Sinusoidal Reaction Formulation for Radiation and Scattering From Conducting Surfaces

8/14/2019 Sinusoidal Reaction Formulation for Radiation and Scattering From Conducting Surfaces

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376 IEEE TRANSACTIONS ON -WAS AND PROPAGATION, VOL. AP-23, NO. 3, MA Y 1975[5] E. N. Fox, n e diffraction of sound pulses by an iniinitely longstrip, Philosophical Transactions of the R oyal Society of London, [13] N. Marcuvitz and J. Schwinger, On the representation of theplane, Quart . Appl . Math. , vol. 25, pp. 193-204, 1967.[6] -, The diffraction of two-dimensional sound pulses incident electric and magnetic fields produced by currents and discon-on an infinite uniform slit in a perfectly reflecting screen, Phil. tinuities in wave guides, I, J . Appl. Phys. , vol. 22, pp. 806-819,1951.Trans. Roy. SOC. ondon, vol. A242, pp. 1-32, 1949.[7] W.E. Ackerknecht, 111, The scattering of pulses by screens, [14]P. E. Mayes, The equivalence of electric and magnetic sources,IR E Trans. Antennas Propagat. (Commun.), vol. AP-6, pp.Ph.D. dissertation, Purdue Univ., W. Lafayette, Ind., 1972.95-296, July 1958.[8] J. B. Keller and A. Blank, Dlffractlon and reflection of pulses by [15] S . N. Karp and J. B. Keller, Multiple diffraction by an aperturewedges and corners, Commun. Pure Appl. Maflz., vol. 4, pp. in a hard screen, Opt. Acta, vol. 8, pp. 61-72.75-94~ 1971. 1161 J. B. Keller and E. B. Hansen. Survev of the theorv of diffraction

VOI. A241, pp. 71-103, 1948.

J. W. Miles, On the diffraction of an acoustic pulse by a wedge,Proc. Roy. Soc. London, vol. A212, pp. 543-547, 1952.Proc. Roy. Soc. London, vol. A212, pp. 547-551, 1952.,On the diffraction of an electromagnetic pulse by a wedge,W. L. Weeks, ElecfromagneficTheory fo r Engineering Applica-G. E. Barr, On the diffraction of a cylindrical pulse by a half-Zions. New York: Wiley, 1964, pp. 345-350.

. - - _,- of short waves by edges, A&a Phys: Pol. , vol. 27, p. 217-234,1965.P. J. Davis and P.Rabinowitz, hlrmerical Integration. Waltham,Mass. : Blaisdell, 1967.V. I. Krylov e f al., Tablesfor the hlrmerical Integration of Func-

Nauk, Belorusskoi, 1963.tions with Power Singularifies. Minsk, USSR: Izd-vo Akademii

Sinusoidal Reaction Formulation for Radiationand Scattering from Conducting SurfacesN A N N . W A N G , MEMBER, IEEE, JA C K H. R I C H M O N D , SENIOR ~ M B E R ,EEE, AN D MELVIN C. GILREATH

Absrruct-A piecewise-sinusoidal reaction technique is developed forScattering and radiation from perfectly conducting bodies of arbitraryshape. This paper presents he theory an d numerical results for scatteringpatterns of rectangular plates and radiation patterns of corner-reflectorantennas. In all cases,experimentalmeasurements are included forcomparison with the calculated data.

I. INTRODUCTIONT O ME TH OD S re available for electromagneticmodeling of continuous conducting surfaces witharbitrary shape : the wire-grid model [l] and the surface-current model [2], [3] using rectangular-pulse bases. Bothmethods have similar limitations with the maximum cellwidth restricted to approximately A/ lO . Unless the conduct-ing body is symmetric or is a figure of revolution, computerstorage requirements have limited the moment-methodapplication to bodies with surface area not exceeding oneor two square wavelengths.In this p aper, the piecewise-sinusoida l reaction techniqueapplied by Richmond [4], [SI to thin-wire antennas isextended t o scattering by conducting bodies o f arbitraryshape. The continuous conducting surface is divided intocells and he surface-current distribution is expanded in

work was supported in part by the NASA Langley Research CenterManuscript received May 1 , 1974; revised November 2, 1974. Thisunder Grant NGL 36-008-138 to the Ohio State University ResearchFoundation.Laboratory, Department of Electrical Engineering, Ohio StateN. N. Wang and J. H. Richmond are with the ElectroScienceUniversity, Columbus, Ohio 43212.Va. 23665.M. C. Gilreath is with the NASA Langley Research Center, Hampton,

overlapping sinusoidal bases in he direction of curren tflow, and rectangular-pulse bases in the transverse directionas in Fig. 3. This a rrangement satisfies Kirchhoffs curre ntlaw without ntroducing line charges. Via an applicationof Galerkins method, he ntegralequation ormulatedwith the zero-reaction conc ept [6] is reduced to a matrixequation. From a physical viewpoint, this ariation alsolution follows from enforcement of reaction tests with a narray of sinusoidal electric test sources. T he curre nt dis-tribution over the conducting surface is determined viamatrix inversion, and finally the scattered field is obtainedby integrating the surface currents. The reactionechniquepresented here is a moment method based on the electric-field integral equation.The rem aining text presents the theory of the sinusoidalreaction formulation for radiationan d scattering problems.The time dependence e is understood and suppressed.Numerical results a re presented for the scattering patternsof rectangular plates an d corner reflectors and the adiationpatterns of corner-reflector antennas.

11. THE REACTION TECHNIQUEThe eaction concept and its pplication ave beendiscussed by Rum sey [ 6 ] , Cohen [7], Harrington [SI, andRichmond [SI.Consider the exterior scattering problem illustrated inFig. l(a). In the presence of a dielectric or conducting body,the impressed electric and magnetic currents ( J i , M i )

generate the electric and magnetic field intensities ( E , H ) .Fo r simplicity, let the exterior medium be free space.Fro m the surface-equivalence theorem of Schelkunoff[lo], the interior field will vanish (witho ut disturbin g the


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WANG et a[ . SINUSOIDAL REACTION FORMGLATION 37 7

0iM iSOURCE

0; * MiSOURCE(b),/

\(-&kJ') \I \ (E . .Hs)\ \\ \II\REE SPACE,'

\' \

\ \( J,,M,),

\'.-/( 4

Fig. 1. (a) Source (&,Mi)enerates field (E&) with scatterer. (b)Interior field vanishes when currents (J,,M,) are introduced onsurface of scatterer. (c) Exterior scattered field may be generated by(J,,M,) in free space.

0

exterior field) if we introduce the following surface-currentdensitiesJ , = A X H (1)

M , = E x A (2)on the closed surface S of the scatterer. (The unit vectorii is directed outward on S . ) In thissituation, illustratedin Fig. I(b), we may replace the scatterer with free spacewithout disturbing the field anywhere.By definition, the incident field ( E i , Hi ) is generated by( J i , M i ) n free space, and the scattered field is

E, = E - Ei (3)H , = H - H i. (4)

When the surface current (J,,MA radiates in free space, itgenerates the field (E,,H,) in the exterior and (- E i ,-Hi)in he nterior region. This result, illustrated in Fig. l(c),is deduced from Fig. l(b) and the superposition theorem.With the scatterer replaced by free space, we have notedin Fig. I(b) that he interior region hasa null field. Asshown in Fig. 2, we place an electric test source J , in thisregion and h d rom the reciprocity theorem thatf i S (J s .E* - Ms 4 ) s

+ I I I ( J i Et - Mi H t) du = 0 ( 5 )

0i ,MiSOURCEFREE SPACE

Fig. 2. Electric test source J, is positioned in interior of scatteringregon.

where (E t ,Hr) s the free-space field of the test source. Inwords, (5) states that he interior test source has zeroreaction with thether sources. This "zero-reactiontheorem'' was developed by Rum sey [ 6 ] .Equation (5) is he integral equation for he scatteringproblem, andou r objective is to use this quation todetermine the surface-current distributions J, and Ms. Toaccomplish this, we expand these functions in finite seriesso there will be a finite num ber N of unknown expansionconstants. Next we obtain N simultaneous linear equationsto permit a solution for these constants. One such equationis obtained from (5) each time we set up a new test source.The magnetic current M , vanishes if the scatterer is aperfect conductor. We assume a finite conductivity anduse the impedance boundary condition

M, = Z,J, X A (6 )where 2, denotes the surface impedance. If M i vanishes,(5) and (6) yield

- J, . [E , - A x H,)Z,] ds = J i . E, ds (7 )S ISwhere (E,,,,H,) deno tes the free-space field of test-source in .We represent the electric current distribution as follows:

N

where the complex constants I, are samples of the functionJ,. The vector functions 4, are known as basis functions,subsectional bases, expansion functions or dipole modes.We employ expansion functions J, and test sources J,with unit current density at th e terminals.From (7) and (8) we obtain he simultaneou s linearequations

N ZC = A,, with m = 1,2.3; . ,AT (9 )n = l

whereC,, = -f/ , . E , - A x H,)Z,] ds = - J,, * E,, d s

(10)n J-1

i IS(11), = 11 i E, ds = J , .Ei ds.

In (10) and (11) the ntegrations extend over the regionwhere the integrand is nonzero. Fo r example, region ?I is


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318 EEE TRANSACTIONS ON A h T N N A S AND PROPAGATION, MAY 1975nf

Fig. 3 . Electric surface dipole and its current-density distribution. Fig. 4. Nonplanar surface dipolewith edges at s1 nd t l and terminalat 0.

thatporti on of the surface S covered by the expansionfunction J,,. Region m covers the interior test source J,.The reciprocity theorem relates the first and second in-tegrals in (10). In the second integral, E,, is the free-spacefield gene rated by J, and the ssociated magnetic current M,,.Forcom putationa l speed an d storage, it will bead-vantageous to have asymmetric mpedancematrix Cm.Furtherm ore, the test source s should be selected to yielda well-conditioned set ofsimultaneous linear equations.For these reasons, we employ test sources J, of the samesize, shape, and functional form as the ex pansion functionsJ,,. Finally we position the nter ior test sourcesasmalldistance 6 from surface S and take the limiting form of theintegrals as 6 tends to zero. (Numerical experiments showedbetter convergence with Galerkin's method than withpoint matching, and with test sources locatedon the surfacetha n with interior test sources.) F or perfectly conductingbodies, the pre sent ormu lation applies equally well toopen a nd closed surfaces. With a n open surface such as athin plate, J, denotes the sum of the curren tdensities on thefron t and back sides.The next section discusses the electric surface dipoleswhich are em ployed as test sou rces and expansion modes.

111. THE SI~XJSOIDALURFACE IPOLFSA planar surface dipole located on th e JJZ plane is illus-trated in Fig. 3(a). This source is an electric surface-currentdensity with height a and width b . The surface-current

density and ts slope arecontinuo us across the terminala t z = z 2 .Fig. 4 illustrates a surface V-dipole. Distan ce along thedipole arm s is m easured by the coordinates s and t withorigin at the terminal 0. The surface-current density isJ = - $ c o s - o n a r m s( 3

Having defined the sinusoidal surface dipole, one is nowin a position to explain ts relevance. The dipole currentdistribution ((14) and (15)) will be used s the asis function s( J , in (8)) for expanding the unknown current distributioninduced on a onducting surface. Furtherm ore, surfacedipoles will be employed as test sources with the reactionconcept to solve the integral equa tion.

IV. THE MFEDANCEATRIXFrom he viewpointof reaction, thecomplexnumberC,, in (10) represents the reaction between two sources.Let (m,n) denote the reaction between sources rn and n ,then (10) can be written as

C,, = - (m ,n > = - J,,, - E, ds.ssn (16)It has been pointed out [6] that the reaction between twosources is related to the circuit param eters bydensity is C m n = Vdmm (17)

where V,, is the open-circuit terminal voltage induced a t mby source n, and Imm s the terminal current of source mwhen it transmits.= cos (n(Z - ; , fo r z 2 I 3 (13) Although the surface dipole described in S ection I11 is a

J = 2 cos ( - for z1 s z 5 z2 (12)-2(z3 - hypothetical source, it is useful t o define itself-impedance

2 ' (1 8)As illustrated in Figs. 3(b) and 3(c), thecurrent densityvanishes at the edges z = z1 and z = z3 , and is uniformlydistributed inhe transverse direction. The surface-current I- 4 l m

with the induced emf formulationZ m m - V m m - C m m


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et a l . SINUSOIDAL REACTION FOILMULATION 379TABLE ISELFMPEDAWCEF CENTER-FEDLAWARSURFACE DIPOLEHOWN INFIG.5

0.2 11.4 8 -69.760.3 23.98 -35.260.4 38.80 -15.36

52.98 - 4.52

Pa-+Fig. 5 . Surface dipole with J = 2 cos (nz/a).

TABLE IIMUTUALMPEDAWCEF CENTER-FEDOPLANARSURFACEI P O WSHOWNN FIG.6a/x. = 0.5 b/). = 0.25 I

0.75 0.8871+j 7.0195.968+j 1.7612.925-j 5.950.076-j 7.913I 0.50 124.43 +j 5.997 110.04 -j 9.321 I -9.568-j 6.65 I -7.194 ij .971 II I I0.25 -15.74 + j 6.4269.468-j19.339.54 - j 9.07053.10j55 .87

0.0 67.09 +j13 .23 139.07 -j22. 49 I -8.659-j26.75 1-19.32 +j 4.263I I I0.0 0.75.50.25

/ YFig. 6. Coupled surface dipoles. /rom ( I@, (18) yields XFig. 7. Plane wave ( E i , H i ) iluminates electric surface dipole.

Themutual mpedance between twosurface dipoles is

Zm, 5 / J , .E, d s . (20)Fig. 6 illustrates a pairof center-fed coplan ar surfaceipoles, andTable I1 fists themutual mpedance Z12.Here sz and sy specify the relative position of the dipoles.h e impedances in Table I and I1 were calculated with aombination of analytic and numericalntegrationtechniques.

iven byI m J m m

V. THEEXCITATIOXOLUMNThe complex quantities A , in (1 1) form the excitationcolumn in the matrixequation C,,I, = A,. Physically,, is th e reaction between the impressed source and testdipole nz. From (]I), A m s given by

A , = / / J , , , Ei ds.m

where (r,Oi,$i)are the spherical coordinates of the sourceand Eo is the incident lectric field intensity at the coord inateorigin 0.ig. 7 illustrates an incident p lane wave illuminat-ing an electric surface dipole located on the yz plane w ithheight a and width b. F or the surface-current density J =2 cos (nzla);,21) can readily be evaluated to yield

A , = (Eo 2)2nab sin ( X i )cos (Yi/2)Xi(Yi2 - n2) (23)where

X i = 0.5kb sin O i l sin q5iYi ka cos Bi.

VI. FAR-FJELDADIATION ND SCATTERINGTh e field scattere d by a perfectly conducting body may begenerated by the electric surface-current density J, in freespace. To obtain he otal field oneadds he free-spacefield ( E i , Hi ) f the electric source Ji.Consider a n electric surface dipole with current densityJ , = 2 cos (nzla) located on th e yz plane as shown in


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380 IEEE TRANSACXIONSON ANTEhWAS AND PROPAGATION, MAY 1975

100

a?a 100t0W

THICKNESS 0.0317xL0.0 I I I I I I I0 0.2 0.4 0.6 0.8 1.0 1.2 1.4EDGE LENGTH 'x

Fig. 8. Backscatter cross section of erfectly conducting squareplate for broadside aspect.

Ill[6

II

Fig. 9. Electromagnetic modeling of plate.

I I I-0 .5 -0.25 0 0.25 0 . 501

Fig. 10. Amplitude of surface-current density induced on perfectly-Conducting rectangular plate for plane wave incident at broadside.

8 I EGREES)Fig. 12. Normalized backscatter cross section in y z plane of rectan-g u l a r plate. oeeS(B,) = 26.25 dB at (90",90").

Fig. 13. Normalized backscatter cross section in xy plane of rectan-gular plate. uss(8, ) = 26.25 dB at (90',90").

-180 -150 120 -90 -60 -30 0 30 60 90 120 150 Ie ( DEGREES ] 0Fig. 14. Normalized backscatter cross section in yz plane of cornerreflector. ooo(B, ) = 10.62 dB at (90',90").Fig. 11. Phase of surface-current density induced on rectangular platefor plane wave incident at broadside.


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et al. :mwsomAL REACTION FOILMULATION 381REACTION- EASUR S =0 .7 5 X (1=12O0=2.75X W = X

Relative gain in E-plane of corner-reflector antenna. G(B,4) =4.05 dB at (90,90).

/ / I I \ \16 . Relative gain ni H-planeofcorner-reflector antenna. G(B,+4)=4.05 dB at (90,90).

7. From reciprocity, the free-space electric fielda distant point ( ~ , 6 ~ , r $ ~ )sj o p sin ( X , ) co s ( K / 2 ) e - j k r

2 X, ( K 2 - 2 ) r E = 8,- b ~ sin 0, (24)X, = 0.5kb sin eSsinY, = ka co s 6;.

he 8- and $-com pon ents of the scattered field with respecthe reference coordinate system 0 can be obtainedVII. NUMERICALESULTS

Fig. 8 presents the backscattering echo area of a squarewith perfect conductivity for hebroadside aspect.the reaction calculatio n, the plate is divided into cells,modes were employedas illustrated9. In this case the ransv erse curren t is neglectednd 45 modes wereused for he current distribution. (It

TABLE IIINUMBERF MOD=USED SCALCULATIONSNumber of Modes

1 0 , l l 3312 551 3 7514 3815.16 61

was found that good results can also be obtained. with amuch smaller numbe r of modes.) F or comparison, Fig. 8also shows the experimental measurements of KouyoumjianThe magnitude an d phase of the current density inducedon a perfectly conducting rectangular plate are illustratedin Figs. 10 and 11. This is the sum of the curren t densitieson the front and back surfaces of the plate. Figs. 12 and13 show the normalizedback scatter cross section of a

rectangular plate. Fig. 14 shows the normalized backscattercross section of a corner reflector. The title of each figuregives the echo areaat he broadsideaspect n erms ofdB = 10 log (~ / i . ) .Figs. 15 and 16 showhe gain f a comer-reflectorantennanheE-plane ndH-plan e, respectively. Fo rcom parison , these figures include xperimentalmeasure-mentsobtained at NASA LangleyResearchCenter. Inthe measurements the receiving antenna was linearlypolarized in he he ta direction. Similarly, the calculatedgain is based on E,. Th e dipole length is i./2 and the radiusis 0.005i..Table I11 lists the numbe r of modes used to obtain theresults given in Figs. 10-16. In each case, the m atrix sizeis equal to the num ber of modes. In Fig. 14, the verticalcurrent J, was expanded in 18modes and he horizontalcurrent in 20 modes. Fo r convenience, the orizontalcurren ts were neglected in the oth er calculations.

[111.

VIII. CONCLUSIONSTh e reaction concept and Galerkins m ethod a re used todevelop a new formulation for perfectly conducting anten-nas and scatterers. Numerical results are presented forscattering and radiation from rectangular plates an d corner

reflectors. The results showeneralgreementithmeasurements.The techniques can be applied to surfaces with finiteconductivity an d arbitrary shape. or arbitrary polarizationand aspect, 12 modes per square wavelength are adequate.If this performance is maintainedfo rcumedconductingbodies, it will epresent a significant imp rovem entoverprevious techniques.REFERENCES

[l] J. H. Richmond, A wire-grid model fo r scattering by conductingbodies, IEEE Trans. Antennas Propagar., vol. AP-14, pp. 782-786, Nov. 1966.[ 2 ] F. K. Oshiro, Source distribution technique for the solution ofgeneral electromagnetic scattering problems, in Proc. FirstG I S A T S y m p . , vol. 1, part 1, Mitre Corporation, 1965.


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382 IEEE TRANSACTIONS ON ANTENNAS AN D PROPAGATTON,VOL. AP-23,XO. 3, MAY 1975[3] D . L . Knepp, Numerical analysis of electromagnetic radiationpropenies of smooth conducting bodies of arbitrary shape inthe presence of known external sou~ces,Ph.D. dissertation,University of Pennsylvania, Philadelphia, 1971.[4] J. H . Richmond, Admittance matrix of coupled V antennas,ZEEE Tran s. Antennas Prop aga t. (Comun.),ol. AP-18, pp.[5] P. K. Agrawal, G. A. Richards, G.A. Thiele, and J. H. ichmond,820-821, Nov. 1970.Analysis and design of TEM-line antennas, ZEEE Trans.Antennas Propagat., vol. AP-20, pp. 561-568, Sept. 1972.[6] V. H. Rumsey, Reaction concept in electromagnetic theory,Phys. Rec. , vol. 94 , pp. 1483-1491, June 1954.[7] M. H. Copn, Application of the reaction concept to scatteringproblems, IRE Truns. Antennas Propagat., vol. -3, pp. 193-199, Oct. 1955.

[8] R. F. Hamngton, Time-Harmonic Electromagnetic Fields.New York: McGraw-Hill, 1961, pp. 340-345.191 J. H. Richmond, A reaction theorem and its application toantenna impedance calculation, IR E Trans. Antennas Propa gat.,[IO] S . A. Schelkun$T, On diffraction and radiation of electro-[ l 13 R. G . Kouyoumjian, The calculation of the echo area of perfectlymagnetlc waves, Phys. Rm., vol. 56, Aug. 15 , 1939.conducting objectsby the variational method, Ph.D. dissertation,Ohio State University, Columbus, 1953.[12] J. H. Richmond, Computer analysis of three-dimensional wireantennas, Ohio State University ElectroScience Laboratory,

Dep. Elec. Eng., Rep. 2708-4, Dec. 22, 1969; prepared underContract DAADO5-69-C-0031 for Aberdeen Proving Ground,Maryland.

V O ~ .AP-9, pp. 515-520, NOV.1961.

Some NewResults on Propagation of Electro-magnetic Waves in Strongly Turbulent MediaR O NA L D L. F A N E , SENIOR MEMBER, IEEE

Abstract-New results for the covariance of the intensity fluctuationsofa plane wave propagatingin a strongly turbulent medium are presented,and the consequencesof thesenew results on aperture averaging, angleofarrival fluctuations, and the temporal frequency spectrum f the intensityscintillations are evaluated. The differencesetween he strong turbulencean d weak turbulence cases aie generally quite s i& w t .

I. INTRODUCTION

CNSIDERABLE EFFORT hasbeen devoted recentlyto he study of the variance and covariance of th e

intensity fluctuations of a w ave propaga ting in a ran dommedium . T he impetus or these studies was provided by theexperimentalmeasurem ent of Gracheva et al. [l],whofound hat he variance o12 of the intensity fluctuationssaturated for very strong turbulence, in disagreement withtheoretical predictions [ 2 ] , [3]. In he ast year severalphysical models ave been pos tulate d o calculate thevariance and covariance of the intensity fluctuations [4], [ 5 ] ,but both rely on the introduc tionof an arbitraryparameterwhich can not e specified xcept y com parisonwithexperimental data . In con tras t to these other theories wehave recently [6] developedanewmod el which s freefrom these limitations. To outline and amplify on ourtheory let us consider the four th mom ent of the electricfieldr4 = (e(zYpl)e(zYp2)e*(z,p,)e*(zYp2)~ ( 1 )

where z is the distance measuredalong the direction ofpropagation and p is the distance measured transverse tothe p ropaga tion pa th. atarskii [ 7 ] has derived the equationsatisfiedby r4, ut it has not proven possible to obta insolutions, except in he limit ofweak turbulence. Ourte-chnique is to obta in an iterative solution to this equation

starting with the solution for infinitely stro ng turbulence.For o12= 03,where o12= 1 .23k ,7 /6C, ,2z11 /6 , k ,s thesignal wavenumber and Cn2 is the index-of-refractionstructure constan t, it c an be seen [ 8 ] that the statistics ofthe field are Ga ussian , since the fi$d at (z,O)consists of theindependent contributions from very-m anyoff axis eddies.We emphasize tha t th e statistics are exactly G aussian onlyfor o12= co.For finite g I 2 we have shown in A ppendix 1that the statistics of the field cannot be Gau ssian. In fact,for 0 I 12I 100 it ap pea rs that the field statistics arevery nearly log-norm al [ 9 ] . Consequently for o12= co wehave (see Appendix 2) from ( 1 )

. .

where r2 s the mutual coherence function andr1 = (pl-If we use (2) as a lirst iteration in the integral equationderive d by Tatarskii [ 2 , eq . (72.1)] for the Fourier transform

of r4 we can show for a Kolmo gorov spectrum that thecovariance

P 2 + P1 - P2)/2and p 2 = ( P I - P 2 - P1+ P2)/2.

of the intensity fluctuations of aplane wave in homo-gene ous turbulen ce s given by [6]

2 - 2 1 5 R+ f3 [r;;;ii.j]The.author is with the Air Force Cambridge Research Laboratories ( ~ , 2 , - 2 ~ 5 g [ ( u , 2 ) 3 1 1 ~ ]btained in [q since this contributes to Br onlyManuscript received October 7, 1974. Notehat for simplicity of presentation we have neglected the term(LZP),,Hanscom AFB, Bedford, Mass; 01731. in the transition region between the fist and second te r m in (4).

Sinusoidal Reaction Formulation for Radiation and Scattering From Conducting Surfaces

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Transcript of Sinusoidal Reaction Formulation for Radiation and Scattering From Conducting Surfaces