AFCRL-TR-73-0351 011758-2-T THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGFtNUR qG DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINlRING Roditioin Laboratory A PROGRAM INCORPORATING DIFFRACTION FOR THE COMPUTATION OF RADAR CROSS SECTIONS By Eugene F. Knott and Thomas B.A. Senior Contract No. F19628-73-C-0126 Project No. 5635 Task No. 563502 Work Unit No. 56350201 Scientific Report No. 2 June 1973 Approved for public release; distribution unlimited. Contract Monitor: John K. Schindler Microwave Physics Laboratory Prepared for: Air Force Cambridge Research Air Force Sytems Command United State Air Force Bedford, Massahusetts 01730 Laboratories 11758-2-T = RL-2221 Ann Arbor, Michigan

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AFCRL-TR-73-035 1 0 117 58-2 -T A PROGRAM INCORPORATING DIFFRACTION FOR THE COMPUTATION OF RADAR CROSS SECTIONS by Eugene F. Knott Thomas B.A. Senior The University of Michigan Radiation Laboratory 2455 Hayward Street Ann Arbor, Michigan 48105 Contract No. F 19628-73-C-0126 Project No. 5635 Task No. 563502 Work Unit No. 56350201 Scientific Report No. 2 June 1973 Contract Monitor: John K. Schindler Microwave Physics Laboratory Approved for public release; distribution unlimited. Prepared for AIR FORCE CAMBRIDGE RESEARCH LABORATORIES AIR FORCE SYSTEMS COMMAND UNITED STATES AIR FORCE BEDFORD, MASSACHUSETTS 01730

011758-2-T ABSTRACT A computer program is described that calculates the radar backscattering behavior of metallic roll symmetric bodies having one or more ring discontinuities in surface slope. The field diffracted by each surface singularity is represented by a line integral of equivalent electric and magnetic currents. The scattered field can be expressed as an integral of the physical optics currents existing over all non-diffracting portions of the surface and the equivalent currents at the singularities. However, since the equivalent currents themselves contain a physical optics component, the original currents must be modified in order to remove it. The predictions of the computer program are compared with experimental data for cones and for a typical aerospace vehicle. Though the program is almost as efficient as one using physical optics alone, the results obtained are markedly superior. ii

011758-2-T TABLE OF CONTENTS ABSTRACT ii I. INTRODUCTION 1 II. MODIFIED EQUIVALENT CURRENTS AND PHYSICAL OPTICS 3 2. 1 Equivalent Currents 3 2.2 Modified Equivalent Currents 5 2. 3 Physical Optics from Surfaces 8 2.4 Comparison of PO Results via (12) and (16) 13 III. A COMPUTER PROGRAM FOR BODIES OF REVOLUTION 16 3. 1 General Program Considerations 16 3. 2 Generation of Circular Arc Segments 19 3. 3 Using the Program 22 IV. RESULTS AND DISCUSSION 25 REFERENCES 39 APPENDIX: PROGRAM LISTING AND SAMPLE OUTPUT 40 iii

011758-2-T I INTRODUCTION The geometrical theory of diffraction (GTD) is a valuable method for estimating high frequency scattering from non-trivial objects, but though it is well suited for analytical and diagnostic purposes, it is much less convenient numerically. Indeed, it suffers the disadvantages common to all ray techniques, one of which is the identification of ray paths from the transmitter to the receiver and the location of "flash points" on the surface. The divergence factors specifying the decay in field strength along a ray must then be calculated, and at points where an infinity of rays come together, the resulting field infinities must be corrected by incorporating caustic matching functions in the analysis. In the case of geometrical optics, these difficulties can be overcome by using ray techniques to specify the surface field and an integral representation to obtain the scattered field. The resulting method is none other than physical optics and since the scattered field is now given by an explicit integral expression, it is quite convenient for estimating the scattering numerically. This convenience is one of the reasons why physical optics continues to be the main tool for radar crosssection calculation in spite of the known deficiencies of the estimates, particularly in regard to polarization. Since many of these deficiencies could be overcome by including GTD predictions of the fields diffracted by surface singularities, it would be desirable to have a procedure bearing the same relation to GTD as physical optics does to geometrical optics. Unfortunately, a surface singularity is a caustic for the diffracted field. GTD does not therefore provide a valid description of the surface field, but since our main interest is in the far field, a knowledge of the equivalent surface field or current would be adequate. The essential requirement for equivalence is discussed by Knott and Senior (1973) who derive the equivalent currents appropriate to any linear element of a surface singularity. The diffracted field is then given by a line integral of the filamentary electric and magnetic currents attributable to the singularity. 1

01 1758-2-T By using this concept of equivalent currents it is possible to combine physical optics and GTD in a single numerical program in which the scattered field, including first order diffraction from all surface singularities, can be found by computing explicit integrals. However, because the physical optics approximation itself yields a contribution from any surface singularity, the equivalent currents must be modified by the subtraction of the erroneous optics portion. These modified equivalent currents are deduced in Chapter II, following a brief description of the unmodified forms for any surface discontinuity in slope. A computer program for estimating the backscattering from any perfectly conducting body of revolution with any number of ring discontinuities in slope is presented in Chapter III. Although the results obtained in any given instance are virtually indistinguishable from those that would have been found by an analytical combination of physical optics with first order, caustically-corrected GTD, it should be emphasized that the program is almost as efficient as one using only the physical optics approximation. Selected data are presented in Chapter IV where some possible developments of the program are discussed. 2

01 1758-2-T II MODIFIED EQUIVALENT CURRENTS AND PHYSICAL OPTICS 2.1 Equivalent Currents The equivalent currents deduced by Knott and Senior (1973) represent a significant improvement in thhe geometrical theory of diffraction and although the equivalent currents produce the correct high-frequency results for edge singularities, they also include physical optics (PO) contributions. In and of itself this fact is of no consequence, but if the equivalent currents are to be combined with physical optics in any generalized high frequency procedure for predicting the far fields scattered by arbitrary obstacles, the PO contributions will be tallied twice. Clearly, if the equivalent currents are to be representative of the effects of edges alone, with physical optics then handling surfaces, the equivalent currents must be modified. When the PO components are removed from the equivalent currents, we then have "modified equivalent currents" which are associated only with edges. This removal can be accomplished quite easily, since Ufimtsev (1957) has explicitly written down both the edge terms and the PO components for the two dimensional problem of diffraction by a wedge. The modified equivalent currents, together with the usual physical optics approximation over surfaces, can then be used in a computational procedure for computing the scattering from arbitrary obstacles with more accuracy than is available from PO alone. Such a procedure has been devised for bodies of roll symmetry and the computer program that does this is described in Chapter IIIm. Knott and Senior (1973) considered the general problem of diffraction by a ring discontinuity in slope illuminated by a plane wave at arbitrary incidence and with arbitrary polarization. Equivalent currents were deduced having the property that a stationary phase evaluation of the integral of the currents around the ring contour yields precisely the first order wide angle GTD express ion. Both electric and magnetic equivalent currents were obtained and they are 3

01 1758-2-T A tE (X-Y) I = -~2 - -- -2 e ikZ sin2/ O (1) tZH it(X+Y) I -2 it m iksin 12 where t is a unit tangent vector at the edge, Eit and Hit are the components of the incident field along this unit vector, 1 is the angle subtended by t and the direction of propagation of the incident wave, and the diffraction coefficients are 1 7T -sinX = (2) coS - Cos n n 1. 7T n n Y -(3) Cos - Cos n n In (2) and (3) n = a/wr, where a is the exterior wedge angle, and? 0 is the angle of incidence of the impinging plane wave while b is the angle in which the scattering is desired to be known; see Figure 1. The equivalent currents (1) and the coefficients X and Y are based upon a two dimensional problem and in order to apply them to a three dimensional body we assume that (1) through (3) hold true locally. When the currents are inserted into the radiation integral, the scattering is then given by S \2 2 eit rt(X Y) +hitrth(XY d (4) 4

011758-2-T a 1 Face 2 FIG. 1: WEDGE GEOMETRY where C is the contour of the singularity, F is the position vector of an edge element of length di, i and s are the directions of propagation of the incident and scattered waves, eit and ert are the tangential components of the electric polarization of the incident wave and a remote receiver, and hi and ht are tanit rt gential components of the magnetic polarizations. 2.2 Modified Equivalent Currents The diffracted waves given by the GTD prescription inherently contain PO contributions, but Ufimtsev (1957) has already removed them for the two dimensional case. Based on his form of results, we may write down two diffraction coefficients, f and g, 5

011758-2-T (X-Y) - (X1-Y1) r(X-Y)-(X -Y), (X+Y)-(X +Y1), g = (X+Y)-(X +Y1)-(X2+Y2), (X+Y) - (X2 +Y2) o < o0 < a- X a-Tr < Q0 <r, 7r < 0 <, O<0 < a-r, ca-7r < < 7, <T<0 0,I..,,.Y (5) (6) where the subscripted terms are PO corrections that must be subtracted from the "standard" GTD diffraction coefficients. These are 1 sin-sinb0 -1 2 coso+cos b0 1 in sin +sin 1 2 cosV+cosob 1 sin(a- ') -sin(a- O ) 2 2 cos (a- +) +cos (ca- 0) 1 sin(a-V )+sin(a-0 ) 2 2 cos(ao-b)+cos(a-i O) (7) (8) (9) (10) where the subscripts 1 and 2 refer to face 1 or face 2 of the wedge. The space exterior to the wedge is divided into three distinct regions, as is indicated in equations (5) and (6), wherein only one or the other or both wedge faces are illuminated by the incident wave. Note that equations (5) and (6) thus invoke the PO approximation that induced surface currents are set to zero over shaded portions of obstacle surfaces, since PO terms are selectively omitted in the appropriate regions. Although V and VI0 coincide with scattered and incident 6

011758-2-T directions for the two dimensional problem in Figure 1, they shall take on a slightly different meaning for the practical cas e involving a three dimensional obstacle. The modified equivalent currents are obtained merely by replacing (X T Y) with f and g, A tE f t E itf I = -2 e 2 ikZ sin /3 (11) tZ 0Hitg I =-2 m iksin / whereupon the far bistatic diffracted field is k ikr- (i-S) sin 2 (feitert ghithrt), (12) JC sin and now represents edge effects alone, since physical optics terms have been removed. The coefficients (2) and (3), and (7) through (10), enter equation (12) via f and g, and since they become infinite for certain values of 0, 7/0 and a, any numerical procedure based on (12) must deal with these singularities. Equations (3), (8) and (10) become singular in the specular directions where rVT, ~ < 7< <T ~+~0 = d 0 2a -, a-lr<0 <, 0 -but fortunately the singularities cancel each other. Specifically, if we let 0 lie within some small e << 1 of the specular direction, then Y! i/e, and becomes infinite when e = 0. However, depending whether face 1 or face 2 participates in the specular reflection, 7

011758-2-T Y or Y 1/, but not both. Thus the difference Y-Y (or Y-Y2) vanishes while the remaining PO term is finite. Note that Y1 and Y2 never become singular simultaneously unless the wedge angle a = wr, in which event the wedge is no longer a wedge, but merely an infinite plane. Equations (2), (7) and (9) similarly become infinite, but along forward bistatic directions instead of the usual s pecular directions. Whenever the difference 0-~0 = r+ E, again where e is a small angle, then X- 1/e and either X1 or X2, but not both, also becomes singular: X1 or X2 1/e. Thus, like the Y's, the X's can become infinite, but in such a way that one infinity cancels the other, and the integrand (12) remains finite. These facts permit us to formulate a simple rule useful in numerical schemes: whenever one of the (subscripted) PO coefficients approaches a singularity, then it and the corresponding (un-subscripted) edge term should be set to zero. While this process may still leave a single physical optics term in the region a-7r < K0 < 7T, that term will remain finite. If it is backscattering that is being reckoned, then b-b0 = 0 and the X's never become singular, hence never need to be tested. The tests must still be applied to the Y's, however. 2.3 Physical Optics from Surfaces The physical optics approximation of the surface current integral is the well-known expression ikr0, ike -ikr ~ si H 2iker sA (xH.)e dA s 27 A 1 where H. and H are the incident and scattered magnetic fields, r0 is the distance between a remote point of observation and some origin located in or near the scatterer, m is an outward normal erected on the surface element dA, s is a 8

011758-2-T unit normal along the scattering direction and F is the position vector of the surface element. We may probe the far scattered field with a linearly polarized receiver whose electric and magnetic polarizations are specified by unit vectors A A A e and h, with the receiver aligned such that e x h = s. The received signal r r r r - is proportional to H h and the PO scattering coefficient s r -ikr A S = kr e H h /Ho where H0 is the intensity of the incident field. The incident field can be represented as A - A ikF i Hi = hiH e so that 2 Ak po ik \A A ikr(i-s), S m h. e e dA. (13) 27r 1 r A According to the usual PO prescription, the integration should be carried out only over the illuminated portions of the body. Although equation (13) accommodates bistatic directions, our immediate A A interest is the special case of backscattering, for which s = -i. In addition, we shall consider only bodies of revolution, for which it will be sufficient to compute only the principal plane scattering in which either the incident electric or magnetic field lies in the plane of incidence; these two cases are H- and E-polarizations respectively. Thus the physical optics backscattering to be computed is 2 ik SE H- + 2 mle dA, (14) A where the upper and lower sign options refer to E- or H-polarization. The surface integral can be written more explicitly as 9

011758-2-T Po - ik ^ ^ i2kr-. = + m ie p(s)ds d, (15) SE, H 7r where s is a surface distance measured along the body profile, p(s) is the radial distance of a surface element from the body axis and A is the azimuthal body coordinate. A numerical evaluation of the double integral reduces to a finite summation, of course, but since m is a function of s as well as $, it is necessary to create an array of p's and ds's, each discrete value being assigned to an element of the body profile. The s-integral in (15) can be carried out analytically for straight line segments and, since many bodies of practical interest contain cylindrical or conical sections, the analytical representation can represent a considerable savings in machine time. The most general surface having straight line segments on a body of revolution is a frustum such as sketched in Figure 2. The frustum A P m. I z h0 z hf z =hr FIG. 2: A FRUSTUM half angle is designated T and the front and rear termini are located at axial stations z = h,h. The incident wave arrives at an angle y from the axis and the surface f r element 10

011758-2-T dA = (z-h0)tanTdzd^/cos. We establish the Cartesian coordinate system (x, y, z) and, without loss of generality, confine the incident wave to the xz-plane. A unit radial p = x cos +y sin i locates the position of a surface element in a plane perpendicular to the body axis and A A A i = -xsin y+z cos y r= z+( cos ^+sin )(z-h0)tanT ^., A ^ m = pcosT-z sin. Thus, for backscattering, po - ik2tanr i2kb(z h0) i2kh0cos' EH 27cos (z-h0) e e dzd, z where h0 is the axial location of the tip of the generating cone and b = cos y - tan T sin 7 cos. The z-integration may be carried out immediately since m- is independent of z: hr i2kb(z-h ) (z-h )e dz= hf 1 { i2kb(h- h) i2kb(h -h 2 {r[i2kb(h -h)e r -[L-2kb(h-hI e f "0 (2kb) r lb = 0 2 (hr-h0o) -(hf-h0) 2 b= Thus, for a frustum 11

011758-2-T Spo SH E,H i2khOcos y - tanTe + 47rcos T i2kh cos y - tanTe + 4r cos T _ r 0 2b2 kb(hr - ho) + e-h) 2b2 i2kb(hf-h)+ikb -2kb(hf-h0])+}e }d, bi0 ik2 (m ) [hr-h) -(hf-h) 2]d, 1 -h h b=0. (16) It might be thought at first that the frustum result (16) can be used for the special cases of the disk and the cylinder, but this is not so. Equation (16) becomes infinite for the disk (r = r/2) and vanishes for the cylinder (r = 0), primarily because of the manner in which the integration was carried out. The correct results for these two special cases may be obtained by returning to the general PO formula (14). The radial integration for the disk may be carried out straightforwardly to yield Spo = E, H Pr i2khOcos y - e 4 + 47r m 1 (2kba+i)ei2kba id b 0 2b6 (17) i2khocos Y + i(ka) m.id, + 4, b=0, where a is the radius of the disk, h0 is its axial location and b = -sinycos. Similarly, we obtain for the cylinder a i2kh cosy i2khfcos y i2kba _ ka m-i r f i2kba + +2 2 o — (e -e )e dp, cosT70, + 27r 2 cos y ~ H (18) E, H, - ka * A i2kba, + - ik(h -h m.ie d, cos =0 12

011758-2-T where a is the cylinder radius and b = -sin ycos ~. The form used to display the results (13) through (14), with constants embedded within the integrand, was chosen so as to simplify the process of incorporating them in the computer program. 2.4 Comparison of PO Results via (12) and (16) Equation (16) gives the PO scattering as obtained from a surface integral in which the z-integration has been carried out analytically. Equation (12) has incorporated the effects of filamentary currents along an edge, with PO terms removed via a modification of the diffraction coefficients. It is of interest to compare the two after the diffraction coefficients X and Y have been removed from (12), since each then purports to give the PO contribution. However, since (12) is written for a single edge, we must apply it twice, once for the front and once for the rear edge of the frustum. Equation (16) includes the entire surface, of course, but the integration has left a "residue" consisting of edge-like terms representing the limits of integration. Since (12) requires the specification of 00 derived from a two dimensional problem, we must decide how to define ~0 for a three dimensional problem in which the incident wave does not arrive at right angles to the local wedge axis. It is obviously the tangential components of the incident electric and magnetic fields that excite the edge currents, thus it must be the component of the incident field direction lying in a plane perpendicular to the edge that defines b0. In order to compute 0 so defined, it is helpful to construct a unit surface tangent 'r along a generator on the frustum and directed toward whichever edge equation (12) will be A applied to. The unit surface tangent r will therefore be perpendicular to a unit edge tangent t at a point on the edge, and at each point on the rim A A A A A A T *.txi m txi sin = cos d =0 sin 0 = sinl3 where sin:3 = t x i |. (Note that in a bistatic case b would also have to be determined and this could be done by replacing i by -s, and sin/3 = txs.) 13

011758-2-T We focus attention on the region 0 < 0 a-rT in equations (5) and (6) so as to simulate the three-dimensional case in which only one surface is illuminated by the incident wave. We consider only backscattering, so that X = 0 and 1 1 Y1 2 tan b0. Since it is the PO component we seek, the diffraction coefficients X and Y will be dropped and, recognizing that f and g are the negatives of PO contributions (the purpose of writing equations (5) and (6) was to subtract PO effects), equation (12) predicts i2kr i 2 2 (hithrt eitert)Y (19) s sin where a is the radius of the ring at an end (front or rear) of the frustum. The bracketed term in (19) is 2 2 2 hithrt -etert = 7 (cos +cos ysin 2) for E- and H-polarizations, respectively, and a - cos y sin T+ sin y cos Tcos 0 + cosycos T - siny sin cos for the front and rear edges of the frustum, respectively. In addition, 2 ^ I2 A2 2 2 2 sin 3 = i x t = cos 2 +cos 7ysin, which happens to be equal to hitht -eitert. Applying (19) to both ends of the frustum,.. r -i2ka sinycos$ i2kh cosy + k cos ysinT +sin ycos C cos f r e r S e e E,H 47r cos co -sin ysin T cos r -i2kafsin cos a i2khfcos 7Y -a e e di, where subscripts "f" and "r" denote front and rear edges. Since af = hftan7, a = h tanT, we may write the integral as r r 14

011758-2-T i\. i2kbh i2kbhf - ktanr \ cos sinT+s inycos Tcos (h e r -he )d^ E,H 4r CosT b r f A (20) where b = cos y- sin ycos t tan. It should be remembered that (20) is based on the edge scattering formula (12). Turning now to the conventional PO result (16) for the frustum, we may insert the explicit values mxi = -cosy sin T - sin y cos cos and h = 0, with the result E, H 47r cos b r i2kbh -(hf+ i/2kb)e ed. (21) Observe that equations (20) and (21) differ only by the presence of an additive term i/2kb, which, in general, is much smaller than hf and h. Thus if kbhf >> 1 and kbh >> 1, equations (20) and (21) become identical. We conclude that the PO corr rections used in equations (5) and (6) produce the desired result. It should be noted that if the body is composed entirely of straight line elements, its scattering may be computed by repeatedly applying equation (12) to all edges, with f,g = X T Y, and by omitting the PO surface integrals given by (15). This process suffers the disadvantage of producing infinities in specular directions, however. 15

011758-2-T II A COMPUTER PROGRAM FOR BODIES OF REVOLUTION 3.1 General Program Considerations A computer program has been written to compute the backscattering from metallic bodies of revolution using the appropriate analytic expressions (16) through (18) for body segments composed of straight lines, equation (15) for those composed of curved lines, and equation (12) for edges (with s set to -i, of course). Although it is possible to produce such a program for a body of revolution having an arbitrary profile, such generality unduly complicates the task without providing much more information than a program restricted to certain kinds of profiles. For simplicity, we permitted the body profile to contain only straight lines or circular arcs, which are then rotated about the body axis in order to generate the body surface. In spite of this constraint, a large number of obstacles of practical interest can be synthesized. The program assumes that the obstacle profile can be generated from as many as 10 abutting segments and that an edge is formed wherever two such segments meet. The locations of the segments are given by the coordinates of the segment endpoints and if there are j segments, then it requires j +1 points to specify the profile and there are j - 1 ring singularities whose edge contributions must be computed. The one dimensional integrals (12) and (16) through (18) are evaluated for edges and straight line segments, but the two dimensional integral (15) is used for curved line segments. A total of 50 sampling points is reserved for these curved portions of the profile, but the program can be very easily modified so as to increase this number. Similarly, the number of segments to be permitted can also be increased by merely changing three statements. The program is listed in the Appendix and it assumes that the input data will describe the profile, segment by segment, via a list of four parameters, one such list per segment. The program constructs the profile as it indexes through the input data, assigns a name, quite literally, to each segment, and computes and stores the characteristics of each edge formed by adjacent segments. Since edges 16

01 1758-2-T represent an important feature of the obstacle, some care has been given to the computation of the edge diffraction terms. Although the local wedge geometry shown in Figure 1 is basic for the computations, the geometry in Figure 3 is slightly more convenient, as may Face 2 6 Face 1 a FIG. 3: LOCAL WEDGE GEOMETRY become apparent. We assume that after the profile has been generated several essential quantities characterizing each edge will have been found: the axial location of the edge, its radius a, and the two local face angles T1 and T2 measuring the slopes of the faces with respect to the axis of revolution. If the segments are straight lines, the T2of one edge will become the T1 of the next. The projection of the direction of incidence onto the plane normal to the local wedge axis is measured by the angle 6 and comparison of Figures 1 and 3 shows that =0 6+ T 17

011758-2-T For backscattering, equations (2) and (3) become 1. 7T - sin - X= cos- - 1 n 1. 7T - sin - = n n 240 7T O0 Cos - -COS n n and from (7) through (10), X = X2 = - 0 Y1 2ttan 1 Y2 =-2tan (a-0O. It is evident from the figure that a = 7r +T - 2, so that Y1 and Y2 are 11 2 + Y = — 2tan(6 +T) Y2 = 2 tan(6+T2). Since / = 6 + - + 6 + Ta-+6+r then 0 1 2' 2~=b0 1 = 7r — = 7 -7 (6 +r -2(6+T.} n n 1- 2 p whence 1. T - sin - n n Cos - +cos - 1-(6+T ) n n 1 - (6 +r2)} 18

011758-2-T Thus the two angles, (6+T1) and (6+ T2), required for Y1 and Y2 may also be used in the diffraction coefficient Y. The angles 1 and T2 are known, of course, and 6 may be found by computing -1 6 = tan (cos A tan y) 3.2 Generation of Circular Arc Segments The computer program expects to read the parameters of each segment endpoint on input, requiring as many cards as there are points. The first datum on each card is an integer code which signifies whether the segment is to be a straight one or whether it will be curved, and if a straight segment is specified, the arc-generating portion of the program is by-passed. The code itself specifies the number of sampling points to be along the arc, and the spacing is taken to be uniform. Figure 4 illustrates the geometry of a circular arc segment. The specific data required by the program is an integer M givingthe number of T1 1 r,h a a r,h c c FIG. 4: GEOMETRY OF A CIRCULAR ARC SEGMENT OF THE BODY PROFILE. 19

011758-2-T sampling points to be distributed along the arc, the coordinates ra, h of the a first endpoint, and T2, the initial slope of the segment. In order to carry out the arc generation procedure, the machine must also read the next card, since the first endpoint r, h of the segment j + 1 is also the second endpoint of the a a segment j. Note that the angles T1 and T2 are precisely those required in order to establish the local wedge parameters discussed above. The coordinates of the center of the circle generating the arc are r, h, and ra, rb and r are all radial distances measured from the body axis; h, hb and h are positions measured along the body axis. The angle p measures the inclination of the chord of the arc relative to the body axis and may be quickly computed from -1 (rb a p = tan (h The total angle 0 subtended by the arc is 0 = 2(T2-p) and if the arc were to be a straight line segment, then obviously p = T2 and 0 = 0. The final slope of the segment is 1 = 2p-T2 and, should a pair of such segments be butted together, the local edge geometry can be determined from the 71 of the previous segment, the T2 of the next, and the common coordinates of their intersection. The radius r of the arc is (rb-r )2 + (hb-h )2 1/2 Lb a a r = — 2sinand the coordinates of the center of the circle which generates the arc is r =r +rt/2 c a ht2 h =h +h/2 c a t' 20

011758-2-T where (h -ha) rt (b a) tan /2 (r -r) ht =(hb a tan 0/2 Since the arc is to be subdivided into M elements, we form an elemental angle e = 0/2M, which is half the angle subtended by a single element. This angle is more useful than the full angle because element midpoints lie at odd multiples of it from the segment endpoints. Thus the coordinates of the midpoints of the M elements can be found from r =r + [hsinq+rt(-cosq) ] a 2 t h=h + 2 ht(l - cos q) -r sin q where q = Ne, with N being an odd integer less than 2M. Equation (15) is used to sum the contribution of profile elements and, unlike straight line segments, curved segments require the specification of AA m * i at each element. The radial and axial components of the normal are r m and h respectively, m r = (htsinq-rt cosq)/2r h = -(h cosq+r sinq)/2, m t t and these components may be used to compute the surface normal m at any point on the body. The quantity needed in the integrand is A mi =h cos y- r sin ycos. As mentioned earlier, only the contributions of illuminated elements are tallied, hence we require 21

011758-2-T h cos < r sinycos?. I - m The PO contribution of any one element of the curved surface is PO - - ik cos )ei2k(hcosy-rsinycos)rAsA, SH + -7 (h cosy-r sinycos )e rAsA6, E, H m m (22) where As = 'i9/M is the (finite) elemental arc length and AO is the angular increment chosen for the k-integration. In the program listed in the Appendix, AO is fixed at 4 degrees but can be quickly and easily set to a finer sampling rate by changing the appropriate statements. 3.3 Using the Program The number of aspect angles for which scattering cross sections are computed is controlled by the user by specifying initial and final aspect angles, and the increment to be used in indexing through the range. If fewer than 53 angles are required, then the total output of the program will occupy two printed pages per input data set. The printed results are packed onto the left side of the sheet so that large, unwieldy computer sheets may be cut down to handy pages 8-1/2 x 11 inches in size. The first page of output summarizes the salient obstacle parameters, including such information as the names and locations of segments, the axial positions of edges, the frequency of the incident wave, and so on. Body dimensions are assumed given in inches, angular information in degrees, and the frequency in GHz. Since the user may misplace the data cards used to run the program, the input data are echoed at the top of the first page for reference. The second page lists the computed scattering cross sections for the aspect angle range specified on input. Only principal plane values are listed (E-polarization and H-polarization), but the list includes the individual returns of the obstacle by class. Four columns are devoted to the physical optics returns 22

0 1 1758-2-T from disks, cylinders, frusta and curved portions (Cogives"), and two columns list the E- and H-polarized returns for all edges. The total cross section is also given. Since the individual returns of each class of scatterer are displayed, the user may examine the characteristics of this class alone, but if the obstacle has more than one scatterer in a given class, then the individual characteristics of scatterers of the same class cannot be separated from one another. The flow of computations is roughly as follows: after reading the input data, the program generates the body profile internally, assigns names to the segments, sets the angle of incidence to the initial value specified and sets the body angle Af to 2 degrees. If the profile contains circular arc segments, the contributions of all elements on such arcs are tallied at once, even if there is more than one arc on the body. The machine then indexes through all straight line segments and sums their contributions, judging which portions of the program to skip on the basis of the actual segment names, and then indexes through all edges to include their returns. The results are added to running sums that are maintained for each class of scatterer. After all contributions have been accounted for, during which shaded elements and segments have been ignored, the body angle X is advanced 4 degrees and the process is repeated until the entire circle has been swept out. The returns are converted to decibels and printed out, and the running sums are then cleared to zero in preparation for the next aspect angle. The input data format is summarized in Table I, and note that each point on the body defining a segment endpoint requires a single card punched according to the description for Card 3. The program assumes that, since the body is a convex one with a closed profile, the first and last points on the profile lie on the axis of revolution. After the program reads the coordinates of the first point, it tests every successive point for zero radial coordinate and, if such a point is found, the reading of input data is suspended and the program commences its main job of computing radar cross sections at the angles of incidence specified by Card 2. When the main task has been completed, the program returns to the 23

011758-2-T TABLE I: INPUT FORMAT FOR THE PROGRAM FORMAT (9A8) HEAD; use up to 72 characters for for title Card 1 Card 2 FORMAT FIRST LAST INK FREQ Card 3 FORMAT M #0 M =0 RA, HA T Card 4 FORMAT STOP (4F10.3) FIRST, LAST, INK, FREQ initial angle of incidence final angle of incidence incidence angle increment frequency in GHz (I5, 3F10.5) M, RA, HA, T number of elements in previous segment the previous segment was a straight one radial and axial coordinates of current segment endpoint initial slope of current segment, in degrees (A4) STOP punched in columns 1 through 4; shuts down the program Note: Card 3 is repeated for each segment endpoint on the body; if the body contains n segments, there must be n+ 1 cards. input stream searching for new data. Thus a sequence of data sets may be con-.catenated, but it is Important that the last card of each set have a floating-point zero (or blanks) punched in columns 6 through 15. A sequence of data sets may be terminated with a STOP card (Card 4). For the purposes of testing the program, computed values were compared with measured radar cross section patterns of two cones of different half angles and a sphere-capped frustum. These data are presented and discussed in the next chapter. 24

011758-2-T IV RESULTS AND DISCUSSION The computer program was tested via a comparison of the predicted radar cross sections of three objects against those measured in the Radiation Laboratory's anechoic chamber. Two of the targets were right circular cones machined from cold rolled aluminum billet and having 15- and 40-degree half angles and base diameters of 3. 937 and 6. 010 inches, respectively. The third object was a spherecapped frustum made of a thin aluminum sheet rolled into a cone, and then fitted with a solid spherical cap in front and a flat base machined from an aluminum plate 0. 25 inch thick in the rear. Its dimensions are shown in Figure 5. 6.62 I FIG. 5: DIMENSIONS OF THE SPHERE-CAPPED FRUSTUM, GIVEN IN INCHES. The cones were measured at several frequencies ranging from 8. 0 to 12. 0 GHz and their patterns have been reproduced in a technical report issued earlier in the contract (Knott and Senior, 1973). For the purposes of comparison, we chose patterns measured at 9. 60 GHz and, since the computer program does not include 25

01 1759-2-T second order effects, an absorber pad was cemented to the base of each cone in an attempt to remove this additional contribution. A detailed study of the cone patterns has shown that the cross base diffraction was reduced by this procedure, but not entirely eliminated, and that the cone patterns still contain second order components with undetermined, but small, amplitudes. No such attempt was made to suppress double diffraction across the base of the sphere-capped frustum, since an evaluation of the predictions in the tail-on aspects was desired. The laboratory measurements were carried out using a CW cancellation scheme in which a sample of the transmitted signal is used to null the residual reflection from the anechoic chamber walls. The target was mounted horizontally atop a beaded foam column and was rotated about a vertical axis. Patterns were obtained for both horizontal and vertical incident electric field polarizations, which correspond to H- and E-polarizations, respectively (magnetic or electric vector perpendicular to the plane of incidence). The deficiencies of physical optics when applied without consideration of edge diffraction are demonstrated in Figures 6 and 8 for the 15-degree cone. The solid datum points are the physical optics values obtained from the program and the solid traces are the measured patterns and, since physical optics is independent of polarization, the computed data in both figures are the same. For both polarizations it can be seen that physical optics is a good approximation only within 15 degrees or so of the specular flash and becomes progressively poorer as the aspect angle swings closer to the body axis. The measured data are much stronger than the PO predictions for H-polarization (Fig. 6) and are much weaker for E-polarization (Fig. 8) in the aspect angle range from 20 to 60 degrees. Although PO predicts a lobe in the axial region, the lobe is some 12 dB lower than the measured values; for H-polarization in particular, PO fails by a wide margin to provide even the gross characteristics of the true scattering pattern. When edge returns are included in addition to the "standard" PO returns, the pattern predictions are greatly improved, as may be seen in Figures 7 and 9 26

011758-2-T for H- and E-polarizations, respectively. The edge contributions elevate the computed axial lobe to within 2 dB of the measured value and in the intermediate aspect angle range for H-polarization (Fig. 7) they are responsible for the pattern characteristics that physical optics fails to produce. There is a marked differential error in the nose-on region for H-polarization, with the computed main lobe being lower, and the computed first side lobe being higher, than the measured values. This disagreement is common to first and second order GTD estimates for narrow angle cones whose electrical dimensions are not very large. It can be attributed to a change in the effective illumination of the rim of the base due to the closeness of the incident rays to the generators of the cone and such "transitional effects" have previously been observed by Senior and Uslenghi (1971, 1973). Comparisons of the measured and computed results for the 40-degree cone are shown in Figures 10 and 11. In this case the nose-on predictions are higher than the measured values, but the sidelobe structure is more accurately predicted than for the 15-degree cone. This is because the base diameter of the 40-degree cone is much larger, with ka = 15. 357 as compared with 10. 060. Aside from the 2dB discrepancy at nose-on incidence, the computed' values agree very well with the measured ones, especially in the intermediate range where physical optics alone is a poor approximation. Figures 12 and 13 compare measured and computed patterns for the sphere-capped frustum of Figure 5 for a frequency of 3.40 GHz. At this frequency the electrical circumference of the base of the frustum is ka = 6. 00 and that of the front edge singularity is ka = 1. 69, a relatively small value. Aspect angle coverage of the patterns ranges from nose-on at the left to tail-on at the right. For both polarizations the specular flashes from the slanted side of the frustum and from the flat base are reproduced quite well (at aspect angles of 73 and 180 degrees, respectively). 27

011758-2-T 20 -10 -10 0 0 0* 0 * 00 * *. *0, oj 30~ FIG. 6: COMPARISON OF PHYSICAL OPTICS (o) WITH THE MEASURED PATTERN (-) OF A 15-DEGREE HALFANGLE CONE AT 9.6 GHz FOR H-POLARIZATION. 28

011758-2-T 20 * W 0 10 -0 00.0 -10 -....I...t.... 0~ 30~ 60~ 90~ FIG. 7: COMPARISON OF COMPUTED VALUES (e), INCLUDING PHYSICAL OPTICS AND EDGE CONTRIBUTIONS, WITH THE MEASURED PATTERN ( —) OF A 15-DEGREE HALF-ANGLE CONE AT 9.6GHz FOR H-POLARIZATION. lB 29

01 1758-2-T 20 10 0 -10 0 0 0 0. 0 0* *0 0.0 00 0* 0 0~ FIG. 8: 30~ 60~ COMPARISON OF PHYSICAL OPTICS (E) WITH THE MEASURED PATTERN ( ) OF A 15-DEGREE HALFANGLE CONE AT 9.6 GHz FOR E-POLARIZATION. 30

011758-2-T - 20 10 10 -10 *. 0. a *. 0~ 300 60~ FIG. 9: COMPARISON OF COMPUTED VALUES (*), INCLUDING PHYSICAL OPTICS AND EDGE CONTRIBUTIONS, WITH THE MEASURED PATTERN ( ---) OF A 15-DEGREE HALF-ANGLE CONE AT 9. 6GHz FOR E-POLARIZATION. 31

011758-2-T I, 20. 10 0 0 * * 0 0 -10 30 FIG. 10: COMPARISON OF COMPUTED VALUES (e), INCLUDING PHYSICAL OPTICS AND EDGE CONTRIBUTIONS, WITH THE MEASURED PATTERN (-) OF A 40-DEGREE HALF-ANGLE CONE AT 9.6 GHz FOR H-POLARIZATION. 32

20 0 10 — 0 lo * 0 O00 -10 - - - ------— i --- —0~ 30~ 60~ 90~ FIG. 11: COMPARISON OF COMPUTED VALUES (e), INCLUDING PHYSICAL OPTICS AND EDGE CONTRIBUTIONS, WITH THE MEASURED PATTERN ( —) OF A 40-DEGREE HALF-ANGLE CONE AT 9. 6 GHz FOR E-POLARIZATION. - 33

011758-2-T -20 I I 0 30 60 90 120 150 180 FIG. 12: MEASURED ( ) AND COMPUTED ( ) CROSS SECTION OF SPHERE-CAPPED FRUSTUM FOR H-POLARIZATION. 34

011758-2-T 20 10 0 1 1 -10 _20 -I I I I 1 0 30 60 90 120 150 FIG. 13: MEASURED ( ) AND COMPUTED ( ) CROSS SECTION OF SPHERE-CAPPED FRUSTUM FOR E-POLARIZATION

011758-2-T There is no single dominant source of scattering in the nose-on region, since scarcely 4dB separates the amplitudes of the edge diffraction due to the base of the frustum and the physical optics contributions from the spherical cap and the slanted sides of the frustum. Although part of the regular lobing structure is due to the characteristics of the base diffraction, the other contributors are responsible for carrying this regularity well into the first sidelobes of the specular return for H-polarization (Figure 12). Aside from slight disparities in amplitude, the characteristics of the entire pattern from nose-on out to, and beyond, the frustum specular echo are modelled quite well. The E-polarized pattern in Figure 13 is less regular than that of Figure 12, but again the lobe structure is reproduced rather well by the computed values. This fidelity in displaying the measured characteristics continues throughout the patterns and into the large specular return from the base of the frustum at 180 degrees. The agreement, however, is better for E-polarization (Figure 13) than for H-polarization in the intermediate aspect angles from 100 to 150 degrees. On the basis of these comparisons, the computer program incorporating the effects of edges via the modified equivalent currents is undeniably a major improvement over any that is based solely on physical optics. It might also be noted that a ray theory such as GTD produces an abrupt discontinuity in the scattering pattern whenever a flash point pops into or out of view. In contrast, the equivalent current representation provides a smooth transition in these aspect angle regions, even though there is no theoretical basis for accepting the precise nature of the transitional behavior. Nevertheless, the program does have some restrictions that should be borne in mind. Since it was designed explicitly to treat metallic bodies of roll symmetry, it obviously cannot be used for asymmetrical targets such as finned vehicles, although it could be modified to do so. In addition, since the body profile must be describable in terms of straight line or circular arc segments only, other profiles, such as a parabolic nose, are cumbersome to model. And although a concave profile can certainly be specified for the scattering obstacle, the program 36

011758-2-T may err because no provision has been made to account for the shadowing of some parts of the body by others, which is an important consideration for concave shapes. All of these limitations can be overcome, of course, but only at the expense of increasing the complexity of the program and the costs of running it. In its present state, however, the program represents a realistic compromise between efficiency and generality for generic aerospace vehicles; all of the computed data shown in Figures 6 through 9, for example, required but 30 seconds of Central Processing time to obtain. It was our objective to produce a relatively simple program useful for predicting the returns from aerospace vehicles and, since first order diffraction is the dominant contribution from the edges of such bodies, only first order effects were programmed. However, the program could be extended to include second order diffraction if a double line integral were to be introduced along each ring singularity participating in the interaction. Such an extension is easily programmable, and would improve the accuracy of the predictions in the aspect angle range from 12 to about 30 degrees in Figures 7 and 9, but would increase the running time by at least an order of magnitude. Alternatively, at least for the base of the cone, a factor can be derived which, when multiplied by the incident field, yields more nearly the correct excitation of the rim. This factor, which is theoretically justifiable, is a Fresnel integral whose argument is a function of the angle between the incident field direction and the nearest generator of the cone. Similarly, the erroneous non-diffractive shadow boundary contribution produced by an abrupt discontinuity in the physical optics surface field could be removed by the insertion of a filamentary current there. For certain simple geometries it would even by possible to choose this current to represent the true shadow boundary effect attributable to creeping waves. Although any and all of these extensions are possible, and would improve the accuracy of radar cross section predictions for specific obstacles, the computer program would be markedly more complex and its generality sacrificed. In its 37

011758-2-T present form, the program is theoretically equivalent to a simple combination of caustically corrected first order GTD with physical optics and in general will be only as accurate as GTD itself. For many practical purposes, however, such GTD contributions are the only significant ones and to consider these alone has enabled us to produce a single program that is little more time consuming that a "standard", but less accurate, physical optics program. We believe the resulting program should be of value to all who are concerned with the routine prediction of the radar cross sections of aerospace vehicles. 38

011758-2-T REFERENCES Knott, E. F. and T. B.A. Senior (1973) "Equivalent currents for a ring discontinuity", IEEE Trans. Antennas Propagat. (to be published). Senior, T. B.A. and P. L.E. Uslenghi (1971) "High-frequency backscattering from a finite cone", Radio Sci. 6, 393-406. Senior, T.B.A. and P.L.E. Uslenghi (1973) "Further studies of backscattering from a finite cone", Radio Sci. 8, 247-249. Ufimtsev, P. Ia. (1957) "Approximate computation of the diffraction of plane electromagnetic waves at certain metal bodies", Sov. Phys. - Tech. Phys. 27, 1708-1718. 39

011758-2-T APPENDIX PROGRAM LISTING AND SAMPLE OUTPUT The program described in this report is listed on pages 41 through 45. It consists of 260 FORTRAN statements and the object code occupies 10316 bytes of storage, or a total of 2579 32-bit words. The output of a sample run for the sphere-capped frustum (whose patterns are given in Figures 12 and 13) is displayed on pages 46 and 47. There are no subroutines embedded in the program and it does not require access to external library functions. 40

01 1758-2-T > PIlPLICIT 3EAL(K) r LL (1),ASI( DAAJ ~O~ I'N L,STOPI. > 1 A iL I( I0L C) If K ~A}E1 1 S020/Fk2 READ (530 1,- (),A(D TJ T(L) LJ*T(J)L L IF( q 0)l 20 rTOP 35 7 IF AI(Di.T.EP) U) T >0 VP:1. II F07~0 2 W(-UiL' WI 20K TO iOTA D) 5 3T -3 0U~ \RJ+D HID - J)i T iL1:DEJ I+ LIT DLL:T 'A/iL DS I) S:DItDe, AL1T 70JI *`1i D ZL SI TOI (LF COS A:CSALF)U11L 41

011758-2-T > 0 > 53 > > 4, > 50 >, > > -,0 > 73 — (I) -HA(L)+0.5*(REX*( I.0-COSA)-T nAY*SI -JA) ( I ) - A ( ) +0.5( r I A+ T Y* C OSA ) H i( I) --.5*(T TE*CC3SA+TRiY*,I SJA)/RADI US (I) 0.3 *(T RE*SI JA-T. AY* COSA)/ RADI US )! TO 50 I: (.-..,( a UJ.).3T). EPS) GO iO 40 Ij IE(L)-CYL 30 TO 50 I- (A.;S(DiL ).LT.EPS);0 10 43 A'.i1L( L) -.':. U LO TO 50 I.<FIE(L) -DISK IF (A-3( RA(J)).T.EPS):o TO 1I LE L- I1 d: I O 55 J-1,LE "J(J) -.0-C0.318309*(T2(J) -T (J)) PI ''(J) -3. 141593/ '(J),P(J) SI '4( PI J( J) ) / (J) CP(J)-=C 0(PT 'r(J) ) X((J) -SP(J)/((J)-!.0) -:IITE (6,500) DO SO Jl,L ITJ+1 '?ITE (,S. 00) J, JA;1E(J), A(I),HA( I;;ITrI (, 700) DO 55 J-1,LEF I-J+1 S$ U-1:D I > C:1PLX ( T1 ( J), TIT2( J)) EL D. 10.0- L ( U ) + 4 I. ( D U M ) KA(.J) K* -A( I) 's.?I rEL ( -., 70, ) J, Hi( I), 1.j(Jl), ',EL G3E, S 'I ITE- (S, 40 ).EAD;'.? TE ( %,50) I LJ - S I JK C0S:3-C:i ( ' ) 3I ',3 "' -1 '.! 3*3I ', C 03 J3 3 C OS 3*C 0S 3J C;IP L ( 0.0.0) DH-3E SC- SE O 10 -, 5l S - S E S:']3-=S 'L SE LH- S~ E D0 130 IFF2, 358 4 ) DLI,~pK I i~ (J)j 42

011758-2-T > SI,J F:S13I\J~ > OSIC0S TEF I>( tI 1.0 0) GO TO 75 > dO iKki LIuO(TIJ) )*SI ThCOSF-S! C Ti(J *CS >7 C i~ (TI GL..0 O O 3 >OI~ (0 J= I() CY)LOTO9 IT( IJ) J~U) G 4 COS 10 T: 1 15 1I O - KAD yK A(J) > I F(H J(F D).LE. PS) GO TO 5 > 30.?3TA K*A(LJ) > G3 (A(CS) LL PS G 0 0 SDU' 0' j D.3C>-'XCO 'I G) -COS( i' S (AJ S < C.r ) 1T C0A50 31. J 2*CSFI I 'Tr I I() 1 I>I )LIES G O G > T *~I J > DU.1:SDA-CFX?. )C LCSAG),I0~k SDfi0. 7- (Li* 7EV7) 43

01 1758-2-T 3 0 TO0 1 2 5 >1 20, S DU I:K*K,* 'T * T-'I+7*TV) *C ",l(.0 1.0) >150D I~ (J.G]T.LE) GO TO~ IE V0 > Y0:1 00 I7(3 L9 0) GO TO 1 33 > AL,I 5 7 079 >13 0 -T~ " 2SI: I 37CO > S S)I '( LT) > A I 2AL17+TI(J ) IF (F0i' EG.EL-3O.W.L.32.131533 GO TO 15 0 IFi-,L (AS( CO2( \J 1)LE. ES) JO TO 1A45 >1 45 YO:0.O. >1 50 Y1=0.0 >1 55 IFAG,2.'L E.Z —Ps o~w2 G E.3.1 31 5 9 3 GO T O I 5 I> (L COS(A4G2).P S GO TO 1 ~O > GO- TO,) 17 0 >1 50 YO:0 0 > 1 55 Y2:.0 >1 70I IF (Y0.E..0 0 Tl O 17 5 > 17 5 y:Y 0 + y I-y > Y TLi 1 Y* (C O.'F+ DS0 *1;:S D U El+SD A i J (C T L X YTC O i) >10 I GOV 0T `Ilij C DLLCLA S(S-L-) IT UL.LDP3)3 D1 > 1 90 C SEA =CA S(SF9{ I> (CS-ER.E.HA 7 GO J0 19 >19-3 DziL) LH L0.0 *ALOG 10 (25kG-E) - 0 05S >2-O05 C SC:C A2( S C IF (CSC.GiI.ZM2Sr) GO TO 20 > C: -9 2 99 44

01 1758-2-T > I10 C-20 0* AL OriIO(C SC) -44 05 >1 C:C A( O T (C '.:3EDLS) r-T) 2 20 T..O 225 L0 I -4 0::)z >L %DC A SSD) > U ( O LI) DE PS) 20 12)i~ 3 0 > ) 12 2 > 3 5 C S FC A, ~3) > j:4O D:2O.0*AL2)1O(CKD+Si)-'O0 > S E: J E -' S D Ul CS E:C S IL') > O TO 25 >2 50 2bL:20O*A L 031 0(CSF-4 4 C5" >2 55 C S i:C A Z3S(S I (CSR.CHE., DtE PS' A 00 T O 200'J 30 TO 2 05 >200O D5H: —2 0.AIOGIO(CS) -4 4.0 53 >,05 4 I T (,50) I, DA, D.1,A FI DPI,16,0,OD 20 TO 10 >i0 70 T 70R! (41P3 >420 - i N, iT ('IA ) >3 00 7i:i4(///'~ 3 ODY p "7 I 3 IC )Oi2 1-:I O S.YLT 2. ')I XJT//) I) ISU IC 1 4 7LES, 5~,~4RK//) r4\ 301ASiLT4LIEJGR JZL, S3 ~1 7154 L2AI CLlQT/1 K, ITOTA ~,1I H~ 1~ hS 0~,T YI &l~i{UPTCS?LTi JS/1 *- 0PL -FL LPL -P >5)FWT I27.,7F2 >75 J,~IT (311FTDT AD E S LOLLC5:/ > / 45

011758-2-T SPHERE-"APPED FRUSTUM INPUT DATA CARDS WERE AS FOLLOWS: r 2C 0 c.( n, Q '. 933 3.312 0.9 r. f 180.C 0 2.000.r 90.00000 50 0.20000 16.75000 50 8.1C940 -90.0000 8. 10940 -90.00000 3.400 BODY PARAMETERS SE MENT 1 2 3 TYPE OGIVE FRUSTUM DISK COORDINATES OF SEGMENT ENDPOINT C. 93350 3. 31250 C.23033 8. 1 940 8. 10940 EDGE LOCATIDN N WEDGE ANGLE SURFACE ANGLES KA 1.69 3 5.996 1 C. 200C 1.27258 130.94 65. 81 16.75 2 8.1094 0 1.59305 MEASUREMENT FREQUENCY, GHZ MEASUREMENT WAVELENGTH, INCHES 73.25 16. 75 -90.00 3.400 3.471 46

011758-2-T SPHERE-CAPPED FRUSTUM BACKSCATTERING CROSS SECTIONS, DB ABOVE A SQUARE WAVELENGTH TOTAL EDGES PHYSICAL OPTICS RETURNS ANG E-POL H-POL E-POL H-POL DISKS CYLINDER f:. J 2. 00 6. n0 8.00 1ic. " 12.C00 14.0 C 1 6. n 18. 00 20.00 22. 00 2 4. 00 26. OC 28. 00 32,90 34. 00 36.00 38.00 4 2. 0) 44. 00 46. 00 4 8. O 0 48. 00 50.00 52. 00 54.00 56. 00 58. 00 6C. 00 62. C0 64.00 66. 00 68. 00 7. 0O 72.00 7 4.00 76. 00 78. 00 8 C.QO 82. 00 84.00 8 6.00 88.00 sc. )c 92. 00 92.00 94.00 9 6. 9 C 96.00 98.00 1 2 Ol. 1 2.00 14.00 2.42 2.32 2. 4 1.62 1.12.56 -0.814 -1.91 -3.29 -4.64 -5.42 -4.15 -2.63 -1.54 -1.33 -2.9 -4.29 -7.32 -9.02C2 -5.62 -2.81 -2.20 -3.42 -6.26 -9.48 -4.61 u.88 3.41 -6.21 -1. 67 2.16 7.36 10.57 12.46 13.38 13.46 12.74 11.32 8.84 5.23 0 74 -1.25.41 C.02 -2.73 -4.81 -4.21 -2.61 -1.75 2.42 1.98 0.67 -1.22 -2.54 -1.92 -0.53 0.24 -6.71 -13.86 - 5.34 -0.82 1.25 1.79 0.87 -1.98 -9.18 -15.82 -3.91 -0.26 0.83 -^.23 -4.15 -14.37 -5.68 2. 14 2.16 -0.16 -5.65 -1.89 4.71 8.62 11.17 12.72 13.43 13.38 12.55 10.86 8.08 3.59 -3.96 -6. 08 -1.11 0.86 0.C5 -2.34 -8.22 -28.21 -11.69 -6.80 -5.26 0.27 0.17 -0.14 -0.65 -1. 40 -2.42 -3.82 -5.75 -8.53 -11.98 -15.47 -20.65 -22.27 -23. 47 -24.29 -22.35 -15.71 -13.28 -10.83 -11.09 -1 1.57 -15.02 -15.35 -11.98 -9. 34 -8.34 -9.01 -11.56 -15. 17 -12.59 -9. 46 -8.28 -9.37 -11.80 -14.77 -12.07 -9.07 -8.12 -9.04 -11.89 -1 4.59 -11.82 -9.36 -9. 42 -11.20 -16.33 -17.03 -13.1 -10.27 -8.73 -8.75 -10.29 -13.32 0.27 -0.35 -2.40 -6.79 -17.16 -10.10 -4.0 5 -1.27 -C. e)8 -2.86 -4.02 -4.46 -5.44 -6.69 -8.73 -11.78 -15.3 0 -14.13 -11.61 -9.88 -1C.28 -12.45 -19.35 -22.42 -13. 42 -10.16 -9.70 -11.89 -18.87 -22.86 -12.52 -9.35 -9.07 -12.02 -20.68 -24.35 -15.53 -13.23 -13.84 -17.18 -24.27 -21.40 -17.76 -17.96 -19.79 -21.08 -38.0 2 -22.59 -17.25 -14.93 -14.81 -16.52 -19.36 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99. 99 -99.99 -99.99 -99.99 -99.99 - 99.99 -99.99 -99.99 -37.55 -31.47 -27.85 -25.24 -23.20 -21.53 -2C.16 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 -99.99 FRUSTA -3.77 -4.35 -4.89 -6.29 -8.21 -10.57 -13.31 -16.57 -2. 13 -18.62 -13.81 -10. 45 -8.39 -7.38 -7. 41 -8.67 -11.71 -16.43 -13.12 -8. 33 -5.22 -4. 13 -4.67 -7.35 -12.78 -9.32 -3. 34 -0.47 0.35 -.97 -5.49 -4.58 3.15 7.32 10.714 12.51 13.38 13.46 12.74 11.18 8.55 4. 40 -2.25 -5.86 -1.33 0.54 0.24 -1.38 -6. 40 -15.59 -1 2. 18 -7. 0 8,.C OGIVES -2.07 -2.39 -2.15 -2. 25 -2.38 -2.56 -2.78 -3.03 -3.32 -3.65 -4. 1 -4. 4 1 -4. 85 -5. 32 -5.83 -6. 36 -6.93 -7.53 -8.15 -8.3 -9. 4 6 -1. 15 -10.85 -11.56 -12.27 - 1 2. 2 7 -12-. 9 8 -13.68 -14.36 -15.03 -15.66 -16.26 -16. 33 -17.35 -17.82 -18.26 -18.67 -19.08 -19.50 -19.95 -20.45 -21.01 -21.66 -22.39 -23.25 -24.23 -25. 36 -26.68 -28.21 -29.98 -32. 5 -34.49 -37.41 -.9 99 - 5. 41 - 40 96 47

U NCLASSIFIED S'icuritv ('l,';'.i.ficintlion I DOCUMENT CONTROL DATA - R & D (Securily cleessi'clific-51),, of tifll,. 11lyt4 4)1 OliN fin.'c wi,1) haifl. I',?irijIfir)(111ol nm-IVID h(' orfelre~td whenf (ten ovralan ripnnf I., clnovalfleIi now 1. OFRIGINA 1ING. AC TIVITY (Corporftle tll,,tr) 2. fEPONOH1 SEICU411 Y CLASSIFICATION The Unive sity of NMichigan lla cdition Laboratory UNCLASSIFIED 2216 Space Research BIClg., North Campus 2b. GROUP Ann Arbor, MRi chig;an 48105 3. REPORT TI IL E A PROGRAM INCORPORATING DIFFRACTION FOR THE COMPUTATION OF RADAR CROSS SECTIONS 4. OESCRIPTIVE NOTES (T'pe ol report nnd Inclusive dateo) Scientific. Interim. r. AU THOR(S) (First name. middle Initial. last namo) Eugene F. Knott Thomas B.A. Senior 6. REPORT DATE 7~,. TOTAL NO. OF PAGES 7. NO. OF RCFS June 1973 4 Ga. CONTRACT OR GRANT NO. 90. ORIGINATORS REPORT NUMBER(S) F 19628-7 3-C-0126 0 11758-2-T b. PROJECT, Task, Work Unit Nos. Scientific Report No. 2 5635-02-01 C. 'ih. OTHER RCPORT NOISI (Ary other,amer M', al,'iy Ise'.''f.nc, DoD Element 61102F h OilHERPORT Nlsy olrecr mc mpy bo rt) ln AFCRL-TR-73-0351 d. DoD Subelement 681305 AFCRL-TR-73-0351 10. DISTRIBUTION STATEMENT A- Approved for public release; distribution unlimited. II. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIV.T Air Force Cambridge Research Laboratories (LZ TECH, OTHER L.G. Hanscom Field.... Bedford, Massachusetts 01730 13. ABSTRACT ' ----..-,, A computer program is described that calculates the radar backscattering behavior of metallic roll symmetric bodies having one or more ring discontinuities in surface slope. The field diffracted by each surface singularity is represented by a line integral of equivalent electric and magnetic currents. The scattered field can be expressed as an integral of the physical optics currents existing over all non-diffracting portions of the surface and the equivalent currents at the singularities. However, since the equivalent currents themselves contain a physical optics component, the original currents must be modified in order to remove it. The predictions of the computer program are compared with experimental data for cones and for a typical aerospace vehicle. Though the program is almost as efficient as one using physical optics alone, the results obtained are markedly superior. Ir) r FORM 1 A 717 M i LJ. I NOVGol 4I / UNCLASSIFIED S.'rc ritv (I;is.li,. io,

UNCLASSIFIED Security Classification 4 Y W LINK A L I N K E L INK C ROLE WT L E WT ROLE T OLE -- IL- — 1 - 1 aerospace vehicles radar cross sections computer program physical optics diffraction equivalent currents i. I I t mm — I mwmm - I I __2 __J UNCLASSIFIED Se-, urltv ('.I s-.,it It t cl I I -w