AFAL-TR-74-119 011764-2-T 11764-2-T = RL-2223 Air Force Avionics Laboratory Air Force Systems Command Wright-Patterson Air Force Bas'e Ohio ' ''.,.' ~' - ''''',...'. " "' '-'~ '''. '- '. '; ~ Scattering from Two-Dimensional Bodies witi Absorber Sheets VALDIS V. LIEPA I 6.J '^: EUGENE.. NE F..KNOTT..tr ' THOMAS B. A. SENIOR.:.;.....,-.. * -;.. ' "..,' ',i,".: ':, ~~ 5~ 'b ob rS e t -'..::''.''-, "'{","..,?'.. " 2'" ".'..'-'.~ " -...-;.....'- ' ~',. ~ -..'i....:..,. ~ ~ i~ '' ' "r;~ ~~ nr~ ~ I ~~r i:: I - ~.i~~~~'~.~~`~ '.: i:,,I ~~ I~:~ r ' ~r": .F~"r* ~~ 'd C~~IYI. 1.:'i', ~~ ~; .~ ''~'r~ ~ '~) 4 h -.t:..;,~... ~ll~IL- ~~ '~~~t-r: t '5 :~, 1,f~l,= ~-t:~ r, ~9~' '-~ ~.Ptri n' i I.~~. r~?~ ~.~ ~ ~ r ~/ ~~.. ~ ~, ~:;:~ "; _.... '~c:' ' I'I. 'r C p ~~ a~'j~ ii' b... V,.. I. r May 1974 Technical.!i;. };i Ii il~ i i. }' ''r i i:...... *- *'. i. '':' t'-" ',"....; ' ~ '"'! '; '-"? " - '. ','. ',4 'r-'.':?.' - *;>. t, ^< 9...,;.. —..; * Report AFAL-TR-7.4-119;r f.....:^.,-; ~., ~......,,... ~,~.......'. '. ';~,-. -. t'.........-,. *,,.,,.:.. ~ ~...... *;r -:.. ~...., ~~ ~,... ~. ',.:.;..... 'i..... '.* '~- ~:.-':~~;,~ i~ r. i' E 1~. i C'.- j i '.r. Distribution limited to U.S. Government Agencies only; Test and Evaluation Data; May 1974 Other requests for this document must be referred to AFAL/WRP Administered through: DIVISION OF RESEARCH DEVELOPMENT AND ADMIISTRATION A N N AR BR

NOTICE When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever; and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications, or other data, is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto. Copies of this report should not be returned unless return is required by security considerations, contractual obligations, or notice on a specific document.

SCATTERING FROM TWO-DIMENSIONAL BODIES WITH ABSORBER SHEETS Valdis V. Liepa Eugene F. Knott Thomas B.A. Senior Distribution limited to U. S. Government Agencies only; Test and Evaluation Data; May 1974. Other requests for this document must be referred to AFAL/WRP.

FOREWORD This Interim Report describes research performed by The University of Michigan Radiation Laboratory, 2455 Hayward Street, Ann Arbor, Michigan 48105, under USAF Contract F33615-73-C-1174, Project 7633, Task 7633-13, "Non-Specular Radar Cross Section Study. " The research was sponsored by the Electromagnetic Division, Air Force Avionics Laboratory, and the Technical Monitor was Dr. Charles H. Kreuger, AFAL/WRP. The computer program described in the report was developed during the time period 15 October 1973 through 15 March 1974 and was written by Dr. V.V. Liepa. This report has been assigned Radiation Laboratory Report Number 011764-2-T for internal control purposes, and was submitted by the authors for sponsor approval on 25 March 1974. This Technical Report has been reviewed and is approved for publication &f ' f-. vJ. —,. Cr; CiL',-,, 'Lc':2'oic iVarfare Dfvisflon ii

ABSTRACT This report describes a program that computes the far field scattering pattern of a two-dimensional cylindrical body (or bodies) treated with absorbing materials. The body surface is assumed to satisfy an impedance boundary condition and the absorber is modeled by equivalent electric and magnetic sheets. The program reduces the coupled integral equations governing the surface currents to a system of simultaneous linear equations, solves for the unknown currents, and then computes the far field pattern from the solution. The integral equations derived and presented in a previous interim report were used as the basis for preliminary versions of the program, but these equations have been found to be in error. The required correction consists of incorporating terms previously omitted and the corrected equations are presented and discussed. Careful attention is given to a description of the input data necessary to run the program and the results of a sample run are included for illustration. The program was developed to explore the effects of absorbent materials on the scattering of electromagnetic waves by edges. Programs used previously in similar explorations embodied a surface impedance boundary condition; for lossy materials covering smooth surfaces of large radius of curvature, such an impedance can be estimated from the layer thickness and material properties, but the relationship breaks down near edges. The program described in this report, however, models the effects of actual materials rather than using the nebulous surface impedance boundary condition. iii

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TABLE OF CONTENTS I INTRODUCTION 1 II MATHEMATICAL FORMULATION 4 2. 1 Amended Sheet Equations 4 2. 2 The Required Equations 9 2. 3 Numerical Procedures 13 III PROGRAM DESCRIPTION 21 IV LIST OF FORTRAN SYMBOLS 28 V USER INSTRUCTIONS AND SPECIAL CONSIDERATIONS 31 5. 1 IBM-CDC Compatibility 31 5.2 Sample Run 32 VI CONCLUSIONS 45 APPENDIX: PROGRAM LISTING 47 REFERENCES 69 v

I INTRODUCTION This report describes a computer program (RAMVS) that solves the two-dimensional integral equations for the currents induced upon the surfaces of cylindrical bodies and thin sheets. The fields on the surfaces of solid bodies are assumed to satisfy an impedance boundary condition and the sheets are assumed to be electrically or "magnetically" resistive. Electric and magnetic sheets can be superposed and, if their resistivities are properly chosen, can be used to model the effect of a layer of absorbing materialihaving arbitrary permittivity and permeability. The program reduces the coupled integral equations to a system of simultaneous linear equations, solves for the unknown currents, and from these computes the far field scattering patterns. RAMVS was developed as a tool to study methods of reducing non-specular scattering and has been used to compute the scattering patterns of simple shapes treated with resistive sheets. As such, it was always a single body that was specified on input. However, the program "understands" only two kinds of surfaces-thin resistive sheets and impedance surfaces-and cannot distinguish between a single body and multiple bodies, as long as all are identified with one class of surface. Thus, although RAMVS has not been used for multiple impedance surface scatterers, it well could have been. Similarly, the program views disjointed resistive sheets as a single surface even though several distinct sheets may be specified on input. These sheets need not be attached to the body or bodies, and in fact the physical separation could be hundreds of wavelengths if, for some reason, such a configuration were of interest. The program allows the impedances and resistivities to vary from point to point along the profile according to a selection of mathematical distributions provided internally. Metallic bodies and sheets may be modeled by simply specifying a constant surface impedance or resistance of zero. Any combination of sheets and body profiles may be specified on input, *but the configuration 1

must be symmetrical. Only half the profile should be specified because the program generates the other half by imaging about the plane of symmetry (y = 0). For running the program on a typical installation (-%60 K word core), the total perimeter of all surfaces should not exceed ten wavelengths or so. This provides for a sampling rate of 8 or 10 points per wavelength and a maximum of a 150x 150 element matrix to be solved. Because of coupling between the integral equations, the maximum number of sampling points must include those on the resistive sheets twice; for example, a body having 100 sampling points, in the presence of a sheet with 25, would represent a total of 150 unknown current elements. RAMVS is based upon the integral equations derived and presented in a previous Interim Report (Knott and Senior, 1974) and the pertinent equations are repeated and rearranged in a form more appropriate for numerical computation. Late in the development of the program, errors were discovered in the previous: integral equation formulation, resulting in the omission of some terms. The necessary correction is discussed in Chapter II, and the corrected equations are given. Both the previous and corrected form of the equations contain higher order singularities which require special numerical treatment, and this treatment has been covered in a previous report. A description of the code is given in Chapter III and provides the reader with an overview of the functioning of each subprogram. For readers who wish to examine the details of the programming, a source listing of RAMVS is given in the Appendix. A list of 'FORTRAN symbols may be found in Chapter IV to assist the reader in relating the program variables to the mathematical description of the integral equations. Detailed instructions for the preparation of input data are presented in Chapter V and it is assumed that the user has some familiarity with FORTRAN data formats. The form of the input deck is illustrated by means of a specific example, and the preliminary calculations and decisions that must be made before the deck can be prepared are discussed. The required content of each 2

card is described and a sample run is displayed to show the results of the program for the specific geometry chosen for illustration. RAMVS has been successfully run on the IBM 360/67 computer at The University of Michigan and on the CDC 6600 computer at WPAFB. It was tested for special cases, such as isolated impedance bodies and isolated dielectric strips, by means of comparison with other programs that have been verified, as well as with measured data. However, the program has not been thoroughly tested for general, absorber-clad bodies due to the lack of any data for comparison, and the user should be cautious and critical in surveying the computed results for such cases. The authors would appreciate learning about any comparisons of computed results with other data that either demonstrate the program's capability or that show its weaknesses. 3

II MATHEMATICAL FORMULATION We are here concerned with the problem of a plane electromagnetic wave incident on a structure consisting of a cylinder of arbitrary profile in the presence of a generalized resistive sheet of infinitesimal thickness. The entire structure is independent of the z coordinate of a Cartesian coordinate system (x, y, z) and the plane wave is incident in the xy plane with either the electric or magnetic vector in the z direction, corresponding to E- or H-polarization respectively. The problem is therefore two-dimensional and the geometry is depicted in Fig. 2-1. The profile of the (closed) cylinder is designated C2 and here an impedance boundary condition is imposed. The surface impedance rl is arbitrary and can vary as a function of position. The resistive sheet profile is designated C and may consist of several disjoint parts, each representing an individual sheet. The boundary conditions at the sheet are specified by the bulk permittivity e, the bulk permeability p and the thickness A of the material being modelled. Special cases are those of electric or magnetic resistivities alone. RAMVS is a computer program for the digital solution of the coupled integral equations for the currents induced in the structure, from which the scattered fields are then determined. The integral equations were derived in a previous technical report (Knott and Senior, 1974), but owing to an error, some of the equations presented there are invalid for the type of sheet now considered. We shall therefore begin by detailing the nature of this error. 2. 1 Amended Sheet Equations In the study of the integral equations for thin sheets which are electrically or magnetically resistive, or which have an impedance boundary condition imposed at the surface, an analytical error was made in Knott and Senior (1974) which invalidates the results for simple nonplanar sheets and for multi-sheets which are not co-planar. 4

#A? 3 - foo FIG. 2-1: Geometry for the composite scatterer. C: Absorber sheet, described by e, p an4 the thickness of the material r C2: Impedance body described by arbitrary surface impedance rl. 5

The error first appeared in the reduction of eq. (2. 24) for a thin impedance boundary condition sheet with H-polarization. Subtraction of the two limits as the observation point p approaches the sheet from above and below does, indeed, produce an identity for J2(s), but on addition we have + + H -s) = l(s)+ L i r (sJs(k)dksI)-1 J2. T (*HL )(k)dks'). (2.27': 2.1) This differs from the equation originally obtained due to the presence of the second integral. As indicated by the slash, this is a Cauchy principal value integral, since the self-cell contributions cancel on summing the limits. Similarly, eq. (2. 29) must be replaced by i 1 lim 1 a Y E (s) - 2 (s)J,(s)+ 4k an J2(s)(' 1)(r)d(s) 1 i (s )J (s)( - r)H(1) (kr)d(ks') q(2. 29':2.2) where the normals are all with respect to the upper surface. The additional terms in (2. 27') and (2. 29') vanish identically for a single planar sheet since then n * r = n' - = 0. In this particular case, eqs. (2. 27) and (2. 29) are correct as they stand, and are decoupled integral equations for J (s) and J2(s) respectively. In all other cases, however, the equations are coupled and both must be solved to determine even the total electric current, proportional to J2(s). The analogous equations for E-polarization can be obtained using duality and are 6

YoE (S) = (s(s)Js)+ (s')H (kr)d(ks') z 2 4 JC( iY A -4 J4(s) (n * r)1 (kr)d(ks') 4 4s ') ' ) 1 JC (2.32': 2. 3) and Z H i (s) 1i 1 (kr)d(iks ') JcA Z0O s - 2t(s) J4(s)- p-C 4kan J4( (kr)(ks C+ iZ 0( + J3(s')(n' H(1)(kr)d(ks') (2. 33': 2.4) C+ where Y0 + J(s) -(E +E ) 3 n(s) z z J4(s)=E -E The added terms in (2. 32') and (2. 33?) couple the equations, but vanish for a single planar sheet. For this reason, the later equations in Chapter II of Knott and Senior (1974) are correct as they stand. These same errors recur in Sections 4. 2 and 4. 3 where generalized (i. e., multiple) resistive sheets are considered, but do not affect any of the equations or discussion through page 33 in Chapter IV. However, eqs. (4. 32) and (4. 33) for a generalized resistive sheet with E-polarization are in error by the absence of the coupling terms and should read 7

Z0H (s) = Z R (s)J (s) J (si(n'r)H (kr)d(ks) -s 0 s — C 4k On s C izo +4 (sA)(. n )H( )(kr)d(ks) C (4. 32: 2. 5) Y E (s) = Y R(s)J2(s) + J (s!)H1)(kr)d(kst) C iY0 2 + - Js'n)('. )H (kr)d(ksr). C (433':. 2.6) These terms must also be introduced into eqs. (4.43) and (4.44) for a generalized resistive sheet in the presence of an imperfectly conducting body, and the equations then become YoE(sl) = YoR(s)J (s+ J (s')H (kr1 )d(ks') C iY0 +(1 (1) + J(s')(n' rH (kr)d(ks') + 1K (s') Jc JC2 -4 (s ')Kzs'' r 2H (kr 2)d(Kks'), (4.43t:2.7) C 8

Y___ S 1s 1 e ->1 4k an1 s 011 01 (H ) d s)H+ (1) + i Jz(s')(. 1r)H1)(kr 1)d(ks')+ 4 Kz(s )(l1 (kr1)d(ks') C 1 1C2 (k )K (SI) - {n r1)H (kr2d(ks') * (4.44t: 2. 8) 2 We note in passing that the incorrect eq. (4.42) is the root of all these errors. The set of three equations necessary for the complete solution of the problem consists of (4. 43'), (4. 44') and one of (4. 45) and (4. 46). For reasons stated previously, the more convenient combination is (4. 43'), (4. 441) and (4.45). The additional terms in (4.43') and (4. 44') vis-a-vis (4. 43) and (4. 44) vanish if A A A, A n.r =n'.r =0 i.e., all the sheets are co-planar. Until recently, this was the only case where the equations had been studied in detail and numerical data obtained. For a single electrically resistive sheet (R* oo, implying J* 0) and a perfectly conducting S body (n = 0), eq. (4.43') reduces to (4. 47), so that the equations (4.47) and (4. 43) used in program REST are not affected by the error. 2.2 The Required Equations The integral equations used in RAMVS for E-polarization are (2. 7) and (2. 8) above and eq. (4.45) of Knott and Senior (1974). To facilitate the programming, it is convenient to rearrange the equations and to change the notation slightly. The equations then read 9

i ~1 C') Y0E (s )z (s )J (s ) +- J W)H 1(kr)d(ks?) 0 z 1 e 1 z 1 4 0 1 iYo 1 C (1) + 'K(st)H0 (krddkkst) + K (s9KH(s9 (krv)H (1)(k dk)s27 2 Hi) 1I( )K( 4*J(s')(n1r)H(1 (kr)d(ks) +Z )J(s )i ( c1 lrn 0O a *x (1), + J n(~)i.r) H (kr) d(ks91 P- — ~ 4k an s1 (S)(n A (1) 2 2 10

i 1 Y E (s ) = O z 2 4 C (1) (1)(kr')+ 0 id(,ks) () Jz(sI) H (krs ks)+ 1 k)dks) 1 1 NC-, 1T s K ( 1 2 "2 z2 '4 K (sH ')t 1) (kr) d(kst1) C 2 - r (s')K (s9)(al - )H1 (k)r)d(ks') c2 (2. 9) The analogous equations for H-polarization can be obtained using duality. This requires the application of the transformation z z z z i I 5 5 J -*-J 5 5 ti-4> 1/tj K-:>K" = rjZ K z z O s to the above equations and gives Z <E-*>Z e m i y0 z 1 m 1 O z 1 4 J" sIH(1) (kr)d~ks) 0 Cl1 -pc j S,. A,.0 (1) 1 C (s n(~r) H 1(kr) d(ks'1)+4 ' c2 YI~s)K (')H(1) (rdk' 5 0 I -I - i K S (9A,.r)H1) (kr) d(kst1) 2 (2. 10) 1 1

iY C s o 1 1 s 4 1 Z lm s e 1 J(s')(nr)H 1 (kr)d(ks') 1 + r4 1 r(s')Ks(s') r)H (kr)d s') s 1 1 )11 C2 - K (s')K (s( )H )H (kr)d( ). (2. 1 4 1 C2 C2 Y H ( J"'(s')H (kr)d(kst) J (s ')('1) + K (s + rI(s')K(S)H + (kr)d(ks') CC2 The three equations in each set are coupled by the unknown currents: J (s) or J (s), and J*(s) or J*(s), representing the total electric or magnetic z s s z currents borne by the resistive sheet, and K (s) or K (s) representing the surz s face electric current density on the cylinder. The normalized electric and magnetic sheet impedances are defined as 12

= i, (2. 13) e 27r(e - 1)A/k m 2 - 1)A/ (24) r where A is the thickness of the sheet and X is the free space wavelength. These parameters are used in preference to the resistivities R and R of Knott and Senior (1974) since they are of order unity for values of c, pr and A/X of practical interest. In addition, they are dimensionless and in this respect are compatible with the surface impedance rl of the cylinder., Since the two sets of equations are related via the duality transformation, it is not surprising that the corresponding equations are mathematically similar. We can therefore describe the manner in which the various integrals are handled by using the equations for E-polarization as an example. 2. 3 Numerical Procedures To develop the computer code, and especially the sub-programs for generating the matrix elements, it is convenient to write the E-polarized equations as YE (Sl) =A1J (s)+A2Js(s )+A3K (s) (2 15) O z 1 lz 1 2s 1 3z2 H (s) =A J (s )+A Js((sl)+A K (s2) (2 16) si 4z 1 5s1 6z 2 where the A's are integral operators whose specific forms are evident from eqs. (2.7) through (2. 9). Thus, for example, AJz ) = Z (s)J (s+ J(sH (kr)d(ks) 7l3

iY f AJ(s ) =- J(s)(n'-r)H1 (kr)d(ks') 2 s 1 4 S Jc and so on. The standard practice for evaluating the integrals is to divide each range of integration into segments (or cells) and compute their contributions assuming a specific type of variation of the unknown current over the cells. In prior programs we have, for simplicity, used the elementary "flat-top" approach in which the current is assumed constant over a cell. The contribution from each cell is then proportional to the sampled value of the current at its mid-point. However, several of the integral operators have singular kernels and the "self cell" containing the point of observation then requires individual treatment. For all except one of the singularities, the method that we employ is relatively straightforward, but in the exceptional case, the higher order singularity involved necessitates a special treatment for adjacent cells in addition to the self cell. The reasons for this are described in Knott and Senior (1974; Appendix A) where the details of the treatment can be found. We now examine in turn each of the integral operators A 1 through A 9 1 9 and show how the numerical computations are performed. The kernel of the operator A has a logarithmic singularity, and by approximating the zero order Hankel function, the self cell contribution can be evaluated analytically. The result is F A(s1) A(s) \fl c 7r A (1 e(s) + L +i (n + 0. 02879837)jJ (s1) 1 z 1 X 2 z, + ( (s9H \kr)d(kst)+ J (st)H ()kr)dks, (2. 18) ^~2A Jc -(A 2A) 14

where the bracketed terms represent contributions by the self cell. The first integral on the right side is evaluated using a three-point Simpson's rule b a f(x)dx - b-a {f(a) 4f +f(b)2 6 \ 2/ and the integral over C1 - (Ai2A) represents the contribution from the remainder of the surface C1. The integral is actually coded as a discrete summation over all the cells i f j, but we have found it just as convenient (and less confusing) to refer to it as an integral in the coding process. The integral in A2 is a Cauchy principal value, and has no self cell. Since H (1) (kr) - H (kr) 1 a(kr) 0 iY0Q A J'(s ) = 2s 4 and a A a a(kr) r)a(ks') J '. A kt (1) ds) J"'(s') r)H (1 (kr)d(ks') s 1 can be reduced to the form iY 2 J(s ) ( )( ' * a(') '_) 'H (kr) d(ks') 2A iY0 + -40 J ' c -(A- +2A) - r *(s ')(n '. )H() (kr)d(ks1). ( 1 (2. 19) The first integral is evaluated over each cell as iY b(A? IiYo A A (1) b() - ^ J'(s.^..'.* (s.r )H0 (kr) 4 s j j ij j ij 0 a(a.) J 15

where a(A.) and b(A.) are the endpoints of the source cell j, and the second J 21 integral is evaluated as in (2. 18). In A3 the integration is over the surface C2 with the observation point on C. There is again no self cell and the integral A3K (s) = K, ) ( kr) (- ir(snrH (kr)} d(ks') (2. 20) 3 z 2 4 K H0 (kr)-i~(s9(.r)H1 C2 can be evaluated directly. The operator A4Jz(s 1) " -A r (s')( r)H(1)(kr)d(kst) z 1 is treated similarly to A2, giving A4Jz(s ) = - J t)(n r r) (S) H1(kr) d(kst) 4 z 1 4 a(ks2A +2A + i 5 J (s)(n r)H(1)(kr)d(ks'). C 1- (A7 12A) (2.21) The singularity in Y 4-,lim y0 1 A5J(s) =Z (sJs"(s)+ l 4m Y0n \ 5 s 1 m 1 s 1 -1 >. 4k n J(s)(n'. r)H( )(kr)d(ks1), s 1 1 I 1 however, is the most difficult one. The differentiation with respect to the normal effectively increases (by 1/r) the order of the singularity of the kernel, but since the same type of operator also occurs in the integral equation for an electrically resistive sheet with H-polarization, we can use the representation previously developed (Knott and Senior, 1974) for that equation. We then have 16

AiY A(s) F A(s) f A J(s ) (S + + +i n +0.02879837 J(S ) mr 1 7r2A/ 0 A r A (1) + 4. J(s') W '. W)H( 1) (kr+ ( d(ks 4 s 0 3(ks') ~2A Y r r 0 C * (1 ) 4 s 0 + - \ J (s) n. r)(n. r)H (kr) C1- (A-2A) + S )(s) - (n )(n. ] (kr d(kst), (2. 22) I1kr ) where the terms in brackets are produced by the self cell, the next integral is the contribution from the two adjoining cells on either side of the self cell and the last integral is the contribution from the remainder of C1. In the computer code the first and third parts on the right side of (2. 22) are programmed as shown, but the second part is expanded further. The contribution of a single source cell A. to an observation cell A., i f j, can be written as Y Y Jn)(n n) H (kr)d(ks') + J"(s )(s. r.)H (kr) 4 s j J 1 4 1 i ( j 1 JA a(A.) J J (2.23) where a(A.) and b(A.) are the endpoints of the jth cell. The integral over A. J J J is evaluated by a three-point Simpson's rule. The operator A6 is relatively simple since the observation point is on C1 and the source point on C2. Only the derivative factor needs expansion and this gives A6 z( 2) 4k r(s')K (n' r )(n H^ (kr) k2 H ((1 (kr) kr ' ' 17

The operators A7 and A8 are programmed without modification: 1 A7J (s2 J (s')H((kr)d(s') (2. 25) iY 0 and with the final operator Ag, the self cell contribution must be removed from the first of the two integrals. The procedure is directly analogous to that for A1 and yields AK(s2 { + 2 L +i(n (2) + 0.02879837) I (s2 A +4 K z(s ) H0 )(kr) - ir(sT)(nt r)H (rdks) C2A (2. 27) The expressions for the far field scattering can be deduced from eqs. (2.7) and (2. 9) for E-polarization and (2. 10) and (2. 12) for H-polarization by replacing the Hankel functions by their asymptotic expansions for large argument. Each of these four equations has the form i t 5 F -F -F z z z where F represents either YOE or H depending on the polarization and the superscript t denotes the total field. Since F is simply the negative of the integral terms in the equations, a far field value can be obtained by selecting those integrals representing C2 contributions observed on C1 or C1 contributions observed on C2. It then follows that for E-polarization Th xresos o h frfel cttrn anb edcdfrmes 18

Y E (p,0) e 4 - j\ J (s)e k(xcosO+Ysi0)d(ks') 1 '0 Y J sJ A -s)( ik (x cos 0 +y' sin 00)d Yg \o e J (sg'te d (ks' -Y JK (st) ((t' ~( ) e-ikxt' cos 0 +y' sin 0)d(ks,) - K C(sO -n s r). e-1^10080^18111 ^ d (2. 28) C 2 and for H-polarization ikpi - c1 C + J ( s') J(s)(' ~) e-ik(' cos O +y' sin &)d(ks,) C 2 where 0 is the scattering direction measured anticlockwise from the positive x axis and r is a unit vector to the observation point. The two-dimensional scattering cross sections are ~~ _ lim 2rp y0E, liY E'($, ) | (2. 30) X p - oo Y andz and 19

_(_,) = lim 2rp Hz() (2 X -, X ^ (2. 31) x p --- oo X i where / is the angle of incidence. These show why it is convenient to choose YOE (0) = H (0) = 1 0 z z 20

III PROGRAM DESCRIPTION RAMVS is based in eqs. (2.7) through (2. 12) and uses the particular mathematical expansions given by eqs. (2. 18) through (2. 31). There are three variables to be determined: the total electric and magnetic currents J and J*, respectively, carried by the absorber sheet, and a surface current K on the impedance surface. If M is the number of cells used for the absorber sheet, and K is the number of cells used on the impedance surface, the total number of unknowns, and hence the number of linear simultaneous equations to be solved, is 2M+K. This is (approximately) thrice the unknowns encountered in cross section computations for impedance cylinders (program RAMD) or resistive sheets (program REST). (RAMD is discussed by Knott and Senior (1974) and REST by Knott et al. (1973).) It is therefore important that in developing program RAMVS consideration is given to writing a program with the least abuse of computer core space. Dimensioning has been simplified by structuring the program in modules, with MAIN used only for starting the program and allocating core space. The user of the program must provide appropriate dimensioning only in MAIN. The FORTRAN source program consists of 11 modules (a MAIN program and 10 subprograms) and contains about 850 statements, including comment cards. The structure of the program is shown in Fig. 3-1, where each block represents a particular module. The arrows joining these blocks refer to a typical execution sequence, discussed in the latter part of this section. The * When linear vectors are dimensioned in the MAIN program and are passed as arguments in the subroutine calling sequence, the dimension of the vector in the subroutine is arbitrary. For two-dimensional vectors, the size of the first dimension entry is passed in the calling sequence along with the vector, while the second entry is arbitrary. Thus in program RAMVS the dimension statements occurring in subroutines are of the form A(1) or B(M, 1). 21

FIG. 3-1: Block diagram and flow graph for the program. 22

purpose of the modules is as follows. MAIN This is a short program (7 FORTRAN statements) and its sole purpose is to start the program and allocate the core space for the dominant vectors (dimensioning of arrays). Whenever vectors must be redimensioned, only the MAIN program need be changed. SUBROUTINE SMAIN MAIN calls subroutine SMAIN, a control program that also performs other operations. These include the reading of control statements, the setting of appropriate controls for the type of computation required, and the calling of various subroutines. SMAIN calls a subroutine GEOM for computation of geometrical and electrical parameters of the scatterer, then calls subroutine MTXEL to generate the matrix and calls subroutine FLIP to invert the matrix and compute the unknown (current) vector. Computations of scattering cross sections and the printing of most of the output is also done in SMAIN. SUBROUTINE GEOM GEOM, as the name implies, reads the input data specifying the geometry and the electrical characteristics of the scatterer and from these generates point-by-point descriptions of the body required for solving the integral equations. The computations include the position of each cell, the components of the surface normal, the cell width, and the electrical parameters for the cell. The technique used for reading in the data is the same as used in our other two-dimensional scattering programs, such as RAMD and REST, developed under this contract. It is assumed that the profile of a two-kdimensional scatterer can be described by an assembly of circular arcs (a line segment is an arc of very large radius) and that the body is symmetric about the plane y = 0. Thus, it is sufficient to specify only half of the body and let the computer generate the image. For the special case of a horizontal line (strip) lying in the y = 0 plane, no image is generated. In GEOM a segment of circular arc is specified by the coordinates of the endpoints and the angle subtended by the arc as measured at the center of the generating circle. While cell coordinates and other parameters are being found, a running distance along the surface is also computed for each line segment. 23

This distance is used as an independent variable in establishing the surface impedance or absorber sheet impedance distributions along the surface. COMPLEX FUNCTION ZFUN This subprogram computes a surface impedance distribution according to one of the following mathematical relations: c* Z = (a-bs) Z = a+bs Z = a+becs where a and b are complex constants and c is a real constant. The distance along the surface of the line segment, is zero at the first (input) endpoint and increases toward the other. Thus, if the left-most endpoint of a line segment is read in first, the distance is computed from left-to-right or, if the right-most endpoint is read in first, the distance is computed from right-to-left along the particular segment. This adds additional flexibility to the possible impedance variations available from the three expressions given above. SUBROUTINE MTXEL1 AND SUBROUTINE MTXEL2 Once the geometrical and electrical parameters have been furnished by GEOM and ZFUN for each cell position, matrix elements are generated either by MTXEL1 or MTXEL2, depend,ing on whether the incident wave is E- or H-polarizedrespectively. The two subroutines are quite similar structurally, and in the earlier development of the code a single subroutine was used with numerous IF statements to differentiate between the two polarizations. The resultant program was elegant but confusing, and for the sake of simplicity and computational efficiency, the subroutine was replaced by two separate codes. In typical numerical solutions of electromagnetic scattering and antenna problems, the matrix possesses certain symmetries that can be exploited in the matrix generation, and even in solving for the unknown vector. In the process of development of RAMVS, various computational possibilities were examined that In the FORTRAN code this equation is written as Z = exp Lc log (a-bs)j 24

would reduce the core storage and the computational time. Because of the coupled integral equations which are involved here, the matrix defined by eqs. (2. 15) through (2. 17) does not, in general, have explicit symmetries associated with typical scatterers that could be readily exploited to save appreciable machine time. To keep the code relatively simple and to reduce programming errors, the matrix elements are computed in a relatively straightforward fashion, with the following modification. Since the electric and the magnetic sheets are physically superimposed and therefore have the same geometrical parameters, the computation of radii between the cells, the Hankel functions, and the dot products need not be duplicated. If A is the entire matrix and A..,A the submatrices as 1 9 defined in Chapter II, the matrix elements are computed simultaneously in Al, A, 1 2' A and A, in A and A, and in A and A. 4 5 3 6 7 8 SUBROUTINE DIST Subroutine DIST is called by MTXEL1(2) to compute the distance between the source and observation points (cells) and five kinds of dot nA.A products, such as (n r). To save unnecessary computations, one of the arguments in the subroutine calling sequence specifies the particular things to be computed. Thus if the computation of a particular matrix element requires the distance r between two cells and (n* f ), only the two are computed by the subroutine. SUBROUTINE HANKC The generation of matrix elements requires Hankel functions of the first kind, and these are computed in HANKC, complemented by subroutine ADAM. For computations of H (x) and H (x), HANKC uses poly0 1 nomial approximations for J 0(x) and Y 0(x), J1(x) and Y1(x) '(Abramowitz and Stegun, 1964). For the sake of efficiency, one of the arguments in the calling sequence specifies whether H (x) or H (1x) is to be computed, or both. SUBROUTINE ADAM The subroutine ADAM complements HANKC and is used to simplify the code of HANKC. SUBROUTINE FLIP As the name indicates, this is a matrix inversion subroutine and is an outgrowth of subroutine ZV08 used in program RAM1A originally supplied to us by AFAL. The technique used is the standard GaussJordan method (IBM, 1966) and the procedure has been explained in detail by 25

Oshiro (1963). | During the inversion process the original matrix A is replaced by -1 its inverse A and subsequently the solution vector (current vector) is obtained -1 by multiplying A by the excitation field. When a new excitation field is given, matrix inversion is by-passed and the solution vector obtained by multiplying A by the new field. BLOCK DATA Numerical constants commonly used in RAMVS are stored in COMMON BLOCK. These constants include Y0, wa, degree-to-radian conversion factor, and others. A typical computation cycle in RAMVS is as follows. MAIN starts the cycle by allocating core space and calling subroutine,^., SMAIN (1). SMAIN, in the first go-around, reads the first three data cards containing a title card and control information, such as incident polarization, bistatic or backscattering cross section to be computed, and others (cards A, B, C, as described in Chapter V). Then GEOM (2) is called, and it reads data specifying geometrical and electrical parameters of the scatterer (cards D through H) which include information such as type of surface, number of cells per segment and type of impedance variation along the surfaces. Using these data, GEOM generates all of the required geometrical parameters and, with help from ZFUN (3,4), the electrical parameters for each cell. Matrix elements are generated next (5,6), either in MTXEL1 or MTXEL2, depending upon the polarization desired. MTXEL first calls DIST (7, 8) to compute the distances and various dot products associated with each observation and source point, and then calls on HANKC plus ADAM (9, 10, 11, 12) for the related Hankel function. Control returns to SMAIN where the incident field vector is computed (13), whereupon FLIP is called (14) to invert the matrix and return (15) the solution vector. SMAIN prints out the geometrical and electrical parameters associated with each cell, along with the current. It then computes either the bistatic or backscattering cross section, and prints out the results. In the case of back"' Numbers in parentheses indicate the position on the flow-graph in Fig. 3-1. 26

scattering, FLIP is called (14) to calculate the new current for each new incident direction, and FLIP in turn acknowledges (15) by multiplying the pre-1 viously computed matrix A by the incident vector. If additional sets of data are provided, the cycle (2, 3,..., 15) is repeated, otherwise the program returns to MAIN (16) and execution is terminated. 27

IV LIST OF FORTRAN SYMBOLS This list contains symbols that have been assigned a specific meaning in the program. Words marked by an asterisk are used as prefixes (suffixes) to indicate a particular modification to another variable. Input parameters are also included in this list. FORTRAN Entry A(I, J) AA AMP"' ANG BJ BY DIG DSQ(J) FAC* FIRST FORM HONE, Hi HZERO, HZ I ID INK IAT II Description Matrix A or its inverse A Dummy variable (complex) Usually indicates amplitude Angle subtended by an arc segment Bessel function J l(x) Bessel function Y0 l(x) Radians-to-degrees conversion factor Cell size Multiplication factor Initial angle of incidence Control selects type of impedance variation desired Hankel function, H (x) Hankel function, H (x) Index, normally assigned to observation point Title, "A" format Angular increment for back/bistatic scattering angles Control in subroutine FLIP If IAT = 1, invert matrix If IAT = 2, bypass inversion Index, associated with I 28

J JJ K KODE L LAST LL(I) LUMP(2, I) M MI MK M1l MM1 MMK MORE NDNP NDR NPDR PHASE "' PHI(I) PI PINK(I) PIPI PIT P(WAVE) R RED RK S Index, usually assigned to source point Index, associated with J Number of cells on impedance surface Bistatic/backscattering control Index assigned to geometry segment Final (last) angle of incidence A vector in FLIP subroutine Array used for cell identification Number of cells on absorber surface Dimension integer passed to FLIP M+K M+1 M+M+1 M+M+K Control integer in input data A A n r At A n.r Phase of a complex number Current vector to be determined 7T Excitation vector 7T * 7T r7T/2 P refers to previous value (of wave) R = r 1 j Degrees-to-radians conversion factor 27rR Distance along the surface 29

SDR s * r SPDR s r SUM*' Sum, as the name indicates (complex) TPI 21r WAVE Incident wavelength X(I) x-coordinate of the cell XA x-coordinate for first endpoint of an input arc segment XB x-coordinate for second endpoint of an input arc segment XN(I) x-component of normal vector, n Y(I) y-coordinate of the cell YA y-coordinate for first endpoint of an input arc segment YB y-coordinate for second endpoint of an input arc segment YN(I) y-component of normal vector, n YZ Yo ZE(I) Electric impedance vector; ZS(I) is stored in part of ZE(I) ZEA Coefficients used in ZFUN for computation of ZEB B ZEXJ ZE(I) ZEXJ ZM(I) Magnetic impedance vector ZMA Coefficients used in ZFUN for computation of ZMB r ZM J ZM(I) ZSA Coefficients used in ZFUN for computation of ZSB surface impedance and stored in vector ZE(I) ZSX Z S Surface impedance constant 30

v USER INSTRUCTIONS AND SPECIAL CONSIDERATIONS The program RAMVS solves the integral equations for currents and bistatic or monostatic scattering from a general two-dimensional body consisting of a closed cylindrical surface with specified surface impedance ZS (stored in ZE) and absorber sheets that are specified by their electric and magnetic sheet resistances (impedances). Mathematical modeling requires that these sheets be thin, and comparison of computed and measured backscattering has shown that for a dielectric sheet reasonable results are obtained when the sheet is as thick as 0. 1X, where X is the free-space wavelength. If thicker sheets are desired, they can be simulated by two or more equivalent thin sheets, requiring, of course, more sampling points. The size of the body or the number of sampling points that can be used depends on the core size of the machine and the funds or computer time available. If K is the number of sampling points on the impedance surface and M is the number of sampling points on the absorber sheet, the total number of unknowns to be determined is 2M+K. To assist a potential user intending to run RAMVS, we illustrate the preparation of input data for a hypothetical/computation. Partial output for the same data is also presented. 5. 1 IBM-CDC Compatibility The program has been compiled and run on the University of Michigan IBM 360/67 computer and the AFAL CDC 6600 machine at WPAFB. Only one discrepancy occurs, and this is in statement 13 (IBM version) in subroutine SMAIN. Depending upon the machine used to run the program, the following (or equivalent) forms should be used: IBM CDC 5 READ(5, 100,END=998)ID 5 READ(5, 100)ID IF(EOF(5)) 999, 998 998 CONTINUE 31

5. 2 Sample Run Consider a relatively thin perfectly conducting ogival cylinder to which are attached an absorber fin at the front and two fins at the back. We choose the body to be one wavelength long with its surface generating arc encompassing 25. The fins are made of material characterized by (complex) permittivity r = 5.0 +i3.0 and permeability p = 2.0 +il. 0, and the thickness of the material is 0. 05 wavelength. The details of the geometry are shown in Fig. 5-1. / / 0"' 0.40.l) z ( o,o) ( # (0.5,0) (0.8,0) =0.+O0. Z\ / 0.382+ i0.509 v25 ZHz 1.59+i 1.59 FIG. 5-1: Geometry used in the sample computation. It is requireed to compute the backscattering cross section pattern at aspect angles spaced 10 degrees apart for both incident polarizations. 32

Preliminary computation: i i ZE = -___ 0.382 +iO. 509 2r( -1)A/X 2T(4.0 +i3.0)(.05) 0 382+i509 r i i ZM = =1 59+il. 59 27M - 1)A/X 2w(1.0+il.0)(.05) = ZS = 0. +iO. (perfectly conducting) Number of sampling points: Using the criterion of approximately 10 sampling points per free space wavelength, we choose: 10 points on each of the upper and lower halves of the ogival cylinder, 3 points on the front fin, and 3 points on each of the rear fins. Dimensioning: If the program is used as listed in the Appendix, the array dimensions are more than adequate. On the other hand, if larger or smaller arrays are desired, the following guidelines should be applied: (a) Dimensioning need be changed only in the MAIN program. (b) Estimate the number of sampling points that will be used on impedance and absorber surfaces. (c) Determine the dimensions of arrays from the following formula: If K = number of points on the impedance surface and M = number of points on the absorber sheet, then the pertinent values MK and MMK that must be entered in the MAIN program are MK = M+K and MMK = M+M+K For particular details, see the comments at the beginning of the program listing (Appendix). 33

Input data formats: Below are the input data formats for the RAMVS program. The letters assigned to each format do not necessarily represent the order of the data cards. Depending upon the geometry used and the computational requirements, some formats may be repeated and others not used at all.,*** INPUlT DATA FORMAT *** A FORMAT (18A4) TITLE CARD: tUSE UP TO 72 cnOLUMNS B FORMAT (I2,I3,4F10.5) MORE,IPOL,WAVFZSFACZFACZMFACZ MORE=O THIS WILL BE THE LAST RUN FOR THIS DATA SET MORF=I THERE ARE MORE DATA TO BE READ AFTER THIS SET IPnL=l F-POLARIZATION I PL=2 H-POLARIZATION WAVE WAVELENGTH ZSFAC MULTIPLYING FACTOR(REAL) FOR ALL ZS ZEFAC MULTIPLYING FACTOR(REAL ) FOR ALL ZE ZMFAC MULTIPLYING FACTOR(REAL) FOR ALL ZM C FORMAT (I2,3X,3F10.5) KODE,FIRSTLAST, INK KODE=O COMPUTFS BISTATIC SCATTERING PATTERN KODE=1 COMPIUTES RACKSCATTERING PATTERN FIRST INITIAL SCATTERING AND INCIDENCE ANGLE LAST FINAL ANGLE INK ANGI)LAR INCREMENT * --- —--------------------------------------------------------— E (DATA n, E, AND F REOIQIRED FOR EACH ABSORBER SEGMENT; AT LEAST ONE ABSORBER AND ONE IMPEDANCE SEGMENT IS REQUIRED, ABSRORBER SEGMENTS MUST BE READ IN FIRST.) D FORMAT (I2,I3,5F10.5) TYPE,NXAYA,XBYBANG TYPE=1 ABSORBER SHEET TYPE=2 IMPFDANCE SUJRFACF N NUMBER OF SAMPLING POINTS ON THIS SEGMENT XA,YAXB,YB SEGMENT ENDPnINTS ANG ANGLE SUBTENDED BY THE SEGMENT, --- —----------------------------------------------------------- E FORMAT (I2,3X,5F10.5) FORM, ZEAZEB,ZEX FORM=-1 ZE(I)=(ZZEA-ZEBS(I ))**ZEX FORM= 0 ZE( I )=ZEA+ZEB*S I )**ZEX FORM= 1 ZE(I)=ZEA+ZEB*EXP(-ZEX*S(I)) ZEA,ZEB COMPLEX IMPEDANCE CONSTANTS ZEX REAL IMPEDANCE CONSTANT 34

F FRMAT (I2,3X,5F10.5) FORMZMAZMBRZMX FORM=-1 ZM(I )=(ZMA-ZMBS( I) )**ZMX FORM= 0 ZM( I )=ZMA+ZMB*S( I )**ZMX FORM= 1 ZM( I )=ZMA+ZMB*EXP(-ZMXS(I) ) ZMAZMR COMPLFX IMPEDANCE CONSTANTS ZMX REAL IMPEDANCE CONSTANT -------------------------------------------------------—. ---(DATA G AND H REQIIIRED FOR EACH IMPEDANCE SEGMENT. FORMAT (I2,I3.5F10.5) TYPEN,XAYA,XBIYBANG TYPE=1 ABSORBER SHEET TYPE=2 IMPEDANCE S(URFACE N NUMBER OF SAMPLING POINTS ON THIS SEGMENT XAYAXBYB SEGMENT ENDPOINTS AlNG ANGLE SUBTENDED BY THE SEGMENT' W --- —---------------------------------------------------------- H FORMAT (12,3X,5F10.5) FORM,ZSAZSB,ZSX FnRM=-l ZS(I)=(ZSA-ZSB*S(I))**ZSX FORM= 0 ZS( I )=ZSA+ZSB*S( I )**ZSX FORM= 1 ZS(I)=ZSA+ZSBREXP(-ZSX*S(I)) ZSAZSB COMPLEX IMPEDANCE CONSTANTS SX REAL IMPEDANCE CONSTANT. f ---- ---- ------ ---------------- -- ---------- ------------------ ---- I FORMAT (I5) INTEGER ZERO IN COLUMN 5 SHUTS OFF READING OF SEGMENT PARAMETERS H --- —---------------------------------------------------------- (USE THIS CARD ONLY IF, IN BR MORE=1) J FORMAT (12,I3,4F10.5) MOREI POLWAVEZSFACZEFACZMFAC Input data cards: The data set may consist of as few as 9 cards but can be many more, depending upon the number of segments used to specify the geometry, the number of polarizations and frequencies desired, and the number of constant multiplication factors for the impedances. The geometry must always consist of two surfaces, an impedance surface and an absorber sheet, though either may consist of only two cells far removed from the main scatterer. The data and explanation follows. 35

2 5 10 20 30 40 50 (column) (card) PI 1 OGIVAL CYLINDER WITH FINS 2 1 2 1. 1. 1. 1. 3 1 0. 180. 10. 4 1 3 0.5 0. 0.8 0. 0. 5 0 0.382 0. 509 0. 0. 0. 6 0 1.59 1.59 0. 0. 0. 7 1 3 -0.4 0. 1 -0. 6 0. 3 0. 8 0 0. 582 0. 509 0. 0. 0. 9 0 1. 59 1. 59 0. 0. 0. 10 10 -0. 5 0. 0.5 0. 25. 11 1 0. 0. 0. 0. 0. 12 0 13 0 1 1. 1. 1. 1. Structure of the input deck for the example of Fig. 5-1.

CARD 1(A) FORMAT (18A4) Title card; use up to 72 columns. This is a title card and is repeated in the output format. The letter (A) after CARD 1 refers to the format used as described above in Input data formats. CARD 2'(B) FORMAT (12,13, 4F10. 5) MORE, IPOL, WAVE, ZSFAC, ZEFAC, ZMFAC This is a control card; the same card is repeated at the end of the data set, but with MORE s 0 (last run for this data set) and IPOL = 1 (E-polarization). MORE ~ 1 Another run will be made for this data set. IPOL m 2 H-polarization. WAVE = 1. Wavelength in units used. If geometry is normalized with respect to wavelength, WAVE = 1. ZSFAC ZEFAC = 1. Relative multiplication factors for impedances. ZMFACJ CARD 3(C) FORMAT (21, 3X, 3F10. 5) KODE, FIRST, LASTJ INK KODE 1 Bistatic pattern is computed. FIRST 0. Initial incidence angle. Only for this incidence are the surface currents printed out. LAST = 180. Final angle. Since the body is symmetrical about the y-plane, running as far as 360 degrees would be repetitious. The program requires that LAST > FIRST. INK = 10. Scattering will be computed at 10-degree increments. CARD 4(D) FORMAT (I2,I3, 5F10. 5) TYPE, N, XA, YA, XB, YB, ANG This and the next eight cards prescribe the scatterer geometry and associated electrical properties. The program requires that absorber surfaces be read in first. In this data set the following order of input is used: (1) frontal fin, (2) rear fins, (3) ogival cylinder. TYPE = 1 Indicates this is an absorber surface. N = 3 Number of sampling points on this segment. XA =0.5 YA Specifies endpoints for the segment. XB = 0.8 YB-0 J 37

ANG = 0. Implies that this is a straight line segment. In generation of the geometry in subroutine GEOM it has been assumed that a body is symmetric about the y = 0 axis, and only the parameters for the upper profile must be supplied. However, when a segment is a straight line and lies on the y z 0 axis, its image is not generated. Such is the case here for CARD 4. CARD 5(E) FORMAT (I2,3X, 5F10. 5) FORM, ZEA, ZEB, ZEX This card prescribes the electrical sheet impedance parameters for the frontal strip. FORM = 0 Selects equation for computation of ZE in ZFUN. ZEA = (0. 382, 0. 509) Electric impedance constants appropriate for the ZEB = (0., 0.) equation selected by FORM 0. Note that in this ZEX = 0. J particular case the value of ZEX is arbitrary. CARD 6(F) FORMAT (I2,3X, 5F10.5) FORM, ZMA, ZMB, ZMX FORM = 0 Selects equation for computation of ZM in ZFUN ZMA = (1.59, 1.59) ZMA = (1. 59 1. 5) Magnetic impedance constants appropriate for the ZM'B = (O.J '. equation selected by FORM = 0. ZMX 0. CARD 7(D) FORMAT (12, 3, 5F10. 5) TYPE, N, XA, YA, XB, YB, ANG This card specifies the geometry for the upper tail fin. Its format is the same as that of CARD 4. The lower fin (an image) will be generated automatically by the program. CARD 8(E) FORMAT (I2, 3X, 5F10. 5) FORM, ZEA, ZEB, ZEX Since the same material is used in the tail fins as in the frontal fin, this card is identical to CARD 5. CARD 9(F) FORMAT (I2, 3X, 5F10.5) FORM, ZMA, ZMB, ZMX This card is identical to CARD 6. 38

CARD 10(D) FORMAT (I2,I3, 5F10. 5) TYPE, N,XA, YA, XY, YB, ANG This card specifies the geometry for the upper portion of the ogival cylinder. Again, the lower portion will be generated automatically in the program. TYPE = 2 Indicates this is an impedance surface. XA =x -0. 5 YA -- 0. Specifies endpoints for the segment. XB 0. 5 YB =0. ANG = 25. Angle encompassed by the circular arc. CARD 11(G) FORMAT (I2, 3X, 5F10. 5) FORM, ZSA, ZSB, ZSX This card specifies parameters for computation of normalized surface impedance ZS. Since the body is perfectly conducting, this card is easy to do, viz. FORM, 1 Selects equation for computing ZS ZSA (O, 0. ) ZSA (0. 0.) Surface impedance constants appropriate for the ZSB (0,0) equation selected by FORM = 1. ZEX 0 CARD 12(H) FORMAT (I5) Integer in column 5 shuts off reading of segment parameters. CARD 13(I) FORMAT (I2,I3,4F10. 5) MORE, IPOL, WAVE, ZSFAC, ZEFAC, ZMFAC With the use of this card (in addition to MORE = 1 in CARD 2), computations for the same geometry are repeated, but for different electrical parameters. Here, only the polarization is changed (IPOL = 1), and since this is the last run for this data set, MORE - 0. Other entries are the same as in CARD 2. MORE * 0 Last run for this data set. IPOL = 1 E-polarization. WAVE = 1. Wavelength. ZSFAC ZEFAC 1. Relative multiplication factor for impedances. ZMFAC 39

Output data: The sample data set was run on the University of Michigan IBM 360/67 computer and the H-polarization results are shown in the following pages. For the shape computed, there are a total of 29 sampling points on the body: 20 on the ogival cylinder and 9 on the absorber fins. The resultant matrix is 38 x 38 (MMK = 9+9+20), and 44. 2 seconds of CPU time were required for both the E- and H-polarization computations. The output consists of three major blocks and in this case each group requires a page of this report. On the first page (p. 41), geometrical and electrical input parameters are given, along with other useful data such as the radius and length of arcs used in generating the profile. Such output is helpful in verifying that all input parameters have been correctly punched and read into the machine. The second block (p. 42) consists of geometrical and electrical parameters for all cells on the body, and includes the computed currents. When the incident wave direction is specified for a range of incident angles, the currents are printed out only for the first angle; if currents are required for other angles of incidence, the input data set must be repeated for the new incidence angle. The third block (p. 43) lists an array of computed cross sections. Since there are three types of current contributing to the scattering, the individual contribution of each type is also given. These include electric current (ELECTRIC) and magnetic current (MAGNETIC) from the absorber sheets, and the surface current (IMPEDANCE) from the impedance surface. The (ABSORBER) column lists the net contributions of the absorber sheet (electric plus magnetic current), and the (TOTAL) column is the sum of the three contributions. Such a breakdown of the cross section into its components is often helpful in determining the sources of scattering (the dominant scatterer), analyzing the performance of the absorber, and studying other scattering effects. The expressions used for computing the bistatic and backscattering cross sections are 40

OGIVAL-CYLINDEP WITH FINS SEG SFG N Um -- E ND POI NV'S NUF TYP CELLS XA YA 1 X BS1 3 * 2 ABS il- fN 1 kB S 3 ~ - Li IMP 1 ) ~.~ 5 IMP 10 -" 5 0' - -7 OF SEGMENTS - XD. YB 9~cy '., --------------- SEG!~FFT PARAMETERS rkNGLE RADIUS LENGTH FORM ZEA (ZMA) 999.10 n?.3C1 '.382.5C19 5,97 1.590) 9 9~.( ~.8 382 n.509 2E~ 2.31 1 18 i ). 2 5.C 2.3 1 1 n08 1 ' ZFB(ZMB) ZFX(ZMX) r.. I.: KEY PARAMETERS INCIDENT POLARIZATION SURFACE IMPEDANCE FACTOR ELECTPTIC IMPEDANCE FACT-OR MAGNETIC IIMPF-DANCE FACTOR TOTAL NUJMBFPP OF POINT'S ON T-HE BODY NUMBER OF SEG(7ME~NTS USED NUMBER OF INCTDEN" FIELD DIRECTIONS NUJVBE7'P C7 BISTATIC DTIIPECTITONS W 7 LI Fr LNG T 1 H 1.1) 1. Ci ~ ) 29 5 19 I 1. -~ -,

OGIVAL-CYLINDEP WITH FINS ABSORBER SURFACF T SFG X Y S DSQ?EOD(JE) ARG(JE) MOD(J!) ARG (J!) 1 2 3 4 5 6 7 8 Q 1 1 1 2.2 2 3 3 3 -o. 65 ^r -'3. 4333 -0. 5? ^0 -. 5667 -0. 4333 -0.5C " -. 5667.t I.1333;. 2 7, 0. 2667 -r 12 3 -0. 2r n -,.2667 I0. 1530 ~" A25 Li ". a71 1. 14 1;C.2357. ra71 n. 1414 r.2357 0.1~ r. i1n r N. n- 43 %. 0943. -0943 0.0943. 094 3 0. n9 43 0. 382 0.382 '*. 382 0.382 0.382 0. 382 0. 382 -.50.509. 50 9 ).509. 50.5C 9 3. 5 9 9 1. 59n 1. 59:0 1.59" 1.590 1.59 1.59" 1. 59" 1. 59 ) 1.59, 1. 59] 1.59) 1. 59 ' 1. 590 1. 59) 1,59 ) ".0015 '.0" 030.0057 0.6425 0.8052 0. 7464 r.6464. 8324 0.7670 1 07.274 81.665 104.946 -56.877 -38.139 -26.0 4 -59.428 -38.695 -2 4.537 175.9988 154.20-7 151.1338 127.6986 143.1452 154.7719 128.3813 132.1762 142.6163 130. 740 95.055 46.245 112.808 141.711 168. 516 120.774 143. 525 162. 476 p? IMPEDANCE SURFACE I SEG X Y S DSQ MOD (JS) ARG (IS) 1! 1 1 12 13 14 15 16 17 18 1 20 21 22 23 24 25 26 27 28 29 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5, -4 07 -.351 4 -5 0504 -2.2515 0,0504 ". 1511 n. 3514 ~. 4507 -. 4507 -0.3514 -n. 2515 -3. 1511 - 0. 054... 05 04 0. 1!11 0.2515 0.3514 0 4597 0. 104 0.0279,,..4 n1 ~.0498 0.0542. 054 2 C. 279 * 0279 0.98 -C.01!4 -C. 0279 -n.hio 1 -r. 0498 -0. 0542 -0.0 542 -r. O410 -n. ~279 -0.01 4 0.05~4,-. 1512 ".2520.35 28 ".4536 r. 5544 r 6552 P.7560.8568 C. 9 76 0.0504 n.1512 n.2520 ".3528 0.4536;. 5544. 8568 ". 9576 0 '. 13 "8.10 C8 0. 1008 0.1008 0. 1008 0. 1lr08 "~ 10 8 0.10 8 0.10 08 0. 1:""8 O. 10T08 0. 1008 0. 10 "8 0.1008 C' I 00 010 00 or 0 00 'I' 01 0 0 0 0 50 5.0 '~) ".0 0 '0.0 10 0 0 1 5 3 CO o.7305 ^.81 0 1, —. 88 1.2416 1. 1703 0.9292 9239 1.1662 1. 2643 0.8253 ^.7342 ".8016 1.0070 1.2464 1.1759 n.9294 n. 9192 1.1644 1.2661 0.824 0 174. 281 131. 381 81. 928 53. 786 28.453 -9.181 -61. 240 -99.740 -125. 7 55 -149. `47 174.316 1 31.735 81. 447 53. 23 28.475 -8. 918 -61.276 -1"0. 13^ -125. 961 -4 8. 968

BACFSCATTERTNG CP')SS S3ECIT1I' 1 \* L OG ( S T A/L A BD A) (9H-P )LAR TZ AT ION) (E LElr PC) 9 P HA SE ( "?- 7GN IFT,' I C) T. I P',-i A SE (ABSO9RB ER) DB PHAS7T -(PI)?SDA NCE) DB PHIASE D B MIAS E S"T TA Cf::3 j4f A) ) -9.4 7 -7. 1 1 -8.35 -8.2 2 -7. 98 -2. 21 -7. "79 -9 3~ -53. 16 -15 1.76 1 62. 19 81.. 8 -16 8. 2C -1 17. 74 - 52.73 3 5.44 -1 17. 9 4 -134.81I 6". 4 2 2. 73 22. 9P -3 9. r'. -1 4 ) ~1 -16.71 - 9 *98 -31.991 - 12.4 7 -23.24 -1 4.13 -14.1 -13.1 3 -14.1 "2 -1i.7 1 2)3.39 111.68 83. 86 49.2 -71 *33 - 13.3 1 -121.87 6 4 *27 6 4.1 6 17 3.19 -113. 48 -98 8 7 52.3 5..16 6.57 -149 5 2 2 -1Ir ~9 4 1 79.97 1 f, 48A -7.1 -1 6. 5; - 19.52 -21.72 -14. 6". - 9.87 -22.55 -1 2.4 6 7If. 15 -2.9?4 -27". `2 -1 7.17 - 15F.2 5 -1.23 - 37 -125.9 ~161 4. 178. 73 -174.35 -153.7 9 -7 2. 15 1 8 * 3 5 -64.59, -146 * 5? -71 *71 32)1 16 17 5~7 '73 - 7.47 2 )3 7" 1(2.89 9 9.9 1 9 3.99 6 4. 53 1 3.2 3 -2 6. r-6 --35.3f) -34. 22 -29.2 2 2 7.1 1 17. 17 1 22.4 5 12 3.81 11 2.67 - 17.6 4 -19.Q7 -4. 82 -2.5 5 -3.7" 4.25 594 1. 52 -11.3 9 -6.~55 -Ii.9 -31.6 9 -19.3 4 2~-I6. 51I 1 24. 17 72. 77 13 2 -45. 26 - 35.4 4 -23. 1 3 -41. 46 13-7, UP 121.653 - 74. 36

2,I lim 2 -ff Es| a/x = -i p2 Y E _ lim 27r, Si o/X- oX H / p-coo E-polarization H-polarization and the associated phases are Arg(E ) and Arg(Hs) for E- or H-polarization, respectively. 44

VI CONCLUSIONS In the previous sections a description of a computer program RAMVS for computation of scattering from a general two-dimensional body consisting of an arbitrary impedance cylinder and absorber sheets specified by equivalent electric and magnetic sheet impedances is given. The sheet impedances must be determined from the permittivity, permeability and thickness of the material, and then read in as input data. Consistent with the core limitations of the particular installation used to run the program, the program has the potential for solving scattering from many two-dimensional problems of interest. These include perfectly conducting bodies, bodies with prescribed surface impedance with or without absorber sheets present, and absorber-clad bodies. The RAM- series programs, such as RAMD (impedance body, E-polarization), REST (resistive sheets, E-polarization), or RASP (electric impedance sheets, E-polarization) developed and used under this contract are special cases of RAMVS. RAMVS was tested against these programs and, for the cases computed, gave the same results. Many other aspects of RAMVS, however, have not been tested. To determine the capabilities as well as the shortcomings of the program, computed patterns should be compared against experimental measurements for structures such as isolated absorb er sheets, sheets attached to metallic bodies, and thick absorber layers, either isolated or when coated on metallic cylinders. Of particular importance would be the determination of optimum cell size in absorber layers. Since there are two unknowns associated with each cell, it would be desirable to use cell sizes as large as possible, thereby minimizing the size of the matrix. On the other hand, cells that are too large may lead to a degradation of accuracy. In retrospect, there are two areas in which the program could be modified. First, a substantial saving in computation time could be achieved by dispensing with the actual inversion of the matrix and solving the system of linear algebraic 45

equations by a direct technique such as Gaussian elimination (Southworth and Deleeuw, 1965). This would require developing, or obtaining if possible, an appropriate code to replace the subroutine FLIP. Additional changes might be required in MAIN and SMAIN to accommodate new variables, if such are needed. Before making such changes in the program, however, an /accurate assessment of the CPU time should be made for a typical computation based on the matrix inversion approach and the direct solution technique. A "rule of thumb" is that it requires almost three times as many operations to invert a matrix as it does to solve the system of equations directly, but this applies only for a single incident vector. When a number of incident vectors are present, as is the case in computing backscattering patterns, it may be more efficient to invert the matrix and perform a simple multiplication with each incident vector, rather than to repeat a portion (back substitution) of a direct solution procedure for each. The other possible modification of the program would be to add a capability to compute the tangential electric and magnetic fields along a prescribed surface and form the ratio of the two to determine the impedance. This would be helpful in relating the material properties of the coatings to surface impedances, especially near edges or other surface or electrical discontinuities, where no known analytical formulas exist. To change the present version to RAMVS to include this capability would require an additional subroutine for computing the electric and magnetic fields along a presecribed surface representing, for example, the outer surface of the absorber. Subroutines GEOM and SMAIN would have to be expanded to accommodate such a surface. The next test for the program RAMVS will be to compute the scattering from the geometries the program was intended to handle. This will encompass scattering by ogival cylinders with magnetic fins (E 3 1), isolated absorber r strips, and perhaps absorber-clad bodies. When possible, experimental data will be used for comparison. The results will be included in the Final Report under this contract. 146

APPENDIX PROGRAM LISTING The following is an IBM version of the FORTRAN source listing of the program RAMVS. To run this program on a machine such as the CDC 6600 at WPAFB (AFAL), two cards should be changed in subroutine SMAIN as described in Section 5. 1 of this report. 47

CCCC CCCCCCCCccCcccccccccCCcccccccccccc cccccccc CCCCCcccccccccccccccc C C C *** RAMVS *** C C (03-10-74 VERSION) VVL C C C C THIS PROGRAM COMPUTES NEAR-FIELD AND FAR-FIELD SCATTERING FROM C C A GFNFLAL IMPFDANCF BODY IN PRESFNCF OF ABSORBER SHEETS; C C TWO-DIMENSIONAL GEOMETRY, EXP(-IWT) TIME CONVENTION. C C C C RE: V.V.LIEPA,E.F.KNOTT,AND T.B.A.SENIOR,"COMPtUTFR PROGRAM FOR C C SCATTFRING FROM TWO-DIMENSIONAL BODIES WITH ABSORBER SHEETS'" C C THE UNIVERSITY OF MICHIGAN RADIATION LABORATnRY REPORT C C NO.O01174-2-T (AFAL-TR-74 ), 1974. C C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC C C C *** INPUT DATA FORMAT *** C C ----------------------------------------------------— C C A FORMAT (18A4) TITLE CARD; UJSE IJP TO 72 COLUMNS C C --- ------------------------------------ --— C --- —---- C C R FORMAT (12,I3,4F10.5) MORE,I POL,WAVEZSFAFA,ZEC,ZMFAC C C MORF=O THIS WILL BE THE LAST RUN FOR THIS DATA SET C C MnRF=l THERE ARE MORE DATA Tn BE RFAD AFTER THIS SET C C IPOL=1 F-POLARIZATION C C IPnL=2 H-POLARIZATION C C WAVE WAVELENGTH C C ZSFAC MUILTIPLYING FACTOR(RFAL) FOR ALL ZS C C ZEFAC MIULTIPLYING FACTnR(REAL) FOR ALL ZE C C ZMFAC MUILTIPLYING FACTnR(REAL) FOR ALL ZM C C --- —--------------------------------------------------------------------— C C C FORMAT (I2,3X,3F10.5) KODE,FIRST,LAST,INK C C KODE=O COMPUTES RISTATIC SCATTERING PATTERN C C KODE=1 CnMPUTES BACKSCATTERING PATTERN C C FIRST INITIAL SCATTERING AND INCIDENCE ANGLE C C LAST FINAL ANGLE C C INK ANGULAR INCREMFNT C C --- —---------------------------------------------------------------— c C (DATA n, E, AND F REOUJIRED FOR EACH ABSORBER SEGMENT; C C AT LEAST nNE ABSnRBER AND ONE IMPEDANCE SEGMENT IS C C REO(IIRED. ABSRORBER SEGMENTS MUIST BE READ IN FIRST.) C C n FORMAT (I2,I3,5F10.5) TYPE,NXAYA,XBRYBANG C C TYPE=1 ARSORBFR SHEET C C TYPF=2 IMPEDANCE SIRFACE C C N NUMBER OF SAMPLING POINTS ON THIS SEGMENT C C XAYAXBYB SEGMENT ENDPOINTS C CANG AG NGLE SUBTENDED BY THE SEGMENT C C --- —-------------------— ~~~-~~~~ --- —---------------- ~.~ ~.~ ~ ---C E FORMAT ( 2,3X, 5F10. 5) FORM, ZEAZFB,ZEX C FnRM=- ZE( I )=(ZEA-ZEB*S(I ) )**ZEX C C FORM= 0 Z( I )=ZEA+ZEB*S( I )**ZEX C C FnRM= 1 ZF(I)=ZEA+ZEB*EXP(-ZEX*S(I)) C C ZFA,ZER CnMPLEX IMPEDANCE CONSTANTS C C ZEX REAL IMPEDANCE CONSTANT C 48

------------------------------------------------------------------------ C F FnRMAT (I?,3X,5F10.5) FnRM,ZMA,ZMR,7MX C C FnRM=-l ZM( I)=(ZMA-ZMRS( I ) )**ZMX C C FnRM= 0 ZM( I )=ZMA+ZMB*S( I )**MX C C FORM= 1 ZM( I )=ZMA+ZM*EXP(-ZMX*S( I ) C C ZMA,ZMR CnMPLFX IMPFDANCF CONSTANTS C C ZMX REAL IMPEDANCF CONSTANT i C C --- —------------------------------ ----------------------------------------------------— C C (DATA G AND H REQUIRED FOR FACH IMPEDANCE SEGMENT.) C C G FORMAT (I2,I3,5F10.5) TYPE,N,XA,YA,XRBYBANG C C TYPE=1 ARSnRRER SHEET C C TYPF=2 IMPEDANCF SIURFACE C C N NUMBER OF SAMPLING POINTS ON THIS SEGMENT C C XAYAXB,YB SEGMENT ENDPOINTS C C ANG ANGLE SUBTENDED BY THE SEGMENT C C0 --- —----------------------------------------------------------------- C H FORMAT (I?23X,5F10.5) FnRM,ZSA, SZBZSX C C FORM=-1 ZS( I )(ZSA-ZSB*S I ))**ZSX C C FORM= 0 ZS(I)=ZSA+ZSRS( I )**ZSX C C FnRM= 1 ZS( I)=ZSA+ZSR*EXP(-ZSX*S() I )C C Z SAZS COMPLFX IMPFDANCE CONSTANTS C C ZSX REAL IMPEDANCF CnNSTANT C C --- —-------------------------------------------------- ---------------- C C I FORMAT (15) INTEGER ZERO IN CnLUMN 5 SHUTS OFF C C READING OF SEGMENT PARAMETERS C C -----------------------------------------------------------------— C --- C (USE THIS CARD ONLY IF, IN B, MORE=1) C C J FORMAT (12,13,4F10.5) MOR,I POL,WAVE,ZSFACZEFAC,ZMFAC C C C CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCcccccccccccccccccccccccccccccccc C C C *** DIMENSIONING FORMAT *** C C --- —---------------------- -------------------------- ---------- ---------------------— C C VECTORS ARE DIMENSIONFD ONLY IN THE MAIN PROGRAM. C C IF K=NO. POINTS(CELLS) ON THE IMPEDANCE SURFACE AND C C M=NO. OF POINTS(CELLS) ON THE ABSORBER SHEET, THEN C C MK=M+K AND MMK=M+M+K C C C C....,MAIN PROGRAM --— A STARTER PROGRAM C C * CnMPLEX A(MMKMMK+1 ),PHI ( MMK ),PINK(MMK ),LL(MMK ),MM(MMK) C C * COMPLEX ZE(MK),ZM(MK) C C * DIMENSION X(MK),Y(MK),XN(MK),YN(MK),S(MK),DSO(MK) C C * DIMENSION LUMP(2,MK) C C * DATA MI/MMK/ C C CALL SMAIN(MIAPHI,PINKZEZMX,Y, XNYNS,DSO,LLMMLJMP) C C FND C C * CARDS TO BE CHANGED WHFN REDIMENSIONI NG C C C ccccccccccc CCCCC Ccccccccccccc cc cc cccccCcccccccccccCC cccccccccccC cccCCCC 49

C..MA IN PRnOGRAm --- —-A STARTER PROGRAM C*~*RAMNIS VERSiON COMPLEX A( 150,151l),PHL( 150),PINIK( 150), LL( 150),MM( 150) COMPLEX 7E( 100) tZM( 100) DIMENSION X( 100),Y( 100),XN( 100),YN( 100),S( 100),DSO( 100) DIMENSION LIJMP(2,100) DATA MI/150/ CALL SMATN(MI,A,PHT,PTNKZEZMXYXNYNI.S-flSOLLMMi-,LUMP) END 50

SIJRROUT I N E S MA I N( M I, A,9 P HII PI N K-, 7 FZ?M-, X,XN ~5s, n0so0.L L M M, LUJMP ) Cl-****, RAMVS VERSIONt F-&H-POLARIZATIONS DIMENSiON LlUMP(2,1) COMPLEX A(MI,1) COMPLEX PHI (1) PINK( 1),ZE( 1) ZM( 1) COMPLEX SUJMDELStJMFSl.JMMSlJMK DIMENSION X(l),Y(I),XN(I),YN(l),S(1 ),DSQ(1),LL(l),MM(l) DI MENS ION Ifl( 18 ),9IPP (2 ) COMMON/PT ES/PI,TPI,PIT,PIPI,YZ,RFO,DIG REAL LASTINK DATA IPP/4HFEFE4HHHHH/ C,,...,REAn INPUJT DATA AND GENERATE BODY PROFILE 5 RFAD(5,,100,FNn=gqq9) iD READ (5,200) mnRF,IPOL,,WA\VE,ZSFAC,ZEFACZMFAC READ (5,210) KOOF,FIRSTLASTINK WRITE (6,150) ID CALL G.FOm( LUMPX,,YXN,,YN,,sDsoZE,,ZMK,M) IF (KODENE.0) Go ro 25 N INC,=I NBIT=l+IFIX((LAST-FlRST)/INK) GO TO 28 25 NBIT=O NlNC=1+IFIX((LAST-FlRST)/lNK) 28 CnNTINUJE C,....CnNSTRUJCT MATRIX ELEMENTS M1=M+1 MK=M+K MMK=MK+M MML=MI,+M DO 35 I=MlMK 35 ZF(I)=ZF(I)*ZSFAC FAC F=ZEFEAC* WAVy FACM=ZMFAC*WAVE XK =TPI /WAVE DO 37 I=1,M 7F I )=ZE( I)*FACE 37 7M( I)=ZM( I)*FACM DO 39 I=1,MK Sf1 )=S( 1)/WAVE 39 DSO(I)=so(i ) /WAVE GO TO 50 40 FAC=ZSFAC/PZSFAC DO 45 T=MlMK 45 7F(I)=ZF(I)*FAC FAC=WAVF/PWAVF FACF=Z FFAC/PZFFAC*FAC FACM=ZMFAC /PZMFAC*FAC 00 47 I=1,M ZF( I)=ZF( I)*F-ACF 47 ZM(I)=7M(I)*FACM FAC=P WAVE/WAVEF ____________ 51

XK=TPI /WAVE DO 49 T=lMK 5(1 )=S( I)*FAC 49 DSo (I )=nS ( I )*FAC 50 CnNTINUE K~~Xq~D~Zv~A IF (IPOL.EO.1) CALL MTXFL1(MlM,1XYXNDOZgMA IF (IPnL. EO? 2CALL MTX EL2(MIMtK,9X,9YXN, YN,Dns oZ EiZ MA) T=LUJMP(2,vMK) _ _~_ - WRITE(6,400) IPP( IPnL-),ZSFACZFFAC, ZMFACMK, IvNINCNBITWAVE C.....,CnMPUITE INCIDENT FIELD AND INVERT MATRIX TFTA=R FD* FIRST CT=cns( TFTA) ST=SIN( TFTA) Dn 60 1=],VM HnLD=-.XK*(CT*X( I )+ST*Y( I) DF L=C MP L X ( CS-( HOL D)SI N (HnLD)) PINK( I )=DE L 60 P INK (I +M )= -D)FL*(X N(I)CT+ YN(I)S T) DO 63 I=Ml,MK HnLD=-XK*(CT*X( I)+ST*Y( I)) 63 PINK(I+Mh:CMPLX(COs(HnLD),SIN(H4OLD)) CALL FLI P (A, MMK OMTLLMM, PINK,9PHI 91) WRITE (6,150) ID C..... *PRINT OUT STUFF FOR THE ABSORBER SURFACE (FIRST ANGLE ONLY) WRITE (69350) WRITE (69375) Don 67 I=1,M DEL=PHIl(I )+(l1.E-50v0.).AMPE=CABS(DEL) PHASEE=DIG*ATAN2(AIMAG(nFL),RFAL(DEL)) nFL =PH I( I +M )+ ( 1. F-50,9 0. AMPH=CARS(nDEL) PHASEH=L)1(;*ATAN2( AIMAG( DFL),PRFAL( DFL) IF (IFP(L.EO.1) WRITF(6,v395) (LU.MP(JI),J=1,2),X( I),Y(I ),S( I), CDSO( I ),ZE (I ),ZM( I),vAMPF, PHASEE, AMPH, PHASEH 67 IF (ipnL.FO.2) WRITF(6,395) (LIJMP(J9l),J=192),X(I),Y(I),S(T)9 ~Dso( I ),?Ff I ),7M( I ),AMPH, PHASFH, AMPFpPHt.SEF C.....PRJNT OUlT STUFF FOR THF IMPFDANCF SURFACE (FIRST ANGLE ONLY) WRITE (6,300) WRITE (6,325) DO 65 I=MlMK I M=I+M D)FL=PHI (IM)+( 1.E-50q0.) AMP=CABS( DEL) PHASE=DIG*ATAN2 (AIMAG(nDEL),REAL(nDEL ) D-FL=ZE( I) 65 WRITE (6,250) (LUIMP(J, I),J=l,2),vX( I )Y(I),tS(I ),DSO( I)? ~DFL,-vA-M P,9P HA SE _________ 52

C.....DnPE OUT THE APPRnPRIATF FIELD FACTOR S WRITF (6v150) ID THF=FIRST-INK IF (KnDE.EO.1I) Gfln TO 70 WRITE (6,800) FIRSTIPP(IPOL) GOn To 75 70 WRITE (6,600) IPP(IPOL) 75 THF=THE+INK IF (THE.GT.LAST) GO TO 105 IF (THFwEO.FIRST) GO TO 85 TFTA=R ED*THE CT=cns( TETA) ST=SIN( TETA) C....,IN THE FOLLOWING LOOP COMPUJTE THE NEW INCIDENT FIELD Do 80 i=i,M HnLDn=-XK* (CT*X ( I ) +ST*Y ( I) DEL=CMPLX(C0S(H0LD),SIN(H-Ln)) PINK( I)=DEL 890 PINK(I+M)=-DEL*(XN(I )*CT+YN(I)"'ST) Do 8 3 I = M 1,M K____ _ _ HnLD=-'XK*(CT*X( I)+ST*Y( I)) 83 PINK( I+M)=CMPLX(COS(HOLD),SIN(HOLD)) IF (KODEEO.0) Go TO 85 CALL FLIP( AMMKtMI qLLvMMPINKPHI 92) 85 CONTINUE C.....ADD) UP THE CURRENTS FOR FAR FIELD SUM=(1.E-25,0.) StJMF= StiM SUJMM=SUJM SUJMK= SUM DO 93 I=1,M IM=I+M DS=DSQ( I) SIJME= SIME-PH I ( I *PTINK ( I )DSl 93 SUMM=SUMM+PHI(IM)*PINK(IM)*DS SliM M= S UMM* V 7 IF (IPOL.EQ.2) GO TO 90 DO 95 I=MlMK IM=I+M 95 SIJMK=SUJMK+(-l.+ZF(I)*(XN(I)*CT+YN(I)*ST))*PINK(IM)*DSQ(I)*PHI(IM) GO TO 99 90 O)FL=SU)ME SUJME=-SUJMM/YZ SUJMM= DEL*YZ DO 97 I=MlMK I M=I+M 97 SUMK=SUMK+(-ZE(I)+(XN(I)*CT+YN(I)*ST))*PINK(IM)*DSQ(I)*PHI(IM) 99 DEL=SIJMF+SLJMM SUJM=DEL +SUMK A MPR =REAL ( SOiME) AMP!l=AI MAG( SlIME) PH ASE F=D I G*ATAN2 (AM PI t AMPR) AMPR=PIT*(AMPR*AMPR+AMPI*AMPI) 53

SCATFt1o.*MJnGiO( Amp-R) AMPR=RFAL (SUMM) AMPI=AJIMAG( SIMM) PHASEM=DTG*ATAN2( AMPI, AMPR) AMPR=PIT* ( AMPR*AMPR+AMPI*AMPI) SCATM=10.*ALOGlO( AMPR) AMPR=RFAL(nDEL) AMPI =AIMAG(OnEL) PHASEO=DTG*ATAN2( AMPI,AMPR) AMPR=PIT*(AMPR*AMPR+AMPI*AMPI SCATD=10.*ALnOlO ( AMPR) AMPR=RFAL (SUMK) AMPI=AIMAG( St.JMK) PHASEK=DIG* AT AN?(AMP I,tAMPR AMPR=PIT*(AMPR*AMPR+AMPI*AMPI) SCATK=l0.*ALOG10( AMPR) AMPR=RFAL( SUM) AMPI=AIMAG( SI.JM) PHASET=DIG*ATAN2( AMPI1,AMPR) AMPR=PIT*(AMPR*AMPR+AMPI*AMPI SCATT=10.*ALnO~o( AMPR) WRITE(6,900) THE,SCATEPHASFFSCATMPHASEMSCATDPHASED, &SCATK,PHASEK,SCATT, PHASET GO TO 75 105 P Z SF AC.=ZS FA C P WA VE= WA VE P7EFAC=ZFFAC PZMFAC=ZMFAC IF (MnRF.EO0) GO TO 5 READ (5,200) MnR E I POL,9W A VEZ SFAC,9ZEE FACZM FA C WRITE (69150) ID GO TO 40 100 FORMAT (18A4) 150 FORMAT (1H1,18A4) 200 FORMAT (1291 314F1.5) 210 FORMAT (12, 3X v3F10 5) 250 FORMAT( 1H,213,w4F8,4ilXt2Fll.*3tF9o4tE9.3) 300 FORMAT( ////18HOIMPEOANICF SURFACE) 350 FnRMAT(17H0ARsnRBFR SURFACE) 325 FnRMAT(8H0 I SEGv4X,IHX,7Xv 1HY,97XIHS,6X,,3HOSO,, &llXv10H ---~ 7S -—,5X,7HMOn(Js),2X,7HARG(JS)/) 375 FORMAT(RHO I SEG,4X,1IHX,97XIHY,7XLiHSt6X,3H0S0, &llX,10H --- 7F --- t13X,10H --- ZM -—,5X,7Hmnn(jE),2X,7HARG(JE),2XI 97HMOD( JM ),2X97HARG( JM),/) 395 FORMAT(IH,213,4F8.492(lX,?F1L.3),2(F9e4,F9.3)) 400 FORMAT ( ////55X, 14HKEY PARAMETERS/I &40X,?1HINCinFNT PnLARIZATInN.22X,lAl // 940X.24HSURFACE IMPEDANCE FACTOR,F 2 0*3/ &40X,25HELFCTRIC IMPEDANCE FACTOR,Fl9 3/I F40Xt25HMAGNETIC IMPEDANCE FACTnRtF19.'3// &40X.,34HTnTAL NUMBER OF POINTS ON THE RODY,,I10// 4 0Xo2 3H N UM-BER O 0F — SYGMJ?-NTS- — US FDp 121/ 54

&i40Xv35HNIiMBFR OF INCIDENT FIELD nIRECTnTONs,I9q// F40X,,29HNU-mBER OF I STAT IC DlI RFC T iONis,II // &4OXq OH WA VFL FNGTH, F3~4.3) 600 FnRMAT (///, 51 X,28HBACKSCATTFR I NG CROSS SECTION,!, &55X,20H1O*LnOG(SIGMA/LAMBflA),/, F57X,lH(,1A1,14H-PnLARIZATION),9///, &31X,1L0H( ELECTR IC),7X,I1OH( MAG7NETIC),7X, LOH( ABSORBER),7X,v cII1H( IMPEDANCE ),$X,7H( TOTAL), /,2lX,5HTHETA,2X, F,5(17H DR PHASE )/) 8900 FnRMAT (///v 49X, 33HBI STATI C SCATTER ING CROSS SECT ION,/ &55X 20HiO*LnG( S IGMA/LAMBnA) I/, &48X,29HFOR INCIDENT FIELD DIRFCTION=9F6,1,/, &57X,lH(,lAl,14H-PnLARIZATinN),///, c3lXIOH(ELECTRIC),7XlOH(MAGNFTIC),p7X,1lOH(ABSORBFR),7X, & H ( I MPFDANC E) 18X, 7H (TOTAL ),v/ 21 X t5HTHETAt2X, &5(17H DB PHASE U)/ 900 FOR MAT (19X,F7. 2,5( 1X,?F8.2 ) 999 RETU-RNI ENn 55

suBROUTINF GEm( LI-IMP,,X,Y -pXN, YNt spDsO, Z 7FtZMKtM) C4**** RAMVS \/FRSinN COMPLEX ZEFA, ZERZ MA Z MB Z FUNpZF ()IM (1) DIMENSION X( i),Y(l),XN(lI),YN(I~)4)so(i),s(l ) DI MENS ION LUMP (2,I)1 cnmMON/PIES/PI,TPI,PIT,PIPI,YZ,RED,DIG K=0 L=0 M=O WRITE (69500) C..... READ INPUT PARAMETERS AND PREPARE TO G7FNERATF SAMPLING POINTS C...,,I F TYPE=1I ABSORBER SHEEFT, M CFL LS TOTAL C.....IF TVPE=2 IMPEDANCE SUJRFACE,? K CELLS TOTAL C.,..,.TVPE=i SURFACE MUST BF READ IN FIRST 10 READ (5,200) ITYPEjNtXAjVAvXBjYBtANG7 IF (N.EO*0) GO TO 120 READ (5,250) IZEFRMIZFAgZEBZEX IF (ITYPF.EQ.1) READ (5,250) IZMFRMZMAIZMBtZMX TX=XB-XA TV=YB -YA D=SORT ( TX*TX+T* TV) IF (ANG.FO.0.0) GO TO 20 T=0.5*RFD*ANG) TRX=TX+TY/TAN( T) TRY=TV-TX/TAN (T) RAD=0.5*D)/SIN( T) ARC =2.0*R AD*T ALF=T/N 010=2. 0*RAD*ALF GO TO 30 20 RAD=999. ARC =0 010D=0D/ N C.....START GENERATING7 30 LAST=2 I F(YA.EOO..0ANDYB.EO.0.0.AND,4NG.EFO,0.0) LAST=1 DO 110 JIM=1,LAST L = L+1 DO 100 J=1,LIM,2 1=1+1 LI.JMP(2,I1)=L LUIMP ( 1,1I ) =I IF (I.Fn.,1000) WRITE (6,t400) IF (JIM.FO.2) GO TO 90 IF (ANG7.FO.0*0) GOn TO 40 STINO=SI N( J'ALF) cOso=COS (J*ALF) X(I)=XA+0.5*(TRX*(1.0*O-COSQ)-TRY*SINO) 56

Y(TI ) =YA+0. 5* (TRX*(SINO.+TRY*(i.0-cnso)) XN( I)=-O. 5*( TRX*,COSO+TRY*S INO,)/R An YN(I )= 0.5*,(TRX*SINO-TRY*,cnso)/RAO GO TO 50 40 X(I)=XA4-0.5*J*TX/N Y( I )=YA+0.5*,J*TY/Nl XN( I)=-TY/n YN(fl= TX/F) 50 ST=0,5*J*Dln S(1 )=ST _ _ _ _ _ _ _ _ _ _ _ C,.....cnmplTE THE ELFCTRIC PARAMETERS lE(ITYPFEO.1) GO TO 60 C.,...ZS IS STORED IN THF ZE VECTOR 7 F(I)=ZEIJ)N(IZEFRMdFAZEBZEXST) GO TO 100 60 ZM(I)=ZFINI(IZMFRMZMAZMF?',MXP,ST) ZEF(1)=ZFIJN(IZEFRMZFAZFB9ZFXST) GO TO 100 C.....EROM HERE TO 100 WE CREATE THE SEGMENT IMAGF 90 K=I-N X (I )=X (K) Y( I)=-Y(K) XN(I )=XN(K) YN( I)=-YN(K) S(T )=S(K) ZE( I)=ZE(K) 7M(I )=ZM(K) 100 DSO0( I )=DI f) LE (JIM.EO.1) GO TO 102 YA=-YA YB=-YB 102 F( IE(TYP F. EO. 1GO TO 105 WRITE(6,300) LNXAYAXf~,YF~ANG,:7RADARC, IZEERMZEAZEBZEX K=I -M GO0 TO 110 105 WR ITE (69 350) L vN vXA,YA tXR YfAN~GvRAf),ARC, TZEFRM IZEAvZEFv ZEX WRITE(6,351) IZMERMqZMAgZMRZMX M=I 110 CONTINUJE GO TO 10 200 EORMAT (129139 5E10. 5 2-50 FORMAT(I?2,3X v5E10.5) 300 FnRMAT(IH,I2,5H IMP,14,1lX,4E9.4,IX,2F7,2,EF7.3,14,1X,2E9.3,2Xv 350 FORMAT(1H vI2,5H ABSI4491X,4F9.49 IX92F7.2,EF7.3,14,1X,?F9.3,2Xq &?E9,*3v El1.3) 3 51 EORMAT(71X,T14,2H (F,E839 1XiF8.393H) (,E8.39 IXi F8.3t3H) (, &F8.3,PIH)) — 40-0 FnRMAT(37HOWARNING: WE'VE -GENERATED 1000 POI-NTS!) 57

500 FnRMAT (13HOSFG SFG NUIIM,3X,6H --,21HENDPOINTS OF SEGMENTS,6H ~ —,SX,27H ----—. --- —----------,18HSEGMENT PARAMETERS,27H --------------------— / TYP CELLS,4X,2HXA,7X,2HYA,7X, &2HXB,7X,2HYB,6X,24HANGLF RADIUS LFNGTH FORM,6X,8HZEA(ZMA).12X, &8HZ FB (7MB ),8X, 8HZEX( ZMX ) / ) 120 RETURN FND COMPLEX FUNCTION ZFUN(I FORM.ZAZBR.ZEXST) C**** RAMVS VERSION COMPLEX ZA,ZB IF(IFORM) 10, 15, 20 10 Z FUN=CEXP(ZEX*CLOG( ZA-ZB*ST)) RFTURN 15 ZFIJN=ZA+ZB*ST**ZEX RETURN 20 ZFUN=ZA+ZB* EXP(-ZEX*ST) RETURN END 58

S(IRRntJTINE MTXFL1I(MIMKXY, XNYN, 050, Z F, ZM A) C***4' RAMVS VERSION E-POLARIZATinN DI MENS ION X( (1)y Y (),oXN( 1 )' YN 1), OSO( 1) COMPLEX ZE( 1),ZM( 1),A( MI 91 ) COMPLEX AA,HZ,HZAHZBH1,9H1A, H1R REAL NPDRNDRNDNP cOMMON/P I ES/PI, TP I9P IT, PIP I?,YZ, RED, DIG MI =M+1 MK=M+I( DO 300 Ti=l,m 1=11 I M=II+M XI=X(I) YI=Y( I) XNI=XN( I) YNI =YN( I) C.....GENFRATE ELEMENTS IN 1,2,4, AND 5 DO 100 JJ=i,m Jj =JJ IjM= + M DS=DS0( J) PDS=1./PIPI /DS DDS=0. 25*Ds*Ds TPI DS=TPI*DS IDS=TP IDS/24. P1ITDS=PTT*D)S IF (I.,EQ.J) GO TO 120 IJ=IABS( I-J) IF ( IJ.LF.2 ) GO TO 110 CALL DIST(XI,YIX(J),Y(J),XNI,YNI1,XN(J),YN(J)949 CR,NPDR,NDR, NnNP,,SO)R, sPnR) RK=R* TPI CALL HANKC,(RK,?,HZHj) A( I,J)=H7*PITns AA=HI.*PT TDS A(19JM)=YZ*CMPLX(0.,NPDR)*AA A( IMJ)=CMPLX(0.,NDR )*AA B=KNPDR*NDfR AA=R*HZ+( spDR*SDR-f3)*H1l/RK( A (I MtJM) = AA*PI TDS*YZ GO TO 100 110 CALL DIST(XIYI,X( J) Y(J),XNIYNI,XN(J ),YN(J)95* CR,NPDRqNDR, NDNP, SDR, SPnR) B=R*R+DDS c=ns*R*SPDR RAK=TPI*SORT(B+C) IF (RAKNE.R8K) GO TO 103 HZA=HZR HIlA=H1 R GO - TO 1-05 __ _ 59

10 3 CALL HANKC(,kA —K,-,HA1A 1 05 RfBK=TPI*SOP.T (RBC R K= R *TP I CALL HANKC (RK9,0*HZ 9H1 CALL HANKC(RRK,2tH7BH1R) AA=Tns* -(H7A+La.,*H7+H7PB) A (1,J )=AA A(IM9,.iM)=(0.25*snR*(HIR-HIA)+NnNP*AA)*YZ AA=CMPLX (0.9-0.25 )*spFR* (HZBW-HZA) A( I,JM)=AA*NPDR*YZ _ A(TM,,J)=AA*NDR GO TO 100 120 AA=CMPL X(P I T ALOG (DS) +0.028798P37)*DS A(T.vJ)= ZELJ)+AA A ( ITJM )= (0.,0.o) A( TMfJ)=(0.,0.) A (I MJM )=( M (J) +CMP LX (0.,9PDS )+AA )*YZ 100 CONTINUEF C,.e*.GFNFRATE ELEMENTS IN 3 AND 6 DO 300 JJ=MltMK J =jJ JM=J+M PITTD S= PI T*Dso(J) CALL DTST(XT,YI,X(J),Y(J),XNI,YNI,XN(J),YN(J),4, CRNPODRvN DR,-vNONP,9 SDRS P DR) RK=R*TPI CALL HANKC(RK,2,HZHl) C ***REMEMBFR ZS IS STORED IN ZE*** H ZA = ZF ( J) AA=HZ-CMPLX (0. vNPDR )*Hl*HZA A( I,JM )=AA*PITDS B=NPDR* NOR AA=R*HZ+( spnR*SDR-B)*Hl/RK AA=CMPLX (0. 9 NOR )*HI-AA*HZA 300 A (TIMJM )=AA*P ITDS DO 500 II=MlgMK 1=11 I M=I+M XI=X(I) YI=V( I) XNI=XN( I YNT=VN( I C.....GENERATE ELEMENTS IN 7 AND 8 DO 400 JJ=1,M ji = J J P IT TOS=P IT* DSO0( J) CALL DTST(XIqYItX(f,J~Y(J), XNIvYNTXN(J),YN(J),0, E R, NIPDRv NDR,,NnN P snR SP DR) RK=R*TPI CALL HANKC(RK,2,HZHl) A(IMJ )=HZ*PI TDS 40-0 A( IM-,J+M)=CMPLXU ~NPR *H1PITDS*YZ 60

C...G FNiFR A TF EPLRTM-E-NTS~ I-N 9 Dn 500 JJ=Ml,MK 3I=33 IF (1,FO.J) GO, TO 510 CALL Di ST(XIYI,X(J),9Y(J),XNI,YNIvXN( J)YN(J),10, &R,NPDRNOR, NnNP, SOR,SPDR) RK =R*TPI CALL HANJKC(RKt?,HZjH1) AA=HZ-CMPLX (O.,NPnR )*H1*ZF(J) A( TMP J+M)=PIT*AA*nSO( J GOn TO 500 510 ls=DSO( J ) ATM, J+M )=0. 5*ZF( J)+DS*CMPLX( PIT, ALOG (Ps )+0.02879837) 500 CONTINUJE RETURN ENnP 61

SIJBROUTINE MTXEL2(MI,M, K,X,Y YXNYN,DSO, ZE, ZM, A) C**'* RAMVS VERSION H-POLARIZATInN C*** RAMVS VERSION DIMENSION X( 1 ),Y(1),XN( l ) YN(1 ),SO(1 ) COMPLEX 7E( 1 ),ZM(1 ),A(MI,1) COMPLEX AA,H ZA,HZZA,HRH1,Hl1,H1R REAL NPDR,NDR,NDNP COMMON/PIES/PI, TPI,PIT,PIP I,YZ,RED,DIG M 1 =M+1 MK=M+K DO 300 II=-,M 1=J IM=T I+M XI=X( I) YI=Y(I ) XNI =XN( I) YNI=YN( I ) C.....GENERATF ELEMENTS IN 1,2,4, AND 5 nn 1oo JJ=l,M J=JJ JM=J+M DS=DSO( J) PDS=. /PIPI/DS nns=o.?5*DS*ns TPIDS=TPI*DS TDS=TPT DS/24. PITDS=PIT*DS IF (I.EO.J) GO Tn 120 IJ=IARS(I-J) IF (IJ.LE.2) GO Tn 110 CALL DIST(XI,YI,X(J),Y(J), XNI YNIXN(J), YN(J),4, ~R, NPDR, NDR, NDNP,SDR,SPDR) RK=R*TP I CALL HANKC(RK,2?,HZHl) A( I,J)=HZ*PITDS*YZ AA=HI*PITDS A(I,JM)=CMPLX (.,-NPDR)*AA A(IM,J)=CMPLX(0.,NDR)*AA*YZ R=NPDR*NDR AA=*HZ+ ( SPDR*SDR-R )*H1/RK A( IM,JM)=-AA*PITDS GO Tn 1O 110 CALL DIST(XI, YI,X(J), Y(J),XNI YNI XNI(J), YN(J),5, &RNPDR, NR,NDNP, SDRSPDR) B=R*R+DDS C=DS*R*SP DR RAK=TPI*SORT( B+C) IF (RAK.NE.RRK) GN TO 103 HZA=HZR H1A=H1B GO TO 105 62

1.03 CA LL H ANK C (RAK,_2 ItH ZApHIA) 105 RRK=TPI*S0RT(B-~C) R K= R* T PI CALL HANKC(RK,0,HZH1) CALL HANKC (RFBKq2HZ8,BH1R) AA=TDS* (HZA+4,*H1+HZB) A (1 9J )=AA*YZ A(TM, JM )=-0. 25* SDR* (H 8-HlA )-NnNP*AA A A =CM PL X ( O0-, 0,.?5)*S P-DR_* ( H ZI3-H ZA-) A ( I,9J M )= AA* N PDR A(lTMJ)=-AA*NDfR*YZ GO TO 100 120 AA=CMPLX(PIT,ALOG7(DS)4.O.02879R37)*Ds A (TIJ )= (Z M (1) +A A*Y A( I JM)=(0.,0.) A(ITM,9J )= ( 0 oI0 AITM J M )= -Z7E (J )-C MP L X ( 0. P DSA A 100 CnNTINIJIF C.....GFNFRATE ELFMENTS IN 3 AND 6 DO 300 jJ=MI,MK J= j J J M = J+M PITTs=PIT*DSO( J) CALL DI ST (Xl YIX ( J), Y( J),XNI,YNT,XN( j), YN(JI)94, ERN P DR N DR.NDnN P,SDR 9 SP DR) RK=R*TPI CALL HANKC(RK,2vHZvHl) C ***REMEmRER ZS IS STnRFD IN ZF*** HZA=Z F(JI) AA=HZ*HZA-CMPLX( 0. NPnR )*Hl A(I1,JM)=AA*PITDS R=NPDR*NDR AA=R*HZ+(SPnR*SDR-BFfl*H1/RK AA=C MPLX (0.,vNDR )*H1'HZ A-AA 300 A (I MvJM) =AA* P ITDS DO 500 Il=MlgMK 1=1! IM=I+M XI=X( I YI=Y(I) XNI=XN( I YNI=YN( I C.....GFNFRATF ELEMENTS IN 7 AND 8 DO 400 JJ=1,M I = I P1ITDS=P IT*DSO(JI) CALL DIST( XI,YI X(I),Y(JI) XNI,YNIXN(J),YN(JI) 0? E~R,,NPDR,,NDR,,NDNP,,SDR,,spnR) RK=R*IP I CALL HANKC(RK,29HZH1) A( IM,PJ)=HZ*,PITDS*YZ 400 A(TM J+M )=CMPLX(0.,NPnR )*Hl1'PITDS 63

C....GNi-R ATF FLEFWFNkTS IJS[ 9 -n0 900 Jj=MLMK IF (I.E~oJ) GO Tfl 510 CALL nIST(XlTYlTX(J),Y(J),XN1,YNTXN(J),YN(J)0,O, &R,9NPDR,9NOR,.NnNP,. S DR, spnR) R K= R *TP I CALL HANKC(RK,2,HZH1) AA=HZ*ZE( J)-CMPLX(0.,NPnR )*Hl A (I M tJ+M)=P IT* AA*DS ( J) GOn TO 500 510 ns=nso(J) i A(IM,,J+M)=0,.5+PS*CMPLX(PITALnG(Os )+0.02879837)*ZE(J) 500 CONTINUJE R FT t-JR N FND_ SUBFRnUTINE DIST( XI Yl,XJYJXNI,YN1,XNJYNJ, 1, &R,,NPDR,,NDR,,NnNP,,SDR,SPflR),;Co*&**.I=0 R*NPD)R C.....il=1 R,9NPODRNOR C.... 1=2 RtNPDRgNDRtNDNP C.I = 3 R NnNP,,SDRSPDR C....=4 RvNPDRgNPR9 SDRSPDR C... = 15 R, NPOR,NnR, NNP SOR, SPDR REAL NP)RvND)RgNONP IF (TIL T. o.ORI,.GT.5) GO TO 50 R X= XI - X RY=YI-YJ R,=SOR T (RX*RX+RY*RY) IF (I*EO.3) GO TO 10 NPOR= (RX*XNJ+RY*YNJ) /R IF (1,FO*O) RETURN NnR=(RX*XNI+RY*YNT ) /R IF (I.EOl1) RETU.RN IF(TFO.4) GO TO 15 10 NONP =XN I*XNJ+YN I*Y NJ 15 IF (1,FO.2) RETUR.N SDR=(RX*YNI-RY*XNI )/R SP)R= (RX*YNJ-RY*XNJ )/R RETUIRN 50 WRITE(6,90) IR 90 FORMAT(31HOSICK DATA IN DIST CALL SYSTEM F ND *O0IjIT* 1=,T1?92X92HR=tE1 1.3 ) 64

SUBROUTINE HANKC(R,N,'HZERn,HONF) C.,.,.HANKEL FUNCTIONS ARE OF FIRST KTNO —J+TY C..... N=0 RETURNS HZERO C..... N=1 RETURNS HONE C..... N=2 RETURNS HZERO AND HONE C.....SU-BROUTINE REQUIRES R>O C.....SUBROUTINE ADAM MUST BE SUPPLIED BY USER DIMFNSION AC 7).B(7),C(7),DC7),EC7),F(7),G(7),H(7) COMPLEX HZEROHONE DATA ABCvflEFGH/1.O,-2.249999791.2656-208,-0,3163866I &O0 0444479, -0. 00394440,0000210, 036746691,0. 60559366,-0. 74350384, &O.25300H17,-O.04261214,0.00427916,-O0.00024846,0.w5,P-0.56249985,P &0,21093573,t-0.03954289,O.00443319,-0-.00031761,-0.OO001109,, &-096366198,O.221209l,2.l682709,-1.3164827,0.3l23951,-0.0400976I t&0.0027873,tO.79788456,-0.00O00077,-0.0055274,t-O.oO0O095 12I &O0O0137237,-O0O00728O5,O.OOOl4476,-0.78539816,-0.041663979!-0. 00003954,0. 00262573,p-0. 00054125,-0.000293339, 000013558, &0.79788456v,O00000569,001659667,O.00017105,-0.002 49511, &0. 001 136539 -0. 000200339 -2. 35619449i 0.12499612.,p0.0000565,p S-0.00637879,0.00074348,0.00079824t-0900029166/ IF (R.LF..0O) GO TO 50 IF CNeLT.0.OR.N.GT.2) GO TO 50 IF (R.GT.3.0) GO TO 20 X=R*R/9.O IF (N.E4O.1) GO TO 10 CALL ADAMCAXvBJ) CALL ADAM( B X,9Y ) BY=O.6366198*ALOGC O.5*R)*BJ+Y HZERO=CMPLX (BJ, BY) IF (N.EO.0) RETURN 10 CALL ADAM(CXjY) BJ=R*Y CALL ADAM(DvXY) BY=0.6366198*ALOGC 0.5*R )*BJ+Y/R HONF=CMPLX( BJ,pBY) RFTU(R N 20 X=3.0/R IF (NEO.1) GO TO 30 CALL ADAM(EvXY) FoOL=Y/SORT CR) CALL ADAM(FjXY) T=R+Y BJ=FOOL*COS(T) BY=FOOL*SIN( T) HZFRO=CMPLX C BJ, BY) I F (No EQv-0) RE-T1JRN — _ _ 65

30 CALL ADAM (GX,Y-) FOOL=Y/SORT( R) CALL ADAM(H,X,Y) T=R+Y BJ=FOOL*COS(T) BY=FOOL*SIN( T) HONE =C MP LX ( BJ,'BY ) R ETUIR N 50 WRITE(6,90) N,R 90 FnRMAT(31HOSICK DATA IN HANKC *QUJIT* N=,I2,2X,2HR=Ell1.3) CALL SYSTEM END SUBROUTI NE ADAM(CX,Y) DIMENSION C(7) Y=X*C,(7) Dn I) I1=1.5 10 Y=X*(C(7-I)+Y) Y=Y+C ( 1 ) RETURN END 66

SUBROUT INE FL IP( A, NM I,9LMgXY, T AT) COMPLEX A( MI 91 ), X( 1 ) 9Y( 1) PBIG7AtHOL D DI MENS ION L(1I)gM( 1) IF (IAT*GT*.1) GO TO 150 D=C MPL X( 10, 0. 90) D0 80 K=lvN [(K )=K M( K)=K BIGA=A(KK) P0 20 J=KN 00 20 I=KN 10 IF (CABS(BIGA).GE.CABS(A(IJ))) GO TO 20 BIGA=A(1IJ) L(K)=t M(K )=J 20 CONTINUE J=[(K) IF (J.LE.K) GO TO 35 DO 30 1=1,N HOLP=-A(K,tI) A(K, I1)=A(J, I) 30 A(JvI)=HOLD 35 I=M(K) IF (I.LE.K) GO TO 45 DO 40 J=1,N H-OLP=-A( JK) A(J,vK)=A(-J,9I) 40 M(JvI)=HOLD 45 IF (CABS(8IGA).NF.0.0) GO TO 50 D=CMPLX( 0.0,0.01 RET URN 50 DO 55 I=ltN IF (I.EO.K) GO TO 55 A( I,K)=-A{ I,K) /BIGA 55 CONTINUE DO 65 I=1,N DO 65 J=1,N IF (I.EQ.KoOR.J.EQK) GO TO 65 A(I,gJ)=A(IK)*A(KJ)+A(IJ) 65 CONTINUE DO 75 J=1,N IF (J.EO.K) GO TO 75 A(KJ)=A(K,PJ) /B'IG.A 75 CONTINUE D=D*BIGA 80 A(KfK)=1.0/BIGA K=N__ 67

100 K=K-1 IF tK.LF.0) GO7 TO 150 I =L (K) IF (I.LF.K) GO TO 120 DO 110 J=1,N HOLD=A( J,K) A (JqK )=-A( J,1) 110 A(Ji)=HOLI) 120 J=M(K) IF (JLE.K) GO TO 100 nn 130 T=1,N HnL D=A (K, I) A(KIl)=-A(JvI) 130 A(Jvl)=HOLF) GO TO 100 150 00 200 I=19N Y ( I ) =CMPLX (0.0 v0.0) nO 200 J=1,N 200 Y (I ) =A ( IJ)*X(J)+Y ( I) RETURN END BLOCK DATA COMMONI/PT ES/PITPIPITPIPIYZREnOIG DATA P1, TP1,P IT, PIPI,YZREDDIG/3. 141 5927, 6.2831853, &1,57079639,98696044.,O.00265258240.rO01745329,57,29578/ END 88

REFERENCES Abramowitz, M.A. and I. A. Stegun (1964), Handbook of Mathematical Functions, NBS Appl. Math. Series No. 55, U. S. Government Printing Office, Washington, D.C. 20402, 369-370. International Business Machines (1966), IBM System/360 Scientific Subroutine Package (360A-CM-03X), Version II. Knott, E. F., V. V. Liepa and T. B.A. Senior (1973), "Non-Specular Radar Cross Section Study", The University of Michigan Radiation Laboratory Report No. 011062-1-F (AFAL-TR-73-70), Ann Arbor. Knott, E. F. and T. B. A. Senior (1974), "Non-Specular Radar Cross Section Study", The University of Michigan Radiation Laboratory Report No. 011764-1-T (AFAL-TR-73-422), Ann Arbor. Oshiro, F.K. (1973), "Computer Programs for Scattering from Two-Dimensional Bodies with Arbitrary Surface Impedance", Northrop Corporation, Aircraft Division Technical Report No. AFAL-TR-73-135. Southworth, R. W. and S. L. Deleeuw (1965), Digital Computations and Numerical Methods, McGraw-Hill Book Company, New York, 216-284. 69

UNCLASSIFIED i 1 [ [ I I Securitv C s si fic';Ition DOCUMENT CONTROL DATA R & D (Securi tyv cin.ssfirfji'onol of f li, ho/ I 1)y.o( t.tro t jitol /ifK)~'Xi4l/ ' 111 uiitiflf niro l l e oil he woit weIr - ie) n Wt 1 vnrnill report 1/. ln AlJflnh) I1. ORIG INA TING A C T I V I T Y (Cororptor le fit1hor) 20. I L PORT 5CC U ll( 1 Y CL A S1 F I C A T0tO Thle Univcrsity of Michigan R:adiation Laboratory UNCLASSIFIED 2216 Space Rcscarch Bldg., North Campus 12b. G OUPI. Ann Arbor, Michigan 48105. NA 3. REPORT TITLE SCATTERING FROM TWO-DIMENSIONAL BODIES WITH ABSORBER SHEETS 4. DCSCRIPTIVE NOTES (7ypo of report -ind Inclusivo dstoe) Technical Scientific Interim (15 October 1973 - 15 March 1974) 5. AU THOR(S) ('irst name, middlo Initial, last namo) Valdis V. Liepa Eugene F. Knott Thomas B.A. Senior 6. REPORT DATE 711. TOTAL NO. OF PAGES 7b. NO. OF REFS!March 1974. 69 6 Oa. CONTRACT OR GRANT NO. 94. ORIGINATOR'S REPORT NUMBR.R(S) F33615-73-C-1174 b. PROJECT NO. 011764-2-T c. Task No. ',h. OTHER RHPORT NOIS) (Any ulicr ntmbhrs et lio many Ie ua.lincJ 7633-13 d'. 76 3 AFAL-TR-74-119 10. DISTRIBUTION STATEMENT Distribution limited to U. S. Government Agencies only; Test and Evaluation Data; May 1974 Other requests for this document must be referred to AFAL/WRP II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Air Force Avionics Laboratory Air Force Systems Command Wright-Patterson Air Force Base, Ohio 13. ABSTRACT This report describes a program that computes the far field scattering pattern of a two-dimensional cylindrical body (or bodies) treated with absorbing materials. The body surface is assumed to satisfy an impedance boundary condition and the absorber is modeled by equivalent electric and magnetic sheets. The program reduces the coupled integral equations governing the surface currents to a system of simultaneous linear equations, solves for the unknown currents, and then computes the far field pattern from the solution. The integral equations derived and presented in a previous interim report were used as the basis for preliminary versions of the program, but these equations have been found to be in error. The required correction consists of incorporating terms previously omitted and the corrected equations are presented and discussed. Careful attention is given to a description of the input data necessary to run the program and the results of a sample run are included for illustration. The program was developed to explore the effects of absorbent materials on the scattering of electromagnetic waves by edges. Programs used previously in similar explorations embodied a surface impedance boundary condition; for lossy materials covering smooth surfaces of large radius of curvature, such an impedance can be estimated from the layer thickness and material properties, but the relationship breaks down near edges. The program described in this report, however, models the effects of actual materials rather than using the nebulous surface impedance boundary condition. rr r FORM 4 A/ "T LiU,,Nov,,, 1 /. UNCLASSIFIED SFi~ urily ('1.iss il ith itiII

UNCLASSIFIED. - Securitv' Cilassificat ion 4. L. IIJ K A L LIN K I LINK C K EY WORDS -.... HO L E WT R OLf E WT f40 L E ' T far field scattering cylindrical body two-dimensional absorber sheets surface currents computer program integral equations impedance surfaces radar cross section radar camouflage radar absorber material K I. -- -1 - -- - - - I.. - t. — - J - -- - -- f -, UNCLASSIFIED S~'ilrlV ('.iv.>I ti,..>ltli