.

Note that some special combinations of wavelength [math]\displaystyle{ \lambda }[/math] and angle of incidence [math]\displaystyle{ \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]\displaystyle{ c=0 }[/math] and considering the case when [math]\displaystyle{ X }[/math] is large (positive or negative). We also note that for [math]\displaystyle{ m }[/math] sufficiently small or large [math]\displaystyle{ i\mu_m }[/math] will be negative and the corresponding terms will decay. Therefore

[math]\displaystyle{ 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]

where the integers [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] satisfy the following inequalities

[math]\displaystyle{ (M_N) \left. \begin{matrix} \sigma_{-M-1}\lt -k\lt \sigma_{-M},\\ \sigma_{N}\lt k\lt \sigma_{N+1}. \end{matrix} \right\} }[/math]

These equations can be written as

[math]\displaystyle{ \frac{l}{2\pi}\left(\sigma+k-2\pi \right) \lt M \lt \frac{l}{2\pi}\left( \sigma+k \right), }[/math]

and

[math]\displaystyle{ \frac{l}{2\pi}\left( k - \sigma \right) \gt N \gt \frac{l}{2\pi} \left( k-\sigma - 2\pi \right) }[/math]

Linton 1998. It is obvious that [math]\displaystyle{ G_{\mathbf{P}} }[/math] will diverge if [math]\displaystyle{ \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]\displaystyle{ x \to \pm \infty }[/math]. Their amplitude and form are obtained by substituting the limit of the periodic Green function as [math]\displaystyle{ x\to\pm\infty }[/math] into the boundary integral equation for the potential. This gives us

[math]\displaystyle{ \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]

where [math]\displaystyle{ \phi^{s} = \phi-\phi^{\rm in} }[/math] is the scattered wave which is composed of a finite number of plane waves. For [math]\displaystyle{ x \to -\infty }[/math] the scattered wave is given by

[math]\displaystyle{ \lim_{x\to-\infty}\phi^{s} (\mathbf{x},0) = A_{m}^{-}\,e^{ik\mu_{m}x}e^{i\sigma_{m}y}, }[/math]

where the amplitudes [math]\displaystyle{ A_{m}^{-} }[/math] are

[math]\displaystyle{ 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]

Likewise as [math]\displaystyle{ x \to \infty }[/math] the scattered wave is given by

[math]\displaystyle{ \lim_{x\to\infty}\phi^{s} (\mathbf{x},0) = A_{m}^{+} e^{-ik\mu_{m}x} e^{i\sigma_{m}y}, }[/math]

where [math]\displaystyle{ A_{m}^{+} }[/math] are

[math]\displaystyle{ 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]

The diffracted waves propagate at various angles with respect to the normal direction of the array. The angles of diffraction, [math]\displaystyle{ \psi_{m}^{\pm} }[/math], are given by

[math]\displaystyle{ \psi_{m}^{\pm} = \tan^{-1}\left( \frac{\sigma_{m}}{\pm k\mu_{m}}\right). (psi_m) }[/math]

Notice that for [math]\displaystyle{ m=0 }[/math] we have

[math]\displaystyle{ \psi_{0}^{\pm}=\pm\theta, (psi_0) }[/math]

where [math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ m=0 }[/math] which correspond to simple reflection and transmission. The coefficient, [math]\displaystyle{ R }[/math], for the fundamental reflected wave for the [math]\displaystyle{ m=0 }[/math] mode is given by

[math]\displaystyle{ 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]

The coefficient, [math]\displaystyle{ T }[/math], for the fundamental transmitted wave for the [math]\displaystyle{ m=0 }[/math] mode is given by

[math]\displaystyle{ 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]

Conservation of energy

The diffracted wave, taking into account the correction for [math]\displaystyle{ 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

[math]\displaystyle{ \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]

The energy balance equation can be used as an accuracy check on the numerical results.