DIFFRACTION BY A PERFECTLY CONDUCTING RECTANGULAR CYLINDER WHICH … · DIFFRACTION BY A PERFECTLY...

N A S A C O N T R A C T O R

R E P O R T

N A S A C R - 2 4 0 5

DIFFRACTION BY A PERFECTLY CONDUCTING

RECTANGULAR CYLINDER WHICH IS ILLUMINATED

BY AN ARRAY OF LINE SOURCES

by R. G. Kouyoumjian and N. Wang

Prepared by

THE OHIO STATE UNIVERSITY

ELECTROSCIENCE LABORATORY

Columbus, Ohio 43212

for Langley Research Center

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • JUNE 1974

https://ntrs.nasa.gov/search.jsp?R=19740020593 2020-02-20T21:10:27+00:00Z

1.. Report No.NASA CR- 21)05

2. Government Accession No.

4. Title and Subtitle

DIFFRACTION BY A PERFECTLY CONDUCTING RECTANGULAR CYLINDER

WHICH IS ILLUMINATED BY AH ARRAY OF LIKE SOURCES

7. Author(s)

R. G. Kouyoumjian and N. Wang

9. Performing Organization Name and Address

The Ohio State University

ElectroScience Laboratory

Columbus, Ohio It3212

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D..C. 2051(6

3. Recipient's Catalog No.

5. Report DateJune 1974

6. Performing Organization Code

8. Performing Organization Report No.

TR 3001-710. Work Unit No.

502-33-13-02

11. Contract or Grant No.

NCR 36-008-11*4

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

15. Supplementary Notes

Topical report.

16. AbstractThe geometrical theory of diffraction (GTD) is employed to analyze the radiation from a

iperfectly-conducting rectangular cylinder illuminated by an array of line sources. The excitation

of the cylinder by a single electric or magnetic current line source is considered first, and a

solution which includes contributions from the geometrical optics rays and all singly- and doubly-

diffracted rays is obtained. A new diffraction coefficient valid in the transition regions of the

shadow and reflection boundaries is employed to obtain a continuous total field, except for

negligible discontinuities in the doubly-diffracted field at its shadow boundaries. Patterns

calculated by the GTD method are found to be in excellent agreement with those calculated from

an integral equation formulation. Using superposition the solution for array or aperture

excitation of the rectangular cylinder is obtained. A computer program for this solution is

included.

17. Key Words (Suggested by Author(s)) 18. Distribution Statement

Antenna, Spacecraft and Aircraft AntennasUnclassified - Unlimited

Applied Electromagnetic Theory

STAR Cateeorv 0019. Security Qassif. (of this report) 1

Unclassified

0. Security Classif. (of this page) 21. No. of Pages 22. Price*

Unclassified 71 $3.75

For sale by the National Technical Information Service, Springfield, Virginia 22151

Page Intentionally Left Blank

TABLE OF CONTENTS

Page

II. METHOD OF SOLUTION 3

A. Geometrical Optics Rays - Direct and Reflected 3B. Singly-Diffracted Rays 5C. Doubly-Diffracted Rays 7

III. NUMERICAL RESULTS 18

IV. CONCLUSIONS 26

APPENDIX ITHE FIELD AT THE SHADOW BOUNDARY OF A THICK SCREENFOR GRAZING INCIDENCE .. 27

APPENDIX IIDESCRIPTION OF THE COMPUTER PROGRAM 31

A. Input Variables 31B. Output Variables ' 31C. Instructions for Representing the Aperture Field

Distribution by a 2-D Line Source Array 32D. Instructions for Using, the "Obliquity Factor' 34E. Instructions for Computing the Incident Field 34F. Sample Programs 35G. Listing of the Computer Programs 40

REFERENCES 69

iii

I. INTRODUCTION

In this report the geometrical theory of diffraction (GTD) [1] isused to treat the diffraction by a perfectly-conducting rectangularcylinder which is i l lumina ted by an array of electric or magnetic currentline sources. The i l lumina t ion by a s ingle l ine source of the array isconsidered first with the geometry of the problem shown in Fig . 1. Thescattered and total fields are calculated at all points exterior to therectangular cylinder, except for the shaded regions around the sourceand edges, which are excluded because of the nature of the high-frequencyapproximation. More wil l be said about this later. The solution isthen generalized to calculate the f ield of linear arrays of such line,sources radiating in the presence of the cylinder.

The scattering by a rectangular cylinder has been considered pre-viously for the special case of plane wave incidence. Mei and VanBladel|2 ,3 j formulated the problem in terms of an integral equation which theysolved numerically to f ind the surface currents on the rectangularcylinder. They give radiation patterns, scattering cross sections, andsurface currents for both E- and H-polarized incident waves. There aretwo disadvantages associated with this method: 1) as the frequencyincreases, difficulties are encountered with the convergence of thesolution (this is particularly true when the technique is employed tosolve a three-dimensional scattering problem), 2) the solution providesno physical ins ight into the scattering mechanism. Morse |4] alsostudied this problem using the ordinary GTD to obtain expressions forthe diffracted field away from shadow and reflection boundaries. Sincethe ordinary theory fails at shadow and reflection boundaries, he intro-duced supplementary solutions there. He employed Oberhettinger's un i -form asymptotic solution [5] near the boundaries of the incident andreflected f ields, and he employed an integral representation of^thef ie ld near the shadow boundaries of the fields of the diffracted rays.Thus, he did not obtain a compact high-frequency solution to this problem.The d i f f icul t ies encountered by Morse at these boundaries may be overcomewith a new edge diffraction coefficient derived by Kouyoumjian and Pathak|6,7|. This diffraction coefficient can be appl ied in the transitionregions adjacent to the shadow and reflection boundaries so that one ob-tains a total field which is valid and continuous everywhere away fromedges and caustics. Their diffraction coefficient is employed in thisanalysis .

Using Keller 's Generalized Fermat's Principle, we include contri-but ions to the total field from the geometrical optics fields ( inc identand reflected), as well as singly-diffracted fields which appear toemanate from the edges. Doubly-diffracted fields have also been in-cluded to describe the interactions between the edges; however, mul t ip ly-diffracted rays of higher order have been neglected because, in general,their fields contribute ins ignif icant ly , for the problem defined here.The incident , reflected and diffracted rays may be shadowed in the geo-metrical optics sense, and hence they contribute to the total f ield onlyin their respective regions of i l luminat ion . Discontinuities in the

SOURCE

Fig. 1. Line source in the presence of the rectangularcylinder.

f ie ld are introduced at shadow boundaries and at the reflection boundar ies ,but they are symstematically removed by employing the new diffract ion co-efficient for the.edge diffracted field mentioned previously. It shouldbe pointed out that there are some residual d iscont inui t ies due to uncom-pensated discontinuities in the f ie ld of the doubly-diffracted rays attheir shadow boundaries. However, these discontinuit ies are so smallthat they are not apparent in the plotted patterns. The patterns calcu-lated from our GTD solution are found to be in excellent agreement wi ththose calculated from numerical solutions.

Due to the high-frequency approximations of the incident and dif-fracted f ie lds , our solution is restricted so that the fo l lowing distancesare greater than 0.7 wavelength.

1) The distance between the l ine source and the closest edgeof the rectangular cylinder,

2) the distance between the observation point P and theclosest edge of the rectangular cylinder,

3) the distances between the edges of the rectangularcylinder,

4) the distance between the observation point P and the-closest line source, when calcula t ing the incident:-ortotal f ields.

2

A computer program based on the GTD solution for the rectangularcylinder in the.presence of an array source has been written and isincluded in the report. With this program, it is possible to obtainnumerical results for both the near- and far-fields of the cylinder underquite general conditions of illumination. Thus, the program is directlyrelevant to both antenna and scattering problems. As an example of itsversatility, one notes that the program may be used to compute thepattern of an array of magnetic line sources mounted directly on therectangular cylinder,, provided the sources are not too close to an edge.

II. METHOD OF SOLUTION

Keller's geometrical theory of diffraction jl] is an extension ofgeometrical optics in which diffracted rays are introduced by a generali-zation of Fermat's principle, the excitation of the diffracted field istreated as a local phenomenon, and away from the diffracting surfacethe behavior of the diffracted field along its ray is the same as thatof the geometrical optics field. The basic idea of GTD is that thefield of the line source illuminates the rectangular cylinder givingrise to a reflected field and an edge diffracted field, which consists ofthe fields of singly and multiply-diffracted rays. The total field U(P)at a point P is equal to the sum of the fields on all rays through P.

(1) U(P) = I Un - (P)rays

which includes the incident field if P is not in the shadow region. Thewave function U(P) represents a magnetic field parallel to the edge incase of a magnetic line source, and an electric field parallel to theedge in the case of an electric current line source. The pertinent raysand their associated fields will be discussed briefly in the followingparagraphs.

A. Geometrical Optics Rays - Direct and Reflected

Let us consider the field radiated from a line source at 0 andobserved at P as shown in Fig. 2. Fermat's principle predicts only .thedirect ray OP. If a line source is being considered, the field along OPis given by

(2)

where SQ is the distance between 0 and P, and C is a conveniently chosennormalization constant. For the configuration shown in Fig. 2, the spacesurrounding the right-angle wedge may be divided into three regions:

+1 ~Jy/ /> /////////////Qp C

'B

v //v/vv/ "'3^ A

HE

SHADOW BOUNDARY AT ir + <f>'REFLECTION BOUNDARY AT TT - >'

Fig. 2. Line source in the presence of a wedge.

IMAGE•0'

RegionRegionRegion

IIIIII

o<TT -

TT +

' iSf< '

4 > ' <

< f > ' ' <

7T -

'*<' < ( ) <

(j,1 ,

'TT + (f ' »

3TT/2 .

Region III is the shadow region, which is not penetrated by the incidentray; the incident field vanishes here.

We know that there is a field reflected from the surface AQc. Todescribe this we introduce an additional class of rays which include ontheir trajectory a point QR of the surface AQ^. Applying Fermat'sprinciple, the distance OQRP along the ray path is a minimum and the lawof reflection results. This simple extension of Fermat's principle whichaccounts for the reflected ray is so natural that we accept it withoutquestion. The field of the reflected ray is readily deduced from imagetheory as

(3) Ur(P) = Ur(A-QE) = ± C,-Jks"

where the positive sign is for Neumann (hard) boundary condition associatedwith the magnetic current line source, the negative sign is for theDirichlet (soft) boundary condition associated with the electric currentsource line, and s" is the distance between the image 0' and the observa-tion point P. The reflected field vanishes in regions II and III, whichthe reflected ray does not penetrate. Let us consider now a further ex-tension of F/ermat's principle.

4

B. Si n gl y-Di ffracted Rays

It is well known that the ray incident on the edge Q£ in Fig. 2gives rise to diffraction. To account for this, Keller introduced aclass of rays which includes the point Qp in its trajectory, this com-pletely determines the diffracted ray path in the isotropic, homogeneousmedium of this two-dimensional problem, so the law of edge diffractionbecomes trivial under these circumstances.

0 isIn terms of GTD, the diffracted field at P for the line source at

(4) Ud(QE) =

where D? is the scalar diffraction coefficient for the accoustically soft(Dirichlet) boun.dary condition and fy is the scalar diffraction coeffi-cient for the.Accoustically hard (Neumann) boundary condition. They arededuced from the general dyadic diffraction coefficient D(<j>,<j>' ,80) ob-tained by Kouyoumjian and Pathak [6,7]. For the special case where theincident ray is perpendicular to a straight edge, the scalar diffractioncoefficients are given by

(5) _ -e-j tr/4

•\ 2nF[kLa+(*-<f>')j + cotf 9n )F[kLa-(<|,^')]

F[kLa+(<j,+ <(,'

where nir is the exterior wedge angle, which equals 3^/2 in this case, and

(6)

in which

F(x) = 2j|v^| ej X dT

(7) a±(<f,±<))1) = 2 cos2

N are the integers which most nearly satisfy the fol lowing equations

(8) = TT

(9) 2irnN~ = -TT + (ft*1)

and kl_ is the large parameter in the asymptotic evaluation of the pert inentintegrals involved in the derivation of the dyadic diffraction coeffi-cient. The quantity L may be viewed as a distance parameter which dependsupon the type of edge i l l umina t ion ; for line source i l l umina t ion , L isgiven by

(10) L =s s'

s + s1

For grazing incidence <f>' = 0, mr, Dh is mul t ip l ied by a factor of h\furthermore, if the diffracted ray grazes the surface in the case ofa soft boundary, D = 0 and the diffracted f ie ld vanishes, as itshould.

The f ield of the singly-diffracted ray is discontinuous at theshadow and reflection boundaries in a way which compensates the dis-continuities in the geometrical optics fields there. This is readilydemonstrated; consider1 for example the incident and diffracted fields atthe shadow boundary, where to s implify the discussion, it is assumedthere is no nearby reflection boundary. Let ir+<j>'-e be a point close tothe shadow boundary, see Fig. 2. In the i l luminated region e > 0 and inthe shadow region e < 0.

(II) U ( < f > ) =

>-Jks'

-jks' -jks

e > 0

-Jks

< 0

For e small it follows from Eq. (5)v that

(12)-j Tt/4

cot *- F[kl_a"U-e)]

+ smaller terms which are continuous at the shadow boundary >

6

From Eq. (9),

N" = 0

Also, as e ->• 0,

(13) cot (f-) = —2n e

(14) a'U-s) = -

(15) F[kLa-U-e)j =jir/4

Substituting Eqs. (13) and (15) into Eq. (12) as e -> 0,

(16)

(17)

sgn e + smaller, continuous terms,

sQ = s' + s

Upon substituting Eqs. (16) and (17) into Eq. (11), it is seen thatthe total field is continuous at the shadow boundary. In an analogousmanner it can be shown that the total field is continuous at the reflec-tion boundary.

C. Doubly-Diffracted Rays

When one face of the conducting wedge is terminated at Qp as shownin Fig. 3, a second order diffracted-ray will emanate from the edge Qp.In terms of the GTD, the doubly-diffracted field at P due to the linesource at 0 can be written as

(18) Ud(QE,QF) = Dc(<J>2 '°),-Jks

,-jkhDs(<f>2,0)

,-Jks

Since Ds(3ir/2,<t>'). = 0, the contribution from the doubly-diffracted raysvanishes for the soft boundary according to the above expression. If ahigher order approximation for the doubly-diffracted field is employed,then this contribution is non-vanishing, as will be explained later.

The field of the ray singly-diffracted at Q£ has a shadow boundaryas shown in Fig. 3; the singly-diffracted ray does not penetrate

S\^2

>.

S' LINESOURCE

•#'/ / / s s / / ^ s s / ss/\ f s f f ' f f / s s / r /

> 1' h

;' / / / s / / / / / / A / / / S / S S / S / / s

I/SB

Fig. 3. Configuration for double-diffract ion.

the shaded region. It will be shown next that the discontinuity in thefield of the singly-diffracted ray at SB(Q^) is compensated by the raydoubly-diffracted from Qp, so that the total diffracted field is con-tinuous at this boundary. Since the field doubly diffracted vanishesin the case of the soft boundary, we only need to treat the hardboundary here.

Consider a point close to SB(QE) so that <|>2 = TT-E, the totaldiffracted field at this boundary is

(19) TD

-jks, e< 0

-jks—

8

where s1,(|»1 are the coordinates of the ray diffracted from Q£, and

a-jkh(20) Un(QF) = "''" Xr> 'Jlt '

When 4 = 0 and the singly-diffracted ray grazes the vertical surface,the second and fourth terms in the expression for the diffraction co-efficient are the same, except for the ± sign of the latter. This isalso true for the first and third terms. As a result,

p-j TT/4 (

(21) DhU-e.O) = "e icot(fr) F[kla"U-e)]

+ smaller terms which are continuous at SB(Q^) I .

As e + 0 , it is seen f rom Eqs . (12), (13), (14) and (15) that

(22) DhU-e,0) = - Jlffi Sgn e;

furthermore,

(23) ' Sj = h + s .

Substituting Eqs. (22) and (23) into Eq. (19) and making use ofEq. (20), it is seen that the total diffracted field is continuous atthe boundary

As we have already noted, in the case of a soft boundary the fieldof an incident-ray grazing the surface vanishes, the edge -diffractedfield is then proportional to the normal derivative of the incidentfield at the edge. The proportionality factor is a diffraction coef-ficient D1 given by Karp and Keller |8J. Thus, for the case of Dirichletproblem, the doubly-diffracted field must be replaced by

d aUMQp) D'U2,0) -jks(24) Ud(QE,Q,) = - f- - _ 5 -

h an 2 £ '

where

(25) D 'U 2 , 0 ) - D U 2 , 0 )

The derivate 3U(Qp)/an is taken with respect to the normal to thesurface QrQp. This contribution is weak in comparison with that of thesingly-diffracted rays; the. contribution of the former is of order (1/k2),whereas that of the latter is of order (I/•IT). In ca lcula t ing the f ie lddiffracted from the soft cylinder, it was found that the field of thedoubly-diffracted rays did not contribute s ignif icant ly , so the contribu-tion from these rays can be omitted in this case.

Let us now turn to the diffraction by a rectangular cylinderi l lumina ted by a l ine source. Depending on the location of the l inesource, the whole domain surrounded by the cylinder w i l l be divided intoregions by the various shadow boundaries and the reflection boundaries.Each of these boundaries is labeled to indicate how it originates. Forexample, referring to Fig. 4, the notation^SB means the shadow boundaryof the incident geometrical optics f ie ld U r ( P ) , RB(A-B) is the shadowboundary of the geometrical optics f ield U (A-B) reflected from the ,surface A-B, SB(A) is the shadow boundary of the singly-diffracted U ( A ) ,which emanates from the edge A, and SB(A,B) is the shadow boundary ofthe doubly-diffracted field U (A,B) which emanates from the edge B. Theshadow boundary of the reflected field i$ referred to simply as thereflection boundary.

R.B.(A-B.) S.B.U.D.)

3TJE

xnz:S.B.(B,C.) R.B.(A.-B.)

LINE I /SOURCE | /

iS.B.(B.)S.B.(A,B.)

S.B.(B,C.)

\/

/

/xn

r s ss / s s / f s S f s / / / r /

^D C^' / //S///S//// S /// //

\\\

n

EZ:\

S.B. (B,A.)

S.B.(A,D.)\\

\3X •yrrr

\S.B. /

/S.B.(A.)S.B.(B,A.)

S.B.(B.)S.B.(A,B)

Fig. 4. Shadow and reflection boundaries of theGTD fields.

10

V

zo3UJcrux

oz

Izoz</>

aUJ

UJX

Z«JIU

CO

•s

ouiXo

00I4

00

00 er

•o=L

u*t

CD

n

In what follows we may locate the line source in the upper lefthand quadrant of the space surrounding the cylinder, without loss ofgenerality. Three cases will be considered.

1) First, the line source is located directly above the cylinder,the whole domain around the cylinder is divided into 14 regionsas shown in Fig. 4. Table I shows the regions covered by eachtype of ray. To demonstrate the use of the table, let usconsider an observation point P in region IX. Examining thecolumn under region IX of Table I, one finds checks(/) forUa(A), Ud(B,A), Ud(A,D), and Ud(B,C). Thus the total fieldat P is equal to UT(P) = Ud(A) + Ud(B,A) + Ud(A,D) + Ud(B,C).

2) The line source is located to the upper left of the cylinderas shown in Fig. 5. In this case there are 18 regions.Table II shows the regions where a particular ray exists andits field makes a non-vanishing contribution.

3) Line source at grazing incidence. Let the line source belocated in the plane which contains the A-D side of the rec-tangular cylinder as shown in Fig. 6. This case is of specialinterest because the trailing edge D lies in the shadow ofthe leading edge A. Also, it lies on the shadow boundary ofthe direct geometrical optics field. The behavior of thefield at the boundary DP must be treated separately, seeAppendix I. As before, the domain surrounding the cylinder isdivided into regions as shown in Fig. 6 and Table III gives theregions covered by the individual rays.

Recall that one or more of the various field components is dis-continuous at each boundary shown in Figs. 4, 5 and 6 but that all exceptthe discontinuities in the doubly diffracted fields are compensated; e.g.,the discontinuities in the geometrical optics field are compensated bythe field of the singly-diffracted rays and the discontinuities in thefield of the singly-diffracted rays are compensated by the fields of thedoubly-diffracted rays.

As a final step in the anslysis, the fields of the individual linesources are superimposed to give the field of a linear array of linesources radiating in the presence of the cylinder, see Fig. 7. A com-puter program has been written to calculate the incident, total andscattered fields once the linear array is specified. Unlike its earlierdefinition for geometrical optics, the term incident field used heremeans the field of the array in the absence of the cylinder, and thescattered field is simply the difference between the total field andthis incident field. The versatility of such a program is evident; thescattering from the cylinder for a wide variety of illuminations can bestudied, and the radiation from antennas in the presence of the rectan-gular cylinder also can be studied. As a matter of fact, the p'rogram

12

was written originally so that the linear array of line sources, whendensely packed, closely approximates the field of an aperture antenna offinite width W. The aperture antenna (more precisely its axis) isdirected toward a point Q on the surface of the rectangular cylinder asshown in,Fig. 7. A description of the aperture radiation in terms of anarray of discrete line sources is discussed in the following paragraphs.

The aperture distribution may be approximated by a discrete arrayof line sources which are properly weighted in amplitude and phase. Thewidth of the aperture denoted by W, is divided into 2M segments; (M =integer). The line sources are positioned at the ends of these segments,which introduces 2M + 1 line sources. In approximating a continuousdistribution, the number M is selected so that 2M + 1 >_ 10 W/x , whereX = free space radiated wavelength.

Three types of line sources are available in this program:

Type I An electric current line sourceType II A magnetic current line sourceType III A magnetic current moment line source.

As described earlier, the electric current line source radiates anomnidirectional electric field which is parallel to the edge of therectangular cylinder, and the magnetic current line source radiates anomnidirectional magnetic field which is parallel to the edge of therectangular cylinder. The magnetic current moment line source consistsof a continuous array of magnetic current moments directed perpendicularto the line of the array and parallel to the aperture in question. Thisline source radiates an electric field parallel to the edge of thecylinder; however, the field has a pattern, j.cos e|, where e is shown inFig. 7. The strength of these magnetic type line., sources is determinedfjom^the equivalent magnetic surface currents in the aperture. Ks =E x n, where E is the electric field distribution in the aperture(assumed known) and n is the outward normal to the aperture.

The field of the two-dimensional aperture can be adequately repre-sented in the forward direction by a densely-packed array of type II andtype III line sources, but such an array fails to approximate the fieldadequately at aspects behind the aperture. This limitation is particu-larly troublesome when calculating the total field. To overcome thisdifficulty an obliquity factor has been included in the program whichmultiplies the pattern of each line source. The obliquity factor isf(e) = cosn e/2, where n = 0, %, 1, 2. When n = 0, the obliquity factoris unity so that the array radiates symmetrically with respect to theaxis of its elements. The case n = 2 occurs naturally in the descriptionof the radiation fields of aperture antennas via the Kirchhoff-Huygen'sapproximation (for the forward region ). The cases n = h, and n = 1 areadded so that the n which best approximates the measured aperture patternmay be used. It is evident that the obliquity factor results in apattern null in the direction directly behind the aperture at e =TT. Inmost practical cases, there is no such null in the backward direction.

13

>X

V

M N

N

0

zooU /Q

K / _ )

o CD

CD CDuCD

o

15

CDI

cc \

<rco^-l«—CD I CDcolco

isIQQIW CD

/CO

O 0•» *»

m oivj

CD CDCO CO

»,CD

Q<

CD CD"CO GC

\ B\\ \

\

M .

UJ0

*S Hnuz3 m£

CD

J/

//

' N/ n/»>\XN >N\

^^K / \ V^Cu Ov

^ NI V

>

k V" \

\v \^ VV N

^rf ^^< Q\

^ V N.S. V V Vs

MM

-Ctf o

< CD

CD CD

\ '

NH ^^°°CD <™^.03 CD(O CO

I IFHPirl- *-»

<

*-~ -Q. <OD

CD CD CDCO CO CO

. M1

_

uQ"

5

toT3

"HJ

O)_G•»->

(U•^

(O13

I

O

0)

«^c-

I03

ID

*

O)

16

M

oztuCLCL

UJUJ

CDI

UJ

<

•o

CO

CO

< co CD

oQ*

17

w

2-D APERTUREREPLACEDARRAY OFSOURCES

{ APERTUREWIDTH = W

(OBSERVATIONPOINT )

Fig. 7. An array of line source in the presence ofthe rectangular cylinder.

On the other hand, the pattern in the forward region is quite satisfac-tory when the obliquity factor is included, and as one moves away fromthe forward region, the pattern drops to a level where the differencesbetween the simulated and measured patterns are unimportant when com-puting the total field surrounding the rectangular cylinder.

The scattered, and the total fields are computed as a function ofe s ( -TT <_ es <_ IT) at a range Rs. The incident field pattern on the otherhand is available for a phase reference at the center of the array(aperture) at 0 in Fig. 7 and computed at a range measured from 0, aswell as for a phase reference at Q with a range Rs (just as for thescattered and the total fields).

The input and output variables for the computer program are des-cribed in Appendix II, and the listing of the computer program is given.Also, a sample case is treated in this appendix to illustrate the useof the program.

III. NUMERICAL RESULTS

To assess the accuracy of the GTD solution described injthepreceding section, it was applied to several simple examples--, where therectangular cylinder is illuminated by either an electric 6^ magneticcurrent line source. Far-zone patterns for the total field are calcu-Jkited by this method and also from numerical solutions of the pertinentinte§r-a.l~eq.uatioj]*._ The cylinders are square with a side length of 1.6

*The computer programs for the integral equation solutions were providedby Prof. J.H. Richmond of the ElectroScience Laboratory.

18

wavelength, and in each case the line source illuminates the cylinderfrom a distance of 0.8 wavelength. These small distances provide astringent test of our GTD solution; also they give us an opportunity toexamine the accuracy of the new scalar diffraction coefficients in asituation where the edges are illuminated by curved wavefronts and wherethe transition regions are relatively broad. The pattern calculatedfrom the integral equation solution must be considered more accuratefor the small dimensions chosen for these examples, since the integralequation method is convergent whereas the GTD solution is an asymptoticapproximation.

Patterns for magnetic current line source excitation (hard boundarycase) are given in Figs. 8, 9 and 10, where the line source is positionedon the diagonal of the square cylinder, on the centerline directly abovethe cylinder, and at a point of glancing incidence on one of its sur-faces. The agreement between the patterns calculated by the GTD andthe integral equation method is remarkable - every detail is the samewithin the limits of graphical accuracy. The corresponding patterns forelectric current line source excitation (soft boundary case) are present-ed in Figs. 11, 12 and 13. Again there is excellent agreement betweenthe two pattern calculations, except in the vicinity region of forwardscatter in Figs. 12 and 13. Note that the level of the patterns is verylow in these regions, so that small errors in the solution become signi-ficant. We hope to look into the reason for these differences at alater time.

The numerical examples considered here confirm the accuracy andapplicability of our GTD solution; this is further demonstrated by anexample treated in Appendix II. As the size of the cylinder and thedistance between the edges of the cylinder and the source (or sources)increases in terms of a wavelength, one can expect the accuracy of theGTD solution to increase, because it is an asymptotic approximation wherek = 2irA is a large parameter.

In the case of magnetic current line source excitation there islittle evidence of shadowing in the forward direction by the small squarerylinder; however, there is distinct evidence of shadowing in the caseof electric current line source excitation, where the total electricfield is parallel to the cylinder and must vanish at its surface. Oneshould expect this.

19

-60

-120

-30

-150

180°

GTDINTEGRALEQUATION

60'

I20e

150

Fig. 8. Pattern, of a magnetic current line source inthe presence of a rectangular cylinder.

20

-GO0/

-120°

o o

-30

-150°

GTDINTEGRALEQUATION

30°

.60'

150°

I20C

Fig. 9. Pattern of a magnetic current line source inthe presence of a rectangular cylinder.

21

-60

-120°

-30

GTDINTEGRALEQUATION

-150150°

180°

Fig. 10. Pattern of a magnetic current line source inthe presence of a rectangular cylinder.

60«

120°

22

-60

-90'

-120

-30"

-150

180'

- GTD• INTEGRAL

EQUATION

150"

Fig. 11. Pattern of an electric current line source inthe presence of a rectangular cylinder.

soc

120*

23

-60

-90'

-120°

• e

-30

-150

180'

GTDINTEGRALEQUATION

30°

60'

I20e

150°


24

-60

-90C

-120

-150

180"

GTDINTEGRALEQUATION

150"


60«

25

IV. CONCLUSIONS

The GTD has been applied to calculate the radiation from a perfectly-conducting rectangular cylinder in the presence of a linear array of linesources, which may be of the electric current, magnetic current or mag-netic current moment type. When densely packed, these sources may beused to approximate the radiation from an aperture. To insure good .accuracy in the fields calculated from the solution described here, theseparation of source and field points from the edges of the cylinderand the separation of the edges from each other should be not less than0.7 wavelength. However, for far-zone pattern calculations, the linesources can be only a few tenth of a wavelength from the nearest edge.

The use of new scalar diffraction coefficients valid in the transi-tion regions makes it possible to calculate continuous patterns in theregion surrounding the cylinder away from its edges. Radiation patternscalculated from this solution and from an integral equation solution arefound to be in excellent agreement for a number of stringent test cases.This demonstrates the utility and accuracy of the new diffractioncoefficients and the overall accuracy of GTD as it has been applied tothis problem.

26

APPENDIX I

THE FIELD AT THE SHADOW BOUNDARY OF ATHICK SCREEN FOR GRAZING INCIDENCE

In this appendix we derive an expression for the field near theshadow boundary of a thick, perfectly conducting screen illuminated bya line source at grazing incidence, as shown in Fig. 14. The solutionnear the shadow boundary in the forward direction is of interest. Inthe following development we employ Eqs. (5) through (10) in ,the text,the subscript h on the hard scalar diffraction coefficient has beenomitted, and it is convenient to use the function

(A-l) f(x) =_ e-jkx

MAGNETIC CURRENTfl LINE SOURCE

Fig. 14. Shadow boundary of a thick screen for grazingincidence.

27

Let the strength of the line.source be such that the incident fieldat P is

(A-2)

The total field at P is the sum of the incident .field plus the fieldof the ray singly-diffracted from edge 1 and the field of the ray doubly-diffracted from edge 2. In the illuminated region", e > 0,

(A-3a) U(P) = f (A o)

In the shadow region, e < 0,

(A-3b) U(P) = ^ -,0,L) f (£9)

Here,

(A-4a) <j>2 = TT - e

(A-4b)

(A-4c)

in which

(A-4d)£• h

+ h

I 2 'P 3 e cot(~) F(2kLp f(h)

Thus for e > 0,

(A-5a) U(P) = f (£ Q )

ff

h) D(<i,2,0;L2)f(£2}

f1-) F(2kLJ)l D(*2,0,D f(A2) ,

and for e < 0,

(A-5b) U(P) = D ( < | > , 0 , L )

-J V4 ,2ircot(f-) F(2kL.J) D(*,,0,L)'fU,)

where

(A-6) L2 -

i + h)

+ h +

arid L i s a distance parameter determined by the wavefront curvature of thefield incident on edge 2 which has been singly-diffracted from edge 1.Since edge 2 is in the transition region of this field, the curvature ofthis wavefront is not simply that of a cylindrical wave emanating from •edge 1; i.e., E^ h Ji2/(h+ji2)' We will determine U by requiring U(P) tobe continuous at the shadow boundary.

As e -> 0,

(A-7) DU,O.L,2n, j ir/4

| e | ' e

sgn e

In a similar manner,

(A-8) D(<j)2,0,r) = - vlTsgn.e

Furthermore, as e + 0

(A-9) cot(:F(2kLl)

Substituting Eqs. (A-7), (A-8) and (A-9) into Eqs. (A-5a), (A-5b) andrequiring U(P) to be continuous at the shadow boundary e = 0, we see that

29

(A-10)2- 3 e .2u, F(2kLl)cot(r>

A- h £2 -2 -J -/4 ,/2lu

F(2kLi)3 e

where h + a2 = Jij at e = 0.

From which

(A-ll)

with

(A-12)

/ hL = m

m =F(2kL;)

30

APPENDIX II

DESCRIPTION OF THE COMPUTER PROGRAM

A. Input Variables

N : is the number of sources in the array which approximates theaperture distribution. Here N = 2M+1, where M 'is an integer

- . (.. N is an odd integer), and M has been introduced earlier insection II. The DIMENSION cards at the beginning of theprogram must be dimensioned as N or larger.

TYPE : is a reference parameter. TYPE is set equal to 1.0 when sourcesof type I (see section II) are used. TYPE is likewise set equalto 2.0 for type II, and is set equal to 3.0 for type III sources,respectively.

AL : is the aperture width (= W of Fig. 7).

AM(I): is the magnitude of the Ith source in the array which approximatesa given aperture distribution.

AP(I): is the phase of the Ith line source in the array (which approxi-.mates a given aperture distribution), in RADIANS.

X : is the point of incidence on the 2-D box and corresponds to Xshown in Fig. 7.

XL : is the length of the box (corresponding to XL of Fig. 7).

H : is the height of the box (it corresponds to h in Fig. 7).

RI : is the incident range (corresponding to Rn- in Fig. 7).

RS : is the scattered range (corresponding to Rs in Fig. 7).

THI : is the angle of incidence (corresponding to e^ in Fig. 7).

XLAMDA: is the transmitted wavelength.

Note that the variables AL, XL, H, X, RI and RS have the same units asXLAMDA.

B.. Output Variables

THS : is the angle of scattering ;(corresponding to es in Fig. 7) indegrees. •

ATAL : magnitude of the total field.

31

DBTAL: magnitude of the total field in dB.

DBSAS: magnitude of the scattered field in dB.

DBGA : incident f ield with phase center at Q (F ig . 7) as a function ofTHS, in dB.

PHASE: phase of the total field in degrees.

TH : angular variable corresponding to (Fig. 7), in degrees

DBHA : incident f ield with phase center at 0 (F ig . 7) in dB, as afunction of TH. Note that DBHA is an output variable in theSubroutine TEST.

C. Instructions for Representing Aperture Field Distribution by a^2-D Line Source Array.

When dealing with the input variables A M ( I ) and A P ( I ) for Ith sourcein the planar array used to approximate a given aperture distribution,the ordering of the array elements is done as follows:

2-D APERTUREOF WIDTH W

2-D BOX

Fig. 15. The ordering arrangement of the l ine sources usedto approximate a given aperture distribution.

Let N = 11, where N is the number of l ine sources approximating agiven aperture distribution over the aperture w i d t h ' W . The orderingarrangement for these sources is indicated in Fig. 15. The source at thecenter of the aperture is the one for which 1 = 1. I. = 2,3,4,5 and 6 forsources to the right of the one designated by I = 1 (as one views the 2-Dbox from the aperture center). Similarly, I = 7,8,9,10 and 11 forsources to the left of the source at the center (designated by 1 = 1 ) .

Let the aperture distribution (assumed known) be represented by thequantity F= jFle 1^ over the aperture. |F] represents the magnitude of

32

the field distribution over the width W, and> represents the phase ofthe field distribution over the width W. Hypothetical plots of |F| and\l> over the aperture are indicated below:

IFI

AMPLITUDEAM ( I )

AM (9)

AM (5)

APERTURE WIDTH=W

A P ( 9 )

(RADIANS)

6 5 4 3 2 I 7 8 9 10 I I

Fig. 16. Field distribution over the aperture.

Fig. 16 clearly indicates the amplitudes and phases of the sourcesdesignated by I = 1, 5 and 9. For example, the magnitude of the linesource strength corresponding to I = 9 is given by AM(9), and its phaseis given by AP(9). Similarly, one can obtain the amplitudes and phasesof all the other line sources. Note that the aperture is divided into2M segments, where M =5. Hence, the number of sources, N = 2M+1 = 11.

33

D. Instructions for Us?ng the 'Obliquity Factor'

The fields radiated by apertures are non-symmetrical on either sideof the aperture, in most practical cases. The fields radiated by the2-D line source array discussed above are symmetrical on either sideof the planar 2-D array. Thus, an obliquity factor of the type cosn e/2(please refer to the discussion in section II) is included for computingthe field radiated by each source in the array. The obliquity factor isdifferent for each n, where n = 0,^,1,2. A function statement FB(SX)computes this obliquity factor for a given value of n. Specifically, thestatement concerning FB(SX) reads:

FB(SX) = ABS(COS(SX/2.0))**2.0

and corresponds to an obliquity factor with n = 2.0. If any other valueof n is desired, the appropriate value must be punched into a new cardwhich replaces the previous one. Note that the value of n directlyfollows the ** symbol in the statement.

The obliquity factor cosn e/2 is plotted as a function of e fordifferent values of n (n = h, 1 and 2) in Fig. 17. When the pattern ofan isotropic source is .multiplied by cosn e/2, it is evident from theresultant pattern that the obliquity factor serves to control the levelof the radiation pattern primarily in the range ir/3< e< 5ir/3. The casen = 0 corresponds to the isotropic case.

E. Instructions for Computing the Incident Field

Two incident field patterns are computed, one .is for a phase referenceat the center of the aperture, and the other is for a phase reference atQ (see Fig. 7). In the former case, the radiation pattern in dB is desig-nated by DBHA, and is obtained as a function of e (or TH as defined inthe computer program). In the latter case, the radiation pattern in dBis designated by DBGA and is obtained as a function of es (or THS asdefined in the computer program). DBGA is computed at a distance equalto Rs from Q. DBHA has been programmed for a range of R-j + Rs from 0(center of the aperture as shown in Fig. 7); however, if the user wishesto change the present range for DBHA, only one card in the program deckneeds modification. A subroutine designated TEST computes DBHA at arange of Rj + Rs from 0; the call statement for this subroutine is

CALL TEST (N, AL, RS+RI, A, TYPE)

If a different value of the range is desired, one must replace RS+RI inthe call statement above by a number which equals the desired value forthe range. Note that the new range should have the same unit as those ofX (corresponding to XLAMDA in the computer program).

34

-30 30*

-60

-120°

-150 150'

180'

Fig. 17. Patterns of the obliquity factor for differentval ues of n...

F. Sample Programs

In this section, we present a sample case which serves to illustratethe-use of our computer program. The example selected involves an array,of three magnetic line sources of unit strength which illuminate a rec-tangular cylinder, as in Fig. 18. We utilize the computer program forcalculating the incident field of the array, the field scattered by the2-D box (rectangular cylinder) and the total field (incident + scattered)surrounding the 2-D box.

35

,SOURCE #3

ARRAY OF 3LINE SOURCES

SOURCE #2

*'

Fig. 18. An array of three l ine sources in the presenceof a square cylinder.

The ordering of the array elements is shown in Fig. 18, where the sourceat the center is labeled source #1. Note that source #1 corresponds to1 = 1 , and sources #2 and #3 correspond to I = 2, and 1 = 3, respectively.For this par t icular problem, AM(1) , AM(2) and AM(3) are each equal to1.0, and AP(1) , A P ( 2 ) and AP(3) are each equal to 0.0, because the l inesources are of uni t strength and zero phase. An obliqui ty factor corres-ponding to n = 2 (i .e., obliquity factor = cos2e/2) has been incorporatedinto the program for the incident field pattern, and the incident f i e ldpattern corresponding to DBHA (phase reference at source #1) is plottedin Fig. 19. Also included in Fig. 19 is the incident f ie ld pattern with-out the obliquity factor (n = 0 case) for the sake of comparison. Thepattern of the scattered field designated by DBSAS, and computed forvalues of es (or THS) which lie in the range -180° <. es 1 ISO plotted.in Fig. 20. The scattered f ield obtained by our method is comparedagainst that obtained from a numerical solution to the integral equationfor this problem given by J. H. Richmond; these results agree perfectly.Finally, the total field (incident plus scattered, each be ing phasereferenced at Q) is also obtained, and is designated by DBTAL. DBTAL iscomputed as a function of 9S (THS in the program) and the results areindicated in Fig. 21 by a dashed curve. The solid curve is added for thesake of comparison; it corresponds to the total f ield when the incidentfield has no obliquity factor in it.

36

-60

-90'

-120

-30°.

•WITHOUT THEOBLIQUITY FACTOR

•WITH OBLIQUITYFACTOR

30°

-.150 150°

180°

60'

90C

120°

Fig. 19. Patterns of an array of three magnetic line sourcesof equal strength, with and without the obliquityfactor.

37

-30

A R R A Y OF 3 -^LINE SOURCES^

-90e

-120°

-150°

GTDINTEGRALEQUATION

I50C

—90 C

I20<

Fig. 20. Pattern of the field scattered by a square cylinderwhich is illuminated by an array of three magneticline sources.

38

ARRAY OF 3LINE SOURCES

-60

-90e

-120

-30

-150°

I80«

FOR ILLUMINATIONWITHOUT OBLIQUITY

FACTORFOR ILLUMINATIONWITH OBLIQUITY

30' FACTOR

60°

120°

150'

Fig. 21. Pattern of an array of three magnetic line sourcesin the presence of a square cylinder.

39

G. Listing of the Computer Programs

of0£<ujX

oH

«Ouj

O

CO

-I^T_Ifvj U J O C 3F~ </)i/)*.i y a t z<-> O UJ UJ aj

zzr

za»

Z XoUJ h- 1— Z^Z — <

»-r>UJ JJ LU

•> UJ .-I- X -J

— x x LU

H-O —

. -

XX UJ_ i<< LU

z z z z z z ^-LU LU UJ LU LU UJ' I/)X X X X X X LULU UJ LU UJ UJ LU «

Uj Uj LU UJ UJ LU UJ

x x r x x x 5»- K. »- K. »- j. k-

^ — _ —. .^ — „<•PI o r> O n O co

*~ *" i- 5

3- LU CO

O O t3 O ;3 O _j LU _JQ Q o O Q Q L U </)LU

x x o — LU u.»_ v- H- >— t" ac

. X X X l - k - t - - ^ O U J t ->->-(_ OOOU.U.U. ' - ' -JZUJ UJZ

u jaOuJ^ -> - ^ * - ^> - ^3 L/?x ^5 ir«

L U I L C C — • u j Q £ a e - * £ £ c £ a c Z x-JQ 0 " •

f- o Q o o . a O Q Q . — Ou,i/»a._j ir\i —oJo i t -x - j - J - i - i - j - J o-J aix< ^

— U.u.LU.^'— ^."— _i— ^Z •-•ujft' O «_Ji LU 3 — U . u . U - x U . u . — O U , c t x Q Z <

^™ ^ ^? CD O S 2 Q CD Q GC *^ ^J H~ uj •>UJ iu C^ h~ UJ m UJ II) '' f 111 t(j ^> ^ ^— 3 ta^ (^ . *"^

! *"^ N^ *^ —3 ^ ^ ^ ^ ^t ^ CD '•/) H* "^ *^ _j *^ ^^i ^ - l A - U i ^ Q Q C ^ O C ^ Q C ^ CO U.<LL *^J

<£U.u . l * - U . u - U . 4 A - U . i ^ * O L U U J ^" ^

Q O*- ~ h- U - U - Q U J C O-*

Orr* — ^ Z a C Q Q a S c C ^ ' a j U J Z u. — —

at *

a. a. </)z z z< LU OJ

a. a a. — — — •o.-»> •— oo

X >-

40

o oo oU Oo oo o

o o o o o o'o 0000000-3 o^p o o o o o o oOOOOOOOOOOOOOOOÔÔÔÔO^j — ^~»~-'—fv^v— w v ^ v u v i ^ v ; j %^ -3 i^f— — — — — — - — — —*— ~* >-* <~* *~- r~* — *~* — rî ~ ^. — O O OOOOOOOOOOOOoôO

_i. *r. "siO Q • »• •• fl

PSI Q* p>PO. Q Z f-

Q O. » pt• Q Z o

~* _l <*) » M * • m •m m _j o • -~ moo -J < o oc ,»• -« oo•* - Q O •> -J Q X — -< -.^ •» » Z Q 1-1 •• -^ ^

.•-• Q < N f M Q • a . O l s J O O » '-^ U.** P S J _ J " » PS( fc.«|.PVJ Oin >_. .4 - J<£ K oi <M o » • • ^— x m Q<3Q _ ) o o ~ " > - o »ro o.** » « . « . o Q; "-> x ^. -• 33

.o: »o -4.<_j) » 3 a j x • *- >"< — • -"M-IQ - ^ O * - ° - - 1 2 ! *Q m - » _ ( < • p j £ m O T : a . a r o. 30 -< O O *: _l » UJ O y to ,

— »^m * "o 2 0 0 — o -~ * * • în o< _ja : -> x u j t - X ' - x _ ~ — »s— u , » o < o " ^ — • • U J » » K J > > —r\j t_ ._, ~ »Q» • u . * - O - J l 0 » > > « jat a >H o <*> • —> o »t~ » I °" <o I * to< ' o m i n f ^ f v i Q ^ x » ^ . ' » ^ - w > i 3 O »Q t,"' ~ » ^ O C » • < « X » - ^ - r ~ O 3 « j

— m< oo_» « « Q Q or d t - > - * . o r - » » ' . |• < o o » i O a s M » » o • o c a c x r o v J - ^ - ~ oi n **«« O Q P O ^ ^ O I / ) * • — ' O ' O ^ j m p ^ ^ ' O t o— « , -« . < • ?< ac 3 3 u. X t o > o a . 1 * ' i n o v 3 > #-* o in » - ^ . Q < » - J JM t - * » s : — • r o p - . r - o ocf u. —. ^_j « > Q u , e L • « o . , » o p p i - j — 4 - < t o «

•^ cj o.^ - ^ X m O P ^ ^ ^ Q **** aîo • t n f N ' a 3 O O ' a #m » ^ » » > O * ^ t o ( ^ a C t ^ J 3 I * » . ^ » - 4 » » f \ J r * > P O c / ) *

-J —1 —I .. -J »>tOr>| » < » ( _ > U CD Ci >— O «• I -O CO 1) 3 —^ i n i n * ^ m ^ ^ U ^ P A J O O X b c ^ j * • • - * » I — * * ^ t o O* <J OLJ oo^ » » ^ P S I o r ^ ^ c o ^ *i_3 » ^ , ^ - t x t o c c o — * * - r s t

— — or O ^> -, _ . . Q Q O X • —- -l» X ( » ' 3 ~ - O I M P S I 3 l «I v

- • _ < — » * - » , O O M C » « » a » > -IJt x — _ i r z _ . » - s c xu ^ . ^ - ^ ' - » * - ^ — . . ^ ^ L J - » t n t n u p s i * ^ t / ) » < • , ^ c ^ - j ô. d . o x ^ - x » O ' O + oto^ • ^ ^ m ^ ^ i n . ^ X k a . * ^ * - • « » • • f x j o c o x 13 ooCi x-x* i^ P\J —j to _j *^-*^ is < *^r t ^ - 1 * — i n * — in^»^- in O O > - C t a ^ * C J * » . > * » • • » H - I _ J P ^ C L ' S * P X * ~ K - ^ * ^_ i _ i ^ j — < — i/it/i«- o o t o O r - j a > » s _ j _ j j > c j a > < ^ > * » ' ; 3 : - » z a c ' 2 i ' > i v i i OC : < : c < O r ; < X t O . O c • •> » a < Q . < < X ' o Q J O ' - * ~ O

. CJ C O 'J * i/) >- X O O x ' M — ) - * » - J " < > — tOI— ^3 »»• S- C (Ni r- — ~- t 0ô3O«-< < O O O - J O C X » - J O < < ^ C 3 ^ i J X Ô it o CL ^ «— II. u -fr i/i to

~ -J n -J o — n ^) 3 • x a - N J i n . •*•'•-* x > 5 » M 3 5 <— — — — — — — — — X X X X X ' X X X X X X X X X X X X X X ^ » . O M L U ^ > - ~ t O l l

^^ - - r ^ i r - i i r - ^ z _ i - i _ i - i _ ) _ i - j j _ j - j _ j _ i - j _ ) _ i _ ) - i _ / - j o5 • — ~ x( n 3 ^ .s < x.-

— -^j — * C3 C « f^ CD ^J tj (w) LJ ^ Cj ^ Li l—J Li tj ( ) *J t i ^J ts ^J *™J lj ^J J ^ «^ L l^ ^ ^ ^ (J '^ y^

41

-i O o O O 3-; O -> "* O O o

t tJ *>

IL

*IIT

IIX

UJ

13

>—ceQ.<

oX

o01

.JjQ.>

UJ t-o »* X3 fA ,O # '(/> (/>

— »—' ••• <t UJ01 . i.

^ LU ^• Q. -I

•f >• U_'

• 3 • *

Z

II

UJO

, ~ ^ 2 ^ ^ To --"< <\I

P«J•~ »

HZ —• »

-« !MII MIP

•-1 »» •-*Z

-X« ina >C< »•• a.— <f— xs o< —i

. » 4>

ou Xu. • mo > -j

* ««? Ht M

N. XU. 1-

X*

>o•o<o

o oI -4

o a.X ._) *Q. II

xa:*

I/Ia

OC

X

X

M »X •

H"l o ^at . u.

I • »-I U. MX « to- * ac

X

- U< •» »00<f00

-i. O -- Iu _j -i « »r. *

u. r- u — vO .

a ~~-. — a-

ry-ac

O —

JJ — X O 03

— (M

< t- <

tc

CD "-

co —CO

<NJ a. i-v t~ cciri «* a

•^ oI jj

< ^ — -u z«; —

, at aa a. .

o — — D I I — -i n du <-«->t-i—.»_-». — ^-s: t- 3 5E

— • <r a, oe n' t- » -I * O M

« . . • vi a.x <r n ^ „ >•

i flt • u u a, ^c

o o33 00

ft. UJu Q

^) ~> O1M

COCOCO

<o i*.

i\i

Oo o j o oO O O O O

OOOOOOOOOQCJ ooooowooooofoooO O O o O O O O O O O O O o O O Q o O O O o O O O O

<

^j ^ -J rK IfN ^<M (M PO fn f ^J- .J- -t •*• -f •*o o o o o oo o o o o oo o o o o w

42

ooooooTooooooooooaoo ooooo oooo°oooo2oooooooooooooooooooooo o o T O O ^ T O O O O O O o O o O O O O O O O O O O O

o o o o o - D O o o S o o o o o o o o o o o o o o o o o o o o o

33.

a

Ia.

0.a

3

<Oa:

u-ilf> _,

.«a

oa.u.oat^J23

o .U i l

s:X O

<U)

X < .u- X _ J - »

- < x rX x * -NII I^ M ™-» JE I

CO

o cr• IT>

-4 u>00-i o.11 t-11X O•- O

' 'JOJQ

»irt< oH- t-OJo o••o

_l< —H- O< ••• o

I/I tX 0»

O 0

35,Z iM F—t- IIZ W»

oxo *-«7«ff<0»0»

• * •o o oII II II

-J_l < << »- oh- CD 23 —< LJ Q LJ

<• • »>_

o o >r <ii u

OO UJ ^- ».< <s> ZOO < —•=o i a; u.

a a -.

uo

ooc(«*

O It •

pg >-I -J

I/O •X i/>

a uo—«CO&•vv

ti :

l<0 'X i

I- ~ o — — >•>•>_ oc * in -s> — —«• I — « D <M r\j^> X ^ i^ U Z. ZX II || * * < <I- — — + I/) •- t-

— — X i* «I <IIII II• y) — — II II . ,.

X ^5 uO Q. Q- JJ ^, (- X >. X >

oo —X —I <"_l XX t

I/I '>>> >

^5iZ < <— I- H-- < <Z II Ma u oO H- —

O(M

OII

iO33

•t -f *tO

-a 'M (*, m> f ' - r > i ^ « r >•$• •& -fO O O OO O O Ooooo

oOo

OINJ

O OOO-.00

o •^- - < 4 ' - f - t j f * T { r \O o < - > O O O O O OO O O O O O O O OC J Q O O O O O O O

i n io i r i i n iA i r i ^ i r > i r i l ' ' i ^> ' ^ i / \ i n

O O O O O O O O O ' - ' O C J O OO ^ O O O O O ^ O i J O O O Ooooooooooooooo

OOo

43

) O o O O O O, - . > ° O O -3 O o

- ^ O r ^ O r j O ^ O o• ^ O o O f ^ ^ O ^ Oo o b o b o o o o

• 00

' o O o ^ o ^ s o o o o o o1 o o o <-> o o o o o o o o> O O O ° O O O O O O O o'oorjôooooooo'oobôooooooo*OO®CD®COGOOOGOC^Ô*(7*

O'o O O O O O

O o>W> O>0>o

o a tr>

*~ i/>-)30<J 0

QQ 0z.~<£•^ U —i) £3• ' 0

_J-< 0

-. UJ-1

<VJ (V •

— — 03. Q. i_

I I I^ cc 3;i— t— >_

552

>- u. a.

o»!>o*o

o

oUJ

•UJ_Jt

ah-

UJt—

S>

J,

UJ

5O

• < QX UJ

• • a.. o- a.52,0* 00

O UJ

0 k-

Z 0— UJ

O

z zQ•Z UJUJ _J

O00 Z

z-x 'UJ 1-1—< Qt— UJ

UJ

— s-^-i Z

»H

UJ CX O>-

^~u. ZO UJ

uj at

5°

<_ixv.

oo

10.

*

«l

• OO

Q.. >•

+

Q. —

* —— OO

00 |

v--'—

_ OO

— X00 |

+ X

— Koo a.

0. II

n —CC. OX cf.

OO OO -*

Q » » — oo

_!- — +— +x — — a. * —« 00 OO >• ~~ >—

_ 1 - 1 # - oo

O * V * O. 00 . ~at ~ # ~ >- X >• x ».« a. -. » -f 1 + -.

— ••— V - ^ C L Q . * — O. XO

o o X ~ X 4 . - f ~ » — t-

—'a* x -) -J _j — a. oo»< X | X X X O O > - I > — OU. , Q. ^ I | X » 4. — illi i » x « ° . o . l o . o . +"• O— — — — xx — >•> -» u>>-

O o O X o O - l _ l — 0-0- X~ t - —X 1 X X X — XX * — « 0.

- ^ ( 0 - 1 1 | 0 0 * # • — >•

Z X — X X X I X X ~ X J X

CD X -^ I »

» -» - f c -> - l - ( _H-»_^ -O i t - — XxO Q C a c o t o c a c t o L c t x c t o o « • !• O r O O O f o O ^ r I O V Q . Q .

. ^ o O O O o O O O ^ O O o O O O Q . 0 0 > - x

J a z z z4, > <3 .j <00 — ( _ ! - , _x5 ;s r2 . z : ' » - s :5 ; f s iT -<< j<

— t - i a t a C a c i o t o t e t Z ^ l i |

- < . t x J c 5 i j £ ^ 3 : j c i i 3 i i n u— — n 'i it fi n n M M Jt M a -* MUJ n, •-» t\i.(f\ * T | / \ > O h » G O X O x I XO — a t c x ^ ^ ^ ^ ^ c t ^ - . ^ S l Q . Q .

UJQ

^-

«

—

X

I >

O 2• <

r- <

ii "PO ryI XQ. H-

0UJQ

0 1-

0 »

^— '# a.

0. X

1 •. 0.

0. X3- 1

z z< <t- 1-< <rl 1a a

n MX X»— H-

X+

>»

X Xh— 1

t —O (N• 2

r*- h-

H uX x(- »-

CT-

O ^1- 0

o o»o

n-« »—o• o

•4 (.9• •

UJ ^

_J O

•^ oo •• Or

O 01CO •w* -*

h— t

v oo

X i»- X

1 ^"" 1

H- X—• t—

'•SI (Sia: a• t

0 0

o o• •

1— UJ_l 0• •

x x

UL -J-I-* •—

0>ooo>ot-

nO

•r>•oOr

•**fc

o J-O ^

•«4

^~

X-

1 0

X 0k- •— or00 UJ

« O

Z O< CC

0 1

0 X• 2.

»- 1 .

. X— Q-

;r u.

•—4

0.

Q.

tiXa.

m•

.— «

*^ it» 3T

' at Q

II -1< <Ii» o

t— t

a.

a.+— <xa.

ir\t

^^

Q 2

"• Xo a_j-i -*< x0 Q

• O O U# ** »

0 <•» i«. fn•• "- N. O-> >" in O~> o o o1 O O Oj o o o O o O O

o o o oO O O Oo o o o

44

OoOoOoOO OOOOO° oOOoOoOoOOOOQOoOoOOoOoOoOODOOO-->~->oOOo0ooao6oooo6ooooo0ooooooooooo.

o o

3a<Q

01QII

^4

<ro

•0•

<N

a.a.IQ0•

or>-Cvl»

<na«

inf»on

a.X f^

a. Q+ Qo *o o• o

0 —r- ooCM u> II

ro —<o r>J» a:

in at

» — oa Q •a a <N

• •^J fB^

•» •>

ooo o-V |C>J <NX XQ. Qu•• »0 ooa•i »in >n

t <

» »aj u.0 Q

u.Q

UJO•J*ooOII

i>Jr\jdc

-, ou. •Q (NJ

0 o

-1 r-l•» *

0 o1 •

0 0« »

•3. ^I -L

•» «•in vn• ••> *

O "^0 D

rvlQ

0Q

LOQII

<VJm0£Q

— Or>l •Q <N

00

— 1 —4^ •*

5?a Q.• «o a00• •in m• •

* „K —5QO

-}a»•*

0

COau

fOCOa:o

-* o—) •O fM

• .

•» *Q. Q.T" T-

\- >-1 4.

—1 <-4X xV- t-

•. •»UJ UJu> 0» »

in in• •» »

x _i0 Q

_l

O

O»•*00QII.•4_)Q

— O_J •O <\l

a. at- V-

I +o oo o• «

0 0r- f-(M f\|

•» nu. u.o-o-

w «,IT* mt t

•* <k

r 2:d Q

7.Q

a:o

01U

• n.(SJ_l0

-~ o2 •O CNJ

00

1-1 "*9> •»

rsj IN)X Xk— V—» •»

O ij0 or» »

in in• •

• >a a.Q 0

Qi.

noQ

l/>au

OJrvj_ia

— oCL •Q IN

0 o

-< -H

•t *

00• .

o o» •

3: 3a. x•. •

in m• •

•» *

O .*D G

a:Q

oQ

00QII

CvlM-JQ

— 0i*S t

0 oj

O

^^»

flX1-»

!_

o»

int

xa

lij •* II U - o ^ m U . U . U - ' U J U . U . O l i . J ' J ' U . U - i — U - l f - U - u * UJ 3 U. U . 5 ; U j a C U - U , G U J U. u_ O UJs r X D O c ^ ^ o o o ^ ^ u o Q ^ c ^ i j o o ^ n r o o O ^ ^ D O C i — o - . o o a z — i o o ' ^ i c..ar-o:.— • • a : a / — » r\i a; a: _ • •• of. ~*. — • •• oc .-y — t • or =£ — • .•*:£_ • , - \ i n : a c — •

o i r ^ u . i - x u j - £ u - u - x u j c x . u . u . l u j ~ 5 u. u- -T- '-" >o u. j. j. o> o u . u . ^ u i T i v j . u . X ' - U - J U - a . - ^ u UQ . c t Q C O Q Q . x o o Q a . o o c i Q a . X Q a o i a Q Q ' - i a . ' ^ Q ' J O Q . x Q o o a . r i Q Q ' - ^ Q .>-— ! )> -_ . II >• M || >• — II >- ~ !!.'> — !| > — II >• I! II >-i - O j _ j - ^ K - t 3 — I _ ) < M > — f - * _ J _ i r \ J t - O _ i _ j m ( - ( j - l _ J l -O_ i - - l~ - '> -O_ l_ l 'MH--<_ l -»<Mi -— II -J _J (M — II -J_| iM—.O_l- l f *~ I) -I -J '^ — II -I _) _4 ~. II _ j _ l ( M — |l _ I_J(NJ — f O _ ( _ l f < 1 —u_33s;,<a:u.. o < t < a c u . c x : < i < a i u . a < < c ^ t ' - u j < < r - ' x f < t < _ j u . o < i < _ ) ' a . _ j < < r _ i L L_ar l j oo ' -«arooo — Q O o O ' - ' . ^ ' o O Q ' — a u o 0 1 — u ' o u o ' — o o o a — ' Q U O " —

o o o o o o o o o o o o o o o - 3 o ° o o o o o o o o o o o o o o o o o o o o o a ao o o o o o o o o o o o o o o o o o o o o o o o o o O o o o o o o o o o o o o o o o o

45

3 C 3 o < - J O ° O O O O O o O O < 3 O O o O o O Q O f 3 O o O O O O O ' 3 O O O O c 3 O ; j O.Q O»O. jM< ' i 0^?OC? > o - *M ' f ^> l r t ' r t f ^cn o < O" J :NJ ' ' r | ^ ' - ^O^ - ' r >O»O-^<MP i>>a > i r t >O° f—*oO- i O ^ - o ^ j ^ j > O ' O > o - O ' O ( > i r ~ i » . i * " i w ^ p - ^ r » p " a o o o c o 0 0 c o * o o w < * < o o % o > o < o * o > o > o » o » ' > O v O

> j f > j ^ c g N r g f \ J ( \ j ^ r > j ^ N c g N r > j N f g ^ p g ^ t \ j ^ r s | ^ < \ | ' N p g ( N I P j ' N J P v l ^ p g i N o a P > l « \ l r » j ^ p j < N ^ f O

— O2 •T O

* °.— OIA —

~ — — • or a.-» ift W O — »•— 0£ 9C — U. U— - — 3E # ||i/» u. u. a. < —X * # » I «- •I M 1M _ « .« _c t - » N " ^ ~ N - « O < / > i M O

• ~ — X » - a : > - i - - J V > » 3 j « - . - •o o - » > _ Q H . i _ a k _ . ^ f O _ j < o ° o• t » t ^ » # # # # » - . - » v a . o * t t^ <-* o 3 Q . i A Q . a i r \ a ^ Q a . 1 o -4 -<» • - — • — »- H. —

> - ^ > v ( C i_ -,„!-•-. — — — * » X O ^ - < X > * > » > .O Q J - ^ I « J — — X 3 : x r « - X 3 e x i n < " O _ i « - J • t ^ - ' M• •> XI «>-^ fcO^>— T " h - ^ > - X f - » X C V J * < 3^ XX •

-• X »- I- <00 > a ; l — I— *^ + O k _ l — Q UJZ >-l- X••Q 1+ f s J O C ~ — — • — U . " - U - ' ~ ' U . — II l N > t O II »O I , - f •>*

rr\ *~ f M O J ^ — ( M L L a ^ Q . * ^ * ^ - . J»» -^ U J I I OO —•X f XX t o U . X Z * * - ^ « - * » ' - » » - < > IA(J — 0.— • • t/>H.. o •->_ o»^ . < ' - | i r » ' M i r > _ ; i r > r ^ m — •. » t • — x— — •— oo Q• II •> . II. ^1 — • • . f c - c £ » K t O . » _ J » ' M lA^HU.33fVJ ^--» °P • * '1

•^ m stst j—i1-1 • ~ I X « O O Q O » O O O . J ~ • • '" — ~_ _ j Q A _ «j^.|^«.#«.»,»,»,»^

«M

o o o o o o o o o o o o o o O o o o o o o o o o o o o o oo oo o o o o oO C 3 O O O O O O O O O O O O O O O O O O O O O O O O O O OO OO O O O O O

46

O O O O fjo o o o oo o o o oo o o o oo o o o o

oooooooooa

ro fr\ CO fO

ooooX-J

•"• iv**• 7* U— IIu. —• -.

•*> 0:3-J . - -IO o <* • o~ o3 — —••- X •— _l O11- 0. O

• r uO •— —•

i r 33 r

o

<r

x

X

a.

— oX '"•f OQ. *-

x xI I

a. Q-x x

« a

o o• •

XX O

I "+o o a.oo —rg rg u• • II

a Q- —)• . a

in m u

o-"t3t

U.*

coa*

OO

. •• 1/1i in M

X Q

a. in. o o

II II

0r^ — 'OC 0

u. o» ••tNJ 0_J •CO 0Q —•» X

X O— II

T cy

CO UJ

OL O

— O

oS •

a -iu •

_!• I — — Q -.

uiin

.• •* — o o—•3 • N.

— — 11 X

X —I IM

Q. Z -~

X < 3

— <T XI- I •'JL **• Q*Of 0. a

O* 0? '* u. o o •

Q. CJ . £ > " • — • « « / > « / > t lL - .OO

O O

o o

O•

IIjiCJ O

ft «kX X

X ~ —VI O O

»••vjac_jQ —•» I-m >-• •*

O Q.X

o

oooX

a.x.ou

z —

in in

M

cc.

Q

I X

if»in

•O I X

3 |. —I —

s: x x— LL 7

U -^ Q.n 3£ *(M _j in3C Q •-J •» OQ -. — .

KS rO

a — *

-» o o

O -* *4 -^

•O * » Q

*o

a.X

o —

O M 3 — — .

x o

omt

o --

o

o

inaec

•— o

*- — —i u. xo. u. x UJ3 — O ti-

_J lL l i .XuJ- | | II — -^ UJ •—

llO ;

OII

1-a.jJa — au_j l-J —

oa —

<\j ^vi

a o Q - a

• i- o— uu. -w

Q Q^ rvl

U. OO ivj

U. XC Q

II

UJ iy

UJ _)0- X

1- O— II 'a. >

^ 'M -M IVJ .-v

0 o o o c

47

O

O•

O

•f a.t£ f

— OLL II

—I-* UJ

X LOa. u- M

o a- •* X O

x xIMX

- -OCin in O• •

u. u. u_ •IVJ (VJ

u. u- u. ujn'-îi^Z-< iT — .i. u. X ifCi Q C Q.

' II >

.o o O —

.•t* — -< -~ o-• a. s. o •_j •» a • oo< in t o «.O • Q • ii* o £• o c?-« — <••o ca • o inac u. o — •— •» • x Û. -• o -J •># « t~ QL x-• — (\i s: o— _i • o ~— ct a; n o< o ^ — Qii ir . „ ^

— -» ~> — x.— - x <— — a. </) -j_i j —a -iJt X >*• O <a a - - i—i *

fVJ -J- -• « —

z x °- a, Q-< > - « « «h- i in• «> &I X O O O

— O i-X O O

GO O * * *

II IIO O|| M

Q. C;

^«> tO O O O O o

>-<ca >-ao >ooo >-o^ o O G O j _ j f c - i / i i / ) _ j _ j h - G O i / ) < / ; j - J > — fâ^- i j — o a j j — o a ^ j - J — QU , Q Q < * I U , Q C < < d . Q O O < —> O**•>- o -t"* -< !>J !M'' •*> m n' "5 O O30 o O

OO

o r«-

O O OO O O

*f>•Nmmoo

O _. (M n>»• * >r «»•fO r^ CO P%o o o oO O 0 O

if> •* rj -H"> <M ,-O îr> in m i/^f*^ r*i f*^ ro0 o 0 0O o O O

(M rr>I O f-• IA in

o o o o oo o o o o

OO

'OOOOOOOOOOOOOOOO

48

o00

0

o00

Q.X

<a4-

#ai

-PH

P*

-TH

P*

i r x4- ^ ,«. h- k-— 3! ~ I *a. t- >- -i -«* I ! A-Jin — — # *• X X m

oo oo <J\ _ •"•o 3 o ;/> <-oO i J <J O O* » * LJ O

<\J ±3 M — •—

ii "iF u ii ii

r\J 03 (M ai -J -~-J _l 35 _l 31 00

O O Q

C300

a.X

ata

o00

LU

a.a.Ii3.

oCD

LUO

a

1

^

o

•Xk-1

CMx

o o o o« , • «

0 O 0 O•• •> • ••o o o o• . • •

<1 O O O

~- — — —X X X X-J -1 _J _J^™ — —* ~*a. a a a£ z -s. s:

I— O ( > O O«- II II II II

o — — — —2 — —

o i/)• X

0 —•• —•o >-• «

O Or— uo

X —-J u.a *O *II-. •- <« — a

— o

«ta.-+

o -i

-~ o n•~ » 4.QC O -^>• • «I O —

c a

o o o o o

. oc »f) O

~ ovfi rn f^ f**1 f*^ O "^. • . • . .x

O _i• a

UJ -1-1 wj - LUa. a. i. a. a

0 <• M

o

00X

oro

O

a.Xa.

IXa.Qz

o

o o

• on •~~ o

X J— 3.

o00

OrLU

a.XCLI

—4IQ.

Q2"

o < < < < o o < ^• O J Q U C J . » O —

o o o —33 -. — - .~.._<

^ o o o o o o o a :

(J O - O - O O — — ^O

-J - * - 4 f \ J ' N J M | | I M tX X X X X x ° - * 2

a ^

i —. o . • . •

O • O O O.o. O . • . t

O o —

-» U (•> J ;3 '-— , 1 1 • • t I

01X

00 >.

— O-,

X QL+ X

— a.

Oa-

O

o

o

a.

o:

a

X

—. a- •— te <x

T "• > z ?» -o ~

- II —-4 Z

OO •

> o«. — (M

00 — ^x < <I OC -»

_l OO -J ^ Q — —•« X • < . _ —

a I a. t- o _4 _jx —• X < • f :£f fNj t | UJ <T O

O -? C — O LJ 4- —.Z < r ^ X • 4- —-«•«

_ C

i; •O j.

• / ; • « - at r\i- a: < _i ca

X — C-k-

a; —u •-•

— . 00_, x

< LU < •* <X -I O — f- I'-ll t H •— II l|

— 3C — ~- — -.

J. <— o —

U. 000 II

— >-

' •••• •—• -will ^ ""*

<r x < Q <a: — *- 4. t-

«

*«

O o O O ^o o o o o

r> oO O

•a<r(M

Oo

OO

oO

oO O O O O O O Q O

OtAin<»•oO

in•*oO

oO

oO

o j)•r.tooOO

rsi

O O O

49

O o .ooooooooooo o-o oooOoOoOoOo°a°ooooooooaoooooOoo O o O O O O O O O O O O O O o O o O o O a O o O o O O O O o O o O o O O O O O t l O oO O O O O O O O O O O O O O O o O o O O O o O o O o O O O O o O O O o O O O O O o O o-^-4(s)<^-*'^\^t;2£<2'ic^'2'i :"12 l^''2<*^-*pgf* l>J '»«'o^"o(? '!^- i rv»'^-t^'O t>-w< ' 'o-<f\jPr>ro (nrom<^rn^<^^>f*^«* l ' r*«*'>t>* '^ i '>i f t 'Ainu>u\ir»!rt ini /No«o>O'O>a>o-o<oo'<>(i.h-| . , !

** * » — - • - » II IIfn -. M M — —. M „ ,, -. -.

<t — <\( <y <vj — (M MM — — iO (M _J af (* pj _i — — < < <* _j < < < _) < < < K i- Q— • « O Q Q < Q > _ > - +- Q f I I Q I „ - . - . - .• * | - . - . - » l — M - « — Q Q M

<\J «•• M M " < ' » l - l » , Q O M t - ' ^

J i-— — — «— «M M t- . •• » « \ j5 — < « * < — < n c » . t t _ , _ o eO - > < t » - » - t - < > - « t - . t _ . J l 9 <4- M H- H H II k- II O -J O • • Q~* — II —• ~ ~ II -» | • t ID UJ |

.j < — < < < — < •— • • a a — i-_ ( Q < » - t - > - ' < l - < a a z Z < —- -^ h-M i I I O Q Z Z O Q t - Q Q Q a > - ^ •-• — .— — ZZ << ^ 7. Z Z Z Z </» »_ z( < j «-<< • • < < — < < ! < < - • O<»"C t _ « « O o » ' O » « « « O OO</)r f < -J_J • • _ / _ ) • - ! _ ! _ ! - ( « * - J #S >- x x oo xx ox x x x — o -» a. ~'

_ J • a . Q . Q . Q . C L Q . Q . Q . a . Q . Q - < Q .ae i- x x xx xx xx x x — . x- Q x

'

OQ QO OO O-»o«-Q —

— a: f\i _i a o c'c Q o o o o o o Q-O o _ i o o o _ i c <- l-K O3 3t >- — > - — » - — I -- »-— — — — •X IS O ^co ujuj UJ-JPJJ — jrg- j^- j" M ~i/)Zt/>

H

ii T II II <Z. II II "I » II " II M • — -••— ~* -- — — — — — I — I — I — I — — Z I ziiJ-«O-«JL ,1 .1 "i — -. — -» — -•-.-•-. a >s> -j\ (N 1/1 "> — t^ ^ (o >^> — — i/> -~ "> — -o — — z i .+ ••*

^ — H. — f- •- — '- — — — — — " x x x * xx — X X ^ X X - H X — X — x — x — *- — --• — — •«•».„„«, ^ — — — — — — — — — — — < — — < — — < — — — — — — — — — — y)Z II M — — — ~

— — x —

s

50

' °. OOoOOOOOOOOOOOO OooOoopooooooooooo

o r>

Q <Q-

<» _J

• < X

i *< -I

X# R

O

T

•z

Ia.

o —ce. •-1 • «t— o1 -. -I

• i aa. 5C

x*

ooc

a:<x

*

o_l x

i

— <— a

i/> *x •—a. —x 3

siu U

< — -* -.O -« Z •--I — Z —u, :0 • O

< ? &* '

t r ~i2 * oJ.— o z — —CO < I- _1

c — • <II O O

— -I t-sr <^ r\j u.— o C — O

x — — < oc- — — — o- -u.

H_ X -i X 1^ 3 i-O » — * — |l -J

O O O

o o oO O O

O O O

X X X

Z II II II

-~ O — 05 ^ - » _ < O — ' - ^

Xtn

-^ . O O ,» II

_ J - J » - * < < l/l » >- '

< O ^ O u O » ^ O * ^

|| II U II < <N4 O II ^

as u. O u. o u.II M

II O

ro

2

N•*UJ

»»j »

< -t^» t -35 !MQ -J» UJ

UJ > «(SI X II^ in ujX • '•"a > <

J> N 5< ^ «LO UJ Xc » mo * .»• M -f

< «J •O < !M

^3 t- •* •C < UJ-# «

-1 » *< x • n+— 'f\ 'SI•o * V-

t—z o"• — 1

a.ofUJ ^0 <

tt —(J (/IUJ

UJ t-0^ OJoc <UJ O>•JJ CfOC ^3

u.z— o

UJ

i/) OH-'Z cr>b^ ^>o -<a. *

^ <->UJ I—

O I

°z:uj Ox yt

</\ i/>JJ X

•^ <X t-I- <II II

o au, u.

QL_

II X

* O O

1 "O o *T in rvj"NJ . >»• in i»-

1 O O o OOOO O

O O IM•* in m000<O «J <OO VJ O000

51

eg-.xo <*00 —

o-ICL

a:<

QCO

Qu.I-Q.§

aVJLt-a.z<~"u

«O

O

#

U.

*

OaiO

O»-

--3O

.

*in•

X' <s:

> V>— <Q Tu. a.l_ *I X

a coU. Q1- »as xa r»

-> n— ZQ Qu. u.

_JUI< </>

CT :O U .CO •

* a. a.o < <

x u. «. «< h- *• x5: a. -> in»- r — * •• < Q -jO O

u. x «s .

O O°o co

Q oLL U.I- l-X a.a. ac• <2 •U. O

'/I h-

£ XQ a. ocII (I 0^-~ -» ^_j -J if

o o o

a.31

LU <

^2$I- X.jr

I i O .-) S Z 3-io. 3 ij z <>-

. -5eg —II U-> a.O "•;n a.o> <0- —ra u.o —

— QQ u.u. t—\— aa. rx << ii— x<u- £""• <

Q. -i < •• t^ IVJ . ~ — || II CJ Q

1/1 UJX 3"- Z.II —X 1-

iioin^<o

— a> x —i -.< H- «i< a z — re _ ac

U.U.U.U-5

a: O < o a. u.OZ P- rf\

in *nc* tn•O in

o

.--v o i / > a . a . v - > - _ J - j

o c a x i r s : z x < <0.0. ^ > - < < a , a . u o

atu.

oe<

z

Q

CDO

ou.I-oo

o.UJCL

O<o

->

O

o I<:T» — . h--• on

3 -•

» u

opo-O >O -Co»-»oo u o

ino-Ooo

^ -30000000oooooooo

o o o o - < ^PO-• -* -^ -* -< -*-i-U O NO -O ~O O -O< _ » o o o < J « J ou o o o o oo

O -O -O -O .0*0 ^ <Wooo<-»ooo«o o o o o o o o

^ >O <O •» -Oo o o o oQ O O Q O

52

X*X*<

r« -» <•+ » -J- > o x-- Uj »a a -. _iu. „ „>- » IA «.* O . iu9 u. Q a.+ « -« >••• n u. t--* Q. -» x •l u. o •» z

~> i- ao •• •— 1/1 -i «\ «ô X -u. * ->U. ' • h- i-4 X OO ULJV- O „ «M . O=0 <j- a: -5 — in atQ so I C3 «>» . ^— in u u. — # f~ pH o< -• -> a: « .y»-»<o - > » - » o o •-> — in o z oo uj~*a a • u- o «< a.Qk- U- O 1- ,, « •>.U. |_ u_ O3

;g -s *^«cot!$ xu ii » Q t- uj 

~ s:— x u - ^ x c i o c ~ u»— x o « < * - — f ^ . o x f

5 « < I O Z c 3 Q * O O *• Ojr - ~ < T ^ - O U - O l i - ' / 1 f) •

— . o - i l . ^ u . i _ » n o_ Q o O » o — ' x x i / i i - u o c » »^ ~ U - « — » OJ i" — oo Q i- < Q-JU.—I 3 t5Ct- I t-UJ < -DUJ CC 2; «3 -^— , _ « - 5 — y_ » >_ . U " «• »— v M • »«>-~- O U.' k-.— Q<M~' -»«-O 1-^.3;— »#• l_U_ .« Z _II O U »- O O U . I I — -JQU. C C I I C : ~ J ^ - V m # _ J » C C i N . «.£« j...u...!! x t-u.i--jn-ii.h- a->. ii — ^x i f t x a . xx^ —=o c.~i-5iiûja X^-1'' u.P'-iâ/ -»OuJiA'VinN« _ /<»>. ^r-uj-''— a ^ - > 3 C C I A C O X C O » - U . 3 3 X 3 JJa> -J^J lA — IA— O S I 3 3 — L jOJ<QLJ>- — Z1/i "D ' - iASjh .^ t -Z rsiiA — f-^ »- >- a.a >- »- zuu,noQ> - 1 -J 2 n n c> a u u n »i — . ^ o c o — >-<»-< « i_< o>->-=.u.a..i_»i j x x , o _ i _ ) x x t - j o i A - o t - ^ s r ^ r _i — ZT j>-«-_i3.< / )s:*-^-T-' jj < i — x. ~i -£ -t ~3Z - U ^ c S ^ — Z Z — ' U . U - S S J z C 3 J O u . o a C 3 - ^ r i < » - 3 C O < 3 ^ < > -

• •- C •— O a c Q O — ' . J Q . u . Q - i J , O»O.u . OJLJ I-J</ 'o -*

«D Z O> >T lA CD ,-\)

lA IA IA IA CO -4o* y» »A IA a -^<O -O (A lA 33 —<

» o # *

. O OOooooo ooooooooo oo«ooooo o o<->

53

O o O o O o o O ' - J O t * O ° O o O o o O ° O OO O ° 3 O o O O £ > - 3 u O ° O ' ; 3 O ' 3 o O O O : 3O o O o O o ^ o O o O o O O O Q O O O O O OooOnOoOoOoO^ooOoOOoooo

na

u.u.oo

o»M

a•>oaja

o u. •< • t-at -. i-

— oO UJT. I

—I t—X

INIo

oa.o

O QEL »i. -Iu >-o •

CL

to -»« CM

ajz

c»

••a.1-t/>

a.o

.X

*o XIto

+ -H a.-< z *

X

*o(M

I(M

uZIIIM Z

a. Q-

o o

O I — —

CNJ rg•f 4-

•» 3C CL-. in ~ x x(T> « <\J | I. O Z

O I — to oo—< <M< <

> V« »

5 a

3 '-«I- l-

UOXX

oa:

XID_l

a.z•o

oI i

II U (Jr\j u MO -" <M

~— a:i/> <SI CDn II •—-1 -"M II

X XII II ~ — -i n

U. UL Z CM•— -< U. X

rsj t- v- 3 o oz o -J 3 + +I t • -J O O

i 'Q_ U. ZUJ >- — u.

a oOO 00II II-i ••N)< <OO VO

a c»</> 00

.11 11X >X >

> —•» >—> O C>- CJII ">

X— Ja.

OII II

<-> O >-» <-> *

o o• -O O

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o

54

ooooooooooooo'ooooooOoooorsooooooooooooooooooooooooooO O O o O O O O ~> O O O O o O O OoooooooooooooooooIN m -f if* -O l*~ CO O> O ™< f\J ,~l .*'iA O ^ 00i A m « r \ i A « r i i n m m w 5 O < 5 - o o o > o - O ' O> O - < O < O < O ' O < £ > O > O - 0 ^ < O < O < O < O - O < O - O

zUJta^

— oI- -*> u.< u. >_O UJ >->s: o z

UJ00 UJuj o: CC

a

zX — —

a- x— X

<Moo»

rg

01 (_-

I «ifi Or• a.

oII II

V^ \J^

a.*ot

o

*^C -I— c.- .I ois

«t O O r\jO — X — Xa. x co x to-» _i _i -vX O. X Q. >— s- x r >•t- O — (_J •—

ct — * — «o Q. — . a. t\iUO X X X X» u: ,_j tu oat o «. o «

-» «. * -j « -»

-j I i / ) ' ~ i /> > -G > ->o — u it u u *-

X •- <Q. h- UJ•• u z s:

Q. < — <— re: tox u. oca u. ai uj•• -< i- i

_i u uj »-X 2." a ** Z

^1 UJ (X >-:X -J <t• _j a x

Q uj i-— vr uj oo <-> za r z uj

UJ < —Z ^ Q

^ I IoCLaD

i QC UJ' O Q -<j UJ —Q z a u -i! Z ~i-: — z z M <» a x x a

uj M i a. "£.i o i- • ~ -J^ < <N Q XUJ > — «s. ••CJ a O CX.-• UJ II **U isi OC. H-Z 03 "J-1 C "J•— O -J *

CD ~ •u, u. z aO o < ~ •- —uj X Q Q. a. ••

j U J u j U J Q . •• <-• L~ ~^.UJ -J _J O — -1 3^> 'J O Q Q a a.< z 7- uj v, «. +_ s < « : s n > - G - <

j I I I I a x aUJ uj I)

 - ' o : D

zX•

<MQ

5 -1U Q

I-I <M_l C< HO U

H- QUJ Z3t: UJ

<-> u u o <J o

, *- J- O (M >O— \» (M r\)

ooooooooooooo

^—-^'^'^Psiro-''O O O O O O O O OuooooooooOOOOOOOOO

55

O O O O ' o P o ° O

i O O O O O P O O o ° P °> O O O O O O O O o ° O °i p O O O O O O O o r a O °- • D O O o O O O O o ° o O

t O O O o O O O O o ^ O ^•^ 1C* ^ ^*™ Jp ^* ^ **J r\j l"* x+* '

O O o O o ^* Or- f- p. I"" r I- f-

O o <-> Po p o o

o o o oo o o o

' b o -4• r~ r- r-

-« Pt-. a:

< ou

X —« o

aj a.CO QC ••••.00

X -4 CJ

u.••o

oo0.oo

Zx a

o. oo ,

Q aa —— ooQ rt <-? <No o X H-

o. »uj a. — — 'z i- Q—• UJ v^h- en o3 UJ'-> I I OQC ujCO K-

23 -H oj Z00 t- h- •-•

O -I- a.

CL £O o. ^*a a

xX LUUJ (_)-I II

Q- —z xo —O Cw

oCL

I'MO

19UJQ

CLH-

<J

a.t—OO

a.oJE

onZo

Oi.

oo

*XK

WO

«:aioo#

rg

x

+B

R

•z.*—(00

^x*CLU2

t-Cfato»<N»•»

ZX

z: >^x i-•v -j- t-« o «•a — <o CL >-JC | UJh- ~ 33II CL II

xig

O —K >-i

CL

O #O Z

P •• eg

O —

•f0.ZII

CLZ

Ifi

CL —•»CL «!•Z O

Z CLI

P**2

#

eg

S1«•zX

»_)x#•xlX<tt

<M

»

»-l *

< —«-r *<4

CL to« I

-< OCL *—I- XQ: _ia aoo 3:

:*rg -»

* •— <M

SQ

R

occaI

2x

CLO3

C£CJOO

rg

CL•f

oc t—co ,•¥ OO— jj Cu ffl a.a. < t- — ?-— u -4 <

X LU I||. — •

—1 LL IUJ "- i

IN •

a: CL x — Q_

I # A. CM ~ -i s: * xto x o -it —

•f x x z Q.• II UJ x O (_)

-H a. (_>•«• u >-II < II • Q.a. xi CL CM O O< x o *- o o

* 1/1 *_J - OX -i ># o

X#

oeg

O —H* *•*a.O •»O Z•~- X

• INP —

X <^* O

< =c >- _i a.* —< CO —. « I— u. I oo atLI ^ >-• 33 s: ca sr

* O CL <j u« *- ^-4 II -* II < II X

* CL u 3£ •" Z ii

<00 . - 'O t -< -<X

CL —

+ T 3; -r

z2f 2if H Z CLz: s; x |z -«c #— -, • OIT. tr> eg «

o o a: 5: xI =Q < _,

I * CL• t — -J y;

S )_ t- 00 X LJZ O _l o » —I « • U * a.

2: -i- x xZ IN eg • ii ujx uj uj -i f. ou — — u < ur\l u. u. s; ^ 5-uj •- i- < x o

(M rg (M•-• r-t "

o o oO O CJo o oo o o

Of>

oOoo

OQOoOoOOOoOoWOooooooPooooooo

in in -< f^ -<o r- o o -<„( _< CvJ (\J INJ

o o o o oo o o o oO P P O P

r ro o o o o o o o o o o oOOOOOOOOOOPOOPOOOQOpOOPO

56

0 0

*— t "s|

•^«•"• X— » «^-<\I ^^J. ^•!<• *-»— * ^>

îx • j41- i>

t r^*\J O— u— «

— *

>rf-

o

0 0

«*

•*

QL

^^•**•*>-•*<\J•i>-fr2X"*•••w-oX<»•

~-* ^r\» x

™ *•— (NJ

M

O -*o <

O

^

•M r\J- 9-

O Oo o0 0

0

*~t-^<I*

"^*/)»

~Î )-M*00

_J_

<Io

r"i

o0o

O O O O O O O O -j O"O O O O

<3

XMl•»-1X

X

<N

—*"— J — .^ -»•«• 1n. oi^ 1

^

— oa. —»~ x<x J

oi X•» o<M -.+ •-» (M

rvj ^_•tt cC* <JZ oiX 1 ' -t Q.— -. O ».* 0 00-1 -f •- 1 Ox a n ii* •- a a. z:^ C- C O QXI — ~ —•tt

1 *^t Q.

Nj- ^ -^ -* ^* <tf

^ S

« *-» • * •

•~ <r C? O Ox •* rv LLJ uj jjj -» a. , S » « f \ j ^XC L i A * Q - O ( X v _ > O 2" t O - ^ ^ ^ j ^ J C ' j C D O O ^ Q CO"^ > | ^ — O ( f - .H-^. i^ j | C X - ^

11 * ^ "-* 1" " ** ^- _ II II G *- O

S* ' fl UJ -Z

2 JC UJ•-^ ' .

-^rg -^ rvj

'J^ ^^ L^ ^ *^ «^ ^5 y^ *O ^^ -"O i 5 ^^•*O r-4 J f\j •"*) r^ *^ ^ ^ ^^ LO .^ sO•^ Irt t/ {f\ Ij ^ l/ Li «^ j tO u^ Ô O O O ^ O ^ J O ^ O t ^ O C J^ ^5 C? 5 >O O ^J O O -O ^3 C5 CÔ ^3 C3 5 ^3 O ^3 O ^3 O ^^ ^3 ^3

JC

CC

OC

C

_»

X,

^—f~t—t>~o— 5

U-

XLU

_Ja.yao

0

i—4

UJXt-

^*^<lX

Q

15Ol>-*e»— «

UJ

X^_>

Xl«

-^

-;

o£0UJt—

H-I

LUXK-

c^LJLL

C_j

^^H~

^ j^C

LU

O^_^

*"002T

<-

^"

0

r>

r\i

lUj

Qa01

OlX

o

0101,-»

1

o

o}_

a:UJQfV'

0

oaOl_Jua

O

o o

o r>

^^<jt—O)a: a^^<l X* 01•^ _J

3.

II •£.O

X O

o

ooooo

1

^^^^X

o•»*

X-J0.2;oQ.

XOl0

.

Xw

a

oCJ

OO

o o

o o

n

a*,

aQ.

fco01Q1—1—

«&t—.

»-4

a.>-ocQ(

01

£V

^Q.1-

*.M-t

Q.v^

1—Ol^p,_J

O

Q.X

UJ

£.i_J

o

0

o0oo

O O O

0 0 0

••n & o

•(NJ

f—ry*

011

oH^

a.

Q.- ^

f^

CLl—oi^jf01•^-^X

a.

*in•_^

x

*CSI-t-^^

X

— *X •* f\i »

01 • -»

» 0 oiO • 1. . •— •,

oi X o(_) — 1 ~-

Q- Xi ^ _j'

_i o a. o

i-4

-^ *N

0 00 00 Oo a

o o

6 o

-( <M

-z(XIJ

Ol 7x oi

—4 f*">

f- r-0 Oo oo oo o

57

O O O O o O O O O O O O OO O O O o O O O O O O O OO O O O . Q O O O O O O O OO O O O o O O O O O O O Oo o o t - » o o o o o o o o o« ^ ^ * $ * ^ ^ N ^ > f l A i A l A IAIAIA

O i O O O O O o Oi O O O O O o O1 o o o o o Q oI O O O o O o Oi O O O O O o O• "O O O —4 (N r«"i N^". N- ^ CO CO OO OD CO

X X

n o

o oO

0 > >+ I

-i ra 3 — —o > — •* o o" . « * . •_J 13 ~ — -< -«X . -O J3 " ••• i/> O o* a. o.sr a r- r- T ix »-« -4 a. a.. x o o i •*•

O, CJ <^> f«l T i3; •. m vn OL a.O ^ a j u / a o * * *• CJ — • „ 1 _J J

X ^ f O r o X Xa. < co ac •• •„ . D IN (M 2T H/J) » « « X X2 rg <O O » •S a — — «* «=

— II n at: < 4

«^X»

CO••o••*COo

LU

(B^H-

Oa:JD13i/>

EL

INT

EG

RA

LS

zLUacu.

UJT*^>

coLU"*

3DO.Z.ijO

TE

RS

AN

T V

AL

UE

C

(X)

AN

T V

ALU

E

S(X

)N

T O

F F

RE

SN

EL

INT

EG

RA

LS

GA

TIV

E,

TH

E

AB

SO

LU

TE

V

AL

UE

IS

U

SE

I

LU *- H> UJ JJs: j -i s z<l 3 -» 3ac co co o co<a LU LU oc 1-1O- QC QC <-

XLL, UJ n i ujo x x x u,

t- i- i- i-.zuM«

KO.,

acO O co xCOMJ

Q

X K

EM

AIN

S

UN

CH

AN

GE

D

LU

_l<t>

•

t-zUJyTVOQC

^

Ul

T^-

natU.

^-

LU

OLUZ

*_)«•

9CCJ

^«

l_lO**•-1

or

LU>-Z•_•II^^xa,

O

aacu.

*-acLU

aLU

5C

>-#

-J«

, ,

2co*^j

(V

OLU

t-

•«f

II

X

co

X

Q

•f-_J•

X

acOu.

a

<

§aca.a**h-

UJacLU

Z U.c u.t- O

D O_l Z4 ta^> COLU -J

ES

NE

L IN

TE

GR

AL

S

BY

BC

ER

SM

A,

S A

ND

O

TH

ER

A

IDS

T

D

CO

MP

UT

AT

ION

, '

o: LULL -1

COu, <O h-

Z *Jo ^~ u

LU I— "-CV ^ >—Z t- <LU r> ^a. a. cuLU s: xU- r~i >"UJ o ^a. 2.

000(*)

4

a»

?^

•oz•ooc-Hi

KJ«

p1

LUpgCO—4•CTi

kA

*

1UJ

(NJ

eg

•fIVl

~4

•|

UJ(ACOr»-

** or*l O

rg + *-^« • *

-^ ^ 'JT\» «« v

•4 •-*• It ***— * •"* fsl -^ «*X • — r>| —

co | :£ o . —tD 'xl O If "f &«5 *•* •-/> "^ ^* On LL n M n nM —• o <" '•g u*

•

LUIA<£>

0

*CO•

•

«

7LU

IAOJ

•

1UJ33

3!

0INCJ

•ft+K^•».LT\

1UJCOonr^-fNJ

tro•f

01

UJ*-4

•

0 rjt 1

IA LU•f O

« V-C

so rg1 =>

LU •00 CO>A *-* rsl

00 —CO f*l• 1

IA LU*• orsj rg

O ~^-< pg

1 ->LU ,

CO *vD >N|^ •»N- »«fl *J"• 1

*G 'JLJ«— sO«-» «J— CO

^* >f•tt- ^l/l *II "N

( J O « - > O r - 5 O O O O

o r- P»J -o o ,$•O O -i -i rg MO O O O O O

roO

58

o <-> o tJ o ° o o o o oooo^ O -3 t> O 0 0 0 0 0 O O O O- . O - j O o ^ O O O O O O O Oo •:> -> --0 o '•> o o o o o o o o• ' J ' J ' 0 ' ' ' 3 0 0 O O O O

I IUJ fj UJ

CO fM (M -HINJ I °O |(T, LU M JJ

^ ro r- r-I 01 + ,4-

ivl O* TNI O"

I I I *Uj M uj MCO * O «,j. -~ C7^ •—&< \f\ O -O

01 — Ir- uj r~ iuO< 03 CvJ -T3• r\j • o>

r- <r p- o>* «o i f-KJ O ISJ O

t I IILU fj UJ ixl

f * s> »a> " o ~<M M ^- (*•>CT- I -i I•O UJ O UJ

—< —< .* O

KJ P- »M -*# ff- #1*1 .•r • ^- •I ir> I o*

UJ + UJ |CO l«sl O is) *"•-•in * rj * *o3isj — c- ~ * *OD -* tr\ •* <£ OvO T MI + Ir- uj o> o IL ** <. | . VA * *

=0 J- UJ O •* C «"_ .-^ -1 I _(_- ' —

— ^ ^ , i r \ o > ~ - r - ' N i * »•M IM . ^ U % O ~ - ' < M — "^ >sJ

Z— — rsi— O-T — O K * * ' 1

- t o o 7 > s > . . ^ - — • x m v r ' Q c .oo— . — -^ >r — «r (J • • >»- u 1/1 -r — I • — i i/> o o »- ooj ii ii ii ii -NI -r it i>j ii ii it uj i* 3 i/> rvl •< » | c O » | v l ^ - ' v 0 ' ^ a j

^ (M —I

O (T|J-\ in '^ l 'nO ^ -H IN !r> ro-• ^ -< -* -rf •*O O O O O O

o o o oO O O O

59

O°OOOOOOOOOoOO O oO O oO O o

r > . C D a o c o c o c o c o c o c o c o c o a o c o

oOO

o o o oo o o ao o o a

zLUt—Z

isl

•O OOX

00

X

LL0

O

idx

0

h-

O

Of.

LU

— 1LU »-Q.>• 00t~ ai• i—

> Q.00 3iX <_)

•> U

< ^

z *y— oh- Ooo txX °-\—

UJLU Zjy •"•

— h-H1 —

c; ^3: J3 00OO

OO

000^4

<

o00-^

» oo

oo O—1 X

O CDo a

0 ">oc ***-1 -<

Q. Q1 X

~ a.

•0 < -1_ -«•- — H-s: m «< oc o. — ' L=

-» — oO Cj •00 X 0

"•* TT *x i- <

»Z 7 Xo ntari *"* >^

UO - UJ-£ ^ -JUJ -^ ^y: T *:_ — ua 3 o

NtX

L.H

.RI

,RS

,TH

I ,D

MA

X ,

F I E

LD

, S

CX

JRC

E,

TP

,ST

P,S

QR

TP

I,T

T,T

TD

EG

,Pia

2,P

ID4

F

INC

IDE

NC

E

MH

ICH

IS

X

IN

M

AIN

P

RO

GR

AM

-, — oa. Q-X «v I—• I- Z

< 00 ~Q Z o

< 0_j — ' 00X Q- — .

^Z -£ ZC3 ^^ "^- > ^>•*-(!.3- s: xa ao <->

0. ,-

6.

2 8

31

85

3 0

71 7

96

* Z

))/S

QK

T (

6 .

28

3 1

85

30

Sl(

Stt

*SS

>S

C*S

C-S

A*S

^)

/( 2

.0*

SB

* S

O)

( S

B*S

B+

SC

*SC

-2.0

*SB

*SC

*CO

S (

AA

) )/2

.0 )

)**2

.0

— o *- xx o a: oo_j ^ cj *••Q. 00 u,z: ii n d0 — ~ 0

0 <_> —— oO V* OOa f • caX CO 35 <tLU 00 00 II

O *" * ""*II < < X

^ :2 2 ~*^ ^ o ^x x u. x

H

Xoo

COX

LU

a

f-4

a-

o>-oX

^*

o""

COoXIX^_

00

"

Xv/)

(M•x,^^^4+z

Mzz

—11z

CJJXN.-J

IIo

-^t<4

•^*•J

o ••

f ^

o o —U II U-_l o —

inI ""*

uK

0

o•

0e-t

LU O-3 uj

O z •

C3 ^M t

II Z —X O X+~ O i-i

^-

•»-^00X

Xtff

ac —^, x f1^ t— -"* II 00 <X — x M

Jtl - -4 •*-•

X Z ^* O>— < X Q

Th-

00ou#

1-4

II

it

o

aM

a.•

k—

t•

^•ofl

<*>•cUL

X'a.>-,

i•4

IV

ri—io.

oa

-4

II

~*

OQ

1-4

a.•i-0

se.xV*

• *^Q -<z< 0t t—

o

<T O

c» — .. 'Jj •— 4

4 t

LU OO. LU

> •

-* — . rvlX X II-. — i_

— XZ •Z 31 »

— -.5— h- Zoo < — « 1

-» X O " •-" -4oo » _/ _ — ,—X Q X M X *-• * • • x » <

O — X oo O* -< • x -iZ « 0 X C3 iz — — •> » <1 i- a. Q — x

— < 1 * -" 4-— no: — I a:

00 O » — -• | h- Hx-'n:o- -" -< —« x X x ~ — .a-4

Q "t- l lx1--) —* C£ I -« -< U- 0— T — ~ooa — z-ii-Q. — X _ | < <| | * X -LL U.

•-• in a — | —— • — » < -X Z,_(Nj . - .Z-*xXZ< a — z -4 i K- •o — u i .1 / u t-_ J Q . X > - - . X ~ O

. 0 ' ~ • *- II — ~C£ u. ~* •- < — O •x ii — • o — z at - ~ X O - J ~ - < Z+ — Z X o <

• X o . < < O OO » « X t .•» — o ^ •*• -~ o o— z c- c t z a aC. 7 • t X Z . •

O (_) I- O -J II ^ O Ozxo • t — . o t •— u • x. r — • i z

z_ ._ j3 — X x u - Z X x Xl_ )X~~ — <— '-^-^

CO

<J O

•> o o

O O O o Oo o o o oo o '-» o oo o o o o

o o f- -i-H ro lf> OO -, D _,<J O O Oo o o oo o o o

60

o —*

*"* «^^^ •

— in~ o ai- z— < #:•) I -. -Z "i II «tt a. < —_ — I in</> </» ca .o o a ou o » -* -»•» * • u. «— -» x x o«i •— • (j, U, uj_ — • -~ QO O .t -» <MQ Q • _j • «ii n in „ '-n .-o-• ** -< -I U. U.-i 1-1 UJ || , »— — • -> >. nO O * •» Qa Q -«i u.

ii -« x —-» — UJ -> ,3 T

u - Q £t 3 Q

<M f\J ^ U. » C f£no — •-• x 3> o1-1 »-• Z ao h- in u,

-»a a. o o » >ot • .4 ' » jj 4

— *->- Z — O O .— _ i O # - » r > i - m— • • • — t- tO"-» X Q— Q O>z —1-1 _jin u-_ i o x o .< — — _ ) — • < • ' M'I— CL < *— Op < _ J I _ ^ Q.J: — 2. u33^-<S i _ < f 3 . Q X < S X < ^U - < < < l f - U O ! _ < » < > » » I U .• » • • - ' H < »•< »x s: -.< —— Q O - M ^ s:^,_iu. o — - < ~> z a.— Z « C O O . l _ I O < » O ~ > # • — 1 - 5 ;•_ i < ! «4 Q «OI 2" - J (_ I I « - I l i J O " * *« - • • > — O O X J < X -»Qh- _ 1 U . X

• H- —• O •• I- -I -> OU. —— • _i — Q •-• < Xo >_» .—. _ j — - _ j i » «< « _ - « - c > - - - i _ i ) - j u . — J Q . s :<• *o^ < o < * - x *-•-' r j i x u . * . — --5 — — — • • z : <s:

oz —• c — > - • » m i<a : o ^ - ' ^ ^ - c c . ' M — oO Q_' - I«I ••— •• *:o<J »-oC3-~ i— no ro — - x Q . u - i i . i i o u.ii sun m*X ,*-»•— — C O 0 « > v - ^ I I — . 1 1 O « " « > . 3 : — ~ - 3 U - _ — < ~ 7 ~ ^ > »

— « • » U J j I I ^ ' J ^ + U J - C ^ O " - — _ / - » U J » l J J O > v » < 2 L T " O . X U J - JLLJ .m^^ u j i ^ j c j m ^ a . ^ — Z 3 3 _ ) 3 O Q . X H - - J a j r - ~ " l 3 | ^ ~ 'u a. Q. u< ^y o < < < ' i o » i i — i- — Q — «; + j ( _ x < w s : < j ^ i i s i o n ^ t-

— > - > • - < O i — o m a u J t

"• u. •*• ii 3 '-J•-••-' *^ — '3 o

in ino in

,»• in ^

• • N. ^ «T" ^ %r >w ^ >^ u* u* 10 *• *•* **" «* ut *• ji ui **• u ul *** m *• *o **J ^ ^^ *n ^^ «jj ^ */ ^* ^* ^ ^^o o < - > o o o o o o o o < J o t > o o o ' - » o o o o o o o o o o o o o o o o o o o o o oo i o a o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o oo o o o o o o o o o o o o o o o o o o o o o o o o o o o o < - > o o o o o o o o o o

61

o o

O O O O O o °o o o o o o oo o o o o o oO O O O O o OO O O O O o O~A s- nft fr» o i rviO* O* CT* 0* O t-$ OCO CO GO CO 0* o* *

0 0 0 0 0 - 3 0 0 0— « (Vj "1 in •<*•

X

*» z*^ ta*

in a.• X

o «•-< <u. nX TE

— — -I* rg x

LU in —iQ — u. <

o -* ~

CO

o*in

oo

XoUJ

Ou.

II ~J O' i-J — U. I

Q Q

X «-!in ajII UO II_i x— s:U. Q

COo

X i<X •

* »r«> •-!U. II

15-Q »

U, —x2>- Q

U.

et —O t*u. o

O^ Qa. u.* •*u a.a s:

x u. <o»t

-< a:II Q-5 II

00 U. CDo> — oin si uo O X

— <o u. s:Q —• Q

>- O ->

X X Q< < U,s: s: —a x•• # »-

in •< . 'i in >• o1 in — ^- '

V. UJ* Q.* >O I-Z •O zu. -

u co* ••X ">f>- «* I"-QL t.O OZ CO

Q ^ —LL LJJ

•~t •• Oa. Q ora. u. 3< — a» CD oox o •m • Q* o -iO u. iuu. — • — •— x u.

o

I••CO

-~ JJ

O* Q» • — •

CO r*i -<M U. -• >- •» >^ O -.

UJ X • >^a i- i

~ o x i .— < i— O — < •_, _, -i . O

-J - Z «. a ^ —

o s: — r r• < z »— •<o x •- - *_/ 7. . -J .UJ •» -> >- t < X -x in z: "- •

oeo

QU.

• O —" oo r<l • •-^ • o -i

_i s:Q. a

O O(^ uII —

CVJ

+

-1 X• t-

II —~ QfM a.

* —

< o. a.

O —

CL

OJ

T O • O • i/l-< - O O »_-< ,00 .a.» o -< • z

33 . | . O »(NJ . t. t -• O• — o • IM • a.» ^, _< (M _. i i_

CVJ | •. -H » l_l Q.• » i » • O> 2:

O o O • O » <. • J> o • O ••o- - - . t 3O • . • ~> -5 a.• • O — -• » «-

e. j- s. •j- ...

U00 Z0*ino

\j

— oo£ o ota. u. a.

>- a x< *-s: -i ••ac _i i/jC < a:u. U -

— „ a a a z

U.

—x — , ELx s:

a. _is: <

,_ Quj z

^ _ ->

o*in-o

ininin

o o o o o o o u o

in <nr^ r^o o o

«-i <M^) or. r~o «J

o o J — —->. - ,—. — ">O O 'J ^ J

o o—*^<-> O

62

oI»

oo

O

-4 <-. X» o-• X

O &

o-UJQ

Q -* -*U- .. _.t— • I<" O »X . .xx°.o ->tz*Iz — •

l/Jf-a

J PO • -JUJ . -> <~, t < r

• o. o »

I —a -<

o au. u.

T 0.

ft. -4a —u. at- u._

a. Xi-

UJz z~ o- •-•3 <s>o z

oI—r

o<«>

I00 »(M «• — M• m _,

rg I ••o o o•• • <r

o ** ••o » «. • o

o o ••UJ

. o - -O m .om •. o •

•• • co _i. o ~< •

o co i <s>oo w > t--• I .0.

!L "2^• .M » CJ

<M -4 • U.— • O >-

** • C^ ^• O >. CL

O • O •• i t Q• IM O U.

U II

1 o < xi z o <t

• • fSJ Xo c ~* »-IA un on UJUJ a) uj Z iX X x —

63

O O O O O o O o O O O O O O o O o ^ O ° O O O O O o O o O o OO O O O O O ^ C ^ O O O O O O o O o O o ^ O O O . O O o O o O ^ O

< j > J o ' ~ ' D - J ' - ' r - ' O O O O 3 o O - 3 o O ' O O 3 O o O o O o O O O O O O o O o O - 3 OO O o O o ^ O O O O O O O O ^ O 0 • 3 O o 3 O o O o O o ° O O O O O ; - j O i - ) O : - > ' 3

000000° o ~o o^3 f^

JJa.

I*•

O

O ^«O >->< —Q i e O

***

OOC

O

ULU

tTv _^ — < M <NJI I I X X x - t X-<rJrMfMf \ j lNJ fNJ<Nii n ii N H ii n H

« a :o —

— II

i ~ >- I -J

C OtL<

ac

t— — a:

* #

O O oO U O

O OO O

r r o r r t - ' < fO o O O O O o O

x i > ' ^ > < > O 'OOoOOoOoO - - — - -OOOOOOOOO OOOOOOOOO

64

o o o o o 'O O O O o 'o o o o o 'O O O O o 'O O O O o 'r*\ -f LO <O r» '00 00 00 OO oo '

'OOOOOOOOOo'OOOOOOOOOQO O O O O O O O O Ooooooooooo'OOOOOOOO OÔô-^<\jr-\.în-or-oo

OoOOOOOOOOoOOOOOOOOoOOOOOOOOOOOOOOOOOoOOOOOOO

IT'OOOOoOOoOOOoOOOOoOOOoOoOQO

or»M o

N.<M

Io

oLU

atLUecoLU0

o <•»- as<M

— II(\| t-» Zt I—«

o or- Q.

•• Q

•33 LJ

Z (M< r-t

• oO |«.

— •• — (V • ~«t\l II —1 • -» II I

UJ I II INJ UJ ••• J • X - -" •• Z • •- • O O

o oc o o o H ^-"VJ CL i i j f\J •»«. LU . J . Q — '

-~ fc- — •-» — oin o* x ^ (^ -^

z <" z: S3 r — zD X D T 3 T 3Z iM Z -< Z IN Z

infM

I +-•U i

— 1 < II

LU

I/)

a:s:0

II IIt-a.

O U t- -Jz —

oin LU

oa <o a.

_j uj UJ LUO _) _l -JZ '.5 O O<l Z Z Zw <r «« «*1/5 *~ ~* —C Z oo Zo _, o —» i>O O i/Ja # # #< Q |_ H-Og ^y ^ n

*• ^ "• SC.X -t- * +a > x >

i; i~ k- t-II O O U-« II II IIX -> (>J <MO > X >- o o j

-H <M>. 3-o o

X X |_O o _J

" ~ a *LUa.

Q.O.LT to Z l— < X X X X 3 C c r-J -/ Z LU UJ II l| II II || II || II |l

O ^j || Z CO _< Cy fo *^ LT\ O " CC 0s

-J _J O Z< < Z O *o o •* o <a

O OLA

» # * * » *

o ^

o o o oJO O Oo o o o

<J O O O O O <J O O(J O o ^ O O a o Oo o o o o o o o o

65

• I • ' .o o o oo o o oo o o oO O o Oo o o o,/>h- no <J»IM <M (N MO O o O

i i. . • :O O O O O o O o O O O O O O o O o « O O O O O o O O O O O O o O o O O O O o °O O O O O o — _ _ _ _ _ _ - _ - _ _ _ _ _ _ _ _ _ _ _ „ _ _ _ _

o o o o o oo o o < J ' - ' O t - ' O < ~ ' O a o o o o t - ' O l - ' r ' c ' l - J i - » < J O O i J O O o < - ' o O O O O o o

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o oO O o O O Q O o O O O Q O O O O o O o O O O O o O O O O O O o O r j O O O O o O

fn fO ^^ rO **^ ro ^^ ro ' ro *^ *^ *^ x^ >^ *^ s^ ^ *4" vt" i^ ^^ *^ irt ^^ ^rt i^ ' ^^ ^^ \rt ^ o ^O SO ^3 *& *A ^^3 CT5 ^3~ C3 ^3 ^j (^ ^^ ^3 f. (^ ^3 CJ ^J CD ^ *O ^^ CO ^ C3 C> ^^ ^3 ^-^ 'O *-^ CJ *3 ^ ' ^3 ^^ CD ? ^3 ^3 Q ^3 C3

! i ' • i • i i ! , : ! or ! : : : ' i • •< i! ' ; . i i i I • ! . i i uJ i : ! . ; ! . s o i

I •

• oO f-r- <sj

O

pg»ii

UJQ.>

.~ !t • !-^' : t i• o• r^ :

II 1 ro • ai

ofN I u. •

o — o —rg ai• a.

. o u • r^

» ^ O* •* —4

O O O O

<tf*

X

I— L1J|_UJ(— UJ t -LU t— |_l—

2Oi»

%&.

• 10X 111< HI

H- O

a o

X

I II

uTUJ

o'.UJa

oegox

X<

oX

x<

uI- ICf.UJ

rtl eg

> >•

X • Xir\ < —• <I* ^ z: H

a a _i < _j

I z + i < - o z x O x > o +> X>- < . Q L < >

fSI

or-

oa<

x>

X<

x

UJ IL K Ih- t- U5 SI,< i- rg < oj <

II II

atO

a:u

**«

II II LU

»#•»

II II II

>

• u

X >

I II II

ana#

**

<QCa»

O o o oOOOOQOOOOOOOO

ooooOOOOOOOQ

• < • • - •o o ^ o o oOOOOOOOOOOOO

oOO

o o o o cj oO O O O O Oo o o o o o

<-j o o o o o oo o o o o o oO O O O O O O

66

I

u v _ » o G o o O O o ° O O O O O OO O O Q Q O o O O O O O O O O OO O O O o ° o O o O O O O O O o O

' O O O O O O O Q O o O O O O O> O O O O O O O o ° O O O O O O

._»vj >-• -^ (_j « • _ > • - < < . . > • — v j ( j a ^ j » - ' o i - > o O o o o o o o O o o o o o oO O O O o O o O o O O O O O O o O o O O O O O O o O o O O O O OoooooooOoOooooooOooooooooOaoooooO»O-(<1>><frt^iA>Or~rOO'O^(\j(*»vtiiA.^f>-eoC

>O_»(\JPr(»>-in-Or-coa>O

,0r-^.r-f».f--r*.r-f>-^r-co<DcocococOcocc<cocoCT'<7>o»c<o*0>o%C*O<o<oOoOOOOoOoOo'OoOOoOoOo°OOOO<->r>OoOO.H

-J<\j-1">.*lA.A^COO>

OoOOOoOOO—•! .J »J -J •< —I ^ ~l

I

00o,a:oo. o

o:.<oa

zLUa.

CO

:Qo

=oao

oCOv-4XCO»

u

or-fsl

M

o

to

or-MX

Oh-PsJ

a

§oooof<_)oo.

30

NV

ER

n<OJ

^5

32

92

5

< rHt-™ IA

X >

-J II II -I< X > <iU <• < O

oo10 O01 <Ia cco xr- a.<N II

X3 <<a.n

*

VA X <O < —

< ~» -I UJ LU

O I- \-< < (- <OC |_ O tA t-I U X • U

O Z + I Z. X >•

lA _l < < —III -I II II -I> < X > << o < < o

»

*«a oo o -f

00 Z h- • ~LU < OL « -)> Z -. -7C£ 00 > -3 —r> —i -< "> ooO LU II — CC

co ~> oo oa: •— ~J o H< O UJ -•

-> x i-> a. o.CD 00 *•O O —O O 00#. * *a -» —< -> ->a T -7» ~- —in m coIN O Qo o o. n n

n -* — i

»»a.ozLUaIv-•-• r3C —O •

Zoo

oa.

a o ~~>IA II —

LU -.3jO X U

oo

01OLUa

oo0LUQ

O

a

X0 O1 H-Ct— OLU O_J —

I > -<

3C «

~ I—O

• II

LU LUD =>Z Z

— x •>- t- ii:o a o z sc0 X X O -

» # *

I « •»* #

* * * »

oo(A

oLU

CO«

*

• -I — UJ .Z Z< LL, Q. Q O

> o -i M o o

o o^< o

LUZL<Ofu.2 Z

ae-J o_J h- Q< LU ZO OC UJ

lA LA 1A IT<O O O «->oo o ooo o o

ro IA o •* -O o^ •$ \e\ *t\ \r\ *QIA IA IA " IA IAo o o o o oo o o o o oo o o o o o

nIAooo

•a r~ r*- r- <-> —t^ fv iIAU> IA IAJ5O^ ) 'OO O U O O U Q Oo o o o o o o oO O O O O O Q O

^oooooooooooooooooooooooo

O -^ (M^*r*-h*>a<o-oooooooooo

67

—. 00O> <M

00 •« -4

• Ol •— .00 —• Ul ~ •• —I- ( 0 3 I• 3 u- x •9 I— ""^ "^ •

o — x «• oo ax —_i s o> r xUJ «g » |_ .4

U. ov _ -> ,

IT- LU > -> > -i Oa o • < • • •Z - -I 3T C3 ^ Ot- O • > - « . »Q Z -J • . O Oa. -^ • . • • •— — x o o o • -•a. -* ^ » r ^ t o o «•z. ~ _. • • -« «o< Q • o • I • OO»LC CO » O » O - ^

O»- 'NJ • • • • . ! •u.a. • — o • (M • Q_«•- » f > ^ ^ ! N J ^ ^ » V^I< ,v I « — . O —

— — O O O » O * 5 -2 - * » « c > o » u <u. — o — . - . . .„ O O » • • -i O Q>— a. . • o O «• • u._ J - 0 0 . - - . —i j ^ * ^ * ^ « « * < N J X

3C_ <

- .- ll lia; — ~> ." -i -I -J-^ r <r — — -i —j - *; r '.; -1 <\s* ^J *— w. ,j vj o

< < -1

-> -5 -w _,

68

REFERENCES

1. Keller, J. B., "Geometrical Theory of Diffraction," J. Opt. Soc. Am.,52, pp. 116-130, 1962.

2. Meil, K. K. and Van Bladel, J. G., "Scattering by Perfectly-ConductingRectangular Cylinders," IEEE Trans, AP-11, pp. 185-192, 1963.

3. Andreasen, M. G. and Mei, K. K., "Scattering by Conducting Cylinders,"IEEE Trans, AP-12, pp. 235-236, 1964.

4. Morse, B. J., "Diffraction by Polygonal Cylinders," J. Math. Phys.,5, pp. 199-214, 1964.

5. Oberhettinger, F., "On Asymptotic Series for Functions Occurring inthe Theory of Diffraction of Waves by Wedges," J. Math. Phys., 34,pp. 245-255, 1956.

6. Pathak, P. H. and Kouyoumjian, R. G., "The Dyadic DiffractionCoefficient for a Perfectly-Conducting Wedge." Report 2183-4,June 1970, The Ohio State University ElectroScience Laboratory,Department of Electrical Engineering; prepared under ContractAF 19(628)-5929 for Air Force Cambridge Research Laboratories.(AFCRL-69-0546) (AD 707 827)

7. Kouyoumjian, R. G. and Pathak, P. H., "The Dyadic DiffractionCoefficient for a Curved Edge." NASA CR-2401, 1974.

8. Karp. S. N. and Keller, J. B., "Multiple Diffraction by an Aperturein a Hard Screen," Optica Acta, 8, pp. 61-72, 1961.

69

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

WASHINGTON. D.C. 2OS46

OFFICIAL BUSINESS

PENALTY FOR PRIVATE USE »3OO SPECIAL FOURTH-CLASS RATEBOOK

POSTAGE AND FEES PAIDNATIONAL AERONAUTICS AND

SPACE ADMINISTRATION451

POSTMASTER : If Undeliverable (Section 158Postal Mnnnal) Do Not Return

"The aeronautical and space activities of the United States shall beconducted so as to contribute . . . to the expansion of human knowl-edge of phenomena in the atmosphere and space. The Administrationshall provide for the widest practicable and appropriate disseminationof information concerning its activities and the results thereof."

—NATIONAL AERONAUTICS AND SPACE ACT OF 1958

NASA SCIENTIFIC AND TECHNICAL PUBLICATIONSTECHNICAL REPORTS: Scientific andtechnical information considered important,complete, and a lasting contribution to existingknowledge.

TECHNICAL NOTES: Information less broadin scope but nevertheless of importance as acontribution to existing knowledge.

TECHNICAL MEMORANDUMS:Information receiving limited distributionbecause of preliminary data, security classifica-tion, or other reasons. Also includes conferenceproceedings with either limited or unlimiteddistribution.

CONTRACTOR REPORTS: Scientific andtechnical information generated under a NASAcontract or grant and considered an importantcontribution to existing knowledge.

TECHNICAL TRANSLATIONS: Informationpublished in a foreign language consideredto merit NASA distribution in English.

SPECIAL PUBLICATIONS: Informationderived from or of value to NASA activities.Publications include final reports of majorprojects, monographs, data compilations,handbooks, sourcebooks, and specialbibliographies.

TECHNOLOGY UTILIZATIONPUBLICATIONS: Information on technologyused by NASA that may be of particularinterest in commercial and other non-aerospaceapplications. Publications include Tech Briefs,Technology Utilization Reports andTechnology Surveys.

Details on fhe availability of these publications may be obtained from:

SCIENTIFIC AND TECHNICAL INFORMATION OFFICE

NATIONAL A E R O N A U T I C S A N D S P A C E ADMINISTRATION

Washington, D.C. 20546

DIFFRACTION BY A PERFECTLY CONDUCTING RECTANGULAR CYLINDER WHICH … · DIFFRACTION BY A PERFECTLY...

Documents

Transcript of DIFFRACTION BY A PERFECTLY CONDUCTING RECTANGULAR CYLINDER WHICH … · DIFFRACTION BY A PERFECTLY...