Integration Quick Response Code in .NET 63-D WAVE SCATTERING FROM 2-D ROUGH SURFACES
Denso QR Bar Code barcode library for .net
use .net vs 2010 qr code printing tocompose qr code on .net
Qrcode recognizer on .net
Using Barcode reader for visual .net Control to read, scan read, scan image in visual .net applications.
E 0.60
Visual .net barcode implementation for .net
using barcode integration for vs .net control to generate, create bar code image in vs .net applications.
~ to>
Bar Code barcode library in .net
Using Barcode scanner for visual .net Control to read, scan read, scan image in visual .net applications.
0.50 0.40 0.30 0.20
Control qr code 2d barcode data for c#
to render qr code iso/iec18004 and qr code data, size, image with c# barcode sdk
of RcaliadoN -:UO -.---275
Control qr-codes image in .net
using barcode printing for asp.net control to generate, create qr code image in asp.net applications.
Control qrcode image in vb
using barcode creation for .net vs 2010 control to generate, create qr code image in .net vs 2010 applications.
.Ii ii
Paint ucc.ean - 128 for .net
use visual studio .net crystal ean / ucc - 14 printer toadd ucc - 12 on .net
Ii 0.10
Qr-codes barcode library with .net
using .net vs 2010 crystal todisplay qr-code on asp.net web,windows application
~ ~ 0 ~ 60 SCattertDI ADIIe (dear"s)
Build 2d matrix barcode on .net
generate, create 2d barcode none for .net projects
Figure 6.1.2 Convergence of bistatic scattering coefficient with the number of realizations for a 2-D rough surface. Four cases are shown: 155, 225, 275, and 310 realizations. The cases of 275 and 310 realizations overlap each other.
.net Framework upc code drawer for .net
using barcode integrating for .net control to generate, create upc barcodes image in .net applications.
~ 8 u
Visual Studio .NET ansi/aim code 93 generating on .net
using barcode integrated for visual .net control to generate, create code 9/3 image in visual .net applications.
"iii 1.00
Control code 128 code set a data in .net
code 128 code set a data on .net
0.80 0.60
EAN-13 Supplement 2 generator for vb.net
using barcode printer for aspx.cs page crystal control to generate, create european article number 13 image in aspx.cs page crystal applications.
0.40 0.20
--this method (310) ..... Kirchhoff (360)
Excel Spreadsheets qr codes integrating in excel spreadsheets
use microsoft excel qr barcode implement toinclude qr code iso/iec18004 in microsoft excel
a 0.00
Control ucc ean 128 image in .net
generate, create ean / ucc - 14 none for .net projects
-~ 0 30 60 SCatterlOI ADlle (dear"s)
Control uss code 39 data for .net
to print code 3/9 and code 39 full ascii data, size, image with .net barcode sdk
Figure 6.1.3 Comparison between the SMFSIA and the second-order Kirchhoff method. The second-order Kirchhoff method result is based on 360 realizations.
Control ean13+2 data in word
ean13+2 data in word documents
Order Number 1 2 3 4 5 6
Draw pdf417 in vb.net
using .net vs 2010 tocreate pdf 417 on asp.net web,windows application
Number of CGM Iterations 27 20 13 7 3 1
Error Norm % 29.31 6.74 2.56 1.04 0.12 0.01
Table 6.1.2 Convergence of the SMFSIA of 2-D surface of a single realization.
1.1 Scalar Wave Scattering
For each order of solution the error norm E(n) can be easily computed as follows. From (6.1.12) and (6.1.15) we obtain
liZ X(n) -
bll =
+ Z(FS))X(n)
_ (b _ Z(w) X(n))11
= Ilb(n) _ b(n+l) II
(6.1.18) Table 6.1.2 shows the convergence of the SMFSIA for a single realization.
1.1.3 Convergence of SMFSIA
In actual implementation of the SMFSIA the iteration stops when the error norm of the original matrix equation has reached the established smallness criterion. In this section we examine the convergence of the SMFSIA. We note that the right-hand side b corresponds to the incident wave and is of the order of 0(1), where 0 stands for the order. The column vector
Z(w) X corresponds to the last term of (6.1.8). It is the original impedance
matrix with the removal of the near field sparse matrix and the flat surface impedance matrix.
_ eXP(ik P)] U(x, y)} JJp>rd 47f(xd+Yd+ z d)2 47fp (6.1.19) where Xd = x - x', Yd = Y - y', Zd = f(x, y) - f(x', y') and P = (x~ + yJ)~. We note that U(x, y) is of the order kO(l) and has a phase that is randomly fluctuating, whereas Zd is of the order of rms height h. Thus in the limit of large r d, with r d h, the first term in the square brackets can be Taylorexpanded. Therefore we have
O(Z(W) X)
[eXP[ik(:~ +;J +:Jl~]
o [Z(W) X] = 0
=0 =0
[ff dxdy exp(ikp) (ikZ J ) U(x, y)] JJp>rd 47fp 2p
dpp f21r d exp(ikp) (ikZ J ) U(x, y)] Jo 47fp 2p (6.1.20)
[1~ d/xP~~kP) ikh2 U(x, y)]
By performing integration by parts of (6.1.20), we get (6.1.21) which is much smaller than 0(1) in the limit of large rd.
Electromagnetic Wave Scattering by Perfectly Conducting Surfaces
1.2.1 Surface Integral Equation
Consider an electromagnetic wave impinging upon a rough surface that is perfectly conducting. Then the integral equation for r above the rough surface is
H(r) =
Letting r have
~,dS' [~ x G(r, r') . n' x H(r')] + Hinc(r)
r + which is a point infinitesimally above the rough surface, we
nxH(r) = nxHinc(r)+JiI!l nx
r J8' dS' [~x G(r,r'). n' xH(r')]
For a constant vector a
~ x G(r, r') . a = ~ x
Using (6.1.24) in (6.1.23), we have
(1 + :2 ~~ )g(r, r') . a
a (6.1.24) (6.1.25)
~g(r, r') x
nxH(r) = nxHinc(r)+ JiI!l nx
r J8' dS' [~g(r,r') x (n' x H(r')]
The integral in (6.1.25) is singular at r = r'. To handle this problem, we perform similarly to the Neumann case of 4, Section 1.5. Let
r = p + On (6.1.26) where p is a point on the rough surface. If 0> 0, r = r +, and if 0 < 0, r = r_. Thus, r + is infinitesimally above the surface while r _ is infinitesimally below
the surface. In (6.1.25) and (6.1.26)
n is the normal to the surface at point
p. Then nxH(p)=nxHinc(p)+ lim nx
r J8' dS'[(~g(r,r')) __- + r-r
(6.1.27) The f8' integration is divided into a circular disk Sa of small radius a about p and the rest which is known as the principal value integral
1 1 +1
dS' =
dS' =
1 +f
represents principal value integral, which is the integration over S
with an infinitesimal circular disk of radius a, Sa, removed from S.
1.2 Electromagnetic Wave Scattering by Perfectly Conducting Surfaces
We next examine
I = lim 71 x
f JSa dS' ('Vg(r, r')) r-r
x 71' x H(r') x (71XH(r))]
[(f dS''Vg(r,r')) - Iss
where {) > 0 for r = r + and {) < 0 for r = r _. Let (p', ') be the polar coordinates of r' of circular disk of radius a centered about p
fSa dS''Vg(r, r') = r-+r J
dp' p'
d ''V
1 1 1 1
411" r - r
1_1 -, I
dp' p'
d ' [-
_ 1 _ 3 (r - r')] 411"lr - r'l r=r
d ' , o pp
d ' [_ (71{) - (p' cos ' X + p' sin ' fJ) ] 411"(p,2 + {)2)3/2
The X and fJ components integrate to zero because of the sin ' and cos ' dependence. Thus
f JS
dS''V (- -')
d ' ,
= -71"2 - (p,2 + {)2)1/2
where the
- (a2 + {)2)1/2 1