I- 11 ob UMIRL 048 v MICHIGAN. I ~ UNI VERS ITY OF 2541-1-F = RL-2058 2Sh1..J.-F 2541.1-F I J I1 - 19875 TH'RADAR GROSS-SECTION OF THE B-57 AIRCRAFT AT X- AND S-SBANDS by Jo W. Crispin and T.* B. Curt z Work performed for the OC~6L AEONAUTICAL lABORATORY, INC,,, Buffalo, New York. 4 & 1pA I~~ O"'n O.A.L. Subcontract No. S-56.-80 under Contract Ai l13(600)4155 2? June )$56 GDS 5Opjllrt.ECUF, 1 ~ 0 op ".'gQcY NAL~2 s 1bjec l (PD APP OVID By -C:ap* -2&eJ6 - 'o we6 Moeve M~. Siegel~, Pr'oject supervisor *0 OSRAgAT 1 2 YEAR INTER VALS) T AJrT!JAL-fl DECLA pEN0~ Go IR 5208.1 ~, ~LLLLL I U piohIbIt by F; fE.C4U ig 17 -- I -IFT-V IV i I I

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__ __ 1_ UNIVERSITY OF MICHIGAN 2,1 -L-F rAlS4.j3 J' Gd OILTSiS: Section TITLE Page List of Figures iv List of Tables v I Introduction 1 II Theoretical Cross-Sections at X- and S-Bands 6 III Comparison of Theory and Experiment 23 Appendix A Geometrical Breakdown of the Aircraft and Computational Procedures 25 A.1 Introduction 25 A.2 The Fuselage 27 A,3 The 4Iing Tanks 31 A,4 The Engines 32 A,5 Wing, Stabilizer, and Rudder Surfaces 36 A,6 Multiple Reflections 45 Appendix B Combination of Component Cross-Sections 52 B.1 Introduction 52 B.2 Shadowing Effects 52 B.3 Summing the Component Cross-Sections 54 Referenoen 57 * j. LM IP L

UNIVERSITY OF MICHIGAN 2541-1-F LIST OF FIGURES Number 1-la 1-lb 1-ic 1-2 Title Side View of B-57 Showing Coordinate System Used In Computations Top View of B-57 Showing Coordinate System Used In Computations Front Fiew of B-57 Showing Coordinate System Used In Computations Coordinate System (Defining the Aspect Angles 3 and ( Page 2 3 4 m 2-1 thru 2-7 2101 2-2 2-2 2-)4 2-h 2m6 2-7 2-6 thru 2-nl4 2-6 2"9 2-10O 2-11l 2-12 2-13 aadar Aaaar Htadar Itadar ftgdar RadafP R~adgr Agsdar Addgr Ar~ddr1 Cross-Sections of the B-57 at S-Band Cross-Seotion of the B-57 I Cross-Section of the Oroes-Section of the Crosn-Section of theI CroamS-ection of theI Crooa-8eotion of theI Cro~s -Setion of the I Urn-Sections of the Oross-Section of the Cross-Section of the Cross-Section of the rose-Sec~tionl of the 1 Oposi on of the i Croses-ection of the I B-57 i B-57 B-57 83-7.B-57 1 B-*57 i ~-Sm 7 84~~7 E 847 e at at at at at at at at at at S-Band I S-Band I S-Band I S -Band I S-BandI S-band i S-Bland I XmB and X-Band I X-sB mid f X-Band f X-Brand f X-Band f or f or for ror ror V or tMI5 — i-m 00 150 30 0 45O 600 750 B-1.4 8 9 10 22 13 14 15.%21 16 17 16 19 20 21 r orA~ ror 4 orf ~/0 =15a i 300 r 600 2-14 Radar Cross-section of the B-57 at X-Band for,/O 7 0 I Ii — IV

C@N FI F IEIN IIAt UNIVERSITY OF MICHIGAN 2541-1-F LIST OF FIGURES (Cont'd) Number 3-1 A-1 A-2 A-3 A-4 A-5 A-6 A-7 A-8 A-9 A-10 2.1 Title Comparison Between Theory and Experiment at X-Band ( f0oo) Coordinate Systems Breakdown of the Fuselage Breakdown of the Radome Section Breakdown of the Wing Tank Breakdown of the Engine The Elliptic Cylinder The Truncated Elliptic Cone The Ellipsoid The Tapered Wedge Double Reflections SIST OF TABLS Radar Creoa-Seetion of the B-7? at X- and S-Bande For p s900 Page 24 26 28 30 31 34 37 39 o4 a6 622 22 I -F W\JFH JENT/A\L

COIIF I IfDEIN/A\L UNIVERSITY OF MICHIGAN 2541-l-F I INTRODUCTION The Cornell Aeronautical Laboratory, Inc. (C.A.L.) contracted with The University of Michigan to determine theoretically the monostatic radar crosssection of the B-57 Aircraft. These computations were begun on 7 May 1956 and were performed under C.A.L. subcontract No. S-56-80 under C.A.L. prime contract No. AF 18(600)-1550. The problem involved the determination of the cross-section of the B-57 aircraft at X- and S-bands for horizontal and vertical polarization as a funotion of the azimuth angle for eight different elevation angles; the elevation angles ranged from 15 degrees above the horizontal to 90 degrees below the horizontal in 15-degree increments. The coordinate system used for these computations is shown in Figures 1-1 (a, b, and c) and 1-2. The first three of these figures show the orientation of the rectangular coordinate system with respect to the aircraft, and Figure 1-2 defines the polar angles a and r used to denote elevation and azimuth, respectively. To determine theoretically the cross-section of a complex shape such as an aircraft we carry out a series of three steps. The first step, necessary for economic reasons because of the extreme complexity of the mathematical expression for the shape of the aircraft, is to select proper combinations of simple shapes for the purpose of computing the cross-section of the aircraft itself. The second step is to determine the cross-sections of these parts, Here it is necessary to approximate the cross-sections for the simple shapes when their results are not known exactly. The third step is to combine the cross-sections 1 D~F fOE1/:L

786 -nii - FIG. 1 -la SIDE VIEW OF THE B-57 SHOWING COORDINATE SYSTEM USED F 1 IN THE COMPUTATIONS (All Dimensions Shown Are in Inches) I - _ NtcI t__ g ^, C) z

I - I m U FIG. I- I b TOP VIEW OF THE 57 SHOWING COORDINATE SYSTEM USED N THE COMPUTATIONS (Al Dimensions Shown Are in Inches) z w C:: U) -i-n w - 1 an:Z ^ _ lime I

CONF FlIIEINTIL/A4L UNIVERSITY 2' OF MICH I GAN 541-l-F 1 z 383.71 I I. 10~ Dihedral 4~ 21' Dihedral I FIG. - 1 c FRONT VIEW OF THE B-57 SHOWING COORDINATE SYSTEM USED IN THE COMPUTATIONS (All Dimensions Shown Are in Inches) ColNFII4 T/AL LOHFM 1Uufi f /AXL --

COI FI EHNI/A\L UNIVERSITY OF MICHIGAN 25l-1 — y x To Radar FIG, 1-2 COORDINATE SYSTEM (Defining the Aspect Angles 3 and ~') of the parts of the aircraft, computed with the simple-shape approximations, in some appropriate manner so as to give the cross-section of the entire structure These three steps and their application are discussed in detail in Reference lo The details of the application of the first two steps to the problem of determining the cross-section of the B-57 are given in Appendix A. The application of the third step is discussed in Appendix Bo The results of the computations are presented in Section IIo 5 C8~U F BDENIkIl/A\L m -,

WCH FI EIN1F IHE II/AL UNIVERSITY OF MICHIGAN 2541-1-F II THEORETICAL CROSS-SECTIONS AT X- AND S- BANDS The theoretical monostatic radar cross-sections, -(,/3,), for the B-57 are presented in this section for X-band (A= 3cm) and for S-band (A= 10om). Both horizontal and vertical polarizations are considered. From Figure 1-2 it is seen that the unit vector, r, defining the direction from the radar to the origin of the coordinate system is given by r = (-cosos/3 oo ) + (-cospsin") iy + (sin/3) i Thus, we can define horizontal polarization to be the case for which the electric vector has the direction A '0\ o P (-sin ix + (cos a) i., for /3< 90~; and h - Ph= - ' fore= 90~. This results in vertical polarization being given by v = (sinjcos ) ix + (sin/sin') i + (os/3) i., for3 <900; and Pv = y for,3 =90~. The computational procedures used are given in detail in Appendices A and B; the results are presented in graphical form in Figures 2-1 through 2-14. Each figure displays CG( /, (') as a function of r for a fixed value of the elevation angle, /. Both polarizations are included in each figure. Since for D3 = 90~ the cross-section is independent of d, a graphical display of the computational results is not required. Thus, the /3 = 90~ results are presented C FI EnI6 IF, IENLA\L

COINIF DEIN1 TI/AL UNIVERSITY OF MICHIGAN251-l1-F in Table 2.1. Table 2.1 also contains the values of the cross-section at other aspects near o = 90~ to illustrate the pattern to be expected in the vicinity of 1 =90~. The cross-section was computed for each value of 3(P = -150(150)750) for (= 0~(10~)1800, and for those values of jat which sharp peaks in the crosssection occur. These peaks are very sharp (in v), and in the graphs we have shown upper bounds for the peak widths (Appendix B.3). The radome located at the nose of the fuselage of the B-57 was assumed to be completely transparent and the interior of the hole was assumed to reflect the incident energy in an essentially isotropic fashion (see Appendix A). Many of the angles involved in determining the cross-seotions for the B5$7 had to be measured with a protractor, and since some of the approximation formulas used are very sensitive to changes in the angle parameters, a slight ditff nore in the measurement of an angle on an airoraft drawing can result in marked differenaes in the oompeuted value of oross~-setion for a particular choice of P, (, and, * Thia faat should be kept in mind in examining Figures 2-1 through 2-14 and Table 2.1, espeoiall in the vicinity of the sharp peaks. CONlFIDENllT/A\L.

COIN FID EINITI/AL U NI VERSITY OF 2541-1-F M I C H I GA N 106 8 6 4 2 8 6 4 2 104 8 6 4 2 Cb b 8 6 4 2 102 101 100 8 6 4 2 8 6 4 2 8 6 4 2 10-1 y (DEGREES) FIG. 2-1 RADAR CROSS-SECTION OF THE B-57 AT S-BAND FOR 8=-15~ 8 ccim01Fi DDJEN Th/k\LL.

COI i F DEIN TTi/AL U NI VE RS IT Y OF 2541-1-F MICHIGAN 106. % 8 6 4 2 S05 v 8 6 4 2 104 8 6 4 2 103 E 8 6 4 2 102 I 8 6 4 2 101 6 4 2 i0o V W 6 4 2 101 y (DEtORES) FIG. 2.2 RADAR CROSS-SECTION OF THE B-57 AT S BAND FOR p9=0' -- 9 LH ET/\ I1_: - in -M i ~ I - - -

COIkf F IDLENTI/Al U N I V E R S I T Y OF 2541-1-F MICHIGAN - 106 8 6 4 2 105 8 6 4 2 104 8 6 4 2 103 - CM b 8 6 4 2 102 8 -6 ~ 4, in 2 101 8 6 4 2 100 8 6 4 2 10-1 Y (DEGREES) FIG. 2-3 RADAR CROSS-SECTION OF THE B-57 AT S-BAND FOR 83=15~ 10 -

CO NFIIDEINIT-/AL -.. U NI VE RS ITY OF MICHIGAN 541-1-F 106 8 6 4 103 0r4 E,, 102 101 1 tO0 1 V (DEIORES) FIG. 2-4 RADAR CROSS-SECTION OF THE B-57 AT S-BAND FOR 8=30' COI- I11 -DE I,I/ ---

CON F1I DEIN l AL U NI VE RS ITY OF MI CHIGA N 2541-1-F 106 8 6 4 2 8 6 4 2 104 8 6 4 2 103 Cb E b b 8 6 4 2 | c -. 102 o 8 m 6 o 4 a. 4 U 2 101 8 6 4 2 100 10-1 8 6 4 2 y (DEGREES) FIG. 2-5 RADAR CROSS-SECTION OF THE B-57 AT S - BAND FOR = 45~ 12 CL j F INFIEL -.

CJBI, F1 [EINIT/A\L UNIVERS ITY OF MICHIGAN A I 2541 - 1 - 106 104 8 6 4 2 E 102 100 loi V (DEOREES) FIG. 2-6 RADAR CROSS-SECTION OF THE B-57 AT S-BAND FOR 9=:60' 13 0ONHF E:NT I/A\L

tUN I V li I4'Y OF MIA. iN 2541J.1 —F 8 6 4 2 jQ05 8 6 4 2 8 6 4 2 8 6 4 2 Io ~1cm -Hor. Pot. - — Vert. Pot. -, -,-, -o LI C a I I I -- 4 —. —f - I I I I I a 57t C4 0 -. o o 0.. I I 1 o2 I 0 1 I I ( 8 6 4 I.. o I LI4-4-L — iII.J- 10 6: ___ — 7 ----. — 2 -----—. — ---- 8 I 1 -4-. — + — i — I --- T — I — lT I i I II - I 1.1- i — + --- I - i - i i i i 101 8 0 O 8 1-4 - 20 160 Y (DEGREES) HIG. 2- 7 RADAR CR0 ZS -,~)ECTION OF THE B-57 AT S- BAND FOR j/ 750 - 1 4 rr ri~ I,-. Eli~ I AL

C0 1 F IIFI D BIFi7/A\L UNIVERSITY OF MICHIGAN 2541- -F 106 8 6 4 2 5 8 6 4 2 1 8 6 4 2 E %I 103 8 6 4 2 102 101 8 6 4 2 6 4 2 100 8 6 4 2 10'1 V (DEGREES) FIG. 2.8 RADAR CROSS-SECTION OF THE B-57 AT X BAND FOR.8- 15 15 C8hiHlF.EN T/A\L.. --

F Fr U N I VE R S ITY OF MICHIGAN 2541-1-F 106 6, X = 3cm 2 - --- Hor. Pol.- - - 105 Vert. Pol. l05 8 --- I 4.8- - 4 --- --- -- --- -- --- -- --- --- 1.8 4 4 ii 104 Io2ESE (DEGREES)S= FIG 2-9 RADAR... 9 0 4 ofI 1 /fI I I V- S --- - -- - -- - -- - - 6 ffi- - _.._ 6 l;=... -,....,( - 6 I L} 102 o ==, = R I o-m II I, 0 40 80 120 160 Y (DEGREES) FIG. 2- 9 RADAR CROSS-SECTION OF THE B-57 AT X-BAND FOR 8 0~ 16 C0NI F DEI T/I\L

C: OFl DN e ENI/A\L VERS ITY OF MICHIGAN 2541.1-F UNI 106 I 1 104 1 4 A AI io3 "4 102 101 10-1 y (DEGREES) FIG. 2, 10 RADAR CROSS-SECTION OF THE B-57 AT IX-BAND FOR ~ a 156 CO FI17

COWF ll EINIfTI/AL r --- —I UNIVERSITY OF MICHIGAN 541-1-F 106 8 6 4 2 8 6 4 2 104 v 8 6 4 2 103 Cb E b 8 6 4 2 102 8 6 4 2 101 8 6 4 2 100 8 6 4 2 10-1 0 40 80 120 Y (DEGREES) 160 FIG. 2-11 RADAR CROSS-SECTION OF THE B-57 AT X-BAND FOR 8 = 30" 18 /-I 6 ~ is E/ CIO)NIFI~~NlnlBI....

OC F1 DE NTIIAL -UNIVERSITY 25 106 8 6 4 26 - 105 8 4 2 -- 104 6 4.21 cii i - 6 4 l:.. - 102 -III I!.I OF MICHIGAN 541-1-F 100 10-1 Y (DEGREES) FIG. 2-12 RADAR CROSS-SECTION OF THE B.57 AT X-BAND FOR p445' 19 e&lF.E1NT/A\LL

UNIVE RS ITY OF MICHIGAN 2541-1-F 106 8 6 4 2 8 6 4 2 104 8 6 4 2 8 6 4 C4 b 2 102 8 6 4 2 101 8 6 4 2 100 8 6 4 2 10-1 y (DEGREES) FIG. 2-13 RADAR CROSS-SECTION OF THE B-57 AT X-BAND FOR 8 = 60" 20 C, 8 IN! Fl flE ITr/A~1L

Co 8l F- I I nD r l\ II UNIVERSITY OF MICHIGAN 2541-1-F 10 6 8 6 4 2 8 6 4 2 104. 'W 8 6 4 2 E 8 6 4 2 102 101 8 6 4 a a 6 4 100 o10' L 0 y (DEROEES) FIG. 2-14 RADAR CROSS-SECTION OF THE B-57 AT X-BAND FOR fa 75' 21 @CNI FII DEINITI /AL

C N FIID EINTII/AL U NIVE RSITY OF MICHIGAN 2541-1-F Table 2.1 RADAR CROSS-SECTION OF THE B-57 AT X- AND S-BANDS FOR A v 90~ cross-section (in m2) /. X-band _S-band ( 7 730 (o 79~ (0k 730 i, 79~ 79~ 8.2 x 103 8.7 x 103 2.5 x 103 2.6 x 103 85~ 7.7 x l1o 8.2 x 103 2.2 x 10 2. x 103 90~ 8.2 x 103 2.5 x 103 i. iii, i.iiiii — M 1 22 ---— 2 J NI FII DEIN TII/A\L

CON FID~IHENrT/A\L UNIVERSITY OF MICHIGAN 2541-l-F III COMPARISON OF THEORY AND EXPERIMENT Estimates of the radar cross-section of the B-57 aircraft have been computed as functions of the azimuth and elevation angles for X- and S- bands, It is believed that the values of cross-section given here are for the most part correct to within 6 db. It is to be noted that for some ranges of azimuth angle (notably in the vicinity of sharp peaks) the values of cross-section given here should be considered as allowing a one- or two-degree tolerance in the azimuth angle. The Telecommunications Research Establishment of England has performed dynamic text measurements of the cross-section of the Canberra B2, The measurementa were made at X-band, and the results of the experiment are reported in teferenoe 3, Their results are for a.radar in the plane of level flight ( O) andi the aspect is given in terms of the bearing of the radar relative to the line of flight of the aircraft ('). The average values of cross-section reported were derived by weighting each run according to its duration. Their results for the Canberra B2 are shown in Figure 3-1, where these experimental data are compared with the theoretical evaluations reported in Section IIo Examination of Figure 3-1 shows that there is good agreement between theory and experiment at the aspects for which experimental data is availableo Apparent discrepancies could easily be due to slight variations in the aspect angles /j and W, The authors would like to express their thanks to Mo L. Barasch and R, E. Kleinman for their assistance in setting up the computations, and to Be Bernstein and A. Nelson for their help during the computational phase of the program0 - 23 C INI FIB E Nl ll/AXL

C8NI F DEIN T/AL UNIVERSITY OF MICHIGAN 2541-1-F 106 8 6 4 - 2 1 I I I I I 1 =1 1 105 8 6 4 2 104 8 6 4 2 C4 E z 0 Io. u i — U 0 s. u 102 101 100 10 -10- 1 y (DEGREES) FIG. 3-1 COMPARISON BETWEEN THEORY AND EXPERIMENT AT X-BAND (=)0~) 24 Dg IF1 f-l7JEJNTDA\If.

OINI FII DEIN l/AIL UNIVERSITY OF MICHIGAN A 2541-1-F APPENDIX A GEOMETRICAL BREAKDOWN OF THE AIRCRAFT AND COMPUTATIONAL PROCEDURES A.1 INTRODUCTION For the purpose of computing the radar cross-sections of the B-57, the aircraft is broken down into twelve components and each component then replaced by a combination of simple (mathematically speaking) shapes. The components together with the cross-section symbol used to indicate the magnitude of the contribution of each to the oross-section of the aircraft are listed below: the fuselage (1), the left wing tank (0C2), the left engine (0G3), the left inner wing (AG)* the left outer wing (s), the left stabilizer (G), the rudder (7), the right wing tank (G~), the right engine (CO9), the right inner wing (lO) ' the right outer wing (ll), and the right stabilizer (_2). Also included is a thirteenth contributor, 13, representing the contributions of multiple reflections. The remainder of Appendix A is devoted to the further breakdown of each component and to the formulas used in computing the Hi se Component coordinate systems are used throughout this appendix, and in each case 25 II IIE 1frl I__________________ _ 25__________________

COlINIlIDE NTI/AL. -U UNIVERSITY OF MICHIGAN. 2541-1-F the relationship between 1 coordinate system is givei are shown in Figure A-1. z the component coordinate system and the aircraft a. The aircraft system and a typical component system ZI. To Radar PI /....,.Y I x' XR Radar x Typical Component System Aircraft System FIG. A-1 COODINATE SYSTEMS The oross-section formulas employed are, unless otherwise noted, taken directly from Reference 1. The derivations of these formulas plus a general discussion of the methods of employing them for the determination of the orosssection of an aircraft are given in Appendix A of Reference 1. The oross-sections were computed at two frequencies (o\ 0.03m and Xa= O.lOm) for horizontal and vertical polarizations. A discussion of the shadowing effects and of the combination of the component oross-sections which yield the crosssection of the aircraft itself is presented in Appendix B, 26 F

CON F LDENT/Al U N I VE R S I TY OF M I CH I GAN 2541-1-F A.2 THE FUSELAGE ( 61)) The fuselage is considered to consist of five sections: the radome nose, a prolate spheroid, a cylinder, a truncated ogive, and a prolate spheroid (Fig. A-2). Denoting the contribution of the first by 0,1, the contribution of the second by 01,2, etc., we find that only one of these contributors is of significance at each aspect with which we are concerned. The radome and front prolate spheroid combined can be approximated by the surface x12 + y2,12 X' Y + 1, b2 c2 with b = 0.99m and c = 6.35m. The cylinder can be approximated by the surface x,2 + y'2 a2, with a = 0.99m; the length, L, of the cylinder is 3.94m. The axis of the ogive section is tilted up approximately 4~ above the z' axis in the x'z' plane, and the ogive is defined by an arc of a circle of radius 64.8m. The upper portion of the fuselage within the limits of this ogive section could be considered as a truncated cone faired into the ogive; this upper portion will not contribute to the cross-sections computed because the angle of elevation is limited by d5 - 15~. The radii of the ogive at the truncation planes are 0.99m and 0.34m. The rear of the fuselage is approximated by a prolate spheroid defined by x, 2 y + y z2 b2 c2 27 This document contains information affecting the notional \defense of the United States within the meaning of the Espionage Laws, (Title 18 U. S.- C., Sections 793 and 794). Its transmission or the revelation of its contents in any manner to an unauthorized person is prohibited by law.

_ _ __ z (-x') st91 lJl -— I o I AZi - C__ C/) 4 I &0 mnril I _ ~ s r n ] 1 - Radome (Prolate Spheroid) 2 - Prolate Spheroid 3-Cylinder 4 -Ogive 5 - Prolate Spheroid 6-Truncated Cone (Cross-section Contribution Negligible) FIG. A-2 BREAKDOWN OF THE FUSELAGE 0 z> Z

UNIVERSITY OF MICHIGAN 2541-1F where b = 036m and c = 0.795m. By consideration of the importance of each contributor we see that 1 1t1' for QQc650 1,. for 650< 9(900) 1$2' 1, for = 90g for 900 (9<980, and ls4$ = 1,5' for 980 Q l800 with oo 0 = 00os Acos a. Formulas for the C'. are li 'Wb4c2/(a2oS2Q+b2sin2@)2, (b = 0.99m anda= 6.35m) 1,2 -2 d 2' O21rr La/X (L - 3.94m and a 0. 99m) 1,93 1,4,; K in0J i ~I 64.8in and h - 63.8m)* 0*1 'l5 1rb402/(o20082g +b2sin02)2, (b 0".36m and c = o.795m), The radome contribution requires a few additional comments. A larger and more detailoed drawing of the radome Section is shown in Figure Ai83, The osur aoe of the radomte is asued to be completely transparent to the incident energy. Under this assmption the surface irradiated by the incident energy at the join of the radme and fuselage can be approximated by a truncated ogive of halPgngle 250 capped by an annulus. The annulua has radii of O.318m and 0o27.me Since the ogive contributes only in the interval 900 ( 9(980, the tilt in the ogive axis is neglected., 29

C 0 IN F II D E INITlII/AXl rE RS ITY OF MICHIGAN ---- A.2541-1-F - I UNIV / L 250 Radome Surface -, / / / F-T ""'-" Metallic Surface of Fuselage /Antenna (APG-31, Prov.) --— Dehydrator (APG-31, Prov.) Converter (APG-31, Prov,) I FIG. A-3 BREAKDOWN OF THE RADOM SECTION The radar equipment covered by the radome and located inside the opening has only been provisionally specified, Thus, because of the uncertainty of the type of the equipment, as well as the size of the equipment, it has been assumed that this equipment and the interior of the opening would scatter in an essentially isotropic fashion. This is approximated by the return from a sphere whose radius is (0.270 cos Q)m. Thus, = [W(R - R )]i + [tRa n J 2 1,1<4 o.r sin.9 COiFI.. - 30 C 01F1 FII DE I~ITII/AL

C 1 FIIIDEINITM11/L UNIVERSITY OF MICHIGAN 2541-.-F 2 [RI+Rlh R Rjtai^Cft S2n2^-d ) g 2 C* = -L- - 1 +, -, -,, )]- + * rR2cos G, 1,1 I8rsingtan2t 8 irsi* for 0 < <o; and 2 [R1 + R2A R tan ( + ) 2 2 =........ +2 +R3cow, for oQ9 < 65~ 8 Irrsinantan2 8 rsing with oos ~ = oos Oooes ". The resulting curve for U,1 is then smoothly faired into the ourve for,2 in the interval 600( Q < 70~. A.3 THE WING TANKS (C2 AND C8) The wing tank is approximated by a prolate spheroid faired into an ogive, as illustrated in Figure A-4. '(-x' ) 0.U (I I <-1.31m FIG. A- BRAIADOWN O THE WING TANK The dimensions of these shapes as well as the relationship between the aircraft and the component coordinate systems are shown on the figure, oF31 I IL CtlJ HFlIIDE ITH 'AX\L

CONFIIEINITII/AL U NIVERSITY OF M I CH I GAN -r I.. 2541-1-F Since at all aspects (disregarding shadowing for the moment) we have 6i'= 't, it will suffice to define only C72, The magnitude of CG2 as a function of /3 and z- is given by G = rb42/(c2cos2 +b2s sin2), for 0~ < 90 where b = O.40m, c = 1.31m, and cosa = ooe/ cos t; and /(1 -f h 9), for 90o< (9go +, /O sing 2 0 = /o2sin2 c/4h, for 90 +0(= G, and A2tan4o, 4'0 mm~~n......................... where /;= h= cosQ = cois3 - ------------— v T --- *V* Vr %.* L U V s 16 Trcos6Q(1-tan2o tan2)3' cos'(h//O),150, 11.4m, 11.Om, and Cos cos.s '. A.4 THE ENGINES ( C ANDo ) From the front the engine is considered to consist of a torus surface with a small prolate spheroid located in the center, The outer radius of the torus is 0.39m and the inner radius is 0.29m. The prolate spheroid has a semi-major axis of 0.25m and a semi-minor axis of 0.20m (only half of the semi-major axis is visible when viewed from the side of the airoraft). Viewed from the side, the engine is considered to consist of a truncated ogive, a cylinder, and a second truncated ogive. (The top of the engine is not cylindrical; however, for C | 32

C INFIIDEOUEINTI/AL UNIVERS ITY OF MICHIGAN 2 541-1-F those aspects for which the top of the engine would be considered, the stationary phase point on the surface of the engine would be inside the wing and thus would not contribute to the cross-section.) The front ogive has radii of 0.60m and 0.39m at the truncation planes, and the truncation planes are 0.82m apart. The cylinder has a radius of 0.60m and a length of 4.48m. The rear ogive has radii of 0.60m and 0.29m at the truncation planes. Viewed from the rear, the engine is considered to consist of three sharp edges (modeled as wire loops) of radii 0.29m, 0.28m, and 0.21m. The structure baok in the interior of the "hole" (in the case of the F-94C this i1 about 6 fent inside the hole) is approximated by three wire loops of radii ranging from 0.l4m to 0.15m. This breakdown is illustrated in Figure A-5. Nogleocting shadowing effects for the moment it is saen from the airotraft drawings that C = a. Thus, it will suffice to restrict ou attention to | With the zi axis coineident with the x axis and the yt axi oQincfidnt with the y axis, the breakdown of the engine desQribed ebove and illustrated in figur~ A-5 leads to the following expressions for s: A + [tbI )40)2] / [(oI)2o0o2Q + (b)2in2'] where A.0Q ab2/, f o' = Oe, 2' Irab/sinW, f or 0~ < <53, and a - O.05m, b' -- 0.20m, b = 0o.34m, o' = 0.25m, and cosG = oos /cos J'. 33 I C1l FllDEEHN T/A\L

U N I V F EI I /A 'ERSITY OF MICHIGAN 2541-1-F SIDE VIEW 0.58m. - REAR VIEW (All wire loops) r = 0.29m r2= 028m r3 0.21m r4=0.15m r5=0.14m 1 - A Truncated Ogive 2 - An Ogive Faired into the Cylinder (Contribution Negligible) 3 - A Cylinder 4 - An Ogive 5 - A Torus 6 - A Prolate Spheroid FIG. A-5 BREAKDOWN OF THE ENGINE 34 __ L — CmNIFI IFINIlTIIIlL

U NI VE R SI TY OF MICHIGAN 2541-.1-F where /10= 1. 7lm, h = 1.1llr, arnd cosQ = 034oos {.O = Tr L a/s 3 where L =4.h8mn, and a X 060ra. whero/0 4,82mn, h 4,22m, and ~ ~ [an2(9 - 220) where a 0*O29mp and ooog a oos,4 oio ( CE. ~~a2(0+ 220) +~ ban2 (Q u220)] where a 0i.29m# and coQ oo a oil w' ra 2tan2(220) +lrZ (rj)2 i)-m LdlloJ ~F4INTV/Ad

COiiN,llBEINIq'TI/AL UNIVERSITY OF MICHIGAN 2541-1-F where a = r = 0.29m, r2 = 0.28m, r3 = 0.21m, r4 = O.l1m, r5 = 0.145m, and r6 = O.l4m. A.5 WING, STABILIZER, AND RUDDER SURFACES Each inner wing ( an and 0 ) is considered to be composed of an elliptic cylinder faired into a tapered wedge. The outer wing ( and ), stabilizer (6~ and 12), and rudder ( G) surfaces are each considered to consist of a truncated elliptic cone faired into a tapered wedge with the t"wing-tipt approximated by an ellipsoid. In terms of their coordinate systems the elliptic cylinder contribution is given by 2rL2a2b2, for 0= 900; (a2cos02 + b2sin2 )3/2 for 9: 90~ the contribution is negligible. The ooordinate system and the meaning of the parameters is shown in Figure A-6. The contribution from the truncated elliptic cone is negligible except at the aspects defined by - 8 tang = tano( sin20 + 20co0s2 36 COI IF D EIDIE II/A\L

CINFIIENTIF -I/AL UNIVERSITY OF MICH IGAN 2541-1-F II // // fL/ // \ /~';. A-6 THE ELLIPTIC CYLINDER 37. - - -

(CION F UIE!NTITl/A\L UNIVE RS ITY OF MICHIGAN 2541-1-F At these aspects the cross-section contribution is given by ar [() 03/23-)+2 2 tan4 9 A >2 COS3 e The coordinate system and the meaning of these parameters is shown in Figure A-7. The spheroid cap contribution is given by Tr(atbtc )2 Gv = "T --- —" --- —----— "" j [(a')2sin2Qcos2$+ (b')2sin2Qsin20+ (c')2cos2] with the coordinate system and parameters as shown in Figure A-8. The tapered wedge contribution is given by L2 cos + A, d (to.)2 where 2 2 A = L tan (o2 ) for o DC 4r L2sin2(2o)0 --- — L2in-2- —.- for 0~40<c, and 4 rcos2 (c -e)cos2(o +( ) = the angle between the polarization vector and the plane formed by the direction of incidence and the edge of the wedge; the remaining parameters and the coordinate system are shown in Figure A-9. Since the contributions from the right wing surfaces can be obtained by symmetry from the contributions from the left surfaces, it will suffice here to restrict our attention to GC, G0, cg', and CO. 3 8 ' COIN1,F ''EINITIfAL

88 M~F -E Nt, TI/~LL UNIVERSITY OF M ICHIGAN 2541- -F tona = a/L2 r/ = a/b y FIG. A-7 THE TRUNCATED ELLIPTIC ( 39 Ct D F DEIHIJ ~

COINIFIIDEINE xIi/AL U N I V E R S I T Y OF MICHIGAN 2541-1-F To Radar (O,Oc') / / / I I I I/ I ' (O,b',O) y 40 - - \ I \l FIG. A-8 THE ELLIPSOID 40 OsIFIFEITI/AIL..I -- I 1 — 3

. --- —- U NIVE RS ITY OF M I CH IGAN 2541-1-F ' A To Radar A I 11', y I LI FIG. A- 9 THE TAPERED WEDG..... 41 @lnl~F -nlrl

UNIVERSITY OF MICHIGAN 2541-1-F The left inner wing The elliptic cylinder of length, L, located in the front is defined by (x'/a') + (y'/b)2 = 1, where a' = 2.31m, b' = 0.348m, and L = 1.69m. The tapered wedge has a length, L, of 1.69m, and a half-angle, <, of 5~. The relationship between the cylinder coordinate system and the aircraft coordinate system is given by sin /3 c cos(2~)sink, tan ('' sin(2~)tan$. With regard to the edge of the wedge, the direction of incidence is normal to the edge if sin <'r tan(2~)tan/3, and thus the relationship /C 9@ is used in relating the component coordinate system to the aircraft system in this normal plane. The left outer wing The truncated elliptic cone is defined by a = 2.31m, L= 3.89m, o = 13.50 b = 38m, L = 9.56m, L= 6.66 The tapered wedge has a length of 6,04m, and a half angle of 5~. The ellipsoid cap is defined by (x'/a')2 + (y'/b')2 + (z,/c')2 = 1, where a' = 1.17m, b' = 0.142m, and c = O.567m. C FDTII/A42 VONIFIERNITM4/L

CGON F DENEIIT/A L UNIVERSITY OF MICHIGAN 2541-1-F For the ellipsoid and the cone the relation between the component coordinate system and the aircraft system is given by sinp3= -sin(20)oos(4021t )singcos0-cos(20)cos(4021' )singsinO+sin(4021t )cosg; oosGcos s= cos (20)sincos-sin(2)s(2)sinsin; oosa sin(= -sin(20)sin(0421')singcos$-cos(20)sin(4~21')singsin0-cos (421')oosO. For the tapered wedge the direction of normal incidence to the edge is given by cos( - 70~) tan(4~21 ) tan / and the relation p = Q in this plane is used to relate the component system to the aircraft system. The left stabilizer The truncated elliptic cone is given by a = 1.4Om, L = 3.00m, oC 12~; b = 0.168m, L2 = 6.58m, 7 8.35. The tapered wedge has a length of 3.58m and a half-angle of 5~. The ellipsoid cap is given by 2 2 2 (x'/a')2 ('/b')2 + (z '/)2 1, where a' = 0.635m, bt = 0.076m, and c' = 0o380mk. For the ellipsoid and the cone the relation between the component coordinate system and the aircraft system is given by --— N Fl- -E —N --- —-.-L

OI FIDEINITI/ALL UNIVERSITY OF MICHIGAN 2541-1-F sin/= - cos(945' )sinQsin0+ sin(905' )cose; tan f Asin(12~)- Bsin(9045')cos(120)- Ccos(9~45')cos(120) Acos (12) -Bsin(9045')sin(120)- Ccos(9045' )sin(120) where A = singcos0, B = sinQsin0, C = cosQ. The direction of normal incidence to the edge of the wedge is given by sin (~ tan(lO0~)tan/, and the relation / = 9 is used to relate the two coordinate systems in this plane. The rudder The elliptic cone is given by a = 1.69m, L = 1.65m, = 27.5; b = 0.147m, L = 3.06m, y= 11.4. The tapered wedge has a length of 1.59m and a half-angle of 5~. The ellipsoid cap is given by a' = o = 0.079m and b' = 0.91m. For the cone and the ellipsoid the relationship between the two coordinate systems is given by sin/3= oos(77.5S)sinecoso + oos(l2,5~)cose; oos/3 aos "= -cos(l2.5~)sineoos0 + cos(77.5~)cose; oos /3 sin = 8.inasin0. For the wedge normal inoidenoe is given by cos {= oot(l5~)tan, and the relation/2 ~ is used to relate the wedge coordinate system to the aircraft coordinate system. CO44DE @/

COINIF IDEBII/ALL UNIVERSITY OF MICHIG A N 2541-1-F A.6 MULTIPLE REFLECTIONS ( 3) Only double reflections are considered in this examination of the effect of multiple scattering on the cross-section of the B-57 aircraft. This restriction is not a serious one since an examination of the aircraft drawings displays a definite absence of "corner reflectors". The methods of geometric optics are applied in this analysis. Extensive use is also made of the material of Reference 2, wherein multiple scattering by N bodies is discussed. Taking N = 2 and approximating each pair of B-57 oomponents in the vicinity of the reflecting points by the surfaces 2 2 xf yi = -. - =-zi, (i= 1,2), il /O i2 where the z. axes are oriented in the direction of the normals to the surfaces (thus iz iz = 0 ) and 2. ~ 1; i. = i. =i *i o, Xl X2 Xl Yl Yl X2 Yl Y2 the material of Reference 2 indicates that the double-reflection contribution to the cross-section, Cd, is given by 2 11 12 21 22o si l[s) in(2)+ cos+ sin 2+ c sin In order for the reflected ray to return in the direction from which it came it is necessary that the normals to the two surfaces at the reflecting points be perpendicular. ----------- ---------— 45 rr 1: 15 L()iH FI BE Nq!AI

COIN F IDE TI/AL UNIVUNIVERSITY OF MICHIGAN 254l-l-F The geometry of the situation as well as graphical definitions of the parameters g and R are given in Figure A-10. R --- —. - - P22?12 % \ / \ FIG. A-10 DOUBLE REFLECTIONS (showing one of the two rays; the other ray follows the reverse path) In the cases corresponding to I = 0 and = 900, one body is in the "shadow" of the other (see Appendix B), or a triple reflection is involved. For these reasons this consideration of multiple reflections is restricted to aspects in the range 15 0 f l 75. With this restriction we see from Equation A.6-1 that d<. 12.22 (A.6-2) R2 This inequality indicates that it is not necessary to consider pairs of B-57 components having radii of curvature which are small with respect to R. It can be observed by examination of the B-57 drawings (Figures 1-la, l-lb, and l-lc) C0N FiDE NTII/A\L

CO-NFI DEINTIIT /AL UNIVERSITY OF MICHIGAN 2541-1-F that many pairs of components can be immediately disregarded because the geometry is such that their normals are never perpendicular to one another, Therefore only the following are considered: (1) the fuselage-engine combination, (2) the fuselage-inner wing combination, and (3) the "fard wing tank-"far" outer wing combination. The fuselage-engine combination This analysis is broken up into four parts depending on aspect. The first deals with aspects involving the front torus approximation for the engine, the second involves the front ogive approximation used for the engine, the third involves the rear ogive approximation for the engine, and the fourth deals with the remaining aspects. It is readily een from Equation A,6-1 that the "flatter" the surfaoes are in the vicinity of the specular points, the larger will be the doublae-refletion contribution. Thus for the first category of aspects (i.e., cos coo (S> oo Hi, with 6 o), we find that the contribution of the fuselage-engine combination is always less than the contribution computed by Equation A.6-1 with /l l1.0Onm, /12 = 40O3m, /21 / 22 = 0O39m, and R S 2m It is found that this upper bound is always less than the - C oomputed by the methods of the preceding sections of this appendix by a factor of at least three, and thus the double reflection contribution can be considered negligible within the accuracy of the aircraft cross-asetion oomputational procedure used here for this aircraft and for the aspects under discussion. C[8 ~-r m --- —--— 7 --- —-- LOLE~F ENI1/\I

U N I V E I V1 IE C I -FG A N UNIVERSITY OF MICHIGAN,, 2541-1-F The contribution for the second category of aspects has been computed for aspects given approximately by <= 90 - E. (Here 1 = 15~ corresponds tok,4 30~; I= 55~ corresponds to/53680; and -67 corresponds to l 2120 Thel t 15 andJ 670 values represent the geometric limitations for doublereflections.) The contribution to the cross-section for these aspects is estimated using Equation A.6-1 with /~ = 40.3m, /~12 = l.OOm, /~ = 1.71m, 22 = 0.60m, and R,2m. Computation of this estimate shows that its magnitude is less by factors of thirty or more than the sum of the single-reflection contributions and thus can be neglected. The third category of aspects is treated exactly as is the second except that the aspects are of the form I= 900 + ~, and /1 = 64.8m, and /~ 21=4.82m. The magnitude of this contribution (computed using Eq. A.6-1) is always less by at least a factor of ten than the corresponding q i. Thus again the contribution can be neglected* Aspects involving the rear of the engine would also involve the small prolate spheroid tail of the fuselage. The magnitude of the radii of curvature is sufficiently small and the values of R involved are sufficiently large to permit the conclusion that the contribution here is negligible on the basis of Equation A.6-2. As can be seen from Sections A.2 and A.4, the central portions of the fuselage and engine are approximated by cylinders. The single-reflection oross-section of a cylinder varies (for normal aspects) as 1/A, and thus geometric optics ( — 0) predicts an infinite cross-section for the cylinder at these aspects. Thus, physical optics rather than geometric optics would be required to obtain a meaningful estimate of the cross-section contribution 48 -8 F FI COINF IDiENIA? L

CO)I FIllDIENT/ALL UNIVERSITY OF MICHIGAN 2 541-1-F stemming from double-reflections from the two parallel cylinders. (This occurs only for <= 90~ and >3 >30~.) However, since it has been established that the double-reflection contribution is negligible (both for "= 90~ - and (= 9Q~ + E ) and since continuity in the ratio of single-reflection contribution to double-reflection contribution is to be expected, it follows that the double-reflection contribution from the two cylinders at this aspect will be negligible in comparison to the sum of the single reflections. Thus, the actual determination of the magnitude of the double-reflection contribution by physical optics is not required, In summary, the double-reflection contribution of the fuselage-engine combination is always negligible. From this it readily follows that the doubli-reflaetion oontributions from such combinations as the fuselage and the wing tank or the ongine and the wing tank are also negligible due to the maller /~ and the larger values of RH Thi~ Nthe ~elainnor wingE_ combination For p # 0O the breakup of the aircraft given in the earlier portions of this appendix indioatte that a double^refloetion @ontribution from the oombination of the fuselage and the innr-wing surfa4oe in impos-ible since the norma ls to the bodies would nePver be perpendioular to ~aeh other For ' = 0~ txaminatlon of the aircraft drawing indioates that doubl^-reflections twill ooqur for this eombination, Study of the airrcaft drawings ind icates that although the breakup of the fuselage g ive n Sin ction A.2 is adequate for oingle reofle-etions (due to thadowing effQtst), the breakup must be mouified for the double-reflaction estimate. That is, in the /5 0 plane the fuselag:a is spheroidal in shape at the Join of the fuselage and the inner wing with the ' —,,, -I —49 @OIOIFIIDEINIT/A\L

UNIVE RS ITY OF MICHIGAN 2541-1-F normals perpendicular at this join. An estimate of the magnitude of the double reflection contribution stemming from this combination for v = 0~ can be obtained from a modified form of Equation A,6-1. Equation A.6-1 can be written in the form y/'11 /'12 21 =.__ t,_...I..l....,,_.,-2,4.1.. _._._._. in(2,) [Rsin(2 g) +Rincos ) +/csin C + +sin + from which it follows that;, - li-., = Oi1ns2 /](/2 4 0 sain (2 )sin /2j, ao +/11 in Using Cd. to estimate this double reflection contribution, we have J r900 -., /11 = 1.Om$, P/2^40O3m, and /21 = 0.052m. This contribution is not negligible in comparison with the sum of the single-refleotion oontributions and thus is added to 0i in obtaining the cross-section of the B-57 in the interval 0 < 'c 550 for = 0~. The far wing tank-far outer wing combination The double-reflection contribution stemming from the combination of the far wing tank and the far outer wing will be relatively large only in the aspect ranger, 15~, 800 < I< 110~. An estimate of the magnitude of this contribution for (= 900 (aspects for which the contribution will be a maximum) is obtained by using the limiting form of Equation A.6-1 when/O -co, 22 CINI FIADENfTfhIA\L

CWINIF IIDEKHi\FI/AIL UNIVERS ITY OF MICHIGAN 2541-1-F Here we have, /~, and /~11 = 4.29m,/~2 = 0.4m, /~21 = 9.65m, with R = (0.4cscJ)m. This leads to estimates of the double-reflection contribution to the cross-section which are less by at least a factor of ten than the sum of the single-reflection contributions for these aspects. Thus this contribution can be considered to be negligible. -— l --- —----— l/E\L

CON FIED I NII/AL UNIUNIVERSITY OF MICHIGAN 2541-1-F APPENDIX B COMBINATION OF COMPOKENT CROSS-SECTIONS B.1 INTRODUCTION As pointed out in Appendix A, the determination of the theoretical crosssection of an aircraft is broken down into three steps: considering the aircraft to be approximated by combinations of simple shapes (mathematically speaking), using approximation techniques to determine the cross-section contribution of these simple shapes (physical and geometric optics), and combining these component cross-sections in an appropriate manner to yield the cross-section of the aircraft itself. Appendix A contains a complete discussion of the simple shapes used to approximate the parts of the aircraft and a listing of the formulas (taken from References 1 and 2) used to compute the cross-section contributions of each of these component parts. Appendix B is devoted to the third step, the procedure of combining these component cross-sections. The discussion is divided into two parts, one devoted to shadowing effects and the other to the actual combination of the component oaoss-eoctions. B.2 SHDAPING EFFBCTS As pointed out in Reference 1, the cross-section of a partially shadowed component is unchanged unless the part of the body which contributes to the cross-section - that is, the part of the body which the incident radiation hits at normal incidence - is shadowd.e In that oase, since the specularly reflecting part of the body is shadowed, the component contributes only a negligible amount -OFUTUA5 OFIfIF DENI~,TI/A\L

W01M FIIEETI/A\L UNIVERSITY OF MICHIGAN 2541-1-F to the cross-sectiono In what follows relative to the computation of the B-57 cross-sections, the term "shadowed" is applied to surfaces which are "insides the aircraft as well as those which are shadowed in the usual sense, A study of drawings of the B-57 led to the following conclusions relative to shadowing effects (for -15~./3 e90~ and 0~0 a 180~): 1. The fuselage, T-lis in shadow for -15~5o/3S15~ at -'=90~ and for 85~< ( <90~ ato/ 0~* 2. The left wing tank, G2, is never in shadow for -150~L/3900. 3. The left engine, 3, is in shadow for -15~0 645~ at es 90~* 4. The left inner wing, C, the left outer wing, G(, and the left stabilizer, ~, contribute appreciable amounts to the cross-section only at very special aspects and shadowing never takes place at those aspects over the range of involved in the computations. 5. The rudder surface, c0, is in shadow for many values of the azimuth angle for values of/O0~; however, because of the nature of the rudder surface, its contribution is negligible at these aspects and thus detailed information on shadowing is not required. 6. The right wing tankG, is in shadow for 77~s/3,<1100 at s= -15~ and for 55~ < < 180~ at/ = 0~. 7. The right engine, 0~, is in shadow for -15~0 d- 45~ at {= 90~ and for 30~0 1600 at /3 0~. 8* The right inner wing, C~0's has a non-negligible contribution 8. Th rigt iner wng 1 only in the vicinity of - 0O~,/= 0~. Thus detailed information on shadowing effects is not required. H A53 C0iNI FiIN I-rllA I

CJWI, NIF lAIENTI/A\L UN I VE RS ITY OF MICHIGAN 2541-1-F 9o The right outer wing, Cll and the right stabilizer, "12' yield contributions which are always negligible. Thus no analysis of shadowing effects was performed. Shadowing was taken into consideration in obtaining the value of 13 13 (multiple scattering); this is discussed in Section A.6. Thus, in determining the contribution of a given component to the crosssection of the aircraft at a given aspect, the value of the component crosssection is computed according to the formulas given in Appendix A and these values are used unless the component is in shadow. In that case the contribution of the component is taken to be negligible. When a simple shape is partially in shadow, the contribution which arises from the shadow boundary is included. However, since the surface currents at such a boundary are continuous, fictitious sharp boundary effects must be omittedo B.3 SUIMING THE CAOPONENT CROSS-SECTIONS In combining the values of the cross-sections of the component parts of the aircraft, it is assumed that the fields, when one averages the phase contributions of all the parts, are such that these contributions can be added in random phase. This assumption is a valid one for the small wavelengths with which we are dealing here because the phase differences depend critically on aspect and on the dimensions of the aircraft. Since in practice the aspect is changing continually and the production dimensions of the aircraft vary enough to seriously affect phase relations, averaging is the most meaningful procedure 54 CDIWIF cIEIT/A\LL

UNI VE RS ITY OF MI CHIGAN 2541-1-F This leads to the simple addition of the radar cross-sections of the various parts of the body in finding the cross-section of the composite body itself. An upper bound on the maximum error involved can be computed from Formula Aol-6 in Reference 1; at X- and S-bands, however, such a calculation is not necessary. Thus, the cross-section of the B-57D C(% {)3 is obtained by the relation 13 iw 1 where the GC are as defined in Appendix A. If shadowing effects take place the value of the C0i involved is replaced by zero. In connection with the step of combining these component eros sections one additional item is worthy of consideration here. That item involves the question of the width of sharp peaks, auch as those obtained at particular aspects for much shapes as cylinders truncated cones, the tonris, and tapered wedges, The physical optics methods employed to derive the formu!la for thefs contributionn yield one expression for "onomal^" incidenee ind a second oxproemion for tnon-normal" incidence. At short wavelengths th e ont ritlttion at normal incidtnce is considerably larger than at non-normal incidenee. It'f the body Is a^harply terminated, the widths of the peaks canm be measured by using tho rolatonnhips given In section A*l*7 of Reforenco 14 and ^xtensitot of those relation.o If, on the other hand, the onds of the tbdy Ar. not sharply tsminatd but inatead ure smoothly roundedi the rwBult to a "pencil sharptt peak. For the purpose of the presentation of the B-47 data in - e,,, C01S FllE:~ ll/\

CONF11EINEfT/kALLL UNIVE RS ITY OF MICHIGAN 2541-1-F Section II, bounds to the width of these peaks are shown; that is, for the purpose of obtaining an upper bound to the width of the peaks, it is assumed that the contributor involved is sharply terminated. Thus the bounding peak widths are determined from the relations given in Reference 1 and the extensions of those relations. CF FDENT/A\L

COINF DENITT/AL UNIVERSITY OF MICHIGAN 2541-1-F REFERENCES 1. C. E. Schensted, Jo W. Crispin, and K. Mo Siegel, "Studies in Radar Cross-Sections XV - Radar Cross-Seotions of B-47 and B-52 Aircraft", The University of Michigan, Engineering Research Institute, Report No. 2260-1-T, August 1954. CONFIDENTIAL 2. R. R. Bonkowski, C, R, Lubitz, and C. E. Schensted, "Studies in Radar Crosse-Sotions VI - Cross-Sections of Corner Reflectors and Other Multiple Soatterers at Microwave Frequencies*, The University of Michigan, Ingineering Research Institute, Report No, UMM-106, Ootober 1953. SECRET (UNCLASSIFIED when Appendix is removed). 3. S. Ratoliffe, "Aircraft Eohoes at X-Band (1)", Telecomunioations Research Establishment, Gt, Malvern, Worcs., England, Memorandum 750 (June 1953), CONFIDENTIAL, c.i.i.nT57 CO ~1 F B.E.IN TI/AXL