014518-1-F NON-SPECULAR RAM by Thomas B.A. Senior and Valdis V. Liepa Radiation Laboratory The University of Michigan Ann Arbor, Michigan 48109 FINAL REPORT April 1977 P. 0. No. 1-83493 (F33615-76-C-1064) 5 April 1976 - 5 April 1977 Prepared for Emerson and Cuming, Inc. Microwave Products Division Canton, Massachusetts 02021 14518-1-F = RL-2275

ABSTRACT Techniques are developed for reducing the non-specular scattering displayed by ogival and wedge cylinders in an aspect range about edge-on. To achieve a cross section reduction over almost a 10:1 frequency band, it is necessary to decrease each type of contribution individually and also to minimize their interaction. The various contributors are discussed and analyses performed to specify the surface impedances necessary to adequately reduce them. With the restriction to a coating no more than 50 mils in thickness, materials have been assessed to determine their ability to realize the desired impedance. The best material found is capable of about 10 dB cross section reduction over most of the frequency range. Computer program RAMD has been used to optimize te application of this material and, in addition, to verify the results of experimental measurements. i

TABLE OF CONTENTS CHAPTER PAGE 1 INTRODUCTION................. 2 THEORETICAL CONSIDERATIONS............ 2.1 Geometries..................... 2.2 Bare Body Scattering............ 2.3 Surface Impedance Concept............. 2.4 Theoretical Approach to Cross Section Reduction 3 SPECIFICATION OF A COATING......... 3.1 Impedance Specification........... 3.2 Coating Specification............ 4 COMPITER DATA...............* 5 CONCLUSIONS....................... 6 REFERENCES.................. 00 00 00 00 0 0 $ * 0 * * * * * * 0 * * 1 6 6 9 28 32 46 46 55 76 119 120 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 * 0 0 0 0 0 * 0 0 APPENDIX A COMPUTER PROGRAM RAMVS (as of August 1976)..... 121 B COMPUTER PROGRAM RAMD (as of January 1977)..... 141 ii

CHAPTER 1 INTRODUCTION This is the final report on a sub-contract (P. 0. 1-83493) from Emerson and Cuming, Inc. covering the period April 1976 - April 1977 and concerned with the theoretical design and evaluation of radar absorbing materials for a specific application. The overall purpose of the prime Air Force Contract F33615-76-C-1064 was to study the availability of broadband absorbers for reducing non-specular scattering and to demonstrate their capability using two generic shapes: a 25 included angle ogival cylinder and a 25 wedge cylinder of length to width ratio 2.810, both of them with particular reference to H polarization in which the magnetic vector is parallel to the front edge, i. e. the electric vector perpendicular to the edge. At the lowest frequency of interest (2. 0 GHz) the bodies are 0.800X and 0.716X long respectively where Xis the wavelength, but their surface lengths are almost identical (0.807X and 0. 803X). This was one of the considerations that led to the choice of these specific shapes under the prior Contracts F33615-72-C-1439 and F33615-73-C-1174 held by the Radiation Laboratory. The three main types of non-specular scatterers are edges (or other discontinuities in the slope and higher derivatives of the body profile), traveling waves and creeping waves and the bodies were selected to exemplify them. Thus, the ogival cylinder supports a traveling wave and, in addition, gives rise to edge scattering from both the leading and trailing edges. The wedge cylinder displays "edge" scattering from the join of the wedge and cylinder (where the curvature or second derivative of the profile is discontinuous) as well as from the leading edge, and in addition supports a traveling wave on the wedge, smoothly matched into a creeping wave on the shadowed portion of the cylindrical cap. All three types of scattering are geometry and polarization dependent and diminish with increasing frequency. Their magnitudes are generally small compared with any specular contribution that may exist, but at the aspects of most interest in this study (~ 300 about edge-on), there is no specular contribution to the backscattering cross section. Non-specular scattering is therefore the only scattering that occurs. 1

Traveling and creeping waves are forms of surface waves whose propensity to cling to surfaces sets them apart from edge-diffracted contributions. The two types of waves are distinguished from one another mainly by the illumination of the surface on which they travel. Creeping waves are 'born" at the shadow boundaries of bodies whose radii of curvature are electrically large. They propagate into the shadowed region following geodesic paths leaking off energy in the direction of the forward tangent to the path as they go. As a result of this leakage, the wave attenuates as it progresses and its electromagnetic properties at any point are almost exclusively determined by the local geometry of the surface. It is evident that such a wave can traverse the shadowed portion of, for example, the wedge cylinder, and on reaching the boundary of the shadowed/lit regions, the energy radiated will be in the backscattering direction and will contribute to the backscattering cross section. In theory and practice, creeping waves are only a significant source of scattering where the incident electric vector is perpendicular to the surface at the shadow boundary, i. e. for H polarization with the wedge cylinder. Once established, they are no longer dependent on the incident field. They are closely bound to the surface over which they travel, and this fact, coupled with their natural rate of decay even on a metallic surface, makes it possible to diminish their effect using lossy coatings. Traveling waves are also significant only when the electric vector is perpendicular to the surface, i. e. for H polarization with either of the given bodies. They are supported by surfaces of large radius in the direction of propagation almost all of which is illuminated, and in contrast to creeping waves, they tend to build up as they travel, being fed by the incident wave. As regards backscattering, a traveling wave is harmless as long as it continues to travel forward, but when it reaches any discontinuity in the surface, a portion of it may be reflected and travel back to contribute to the backscattered field. The rear edge of the ogival cylinder provides such a discontinuity, and the resulting backscattered field has the fan-shaped pattern characteristic of a traveling wave antenna. Under the afore-mentioned prior contracts it was shown that traveling wave contributions could be significantly decreased by appropriately loading the surface. Not unexpectedly, the largest reductions were obtained when the entire surface was loaded, with the loading greatest near to the trailing edge where the reflection takes place. 2

The third source of non-specular scattering is edge diffraction and this is also strongly polarization dependent. With E polarization, for example, the leading edge is the primary source and methods for the reduction of this type of scattering were developed under Contract F33615-73-C-1174. For the present study, however, H polarization is of most concern. Although the rear edge of, for example, the ogival cylinder is now the dominant source, and its scattering can be suppressed by the same treatment used in regard to the traveling wave, the front edge does contribute a small amount. Its scattering is a function of the wedge angle as well as the incident field direction, and for a 250 total angle wedge, the front edge contribution at an angle of 250 from the symmetry axis is only 15 dB below that attributable to the rear edge. To reduce the complete scattering cross section by an amount greater than this therefore requires a front edge treatment as well. In the aspect range about edge-on, two or more sources contribute to the backscattering cross section of each body, and to devise a cross section reduction scheme which has the potential for being broadband, it is essential to reduce all sources of scattering individually. Some progress was made under the prior contracts referred to above. It was found, for example, that a material presenting a given surface impedance could significantly reduce the traveling and creeping wave contributions and, provided the impedance was chosen appropriately, the front edge scattering as well. A computer program designated RAMD was developed to solve the integral equations for the currents induced on the surface of an arbitrary twodimensional body subject to an impedance boundary condition and illuminated by either an E- or H-polarized plane wave incident in a plane perpendicular to the z-axis. The impedance may vary in any prescribed manner over the surface, and from a knowledge of the currents, the backscattering cross section is computed as a function of the angle of incidence. The program is described in Appendix B which also contains listings of two subroutines for computing the surface impedance of one and two layer metal-backed coatings. Valuable as the concept of surface impedance is, it does have some shortcomings for an investigation such as this. A knowledge of the impedance does not 3

indicate the electromagnetic properties of a coating that will produce it, nor is there any assurance that a material can be found to yield an impedance which is independent of both the incident field and the surface geometry. For these reasons, our original intent was to rely heavily on a program called RAMVS which computes the scattering from a two-dimensional body in the presence of a number of resistive, conductive and/or combined (i. e. resistive and conductive) sheets. The program had been successfully applied to single and multiple (but well separated) sheets extending out from a body, and was a vital tool in our development of techniques for reducing the E-polarized backscattering from ogival and wedge cylinders under Contract F33615 -73-C-1174 (Knott and Senior, 1974). By placing one or more sheets close to the surface of a perfectly conducting body, we hoped to simulate the effect of a coating whose properties could vary in depth as well as laterally and would be explicit in the formulation, in contrast to the situation when a surface impedance is used. Knowing the permittivities and permeabilities which different materials possess, we hoped then to arrive at the specification of a coating which would be effective in reducing the backscattering over the entire frequency range. Unfortunately the program was not equal to the task. Because of the formulation adopted, numerical errors increase rapidly as the spacing between two sheets or between a sheet and the body becomes comparable to the sampling distance (or cell size) along either. Since the sheet must be located at the mid-point of the layer it is designed to simulate, a coating 50 mils thick would require a minimum sampling distance of no more than 25 mils, and significantly less if high accuracy is required. Even a body only one inch in length would now strain the core storage of the computers available for running the program, not to mention the financial resources on hand. A considerable amount of time was spent in attempting to overcome the spacing limitation, and some of the approaches tried are documented in Appendix A, which also contains a listing of program RAMVS in its most recent form. It was ultimately apparent that only a major restructuring of the program could suffice, and since the time and funds available would not allow this, we were forced to seek another tool to help in specifying an effective coating. 4

Meanwhile, the electromagnetic properties of some of the materials being proposed for a coating had been measured at the Avionics Laboratory of Wright Field, and in all cases the impedance of a layer 50 mils or less in thickness was virtually independent of the angle of incidence. This gave some confidence in returning to a surface impedance approach, and since the results obtained using program RAMD were also in good agreement with measured data for an ogival cylinder with material OG-C-1 used as a coating, we transferred our allegiance back to our old and welltrusted program. Rapid progress was now made. Having delineated the scattering produced by the bare body at the four designated frequencies throughout the band (see Chapter 2), we exercised RAMD to determine the cross section reduction provided by impedances whose largest magnitudes were comparable to those of a 50 mil layer of the materials available, but which varied in some chosen manner over the length of the body. It was immediately apparent that to obtain any reduction at all at the lower frequencies, it was necessary to put as much material as possible on the body, i. e. to cover almost the entire surface with a coating 50 mils thick. The only departure from the uniform coverage was near to the front edge where the impedance and, hence, the layer thickness must be tapered to control the edge scattering. The procedures that were followed to arrive at a coating which is 'optimum' subject to the constraints provided by the materials available are described in Chapter 3. It should be noted that because of the difficulties posed by the lower frequencies (where the bodies are less than a wavelength long) and by the modest capabilities of the materials, there was little scope for a more subtle deployment of material. Indeed, the resulting prescriptions for both the ogival and wedge cylinders are identical in spite of the very different types of scattering exemplified. Finally, in Chapter 4, computed data are presented for some of the bare and coated bodies that have been considered experimentally by Emerson and Cuming. 5

CHAPTER 2 THEORETICAL CONSIDERATIONS 2.1 Geometries The two bodies to be studied are shown in Figure 2-1. The first is a 25 angle ogival cylinder whose relevant dimensions and the symbols used to denote them are as follows: wedge half angle: surface radius: overall length: surface length, top: maximum width: n2= 12.5~ R= 10.914 inches I = 4. 724 inches (= 2R sinfg) s = 4.762 inches (= TrR/90) max 2 w = 0.5174 inches (= 4R sin Q/2) The other is a wedge cylinder (the two dimensional analogue of a cone sphere) with the same wedge angle as the ogival cylinder: wedge half angle: Q = 12. 5 base radius: a = 0. 752 inches overall length: t = 4.226 inches (= a [1 + cosec]J) surface length, top: sm = 4.737 inches (= a [(1 + 2/90)7r/2 + cot2]) We observe that the surface lengths of the two bodies are almost the same, and would have been identical if, for example, the base radius of the wedge cylinder had been chosen as 0.756 inches. The four designated frequencies throughout the band are 2.0, 3.75, 7.5 and 15. 0 GHz, with wavelengths 5.901, 3. 147, 1. 574 and 0.787 inches respectively. Some corresponding electrical dimensions of the bodies are: f(GHz) = 2.0, = 3.75, = 7.5, = 15.0, ogival cylinder s a/ = 0. 807, max = 1.513, = 3.026, = 6. 052, wedge cylinder a/X = 0. 127 = 0.239 = 0.478 = 0.956. 6

y z x y x Figure 2-1: The bodies (not to scale). 7

At 7.436 GHz the surface length of the ogival cylinder is just 3A, and the body is then electrically identical to that considered under Contracts F33615-72-C-1439 and F33615-73-C-1174. In analyses and computations of the scattering we assume a plane wave incident at an angle 0 to the symmetry axis of the body and with its phase zero at the o center. The scattering angle 0 will also be measured from the same axis so that 0= 0 represents backscattering with 0 = ir (= 180 ) for edge-on incidence. We shall treat only the case of H polarization in which the incident electric vector is parallel to the generators of the cylinders and to the front edge, i. e. in the z direction. Since the bodies are two dimensional it is convenient to express the far zone scattered field in terms of the complex amplitude P(0, 0 ), i. e. the coefficient of kp)1/2 ei(k - 7/4) in the far field, where p is the distance from the origin of coordinates at the front edge (Bowman et al., 1969; p. 6) and a time factor e has been assumed and suppressed. The two dimensional scattering "cross section" is then (0, 00)= IP(0,00)I implying 2 =72 IP (2.1) which is independent of frequency if IP is. In presenting the results of our analyses and computations of backs cattering, we will generally display cr/X in dB, i. e. 10 log a/A. However, the experimental measurements performed by Emerson and Cuming, Inc. were carried out using cylinders of width L = 18 inches. Insofar as predictions based on two dimensional bodies are applicable, the connection between the measured (three dimensional) cross section ar3 and the two dimensional cross section given above is 3=2L 2 a(2.2) 3 = 8

(see Knott et al., 1973; Appendix C). Thus, a3 would also be independent of fre2 quency if /A were. For L = 18 inches (= 0.457 m), 2L = -3.79 dBsm, implying a3 =10 log - 3.79 dBsm, (2.3) 3 AX and in all comparisons of theory and experiment we display a3 in dBsm as a function of 0 = r- 0. 0 2.2 Bare Body Scattering To fully appreciate the cross section reduction task confronting us, it is important to have some knowledge of the type and magnitudes of the contributions to the backscattering of the bare body, with particular reference to the aspect range 150~ S 0 < 1800. This will also indicate the most effective deployment of an abo sorber, but it should be emphasized that the idea of individual portions of a body scattering relatively independently is a high frequency concept. As the frequency decreases, interaction between the individual scatterers becomes more important and, in addition, each scattering contribution becomes less localized in origin, with the decomposition breaking down completely when the wavelength becomes greater than the relevant dimensions of the body. For the cylinders of interest in this study, the concept is quantitatively valid at the two highest frequencies at most, but even at the lower frequencies the concept can still provide some useful qualitative information. The front edge is the same for both bodies, and its scattering can be estimated using the formula for diffraction by a wedge of included angle 22Q: Pf(0 0)= sin (cost - 1 + (cos +cos(7r-0 )) (2.4) where v = 67/36 (Bowman et al., 1969; p. 263). In particular, for edge-on incidence, P = -i 0. 0633 giving a=/X -25.94 dB. The ogival cylinder has an identical rear edge and if it is fully visible the (direct) scattering which it contributes is 9

-2ikI cos 0 P ( j P O -0,w-0 )e 0r0 ( 0 0) = where P is given by (2.4) and the phase factor has been introduced to maintain the phase reference at the origin. In addition, the rear edge reflects a wave which travels back along the surface of the cylinder to produce the scattering pattern characteristic of a traveling wave. The precise manner in which the rear edge diffraction merges into the traveling wave effect as 0 -> x - Q is not well understood, 0 but if, for the moment, we ignore the traveling wave contribution, the backscattering patterns of the ogival cylinder at the four frequencies computed using o(0 0f(o. 0 +Pr ) I'. (2.5) i.e. on the basis of front and rear edge diffraction alone, are as shown in Figures 2-2 through 2-5. On each figure we have superimposed the cross sections attributable to the two edges individually, but it should be remembered that at least for 0 >160~ the rear edge contributes primarily via the traveling wave which has not been included. We therefore expect the patterns to be meaningful only for 0 < 160 and at the higher frequencies. The backscattering from the wedge cylinder is more complicated. In addition to diffraction by the front edge, there are contributions from the creeping wave supported by the smoothly-rounded base and from the discontinuities in curvature where the wedge and circular cylinder join. The front edge contribution is again given by (2.4). The upper join where a cylinder of radius a meets a cylinder of infinite radius (the wedge) yields -(u), 1 r 2 -2ikacotScos(0 - o P(00o ) 8ka sec(0o -4)31 + sec (o0 -Je (2.6) (Senior, 1972) which decreases with increasing frequency, and similarly 10

-5.0.. 'o a" V rear.... - -10 b b0 0 C) V-e I I I I I I I I -15 I I I 1 I I I I I I /,front I I 94 I / I I I I -20 1 I A 180 160 140 120 0 (degrees) Figure 2-2: Ogival cylinder, 2 GHz: high frequency approx. 11

0 r -5 total r o- - I I ^b -10 - \ 0 I \ I I I / -15 - / I / I I I front / 1 o -20.... 180 160 140 120 O (degrees) Figure 2-3: Ogival cylinder, 3.75 GHz: high frequency approx. 12

-5 total -10,'I front I 180 160 140 120 00 (degrees) Figure 2-4: Ogival cylinder, 7. 5 GHz: high frequency approx. I I fo I I I I -20,/I I 180 160 140 120 ~o (degrees) Figure 2-4: Ogival cylinder, 7.5 GHz: high frequency approx. 13

-5 total I4 / \ / rear O/'front -20 - 180 160 140 120 0O (degrees) Figure 2-5: Ogival cylinder, 15 GHz: high frequency approx. 0 / I / I -20 14

< ^2 - -2ikacotgcos(0 +) pj( )(0o ) - 8ka sec (o +Q) + sec (0 +2)t e (2.7) valid as long as the lower join is fully visible, i. e. 0 > T - o. For this same aspect range the creeping wave contribution can be estimated using the known solution for a circular cylinder of radius a. Naturally, the result is independent of and from the formula in Bowman et al. (1969, pp. 110-111) cw ir(v+ 1/3) -2ikacosec2cos 0 P TCe e (2.8) 1/3 where T= (ka/2), and v= v1(T), C = C1(T) have the asymptotic expansions given in this reference. If ka > > 1, P decreased exponentially with increasing frequency, but for ka comparable to unity P is relatively independent of frequency. For 0 < o - Qthe lower join and the creeping wave both contribute through an interaction with the front edge, but there is no simple method for estimating the scattering in this case. The combination of the four contributions leads to a rather complicated backscattering behavior in the aspect range about edge-on, and instead of attempting to predict the net return at each of the four designated frequencies, we show only the individual cross sections aA/kas functions of 0 at 7.5 GHz. The important conclusion to be drawn from Figure 2-6 is that all the contributions are similar in magnitude, implying that all must be reduced to obtain a significant and broadband reduction in the scattering. Using the computer program RAMD (see Appendix B), we have computed the backscattering cross sections of the two bodies at each of the four frequencies. The results are tabulated in Chapter 4 and plotted for the aspect ranges 120~ < 0 < 180~ in Figures 2-7 through 2-10 for the ogival cylinder and Figures 2-11 through 2-14 for the wedge cylinder. We remark that the wedge cylinder always has a local maximum in the scattering pattern at edge-on incidence with a cross section in the range 15

-5 -10 near join b bt o -15 / 0 front edge -20 - cw / far / / -251 I 180 160 140 120 0 (degrees) Figure 2-6: Estimated wedge cylinder contributors at 7,5 GHz. 16

0 -5 01 0 1-I -15 -20II 180 160 140 120 0k (degrees) 'Figuire 97: Bare ngival cyxlindrp 2T0Hz: R AMD. 17

b bD 0 0 r-4 C) -10 -15 180 160 140 0 (degrees) 120 Figure 2-8: Bare ogival cylinder, 3.75 GHz: RAMD. 18

0 -5 b 0 r-4 -15 -20' 180 160 140 120 0 (degrees) Figure 2-9: Bare ogival cylinder, 7. 5 GHz: RAMD. 19

-20 T —4 -20 ' --- —180 160 140 120 0 (degrees) Figure 2-10: Bare ogival cylinder, 15 GHz: RAMD. 20

0 -5 b b - -10 0 -15 -20 - | 180 160 140 0o (degrees) Figure 2-11: Bare wedge cylinder, 2 GHz: RAMD. 21 120

0 -5 -10 b bD -15 -20 --- 180 160 140 120 0 (degrees) Figure 2-12: Bare wedge cylinder, 3.75 GHz: RAMD. 22

5 I-e- b -10 bD 0 -20 180 160 140 120 0O (degrees) Figure 2-13: Bare wedge cylinder, 715 GHz: RAMD. 23

0 -5 -10 b bD 0 — 4 CD r-4 -15 -20 180 160 140 120 00 (degrees) Figure 2-14: Bare wedge cylinder, 15 GHz: RAMD. 24

-10 to -20 dB X. The ogival cylinder, on the other hand, has a minimum edge-on and here the cross section ranges from -20 to -40 dBX. Comparison of the curves in Figures 2-7 through 2-10 with those in Figures 2-2 through 2-5 shows no agreement at the lowest frequency where the ogival cylinder is less than a wavelength long, but quite good agreement at the higher frequencies, particularly for scattering at aspects well away from edge-on where the traveling wave is no longer a significant contributor. Of the two bodies, the ogival cylinder is the simpler, and it is convenient to concentrate on it. The only significant contributors to its backscattering cross section in an aspect range about edge-on are the front edge and a traveling wave. The latter appears by reflection of a forward traveling wave at the rear edge of the cylinder and gives rise to a lobed pattern with a minimum edge-on. Traveling wave theory has been fully developed only for a thin wire and its application to a two dimensional body having non-negligible thickness is difficult to justify. Nevertheless, the concept is a valuable working tool and proves surprisingly accurate even for a body like an ogival cylinder. From an examination of measured data for the backscattering from a wire of length I, Senior and Knott (1968, Appendix A) showed that if X < < I the first traveling wave peak occurs at =01=49.9 degrees (2.9) and the second at 0=0 =98.1j- degrees (2.10) 2 where 0 = - f. For increasing X, 0.6 1, (2.9) progressively underestimates 0 the angular position 0 of the first peak by an amount as much as 30 for \X l; but since the magnitude of the peak decreases with increasing V/, the peak is seldom evident for X >. Unless X < < I, the second peak is not usually evident either, 25

being swamped by other scattering contributions. To see how these findings apply to the ogival cylinder, the angles at which the first maximum occurs in the computed data are compared with the predictions of (2.9) in Table 2.1. The agreement is reasonably good. Table 2. 1: Locations of first traveling wave peak for ogival cylinder. O C0 Frequency 01 (computed data) 01 (predicted) (GHz) 2.0? 55 3.75 37 40 7.5 26 28 15.0 18 20 Since there are two contributors to the backscattering at edge-on incidence, it does not follow that an absorber which reduces one of them will automatically reduce the cross section. Using program RAMD we have computed the backscattering patterns of the bare ogive at the frequencies f = 0.1 (0.1) 4. 0 GHz, and the values of (o(1r, lr)/)l/2 are plotted in Figure 2-15. The dashed line is based on the empirical formula o()(r, r) = -i(. 0465 + iO. 0345 e2i) 2 A)l (2.11) obtained by curve fitting, where I is the length of the cylinder. We remark that the first contributor, -iO. 0465, is close to the high frequency prediction -iO. 0505 for front edge diffraction, suggesting that the second term is the traveling wave contribution. We note that the two contributors are almost out of phase at 2. 0 GHz and are precisely in phase at 3. 75 GHz. If the above empirical formula were to hold at 26

0. 1 Y I I I I I I I I I ' / I I I I K" 0 1 2 3 4 frequency, GHz Figure 2-15: Edge-on backscattering from bare ogival cylinder: computed, ---- empirical (eq. 2.11).

all higher frequencies (and there is evidence to suggest that it does), the contributors would be almost in phase at 7. 5 and 15. 0 GHz as well. It therefore follows that an absorber which acts only on the traveling wave must increase the edge-on cross section at 2. 0 GHz and can provide no more than 6 dB reduction at any frequency. As evident from the formula, both contributors must be decreased in magnitude to achieve a reduction in the edge-on cross section over a broad frequency range. 2.3 Surface Impedance Concept Given a body whose geometry is fixed, the only feasible technique for cross section reduction is the judicious application of absorbers to suppress those forms of scattering which are responsible for the dominant return. According to the Contract specification we are limited to a coating no more than 50 mils in thickness applied to the bare (metal) body, and the task then is to choose the layer thickness and composition in such a way as to achieve the greatest cross section reduction over the entire frequency range. Unfortunately the problem as posed does not have a unique solution even within the bounds created by the properties of the available materials, but nevertheless progress can be made towards an 'engineering' solution. As noted in the Introduction, our analyses and computations are based on the use of an impedance boundary condition to simulate the effect on the scattered field of a coating applied to the surface. It is assumed that the (normalized) surface impedance rl is a function only of the local properties of the coating regardless of the field and the geometry of the surface to which the coating is applied, and this is an assumption which is incapable of prior validation. In particular, it is not evident that a coating will present the same impedance close to the edge of a body as it does when over a broad face, but in spite of this the results obtained using the impedance boundary condition have been found applicable under a wide range of circumstances, and the agreement between theory and experiment provides confidence in the validity of the procedure. Although this approach was not our first choice for meeting the objectives of the Contract, it does appear to be the only one 28

which is feasible at this time, and it enables us to predict the performance of a coating using a computer program which is reasonably efficient. For a surface whose impedance rl is specified at every point, the impedance (or Leontovich) boundary condition can be written as E-(n. E)n = rlZnA\H (2.12) where n is the unit vector (outward) normal and Z is the intrinsic impedance of the free space medium above the surface. Yr may vary over the surface and is zero for perfect conductivity. The derivation of (2.12) has been discussed by Senior (1960) and we remark that in contrast to the analogous boundary condition involving the normal derivative of the fields, (2.12) does not contain any tangential derivatives of Y). By trivial manipulation (2.12) can be expressed alternatively as H -(n. H)n -- AE (2.13) (Senior, 1962) where Y = 1/Z, and this is the dual of (2. 12) under the transformation E->- H, H- > -E, Z <- Y,] 1/ri. For a field incident on a surface at which the condition (2. 12) is imposed, the resulting boundary value problem can be solved either analytically or numerically to yield the scattered field. In the case of a two dimensional surface of profile C illuminated by an H polarized plane wave, the scattered magnetic field is p) = A r A H1( )(kr) + ir(s) H (kr K(s') ds' (2.14) where K (s ) is the current induced in the surface. H (1) and H (1) are the Hankel o 1 functions of the first kind of orders zero and one respectively, and r = p - p' (Knott and Senior, 1974). The integral equation obtained from (2.14) is the basis of operation of computer program RAMD. 29

To determine the impedance presented by an absorber coating applied to the bare body, consider a plane wave incident on a homogeneous layer of infinite extent backed by a metallic sheet. If the plane wave is H polarized and incident at an angle 0 to the normal, sin tan (kd - sin2 ) (2.15) rl 7E= - sin t where e andji are the complex relative permittivity and permeability of the material, d is the thickness of the layer and k = 27 /Xis the free space wave number. Similarly, for a coating consisting of two homogeneous layers of thicknesses d1 and d2, we have 2 1- l-Aexp(2ikd E1i -'sin 0) r=1 21 - sin 0 8 r (2.16) E S +Aexp(2ikd1 1 1 - sin 0 where A + B (2.17) 1-B e1 E2 s- B ) with B i 2 tan kd/ - 2 s(2.18) 2 C11 - sin e (see Figure 2-16). Programs have been written to compute the surface impedances given in (2.15) and (2.16) for any material constants, layer thickness and 0. Versions appropriate to the particular case of grazing incidence (0 = 90 ) have also been coupled into program RAMD to study the effect of a material having a specified thickness variation over the surface of the body. The above impedances are functions of the angle 0 and therefore violate our requirement that r7 be independent of the incident field direction, but if > >, (2.19) 30

y (a) 0 I I. d $f E 1 x Z Z Z Z Z Z Z,.ool vvAmi-n I W 77 77 77 7 7-77,o,,,, -,., / / mutai.4,.0, / 'o, /' - e - e 00, z y e (b) I I -- , - -1 I,., , f Z, I Z, I P', I I x #','/ff/f/ metal f fffff Figure 2-16: Geometries for (a) one layer and (b) two layer impedance calculations. 31

the angle dependence disappears to a first approximation. Equation (2.15) then becomes 7 = -iftan(kd ), (2.20) and for a two layer coating, = 1 - A exp(2ikd 1j) (1) 1 1, i 221 9 4lc 1+ A exp (2ikd/FeL (2.21) where A is again given by (2.17) and B = i tan (kd 2 T). (2.22) 2. 222 The condition (2. 19) is satisfied to an adequate degree by all of the materials considered. This is illustrated by the results in Tables 2-2 through 2-5 for the surface impedance of a layer of thickness d = 0(0. 005) 0.050 inches of material OG-C-1 at each of the four designated frequencies. This material figured prominently in our investigation and the measured values of its permittivity and permeability are as follows: f(GHz) = 2.0: = 22.46+i0.17, = 5.29+i2.99 3.75: 23.38 + i.87, 3.57 +i3.15 7.5 20.53 + il.80, 1.72 + i2.66 15.0: 22.08+i0.39, 1 +il.68. The left hand columns in each of the Tables 2-2 through 2-5 are for normal incidence (0 = 0) and those on the right are for grazing incidence (0 = 90 ). We observe that the impedance rl differs by no more than about one percent between these extreme values of 0. 2.4 Theoretical Approach to Cross Section Reduction In seeking to choose impedances and, hence, material coatings that are effective in reducing the scattering from the bodies at aspects close to edge-on, 32

9 di-vta ar-otl:=(. YrJ ---::O. F-f:=2* vmu::::(5#29v2* 99) reps=(22.46p 0. 1.7) &el,-I(.-i Rdirtza ci=), 0 aris':90, Lend ANGf E II: F 0.0 0.0 2.000 &data3 d=0.005 &end ANGLE Dl F 0.0 0.005 2.000 Lda'ta d=0.010 Lend ANGLE DE F 0.0 0.010 2.000 &drtaL3 d=0.015 Xend ANGLE Di F 0.0 0.015 2*000 &lda t, dr=0.02 Send ANGLE Di F 0.0 0.020 2'.000 XrJa't,. d=0.025 &end ANGLE Li F 0.0 0.025 2*000 &d'tla d=0.03 &end ANGLE D F 0.0 0.030 2,000 &dta d=0.035 Send ANGLE Di F 0.0 0.035 2.000 dd'ita d=0.04 &end ANGLE D F 0.0 0.040 2.000 &data d=0.045 &end ANGLE D F 0.0 0.045 2.000 Xdata d=0.05 lend ANGLE Di F 0.0 0.050 2*000 MU 5.290 2.990 MU 5.290 2.990 MU 5.290 2*990 MU 5.290 2.990 MU 5.290 2,990 MU 5.290 2,990 MU 5.290 2,990 MU 5.290 2.990 MU 5.290 2.990 MU 5,290 2*990 MU 5,290 2*990 22*460 0,170 EPS 22.460 0.170 EF'S 22.460 0.170 EPS 22.460 0. 170 EPS 22*460 0*170 EPS 22.460 0*170 EPS 22.460 0.170 EPS 22,460 0.170 EPS 22.460 0.170 EPS 22.460 0.170 EPS 22.460 0.170 ETA 0.0 0.0 ETA 0.016 -0.028 ETA 0.032 -0.056 ETA 0.049 -0.085 ETA 0,066 -0*114 ETA 0.084 -0.143 ETA 0.104 -0.173 ETA 0*125 -0*204 ETA 0.148 -0.236 ETA 0.174 -0 268 ETA 0.202 -0.302 ANGLE I:l F 90.00 0.0 2.000 l&data d=:0,005 &end ANGLE Li F 90. 00 0.005 2.000 Adata d=0,01 &end ANGLE El F 90.00 0.010 2.000 Xdata d=0.015 Lend ANGLE Di F 90.00 0.015 2.000 Lda'ta d=0.O29&end ANGLE Di F 90.00 0.020 2.000 da<3ta d=0.025 Lernd ANGLE DE F 90.00 0.025 2.000 &data d=0.03 &end ANGLE Dl F 90.00 0.030 2.000 &data3 d=0.035 &dgada ANGLE D F 90.00 0.035 2.000 &data d=0.04 &end ANGLE Di F 90. 00 0.040 2.000 &d3'ta d=0.045 &end ANGLE Di F 90.00 0.045 2.000 &data d=0.05 &end ANGLE D F 90*00 0*050 2.000 MU 5.290 2.990 MU 5.290 2.990 MU 5.290 2.990 MU 5.290 2.990 MU 5*290 2.990 MU 5.290 '2.990 MU 5.290 2.990 MU 5.290 2.990 MU 5.290 2.990 MU 5,290 2*990 MU 5.290 2.990 EPS 22.460 0.170 EPS 22.460 0.170 EPS 22.460 0.170 EPS 22.460 0.*10 EPS 22.460 0.170 EPS 22.460 0.170 EPS 22.460 0.170 EPS 22.460 0.170 EPS 22.460 0.170 EPS. 22*460 0.170 EPS 22.460 0.170 ETA 0.0 0*0 ETA 0.016 -0.028 ETA 0 032 -0,056 ETA 0.049 -0.084 ETA 0.066 -0.113 ETA 0.084 -0.142 ETA 0.104 -0.172 ETA 0.125 -0.202 ETA 0.148 -0.234 ETA 0.173 -0.266 ETA 0.202 -0.299 Table 2-2: Surface impedance r1 for material OG-C-1 at 2. 0 GHz. CO C0

&da; ta aar',it-00,- r=o, vf3.75vau=(3.57,315) rep:(23,38,1.87) gend ANGLE D F Mu EPS E'r &data d=0,,ar=90. &send Ai~OLLE E' F A mu 0.0 0.0 &d'tat d=0.005 ANGLE EL 0. 00)5 Sd3ta d-i0=010 ANGLE iD 0.0 0,010 Sdat;3-i d=t0.15 ANGI.HE El 0.0 0.015 &'it~ d=0.020 A N G) L. E El 0.0 0.020 &data d:0.025 ANGLE El 0.0 0.025 Sdita d=0.030 ONGLE D 0,0 0,030 8data d=0.035 ANGLE D 0.0 0.035 ~da-ita d=0*040 ANGLE E 0.0 0.040 Zdata d=0.045 ANGLE D 0.0 0.045 3.750 3.570 3.150.- I I 23.380 1.870 0.0 0.0 90.00 0.0 3.750 3.570 3.150 &end F 3.750 &ernd F 3.750 &end F 3.750 &end F 3.750 &end F 3.750 Send F 3.750 Send F 3#750 Send F 3.750 Send F 3.750 F 9da-ta d=0.005 gend 3,5; 3.57; 3.57 3.57 3,57 3.57 3,57 3.571 3,57' MU EPS 70 3.150 23.380 1.870 MU EPS 70 3.150 23.380 1.870 MU EPS '0 3.150 23.380 1.870 Mu EFS P0 3.150 23.380 1.870 MU EPS '0 3.150 23.380 1.870 MU EPS 0 3.150 23,380 1.870 MU EPS 0 3.150 23.380 1870 MU EPS 0 3.150 23.380 1.870 MU EPS o 3.150 23,380 1.870 MU EPS 3*150 23.380 1.870 ETA 0.032 -0.036 ETA 0,064 -0.071 ETA 0.099 -0.107 ETA 0.138 -0.142 ETA 0.181 -0.177 ETA 0.231 -0o210 ETA 0.289 -0.240 ETA 0.358 -0.263 ETA 0.438 -0.274 ETA 0.527 -0,268 ANGLE 1:1 90.00 0.005 &data d=0.010 ANGLE ED 90.00 0.010 &data d=0.015 ANGLE El 90.00 0.01.5 &data d=0,020 ANGLE El 90.00 0.020 8dat3- d=0.025 ANGLE D 90.00 0.025 &data d=0.030 ANGLE ED 90.00 0.030 &da~ta d=0.035 ANGLE l 90.00 0.035 9data d=0.040 ANGLE El 90.00 0.040 &data3 d=0045 ANGLE E 90.00 0.045 Sda-ta d=0.050 ANGLE El 90.00 0.050 F 3.750 Send F 3.750 Send F 3.750 Send F 3.750 &end F 3.750 Send F 3.*750 &end F 3.750 Send F 3.750 Send F 3.750 Send F 30750 MU 3.570 3*150 MU 3.~570 3.~150 MU 3.570 3.150 MU 3.570 3.150 MU 3.570 3.150 MU 3.570 3.150 mU 3.570 3.150 MU 3.570 3.150 MU 3.570 3*150 MU 3.570 3.150 EPS 23.380 1.870 EPS 23.380 1.870 EPS 23.380 1.870 EPS 23.380 1.870 EPS 23.380 1.870 EPS 23.380 1.870 EPS 23.380 1.870 EPS 23.380 1.870 EPS 23,380 1.870 EPS 23.380 1.870 EPS 23.380 1.870 ETA 0.0 0,0 ErA 0.032 -0.035 ETA 0.064 -0.070 ETA 0.099 -0.106 ETA 0 137 -0.140 ETA 0.181 -0.175 ETA 0.230 -0.207 ETA 0.288 -0.236 ETA 0.356 -0.258 ETA 0.435 -0.270 ETA 0.523 -0.263 ANGLE 0.0 El 0.050 3.750 3*57C Table 2-3: Surface impedance ri for material OG-C-1 at 3.75 GHz.

Idata~ a I) Fr; M EF'S &TA=7#50 rIlhl: (1#72 2.66 1eps —(20.53I SO) &end A N G L I E D F m u EP'S ETA 9d;i-3ta anv:90. vd=O,, Lend ANCGLE l F MU 0.0 0,0 7.500 1.720 2.660 20.530 1.800 Llata d=0.005 ANG LE ri 0.0 0.005 &data d=0#010 ANQLE El 0.0 ( 0010 Ldata d=0#015 ANGL..E El 0.0 0.015 8data d=0.020 ANIGL: LI 0.0 0.020 Ad t, d=0 025 ANGL-E El 0.0 0.025 Ida3-tL d=0.030 ANGL-E El 0.0 0,030 %data 'P0. 035 ANGLE El 0.0 0.035 glata d=0#040 A (.LE El 0,0 0.040 &dLta d=0.045 ANG L.E Ye 0.0 0.045 Lend F 7.500 Lend F 7,500 Lend F 7,500 end F 7.500 Lend F 7.500 Lend F 7.500 F 7.500 &end F 7.500 Lend F 7.500 1,72( 1*72( 1.72( 1 72( 1,72( 1.72( 1,*72( lo72( MU EFS 2 2.660 20.530 1.800 Mu EPS 2 2.660 20,530 1,800 MU EPS 2 2,660 20.530 1,800 MU EPS ) 2,660 20,530 1.800 MU EPS ) 2.660 20,530 1.800 MU EPS ) 2.660 20.530 1.800 MU EPS ) 2*660 20.530 1.800 MU EPS ) 2,660 20.530 1.800 MU EPS ) 2.660 20.530 1.800 Mu EPS D 2i660 20.530 1.800 0.0 0.0 ETA 0.054 -0.034 ETA 0.110 -0.066 ETA 0*172 -0.094 ETA 0.242 -0.114 ETA 0.320 -0.119 ETA 0.402 -0.103 ETA 0.478 -0.057 ETA 0.* 531 0.014 ETA 0.549 0.097 ETA 0.532 0.170 90.00 0.0 grata d=,,005 ANGLE El 90.00 0.005 8r3d~1;a d=04010 ANGL..E El 90.00 0.010 Lda-ta d=0.015 ANGLE El 90.00 0.015 9data d=0#020 ANGLE El 90.00 0.020 Sda-ta d=0#025 ANGLE El 90.00 0.025 &data d=0#030 ANGLE El 90.00 0.030 gdalta d —0#035 ANGLE D 90.00 0.035 Zda-ta &=0.040 ANGLE E 90.00 0.040 Sdati3 d=0#045 ANGLE E 90,00 0.045 Ldata d=0.050 ANGLE D 90.00 0.050 Len Len Len Ler Len &en ern Len ter Ler 7.500 1.720 2.660 F mu 7.500 1.720 2.660 -id F MU 7.500 1.720 2.660 id F MU 7.500 1.720 2.660 F MU 7.500 1.720 2.660 F Mu 7.500 1.720 2.660 id F MU 7.500 1.720 2.660 7.50 1.720 u 2.660 id F MU 7.500 1.720 2.660 -id, F MU 7,500 1.720 2.660 F mu.7*500 1*720 29660 -id F mu 7,500 1*720 2.660 * EPS 20.530 1.800 EPS 20.530 1.800 EPS 20.530 1.800 EPS 20.530 1.800 EPS 20.530 1.800 EPS 20.530 1.800 20*53 1 0.800 EPS 20.530 1.800 EPS 20.530 1.800 EPS 20.530 1.800 EPS 20.530 1.800 ETA 0.0 0.0 ETA 0 054 -0.033 ETA 0.110 -0.064 ETA 0.172 -0.091 ETA 0.241 -0.110 ETfA 0.318 -0.114 ETA 0.398 -0.097 ETA 0.472 -0.052' ETA 0.523 0.017 ETA 0.541 0.098 ETA 0.526 0.169 &data d=0*050 &end ANGLE E F 0.0 0.050 7.500 1*72( Table 2-4: Surface impedance rO for material OG-C-1 at 7.5 GHz. C~3 01

Ida-La d=,,yang=04,,f=15.0vnuilj(1.0,168),eps=(22.08,0,39) &erld Schta.3 d=0,,firfz-.90. &enad )N(3'l-.E Li F -MU — --- EF8 --- --— ETA --- A N GL E: ' 0,0 0,0 15.000 1.000 1.680 22.080 0.390 0.0 &data AIN(LE 0.0 &dat~a ANGILF 0.0 g da t 0.0 ANGLE 0.0 0.0 % da fta ANGLE: ANGLE 0.0 d=0,005 gerd 0.005 15 d=0*010 &ernd 0.010 15~ d=0. 1It~5&add 0.015 15~ d=0.0.20 8 e nid 0.020 154 d=0.025 &erid Li 0.025 15. d=0#030 &ierd 0.030 15* d=0.035 &en~d 0.035 15. d: 0040 & c3 nd li 0.040 15* d=0.045 &en~d Li 0.045 15. F --— mu --- — ES-.000 1.000 1.680 22.080 0.390 F --- MU --- — FS-.000 1.000 1.680 22.080 0.390 F --- mu --- — ES-.000 1.000 1.680 22.080 0.390 F --- mu --- — ES-.000 1.000 1.680 22.080 0.390 F --- mu --- — FS-.000 1.000 1.680 22.080 0.390 F --- mu --- -ES - 0000 1.000 1.680 22.080 0.390 F --- MU --- — ES1000 1,000 1.680 22.080 0.390 F --- MU --- — FS1000 1.000 1.680 22.080 0.390 F -— mu --- -— EF8 --- 1000 1.000 1.680 22.080 0.390 F - -— MUi --- -ES - 1000 1.000 1.680 22.080 0.390 0.0 0.0 -— ETA ---0.069 -0.0359 --- ETA --- 0.146 -0.071 0.240 -0.083 --- ETA --- 0.342 -0.051 --- ETA --- 0.413 0.038 --- ETA --- 0,410 0.142 --- ETA --- 0. 356 0.*203 --- ETA --- 0.300 0.215 -— ETA --- 0.262 0.203 --- ETA --- 0.243 0.185 90.0() 0.0 15.000 1.000 1.680 gdata(i3 dO0.005 ANGI... E El 90.00 0.005 &data3 d=0.010 ANGLE ' Eli 90.00 0.010 &d3iat d=0.,01.5 ANGLE Eli 90.00 0.015 Agd (.,ad d0 0 '2'0 A NGCLE Li 90.00. 0.020 &daita d=0.025 ANGLE Li 90.00 0.025 &data d=0.030 ANGLE Ei 90.00 0,030 9data d=0*035 ANGLE Li 90.00 0.035 %da-ta d=0.040 ANGLE El 90.00 0.040 &data d=0#045 ANGLE Eli 90.00 0.045 Rdiata d=0.050 ANGLE Di 90.00 0.050 Re( ger Zei get Rei Set gei Set Set rnd F --- mu --- 15.000 1.000 1.680 n d. F --- mu --- 15.000 1.000 1.680 rid F -— mu --- 15.000 1.000 1.680 rid F -— mu --- 15.000 1,000 1.680 rid F — MU --- 15.000 1.000 1.680,rid F --- MU --- 15.000 1,000 1.680 rid F --- mu --- 15.000 1,000 1.680,rid F --- MU-. — 15.000 1.000 1.680,rid F -— mu --- 15.000 1.000 1.680 rid F --- mu --- 15.000 1.000 1.680 2208 0.390FS22.080 0.390 22.08 0.390 22.080 0.390 22.08 0.390 22.080 0.390 22.08 0.390 22.080 0.390 22.08 0.390 22.080 0.390 22.080 0.390 0.0 0.0 --— ETrA --- 0.069 -0.037 -— ETA --- 0.146 -0.067 --- ETA --- 0.238 -0.077 --- ETA --- 0.336 -0.045 -— ETA --- 0.404 0.040 -— ETA --- 0.402 0.141 0.352 0.200 -— ETA --- 0.298 0.214 --- ETA --- 0.262 0,203 -— ETA --- 0.242 0.186 %da~ta d=0.0150 Send ANGLE Li 0.0 0.050 15. Table 2-5: Surface impedance Yj for material OG-C-1 at 15. 0 GHz. w~

some insignt can be gained by looking at the integral expression (2. 14) for the scattered field. At large distances from the body, i. e. in the far zone (1) (1) i(kp- 7r/4) e-ik p H1 (kr),- iH (kr) - i ek e 1o i rkp and since r A -p, we then have s (p) 2 i(kp - 7r/4) z- rkp e P z - ' TP where the far field amplitude is k A ~ 0 P P= 4 nJ- r (s )K(s)e- - ds. (2.23) The form of this expression suggests three possible procedures for reducing the scattering in some chosen direction p, and these could be employed singly or in combination: (i) reduce the amplitude of the surface current K(s ) over the entire surface of the body, (ii) adjust the phase of the current to ensure that there is no stationary phase point in the range of integration, and (iii) choose the surface impedance so that ' ~ p - rs(s ) is small over the surface. The first two are the bases for the more conventional approaches to cross section reduction. An absorber does serve to reduce iK, but not always by a sufficient amount to rely on this technique alone. Shaping of the body is directed at the second method, and the elimination of all specular reflections is, in fact, equivalent to the elimination of the stationary phase points associated with the incident field phase. Unfortunately, such shaping may also accentuate surface wave effects. In the case of the ogival cylinder, there is a stationary phase point created by the (backward) traveling wave, and it is then necessary to reduce this portion of the surface field as much as possible. 37

The third method is more novel. If rl(S )= *(2.24) at every point of the surface, the scattering will be zero in the direction of p and (presumably) small in some range of aspects about this direction. The required impedance depends neither on the frequency nor the angle of incidence and is always real, but there is a difficulty: for any closed surface the specification demands a negative real impedance over a portion of the surface. Thus, to suppress the edge-on backscattering from the ogival cylinder, r1 must vary from 0. 21644 (= cos 77. 50) at the front, through zero at the middle, to -0. 21644 at the rear. We have verified the validity of the scheme by using program RAMD to compute the bistatic scattering from the ogival cylinder for edge-on incidence (0 = 180 0) at a frequency of 2 GHz. 0 The results are shown in Figure 2-17 along with the corresponding curve for the bare body. We observe that the optimum impedance has reduced the scattering at angles within 180 of edge-on, but has increased it by as much as 4 dB at wider angles. We also show the curves for the modulus of the optimum impedance and for a treatment in which the cylinder is left bare where the optimum impedance is negative. Neither impedance is effective and the scattering is even larger than that of the bare body. Although the optimum surface impedance is non-physical for any finite body of non-zero thickness, one geometry for which it is both realizable and simple is the infinite wedge, and for edge-on incidence in particular, the requirement is that ii = sin 2 where 2Q is the included angle of the wedge. As we have already noted, the scattering from the front edge of the ogival and wedge cylinders can be analyzed using the infinite wedge as a model, and it is therefore of interest to examine the edge diffraction coefficient for a wedge with arbitrary but constant surface impedances the same on both faces. 38

0 B / /,, /.-S / / - -10 /, / 1/ Ix -20 - bI o - / 0-40 /(degrees) I / - I I I I -40 I. I 180 160 140 120 0(degrees) Figure 2-17: Bistatic scattering from the ogival cylinder for 0 - 1800 at 2 GHz: ( ) 0 ( ---)rlt ~, (.-) and(..) rlmax(g', 0). 39

Using the method of Maliuzhinets (1959) it can be shown that for backscattering i 2 21,7 cos (7'r - 0) cos -2 vsin cos v ( r- 0) cos -(3 _ 0) 2v v 2 V 2 (Cos - 0)cos - 0) - sin 2 v2 v2 * F (0,X) (2.25) (Senior, 1977) where v= 2 (1 - n/r) (2.26) r7 = coS X and F(0,X ) is expressible in terms of functions defined by Maliuzhinets. In the special case rl = 0 of a perfectly conducting wedge i P(0,0,0): -2 2 7' Cos C 1 C1 1 37. cos(- 0)cos( --- ) - sin 2 v sin os ( - c) coso - - 2) (2.27) and hence 2 1cos v )cos v 2 0) - sin P0,0 1 =cos( 0) - - 2 F(0,)P(0, 0, ) 2 v2 20 (2.28) Apart from the infinity of P(0, 0, 0) in the specular direction 0 = 2 -, the ex2 pression for P(0, 0, rq) is finite for all 0, Q < 0 <, and is zero if sin = {cos(1 - 0cos- ). (2.29) sinU: Osos - a) oo (2.2) v v 2 v 2 - 40

This can be solved to find X and hence rl for any given value of 0, and for a wedge of half angle 12.5 (v = 67/36) the impedance necessary to null out the edge-diffracted backscattered field is listed as a function of 0 in Table 2-6. Over a range of 30~ about edge on, the optimum impedance varies from 0. 216 to 0. 357 and is, of course, real. 0(deg.) r| 0(deg.) rl 0 (deg.) r 180 0.21644 120 0.72613 65 1.57259 175 0.22046 115 0.80194 60 1.63633 170 0.23247 110 0.88005 55 1.69527 165 0.25234 105 0.95976 50 1.74895 160 0.27986 102.5 1 45 1.79690 155 0.31472 100 1.04038 40 1.83876 150 0.35655 95 1.12120 35 1.87417 145 0.40493 90 1.20154 30 1.90284 140 0.45936 85 1.28070 25 1.92456 135 0.51929 80 1.35446 20 1.93913 130 0.58415 75 1.43286 15 1.94645 125 0.65331 70 1.50458 12.5 1.94737 Table 2-6: Impedance for zero backscatter at angle 0. To compute the width of the null and/or the effect of an impedance which is not real, it is necessary to know the function F(0,X). Although this can be determined for a wedge of any angle, its expression is tractable only if v is the quotient of two integers whose numerator is odd. An example is a wedge of half angle 2= 15~ 11 for which v 6, and since this is similar to the wedge appropriate to our cylinders, it is worthwhile to examine it in detail. As shown by Senior (1977), we then have 17 17 5 f( 2 F ( X)= + A 7' _'(-=z 7 -F f12 12 12 12 12 12 ( nfX~ 12 )(+~-1 (2.30) 41

where 3T a 2p-1 6 a 6p-10 (I os + cos11\s sll os +cos 33 -r The impedance necessary to null out the backscattering in the direction 0 is 1 1/2 (1 = Cos 6 sin cos 1 2 cos 2 0 (2.32) and the values for 0 = 180 (-5 ) 150 are as follows: 0(deg.) = 180, r = 0.25882; 0(deg.) = 160, r1 = 0.32267 175 0.26287 155 0.35771 170 0.27497 150 0.39973 165 0.29498 Over this range of 0 the average backscattering is a minimum if the null is located at 0 = 165. The corresponding impedance is 0.29498 and the edge diffraction coefficient is plotted as a function of 0 in Figure 2-18. For 1800 > 0 > 1500 the cross section reduction averages 16.8 dB, but much of this effect is lost if the impedance is complex. This is evident from the curves for rY = 0.29498 exp (t i ) with X= Xr/6, T /3 and ir/2 which are included in Figure 2-18. If y= X /6 the average cross section reduction over the same range of aspects is only 9.8 dB and if g= r /2 (reactive impedance) the scattering actually exceeds that of the bare body. In contrast to edge diffraction, reduction of a traveling wave contribution is not so critical in its specification of the required impedance. Indeed, there is no optimum impedance per se and it matters little whether the impedance is real or not. Since the wave travels over the surface of the body before and after reflection at the rear edge, any coating that will adequately attenuate the wave will suffice. For ogival cylinders with a uniform surface impedance r, computations have shown that the backscattering cross section in the direction of the first traveling wave lobe 42

0.4 I ' I I I ' - /, / 9aI I f I i I I; I I I I I I hvn I I 0 I I I I I I I3= II I / I II / I I I I r I 0 / I I/ / - /, I ' / I I I,' / / I ' /s # I./ / I ~- / /,, / / 1' / / / 180 160 140 120 0 (degrees) Figure 2-18: Edge diffraction coefficient for backscattering from a wedge having 2 150: (-)I bare wedge, ( ---)/ 0, (- - -) -' r/6, (-) r-) 7/3, ( ---) y7r/2, where n7 0. 29498 exp(+t i ). 43

is roughly proportional to exp(-0.7ks Re. Ar) max and to achieve a 20 dB reduction requires only that Re r- >Vs. (2.33) 00max The imaginary part of rj appears to have little effect apart from a slight change in the electrical length of the body with a resulting change in the phase of the traveling wave contribution. If the condition (2. 33) can be satisfied at all frequencies of interest, it may then be possible to coat only a portion of the body. Coatings applied to the rear half of an ogival cylinder were examined by Knott and Senior (1973), They showed the importance of extreme care in fairing in the leading edge of the absorber and arrived at an 'optimum' impedance specification of the form r oi 2 s ) where 2 max s is the surface distance measured from the front of the cylinder. It will be observed that this places the maximum impedance at the rear edge where the traveling wave reflection occurs. To reduce the front edge contribution as well requires an impedance of about 0. 3 in the vicinity of this edge, and an impedance which would be effective in reducing both types of scattering is one which starts at 0. 3, increases slowly over the front half of the body, and then more rapidly over the rear. Indeed, for a body of adequate electrical length for which the two main contributors to the scattering are (or can be made) substantially independent of one another, it is possible to pose a true optimization problem in which the task is to determine the impedance variation for a maximum average cross section reduction over some range of aspects subject to a limitation on the maximum value of the impedance available. Unfortunately, such considerations are irrelevant as regards the present problem. At the lowest frequency of interest, the ogival cylinder is only 0. 807 Xin length, and to satisfy the condition (2. 33) now requires that Re. r t > 1. 2. Even for a coating of the maximum allowed thickness, none of the materials available provide 44

an impedance greater than about 0. 2 at 2 GHz, and complete coverage of the body would now reduce the traveling wave return by only 3 dB. The constraints provided by the lower frequencies effectively prohibit anything other than a coating of maximum thickness over almost the entire surface of the body, the only departure being in the immediate vicinity of the front edge. Even for the front edge contribution precise control is no longer possible because of its secondary excitation by the residual traveling wave. The combination of the direct and secondary excitations now produces a front edge return which is frequency resistive for all of the impedance values available to us, and provided the edge impedance does not exceed about 0. 3, there is no evidence that one value is better than another. 45

CHAPTER 3 SPECIFICATION OF A COATING To describe the procedures that were followed to arrive at effective coatings for the bodies, it is convenient to follow a chronological approach. At the outset of the program we had no precise knowledge of the impedances presented by the materials available, and our first task was to explore the performance of various impedance specifications to determine that which would be most effective. Then, as the properties of the available materials were measured, it became apparent what impedance magnitudes could be realized, and our objective changed from a specification of impedance to the specification of a layer of the most promising material. The entire study was numerical and based on an ogival cylinder whose length was extended to simulate the increase in size resulting from the application of a coating of about 50 mils thickness. The dimensions of the extended body are as follows: wedge half angle:: = 13.065~ surface radius: R = 10.942 inches overall length: = 4.947 inches surface length, top: s = 4.991 inches. max 3.1 Impedance Specification Four types of surface impedance variation were initially considered, all of them at a frequency of 3. 75 GHz. They were m rl s rm r max xs max for m = 0 (uniform coating), m = 1/2, m = 1 (linear variation) and m = 2 (quadratic variation), and for r7 = 0.2(0. 2)1. 0, where s is the surface distance along the ogive measured from the front. Backscattering patterns were computed in each case, and the values of 10 log c/Xin the broadside specular (0 = 90 ), peak traveling wave (0 = 144 ) and edge-on (0 = 180 ) directions are plotted as functions of rm in max 46

Figures 3-1 through 3-3 respectively. We observe that in the broadside and traveling wave directions the uniform coating which "puts most material on the body" is by far the best. At edge-on, the uniform coating is best only for r7 < 0.33 (we did not explore the behavior for r77 < 0. 2). With increasing rmax > 0.33 it becomes inferior to all the other coatings (and actually raises the edge-on return if rm > 0.57), and 1/2 a the s variation is then the best. These are quite different from the findings of Knott and Senior (197 ) at a frequency equivalent to 7. 5 GHz, but it must be recalled that the coatings were then applied only to the rear half of the ogival cylinder. To avoid the potentially large scattering from the leading edge of the absorber, it was found necessary to have the 2 first and second derivatives of rn(s) zero there, forcing us to the s variation of the impedance. By covering the entire ogive we are able to circumvent this difficulty. Since all of the coatings used are reasonably effective in reducing the traveling wave contribution, particularly for the larger values of r,. attention was now max directed at the front edge or 'tip' contribution. From an analysis of a uniform impedance wedge (see Section 2.4), the optimum impedance for edge-on incidence is 77 = sin2, implying r7 = 0.21644 for n= 12. 50. To see if there is any improvement in performance by starting all of our tapered coatings with this non-zero value of r7 at the edge, we considered r = rtip+ (max ' rtip )s m max for m= 1/2, 1 and 2 with 77rmax = 0.4(0.2)1. 0 and 77tip = 0.216. The results are shown as broken curves in Figures 3-1 through 3-3, and are not particularly surprising. The greater amount of material on the body reduces the broadside and traveling wave returns compares with the values previously attained for the corresponding i max; and at edge-on the scattering is now almost independent of rmax, suggesting that the tip is the dominant contributor. However, in the important traveling wave direction the uniform coating is still superior. 47

tip + s bD \ x " tip+s o 0 \ \ s 1/2 t s -10 - 0 0.5 1.0 lmax Figure 3-1: Broadside backscattering for different coatings. 48

0 b bD 0 — 4 oC r-q -5 -10 -15 2 s -20 0 0.5 1.0 Tmax Figure 3-2: Traveling wave peak returns for different coatings. 49

-101 + s 1/2 b. 0 tip ~1/2 -40 0 0.5 1. 0 77max Figure 3-3: Edge-on backscattering for different coatings. 50

It seemed likely that we could improve the edge-on capability of a uniform coating whilst retaining its performance in other directions by tapering the impedance down to a pre-assigned level at the tip. To pursue this idea, we selected the case r1max = 0.4 and replaced the impedance over the first inch of the surface by a sinu'max* J soidal taper. The actual impedance variation used was ti (r 7 -+ ) sin 0 < s < s tip max tip 1_ 1 =n, s, s <s max 1- - max with s = 1 inch and nma = 0.4. We first tried rti = 0.214 and then progressively max tip reduced tip in steps of 0.01 until the edge-on cross section had bottomed out. The reduction in rtip produced a very slight (0. 09 dB) increase in the broadside cross section, a somewhat larger (0.73 dB) increase in the traveling wave peak, but a substantial (8.95 dB) reduction in the edge-on return. The edge-on and traveling wave cross sections are plotted as functions of tip 0. 094 < -? < 0.214, in Figure 3-4, tipwith the values for ti = 0 and 0.4 (uniform coating) included for comparison. tip The backscattering pattern for the optimum tip (rtip = 0. 104) is plotted in Figure 3-5 as a function of angle, 1300 < 0 < 1800 along with the corresponding curves for the bare body and the uniform (rn = 0.4) coating. The toptimumt tip would certainly seem to have had a desirable effect and the average cross section reduction (straight dB averaging every 5 over a 30 range) is 14.6 dB compared with the 7.2 dB provided by the uniform coating. However, these figures are a little misleading since most of the energy is in the traveling wave lobe. If, instead, we compute the total scattered power over the 30 range by converting the dB to actual powers and then sampling every 4~, we find that the optimum tip provides a reduction of 10.8 dB in the bare body return, while the uniform coating gives 9.4 dB. On this criterion, the optimally-tapered tip provided only a small improvement, most of which could also have been obtained by tapering the impedance to zero. The 0-tip pattern is included in Figure 3-5, and though the cross sections in the edge-on 51

-25 r -30 / -10 /I / I x- Traveling wave I I \ / I -35 -15 o A 40/ Edge-on -45 L 0 0.1 0.15 0.2 0.4 Stip Figure 3-4: Effect of tip taper on uniform coatings (a x 0. 4). -40 % de'o Figure 3-'4: Effect of tip taper on uniform coatings (r = 0.4). 52

-5' Bare body -15 / \ / Uniform / /coating -25 / / Optimally-tapered coating -3 / 0-tip coating -45 ', I 180 170 160 150 140 130 0 (degrees) Figure 3-5: Scattering cross sections for ax 0. 4. maxL~ 53

and traveling wave directions both exceed the values provided by the optimallytapered coating, the total power reduction of the bare body scattering is only 0.3 dB less. Apart from the difficulty of fabricating an optimally-tapered coating, it was apparent that the optimum tip value is highly frequency sensitive because of the phasing between the direct tip scattering and that which is traveling wave induced. This conclusion was reached from a consideration of the near edge-on scattering behavior and was reinforced by the results of computations for complex surface impedances. For these tests we chose a complex impedance suggested by the values for material OG-C-1 whose properties had just become available. By scaling the impedance in Table 2-2 for a coating 50 mils thick at grazing incidence so as to obtain an impedance whose real part is 0.4, we have r7 = 0.4-i0.201 = 0.4(1-ioc) with ~ = 0. 5025. When the previous tapering experiments were repeated with this impedance, we were unable to locate a value of Re. tip at which the edge-on cross section 'bottomed out'. Indeed, the minimum for Re. tip > 0 was obtained with tip = 0. In Table 3-1 we list the cross sections for edge-on, traveling wave peak and broadside directions for a series of Re. n tip along with the total scattered power reductions computed as before. Re. 77tip Edge-on Tr. wave Broadside Total power 0 -33.51 -12.83 4.70 -10.2 0.05 -31.47 -12.96 4.63 -9.6 0.10 -29.25 -13.09 4.57 -8.9 0.15 -27.31 -13.20 4.51 -8.3 0.4 -21.11 -13.68 4.21 -5.6 bare -22.44 -3.88 11.67 - Table 3-1: Effect of tapering the impedance rn = 0.4(1 - i a), = 0. 5025. 54

The 0-tip coating is now markedly superior to the uniform coating. It is, in effect, the optimum coating for this material, and yields a total power reduction which is almost the same as the best that was achieved with - real. The scattering patterns for the uniform and 0-tip coatings of this material are shown in Figure 3-6. 3. 2 Coating Specification From the expression (2. 20) for the impedance of a single layer, it is easy to discern the properties desired in a coating material. If jkd E7Z< < 1, r - i kdpL and to get Re. ri large the material must have high magnetic loss. This is particularly important at the lowest frequencies of interest, and to have Re. r, > 0. 5 at 2 GHz with a layer only 50 mils thick requires Im.p > 9, 4. To maintain the same cross section reduction capability at all frequencies (as regards, for example, the front edge contribution), r should be constant over the band and this would be true if e and twere each inversely proportional to frequency. In practice, however, we could accept some reduction in Re. ri with increasing frequency because of the increasing electrical size of the body, and this suggests the possibility of seeking a material which resonates at the lowest frequency. Unfortunately these properties are hard to achieve, and of the materials that were measured, none had values of Re. rl in excess of 0.6, with even 0.2 being difficult to attain at 2 GHz. All that we could now do was to select the best of the materials that were available and attempt to optimize its application to the cylinders. Because of its magnetic loss the material chosen was OG-C-1 whose relative permittivity and permeability at each of the four designated frequencies are listed on p. 32. Henceforth our optimization procedures were aimed at specifying the thickness d of a coating of this material and we started by paralleling our previous investigation of real impedances at 3. 75 GHz. 55

-5 -15 b bD 0 — " 0-tip coating -25 Uniform coatinE -35 180 170 160 150 140 0 (degrees) Figure 3-6: Scattering cross sections for rm = 0.4 - i 0. 201. max 130 56

Program RAMD was run for coatings of the OG-C-1 material having the following thicknesses: / )\2 max s max i d = d max sin s max 2s" d = Id.max = 0. 050 inches O < s < s -<s s for s5 <s <s 1- - max where d max and si = 1. 0 inches. 2 The three coatings will be referred to as the s, uniform and 0-tip coatings respectively. The thickness variation was specified as an input to the program, and the corresponding values of the surface impedance were determined using the single layer subroutine described in Appendix B. The resulting scattering patterns are plotted in Figure 3-7 along with the pattern for the bare body. The cross section reductions achieved are summarized in Table 3-2 and we note the superiority of the 0-tip coating. Coating Edge-on Tr. wave Broadside Total power s 1.90 -2.20 -2.19 -2.2 Uniform 2.66 -11.11 -9.87 -6.2 0-tip -12.74 -10.90 -8.92 -13.1 Table 3-2: Performance of material OG-C-1 at 3. 75 GHz. 57

-5 Bare 2 s coating -15!^ Uniform coating -25 0-tip coating -35 180 170 160 150 140 130 0(degrees) Figure 3-7: Scattering cross sections for material OG-C-1 at 3.75 GHz. 58

To explore the effect of a change in the length sl of the tapering distance, we also ran the program for s = 0(0.25)1. 5 inches. The case s = 0 is, of course, the uniform coating. The cross sectional changes relative to the bare body are plotted in Figure 3-8. The traveling wave peak for the bare body occurs at 0 = 144, but since the presence of the coating displaces the peak (from 152 for sl = 0 to 1420 for s1 = 1. 5), we have simply compared the peak return wherever it occurs with the bare body peak in computing the cross section change. The calculation for the total power was carried out as previously described. We conclude from Figure 3-8 that a taper length of about one inch is optimum at this frequency and provides between 11 and 13 dB reduction in those features which are significant in the aspect region about edge-on (see Figure 3-9). To see whether this choice is also the best at other frequencies we now repeated the experiment starting with the frequency 2. 5 GHz at which the edge and traveling wave contributions to the bare body scattering are in phase for edge-on incidence. The relative permittivity E and permeability p used were =22.46+i0.31, L =4.80+i3.24 in accordance with the data measured by the Avionics Laboratory. For the bare body the traveling wave peak occurs at approximately 136 and Figure 3-10 shows the change in the cross sections at edge-on incidence (0 = 180 ), in the direction of the traveling wave peak(0= 1360), at broadside (0=90 ) and based on the total power, as a function of the taper length sl in inches. The traveling wave peak is a minimum for s 0. 75, but the minimum of the edge-on return does not occur until sl 1. 30. Since we have computed the total power only over a range of 300 from edge-on, the total power is more affected by the edge-on return, and its minimum occurs at sl.1.55. However, this is probably due as much to phase cancellation as to the actual reduction of the two contributors involved, and there is no more than about 1 dB improvement resulting from the use of any taper lengths other than one inch. From s = 1 inch, the total power and traveling wave reductions obtainable are between 4 and 5 dB, and the scattering pattern for this coating is compared with that of the bare body in Figure 3-11. 59

0 -5 - dB change -10 -15 -20 - 0 Broadside Tr. wave Total power Edge-on s1 (inches) Figure 3-8: Effect of taper length for material OG-C-1 at 3. 75 GHz. 60

/ \ / \ / / \ / / / ' / / loI1%1 /4 0 Bare -, ---- /.0, / f 0. /\ / o / Coated -020 /'/ -40..... *.I......I.... I 180 150 120 90 0(degrees) Figure 3-9: Effect of OG-C-1 coating with s1 X 1 inch at 3.75 GHz.

5 0 dB change -5 Edge-on Total power ' rTr. wave Broadside 0.5 1.0 1.5 2.0 s (inches) Figure 3-10: Effect of taper length for material OG-C-1 at 2. 5 GHz. -10 62

5 Bare Coated b / - / \ I o / I / I -15 25 180 150 ( 120 (degrees) Figure 3-11: Effect of OG-C-1 coating with sl ~ 1 inch at 2.5 GHz. 90

At 2 GHz, on the other hand, the same coating is almost completely ineffective (see Figure 3-12). Because the two bare body contributors are nearly out of phase at edge-on incidence, the coating actually raises the edge-on cross section by 3 dB and the total power reduction over the 0 to 300 aspect range is only 0.4 dB. Similar computations were also performed at 7.5 GHz for which the edge and traveling wave contributions are again in phase for edge-on incidence on the bare body and the cross sectional changes for sl = 0(0.25) 1. 0 inches are shown in Figure 3-13. The traveling wave peak now occurs at 0 = 154. The decrease in the broadside return with increasing sl is due to the fact that the specular and edge contributions to the bare (extended) body scattering for 0 = 90 are almost out of phase at this frequency (see Knott and Senior, 1974): the precise out of phase situation occurs at 7. 820 GHz, and it is therefore not surprising that the increase in the specular contributor resulting from the increase in sl (less material on the surface) can actually produce a decrease in the scattering at 0 = 90. The improvement in behavior produced by the taper is somewhat less than at 3.75 GHz. The various curves have minima that occur at smaller values of s and for the traveling wave the minimum is actually at sl = 0 (uniform coating). The best overall performance is obtained with sl 0. 5 inches, but it is not significantly better than that achieved with s = 1.0 inches. Although the edge-on reduction for s = 1.0 inches is only 5 dB, we remark that the edge-on return for the bare body is itself low, and this choice of taper length is sufficient to produce 14 to 16 dB reduction in those features of the scattering which are of most concern. The patterns of the bare and coated bodies are shown in Figure 3-14. As evident from Figure 3-14, there is still a significant traveling wave contribution, but to see whether the entire lobe centered on 0 = 154~ is due to the traveling wave alone, we have examined the surface fields for the above series of coatings. By looking at the magnitudes of the currents for 0 = 154~ over that portion 0 of the upper surface where the rearmost oscillation occurs, the forward traveling wave amplitude can be deduced from the levels of the maximum and minimum in the 64

5 -5 Bare Coated b bI, bO 0 ) -0 I-I -15 180 150 120 90 0 (degrees) Figure 3-12: Effect of OG-C-1 coating with sl = 1 inch at 2 GHz.

5 dB change -15 / Edge-on -- Broadside - Tr. wave "" Total power -.j 1.0 -25 0 0. 5 s1 (inches) Figure 3-13: Effect of taper length for material OG-C-1 at 7. 5 GHz. 66

20 \N I / I I rI ' I^ Bare -IL / I I,, I 0 I Coated 'I / ' / ' f' " J / / \\ / o\! \ /0 -20 / -4/, / 9-40 180 150 120 90 (degrees) Figure 3-14: Effect of OG-C-1 coating with sl 1 inch at 7.5 GHz.

oscillation. Since the traveling wave contribution is proportional to the square of this amplitude, comparison with the corresponding value for the bare body enables us to determine the reduction which has actually been achieved. The reductions are listed in Table 3-3 along with those deduced from the backscattering, either on the basis of the scattering at 1540 or by selecting always the peak return in the vicinity of this direction (we remark that for the uniform coating the peak actually occurs at 1480, with the peak moving progressively back to its bare body position of 1540 as sI increases to 1 inch). Three conclusions that can be reached are (i) the traveling wave reduction deduced from the surface field data is independent of s, as expected; (ii) consideration of the cross section at 1540 as done in Figure 3-13 exaggerates the reduction provided by the uniform coating, and by the coatings with small s; and (iii) the actual peak value is affected by s1 and, hence, the edge scattering. In no case does the actual reduction in the traveling wave as indicated by the surface field fully manifest itself in the backscattering pattern. 025 0. 0.5 0.75 1.0 surface field 16.7 16.7 16.7 16.7 16.7 0= 154~ 22.9 16.6 14.3 13.9 13.8 local peak 15.9 15.0 14.0 13.9 13.8 Table 3-3: Traveling wave reductions in dB at 7.5 GHz. In an attempt to realize the full reduction which the coating is capable of, we have examined the effect of starting the one-inch taper with a non-zero thickness d = dI at the front edge. The thickness variation used was d=d +(d -d ) sin 2s 0<s <s 1 max 1 2s - - 68

and data were obtained at 7.5 GHz for d = 0.005 inch, implying tip = 0.054 - i0.033 = 0.010inch, = 0.110-i0.064 = 0.015 inch, = 0.172 - i0.091 all for s 1 inch. The patterns in the aspect range 140 < 0 < 180 are plotted in Figure 3-15 along with those for the coated body with d = 0 and for the bare body. Judged from the surface field data, the value of dl had no effect on the traveling wave. The broadside scattering varied by no more than 0. 1 dB, but the choice d = 0. 01 inch reduced the edge-on backscattering by an additional 10 dB. It also reduced the traveling wave peak by almost 2 dB more than the O-tip coating, bringing the peak down closer to the level indicated by the surface field data, and had a significant effect on the total power (see Table 3-4). Increasing d to 0. 015 inch reduces the peak by a further 1 dB and also reduces the total power a little more, but the edge-on scattering is now on its way up. It would appear that this is close to the optimum coating at 7.5 GHz, but since the mechanism responsible for this type of tuning is almost certainly phase cancellation, it is questionable whether the results obtained are practially significant.. 0 0.005 0.010 0.015 surface field 16.7 16.7 16.7 16.7 tr. wave peak 13.8 14.5 15.3 16.3 edge-on 5.2 9.7 15.6 12.3 total power 14.4 15.5 16.0 16.3 Table 3-4: Cross section reductions in dB at 7. 5 GHz for various tip thicknesses. 69

Bare o /.I ' t -/ n1 - p 015 ((degrees) Figure 3-15: Effect of OG-C-1 coating at 7.5 GHz for sI 1 inch with d 0 (-), 0.005 ( --- —), O.010 (- - -) and 0. 015 (- -- -) inches. 70

We also carried out calculations for the 0-tip coatings at the highest frequency 15 GHz using a total of 150 sampling points on the body. Since the surface impedance provided by the full 50 mils of coating is now only r7 = 0.24236 + i 0. 18578, the performance is poorer than at 7. 5 GHz. The effect on the broadside (0 = 90 ), traveling wave peak (0 = 160 ) and edge-on (0 = 180 ) cross sections of the taper lengths si = 0(0.25)1.0 inches are plotted in Figure 3-16, along with the total power reductions computed using the data at every 2 from 180 to 150. We remark, however, that for the smaller taper lengths the actual traveling wave peak occurs at a somewhat smaller angle than 160, and the true reduction never exceeds the 12. 6 dB computed from the surface field data. Judged by both the traveling wave and the total power, the uniform (s = 0) coating is the best, but the case s1 = 0.25 inches is best as regards the edge-on return and there is no more than about 1 dB difference for any of the values of s1 considered. The patterns of the bare and coated (with s = 1 inch) bodies are shown in Figure 3-17. In addition to these investigations we also examined the effect of different thickness variations, e.g. an S-shaped taper, in the vicinity of the front edge. None proved superior to the 0-tip coating and subject to the constraint on the maximum layer thickness it would appear that the optimum application of the OG-C-1 material is in a layer whose thickness d starts at zero at the front edge and rises to a maximum of 50 mils in a distance of one inch. At the four designated frequencies such a layer produces the cross sectional changes shown in Table 3-5. f (GHz) 2.0 3.75 7.5 15.0 tr. wave peak -0.3 -10.9 -13.8 -12.4 edge-on 3.2 -12.7 -5.2 1.1 total power -0.4 -13.1 -14.4 -11.3 Table 3-5: Cross section changes in dB for optimum application of material OG-C-l. 71

Edge-on 0 dB change -10 - -20 tO - Broadside - Total power Tr. wave ractual t0 1600~ 1.0 0.5 s (inches) Figure 3-16: Effect of taper length for material OG-C-1 at 15 GHz. 72

20 Bare o., i \ yV4 0 - -,-. I',: '^ ' /! II { Coated I F \ I I \ Ii j ^ ^\ r i \ ^ /A //-\if \. '-4I'0"' /,I/ \ / \I' / "/ \I IV -40 - —,-..... ----.............................. --- —....... — I 180 150 120 90 (degrees) Figure 3-17: Effect of OG-C-1 coating with s x 1 inch at 15 GHz.

Towards the end of the Contract a material was proposed whose properties were attractive as a coating material. From a knowledge of its composition, the relative permittivity and permeability were predicted at a number of frequencies and these are listed in Table 3-6. Two versions of the material were manufactured and designated TCPER-25 and TCPER-26. They had somewhat different dopings and their electromagnetic constants measured by the Avionics Laboratory are included in Table 3-6. Backscattering patterns were computed for 0-tip coatings of each material applied to the ogival cylinder. The cross section changes obtained are given in Table 3-7. Comparison with the results in Table 3-5 shows that a material having the predicted properties would have been superior to OG-C-1 as a coating, but that neither TCPER-25 nor TCPER-26 is, the poorer performance being attributable to the inflated values of the permittivity. f I e__ 1I____ ___ __ (GHz 'predicted' TCPER-25 TCPER-26 'predicted' TCPER-25 TCPER-26 2.0 36.0+il. 0 62.4+i55 16.4+ i0.68 6.1+i4.0 5.4+ i4.2 5.0+il.77 2.5 36.0+il. 0 80.5+i68.2 16.0 + i0.75 6.1 + i4.0 5.6 + i5.1 4.7 + i2.16 3.75 35. 0 + i 2.2 66. 5 + i32.3 16. 3 + i 0.88 4. 5 + i3.8 3.0 + i3.8 37 + i2.39 7.5 33.0+i2.5 36. 0 + i12.2 16.1+ i 0.38 3.2 + i3.6 1.8+i2.2 2.3 + i2.10 15.0 32.0 + i3.5 28.6 + i7.8 19.0 + i3.56 2.0+ i3.4 0.2 + i0.9 1.5 + i1.16 --. Table 3-6: Properties of three candidate coating materials. 74

2.0 2.5 f (GHz).............. 'predicted' TCPER-25 TCPER-26 'predicted' TCPER-25 TCPER-26 tr. wave peak -2.0 -2.9 2.0 -11.0 -5.8 -2.5 edge-on 4.0 2.2 2.9 -3.4 -8.1 1.7 total power -2.2 -1.4 0.5 -8.2 -5.7 -0.8 3.75 7 5 f (GHz) | 37 75 ____ 'predicted' TCPER-25 TCPER-26 'predicted' TCPER-25 TCPER-26 tr. wave peak -12.5 -7.1 -8.7 -12.1 edge-on -20.0 -11.1 -4.0 -8.4 total power -15.4 -7.3 -7.9 -12.9 Table 3-7: Cross section changes in dB for three coating materials. 75

CHAPTER 4 COMPUTER DATA During the course of this study, a large volume of data has been generated using programs RAMVS and RAMD, primarily the latter. The data are all for H polarization and include the surface and backscattered fields of ogival and wedge cylinders, ogival cylinders with various impedances specified over the surface, and ogival cylinders with coatings of several materials and thickness variations. Because the cross section reduction tasks for the ogival and wedge cylinders turned out to be almost identical, most attention was given to the former shape. The data presented here are only a small fraction of the total obtained and limited to those situations for which experimental measurements have been performed. On the following pages we show backscattering data for wedge and ogival cylinders at 2. 0, 2.5, 3.75, 7. 5 and 15.0 GHz, specifically 1. Bare wedge cylinder (for dimensions, see p. 6 ) 2. Bare ogival cylinder (for dimensions, see p. 6 ) 3. Bare ogival cylinder, extended size (for dimensions, see p. 46) 4. Ogival cylinder with uniform coating of OG-C-1 material, 0.050 inches thick 5. Ogival cylinder with linearly tapered coating of OG-C-1 material, 0 to 0. 050 inches thick 6. Ogival cylinder with OG-C-1 coating, cylindrical (i. e. 0-tip) taper from 0.050 inches to 0 over 1 inch (see p. 57). Then angle labelled theta is the angle 0 of the text, with 180~ corresponding to edgeon incidence. The tabulated quantity is the two dimensional cross section o/X in dB and is related to the three dimensional cross section of an 18-inch section of the cylinder by equation (2. 3). The following table may assist in locating a particular data set. 76

Frequency, GHz...... - - ___l....... v~I.... ~ — I.. I Model 2.0 2.5 3.75 7.5 15.0 1 Bare wedge cylinder 4.1.1 4.1.2 4.1.3 4.1.4 2 Bare ogival cylinder 4.2.1 4.2.2 4.2.3 4.2.4 3 Bare ogival cylinder, extended size 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4 Ogival cylinder with uniform coating 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 of OG-C-1 material 5 Ogival cylinder with linearly tapered 4.5.1 4.5.2 4.5.3 4.5.4 coating of OG-C-1 material 6 Ogival cylinder with OG-C-1 coating, 4.6.1 4.6.2 4.6.3 4.6.4 4.6.5 cylindrical tip Table 4-1: Index for location of data presented.

Table 4. 1. 1: Bare wedge cylinder; 2. 0 GHz. B__.ACKcCAT~rERING CROSS SECTICN__ ___ T HL7T 1~ -LOrG (S1~JG-1A/LAmnDAIZP-H —.zNS 5.0-11.41 220. -(I._(is __ 5.9 3-. 2t 2.*4._____ i.5.- - 216.3 ioCO -5.93 2GC2.94 6i,0(, -9.27 214. C -aI ___3__ 56 92 ______.22. a _E- 62C0-.2 ---- 214.3.o0 3.0 C -E. 9c 2L2.8 5 3.3" -7.21 21 2.9 4. QC -..P2 C 64.fa. I8,7 213 E5. 0 -5. 85 211r3.e 65.00 -i5.3 9 214. 9 ___ __ __ _ ___ o266 I4 5621 6. 7.-.82.r4. 8 Ea7,~ -3.7 9218 -800- 5.7.4 _2.6oE.____ 3. r - 3 *.28.22 1.3 -9 - 5. 69 2 C6.'.4 6 9.Q - 2.41 224.0 1 3.00 4E'+ 2 1 3.6 7 3., 0~ -. 236.6 ______ 211. _______74.0f______. __34__ 24 L, - 55.3 3 213.2 75. 00.79 243.7 i7.J 3 216.1 7 7, I 16 IL 25 1.1 1. _____.-.'__21.7. E _______~78.O.: 1. 96 ___-25 4. 9 521J&9.-2 79.0' 2.3c 258.8a 21.1 -4. 96 2 22.6.1 I 2 9 C 266.7 2'2 ___ 4___ +91 ___ 2 24 0. "3i7 ______272 9 3. -". 8 7 22 6.3 P13C. " 3.4 2 2 7 Z..8 2:+. _ 2268.2 _ ___ 84 0 - 364.-.278. S "5"-.7 3.1953J 3. 3 5 2 83.1 29 475 238.3. PA.> I?2 33. 9475 27. 3.o3 4. L3 31.2 3537-.92 1. -9.'4,073E. 36. 1 n ___ 5.89 C. 4. 92... 3 37C' 53 2L62 797.> 4.Q 3 3.8E 3 _____ 4. - 2 8.2.. I __ -__~ -__~~3~... _ 4.9 7. 339.2 3 3 3 -4.7'1 2:-.41 99f3 941.3 3 3L I.6 -5*91.2625.LO..1 __~ 4t).19 26. 11T 4. 4 352.5L -64..266 _____A2. _ -6. or 43>-.1268. 3 3.39 495 1 3.4 3b7.>i -.359 271.7 97 2. 4.3 10.3E 460 ~ 52782. 46.0 A.9 ____2 9-8.571 273. 47' 4 * 319.36 49.0 -.78 27. ).;36 83 7 290 1.7____ 236 E_______EA JC__- 6.7 28 51.3? -1.2 27184 11.0 32 37. 452.311~5 268.8 ~1 4. 3.3 IE1.3 53O 1.7264.2 11300.7846.3 -1.5 25. 17.0 2.5 ____2_3_ 55.0 a'I -13.59 2 257.1 115iI. 0 0 2.25 553*7' 56.7 -13.56 __241.4 1601.6 5 9 5 7.C -13.1t4, 2 32.o9 I 17. OC 1.o6 6 6 4.4 5 8. 00.-1 2 38 2 2 5.6 _____i1C0.3 9.C, 78

Table 4. 1. 1 Continued 1.30 or73.6 123* -.I -5-_____78. 2a i 2 1.f)%"I.29 8 2.98 -12 2.v00 G-,9 ____4 -i23.OC.48a 9 2.1 1 24.O a..-89 96. 8 127o0' -- -2.66- - 160i 129. 00 - 3.o14 12 1 jjr. -3.-63 1 2 6 i31.0C1 -49i4 131.1 i 3 2. __ - 4 a l- 6136.3 3 3.a0 ) - 5.018 141.5 3 5.0 0-2 -6.*27 1L5 2.*5 JJ 7 1._8 n2. 15 8.2 1 37.22 '- -7. 37 164.0.13 8.a0 -__ -7. 92 _______ 17Is.L 1 3 9. V -8.a46 176.92 - i 4.1..3.1e2.6 -1 4.'. -9.5 1181 49.i * -11.29217.1 i.4 6.0 1. S3 224. 3 17 —:1I.9 1 2 31-. 6 c 14~?23 1.1ffi __ I -2, 45 25 2.9 13t I. 0 - 17).53 25 9.6 1-5 2.3 -U -2. 57 ~ 2 E-.1 ~' -~ c: ~272.3 2331 - -7 -12.3 23.7 2 3.3-1.4 9.3 -:3.1I6I 0C. IJ 1.0C 7 & 627 31.4 JL67*-1 1.79 317.54 17 -11.62t 323.4 167 >-11.48 328.4 -11.21 -7 _ 337.5. -11.17!' 33.87 17 5.C -1111340.8 -i76o I- ~.J9 _____ 31s*5 -1 77.o0 0 -1.'7 34 2.1 _ - -7 I-06_____ __342.5. 179.07 -11.3-5 342.8 79

Table 4.1.2: Bare wedge cylinder; 3.75 GHz.....- -3ACKSCATTERING CROSS S-ECTIOI -- TH-_IA_ 1kLOG (S IGMA/LAHMCA) PHPASE 3.:_;.. -.98 - -. 134.3 $.ir -.98 134.5 - 2. ___ -.97 134._ 3. * -.95 135.6i _, ^ -___ 93 ____ 36.5 5.O. -.89 137.8 -6.^__ -.85 __ - 139.3 7.00 -.80 141.: -8____a- -.74-_- -143.C 9. 0 -.68 1L 5.2 i-.: -. 62 147.6| 11.00 -.52 1 0.3 12. 4 -.43 _ _ 13. c 13.09 -.33 156.2 14. ____-.2-___ — i59.5 1.S. -.12 162.9 o 16.0 -.- 6_ 5 17,3:.12 1.-.2.. 24 - - - 1' 9. 3.,37 2. 3- - -. 4 9 -- - - - 21.'.62 22. 1____-____ <_-_____2..3:.85 24,. 9 - 25.0 i1. '5 - - -.; - 3 - - 3-. }...... - 1 7. 7 r i. 2 -- _ _,.. *2 3t."~ 1.27 32._.. 1.23 4 33.C: 1.16 17u.2 174.1 178. 1 182. 186. L 19,.7 195 * * 199. 4 a7. 213.P.216.7 225.' 273. 239.1 244.4 59.00 -2.32 199.9 -60Q.00 -1.21 -— 208.3 61.:' -.53 216.5 62.OC 2. --- —- 224.5 63.0C.44 232.4 -54.0 ____.7 6- 240.2 65. '.97 248.66.0? -. — - - 1. 7 255.8 67.0C 1.r6 263.6 -68.C" - -—. 271.4 69.00.69 279.2 7- ___C __ 2871 -71.0: -.26 295.0 — 72. O' — 1.CC- 3 3.' 73.00 -1.99 311.2 74. -3.29 —2a 319. 75.0C -5.C6 328.2 _76. C -.-7.55 ___ -- 337.5 77. OC -11.45 348.9 — 78. 00 — -1q.28..... 11.~ 79.3' J -2 t. '2 144.3 --—. 0' — 11.55 - 172.9 81. O - -6.35.185.C __2Z.3 C -3.7?3 -194.6 83. ' -1.4 2 3.5 _-8 4.C. J C_-. 4 5 --- 212.S0 85. 0 1.97 22C.4 _-86.___- 3.25 —___ 228.6 7. OC. 35 2 6.8 _8.____ _ 5.29 _____ 244.9 83.C u 6.12 253.1 _9 9.3 - 6.31.... - 2e 1. 91.n0 7,. 269. 92. —.. 7. - - 77 93.37 8.38 2..3 __94. _______.76 — -. 293.4 95. 9..6 3 1.4.- 9.C- - -- 9.31 - 3.9.4 97.0C 9.49 317.4 - 93.3 -.. 9.52 325.*4 99.3' 9.7C 333.4 _1i. 0 C 9. 72 __ -- 341.4 I'1. ] 9.69 39.4 _1C2.C- 9.61. -- 357.3 103. C 9.47 5.3.1 4.3r --- - 9.29 13.2 105. 0 9.;6 21.1 _.5. "u 8.78___.7 - 2 9.C j17, 3.aL44 36.9 _-l.:_. [ _ 8.:6..- 6 -. —.-. 44.8 109.00 7.61 52.7 11. —.. 7*.1 2 6L.6 E 111.0 6.5*7 68.4 -112. -— 5.935 -__76.3 113*.1 5.77 84.2 -114. 0OC 4.522 -- 92.1 115.30 3.7C 103., -116..C.... - 2.79. 16 8. 117.0! 1.79 116.1 — 113.00.. 68 -- 124. 3.3. 0'.55 2 9. 35. - '.: 255.4 -36. __ __-.71.___...- 2 cl. 37.,.7 2 66.8 38*. 0___ - 18 -.- 272. 39.V r' -.17 278.9 __._ -_.-_ -.______.285 L.1 41.0: -1. 8 291.6 3,. r -2.-36 35. 44. 2 -__ — 3.18 ___. 312.9 45. C -4.15 321.C. -5__. - 5,31 329.8 47.00 -6.70 339.9 -43. 9 8.- 8,37___ 352. 49 0 -1i.36 7,7 5___. _' -12.5 2___ __ 29.9 51. ' -14.02 61.6 52.__ -13. L44 97. 3 53. 0 -11.49 125.5 54. 0__ -9.23 - 145.1 55.0C -7.24 159.7 56.00 -5.56 _ ____171.5 57.0 5 -4.17 181.7 5.C. -3.CO 191.1 80

Table 4. 1. 2 Continued -__2_____ _.-I -1 93 -- 1 Is3' ~ 2 ior -3.*48 5 a. 3 1 23.'1 -7.32 17C. 1p-5.c C -12.65 A 2-5-s D ~ 15.*8 2_____ 226.9-.. 1 2 7.0 0 -1 7.98326 1280 J a1682 __ 3C 6,2 i29*CC, -14.53 332.4~ I 3 I9 -19 2* A 31.7 4109 2.03_ 1 3 3.0 -8.83 21.3 -8.1 1 I5 29Q.4 i35.3)0 -7.6:4 37s.~ 1-36.0n - 7.Z8 L.Ljl 1 3 7.31' -7 CE4 50.9 A3 a i)0C ~ 6 *91 T 1 3 9, 90 -6.686 63. 8 A,4 -0i0 -6.090 ___ __ __ __ 143* 3-2 -7.44 8 6. 8 145* 2' — 8.1 97. 1 -14.6.2____3 ___ -53______12 1 47.50' -O10. 1 49. 00 -1011115.')I A5 I 7 4 _________~8 *7 --11. J4 3 122.1 153.0 1 20 9 ~ 127.7 A1~X-43. 3" -12 09 - 14. 1 58.:a __ 7. 2 I~2 9 2 -1 5 1 3.,71 1 26. 1 16 i 2r -2.111a 1 S2.05 __ 2-r* 77 I ~ C.5. A 46 I - 94 a 1 -83. 76. 1690"-1 B32 74.9 1711- V7, 3~ 3___ ~73* E 1 71 00 -17.e3 9 72.7' IL73.27. -1.6 7. 17500-1.12 71.72 -175366 7 18 1 779.0 -1 5, 6 7 1. 18.0 -1I5.*5 8 71.9 81

Table 4. 1.3: Bare wedge cylinder; 7.50 GHz. 1.o21i 258.C _2.fl-2 to21 _____ 25 8,. 3.C C 1.I22 259. 5sGIi*?4262.5 7. OC 1. 25 267,1 8-t-0 LI1.24-.S 9.0 0 12 2 273.2 i1sot. 1.14 281~1 i.2 - I *a6 2 85 L7 1 3. 00.9 6 29 e. 8. 0r67 3%"2 7 _________ _______3C.9 E 1 0 0.25 3&7, 1 i 9 J, C 033.7 -.6 1 -344.9 21.51 qj 35 6.1J _ _ _ _ _ _ _ - 3 6.7 25. 29.-1. 2C ~. ~3. ___.3 2. 33.3 2 I67. 37 3.7 5 217. 7 * _ _ -.7 3 -__ - 3. 3.95 6 24 3.-2.2 3 26 F o5 4o 2.73 27.o4 ______2.7 ~2 P 5-. *44.o -.1 2 -___u~1. g45r 1.5 34.. _ _-3., 6 _ _ _ _ _ __5. 9 4 7.22 -4. -16 37.9 -43.O-0r ____-4.11i 7___ _ 73.6 4 9 *OC - ir 6 3 1. 4 __ _ _ _ 33.2 51.0.3315. 52. 1. 64 175. 5 3.00C 2.6 5 19. 3.38 2f 9.8 550O 3.8 4 226.3 6. al..C 3 242.7 57.O 3.94 259.3.5 8 0 a' 3. 58 276.1. 59. 00 2.'91 9. 610!.55 335.6 1 83.0DC -2.68 3 4.a9 -3.2&..14.5'~ 65.E~ -2.33 1,13.3i -65.C ____ -6 3 1 45.1I _____ ____32_ 192.64 69.'a02 3.2~ 2&2.5l 71.00 - 3.3 729. 7 C3.53 3_____ 6. '73.00 2.76 285.4' 274s 0 ______ 1.4 2__3__4____ 75.00 -.72 325.4: 7 7.00 -8.78 39.2, _ _ _ _ _ _ _ __I_ _ _ 1 7.6 -,03. 27 1.61. __2 0C- 5 187.7A 8.C2.5 9.2.8.o3 PJ 3.j 5 9244.7 — 84.0C_______ 5.52 -2 62-.1 83.nV 5*l 1 279. 8 7. 3.5 1 31 5.3 -9.33 3 7.4~ 2 51 _ 2.7, 2f.5 9 7 G 1 3.e33 2 7,.21 -9 9. 2 __ - _ 14.01 9 0 - -1 2 92.:1 ~2.'15.9- 3 3.5 L3.Oa 15.21 83.9 1 4oI114.5 84 24.C 50a1 4.2 9 3 8.*9 12r- -.I58 68.4 11 9.02 q. 92 97.4 1110 00 5.o9 3 125.5l -t12-.1i3 9L 113.03 -63 1C I051. A1 4.3 ~ -634 ___ I161.1 115. 00 -19.06 8 1'-2* 8 * -— 12.1 8 ____ ____ 117.00 a.1 4-.6 I i -1 C —1- 3.o 4-6 - 82

,Table 4.1.3 Continued 1 9 0 C -2.8 63.7 1 21.0a"7 -1.4 7 8 8.08 2 3.O0Z -2.8s3 I113. 2 174, u. n -Ld. Pnl i12 5. -6.1i2 135.5 as. ___ 879- _____i44.f.i 2 7 C -i.2.59 148.g 1 29.0 C -'2,. 6 8 85.5 i 3 2 %0 ____ I2.37-_____ 6717 — _133.00 - 9.15 77.00j _1 IC-8.51 - 8702_~ 1 3 7. 02 -90.11 119.9 tU3 3.0 a '-9.389 ~1 1 42.2-6G --— I 2 o 4 __ _- __ 4 4.C. 1 4 C 1.22 -15.15 166.1I I 4 3.jr -2 3.44 i19 7.07 1 45S.0 G -2 7. 6 1 r J4'? OC-180; j29.1 J. Y-1 6. 49 -4 ~C,.1 9 1-15.17 50.5 i 5 C E9.9. - r1 3. 76 E 8.07 V-5 3 o-13.L-4 1 3. Q7 99.1 I 7 C-1 4.C33 1~ -1.66115. 1a -222 113.92 A6729 -19.4 39.3 1.6.1 ______-924 ____-7.53 73.7 17300 1578 8.7 1 75.' "A4. 98 86.53 177)r-1.4 88. 179.0 -1.22 88.9a 1 715. 31 89. qL 83

iTable 4. 1. 4: Bare wedge cylinder; 15. 0 GHz. 0 B-.-ACKcSZ'JTTERING CROSS, SECTION THUEL I ~LOG(S-IG'MA/L.AtI8DAL2H —ASE.I ______ 4. i i32.6 1.0 " 4. 33 133.1i 0.Of -4*37- _____ 134. 8 3 4.45 1 3 705 5.00L' 4.68 145.9 6.0 4. 8 _. 4.51 * 74.99 157.8i ___ __ __ __ 5.1 6 __ _ _ _ i65.aZLL 9 u C. 5.32 1 7 2,.9 -qn". n ______4_____ 81. 4j iso5.58 1910.71 5.66 211.3 ________ _ 5.6 i _ ___ __ 2 22. 8 a5. 47 235.3, I 5______2L58 J'O 4.95 263.6~ 1.y4. 15 - 298.3 Ve.0C 3 5. 31I3.a7 2 I ~~3. 4 9 LL. 3. *4.6 -254. 99 79.4 2.*;.,-6.35 29.00 70:3 6. 3 1. 0,6 212.: 3.25 55 2 33.5 133.00 259. 3 7.0I145 6 0.,4 3 5 'V __ __ __ __.17 99.e 4.73 132.7 41.27- 6.2219. -42.O __ 6.c 7 - - 219.5 43.27.2 9 2 5L.1 -44* C_ - 284.7~ 45.O 2.12 2 7. -46D — IC ________ 47'.0 2.11 73.2~ 48O- _______3. 95 IS__ 17 8 4 "15. 47 1 55.3 ___________6.6 __ 18 9.7 51.00 6.29 2 2 3.o9; -5i2.~ It. 77 26. 53.30 4. 21 3 t1I.a8 _ _ _ _ _ _ _ _ _ _ _ _ a 8 8- 2.8 8 5 5 a.0 3. 06a.4 __6 4,j7 3_____ 12.1i 5006.43 144.9 5 aI.0 7.45 183. 590%7.63219.5 -6.9 4 -- -269 -6 ia1.27 5.,3 5 29 8.5 — 6-2 —3. 21 — 3- - 53.a 2.2 53. -2fl 3.3 - 6 7.. 5.86 243.1 -68. 07 -4.,2 4 -2 69. 00 1.L31 338,.2~ 7 1.I 2.)?1 125.1 _4.81 174.-7 -7 3.00 6.08 216.1 a5004.9 299.6.-7 6.J27 *-97_____ 7oc -.46 76.4 7i5.Op -2.*8 5 t -15.,8 -79 o.2E 5.52 193.13 6.51- - 233.03 -5.82 272.2~ -32. C — 3.~ --- 1 14 o-7-~ 5.2-2.351 18.3! i 7 1 26.8 83.27' 3.6 918. 455.2 _ 6.9 A - 3226.3 -1.29 35 1.9 I- 2 7. 7 93. I o.: e 263. --- 4 j, ___ 9 72977 -95.00c 5q~7 3 76.4 ~ 3.71- E 67.7 ~ 3.5 a 2 23. 4 C92 1'70.Z 2 IF.5 2t 3 a.73 17.o2 17.oV9 8 7 4.C 11.15 125.2 A 5 0fi 4 3 714 3.7 2L9.0 -5. 94 9. -112.00a 16 9 ~ 5 8.8 111 oi.0 4.69 76. P At4.o 0 -.50 135. 3. l15.00 -7.02612. 170-2.50 78.0, 537 84

Table 4. 1.4 Continued 1 2 1.00 -65 136.8 123.00 9.990.2 124.01' -6.33 91.51 1250.0. -i.2__ - 4. 1 23.C -8.85117. 1325.00 -11.67U 1276.6 -i 35._Ofl - ~4.a30 43.I 1 37.OV-1%*62 125.3 i13,9.OC -1.0'. 111.0 1 1 35. 0 -221.2 67 I3.8 i 3. -1 3.$C — ____ __ 4 I 27 926. 143 - 17.5 16I5.18, A 297 ____J L.: __7__. 8g.3 163. 0 - 3-22.7 118.6 16 3-253.72 71164 J 169A 22.6776.6 173.L - I. 9.6 Jj8.C'~ -L6.7 I5.. 85

Table 4.2. 1: Bare ogival cylinder; 2. 0 GHz. C7 )Y,- -T T O N ~4 Ti IO)4 LO (0, 4X/AIT )~' ~? -7 7 O -r ~pL 1.4?.. -- - - -?. l - I1L,,217 -- 0I —?1 I 4 0 q7, 1? ~ )i 7)C 1~2 -. - j~ ~77 73 A. A 7 86

Table 4.2. 1 Continued LL 'I7 47 * I 7~ 9 n9 S. J 7-44 T - I 3.'- --- 7 -A 17 7,3 - (;1p'-4 9 --- - 7?7. 0I *. 7.3 C) 7~ - 21* 2 'A 7, 7 -79 9 -1 87

Table 4. 2. 2: Bare ogival cylinder; 3. 75 GHz. )V Q' rT T p; 'f MCs;TTe i:;- - A n --- ___ -* - ~,i ~ —44 4 * ~ ~)7!4 A, - - - - - - 4' -4 88

Table 4.2.2 Continued i, z,, o - -, 7:;. 2 G g. 4 i' -,.O.. - - -.. ---? - 14;,, n.. >.. 5.? 7.... f-j ^. -' ^^n0 1 c7. on -7, 3 70.i:. 1,,.,"i - —. ---. — - - ' -;t > —.-, - -, ---- -. -...o. r; ~rhr, On,-^ 13 i?, 4 - 1,.._ 7 --.- _,,-, ---- --. ~,t?. ----. — - 74.3 P.!3n o -^ 7 t7'^ '. r.. -(, (]2 I 7Q,, 'if, 8, '- -. I 7h, 7.* 5- - 7, ---- -- - --. - r:4, r -, ',_ -'t @ 7 - 7 ---- -- -- 4 -t- --- - -- - - Q-,7 4. 7. 7? 7n, - 7 t -47.7 - 1 7 - 3 7*.:., n.. i 7 - -- 1 7, 7:,:7 q 7, c - j) 7I tjt';, -a - ---- - - 7 - -- -.- -?..,4,- r - 4.. ---- —,'.-i~.... 7 *, n fI i t',. 7 o 7 ^~ pt 1. i.7 89

Table 4. 2. 3: Bare ogival cylinder; 7. 5 GHz. r~Af~V(TT T) T~f7J. rT,T- 0' 7?V P -.iu.-I 17 - ' '7fCIT — -- -- — 1377 4 7-,-~r I *7 4 T 1.4 1 3q 7 1 5f. 13342ell 90

Table 4. 2. 3 Continued rtr 14 r7. ~t.7 74 -% 4 7 -- -- - __7- - -- - 79 1,41 a7f -- ----- -- )~ 7F)-* *-? -0 k r C7 - -------------- '3 - I 75 42 S14~ 4 I 4.7;I? -7. 91

Table 4. 2. 4: Bare ogival cylinder; 15. 0 GHz. 7 C~~ '~ T, 4, 1 17 4 -j f.t * t~1i~ 7 ~ ~'T t. 7 V' t 11,. 3 41 7113. 5 4 l 7,3fln 777?3. 77 -7,S 2 45 * 0 7 1N 7w P --- 27 0~_ 7 ) on~2 -71 4 L 170.5 - - 2 o - - - )J ~-~J'~ 4 - 92

Table 4. 2. 4 Continued 7 79 c;-, -7 IV~43(It? C; P? -2.1 c7 I - -; 7 -~ a — a~j' 74 - - - -7 7f A F. (3G4 1 &4 Q) t 7 G 71. n7 T 79,y 73 ~ ~ 7"17- P 93

Table 4.3. 1: Bare ogival cylinder, extended size; 2. 0 GHz. F., I C3 4.0 0 73 94 0();1`.),9 L8~( 15 4 ) 2 9 9* C 6 1.06 00 46 2 91:14 O0 0 5:"1.0,J. ~41 11 3.. 0() C% I. C) 6 0( 0: -- ) 6 1 0()0 /"7 8.. I3 7 2 0..~ 4/ 1.:1 1 f 9).69.0. ():1 c3 88 *:1 00:1.0 60 9) 9 1.0 91 41 7 6:1 J.00 9 (f.)1 C3 A (4N' 9 9 * 0. IY J J0). O ), 0 Ji 00 'J0 ~19 C)1.02 ~00 Ie.0()2 1 1.0:~ 7 LI -e k00 J.. 4,+:13a X.00 02, i. (-: 1) `) 11 1) ().... p I: 5 I,!...z I k I l.. 'I -> C) %. 'I. C." "". -.. 94

Table 4.3.2: Bare ogival cylinder, extended size; 2.5 GHz..'B A C. C A T T::;1:?'- (3 C R). S.S i: C ]:.) 1 I'I:' A: 10 i.. 0 G ": S1G Miv A/1I....q,i B: A ) o: H A S lE 8 0.)0 0 5.44 3 2"3:) 2, 00 0? 840 6.5 32:1.3 00 7.0. ):1.,3 6. 00 7 23 1 2 1. 90.00 7.30 3 1.1 9 4,0 0 7. 0:1. )1 3 96.00 665 321. 5 9 a " 6., 13::":L i:.100 00 4:". 4 3'2.) 3 1.02.00 ': 4 7 3 >2 9:1.0 4 0 3. 4 9;32 Z3 o 8:1.06.00 2.:: 8. '4 9:10 8000.56 3:26.5,:1. 10 0 2.4,.9.1.1 -) O. -— 3.96 32 6 1.:1.4 o.00 -7.29 3 9 5:1.:1.6 00 -1:1.. 7 5.3 L 18 00 -!. 6. 50 40 4 '"20.9200.1. 2 93.9:.00.-:1.0.00:1. 1.:.; 7 12. 00 7.,39 1 J.9:1. 30 0 0. 77 1 21.0 1 ' 0 0.... + 7...):32 ( -.. 3:. 32, 5:134.0 0 -- 3.09 1.33. 7:136.00 -3.06 1.34.8 *:1. 8 0 -3.2..3 0 135.6:1 ) ) *4. -3.4 136.4 l f o ",0....3~ 1. iS6 o 144200 - 3.89 1.377 0:144.0 - 4.43.37.5 146.00 -5.07 1 7.8 1. -48 00 ' O-.).1 1. '50.00 -6.67 1 38:1 1, 2.00 -7 6:1. 137.1.5.00 -:1.0,99 175.5 160.00.. 62 1 3.:1. '0 4:106 1.0..: J1. 64 00.:1. 32 0 1 J6,, 0 0..... J.- c 16. 0 0. -17..4 z.9:1. 70...0 -:,09:i. 2 8:1. 6 6 0 0..... 3 99,7:1 76.00...-21..' +45 94.4:1. ) 8 00 )...-) 1. 7 S 90 -, ':18 0.0- 0 7:1. 2 - ' C); 95

Table 4.3.3: Bare ogival cylinder, extended size; 3.75 GHz. -1 1 F.T A:1. L> C.. C0C0 ( "I:r AM, 1 I... A r13 Tff PfAS 82.09.19 305,5 84.00:J. 0. 3 1. ___ 0.6__ 81 t6.00 1:1. * 07 350 7.*2 C)s: +0 *. + ~ _ _ _ _ _ _ 3C o.6 90.00 11.,67 307.7 92.0:1.(I-L, [52 ___307.6 __ 94.00:i:L.07 307.2 96.000:1. 0 -31 __ 306,6.1400 9,:1.9 3%.10 +5.:I0.( 0 7.(O/68 3 03.*8 -IO12 I 02.0 5.,70 %3'O II 1.04 -.0. +0 3.:1.i.__A______ 9 6. 3 1 0.00 7 -2 1Q. 0 0...._ ______4 7:110.00 / —70 2 -7 / '7 ) t 0 -___ ___5___ 9 1. *9 111.4.00 3+46 L7 4 I L6.00 __20 _ 1 68.15 I I EQ); 0 0 -__76I_6_ 1. ~0 288 167 5 I. 0 `____-.9 ______:1-7:1..9 1. 6'.00 — 6 5 7:1.80.C) 3 1.3 %.00 — 5,49 3).5 1.4.0 0..4.07 3:L1..6 4.46+0% / 6 3163 I. leikG ) 0 4 i() 1(. 1 l 0 4 5 4 00-9 %31 7 5~00)?9 3_____ ')l 2 1. 6,00 4,' 01 I 8 C 0 _ _ _..v__ __ 31 0 2 I. 5 (k 0 k3l 59 031 7 -172.002 3- -- -:1'.74 0 00 21 * 13 286 * 6:1.76.0 *-21._92 _.1. 78 *Z 00 I22. o33 22 96

Table 4. 3. 4: Bare ogival cylinder, extended size; 7. 5 GHz. PUT A 10LS(~v/ -32. 00 ~ 1525. 2. a U.00 1.3337.2 35.00 ~12. 5934, 13 6.00 33. 790 14. -12 9 9,30.09 1.6.27 4 2 9 1.)1.2 93.0 14.2 -942.QQ 13.8 7 8 95. 00 1256L 960.00 1.3 8 7 97.09 59.2'5 99.0 9.8 12 7. 1.0.1.0 2913 6. 4 ( 0 1-. -74SC 106.0) 501 109.0 '63.5 044 112.0 01 1 7 1 11.00 1I4 11.9 C09 6 0 118.0 -9.2 326.5 1203.001 4 37. 121.0) -632151 124.0..0 761.2 97

Table 4.3.4 Continued 1 25.0 126. 0V) 1 27.0') 12 3. 00i 12 37. 00' 1 3 0.00 1.42.0O0 1 3 3. C) 1345. 00 1347. 0 1,3 8. C 0. 1 '41. 00 1 42.00 1 LI 2- 03 1 45.4.0 0 I 46. 00, 1 47 1-.(0) -1 50q. 0 0 15 2.00 1 53. 0 0 15 * 00 1 C)54.0 9 156 5.00r 1 5 7.0 0 1569. 00C 1670.001 1 1 7 1. 00 1 0 300 165a.00 1'66. 00 16-17.00 16I.00 1 7 1. 00 1 -16. 00' 1 n7o C 0 -1.0 ~-'. 45 3q37 5.8 1 29, 7 -12: 93 i17 33 -".3 -6 * 92 11 326 -1.0 8 7 2 6. 1 2 2. 6 92 — 25; 34 162.4 1 7 2 3 1 7 Q*' 1.27 ' 7 ' 34 '4.1 LoL it4 c0.1 L. 4 7 39. 1L27 '1 '49.7 1 5 5.' 1 64 4 83 3 67 1 167P.2 166.91 98

Table 4. 3. 5: Bare ogival cylinder, extended size; 15. 0 GHz. T1-1 E.'T J1 () I... 10( c T (3I fI,, /... A )31 D P 1,SI:1. 3 8.434 9 81..00 1.6 2 2.4 (: 00:' 1 7.6.2_. W,4 8700.J:. 7 6 I3 el 88.0 1.6E 3%.54:14 0 s 00:1.67,:1.7:.5 C) 000:1. 6 1:1.6:1.637 J:.. 00 16.:1.7.158-.2-()~7.0 1. 6.4:1. 44.0 9 3 00(:1. 7 38:1.2 7 I 94 *00 1.8:18 1 I. * 1. 'LB 9 J5. 99.0 1.3.76 2.4:10:. 0 1.3,45 0 '6,:1.02,0 1:. 475 3,::101.0 3 63 1" * 4 11.04,00 6.9534 9:1 0) 7.90 1.0, k.0 00 8 2? 58:10/006 )C216.4(. 6.9!0.02:i1.:.9 (I( -1. *30 /*3:1J.() 0() 1. 9.91. 2.9:J.1.1.. 00 3 98 3 42. 8E 1.:1 00 3 17 3 () * 1 +1..(O0 -.90, 611.4.8 o1:15J0:.. 9 J3:1.2. 8:1.:16 + C(:~1..2 1.68 1 11. 7. 00-2 08 1/. 6 2 ''2 * 0 -1. QJ 76:1.24 o 000.16 ~3 1 *1 2-:.4:1.2 99

Table 4. 3. 5 Continued:1.I 00:1 2~0:1 20 4. 00.1 9i 3, o 0 1. 3. C.00 I.3 4 0( J. 3 ) 00 I A 0.1 1.4 / U.) i. A 00R ILr 1. / U'. I' 4.'0(:1. 64 0 J1. D k/ 4. J. **'..:J. 6 9 00:1. 7 () 0:1. 71 70 I1.77 C. ) IWI 0 3) — I.+ 94 3-1.3.4:1 -— 4.67 -4. 1..4 O C) C -— 6-3,9 — 3.+98 --- 6-,4 6 -8.+65 7 * 6 1. -6. 9 9 — 72 -- *.4 71 -7, 94 7 —7.0 7 — 677 — 7 A. 8, — 97.76:1. J* 3 0 16. 64 C. C Cs 4. 4..) 2 -.7 74 -.-27,, 74 -4.. A.1.. 1:1. 6 8I:1.67, 23:1.74 + 0 1. 92,4. A 264.* 4 3:1. 3 4. 9 327. 9 344, 4 1) 4:1. 50.4 2:1. 79 0 2:1 8, 3 () 3 3 3 c. 7, 0I:1 4 1. J1 4:1. 66, 3:1. 72 o 9:1 79 C':1 2 -. 4,I. 9x., 205. +;. 2 * 5. 2 639* 1 2_6) 1. 29 J.3 30w 3 1 J 100

Table 4.4. 1: Ogival cylinder with uniform coating-of OG-C-I material; 2. 0 GHz. O: * 0) 1. 7 6 4 12*,l 6 9? 6.4 2.40 668 9)* 1 O I0I1 2% / C:t0.()0** '9.9:1.04N4. 0() ())%:.: 5:10 0:1.:; L1 1 0 00 / 6 1. 3 '* 3:1 1.0) O2. I108 ~0 5 9 ]' 1%) IJ~( *~ 0 2. A. A.6 * A + * 0) /):1' I I AN* ' V ' O.... - J. 2%X `4 0', O 7i:1.4 21. 6 I. C ' 3 / P1 N(.1.. 1. 00 4 (I 101

Table 4.4. 2: Ogival cylinder with uniform coating of 0(3-C-i material; 2. 5 GHz. T' - T C(i I:::i:; -I1 j~ - s'i;, s f cr Oi (I 13O O( — 0.*24 0.5 82)0.*4:1 0.*4 86 001. 2 0 *4 813. 0t)0.1.416 0.3 920.00 1.5 3 0.3 92,00:1..46 0.3 94 00 1,.25 0.44 96400' 0.91 0,4 9 %S00 0.4:1. 0 4:1. 00.0K) — 0. 24 0 5 10 O2 0 0 I:.0705:1.04,.00 — 2 0905:106."00 -3 3,%:)4 0.4 10.0C0 ( -6.76 3%")59,?7 11.,2, 0() ~ 9 l.16 3 58I *7:1. 14 *00 -12.3 351)6 *5:1.6,0 7:17.13 350.,6:1. 18 G 00 2-2*5,57 32 2,:!.:200 -2 4.02`213 *4:1. 22.0 O0 -:1 7 o381 20.4.0:.4 00 -- 1.3,8.37:1. 99 5:12 0 00 -. 1.:1. 68:19 *L 0:1.21 S.0 0 -:10. 2:1.:.7t.6:1. O0,00 — 9 20():1.97 8:1.2 * 00 -8 1:1.98 S 2:1. 4 (00 -8. 7 98, 9 [36 0 0 I 81 19 96 138.)0 )0 -7 72- 200.4:1. 40.00O — 7,4 76 20:1..2:1.42,0 0 -79 22(:1.44.00 +8 17 202.":.6:1.4.00 -82 2?03.2:14 8 * 0 -8 *94 20 *:!O,( 9* 94 4 203.9:1%.5i4 *00 -:I. 0,62 203.6 16,( -:1. )2 20.0:16 0.13 + 44:198 +9 I:164.0( I4 1. 6:1 O~:16 00 - 4 '1. 9 4 + 3:. 1.8 * 0().15 '48:1.9:1..5:1. /0 *0 1 6, 03:13I.:17 2.00 1 6 4 9:16 0:1. 74 C 00.. 1.6.85:1.33,5:17O.0 21 26:18() * 4.1 Io * ( -1 1:1 00 102

Table 4.4.3: Ogival cylinder with uniform coating of OG-C —1 material; 3.75 GHz. 1i ( C" I-' ~,3 C: '.) T E FRll1 C: (3, F? O S SEFCT 1-OC)N T I I. IT J. () L~1. 0( ( T C-l 1ii n~/ L.. A WE ~D Ac~ ~ p E 0(). -1 6( 5 *6 8.()-0. 3 2 346 2 04.00 0.63 346 6 86-00J.2 3469 88. 0(1 ) 0I (I 68kc1 3.47.0 92.0() 1.6 3.B.47.0O 94. 0( 1.29B 346.9 96.00' 0.53 3'4 6.6 9 El00 k-e3 346.2, J0~.(0() - 1.6 345.6 I 104 * 0 0 t-5 *4:. 342,. 1.0 6 0() -3,2 -)1 339,4 10.'S00 -1.1 * 4 3 32JL.5 J.:1.2 0 0 -2 1. 10( 26 5.1 1.14 l0(p) *~1. Z:1 220.5 11.0J e.$+0 -1.3 205.4:1.20 +00 O:.30:1. 97C9.C 7.122.0( —:1.3V.15i 1.97.6:1. 2.4 0 0 C.) * Y 1.98,2 - 12 6 0 31.9 9.:i 4.01 / 1 'I201..6 1~.0 2.).2( 20,.4.4 13.0:2 209.():L 44.0() -16.94A 37.2 1.600 -:..0 40, I ~' 0 1 4. 1.461.6 ' 40:1 1. A1, '.6 ('-:1. 5.448.2.1/~ 0 "' D ~ 449.1 C C1.) C)46.1 6*-1.9. 4 E396J.1.7.00 o 3 + ( (6 00-:1. 6 4 38,3 103

Table 4.4.4: Ogival cylinder with uniform coating of OG-C-1 material; 7.5 GHz. B3CKSCT.~R3i.'iG c.'>rSs snr:co" T! ETi 1 LCG (C' G:; 'A/LA: iDA) PSA3 - 90.00 F. I 86!14 4. 3 32.00 -1. 1 2 177.7 34,.00 1.?6 212.3 36. 0 o 31 230.9 83.00 6.26 233.9 90.00 6.90 241.1 9 2.0) 6.26 233.9 94.0)0 4.31 230.9 c96.00 1.26 212.3. O0 - 1.12 177.7 100. 0 0.86 14 4.3 102.00 ~-0.59 125.6 1'4.00 1,78 11,5 106.00 -5.8 10 3.5 1 3.030 -13.36 73.1 110.00 - 13.50 337.7 112.00 8.69 3!5.2 114.00 5-3.0o5 310.3 116. 0 -10.A45:.n 118.00 -1 7.37 317.4 120.00 '26. 42 9 q 4 122.00 "-15.52 122.1 1 2..0 - 3. 12 130.4 126.00 -13.7 0 19.5 128.00 16.33 139.1 130.00 -t9.43 157.0 132,' 1.89 2-l3. 7 146 00 - 17.2 2 4.1 133.00 -18.69 26 9 9.5 140.00 -21.92 3 2.2 142. 00 -24.60 50 1 4.00 -2 3.28 i'.Q 146. 0 -"1.30 70.2 148.00 -2. 65 221.7 1514.00 -27.63 1C2 159.00 - 32.5 223. 160. " S22^ 10 5 0 12. -00. -: 3 256.5 174.0 "2.39 ',. 176.00 - '?2.16 25Q.9 1 8.00 -21.9O 251' 179.00 -21 95 "s'?. 172 2 - 2 1 7 256' 5 1 74.0!L ^ -. 176.00.7.71 2 5'. o 178.00.22o.9 5 1.7 180. 0 -22. 96 25. 3 104

Table 4.4.5: Ogival cylinder with uniform coating of OG-C-1 material; 15. 0 GHz. f~? ~ fIc F' rri N1( C' CF.1..* * T I IE. T~ 1. 0>L.OCG r G;:m B1. i1 H~) 3000 9.*77 313:1. 7 31 0( 9 * b56 4): S33,0(:1:..96 4 1. *: 3(:)4 o 001 6 60 4 C5,0 1. 4 235.( 33.00C J.2.3PI9:10. 3900:1.2 *: 3. 90.0 1. 2%1.97 1 '0.3 9:1.00) I 2,+0.1 1 52 9200(:.39 7. 93 00 1.3 26 1A~, 9 400: O1. 4 O03 36 9,5 k 0):1 f 2 ' * 0 97.+0 0:1:7%. 96 41 1 99.0 9 6/ 4)::1.0.0)9,7 A:.,:10:O1. * ( 9 *35 293 *-i 0:12*007*5 27 9:10C + 00 4e.:17 246.0:1.04. 00:1W6:.97 I J. 0)"5 *1 00 2 *2 9E:154*:06O. 0) '.9..: 3:2:1 070) 2.7 2:.1:1. 4:10.00~o +,97 90 ) l1-191.0...,9` 20 3:1:12,00 9 1.323.:140 0 -:..93.:1. J.0() C _3,7:1:13006.34 1.'J * 6:1240 -:.1 5. 15 2:21 J.00 -5 6 736.:12,0)-5.641:125.00 16 06 105

Table 4.4.5 Continued J~.I ()'l!) %:1. 3 4 00 1550 J1. %, 3 + ():1.39. +00 1A 1.00 J. 4.00 1, 4 0 0 J.-1 it' 00 J.4I) 00 -1 9 00.1 4 O0 (1'1 9 0() 7()+00:1. 64:1 00 '1. 3) 00:1. 67 4.00 1. 68.- * 00 i. _7(, 0 5j 00:1. ('1 0 00 C.0 8 * 3 2 1. 7 53 -1.403 -. 6 (). — 1.6,937 *:.1. I4.96) — I.6,0:'Il.. -~ A... I, 4 7 9:1.5 4 — 1. 60 -2 (i')1 -2 E 1 -— 23.836 3153 -28 4 87 5 * C25.10 414 94 9 1.I:. 6) o 7 123 4 I 6)C 5~ 3 1. 365 3: 4 6 3 1 I ' 10i 4:1. 61 97:1. 96. 2 4 42 5 2.. 9., 6 23 2.7 3 1.1 5 324 *3 354 6. 3 639 98 * 1. 'I. 0 6 %..1.:1.3.;;4:1. 28' 4 7. '" 1. 56 7:1 4.. 2:1 5 26:1. 273 *2 273,") +:1. 27:1. +. 2.69. 3 267.. 6: ~' 44 62 v 2. 106

Table 4.5. 1: Ogival cylinder with linearly tapered coating of OG-C-1 material; 2. 0 GHz. J.<1. Cit 6 3 I C;It /. it:H 82 O 00 -1 75 * 0() J. 4 56. 33. 0 1.46.O 4 9 ()O *, I2 0 6 2,: '. '0 0 * 55.,*1 J. 0)f 0.42 98.00:1X0 I:: J10.0 J% 3 I I9 % 1 C 00 1. 1 *.1 2 1.-:1 C'' 1. C))*..1 * 1 L - 1. k6 -1.0.3 1 9:1. J.0): O43 1:1. S 00) 23:1 3 I5. ()6 9 J. 04.9 415.0)1 ""J (:1 I '4 ()1I 99 1 4.0 1 ~. A..% IL ~* ~> ~() ~0. ~9.~ ~:i~ 6.1:)1 Co 3 ~ ) fV)II.13-%3 0 1. 8K) QU I. JS 9 94:.(i 107

Table 4. 5. 2: Ogival cylinder with linearly tapered coating of OG-C-1 material; 3, 75 GHz. 1..~ 7.6., Q' 0) d 1.4 Ca) A I) T) A.~~; ii5 7 V"Z 9 r\,1.. '?,z:::iC' ''0.' -. 6. 1 + 2Y -vc I - 1.. ID:1. 1. 4~ 14 I: A... u.. A c C(% J. 3 X.1...1.`) 00 1~;) 0(::I 2.7 ~ r' 'NC A ' 11 <i..t G A.~ j2 4..; i) 407 IA ' 21 i 1 1 I iC). iC) *-i.4...73i.,. -, "I' 1.6.47~-:1. 2\ 'u. 0 3' *1.. ~...t/:15 P I' 'I J*.z, hi 108

Table 4. 5. 3: Ogival cylinder with linearly tapered coating of OG-C —1 material; 7. 5 GHz. 30t'. 00 6.6(C). T O H I-: G 1 Gi If~, j9 I3 k. 0 Al NI 9 0( ) 6A. " 9 Q()346 (k:1 9*~ 9:1.2 V (). 4) 0 1 9.*. 24 2'`5 -4 3/KA I. C)-7 t( 6 0 2 1...:1.0000 0 /9'4 4A~** V..J 1. 1k ) 9 I J( 1.07 Qf). f n.; 3 A. 9(~) 0:Lt'~ I.J J. '.1, k.. 2.1~ Q!) I ~,4 4 0 1. 01. A.. (%,... 109

Table 4.5.3 Continued "), o C, (5'...,1:.... ~ ~,9 147, ) i.1.3O! Ji1. 14. 0.3 i I C. 0.... ii...**.;: 1.. I,"'..... - ' 138.00 4307 i;1;.2.: 1 0....F 1 i...... + 4 "o':}....4. 3i 3,:1...., '.: 4 -:. ). 7., - 'J....,;,1. '1. (4.......:; 7 '::0 J. -'1.; K 15.. 9. iI00..... 1 ". i... 00 -:1. 0 5.,58 I* ***93 * 9 1.'... ^ *.. 7. 5. 6 *; 00 i --- 1 9 0" 1'444 ~.'. 7. 02! 14 )0*3. t.., j-. j;O.00:;;.a 16..:'x s 0 14 o.00 -3..04 114., I 3...........,.j. ) } Q A, / 9A!*4 I.7 000 34.). 1. i" 00 -v': * 0....'' 7 "',; 17,,00....8'0 3:. * / 1 ', 0! 17.00...........;......,; I /,..). "8.00 -930 ':"'.7,, ^"4. 0 C,1 0/9i -'29 8.9 0 4 A.. i..00.-2...2...,. 0.1. j.,,.,... 110

Table 4.5.4: Ogival cylinder with linearly tapered coating of OG-C —1 material; 15. 0 GHz. B -(F -F 'I)( - ' 9 1 8: I 400 J. 2652 200 2 92 4 4 14 009? C. n. 99 9 9,311 -514 9:1 00 71 1 V: I) () 2L -1 1 I %)( 9 *0 I 74., V,. 1. 1. O 0() 0 -:1 J() *1 3 %) 5:10:1. c''i 1 CD 6: 1.0 0~c:1.1.-0- 'I 318 6':1:1..00 1299 1 1 00-1.i~0( *..0l~''7 3 0) '2 91 1 V *tj0 % 4 1+~0 I ~~QQ.. C C. )5 ill

Table 4.5.4 Continued 1. 0() I 40 I" I )04(1) I 4 o lV. (.53 -19 -:1.C 'I 1!5 45~ ~.L j '6'6. 5d T.j) 0 J.44.7 ***' A. C "T — I cl Jli 112

Table 4.6. 1: Ogival cylinder with OG-C-I coating, cylindrical tip; 2. 0 GHz. Q() C0( Z. '4.00. 6().0.0) 0 04 I~.0 440 1.0:1104.44) -*0.2 80 0 O. 9 9, 0:1) 0K - A 72 3S 4 Ai 3. / o I1.7i3 O. 1 I.0() 2 1 2 I %.) 9 1 9 ). [ J. )( 920'.1 I 00 *-69:,j6[1.. 1.0J-::. 507:1:. '5 e,-!1i JK0 E- 11 4 11. e).0 -6 17.6 00 9 Y 4:11 V 00. J1 + 1 L 4 '. I / 1,( - I 6 1. Al. 9:7. 113

Table 4.6.2: Ogival cylinder with OG-C-1 coating, cylindrical tip; 2.5 GHz. p.~ A (I GE 3 )~ Cf ~ O) N 1(30,00 0.31 2. C32 O 0 0.94:1. 2 3680.*7 3159 * a 9l380 0 0:1..89 353 C),. * J' 176 353.A2 94. 00 1. *4.3 357, 8 960():1.0*ID4 35%J 4 9k8 0( 0.+44 J5.*::1.00 * 00 -0 *33 356. -'L8 J.0O'4 +0() — 2.50 3%6 C 44 I. C, 6* 0 0 9-.6 %356 % 1 0:2 00 S.73561 1-:10.00 **3 * 07 36. II:14.00 — 1L5,-;49 4.7:I:1. 6,o00 -- 2 -3,6 8 M0 1:1.0 0 — 3 0.3 6 1.96.o6:1 )0() —:1399 1'4 +:1.:122,() -4J701C3.+0 I.0 0 I10.613 1 V(L337 1,0-9.+64 1. 9 Vr-5 * 00.5 6 JI 9'0 %f 7 I31 4.0(1) I -3.+317:.lE t1 9 4.O -1.3 6o0 0.34719L I. 00 *** A72:192.*3 1A2.00....9.07:193.9 00 4 ( -— 9.+52 1.95. 4 1 1. (,00 0 I.05:1.96.9S 1 A 3 0 00 -:1.0.66:I.?%8.3 J. O 0( -:1.1.o34:1.99.6 I1 V.0 (0 I.- +:1.2.9 200.7 4 O,' 00:1.2,90 20:).-1. %5 (O ~ -:1. 3 77 '202.1 I1 (0 * 0 UK.) ~66 ''0''.0 ~013 I4.0) -:17.6 3 IJ. 6 00 -:1.3+7 0 19 J.:. 3,0( -:9 4.675.7:17400 22.0:1.366:1..0 -22.7:1.(34 *2:1. 7(3 0 (I 00- 2 361 ~ ' 114

Table 4.6.3: Ogival cylinder with OG-C-1 coating., cylindrical tip; 3.75 GHz. > S&; " A T T -: F. j:; N r; (C' F~ O) %; (S 1":, C%, C) N. T Fi F.:, A I 0l I... C C;' ( J' GI M I... P, N 1'9: D 'Il( 3( 0:1.7 336 2. 00 0. 4A2 3 52 2 84,,-~0. 7 % 1)06,' - -) 3 7 A 88.0() 2 64 34 7 6 90.00 2. 75 345 * 'j9 2.7. + 00O 2.. 56-4 9?4 *0 0 A: 406 342.+5 96.0.* 1.. 2.4 340.p "+6 98.00)+ l, 0.,07 '33(.5 10.0 15 336.2_:1. 02,00D -3.+63 33. 2:1.04.0 ")0 *-*6 43 39 1:1.06.00 — ( 2 2.2:1.0 J.00 -5 9 92 30.' 9:.0 -2: 94 24 8.)?:1.102. I 003 —..02)1~.:1. I114 +0() -:1'4.5 9 -1 X':1.:. 1.l61+0( -'I. 3 +02 I 9 +5 1. -1.8.00 -- 26 3 I(1'*19.7:120,00:.3 I17:1 K:'1. 2 2.00 -:1. 4.58 'I -u 7 *1 P.00 ' -:7 07 177.9 V'.-21_530 10,IJ%%9:14 40 'I..0:144,002- I50 371:1.416,0 0 J 5,.7 2 1 0.0 C) 0,"( + 693 I. 4 D 40()2 -:8 31 1:1.. 1. 4~.0 0 -1.55:1.6 *7 3 I 10.4400 -I2.71 35. 7: I 5 0 00 — 2if5 49..2+ I 52 + 00 N-1.E3 30 1 1. J. '% —.00 Ok29,44:1 62 %.JC70,00...()3 690 1 5k ) 00 **E X2 4. * 6 "V. 7:18 * 00-5 2188 7 115

Table 4.6.4: Ogival cylinder with OG-C-1 coating, cylindrical tip; 7.5 GHz. -k T +7'l 3f3 1.63, 32.0)7 1 12 ~ 2 1 '4 1 1 ~ir.~ I)?,') r (2~~-7 303.0 "1,1 L4 7? 6* 12 7, 3 0 3 26.07 i! 6 7)~,3. 0".6 "i 0; 1'76.0' 0.7 1. '6.50 1I 6. C3 Sir 2: 110: 3 1"! 13 73.5S 12-.0. h3'r 17 121~ -3 5 140*)3!1;17 3 142 ) ~1 ~1 6~ ' l i~, 1LJw.>) 7 -. I~~,, 7 5. 152 -'1 ' ~2~00., ' 1 J~1~ri 4. 7,1 170.0')( '3, 03 7) '~~~I! I'w' 1Iti. \'k -1 LV' 116

Table 4.6.5: Ogival cylinder with OG-C-1 coating, cylindrical tip; 15. 0 GHz. B. A C ) S; A T ' ' E R 1 N G C R (::i.' S E C T:t 0 N T i: T fA 10.: e I. 0 G ( S I G M A /I... A i B IA ) F' it A S E 80.00 1.0 *06 3 21.5 81* 0 10 7 06 342, 8 82 *00. 1. 0 '.5 00 1.:L.65 34.4 4.00 12 68 58 7 85 00 1.3*60 78 * 1. 86 00(): 1 4 * 0 93* 9 87,00 1. 3, 85 1.07.9 88.00 1:.31 1.20,4. 8900.1. 262 130. 3 90*00 11.96 134.2 9.1. 0O0 11.50 129. 5 9.2.'00:.1*63 11. 70 93 o (00 12.45 101 9 94.00. 13. 33 88.6 95.00 1.363 76.8 96,00 13.03 63.8 97 0 1..1 54 46. 0 98.00 9.82:19 2 99.00 9,:))2 346 + 5 10000 9 58. 319.3:101.; 00 9 *41 299.3:10., 00 7 * 89 280. " 7 103 4.86 24 54,9 1.00 2 4 20..7:105.00 3,:1. 2: 164 7 106 * 00 4 *26 139.5.i. 1 07 00 346 12:1. 6:1.0800.*36 9 8 109.00 3 63 49.8 1..10.0) 1. 75 353.8 7.11: 00.51.29.7 1:1200 I. 31 5,8 11 3.0 -2. 93 006 ~..1. -,, 1 '.. '' + 116. 00 — 3.14 J1. '50() 2 I I 7.0 - 2.54 1. 40 0 1, 0))...f." 00 5.05' 09, 8.:1 ' 0 X 0 0 -- 1 1. 2..-` '5 i, ' 2 1 -.00 -.1.2-.:9 1 00 1".00...1 3 3 8,2 1. -4 J. ()4 3 i 24 00 -7..0....'. 0 " -1...5. 1..3 117

Table 4. 6. 5 Continued 2 0 J. 0 I )",Q(),I XI, 0()( J. 4( 'I 0() J. 43.,00:1. 44. 00 J. 46 0 0 I 9.0() 0 0.00 00 00:1.60 * 00 'I. 6A2) F 00:1.6~4 * 00:t. 65,0 J. 6 0 ( I 6 -7 00 7' 00 A.0 I6 00 9 0 '.,1. 4 I A,1149. (:10732 — 1878 o 1. 6 -3:.5 239 -.2:1.:1.72 -2 223 23I 3. 02 26 978 6.9 -7 )/ 6.3 2 1-' t('3 %:Y)138>I4 1.4 3.1.:1. 49 */ 9:1. 6 r-fj 44 '3 02 + 9 30()9 2 3320 + 4 113 9 I, 1 2 /, k 1 'C": ) ".":1.44), 9: 1I.4 6 1.6 1.61.46 I 7 320. 321. 7 327 ( 9 323 9 3"29.7:30 118

CHAPTER 5 CONCLUSIONS Our analyses of the types of scattering contributions that the ogival and wedge cylinders present has shown how it is possible to specify the impedances that would be most effective in reducing the scattering; and given the impedances it is then possible to define an impedance variation over the surface that would be optimum in reducing the backscattering cross section over a given range of aspects subject to some constraints or (say) the maximum allowed value of Re. r-. This knowledge can be used to specify the desired properties of a coating material and program RAMD then enables us to compute the cross section reduction that would be realized with any actual coating material. Of the materials available for assessment, that designated OG-C-1 proved most effective, and the optimum application of it has been determined. The resulting coating should produce approximately 10 dB cross section reduction at near edge-on aspects and over most of the frequency band, but though the performance is quite impressive considering the small thickness (< 50 mils) of material allowed and the broad range of frequencies to be covered, it is not felt to be the best that is attainable. Indeed, no use has been made of the existing technology of 'thin film' materials, and judged by the capability that existed even in the late 1960ts, thin films would appear to have just the properties which are desired for this type of application. 119

CHAPTER 6 REFERENCES 1. Bowman, J.J., T.B.A. Senior and P. L.E. Uslenghi (1969), "Electromagnetic and acoustic scattering by simple shapes", North-Holland Publishing Co., Amsterdam. 2. Knott, E. F. and T. B. A. Senior (1973), "Non-specular radar cross section study", The University of Michigan Radiation Laboratory Report No. 011062-1-T. 3. Knott, E. F. and T. B. A. Senior (1974), "Non-specular radar cross section study", The University of Michigan Radiation Laboratory Report No. 011764-1-T. 4. Knott, E.F., V.V. Liepa and T.B.A. Senior (1973), "Non-specular radar cross section study", The University of Michigan Radiation Laboratory Report No. 011062-1-F. 5. Liepa, V.V., E.F. Knott and T.B.A. Senior (1974), "Scattering from two-dimensional bodies with absorber sheets", The University of Michigan Radiation Laboratory Report No. 011764-2-T. 6. Maliuzhinets, G. D. (1959), "Excitation, reflection and transmission of surface waves from a wedge with given face impedances", Sov. Phys. -Dokl. 3 (4), 752-755. 7. Senior, T.B.A. (1960), "Impedance boundary conditions for imperfectly conducting surfaces", Appl. Sci. Res. 8 B. 418-436. 8. Senior, T.B.A. (1962), "A note on impedance boundary conditions", Can. J. Phys. 40 663-665. 9. Senior, T.B.A. and E.F. Knott (1968), "Research on resonant region radar camouflage techniques, The University of Michigan Radiation Laboratory Report No. 8077-9-T (CONFIDENTIAL). 10. Senior, T.B.A. (1972), "The diffraction matrix for a discontinuity in curvature", IEEE Trans. AP-20 (3), 326-333. 11. Senior, T.B.A. (1976), "Cell curvature effects", The University of Michigan Radiation Laboratory Memorandum No. 014518-502-M. 12. Senior, T.B.A. (1977), unpublished notes. 120

APPENDIX A COMPUTER PROGRAM RAMVS (as of August 1976) Program RAMVS was developed during the latter stages of Contract F33615 -73-C-1174 to compute the field scattered by a two dimensional body in the presence of (lossy) electric and magnetic sheets. With an impedance boundary condition imposed at the surface of the body and the usual transition conditions at the thin sheets, the program solves the set of three coupled integral equations for the currents induced. The electric and magnetic sheets can be superposed to simulate the effect of a thin layer of absorber of arbitrary permittivity and permeability and it had been our hope to do this for an absorbing coating applied to a body. The original program was documented by Liepa et al (1974), but since that time a number of minor errors have been found and corrected and we have also modified the expression for the self cell contribution by including higher order terms. In order to save space we will not repeat the entire formulation of the present RAMVS, but will list the differences from the version in Liepa et al by citing the equations from that reference: on p. 10, equation (2.8) should read tA AH(1) H(s) =- J (s)(i- r)H (kr)d(ks) )+Z (s)J(s(g) The free space impedance circled was left out. on p. 14, equation (2. 18) should read A1 J (s1) =... + ( J (s')H ( ) (kr) d(ks') + ( J (s')H ()kr) d(ks'), 2z 1- z 0) +2A C-(A-T-2A) -- 1 121

1 the factor having been omitted; on p. 17, equation (2.22) should be iY0 A(s) s A J(s )= Z (s )++Y -— +. J(S 5 s l m 1 A/2 0 02 A 2 s 1 +.... The factor 2 appears in the denominator as the result of retaining higher order terms in the expansions of the Hankel functions. on p. 33, the equations at the top of the page should be (ZE i i \) 2(e, - ) 27r(4. +i3.)(.05) = 0. 382 + i 0.509 X 27r(r - 1)A 27r(1. + il.)(. 05) =1'59+l159 r (Z i =i 1.59 + il.59 The use of electric and magnetic impedances which are normalized with respect to the wavelength somewhat simplifies the computer code and eliminates one hand computation in preparing the input data. In addition, quite extensive changes have been made in the RAMVS source code to incorporate the above modifications and to make such other corrections found necessary, and instead of citing them individually, we include the complete program listing later in this Appendix. This latest version of the program has been applied to perfectly conducting cylinders, cylinders with specific surface impedance variations, and thin electric and magnetic cylindrical shells. It presumably works (but has not been used) for cylinders surrounded by electric and/or magnetic sheets provided the sheets are not too close to the cylinder or to each other. It has been shown to work for single and multiple (but well separated) sheets extending outwards (like fins) from an ogival cylinder. 122

The words "not too close to" and "well separated" hint at the difficulty experienced in trying to use the program as a tool in our present work. By placing one or more sheets close to the surface of the perfectly conducting ogival or wedge cylinder, we had hoped to simulate the effect of an absorbing coating whose electric and magnetic properties could vary in depth as well as along the surface, and would be explicit in the specification of the sheets. Unfortunately, the program now ran into difficulties. Because of the formulation used here and in most other programs based on discrete sampling to derive a system of simultaneous equations from an integral equation, RAMVS fails when the distance between two surfaces becomes comparable to the cell size or sampling distance. This was recognized when we examined the effect of changes in cell size on the solution for a sheet close to the surface of the body, and though we immediately began an intensive investigation in an attempt to work around the difficulty, our endeavors were not successful. In the hope that there was a cell size or sampling distance that would be adequate to produce the desired accuracy, a numerical experiment was performed using an ogival cylinder surrounded by an absorber sheet. As the sheet-body spacing h and the cell sizes were varied, the changes in the backscattering cross section averaged over 30 about edge-on were recorded, the expectation being that when the sampling rate was sufficient, the results would be substantially independent of any further decrease in the cell size. To minimize the computer costs, we chose a wavelength of 10. 7 inches for which the ogival cylinder is only about 0. 44A in length, with 12 (i. e. 6 times 2) sampling points on the sheet and either 12 or 16 on the cylinder. The sheet-body spacings considered were h = 0.5 (0.25)1.5 inches. Figure A-1 shows the changes in the average cross section as a function of h/A where A is the cell size on the cylinder for either 12 or 16 sampling points there. From the amplitude and phase data, it is evident that the accuracy rapidly diminishes for h/A < 0.8 and, in effect, the program fails if h < 0.8 A. For h > A, the differences of 0. 5 dB in amplitude and 0.5~ in phase could doubtless be reduced were more sampling points used. Having determined the minimum (practical) spacing h in terms of the cell size, program RAMVS was now run for a sheet spaced 0. 2 inches from the surface 123

N 6 N 6 and 8 4r 3 dB 2 1 ".. I I A/XX C I ZE =0 ZM 2 I I \ AMPLI A --- - — * --- —- _ ). 0773.011 + i1.468.350 + i 7.049 TTUDE 0I 0 I I I I 1.0 2.0 3.0 h/L 10 degrees 5 k \ I I I I I I I I I I I PHASE \ 0 I. I I 0 1.0 2.0 3.0 Figure A-i: "Failure" of RAMVS as the body-sheet spacing decreases. 1 9A

of the ogive. The sheet parameters were chosen in an effort to simulate a layer of the OG-C-1 material 0. 05 inches thick, but the results were substantially different from those measured by Emerson and Cuming. To some degree this was not surprising since it is known that to obtain an accurate simulation the sheet must be placed at the mid-point of the layer, but because the discrepancies were much greater than we had hoped, we now ran the program for a circular cylinder with a surrounding sheet and compared the results with data obtained from a Mie series computation for a cylinder with a uniform coating. The comparison left no doubt that the sheet must be located within the layer it is designed to simulate, and only if the sheet is about 0. 025 inches from the body could we hope to reproduce the effect of a coating 0.050 inches thick. In view of our previous findings we were now faced with the need for a cell size (or sampling distance) not exceeding (about) 0. 020 inches, a quite intolerable requirement for the size of body to be considered. Various approaches were tried in an attempt to overcome this limitationi, for example, by choosing sampling points not directly opposite each other on the two adjacent surfaces. By moving the points around, we sought the most effective arrangement, and though it did prove beneficial to have the points staggered or interlaced, the allowed reduction in the minimum sheet-body spacing was no more than 10 percent. This finally convinced us that there was no simple remedy to the problem. We did look at the possibility of reformulating the program to permit its use for the purposes of this Contract. As a minimum, the reformulation would require the special treatment of all pairs of adjacent cells on the two surfaces and because of the magnitude of this task, it was felt impossible to accomplish it with the time and funds available on the Contract. We therefore had to abandon RAMVS as a tool for use in our investigation. A complete fortran listing of RAMVS is as follows: 125

C -CCCCCCCC CCC CCC CC CCC C CCC CCCC CCCCCrCCCCCCCCCCCCCCCC,CCCCCCCCCCCCCC CCCCCCCCC C C C **RAMVS * C C (10-03-714 VERSIONT) VVL C C (MODIFIED 30-08-75) VVL C C (MODIFIED 24-11-75) VVL C C (MODIFIED 25-5-76) VVL C C (MODI)FIRAD 1-10-76) A.t. C C 'THIS PROGRAM COMPUTES:E"l2-FIELD ATID FAR-FIEILD SCATTERIING FROM C C A GENERAL INPEDA1NCE DOD" INl PRES'ENCE OF ABSORBER SHEIETS; C C TWO-DIMENSIONAL GEOMETR't, EXP (-IWT) TIME CONVENTION. C C C C RE: V.V.LIEPAfE.F.KNOTTgANlD T.B.A.SENIORp"COtMPUTER PROGRAM FOR C C SOCATTERING FROM TWO-DIMEIISIONAL BODIES WITH ABS)02DER SHE2ETS",p C C THE UNIVERSITY OF MICHlIGAN RADIATION LABORATORY Qv'PORT C C N0.0117564-2-T (AFAL-TP-74 ) 1974. C C C C CCCC CCCCCCCCCCCCC CCCCCC CCCCCCCCCCCCCCCC CCCCCCCCCCCCC CCCrCCCC CCCCCCC CCCCC C C C * INPUT DATA FORMAT** C C A FORMAT (18A4) TITLE CARD; USE UP TO 72 COLUHNS C C B FORMAT (I2,I3,14F10.5) MORIOAEZFCZFCZFC C C MOPE=O THIS) WILI BE THE LAST RUN FOR THIS DATA SET C C MORE1l THERE ARE MORE DATA TO BE READ AFTER THIS SET C C- IPOL~l E-POLARIZATION C C IPOL=2 H-POLARIZATION C C WAVE W AV E LElG TH C C ZSFAC MULTIPLYING FACTOR (REAL) FOR ALL ZS C C ZEFAC MULTIPLYING FACTOR (REAL) FOR ALL ZE" C C Z MF AC MULTIPLYING FACTOR (REAL) FOR ALL ZNl C C C FORMAT (I2.,JXr4F'-10.5) KODEFAIRSTLASTINKCANG C C KODE=O COMPUTES BISTATIC SCATTERIING PATTERN C C KODE1l COMPUTlES DACKSCATTERING PATTERN C C FIRST INITIAL SCATTERING AND INCILDEINCE ANGLE C C LAST FINIAL ANGLE % C INK ANGULAR INCREHENT C C CANG ANGLE FOR SURFACE FIELD COMPUTATIONS C C (DATA D, Eo AND F REQUIRED FOR EACH ABSORBER SEGMENT; C C ATA LEAST ONS ABSORBEPR AND ONE IMPEDADICEA- SEGNENT IS C P EOUIRED. ABSRORBER SEG3MEANTS MUST B-1 -RE.'AD IN FIRST.) C D FORMAT fI2vI3*5?`-1O.5) TYPENrXA,YAvXB,rY3,A?.G C C TYPE~1 ABSORBER SHEET C C TYPE=2 IM,1PEDANCE SURFACE' C C: N NUNBER OF SAMPLING POINTS ON THIS SEGIENT C CXAYXBY SEGI1ENT EN4DPOINTS C,C ANG ANl" SUBTENDED BY THE SEGMENT C, C E FORMAT (12,3X,5Fl0.5) FORn,Zt-EA,ZEB,ZEX C C FoRm=-1 ZE (I) = ZAZB~(I) )**ZEX C CFORM- 0 ZE (I" ='Z7EA+ZE'QBl~S (I) *-'.ZEXC C FORM= 1 ZE (I =Z1EA +ZRB * E X PZ (-E X *S ()C C ZEAvZEP COMP.LE,:X IMPEDAINCE CONSTANTS C C ZEX REAL INPEDANCE CONSTANT C C F FORMAT (I.2,3X,5-FlO.-3) FoDLtiZnAoz!1BZNm C _Z N (I) ~Z H1,A -Z t3* S (T) *Zc.4( C 126

C FORM= 0 Zm (I) =ZMA+ZMB*S (I) **ZMX C C FORM= 1 ZMi (I) =ZtlAZnB*EXP (-'ZMX*S (I)) C C ZMAtZMB COMPLEX IMPEDANCE CONSTANTS C C ZMX REAL IMLPEDANCE CONSTANT C C (DATA G AND H REQUIRED FOR EACH IMPEDA11CE SEGMENT.) C C G FORM-AT (12,I3s5Fl0.5) T YPE# N, XA, YAs X B YB, ANG C C TYPE=1 ABSCRB3EB SWEET C C TYPE=2 IMPEDANCE SURFACE C C N NUMBER OF SAMPLING POINTS ON THIS SEGIENT C C XAYApXBYB SEGMENT ENDPOINTS C C ANG ANGLE SUBTENDED BY THE SEGMINT C C H FORMAT (12,3X,95SF10.5) FORMjZSAZSI3,ZSX C C FOR M=-1 ZS (I) =(ZSAt-ZSB*S (I) )**ZSX C C FORM= 0 ZS(I)=ZSA+ZSB*S(I)**ZSX C C FORM= 1 ZS(I)=ZSA+ZSB*EXP(-ZSX*S(I)) C C ZSAvZSB COMPLEX IMPEDANCE CONSTANTS C C ZSX REAL ItlPEDANCE CONSTANT C C I FORMAT (15) INTEGER ZERO IN COLUMN 5 SHUTS OFF C C READING OF SEGMENT PARAMETERS C C --- —-----------------------------------— ~ --- —~ — C C (USE THIS CARD ONLY IF, IN Br MORE=l) C C J FORMAT (12,13,r4F10.5) MOREIPOLWAVEZSFACZEFACZMFAC C C C CCCCCCCCCCCCCCCCCCrCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC-CCCCCCCCC*CCCCCCCCCCCCC C C C** DIMENSIONING FORMAT** C' C VECTORS ARE LIME"NSIONED ON-LY IN TEE MAIN PROGRAM, C C IF K=11O. POINTS(CELLS) ON THE IMPEDANCE SURFACE AND C C I=N0. OF POINTS3 (CEALLS) ON T HE ABSORBER SHEET, THEN C C MK=~+K AND NMK=.I+ii+K C C C C...MAIN PROGRAM --- —A STARTER PROGRAM C C* COMPLEX A (MMKrM,MNK+1),PIII (MMK),PINK (MMK),LL CMM'K),IMM(MMK) C C* COMPLEX ZE(!iK),.ZM(M1K) C C* DIMEINSION X(HKF),Y(MK),XN(MK),rYN(UlK),pS(MK),DSQ(M-I() C C *DILMENSION LUMP(2,NK C C* DATA MI/MHK/ C C CALL SMAIN MHI. APHIrPINKrZEr Z MXs YoXN *YN ISoDSQjLLrMMHLUMIP) C C END C C* CARDS TO BE CHANGED WHEN REDInENSIONING C C C CC CCCCCCC CCC CCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCC CCC CrCCCCCCCC CCCCCCC C C..MAIN PROGRAM —A STARTER PROGRAM C**** RAMVS VERSION COMPLEX A(150,1l5l),rPHI(150),,PINK(150),LL(150),rtjMi(150) COMPLEX ZE (100),Zl (100) DIMENSION X (100)#Y (100) gXN (100) #YN (100)#S (100) IDSQ (100) DIMENSION lUNP (2,100) DATA MI/150/ CALL SMAIN (MI, A, PHlPINK,,ZE, ZMrX, YXNYN,, S, DSQ, LL Nk,MMLIJMP) END 127

SUBROUTINE SMAIN (MI,APHI.PINK,,ZEvZMX.Y,XNoYN, &SIDSQv LL, MMFLUMP) C**** RAMVS VERSION E-&H-POLALIIZATIONS DIMENSION LUMP (2,1l) COMPLELX AUIl,) COMPLEX PHII(1),PINK (1),ZE (1),rZK(1) COMPLEX "qUMDELvcLUMEoSUIMrSUMK DIMENS IO N X (1), Y (1),XN (1),YN (1) oS (1), DS Q (1) oLL(1),MMl (1) DIMENSION ID (18),IPP (2) COM MON/PIES/PI vTPI o PIT o PI PI*,YZ IRE'D DIG REAL LAS'A-vINK DATA IPP/4iIEBEEv4HIHHHH/ C...READ ITIPUT DATA AND GENERATE BCDY PROFILE 5 READ(5,100,END=999) ID READ (5,20C) M-OREIPOLgWAVEZSFAC1ZEFAC1ZMFAC IF (ZSFAC. EQ.O0) ZS PAC= 1.E- 10 IF(ZEFAC.EQ.O) ZvEFAC=1.E-10 IF(ZMFAC.EQ.O) ZMFAC=1.E-10 READ (5o,21C) KODEFIRSTvLASTIUKvCANG WRITE' (6,150) ID CALL GEON (LL'MiP,,X,YXNYUSDSQZEZMKp1) IF (KODE.11E.0) GO TO 25 NINCIl NBIT=1+IFIX ((LAST-FI'RST) lINX) GO TO 28 25 NBIT=O N4INC=1+IFIX ( (LAST-FIRST) lItK) 29 CONTINUE C...COINSTRUCT MATRIX ELEIMENTS I'l H + I MK=t4+K 14M'!K= K +M N 111= N + DO 35 I=Ml,MK 35 Zr j(I)=ZE (I) *ZSFAC FACE=ZEFAC*WAVE FAGCM=ZMF AC *WAVE XK=TPI/WAVE DO 37 I1lM ZE (I) =ZE (I) *FACE 37 ZI (I) =ZM- (I) *FACM DO 39 T=1,MK S (I)-S(I) /WAVE 39 DSQ (I) =DSQ (1)/WAVE GO TO 50 40 FAC=ZSFAC/PZSFAC DO 45 I=MlvMK 45 ZR (I) =ZE (1)*FAC FAC=WAI YE/P WAVE FACE.=ZEFACIPZEFAC*FAC FACG=ZMFAC/PZM~FACi1?FAC DO 47 11,rM ZRE(I)=ZE (I) *FA.CE 47 ZNM ( I) =ZN"- ( I) *IF ACM —l FAC=PWAVYE/WAVE X K = PI-L/ IIA VE DO 49 I~1ltMK S (1)=S (I) *FAC 49 DSO(I)=03Q(I)4cFAC 50 CONTIN1UE 128

IF (IPO'L.EQ.1) CALL MTXELI (MIMKXKXYXNlo-,YN',DSQZEZM,iA) IF (IEPOL.EQ.2) CALL M4TXEL2 (M14,MK,XKXYXN.YN,DSQZEZHOA) WRITE (6,400) IPP(IPCL).ZSFACZEFACZMFACMKI,I,NINC,,NBITWAVE C...COMPUTE INCIDENT FIELD AND INVERT MATRIX TET A=RED*CANG CT=COS (TETA) ST=SIZI (TETA) DO 60 I=10,M, HOLD=-XK* (CT*X (I) +ST*Y (I)) DEL-CMPLX (COS (HOLD),SIN (HOLD)) PINK (I) =DEL 60 PIN K (I+ H) =-DEL* (XN jI) *CT+YNl (I) *ST) DO) 63 I=I11,MK HOLD=-XAK* (CT*X (I) +STI*Y (I)) 63 PINK& (I+M) =CMPLX (COS (HOLD),SIN (HOLD)) CALL FLIP (AMMKrMIrLLsMMrPINKrPHIol) WRITE (6,150) ID C...PRINT OUT STUFF FOR THE ABSORBER SURFACE (SPECIFIED ANGLE ONLY) WRITE (6,350)CANG WRITE (6,375) DO 67 I~lM DTr.L=PHI (I) + (1. E-50,0.) AMP E=CABS (DEL) PHfASERE=DIG*ATAN2 (AIM'AG (DEL) p REAL (DEL)) DEL-::?II (I+N-) 4' (1.E-501O.) AMP11zCABS (DEL) PHASEH=DIG*ATAN2 (AIL'IAG (DEL),REAL (DEL)) IF (IADOL.EQ.1) W.-RITEN(6,395) (LUMP0(,I),J=1,2),X (I),Y (I) fS (I), &DSQ. (I),ZE (I),Ztl (I) v TIE110PHASEE3, AMPHPHASill 67I? (IPOL.EQ.2) WRITE(6#3951) (UPJI J12 XIYI SI &DSQ (I) o ZE (I,Zr VI)M o,AIMPHtPHIASEHrAMPE, PHASEE C...PRINT OUT STUFF FOR TH!E IMPEDANCE SURFACE (SPECI-FIED ANGLE ONLY) WRITE (6,,300)CANG WRITE (6,P325) DWO 65 I=MltMK DEL=PHI (IM) + (1. E-50 f0.) AMI-P=CABS (DEL) PH ASE=DIG*ATAMl2 (AIM AG (DEL) 0,REAL (DEL)) D7EL=ZE (I) 65 WRITE (6,250) (LUMP(JPI),J=1,,2)sX(I) rY(I) oS(I)DSQ(I) r &DRLvAMP, PHASE C..... DOPE OUT THE APPROPRIATE FIELD FACTORS WRITE (6,150) ID 'T'HE =FIRST-I ~lK IF (KODE.EC.1) GO TO 70 WRITE (6,800) CANGIPP(IPOL) GO TO 75 70 WRITE (6,600) IPP(IPOL) 75 TlfEs=THiE+INK IF (THE.GT.LAST) GO TO 105 IF (THE.EQ.FIRST.AND. CANG.EQ.FIRST) GO TO 85 TET.A=.RED*TUjE CT=COS (TETA) ST= ST 11 (TETA) C'...IN THE FOLLOWING LOOP COM""PUTE THE NEW INCIDEINT FIELD DO 80 I=1,M HfOLD=-XKK* (CT*X (T) +ST*y (I)) DEL=C MP LX ('-OS UICLD),ZI (12OLD)) 129

PI1NK (I) =DZEL 80 PINK (I+M) =-DlEL* (XN (I) *CT+YN (I) *ST) DO 83 I=Ml,MiK HOLD=-XK* (CI*X (I) +S1*Y (I)) 83 PIN11K (I + ) = C?P LX (CO S (HO LD),SIU1 (HOL D)) IF (KODE.EQ.0) GO TO 85 CALL FLIP( (A, IKMILL,N,MrPIU~K.PHI,2) 85 CONTINUE C.ADD UIP THE CURREENTS FOB FAR FIELD SUM=1l.E-25,0.) SUM E SUM SUMK=SUM DO 93 I1,?1# DS=DSQ (I SUMEStME-PHI (I) *IPINK (I) *DS 93 SUMM=SUMM+PHI (IM) *PIN'K (IM) *DS SUM MSUMM*YZ IF (IPOL.EQ.2) GO TO 90 DO 95 I=MltMK IM1I+M 95 SUMiK=SUM1K+ (- 1. +ZE (I) * (XII (I) *CT+YN (I) *ST)) *PINK (IN) *DSQ (I) *PHI (IM) GO TO 99 90 DEI=SUME SUME=- SUM M /YZ SUNMMDEL*YZ DO 97 I=N1,tlK 97 S3UMKiS(JMK+ (-ZE (I) + (XN (I) *CT+YN (1) *ST)) *PINK (IM) *DSQ (I) *PHI (IM) 99 DEL=S~uME+SUMtI SUli =DEL+ S UNK AMPR=VEAl (SUIM) A.MPI=AIMAG (StUNE) PHlA~3iEE=DIG*ATAN2 (AtIPIAMPR) AHP!.I=PIT* (ANlPR*AC4PR+AMPl*AMiPl) SCATE=10.4'ALCGIO (AMPR) AtIPR=REAl (SUMM) ANlPIAIMlAG (SUtIN) PFIASE'M=DIG*ATANll2 (AIPI,ANMM) ANIPfl=PIT* (AMPR,;,AMPR +AMP.I*ANMPI) SCATM=10.*ALCGiO0 (AMPR) A IP hR =IZ 7AL (DEL) AMPI.DAIMAG (DEL) PHASzHD=DlG*ATAlT2 (AMPI,AMPR) &.1P I= P I T (.AQiPR* IPR +AMPI-'.AMPI) SCATD=10.*ALCG1O (AMPR) AlPR= REA~L (SUMK) AMlP I=AINAlG (StUNK) Pl'l.AS]jK=DIG*ATAT12 (AMPI,ANlPR) A.MPR?=P&IT* (AtiPR* AMPR *AMPI*AMPI) SCATK=10.*ALCGIO0(AllPB) PEISET=DlG*ATAN2 (AMPI, ANIPR) Alr9=?T* (A?.iR*AlIPR+AMPI*ANPI) SCATT=1O.*ALCG1O0(AMIPH) WRIT_'Z(6,90OO) THEvSCiATEPiASEESCATM,,PIIASEMsSCAT&DPHASED, &SC%.ATK, PHASEK, SCAT'ZPHASET GO TO 75 130

105 PZSFAC=ZSFAC P WA VE= WA VE PZEFAC=ZEFAC PZM FAC=ZMFAC IF' (MORE.EQ.0) GO TO 5 READ (5,,20C) MOREIPOLWAVEZSFACZEFACZMPAC IF(ZSFAC.EQ.0) ZSFAC=1.E'-10 IF(ZEFAC.EQ.0) ZE-FAC=1.E-.10 IF (ZMFAC. EQ.0) ZHFAC 1. E-10 WRITE (6,150) ID GO TO 40 100 FORM1AT (18A14) 150 FORMAT (1lHl1,18A4) 200 FORMAT (12,394F10.5) 210 FORNAT (I2v3X,'4Fl0.5) 250 FORMAT(1H,2I3,L4F8.4I,1X,,2Fll.3,rF9.4,F9.3) 300 FORM-AT(////18H0IM-PECA3CE SURFACEl;INCIDENT FIIELD DIR-ECTIONU',F7.2) 350 FORMAT(171HOAB-SORBER SURFACE,';INCIDENT FIELD DIRTECTION=,rF7.2) 325 FORMAT(8H0 I SEGL4X,1HX,7X,1HY,7X,1HlS,6X,3HDSQ, 375 FO R MAT (8 H0I SEG,?4X,1HXv7X,1HY,7X,lHS,16X,3HDSQ, &11Xf10H —" ZE.I -#13X,10fl-'-' ZiI —,r5X,7HMOD(JE),r2X,7HARG(JE),2X, &7H4'6iOD (JII) v 2X,7HAR.G (J?9.)/) 3 95 FORMAT (1 H, 2I3,LsF8. 4,2 (l:,2Fl11.3)v2 (F9. 4, F9. 3)) 400Q FORMAT (//2S.Xg14HKEY PARAMETERS// &lOXs2l HINCIO)EIT POLARIZATICNr 18Xv,IAl/ &lOXr24lHSURFACE IMPELANCE FACTORoF22.5/ &10X,25HELECTPIC.- IMPEDAYCE FLCTOR,F21.5/ &lOXe25HMAGNtJETIC IMPUEDAINCE FACTOPF21.5/ &10X,K~3LHT0TAL NUMBER OF POINTS ON THE BODYv16/ &l0Xs23iHNUMBER OF SEGMIENTS USEPDo117/ &io:~l,r35HNU;'!B ER OF INCIDENT FIELD DIRECTrIONSI5/ &lOX,29HNUMB3ER OF BISTATIC DIRECTIONSI11/ & lOX, 1OHWAVELENGTHoF34. 3) 600 FORMIAT(///,33X,28HBACKSCATTERING CROSS SECTION,!,/ &37X,,20H10*L.CG (SIGMA/LAMBDA) o/, &39X,lH(,1A1,1l4H-POLARIZATION),///, &16X, iON (ELECTRIC),7X,1OI! (,AGN"ETIC),7X,1lOH (ABSORBER),7X,t &1IH (IMPErDANCE), 8X,,71i(T-OTAL),/,,6X,5.SilITETAo,2X* &5(17H DB PHASE )/) 800 FORMAT(///,r3lX,33EIBISTATIC SCATTERING CROSS SECTION,!, &37X,2OHlC*LOG (SIGMA/LAMBDA) 0,/,, &30X,29HFOR INC-IDENT FIELD DTRECTI0N~vF6.1,/, &39Xo,lH(,1 Alf 14H-POLAPIZATIOV),// &-16X,1OH(ELECTRIC).7Xa,10HI(lAGNETIC),7X..l0H(ABS0ORflER),7X, &11H (IMPEDEANCE),8X,7H(T0TAL) v/,6X,.5HTHETLA,2X,, &5(17H DB PHASE )/) 900 FORMAT(OXF7.2,5 (lX,,2F8.2)) 999 RETURN END 131

SUBROUTINE GEOM (LUMIPXYXNYNS,S,DSQZEZMK,MH) C**** RANVS VERSION COMPLEX ZE1~,ZEBvqZMAZMBoZFUNrZE (1) s,ZM(1) DIM ENS ION X (1),Y (1),XN (1),YN (1),D SQ (1),S(1) DIMENSION LUMP(2*1) COMMON/ PIES/ PITr l,PIT,PIPItYZ IRED sDIG I=0 K=0!=0 m=0 WRITE (6,500) C.. READ INPUT PARAMETERS AND PREPARE TO GENERATE SAMPLING POINTS C.....IF TYPEP=l ABlSORBER SHEETe M CELLS TOTAL C...IF TYPE=2 IMPEDANCE SURFACE, K CELLS TOTAL C...TYPE7=1 SURFACE MUST, BE READ IN FIRST 10 READ (5,200) ITYPEvNXAYAXBvYBANG IF (N.EQ.0) GO TO 120 LIM=2*N-1 READ (5,25C) IZEFRMZEAvZEBZE~X IF (ITYPE.EQ.1) READ (5,250) IZHFRM,,ZMAZNBoZHX TX=XB-XA TY=YB-Y~A D=SQRT (TX*TX+TY*TY) IF (ANG.EQ.0.0) GO TO 20 T=0. 5*RED*ANG TRX=TX+TY/TAN (T) TRY =TY-TX/"TAN (T) RAD=0.5*1751IN(T) ARC=2. 0*RAD*T ALF= T/N DID=2. 0*RAD*ALF GO TO 30 20 RAD=999. ARC-D DID=D/N C...START GENEPATING 30 LAST=2 IF (YA. EQ.0.O.AND.YB.EQ.O.0.AND.ANG.EQ.O.0) LAST=l DO 110 JIM=1,LAST L=L+l DO 100 J=1,LIMr2 1=1+1 LUMP (2,1)=t LUMP (1,I)=I IF (I.EQ.1000) WRITE (6,400) IF (JIM.EQ.2) GO TO 90 IF (ANG.EQ.0.0) GO TO 40 SINO=Sl"I (J*ALF) COSQ=COS (J4 ALF) X (I) =XA4O. 5* (TR10X* (1.0-COSQ) -TRY*SINQ) Y (I) =YA+0.5* (TRX*SINQ*TRY* (1.0-COSQ)) XN(I)=-0.5*(UTRX*COSQ+TRY*rSINQ)/RAD YN (I) = 0.5*.(TRX*SINQ..TRY*COSQ)/RAo GO TO 50 40 X(I)=XA+C.5*J*TX/N Y (I)=YA+O. 51J*TY/N XN (I) =-TY/D YN(I)= TX/D 50 ST=0.5*J*DID 5(I) 5t 132

C....COMPUTE 1IHE ELECTRIC PAPAMETERS LF(.TTYPE.EQ.1) GO TO 60 C...ZS IS STORED IN THE ZE VECTOR Z?(I)=ZFUN(IZEFRM,rZEA,pZEi,ZEX,,ST) GO TO '100 60 ZH (TI)=ZFfUU1 (IZ,.!F'&RiM, ZtlAD N,ZM1ZiX,,ST) ZE(IV)=ZFUN(IZI-I'rN,vZtA,ZE2,OZEX,,ST) GO TO 100 C...FROM HERE T10 100 WE CREATE THE SEGMENT IMAGE 90 K=I-N X (I) =XC (K) Y (I) =-Y (K) XN(I)=XN(K) YN (I)=-YN (K) S (I =S (K) ZR (I) =ZE (K) ZN (I) =ZM (K) 100 DSQ(I)=DID IF (JIM.EQ.1) GO TO 102 Y A=-Y A YD'=-YB 102 IF(ITYPE.EQ.l) GO TO 105 WRITE (6,300) LN,t,XA,,YAXBYBANGRADARCIZEFRMZEAZEDZEX GO TO 110 105 WPITE(6f350) LNXA,,YAXB.,YBANGRAD,ARCrIZERM,ZEAZEBZEX WRITE(6,351) IZMFRIrIZNA*ZVMDZMX M= I 110 CONTINUE GO TO 10 200 FORMAT (I2,I3r5Fl0.5) 250 FO R -NAT (2, 3 X, 5 F10. 5) 300 F01,11AT0(11 r12t5H1 IMPI~slIX,4F9.L4,IX,2F7.2,.F7.3,114,IX,2F9.3,2X, & 2F 9.3, 1O. 3) 350 FOlRMAT&(111 j,12,511 ABSI~,1XL4F9.4,1X,2F7.2,F7.3,I4,1X,2F9.3,2X, & 2 F 9.3 r F 1 0 3 ) 351 FORtMAT(7lXo,14,2H (,F8.3,IlX,,F8.3,311) (,vF8.3,1lXF8.3,r21) (I &F~3. 3,111)) L400 FOIUMAT (37HOWARNING: WE'VE GENERATED 1000 POINTS/) 500 FORI 3A T I1 3 HO0SEPG SRG NUM,3X,,611 -,21HBNDPOTJ!TS OF SEGNENTSp6H &- 5X,27H,18UISEGMENT PARAIElTlERs.,2711 /1411 NUll TYP CELLS r4X, 2IiXA, 7X r2HYA, 7X, &2flX13o7X,2HYs,6X,24fIANGJE RADIUS LENGTH FORli,6,X,8HZZA(ZI4A),12X, &8HZE B (ZMB), 6X, 8HZEX(%NX) A 120 RE-,TURN END C**** RAMVS VERSION COMPLEX FUNCTION ZFUN(IFORFiZA,7ZB,ZEXvST) COM4PLEX ZAZB IF(IFORM) 10,15,20 10 ZFUN=CEXP (ZEX*CLOG (ZA..Z1*ST)) RETURN 15 ZFUN=3 A4'ZB*ST**ZEX RETURN 20 ZFUN=ZA+ZD*EXP (-ZEX*ST) RETURN VE ND 133

SUBROUTINE MTXEL1 (MI,M,K,XKX,Y,XN,YN,DSQ,ZE,ZM,A) C**** RA1MVS VERSION E-POI ARIZATION DIMENSION X (1),Y (1),XN (1),YN (1),DSQ (1) COMPLEX ZE (1),ZM (1),A(MI,1) COMPLEX AA,HZ,HZA, HZ, H1, H1A, H1B REAL NPDP,NDR,NDNP COMION/PIES/PI, TPI,PIT,PIPI,YZRED,DIG M1=M+1 MK=M+K WAVE=TPI/XK DO 300 II=1,M 1=11 IM=II+M XI=X (I) YI=Y (I) XNI=XN (I) YNI=YN (I) C.....GENERATE ELEMENTS IN 1,2,4, AND 5 DO 100 JJ=1,M J=JJ JM=J+H DS=DSQ (J) PDS=1./PIPI/DS DDS=0. 25*DS*DS TPIDS=TPI*DS TEST=TPIDS*2.5 TDS=TPIDS/24. PITDS=PIT*DS IF (I.EQ.J) GO TO 120 CALL DIST (XI,YI,X(J),Y (J),XNI, YTI,XN (J) YN (J),5, &R, NPDR, NDR,NDNP,SDR,SPDR) RK=R*XK IF(RK*.LE.TEST) GO TO 110 CALL HA1KZ1 (RK,2,HZ,H1) A(I,J) =HZ*PITDS AA=H1 *PITDS A (I,JM) =YZ*CMPLX (0.,NPDR) *AA A (IM, J) =C.PLX (0.,ND) *AA B=llPDR*tDR AA==BL*[IZ (SPDP*SDR-B)*H1/RK A(IM,JM)= AA*PITDS4YZ GO TO 100 110 CONTINUE R W=R/UAVE B=R W*R W+DDS C=DS*tl *SPDR RAK=TPI*SQRT (B+C) IF (RAK.NE.BBK) GO TO 103 HZA=IIZB GO TO 105 103 CALL HANKZ1 (RAK,2,HZA,H1A) 105 RBK=TPI*SQFT (B-C) RK=R*XK CALL tlAI'KZ1 (RK,0,HZ,H1) CALL HANKZ1 (RBK,2,HZB,H1B) AA-ITDS* (HZA+4. *HZ+HZB) A (I J) =-AA A (IM,Ji)= (0. 25*SDR* (H1B-H1A) +NIDNP*AA) *YZ AA=CMPLX (0. —0.25) *SPDR* (HZB-HZA) 134

A (I,JM) =AA*NPDR*YZ A (IM,,J) =AA*NDR GO TO 100 120 AA=CM4PLX(PIT,,ALOG(DS)+0.02879837) A (I,J) =ZEW()+AA*DS A (Ij,JM)=(0..0.) A (IlloJ) =(0..O.) A ( I, J L) =(ZM (3) +CMPLX (0. vPDS) + (AA'-CtIPLX (0.,0.5) )*DS/2.) *YZ 100 CONTINUE Co..GENER~ATE ELEMENTS IV 3 AND 6 DO 300 JJ=M1,rK JM=J+M PITDS=PI'E,*DSQ (J) CALL DIST (XIYIrX (J),Y (J) XNIYNI,~XlN(J),YN (J)u4,r &RNPD~vE. NlNDIIPSDESPDR) RK=R*XK CALL HANKZ1(BK,2,.HZoHl) C ***REMEM1BER ZS IS STORED IN ZEI*** HZA=ZE (J) AA=HZ-CMIPLX (0.,NPDR) *H11*HZA A (IJMf) =AA*PITDS B=I PD R *ND R AA,=8'HZ+ (SPDPRcSDR-B)*Hl/RK AA=CMPLX (0.,NDR) *Hl-AA*HZA 300 A(IM,Jtl)=AA*PITDS DO 500 II=MlMK X=X (I) YI=Y (I) XIII=XN (I) YNITYN (I C.(..EINERATE ELEMENTS IV 7 AND 8 DO 400 JJ=1lii J=,3 3 PITTDS=PII*DSQ (3) CALL DIST (XIYIpX (3),Y (3),XlIIIYNIXN (3),YN (3) o0, &RrNPDRf`NDRvNDNPpSDRvSPDII) RK=R*XK CALL HANKZ1 (RK,,2,,IIZpHl) A (IM4,J) =BZ*PITDS L4QQ A (IMJ+,M)=CMPLX (0.,rNPDR~) *,H*PITDS*YZ C...GEIIJERA'IE ELEMENTS III 9 DO 500 JJ=MlMK IF (I.EQ.J) GO TO 510 CALL DIST(XIYIX(J) 4Y(3) vXl~lIYNIrXN(J),YN(J) t0, &R, NPDR, Nr~2NDNPvSDRpSPDP) RK=R*XK CALL FHAlKZ1(PK,,2rfHZB1) AA=1]Z-CI"PLX (0.,NPDR) *Hl*ZE (3) A (IM,J+Ui) =PIT*AA*DSQ (3) GO TO 500 5 10 DS = SQ(J) A (I M, 3+ M) =0. 5 *ZE (J) +DS*CMPLX (PIT, ALOG (DS) +0.02879837) 500 COINT IN UE It ET UBN ELID 135

SUBROUTINE MTXEL2 (MIMKeXK.,XYXNYNDSQZEZiIoA) C**** RAMVS VERSION H-POIARIZATION C**** RAMWVS VERSICN COMPLEX ZE (1).,ZX' (1),A (MI, 1) COMPLEX A-AHZHZArliZE, H1,HlA, fiB REAL NPDRNDRNDHP` COMMON/PIES/U, TPI,PIT, PIPIYZ,RED, DIG M1=M+1 MK=M+K WA VE=TPI/XK DO 300 11=1,11 XI=X (I) YI=Y (I) XNI=XN (I) YNI=YN (I) C...GENERATE ELEMENTS IN 1,2,11, AND 5 DO 100 JJ1lM JM=J+Ml DS=DSQ (J) PDS=I1./PIPI/DS DDS=0. 25*DS*DS TLPIDS=TPI*DS TEST=TPIDS*2. 5 ~EDS=TPIDS/241. PITDS=PIT*~DS IF (I.EQ.J) GO TO 120 CALL DIS.-T(XIYIX(J),FY(J),XNIYNIXN(J),FYN(J),5,, &R, NPDRoNfLRrNDNP,,SDR,jSPDR) RK=R*XK IF(RXK.TLE.~EST) GO TO 110 CALL HANKZ1(RK2,2,HZoHl) A (I,J) =IIZ*PIIDS*YZ AA=1H11 *PITDS A (I,JM) =CMPLX (0.,-NPDR) *AA A (I1J CHFLX (0.,NDR) *AA*YZ B=NPDP *NDR AA=B*HZ+ (SPDR*SiDR-B) *Hl/RK A (I M, JM) =~-AA*PITDS GO TO 100 110 CONTINUE RW=R/WAVE B=R W * fW-+DD S C=DS*RW*SPDR RAK=TPI*SQQRT (B+C) IF (RAK.NE.IRIK) GO TO 103 HlZA=FZB l 1 A=Hi1 B GO TO 105 103 CALL HIAZKZI(RAK,2,H1ZAH1A) 105 RBK=TPI*SQEiT(B.C) RK=R*XK CALL HAN.KZ1(RK,,O,,HZ,,H1) CALL HANKZI (PBK,2,rHZEH113) AA=IDS * (HiZA+4. *IiZI.IIZB) A (I,J) =AA*YZ A (IMl,JMi) =-0.25*SDR-* (HlD-lilA) -NDNP*AA '136

AA=CMIPLX (0. 0.25) *SPDR* (HZB'-[ZA) A (IJ,JM) =AA*NIPDR A(1f4,J)=~-AA*ND'R*YZ GO TO 100 120 AA=CMPLX(P1T,ALOG(DS)+0.02879837) A (I.J) = (Z:1- (J) +AA*DS) *YZ A (I,J11) = (0. g.0.) A (IM,J)=(0.,.0.) A (IM,JM) =-ZE (J) -CMPLX (0.,PDS) -(AA-CMPLX (0. 10.5)) *DS/2. 100 CONTINUE C... GENERATE ELEMENTS IN 3 AND 6 DO 300 JJ=ElMK JM=J+M PITDS=PIIt*DSQ (.) CALL DIST (XIYIX (J)vY (J) XNI,YNI.,XN (J) YN (J),t4, SR. NPDRI1DR,NDNPvSDRrSPDR) RK=R*XK CALL EIIANKZl(PK2,2HZ,Hl) C ***REvMEMBER ZS IS STORED IN ZE*** HZA=ZE (J) AA=HZ*HZ.A-CMPLX (0.,NPDR) *H1 A (I,J-) =AA*PITDS B =N PD P*NDR AA=B*H~Z+ (SPDP*SD.R-B)*~Hl/RK AA=CMPLX (0..NDR) *Ul*HZA-AA 300 A(IMI,J:1I'=AA*PIT-DS DO 500 1I=Ii1,MK XI=X (I) YI=Y (I) X NI=X N (I) YNI=YN (1) C*.....GENERATE ELEMENTS IN 7 AND 8 DO 400.J31,M PITDS=PI~*DSQ (J) CALL DIST (X"&I,YI.X (.3),Y (.3),XITI,YNIXN (.3),YN (.3) o0, SR. NPDRNDRmIIDNP*SDR,SPDR) RK=R*XK CALL HANKZ1 (R.K*2siZ,H1) A (ILI,J) =[iZ-*PITDS*YZ 400 A (MJ+N1) =CMPLX (0.,NPDR) *1I1*PITDS C...GEWNLIRATE ELEMENTS IN 9 DO 500 JJ=M1,MK.3=.3. IF (I.EQ.J) GO TO 510 CALL DTST (XIYIX (J),Y (J),Xll,YNI,XN (J),YN(J) r0, SPrNPDR, NDRvNDNP,SDL~,SPDR) RK=R*XK CALL 1fANKZI1(RK,2,[IZ,H1) AA=HZ*ZE (3),-CMPLX (O.,tNPDR)*111 A =I'J~)PIT*AA*DSQ (J) GO rO 500 510 DS=DSQ(J) A (ILMi,J~i'!)=0.5.-DS*CM-PLX (PIT,ANLOG (DS) +0.02879837)*~ZE.(.3) 500 CONTINUE RETURN END 137

SUi3-ROUTI&NE DIST (X1,YIXJYJsXNIvNIXNjsyNJI, &R, NPDR, NDRvNDNI? SDR,SPDR) C I =0'RvNPDR C....1=l R,NIPDR,NDR C....1=2 R,NP1AiNERvNDNP Cas* 1I=3 R NDNPvSDR,SPDR Coeo*9=4 R*NPDP,NDR, SDRvSPDR C....1=5 R,NPDR,NrR,NDNP,SqDRvSPDR RE-TAL UIPDB,,NrR,UIDNP IF (I.LT.O.OR.I.GT.5) GO TO 50 RX=XI-XJ RY=YI-YJ R=SQRT (RX*RX+RY*RY) IF (I.EQ..3) GO TO 10 NPDP= (RX*XNJ+RY*YIJ) /R IF (I.EQ.0) IiETURN NDR= (RX*XNI+RY*YNI) /R IF (IE.1 ETURN IF(I.EQ.4) GO TO 15 10 NDNP=XNII*XNJ+YNTI*TNJ 15 IF (I. EQ. 2) RETURN SDR = (RVX*YNI-R Y *XNI) /11 SPDR= (PX*YNJ]-RY*XNJ)/RRET URN 50 WiRITE(6,9O0) ER 90 FORMAT (31H0SICK( DATA IN DIST *QUIT* 1=,12e 2X,2T1R=,-E11.3) CALL SYSTEM ENiD SUD-PCUTI1NE HfAi4KZ ICR,U, EIZERO, HOllNE) C..... HANKEL FUNaCTIONS ARE OF FIRST KIlJD —J+IY C........N=O RETURNS HZERO C........N=l RETURNS HONE C........N=2 RETURNS HZERO AND HONE C...SUBROUTINE REQUIRES R>0 C...SUBROUTINE ADAM MUST BE SUPPLIED BY USER DIMENSION A(7),B(7),C(7),D(7),E(7/),r?(7),G(7),H(7) COMPLEX II'ZEiROUlONE DATA ABCDE,?,GH/l.0v-2. 24999097, 1.2656208,-0.3163866q &0.0i444479,-O.00394144,O.00021lO.36-746691,O. 60559366,-0.74~35038'4, &0. 253001-17,-0.0426 12114,0. 004279 16,-0.0 000248461,0. 5,-0. 562149985, &0.21093573,-0.-03954289,0.004L43319, —0.0003176lO.00001109I &-0.63-66198,0. 22l2091,2. 1682709,-1..31614827,O. 3123951,-0.0400976, &0.002787-3,C.797881456,-0.00000077,-0.0055s274,-o.00009512m, &0.O0137237,r-O.00072805.,0.00014476,-0.785309816,-0.04166397, &-0.00003CL540.002602573,-0.000514125,o-0.o0029333,0.000 13558,. &0.7 9788456,. 00000l56, 0. 0l659667,0. 000l7l05,-0. 00249511, &0. 0011365 3,-0. 00020033,-2. 356 19449,0. 124996 12,O.0000565, &-0.00637879,0.00074s348,0.00079824,-0.00029166/ IF (R.LE.O.O) GO TO 50 IF (1N.LT.O.OR.N.GT.2) GO TO 50 IF R. GT. 3. C) GO TO 20 X=R*i1/9. 0 IF (N,.EQ.1) GO TO 10 C A.L L A D AM (ArX, B J) CALL ADAM (BXY) BY=0. 6366 198*ALOG (0.5*R) *BlJ+Y HZ7ERO:)=C[IPLX (f3J, BY) IP' (Li.PQ.0) RETURN 138

10 CALL ADAMICXY) BJ=R*Y CALL A DAM (D, X, Y) BY=0.6366198*ALOG(0.5*,R) *BJ+Y/R HONE=CMPLX (13J,BY) RETURN 20 X=3.0/Rl IF (N.EQ.1) GO TO 30 CALL ADALM4 (ErX,,Y) FOOL=Y/3Q1'lT UR) CALL ADAM (FXY) BJ=FOOLAr*COS (T) BYPFOOL*STN~ (T) HZERO=CM.PLX (BJ,,BY) IF (N.EQ.0) IETUIIN 30 CALL ADAM (GXoY) FOOL=Y/SCPT (R) CALL AD AM~ (HU, XY) T= R +Y BJI'OO0L*-COS(T) BY=FOOL* SIN CT) [iOTE=CrAPLX (BJ,BY) RETOR N 50 WP IT E(6,p90) N s R 90 FOR!,IAT (32H0SICK DATA IN HANI(Z1 CALL SYSTEM ENi' D *QUIT* N=,I2,2X,2HR=,Ell.3) SUBROUTINE ADA'M (C#X*Y) DIMENSION C(7) Y=X*C (7) DO 10 I=1,5 10 Y=X*(C(7-I)+Y) Y=Y+C (1) RETURN END SUJBROUTINE FLIP (ANNIvLMXrYXAT) COMPLEX A (MIIel),X (1),Y(1) DBIGA, HOLD DIMENSION L (1),M(1) IF (IAT.GT. 1) GO TO 150 D=CMPLX (1. 0,0. 0) DO 80 K=1,N L (K)=~K M (K) =K BIG AA KK). DO 20 J=K,N DO 20 I=-KrN 10 IF (CAB3S(BIGA) GE. CABS (A (IJ)) GO TO 20 BIGA=A (IoJ) L (K) =I M (K)=~J 20 CONTINUE J=L (K) IF (J.LE".K) GO TO 35 DO 30 I~lN HOLD= —A(K,I) A (K,I) =A (J,I) 30 A(J,TI)=HOLD 35 I=K4(K) IF (I.LE.K) GO TO 45 DO 40 Jz1.N HOLD=A (J,K) 139

40 A(J,I)=HCLD Li 5 IF (CABS (fIGA). NE. 0.0) GO TO 50 D=Ct',PLX (0.0,0.0O) 5 0 DO 55 I=1,N IF (I.EQ.K) GO TO 55 A (I,K)~-A (I,,K)/BIGA 55 CONTINUE DO 65 I1,N DO 65 J=loN IF (I.EQ.K.OR.J.EQ.K) GO TO 65 A (I, J) = A (I, K) *A (K,.J) + A (I, J) 65 CONTINUE DO 75 J=lN IF (J.EQ.K) GO TO 75 A (K,.J) =A (K,.J) /BIGA 75 CONTINUE D=D*BIGA 80 A(K,,K)=1./BTGA K=N 100 K=K-1 IF (K.LE.0) GO TO 150 I=L(K) IF (I.LE.K) GO TO 120 Do 110 J=1,N HOLD=A(J,K) A (J,K)=-A (J,I) 110 A(J,I)=HOLD 120 J=M(K) IF' (J.LE.K) GO TO 100 DO 130 I=lN HOLD=A(KIl) A (X~,)) =A (J, I) '130 A(%j I)HOLD GO TO 100 150 DO 200 I~ltN Y (T) =C riPLX (0. 0,0. 0) DO 200 J~lpN PETUR N END BLOCK DATA C0LntON/PIES/PITPIPITPIPIYZ,R1ED,DIG DATA PITPOIPITPIPI,YZREDDIG/3. 1415927,6.2331853, &l1.5707963,9.8696044,0. 002652582'b 0.01745329,57.295-78/ END 140

APPENDIX B COMPUTER PROGRAM RAMD (as of January 1977) Program RAMD solves the integral equations for the currents induced on a two dimensional body subject to an impedance boundary condition when illuminated by a plane E or H polarized electromagnetic wave, and then computes the scattered field. The equations on which the program is based are (Knott and Senior, 1974) i ~rf? (1) t A H (s) = K(s)+ lim 1 r (s)r)H ( kr)-i(n' r) (kr)K (s)d(ks) (B. 1) Z 0 1S for H polarization and i 1 (1' ~ Y E = r(s) K (s) + lim 1 fH)(r) - ir( H )(kr ))K (s')d(ks') z z p 4 k -o - (s) z (B.2) for E polarization. When the limit and integral operations are interchanged, the integrands become singular when the field and integration points coincide, and although the singularities are integrable, for a numerical solution it is still necessary to evaluate the self cell contributions analytically. The forms used in the digital solution are actually as follows: H(s).= 0 + 5 ++ i (0. 0287985 + In n(s) K (s) Z 47 A 42 7r ()HdsI A' kr) ds' ()3) + - n(s) K (s ) H( (kr)ds' - Ks( )( r)1(kr)ds (B.3) C- 'C-A and Y Ez 5 )ri(s) + + i(0. 0287985 + In K (s) 0 z 47 2 K(5)H + K ( H kr)ds - (s )K (s )('. ) H(1(kr) dsr C-A C (B. 4) (B. 4) 141

for H and E polarizations respectively. In each equation the first term on the right hand side is the contribution from the self cell of size A, and the integrals then represent the contribution from the rest of the body. The only difference from the equations used in the original program RAMD is in the self cell terms, and instead of writing a new program we have chosen to modify the old one. The changes made are (i) inclusion of a curvature effect contribution (ii) addition of a number of impedance generation subroutines designated ZFUN and (iii) some cleaning up to make the program simple and more versatile. The added term representing the effect of the curvature of the self cell is 4~ where a is the angle in radians subtended at the center of curvature. Although its expression is simple the need for such a term became apparent only when the results obtained with RAMD for a perfectly conducting circular cylinder were compared with those computed from the Mie series expression. An analytical derivation of this term is given by Senior (1976), and its inclusion in the program markedly improved the accuracy achieved for a given number of sampling points. Since the magnitude of the correction increases with increasing curvature, the improvement is most dramatic for bodies of small radius, for example, a cylinder having ka = 1. It is relatively unimportant for a body such as the ogival cylinder of interest in this investigation, and the results previously obtained for this geometry are in fact still valid. The second modification to the program was the addition of numerous ZFUN subroutines to generate the surface impedance at each point when either a specific surface impedance variation, or the thickness and material properties of a homogeneous coating is given. In the main program there are options to generate an impedance having either a power law variation with distance s or an exponential form, and subroutine ZFUN is used only when a more complicated variation is 142

required. As many as 8 ZFUN's were used in the course of our study. Some compute specific impedance variations and others are directed at the case of a coating in the form of a layer. Three ZFUN's are included in the program listing to compute the impedance for (i) a cylindrical tip' impedance (ii) a 'cylindrical tip' layer thickness and (iii) a two layer coating. Of these three, only the first complements RAMD and derives all the input information from the input data. When the impedance is computed from the given thickness and material properties of a layer, more input parameters are needed than can be assigned in the input data list. Instead of changing RAMD to accept all input information, we chose to input part of the data via the original data input and the rest through the source statements in the subroutines. Thus, in the second ZFUN for a cylindrical tip layer thickness, ji (MU), c (EPS) and frequency (FREQ) are inputted as data, but the tip thickness (TTIP), the taper length S1 and the maximum layer thickness go in as source statements. Although a change in the data then requires recompiling the subroutine, the procedure is not difficult when running the program from a terminal. The other change made in the original RAMD was to remove the option for reading in the geometry and surface impedance on a point by point basis. This seemed a worthwhile option when RAMD was developed but there has proved to be little need for it. All the geometries and impedances for which RAMD has been used so far have been generated internally and in the subroutines ZFUN. A complete Fortran listing of RAMD is as follows: 143

2 C INPUT FORMAT FOR PROGRAM RAMD-B~-~ —VERSION OF AUG 5,1976 C 3 C MODIFIED NOV 18,1976 C 14 C PROGRAM NOW COMPUTES SURFACE FIELD FOR SPECIFIED ANGLE CANG C 5 C***~***~***~************~********* 6 C CARD 1 FORMAT (18A4) TITLE CARD; USE UP TO 72 COLUMNS C 7 C%.I* * ********* ****** ** *-*** *** c**** *** * ** ** ******** l* * * ** ***- * * * *z* * * -,** ** ** ** * C 8 C CARD 2 FORMAT (12vI3v2F10.5) MOREKODEZFACWfAVE C 9 C MORE=O THIS WILL BE THE LAST RUN FOR THIS DATA SET C 10 C MORE=1 THERE ARE MOP~E DATii TO BE READ FO R T11IS SET C 1-1 C KODE=O COMPUTAES BISTATIC SCArTTERING P'A'16TERII C 12 C KODE=1 COMPUTES I3ACKSCATTJ"ERING PATTERN C 13 C ZFAC A FACTOR MULTIPLYING ALL ELEPaENT IMPEDANCES C 14 C WAVE WAVELENGTH C 15 C*************4********~*********~** 16 C CARD 3- FORMAT (12vI3,4F10.5) IPPIOPToFIRSToLASTsINK,,CANG C 17 C IPP=1 E-POLARIZATION C 18 C I.PP=2 H-POIARIZATION C 19 C IOPT=O SURFACE PARAMETERS NOT PRINTED C 20 C IOPT11 SURFACE PARAMETERS PRINTED C 21 C FIRST INIETIAL SCATTELRING AND INCIDENCE ANGLE C 22 C LAST FINAL ANGLE C 23 C INK ANGULAR INCREMENT C 24 C CANG ANGLE FOR SURFACE FIELD COM1PUTATIONS C 26 C CARD 14 FORMAT (I2,13,,5F10.5c) NIMPXAYAvXBYB,.AT1GLE C 27 C N NUMBER OF SAMPLING POINTS ON THIS SEGMENT C 28CImp=-1 1ZPFDAN,1CE GIVE"Nl BY USER-SUPPLIED SUBROUTINE C 29 C IfMP=0 ZS (I) =ZA+ZB*S (I) *ZEX C 30 C IMP=1 ZS(TV=zA+zB*Exp(-zRX'*S(I)) C 31 C XA*YAvXBoYB SEGMENT JENDPOINTS C 32 C ANGLE ANGLE SUBTENDED BY THE SEGMENT C 33C******************-****** ***********C 34 C CARD 5 FORMAT (SX,5F10.5) ZArnZBE C 35 C Z A vZB CCPE. IPDNC OSTNSC 36 C ZEX REAL IMPEDANCE COUSTANT C 37C**********4************************ 38 C CARD 6 FORMAT (12) INTEGER ZEvRO IN COL-UMN 2; SHUTS C 39 C OFF READING OF SEG~IENOT PARAMETERS C 140 C**************************c********'**C 41 C CARD 7 FORMAT (12,13,2F-10.5) MOREKOOEoZFACWAVE C 142 C THIS C."ARD IS USED ONLY IF, Oil CARD 2, MORE=1 C 43C********************************c***'C 44 COMPLEX A (1OO,101) sPHI (1OO) PINK (1OO) vSU:.,DELB 13,B2tZS (10O) 145 COMPLEX ZSFAC (100) 146 REAL LASTIm( 47 DIMAENSION X (100),Y (100) X,XN(100),Y11(100),S (100),DSQ (100),rANG (100) 48 DILMENSION ID (18),LUMP (100, 2) vIPOL (2) 49 DATA REDDIGPIFrIPOI/0.017145329, 57.29578,O.07957747l5l,4HEEEE,4HHHHH/ 50 C...READ INPUT DATA AND GENEPATE BODY PROFILE 51 5 READ (5,100) ID 52 READ (5,200) MO1'RErKODEvZFACWAVE 53 READ (45,20C) IPPIOPTFIHSToLASTvINKCANG 541 WRITE, (6,150) ID 5 5 WR I TE (6, 3 00) 5 6 CALL GEOM1 (LTJMiP,,XY.XNi,YUlSDSQANGZS,MLL) Sd7 20 IF (KODE.NE.O) GO TO 25 5 8 NINC=1 59 NBIT. =1 +1FIX ((L AST- FIR S.) /INK) 60 GO TO 30 144

6 1 25 NBIT=0 62 NI.NC=] I+IFIX ( (LAST~-FIRST) /I11K) 63 30 WRITE (6, 400) IPOL (IPP) 0,ZFACLL,H,MNINIC, NBIT, WAVE 64 XK=6. 2831 85/WAVE 65 DO 35 1=1,?! 66 ZSFAC (I =ZS (I) *ZFAC 67 35 DSOQ(I) = DSQ (I) /WfAV 68 C...CONSTRUC~ MATRIX ELEMIENTS 69 3 8 DO 55 I=1,M 70 DO 55 31lM 71 IF (T.EQ.J) GO TO 40 72 TX=X(I)-X(J) 73 TY=Y(I)-Y(J) 714 P=SQRT (TX *TX+TY*TY) 75 RPQ=P*XK 76 Co....CNR= (N-PPIME DOT R) 77 CNR~- (TX *XN (J) +TY*YN (J) )/P 78 CALL HANK(PPQ,1,BJ,8BY) 79 Bl 1. 570796 *DSQ (%J) *CIIR*CMPLX(.-BY$,BJ) 80 CALL HANK ((PQaO,[3JBY) 81 B2=1. 570O796*DSQf(J) 4CflPLX(B3,DY)82 GO0TO 45 83 40 B1=Ct1PLX (0.55,0.0)+AUiG (J)*PIF 84 B2=DSQUJ) ~',CHPLX(l.570796,O.0287985+ALOG(DSQ(J))) 85 45 IF (IPP. Q. 1) GO TO 50 86 A (IJ) =B 1+82*ZSFAC (.3) 87 GO/TO 55 89 50 A (1,J3)=D2+B1*ZSFAC (J) 89 55 CONTINUE 90 C.. COM1PUTE 111CIDEENT FIELD AND INVERT ZMATRIX 91 TfETA=?EDC.-CABG 92 CT=COS('fETA) 93 ST=SIN (TiETA) 94 DO 60 I=1loM 95 -H0LD=-XK —k(CT*X (I) +ST*Y (I)) 96 60 PINK (I) =C&'!PLX (CCS (HOL,0D).,SIN (HOLD)) 97 CALL "FLIP (AM,,PIITK,,PfI, 1) 98 C...PRINT OUT CURRENTS AND ELEMENT PROPERTIES FOR FIRST ANGLE 99 IF (IOPT.EC.O) GO TO 63 100 WRITE (6,151) ID 101 WRITE(6,350) CANG 102 WRITE (6,500) 103 63 DO 65 I~lM 104 A!MP=CA3S (?HI (I) 105 Pf1ASrE=DIG*ATAX2 (AI1AG (PRI (I),REaAL (PHI (I)) 105 IF (IOPT.EQ.O) GO TO 65 107 WRITIE (6,250) (LUNP (IoJ).*J~l,2),X (1),Y (I),S (I)rDSQ (I),ALP,`P11ASE, 108 &ZSFAC (I) 109 65 CONTINUE 110 C...DOPE OUT THE APPRCPBIAITE FIELD FACTOIRS 112 I? (KODE. AzC.,l) GO TO 70 113 WRITE (6,800) CANG 114 GO TO 73 115 70 'WRITE (6,600) 116 75 T~IE=THE+I11K 117 IF (THE.GT.LAST) GO TO 105 '118 IF (THE.EQ.FIRST.AND. FIRST.EQ.CANG) GO TO 85 119 TETA=RrD*'THE 120 CT=C0S(T7TA) 145

121 ST=SIN(TETA) 122 C.....IN THE FOLLOWIiUG LOOP, PINK IS NOT NECESSARILY THE INCIDENT FIELD 123 DO 80 J=1,PM 124 HOLD=-XK* (CT*X (J) +ST*Y (3)) 125 80 PINK (3) =CVPLX (C05 (HOCLD),SIlN (HOLD)) 126 IF (KODE.EC.0) GO TO 85 127 CALL FL P (A, MPI NIK,PHI, 2) 128 85 SUM =C MPL X (O.OO-.O) 129 C. ADD UP THE CURRENTS 130 DO 95 J=1#M 131 DEL=DSQ (3) *Pfll (J) *PINK~ (J) 132 IF (IPP.EQ.1) GO TO 90 133 SUM1=SUMl+DEL* ( ZSFAC(.1) -CT*XN (J) ST*YN (3)) 1341 GO TO 95 135 90 StJM=SULH-+DEL* (1.0- ZSFAC (J) *(CT*XN (J) +ST*YN (J)) 136 95 CONTINUE 137 SCAT=20.C*ALOG10(CAt3S(SUM~))+1.9612 138 FASE=l180O0+DIG*ATAN2 (AINIAG (SUM),-REAL (SUM1) 139 WRITE (6,900) THESCATeFASE 140 GO TO 75 141 105 IF (MOPE.EQ.0) GO TO 5 142 DO 108 I=10H 143 108 DSQ(I)=DSQ(I) *WAVE 144 READ (5, 200) MORErKODEoZFAC,9AV.E 145 WRITE(6,150) ID 146 GO TO 30 1147 400 FORMAT (18A4) 148 150 FORMAT (lHl,18A4) 1 49 -151 FOIRMAT (1H0, 18A4) 150 200 FLORnAT 12,,13,5F10.53) 151 250 FORMAT (2l5,5Fl0.5,Fl0.3,2Fl0.5) 152 300 FORM~AT (lOHOSEG Nt~elll1X,2L4HENDPOINTS OF THE SEG~JENTI1X, 153 & SrEGnENT PARANETERS 154 6~ ---'18NUM CELlS,6X,2HX~,8X,2HYA,8X,2HXB,8X, 155 &2HY8,6X,2lHANGLE RADIUSLEGT12FO3,0,A'6i 156 &'ZB',14Xp'ZXI) 157 350 FORMAT (IHO v ISUR FACE IMPEDANCE; INCIDENT FIELD DIRECTION=IsF7.2) 158 400 FORMAT (//25X,14HfKEY PARAMETERS// 159 & 1OX, 2 1 1INCIDENT POLARIZATION o2 2X r1A 1/ 160 &lOXo'SfURFACE IMPEDM~CE FACTUR'l,?20.5/ 161 0 IOX,23HfNtMBER OF SEEGNENTS USED, 121/ 162 &lOXr3!!HTOTAL NUMBER OF POINTS ON THE BODYtIl0/ 163 &l0X,,35HiNUM'BFR OF? INCIDENT FIELD DIRECTIONSs,19/ 164;S10X,,29fINUM DER OF BISTATIC DIRICXOS,15 165 & 10X,I OUfW A V ELBN G T HF 3 4. 5) 166 500 FORNAT (lIHO0 I SEGo,4XL$HX(I)j,6X,4HY(I),r6X,4HS(I),r5X.,r6HDSQ(I),, 167 &4X,,6H-lIOD(J),'4X,,6HAPG(J),4X,5r-'RS(I),5X,,5HXS(I)/) 168 600 FORMAI! (///,,20X,K28[BACKSCATTE-RING CROSS Sr.ECTIOti/17X, 16 9 &36HTHETA 10*LOG (SIGi3,A/LArMDDA) PH1ASE/) 170 800 FOR"IAT (///,,19Xo33JHlISTATIC SCATTERING C2OSS SECTIOtl/18X, 171 &29HFvOR I M'JIDENT FTIED DIRECTION=,722x 172 &28EiTHETA 10*L0G (SIGlA/LAMI3DA)/) 173 900 FO R iiAT (9Xa, F 13. 2,rFI. 2,rF16. 1) 174 END 146

175 SUBROUTINE GEOM (LUcIPXYXNYNSDSQsANGZS,U,Ll,) 176 COMIPLEX ZS (100) IZAoZB 177 DIMENSION X (100),Y (100),XN (100),rYlH(100) tDSQ (100),S (100),'ANG (100) 178 DIA1ENSION IJ~IP(100,2) 179 DATA RED/O.017L45329/ 180 I=0 181 L=O 182 C...READ INPUT PARAMETEBS AND PREPARE TO GENERATE SAMPLING POINTS 183 10 READ (5,200) NIMPXAaYAXBvYBANGLE 184 IF (N.EQ.0) GO TO 120 185 LIM=2*N-1 186 READ (5,250) ZAeZBpZEX 187 TX=XB'-XA 189 TY=YB-YA 1.89 D=SQRT('rX*TX+TY*TY) 190 IF (ANGLE.EQ.0.0) GO TO 20 191 T=0.5*REE*ANGLE 192 TRX=TX+TY/TAN(T) 193 TRY=TY-TX/TAN (T) 194 RAD=0. 5*D,/SIN' (T) 195 ARC=2.0*RAD*T 196 ALF=T/N 197 DID=2.0*RAD*ALF 198 GO TO 30 199 20 RAD=999.999 200 ARC=D 20 1 DIED=D/N 2 02 C...STAflT GENERATILNG 203 30 DO 110 JIH=1,2 2104 L=L+l 205 DO 100 J~1,LIMo2 206 1=1+1 A2,07 J.UNlP (I,1) =1 208 UJ1P (I, 2) =I 209 1? (JI~i.EQ.2) GO TO 90 210 I F (A N GL Z.E.B.0.0) GO TO 40 211 StiNQ=SIlN (J*ALF) 212 CosQ~cos (J*ALF) 213 X(I)=XA+0.5*(TRX*(1.0.'COSQ)-TRY*SINQ) 214 Y (I) =YA+0.54 (TRX*SIN'Q*TRY*(1.0-COSQ)) 215 XlkN(I)=-.0.5 * (T7RX*C0SQ+TFY*SINQ) /RALD 216 YH (I)= 0.59*(~TRX*SIN&'Q-TRY*COSQ)/RAD 217 ANIG (I) (AN11GLE/1l) *RED 218 G0 TO 50 219 40 X(I)=XA+0.5*J*TX/N 220 Y(I)=YA+O.5*J*TY/N 221 X14(I)=-TY/D 222 YN(I)= TX/D 223 AVG (I) =(AIIGLE/N) *RED 224 50 S(I)=0.5*.J*DID 225 C.. COMPUTE THE IMPEDANCES 226 IF (I~IP) 60,700,80 227 60 CALL ZFUN(ZAoZB,9ZEXS(I),ZS(I)) 228 GO To 100 2 29 70 ' —S (I) =Z A + Z.9(I)*ZEEX 230 GO TO 100 231 80 ZS(i-)=ZA+ZB*EXP(-.ZEX*S(j)) 232 GO TO 100 2 33 C......-RO M HE RE TO 10 0 W E CR EATE TIB"C S 7:0MCEZT IMiA%'EL 234 900 K=XIbN 147

235 X(I)=X(K) 236 Y(I)=-Y(K).237 XN (I) =XN (K) 238 YM(I)=-YN(K) 239 AUNG (I) =ANG (K) 240 S(I)=S(K) 241 ZS(I)=ZS(K) 242 100 DSQ(I)=DID 243 IF (Jr-I.EQ.l) GO TO 110 244 YA=-YA 245 YB=-YB 246 110 WRITE (6,300) LNXAYAXBYBANGLE,,RADlARC, 247 &IMPZAZSZEX 248 GO TO 10 249 120 M=I 250 LL=L 251 200 FORMAT (I2o13,5F10.5) 252 250 FORMAT (5Xr5F10.5) 253 300 FORMAT (I3,63X,4F 10 5,F8.2, F8.3, F8.4I5,2X, 254 &2(FB.3,F9.3,2X)*F9.3) 255 RETURNI 256 END 257 SUBROUTINE FLIP (AU,X,YIAT) 258 COMPLEX A (100,10),X (100) rY (100)DBIG AHOLD 259 DIMENSIOt L (100),N (100) 260 IF (IAT.GT. 1) GO TO 150 261 D=CM PLX(1. O,0*.0) 262 DO 80 K=1,N 263 L(K)=K 264 tl(K)=K 265 BXGA=A(K,,K) 266 DO 20 J=KN 267 DO 20 I=K,11 268 10 IF (CABS (BIGA).GE.CABS (A (IJ))) GO TO 20 269 BIGA=A (1,J) 270 L(K)=t 271 li (K) =J 272 20 CONTINUE 273 J=L(K) 274 IF (J.LE.K) GO TO 35 275 DO 30 I=1,1 276 HOLD=-A(KI) 277 A(K,I)=A(J, I) 278 30 A(J,I)=HOLD 279 35 I=M(K) 280 IF (I.LE.K) GO TO 415 281 DO 40 J1=,N 282 HoLD=-A(JK) 283 A(JK)=A(JI) 284 40 A(J, I)=IICLD 285 45 IF (CABS(BIGA).NE. 0. 0) GO TO 50 286 D=CN PLX (0.0,o0. 0) 287 RETURN 148

288 50 DO 55 I-1,N 289 IF (I.EQ.K) GO TO 55 290 A(I,K)=-A {I,K)/BIGA 291 55 CONTINUE 292 DO 65 i=1,N 293 DO 65 J=1,N 294 IF (I.EQ.K.OR.J.EQ.K) GO TO 65 295 A(I,J) =A (I,K)*A (K,J)+A (I,J) 296 65 CONTINUE 297 DO 75 J=1,N 298 IF (J.EQ.K) GO TO 75 299 A(K,J) =A (K,J)/BIGA 300 75 CONTINUE 301 D=D*BIGA 302 80 A(K,K)=1.0/BIGA 303 K=N 304 100 K=K-1 305 IF (K.LE.O) GO TO 150 306 I=L(K) 307 IF (I.LE.K) GO TO 120 308 DO 110 J=1,N 309 HOLD=A (J,K) 310 A (J,K) =-A (J,I) 311 110 A(J,I)=HCLD 312 120 J=M (K) 313 IF (J.LE.K) GO TO 100 314 DO 130 I=1,N 315 HOLD=A(K,I) 316 A(K,I) =-A (J,I) 317 130 A(JI)V=IOLD 318 GO TO 100 319 150 DO 200 I=1,N 320 Y (I)=CPpX (.0,0.O) 321 DO 200 J=1,N 322 200 Y (I)=A (I,J) *X(J) +Y(I) 323 RETURN 324 END 325 SUBROUTINE HANK (R,N, EJ, BY) 326 C.....SUBROTITIlE REQUIRES R>0 AND N EITHER 0 OR 1 327 DIMENSION A (7), B (7),C (7),D (7),E (7),F (7),G (7), H (7) 328 DATA A, B,C,D,E, FG,G / 1. O0,-2. 2490997, 1. 26562031-0.3163866, 329 &0. 0444479,-0.00394444, C.00021,0.36746691,0.60559366,-0. 74350384, 330 60.25300117,-0.04261214,0.500427916,-0.00024846,0.5,-0.56249985, 331 80.21093573 -0.03954289T,0.C0443319,-0.00031 76 1,0.00001109, 332 S-0.6366198,0.2212091,2. 1682709,-1.3164827,0.3123951,-0.0400976, 333 80.0027873,0.79788456,-0,00000077,-0.0055274,-0.00009512, 334 80.00137237,-0.00072805, 0.00014476,-0.78539816,-0.04 166397, 335 &-0. 00O03954,0.00262573,-0.0005a125.-0.00029333,0.00013558, 336 &0.79788456,0.00000156,0.01659567, 0.00017105,-0.0024951 1, 337 &0.001 13653,-.0 C020033,-2.35619449,0. 12499612,0.0000565, 338 &-0.006 37879,0.00074-348, 0.00079824,-0.00020 166/ 339 IF (R.LE.0.0) GO TO 50 340 IF (R.GT.3.0) GO TO 20 3t41 X=R*R/9.0 342 IF (N.NE.0) GO TO 10 343 CALL ADAM (A,X,BJ) 344 CALL ADAM (B,X,Y) 345 BY=0.6366198*ALOG(0.5*R) BJ+Y 346 BETURJ 149

347 10 IF (N.N1) GO TO 50 348 CALL ADAM1(CpXvY) 349 BJ=R*Y 350 CALL ADAi4(D,,XY) 351 BY=O.6366198*ALOG(O.5c*R)*BJ+Y/R 352 RETURN 353 20 X=3.O/R 354 IF (.EO GO To 30 355 CALL ACAN4(E,,XY) 3 5 6 FOOL=Y,/SQRI (R) 357 CALL ADAM4(F,X,Y) 358 GO TO 40 359 30 IF (NI.NE.1) GO TO 50 360 CALL ADAM (GoX*Y) 361 F0OL=Y/SQflT ().36 2 CALL ADAM (IIXvY) 363 40 T=R*Y 364 BJ=FOOL*C0S IT) -36 5 BY=FOOL*SIN (T) 366 ]RETURN 367 50 N=100 368 RETURN 369 END 370 SUBROUTINE ADAtl (CX,X,Y) 371 DIMENSION C(7 372 Y=X*C (7) 373 DO 10 I~1,5.3p7 4 10 Y=X*(C(7-I)+Y) 375 Y=Y+C(1) 376 RETURN 3 77 D 378 SUBROUTINE ZFUN(ZAvZBZEXsSS,.ZZ) 379 C IMPE0DANCE FOR CYLINDRICAL TIP CONTOUR 380 C ZA- TIP VALUE (COMPLESX) 381 C ZB - MAXIMUM VALUE (COMPLEX) 382 C ZEX- JOIN POINT (REAL) 383 COMPLEX ZAoZDZZ 38 4 DATA PIT/1.50707963/ 385 ZZ=ZB 386 IF (SS.LE.ZBX) ZZ=ZA+(ZB-Z.A)*SIN((PIT*SS)/ZEX) 387 RETURN 388 END 389 SUB.ROUTINE ZFUN (MlU4,bPSFv,SZZ) 390 C IMPEDANCE FOR A CYLINDRICAL TIP 391 C MATERIAiL THICKNESS VARIATION 392 C TAPER 1 INCH LONG, MAX THICKNESS 0.050 INCH. 393 C DATA INPUT INFO: 394 C ZA =MU 395 C ZB = BPS 396 C ZEX= F, FRTWEQ IN GLIZ. 397 COMPLEX MUEPSZZXPLUSvXMINUSETA 398 DATA TOLD/-I./, PIT/1.57107963/,ANG/90./,TPI/6.283185/ 3 99 DATA RAD/.017'4533/ 40 0 C COM'-PUTE LAYE'-R THICKNESS T 150

* 40 1 S 1= 1.0 402 TTIDP=0.0 $0 3 T=0.050 40 4 IF (S. LE. SI) T=TTIP+ (O. 05-TTLIP) *SIN (PIT*S/SI1) 405 ZZ=ETA ~406 IF (TOLD.EC.T) RETURN 407 C COM4PUTE IMPEDANCE FOR GIVEN T 408 ARG=ANG*RAD 409 SIN SQ=SIN (AEG) 410 STNSQ=SlNSO*SINSQ 411 ETA=TLPI*T*F* (O.084667) *CSQRT (NU*rwPS-SI11lSQ) (0.,,-1.) 412 XPLUS-CEXP (ETA) 4 13 XMINUS= (1.,0.) /XPLUS 414 ETA=CSQRI (MU/EPS) *(XPLUS-X?1INUtS) /(XPLUS+XMINUS) 415 ETA=ETA*CSORT(l.~-SINSQ/(MtJ*EPS)) ~416 ZZ=ETA 417 TOLD=T 418 RETURN 419 END 4 20 SUBROUTINEJ ZFUN (IIUjEPSpFSZZ) 4 21 C IMD-FDANCE F'OR S**1 MATERIAL THICKNESS VARIATION,, 422 C MAX THICKNESS 0.100 INICH. 423 C DATJ'LA INPUT INFO: ~424 C ZA = MU 1125 C ZB = EPS 426 C ZEX= F, FREQ IN GHZ. 4 27 COMPLEX MUEPS,ZZ XPLUSX~lINUSETA 428 DATA TOLD/-I./, PIT/1.5707963/,ANG/90./,TPI/6.283185/ 4 29 DATAA PAD/.0174533/ 430 C COM'PUTZ THIlCK"NESS 431 T=O.1*S/4.9912 4 32 AllG=A N G PA D 433 C CO:1PUTE ItIP"ADANCE 434 SIN SQ=SIN (APG) 435 SI, SQ=-SIU~SQ*SINSQ 4 36 ETA~=TPI*T*F* (0.084667) *CSQRT(MU*ES-SINSQ)*(0.jp-l.) 437 XPLUS=CEXP(ETA) 438 XklINUS=(l.,0.)/:(PLUS 439 ET-rA=CSQRT (MU/"PS) *(XAPLUS-XUIl1TUS)/(XPLUS+XMINUS) 440 E&7TA=EFTA*CSQRT (1.-SINSQ/ (tIU*EPS)) 441 ZZ=ETTA 442 TOLDZT 443 RETURN 444 END 151