015224-1-T ANALYSES PERTAINING TO THE REDUCTION OF NON-SPECULAR SCATTERING Technical Report by T. B. A. Senior Radiation Laboratory The University of Michigan Ann Arbor, MI 48109 Prepared for Air Force Office of Scientific Research under Grant No. 77-3188 ModA December 1977

015224-1-T ABSTRACT An important problem in connection with radar absorbers is the design of absorbing coatings to reduce 'non-specular' scattering contributors such as edges, traveling and creeping waves. The present study is directed at the first two. The surface treatment is simulated using an impedance boundary condition and the main tools employed are high frequency diffraction analyses and a computer program for the direct digital solution of the integral equation for the induced current. For simplicity attention is confined to a two-dimensional problem, and the body used to illustrate the techniques is a 15~ half angle ogival cylinder illuminated by an H-polarized plane wave incident in a plane perpendicular to the generators of the cylinder. By a combination of analytical and numerical techniques, surface impedances are specified providing significant and broadband cross section reductions over given aspect ranges. i

015224-1-T TABLE OF CONTENTS Section Page No. 1. INTRODUCTION........................ 1 2. GENERAL CONSIDERATIONS................... 4 3. SCATTERING BY EDGES AND OTHER DISCONTINUITIES...... 9 3.1 Backscattering from a Wedge.............. 9 3.1.1 Half Plane................... 14 3.1.2 Wedge of Half Angle 5~............ 119 3.2 Impedance Discontinuities............... 26 4. CROSS SECTION REDUCTION OF AN OGIVAL CYLINDER...... 35 5. ACKNOWLEDGEMENT...................... 73 6. REFERENCES......................... 73 Appendix A. EDGE DIFFRACTION COEFFICIENT................ 75 B. ANALYTICAL APPROXIMATION.................. 80 ii

015224-1-T 1. INTRODUCTION The problem of reducing the backscattering cross section of a target is a rather old one, and the specification, manufacture and application of radar absorbing materials is now a well-developed art. These materials are customarily designed to reduce the backscattering over a range of frequencies for normal (or near normal) incidence on a surface, but the fact is that many targets of concern are so shaped that in the aspect ranges of most interests no surfaces are viewed at normal incidence. This is true, for example, for many aircraft at front and rear aspects, but even in the absence of specular contributions, the scattering from the wings, fuselage, etc., can still be significant. It is therefore important to seek absorbing materials which are effective in reducing 'non-specular' scattering and the present study is directed to this end. The three main types of non-specular scatterer are edges (or other discontinuities in the slope or higher derivatives of the surface), traveling waves and creeping waves. All are aspect and polarization dependent and diminish with increasing frequency. Their magnitudes are generally small compared with any specular contribution that may exist, but they can be important under all circumstances and most particularly when there is no specular scattering. Traveling and creeping waves are forms of surface wave whose propensity to cling to surfaces sets them apart from edge-diffracted contributions. The 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 1

015224-1-T of bodies whose radii of curvature are electrically large. They propagate into the shadow 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 rear of a smoothly curved body such as a circular cylinder, and on reaching the boundary of the shadow and lit regions, the energy radiated is in the backscattering direction. For a metallic body creeping waves are only an appreciable source of scattering when the incident electric field is perpendicular to the surface at the shadow boundary, but 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, combined 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. They are supported by surfaces of large radius in the direction of travel almost all of which are illuminated, and in contrast to creeping waves, they tend to build up as they travel, being fed by the incident field. 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 or its electrical properties, a portion may be reflected back to contribute to the backscattered field. The resulting field has a fan-shaped pattern characteristic of a traveling wave antenna. As in the case of creeping waves, a traveling wave contribution can be reduced by appropriately loading the surface. 2

015224-1-T The third source of non-specular scattering is edge diffraction as, for example, by the front and rear edges of an ogival cylinder. This also is strongly polarization dependent. Thus, for a plane wave at near edge-on incidence in a plane perpendicular to the axis of an ogival cylinder with its electric vector parallel to the edge (E polarization), the front edge is the primary source. On the other hand, for H polarization the rear edge is dominant, and its scattering can be reduced by the same method used for the traveling wave, but even so the front edge still contributes a small amount. It may therefore be necessary to treat the front edge as well to produce a substantial decrease in the scattering from the entire body. In three previous studies (Knott et al, 1973b; Knott and Senior, 1974; Senior and Liepa, 1977) techniques were developed for reducing non-specular scattering using surface treatments and/or resistive sheets extending out from the body. The present work is a continuation of these but is exclusively concerned with the specification of a surface treatment to reduce edge and traveling wave contributions to the backscattered field. The surface treatment is simulated using an impedance boundary condition and the main tools employed are high frequency diffraction analyses and a computer program for the direct digital solution of the integral equation for the induced current. For simplicity attention is confined to a two-dimensional problem and the model used to illustrate the techniques is a 15 degree half angle ogival cylinder illuminated by an H-polarized plane wave incident in a plane perpendicular to the generators of the cylinder. The task confronting us is to devise schemes for broadband reduction of the backscattering cross section with particular reference to a range of aspects about edge-on. 3

015224-1-T 2. GENERAL CONSIDERATIONS Our analyses and computations are based on the use of an impedance boundary condition to simulate the effect of a coating applied to the surface. It is assumed that the (normalized) surface impedance n is a function only of the local properties of the coating regardless of the field and the geometry of the surface to which it is applied, and this is an assumption which is incapable of prior validation. In particular, it is not evident that an actual coating will present the same impedance close to the edge of a body as it does when over a broad face, nor is there any assurance that a material can be found to yield a desired impedance independent of both the incident field and the surface geometry. In spite of these shortcomings, the concept of a surface impedance is a valuable one especially for a study such as this, and the fact that for bodies with a variety of different coatings good agreement has been found between measured scattering patterns and those computed using the appropriate surface impedance provides confidence in its validity. For a surface whose impedance is specified at every point, the impedance (or Leontovich) boundary condition can be written as E - (n ~ E)n nZnH (1) where n is the unit vector outward normal and n is the surface impedance relative to the intrinsic impedance Z of the free space medium above the surface. n may vary over the surface and is zero for perfect conductivity. The derivation of (1) has been discussed by Senior (1960) and we remark that in contrast to the analogous boundary condition involving the normal derivative of the field, (1) does not contain any tangential derivatives of n. 4

015224-1-T An immediate consequence of (1) is that the surface supports electric and magnetic currents whose densities K = n^H and K+ = - n^E respectively are related by K* = - nZn^K, and in general both are non-zero. By trivial manipulation, (1) can also be expressed as H - (n * H)n = - - n^E (2) (Senior, 1962) where Y = 1/Z. This the dual of (1) under the transformation E - H, H + -E, Z + Y,n + l/n, implying K + K* and K* + -K. It follows that in spite of our concentration on the case of H polarization, the analogous results for E polarization can be obtained by replacing n by 1/n. If an H polarized plane wave is incident in a plane perpendicular to the generators of a cylinder whose profile is C, the problem is a scalar one for the component Hz parallel to the generators. The electric current is now in the direction of the tangent to the profile, i.e. K = sK, and the scattered magnetic field is Hs = ZHS with Hs ) = ik i' r H((kr) + in(s')Ho(l)(kr) K(s')ds' (3) _- 4 4 1 where r (= rr) = P- P' is the vector from the integration point to the point of observation and H ), H1 are Hankel functions of the first kind of orders zero and one respectively. A time factor eit is assumed and suppressed. 5

015224-1-T On taking the limit as the observation point approaches C, we have Hi(s) = K(s) i- ' * r H )(kr) + in(s')H(1)(kr) K(s') ds' (4) This is an integral equation for the surface electric current density K(s) and is the one on which the computer program RAMD (see Senior and Liepa, 1977) is based. The first terms in the integrands of (3) and (4) are contributed by the electric current and the second by the magnetic current. In the far zone of the body the Hankel functions can be replaced by the leading terms in their asymptotic expansions for large argument to give Hs () ei(kp - /4)p (5) in which the far field amplitude P is P 4 | ' - n(s') K(s')e ik Pds' (6) 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 current over the entire surface of the body, 6

015224-1-T (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 n' * p - i(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 IKI, 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 an 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. The third method is more novel. If n = n ~ p (7) at every point of the surface, the scattering will be zero in the direction 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 (active) impedance over a portion of the surface. Thus, to suppress the edge-on backscattering from an ogival 7

015224-1-T cylinder of half angle Q, n must vary from sinO at the front through zero at the middle to - sinQ at the back. We have verified (Senior and Liepa, 1977) the validity of the scheme by using program RAMD to compute the bistatic scattering from a cylinder. The impedance did null out the scattering in the backwards direction, and reduced it below that of the bare body within 18 degrees of edge-on while increasing it at larger angles. In an attempt to avoid the negative impedances, we also computed the scattering for an impedance equal to the modulus of the optimum, and for a treatment in which the cylinder was left bare where the optimum impedance is negative. Unfortunately, neither was effective, and the edge-on backscattering was actually larger than for 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 n = sinQ (8) where Q is the half angle of the wedge. This is not irrelevant to our task since the edges of the ogival cylinder resemble that of a wedge, and we shall now examine the scattering from a number of discontinuities in the surfaces of coated bodies. 8

015224-1-T 3. SCATTERING BY EDGES AND OTHER DISCONTINUITIES The problem of immediate concern to us is the backscattering from the front and rear edges of the ogival cylinder, and this can be analyzed using the infinite wedge as a model. We treat first a wedge of arbitrary angle and then particularize the results to the case of a half plane and a wedge of half angle 15 degrees. We then examine several types of impedance discontinuity on an otherwise smooth surface and compare the scattering with that provided by a geometric discontinuity. 3.1 Backscattering from a Wedge A wedge of half angle Q having the constant surface impedance n on both faces is illuminated by an H polarized plane wave incident at an angle 00 to the symmetry plane of the wedge (see Figure 3-1). As shown by Maliuzhinets (1959) the far field amplitude of the edge diffracted wave in the direction 0 is then )osS ( - 0o) (- 0) P(O, 0o'n) i ( T () 2 ~ (sin - + sin 1 (C -0 o) + T (2f -0) sin 1 (2r - 0) - sin (r - 0) (9) where v = 2(w - )/v = 2 2 /f. (10) 9

015224-1-T Figure 3-1: Wedge geometry. It is sufficient to take Q < 0, 00 < T and in the particular case of backscattering (0 = 0) P (0, 0, n):- /- - - 0 Vos 1(I1 T i 4 sin 2os () + 1 V 2(2rr -. (11) cos -r( ( The function v is given as the solution of a difference equation and can be expressed in terms of the special functions T introduced by Maliuzhinets:, (a) = ~+(a + + ) ~A (a + 0 - x) A (a -: + X) ~, (a - - X):(A ()))4 +/2 (a + x) - /2 ( - X) where n = cos x (12) 10

015224-1-T Hence r (r - 0):(%> (m))4 >/2 ( - ( + x) >+/2 (" - 0 - x) and (-0) =(Q ($))4 2 (-0 + x) (/2 (-0 - x) c 1 ~T_ - ( ) '/2 (T/2)4 cos v ( - 0 + x) cos v ( 0 - x) ( %/2 (r - 0 + x) 0/2 ( - 0 - (Maliuzhinets, 1959), giving T( ) os 2 ( =-cos (- I 0) + cos g (Tg - 0 + X) g (r - 0 x) (13) g (a) =? ( (14) Similarly '(:27r 08 = fcos 2 (T - 0 + X) g ( - 0 - x) (15) PiT =(Tr - 0 2 +(70) and when (13) and (15) are substituted into (11) we have cos - ( ) - 0) cot 2 -P (0, 0, n) - 2 cos1 1 372 c2os c (T- 0) cos-( - ( 0) * cos ( 0) - in x g ( - 0 + X) g ( - 0 - x) (16) 11

015224-1-T Certain conclusions can be drawn which do not require a knowledge of the function g (a) and are valid for wedges of all angles. In general P is infinite when cos - (3 0) = 0, corresponding to the direction 0 = r/2 + Q of specular reflection from the upper face of the wedge. For angles in the immediate vicinity of this direction we may replace 0 by r/2 + 0 in all factors except the cosine to get 2 sin X P,, 4r cos 1 (3 - 0) + x - ) But /2 ( - X)2 - "2 - x) cot X and /2 ( - Q + x) %+/2 ( - - x) = (+/2 (/2)) cot x sin v (Maliuzhinets, 1959). Hence 2 2 9 (j - Q + x) g ( — - - x) = tan cosec X implying 2 () 2tan il. - 1 3, T P 0 'IV n } _ " sec - 0) (0, 0, n) 4v c 1 (3+ n 4v1 + s v 2 The diffracted field is therefore infinite when 0 = i/2 + Q except when n = 1, and since P(H"gUn)^ }^ ~.,);,., (17) P (0, 0, n) - /2 - (17) for 0 X T/2 + a, the singularity is identical to that displayed by the physical optics approximation. 12

015224-1-T For grazing incidence along the upper face 0 = Q and cos - (r - 0) = 0. P (0, ', n) is then zero provided n:= O. However, we caution that this is a situation in which a traveling wave may be excited and, in addition, plane wave illumination is not possible unless n = O. Also cos -, cot - s i 2 s P (, n) = - - cos cEcos - sin g ( + x) 9 (g - x) 2" 21 2V cos = - V 2 2 V which can be reduced to ^p (iT. 1 (1 + cos 2 )2..... 2).... sec x (18) (-2 2' s) sin * - Vsu using the expression (14) for g (a) and the relations given by Maliuzhinets P (2? 0) - - 4 (3 - sec t) cot sv (Bowman et a., 1969; p. 263). For = 15 (ve = /6), P (j-, -, 1) = i 0.10134 compared with P (j- j2' 0) = - i 1.18475. sin - whereas for a perfectly conducting wedge P (0,, n) is zero if ) (3 - sec 13) cot (Bowman et al., 1969; p. 263). For n = 150 (v: 11/6), P (' ' 1) = - i 0.10134 compared with P (" 2 O )=- i 1.18475. Our final observation is the most important of all. As evident from (16), 13

015224-1-T sin = cs - 0) cos s ( 0) (19) and for any wedge and any angle 0 it is now possible to determine x and hence n such that the edge diffracted field is zero in the backscattering direction. Results for 0 = 0(5)1800 and three different wedge angles are given in Table 3-1. If 0 = T (edge-on incidence), (19) reduces to sin - = cos v 2v implying n = sin Q (cf equation 8). In all cases n is real and positive, corresponding to a purely resistive coating, with n < (>) 1 if 0 > (<) 7r/2 + Q. For a half plane (v = 2) the requirement is simply n = 1 + cos 0. (20) To proceed further, however, it is necessary to determine the function g (a) for the wedge in question and we shall now examine two particular cases. 3.1.1 Half Plane If: = 0 ( = 7) so that v = 2, (16) yields P (0, 0, On) = i 1 - cos 0 (1 + cos 0 - n) g (7 - 0 + x) g (X - 0 - x) P cosO (21) where, from (14), J r/2 (Tr/2) ( 9 (ct) =[1 4 (22) 14

0 r (degrees) _= 0 m_ m= 450 0 1 im 12 0 |5 0 450 | - ~ - degrees...^-~ —.,~ X(degrees) n = o n = 15~ = 45~ 180 0 0.25882 0.70711 - 175 0.00381 0.26287 0.71124 85 1.08716 1.31472 1.57719 170 0.01519 0.27497 0.72357 80 1.17365 1.38991 1.62122 165 0.03407 0.29498 0.74382 75 1 1.25882 1.46240 1.66037 160 0.06031 0.32267 0.77157 70 1.34202 1.53155 1.69422 155 0.09369 0.35771 0.80628 65 1.42262 1.59679 1.72241 150 0.13397 0.39973 0.84725 60 1.5 1.65758 1.74464 145 0.18085 0.44825 0.89375 55 1.57358 1.71340 1.76069 L^ 1 140 0.23396 0.50276 0.94495 50 1 1.64279 1.76380 1.77038 135 0.29289 0.56269 1 45 1.70711 1.80834 1.77363 130 0.35721 0.62743 1.05802 40 1.76604 1.84668 --- 125 0.42642 0.69634 1.11814 35 1.81915 1.87848 - 120 0.5 0.76874 1.17950 30 1.86603 1.90349 - 115 0.57738 0.84396 1.24125 25 1.90631 1.92150 - 110 0.65798 0.92128 1.30258 20 1.93969 1.93236 ----- 105 0.74118 1 1.36271 15 1.96593 1.93600 ----- 100 0.82635 1.07941 1.42091 10 1.98481 ------- 95 0.91284 1.15879 1.47647 5 1.99619 ------- ---- - 90 1 1.23746 1.52876 0 2 --- --- Table 1: Impedances for zero edge diffraction.

015224-1-T But /2 (c) =exp [ 2u - r sinu du /2 ( - =exp Tr Cos u } (Maliuzhinets, 1959) and since p (a) = COS 'J2 exp 2u - w sin u du Cos + cos u ] (Bowman, 1967), it follows that v cos a + 1 -2 r/2 (a) = c + 1 ) giving g (a) = - 2 2 4, (23) i (-7r (23) Vcos- +2 K 1C The function (a) has been tabulated by Bucci (1974) for a range of complex a. Alternatively, in terms of the function f () (1 + sin)-1/4 exp s! 2inu (24) 0 introduced by Senior (1975), g () = ( + sin ) / (a) (25) 4 (.cc) -1 17T-/?r C) = (~))l (.25) cos 2- - a) and when this is substituted into (21), we have P(, 0, n) = i 1 cos (1 + cos 0 -n) j r - 0 + x) f ( - 0- x)). (26) 16

015224-1-T In the particular case of edge-on incidence (0 = 7), 4 P (w,, n) = - (f(x))2 (27) where we have used the fact that f (a) is an even function, as evident from its alternative representation f = (cos a)-1/4 exp 1 vv. (28) f cos v' ( Equation (27) differs from the analogous expression in Senior (1975) only in having n replaced by 1/n, as necessary by the change from E to H polarization. A program has been written to compute f (x) for all real n, but (24) and (28) can be used to provide asymptotic approximations to f (x) whose accuracy is adequate for many practical purposes. Thus, if [n<< 1 then x = r/2 - n and (24) yields f(x) 2-1/4 exp (- -) (29) so that P (r, T, n) - - exp (-2 ) (30) implying P (XX, TT, 0) 0. (31) On the other hand, for I|n >> 1 f (x) - 1/4 exp 2- L [1 + n (2n)] (32) giving P (r, a, n) - 2 exp {_ 7 [1 + In (2n) ]) (33) 17

015224-1-T and implying P ('v, r, A) = - 2 (34) corresponding to a perfectly conducting half plane with E polarization. It now follows that if* n = I|neie, |e < T/2 |P ( 7, w, n)l - Inl exp Intlcos e (Inl << 1) (35) IP (E, X, n)| - 2 exp - T-r [ (1 + In (2lnl) cos e + e sin. 8n (tnt >> 1) (36) and these are independent of the sign of e. Between them the two expressions provide rather accurate coverage of the entire range 0 < Inl < co, with an error of no more than about 5 percent with the possible exception of the interval 0.8 < Inl < 2. * With n defined in this manner and n = cos x, X = X - Xi where, for Intl 1, sin xr = 2-1/2 (1 - nl2)2 + 4 tn2 sin2 + 1 - in12)112 sinh xi = 2"l/2 -(1 - tn2)2 + 4 int2 sin2 - 1 + Inl21/2 and for Inl > 1 sin xr = + 2-1/2 '(1 - ltn2)2 + 4 In2 sin2 6 + 1 - Inl21/2 sinh 21/2 (1( - |n|2)2 + 4 Inl2 sin2 0 - 1 + Inlt1 }2 with the upper (lower) sign for o > (< 0). 18

015224-1-T We have used (35) and (36) supplemented by the tabulation of Bucci (1974) to compute IP (r, 7r, n)l as a function of Inl for different values of e. The results for e = 0, r/4 and Tr/2 are shown in Figure 3-2 and it is seen that arg n has most effect for Inl somewhat greater than unity. If e = r/2 corresponding to a reactive surface, the exponent in (24) is purely imaginary, and as noted by Kay (1957) the modulus of the scattered field then has a simple algebraic expression. From (24) If (x)i = (1 + 1 + In2)-1/4 giving |P ( n)| r |n) (=1 + + 1 + n(2)37) for arg n = r/2. Since P (fr, rr, 0) = 0, the metallic surface is actually the optimum for minimum backscattering at edge-on incidence, and IP (0,, 0 )1 is plotted as a function of 0 in Figure 3-3. Alternatively, we can null out the backscattering in any direction 0 Us f/2 by choosing n in accordance with (20), and Figure 3-3 also shows the curve for n = 0,13397 for which the null is at 0 = 150~. Both fields are, of course, infinite in the specular direction 0 = 90~, but the angular range where the field is large decreases with increasing n < 1. 3.1.2 Wedge of Half Angle 15~ For a wedge of half angle = wT - < with 0 = 4 where m and n are integers with n odd, Maliuzhinets' function,i(a) can be expressed as a product of trigonometric factors. A particular case is a = 15~, i.e., o = 11r/12 implying v = 11/6, for which m = 3 and n = 11, and we then have,(a) = hfot where h(c) is the even function 19

0.5 0.4 0.3 ro~~N)~/ P - -- /.-/ 0.2 O.1 -, ro 1i 0-2 1 10 10 nl Figure 3-2 Edge-on backscattering for a half plane with impedandce n = nie-e with e = 0 ( ), ~r/4 ( --- ) and 1r/2 ( ----).

1.0 I II~ \ f~~ \ 0.75 I 1 0.5 1 I \ N.)~~~~~~~~~~~~~~~~ I NN P~~~~~~~~~~~~~~~~~ I N I \ I 0.25 0.25 0 180 4^120 (degrees) 60 180 if plane wt i1d97 (-n- -=-) a0d Figure 3-3: Backscattering from a half plane with impedance n 0 (-) 0.13 (~

015224-1-T 3I r ~ 6p ~ r _6p-10 os OS + cos TI ( os I + cos 633 h(a) = sec 22 -2 r ) — ' r 6 T[n os 1T- + cos os + cos 37 p=l p= I 33 It follows that 4(a + ) ~( - A) h(a + $) h(a - ~) D/2() = () () ) - h( ) h($) giving h 11 n 2 rh 11 (2 Fg(a~) - 1(2 2 t '~ ^ h) + 1 h(a - 1 ( 12 12 and hence, from (16), 2 6 cos 1 ( 0) P(0, 0, n) = - i 0.23632 6 11 6 37T cos 2 - 0) cos ( * (cos (- 0) cos _ ( _ - 0) - sin 11 _ h(_ _ _ ) h( ) h h(T ) h(- 7 - 0 + X) h(- -- 0 + X) h( 23 ' - 0 - x) h(- - x (38) We have computed this expression as a function of 0 for a variety of n and th.e results for n = 0(0.25)2.0 are plotted in Figures 3-4 and 3-5 (note the expanded scale of the latter figure). In contrast to the situation with a half plane, the edge-on backscattering is no longer zero for perfect conductivity, but can be made zero by choosing n appropriately. From Table 3-1 the required impedance is n = 0.25882 and the resulting edge diffracted field is shown in Figure 3-6. It is essential that the impedance be real to obtain the effect. If arg n 1 0 (or a), there is no longer a zero in the backscattered 22

015224-1-T 1.0 0 \0 0.25 0.7 I I \ 1 I 0.5 \ /I \ \ I 0.75 \ I I I \\ IPI I I \ I I / \I I \\ PJ2,1 I \ 180 120 90 ( (degrees) Figure 3-4: Backscattering from a 15~ half angle wedge having the real impedance shown. 23

015224-1-T 0.4 I i I I I 0.3 I II I I 1I 7 I III ( (degrees) Figure 3-5: Backscattering from a 15~ half angle wedge having the real impedance shown.

01 5224-1-T 1.0 0.5 11 o.s o -I ii /_ /3 2 'ff/2 / ~ 0 180 135 90 45 ( (degrees) Figure 3-6: Backscattering from a 15~ half angle wedge with impedance n= 0.13397ei for e = 0, T/6, 7/3 and ~i/2. The broken line is for a perfectly conducting wedge (n = o). 25

015224-1-T field and this is illustrated by the curves for n = 0.25882 e i with o = r/6, 7/3, and f/2 included in Figure 3-6. For the purely reactive impedance (e = 7r/2) the scattering is comparable to that from the perfectly conducting wedge. If the desire is to reduce the backscattering over some range of aspects about edge-on, it is more effective to null the field at an angle less than 180~, even at the expense of an increase in the edge-on return. The impedance necessary to produce a null at any given 0 in the range 150~ < 0 < 180~ is shown in Figure 3-7, and the edge-on backscattering as a function of n, 0.1 < n < 0.4, is plotted in Figure 3-8. The average cross section reduction which can be achieved can be judged from Figure 3-9 showing the backscattered field for the four impedances creating zeros at 150, 155, 160 and 165~. Over the range from 0 = 150~ to 180~ the average power reduction is greatest for n = 0.32267 (null at 0 = 160~) and is -17.7 dB, although the edge-on reduction is then only -13.8 dB. Once again, much of the performance is lost if n is not real. If, for example, n = 0.32267 eia with 0 f O, there is no longer a null at 0 = 160~. The entire curve is now raised above that for e = 0, but as seen from Figure 3-10 the curves for small o still have local minima at aspects determined to a first approximation by Re, n. 3.2 Impedance Discontinuities To minimize the backscattering cross section of a body over a range of aspects it may be necessary for the impedance to be a function of position on the surface, and this could result in discontinuities in the impedance or its surface derivatives. It is therefore of interest to examine the scattering from such a line singularity. We can treat the case of a jump discontinuity in n by considering a wedge of half angle s = 7r/2 with different face impedances (see Figure 3-11). From (9) with v = 1 we then have P1o4 s:ls 0, O ( - ) (39) P (0. 0) Zsin0 + sin 00 ( -I I - 0 )(3 26

01 5224-1-T 0.4 0.3 0.2 180 170 160 150 c (degrees ) Figure 3-7: The impedance for zero backscattered field in the direction q. 27

015224-1-T 0.04 0.03 0.02 0.01 0 - I.- I _ --- — 0.1 0.2 0.3 0.4 n Figure 3-8: Edge-on backscattering as a function of rn. 28

015224-1 -T 0.2 / / / / 0.15! / / /! / 0.15 // / 0.05 / 180 170 160 150 140 ~ (degrees) Figure 3-9: Backscattering from 15~ half angle wedges with impedances n = 0.29498, 0.32267, 0.35771 and 0.39973, creating zeros at q = 165, 160, 155 and 150~ respectively. The broken line is for a perfectly conducting wedge (n = 0). 29

015224-1 -T 0.04 0.03 200 15~ 0.02 030 0.01 - 0\ 0 0 180 170 < (degrees) Figure 3-10: Backscattering from a 15~ half angle wedge with impedancen = 0.32267 e for the o shown. 30

015224-1 -T y 12 1 Figure 3-11: Geometry for impedance discontinuity. where ) = ( + X2),/2 (a +2) + - X) X / 2 (a 2 - X1) (40) (Maliuzhinets, 1959) with f/2 < 0, 00 < 3 r/2. In (40) n1, 2 cos x1, 2 (41) Since ~/2 (a + ) /2 ( - 2) = [ /2 () cos (a + (7 - o = 2 + /2( - 00 - x2) /2(- o + x1/ 0o - X1) -1 0 r4l / - o + x2))/2p - 2 - x2 - (cos ) - 21 [ /2( 2 0 + Xl) /2( 2 0 - x1) A 31 - o + ) / - o - )

015224-1-T Similarly -f:-5w 5- 5 37r 37 - -;/(22 / 2 - 0 + X2) /2 - 0 - X) 2 2 - X + X1) /2( - 2 0 - X,) 1 (cos 0 - fcn2 ' W/2) 2 -0 + x1)/ 2( -2 - Xl) /) w2( - 0 + X2)>P/2( 2 0 - x2) and r/ 2 0 + X2)w/2(2 - 0 - X2) 2(- - 0 + Xi)2/2(- 2 - 0 - X1) (cos 0 + nl)(CO s 0 - n2) (cos 0 - )(Cos + n2) (2 0) on using the fact that,/2(a - w) = tan 1 (a + 2),w/2( + ) Hence cos 0 cos 0 1 P (0, 00o) = i(n2 - n) sin 0 + sin 00 (cos 0 - n1)(cos 00 - n) 'r2( 2 + x- X)w /2( - 0 - X 0 + x- 0I - x2 0 / -, ) ' /2( - -- 0 + X2)'w/2( 0 - X2),/2 0 + X2)w/2( 3 - 00 -x2 (42) and in the particular case of backscattering (00 = 0) P (0, 0) = 2 (2 I n)1 sece si g( 0 + n-')- g(w - 0 +,1YjTTr - e x) (43) 32

015224-1-T where g(a) is given by (14) and e = 0 - 7/2 is the angle measured from the surface With impedance n2. For a small discontinuity in the impedance such that n2 = n, nl =n + A we have, to the first order in A, ~ = 1 sine P (0, 0): A sec e in et (44) 2 sse 8 +sin 0+ since the product of the g functions is unity to this order. The expression is identical to that obtained by Kaminetzky and Keller (1972) by applying boundary layer techniques to a surface of large but finite radius, and is qualitatively similar to that for the edge-diffracted field of a metallic wedge whose angle is close to f. The scattering increases with increasing e < f/2 and for grazing incidence P = - 2 (45) if n = O. This shows the importance of having a surface treatment commence with a discontinuity in the impedance which is as small as possible and, hopefully, zero if we are to avoid introducing another source of scattering. On the other hand, if n f 0, for e = 0 P = 0 (46) but it is then no longer possible to have a plane wave at grazing incidence, and for any surface wave that may exist the reflection coefficient is not necessarily zero. If the impedance is a continuous function (perfect fairing of a coating), there may still be discontinuities in its first derivative and these could now produce diffraction. The asymptotic analysis of Kaminetzky and Keller (1972) also provides an expression for the scattering from a small discontinuity of this type, and in the backscattering direction P (0, ) = se e sin(47) 33

015224-1-T a y=O+ where A' = [aY-]y0 The result is somewhat similar to that for a line discontinuity in curvature of a perfectly conducting body (Senior, 1972). For grazing incidence (e = 0) with n = 0 P (0, 0) = - (48) and because of the k1 factor the practical significance of this type of scattering is much less than for a discontinuity in n. If n f 0 P (0, 0) = 0 (49) but we again caution that no account has been taken of any surface waves that may exist. Even a discontinuity in the first (or higher) derivative of the impedance can reflect a surface wave, and in previous numerical studies of cross section reduction techniques it was found necessary to eliminate discontinuities in the first derivative as well as in the impedance itself. 34

015224-1-T 4. CROSS SECTION REDUCTION OF AN OGIVAL CYLINDER For surface treatments which can be simulated by a surface impedance, we can use the foregoing analyses to aid in the specification of an impedance for maximum cross section reduction. To illustrate the concepts and procedures involved, we consider the case of an ogival cylinder of half angle 15 degrees (see Figure 4-1) having the following dimensions: half angle (Q) = 15 degrees surface radius (R) = 34.3536 cm surface length, top = 17.9876 cm (= 2RQ) overall length (k) = 17.7828 cm (= 2RsinQ) y Figure 4-1: Ogival cylinder geometry. 35

015224-1-T A plane electromagnetic wave is incident at an angle 00 to the negative x-axis in a plane perpendicular to the generators of the cylinder with its magnetic vector parallel to the generators (H polarization), so that i -ik(xcos0 + ysin0o) (50) H =e 0 0 The far zone scattered field Hz5(0 00) %f' ei(kp - /4) P(0, 00) is observed in a direction 0 where 00 = 0 for backscattering, and the task is to minimize the backscattering cross section.(0) 2 IP(0, 0)12 (51) with particular reference to the range of angles ~ 30 degrees about edge-on: 0 = i (= 180 degrees). Obviously the impedance should be chosen the same on the top and bottom surfaces of the cylinder, and we can therefore confine attention to 0 < 0 < 180 degrees. Our aim is to reduce the backscattering over a wide range of frequencies from (about) 5 GHz on up on the assumption that the surface impedance is frequency independent and can be chosen at will. At 5 GHz A = 5.99585 cm and since the body is then 3X long it is reasonable to expect that high frequency asymptotic techniques can be used to analyze the scattering. Two of the main contributors to the scattering are the front and rear edges, with the latter 36

015224-1-T contributing either directly or via a surface wave on the body. As the frequency and/or the aspect changes, constructive and destructive interference will produce a backscattering cross section which varies, and to achieve a broadband reduction it is necessary that the contributors be individually reduced. The primary tools in our investigation are a computer program for the numerical solution of the integral equation (4) by the moment method and an explicit analytic/integral expression for the backscattered field developed using asymptotic techniques. Program RAMD solves (4) for any specified surface impedance n(s) and, hence, determines the far zone scattered field. All our computations used 96 samplint points distributed uniformly over the two sides of the cylinder, implying a cell size x/16. For the three constant impedances n = 0, 0.5 and 1 the backscattering cross sections a(0) at 5 GHz computed in this manner are shown in Figure 4-2. The case n = 0 represents the bare or perfectly conducting cylinder. The cross section decreases via a series of oscillations from broadside (0 = 90~) to edge-on (0 = 180~), and we observe the rather broad lobe centered on 0 = 1580 which can be attributed to a traveling wave. Increasing the impedance to 0.5 decreases the broadside return by about 10 dB, but decreases even more the scattering for 0 X 1400 and effectively suppresses any traveling wave return. By increasing n to 1 we achieve a further 22 dB reduction in the broadside cross section. The pattern now consists of a series of lobes of almost equal amplitude, but for 0 - 1800 the scattering actually exceeds the bare body value. 37

20 ()/ 02 / o/ v 180 1 0 + (degrees) 1. 0 9 IIt o i \! J.I I,,110 Il 1I I\ / n = 0 ( —), 0.5 ( ---) and 1( ----) at 5 GHz. ii II I I Il II Figure 4-2: Iackscattering cross section of the 150 half angle ogival cylinder with uniform impedance n = 0 ( -), 0.5 (- ) and l (- --— ) at 5 GHz.

015224-1-T The amplitudes and (normalized) phases of the corresponding surface fields K = Hz are shown in Figures 4-3 and 4-4 respectively. The physical optics approximation is KP.- = 2 n i ek i r (52) n. 1 - n where the notation is as defined in Appendix B. As n increases, the increasing suppression of any (backward) traveling wave component is clearly evident from Figure 4-3, and for n > 0.5 the physical optics approximation is remarkably accurate. This is true for the phase as well as the amplitude, and in Figure 4-4 we have plotted arg K - arg KP~. In view of this agreement, an analytical approximation which combines the physical optics expression for the backscattered field with the edge waves of Chapter 3 should provide a reasonable estimate of the scattering. The necessary formulae are derived in Appendix B and a program has been written to evaluate the physical optics integral numerically for any specified (continuous) surface impedance n and hence compute the backscattering cross section. For 150 < 0 < 1650 only the upper surface of the ogival cylinder contributes and the resulting cross sections for n = 0 and 1 are plotted as functions of 0, 900 < 0 < 1650, in Figure 4-5. Since the theory ignores any surface (or traveling) waves that may exist, the accuracy of the prediction could be expected to improve as n increases towards unity, but in actual fact the curves for both n = 0 and 1 agree almost equally well with those computed using program RAMD and have a common failing: the maxima and minima are progressively displaced. 39

2.0 A —O. iK r X n= 0.5 1.0 l0 surface distance s surface distance s - -4 Figure 4-3: Amplitudes of currents induced on the 15~ half angle ogival cylinder with uniform impedances shown when illuminated at f = 150~; f = 5 GHz. The broken lines are the physical optics approximation (55).

90 0/)? 60 30) 0) 00 J0) L. 0 30., i~~~~~~~~~~~~~L I) 1 0 ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~~~~~~~~~ -30 1 -9~ surface distance s -— * Figure 4-4: Relative phase of currents induced on the 150 half angle ogival cylinder with the uniform impedance n = 0 ( — ) = 0.5 ( ----) and n = 1 (- - -). Illumination at t = 1500; f = 5 GHz.

20 rO I~~~~~-20~~~~~~~~~~~~~~I 0 I ~180 i0 4/ (degrees) 120 90 I I I ~ ', I Figure 4-5: Backscattering cross section of the 150 half angle ogival cylinder with uniform impedance shown computed at 5 GHz using RAiD (C ) and the analytical approximation (....).

015224-1-T The reason for this displacement has not been found. With the analytical approximation the minima occur at almost precisely the angles corresponding to destructive interference between the two edge waves regardless of n, i.e., at 0 =cos (- ) where m is an odd (even) integer if the two edge contributions have the same (opposite) sign for incidence at the angle 0. With the output from RAMD, however, the locations of the minima are consistent with the destructive interference between two edge waves emanating from the ends of a body whose length a' is slightly less than a. From the computed backscattering cross sections for f = 4.5(0.1)5.5 GHz and n = 0, 0.5 and 1, the 4 or 5 clearly defined minima in the range 950 < 0 < 1500 have been read off and the equivalent lengths a' deduced. For any one n the results are virtually independent of frequency and show only a weak dependence on the minima locations, with a' ranging from 0.97k for n = 0, to 0.94 for n = 0.5 and back to 0.96Q for n = 1. Although it is believed that the effect is real, there is no obvious explanation, and we are forced to conclude that the minima are not exclusively determined by the edge waves from two identical wedges located at the ends of the cylinder. In spite of this shortcoming, the analytical approximation appears to accurately predict the main contributors to the scattering for aspects 0 < 1650 and can be used to devise and evaluate techniques for cross section reduction in this range. As an illustration, consider the problem of reducing the backscattering cross section at broadside (0 = r/2). As seen from (B.8) there 43

015224-1-T are three contributors to the far field amplitude: the two edges and the (unterminated) surface, and since we are not interested in phase cancellation as a means of cross section reduction, it is necessary to seek a surface impedance which will reduce the individual contributors. Clearly this should be symmetric fore and aft, i.e. n(-a) = n(a), in which case P(7/2) = 2PW(f/2, f/2, n2) + Psurf(T/2) (53) and a convenient measure of the backscattering averaged over frequencies is then = 1 + a2 where 1 - 2 12PW(r/2, r/2 n 2 and 02 2 Ipsurf(/2) 2 This enables us to restrict our use of the analytic approximation to the single frequency f = 5 GHz. For the bare body (n = O)1PW: = 1.1846 and Ipsurfl _ 5.2236, giving o/x = 12.62 dB. As evident from Figure 4-5, there is a marked improvement on going to the uniform impedance n = 1: IPW = 0.10134 and psurfl = 0.02454, producing the much smaller cross section o/X = -15.76 dB, and this impedance was the starting point for our investigation. We can null out the surface contribution by choosing n = cosCa 44

015224-1-T (see B.9), but then jPWI = 0.11816 and 3/X = -14.49 dB. To reduce pW and still keep surf small, we first examined the effect of adding 'tips' to the cosine variation so as to raise the impedance to a value n2 at the edges. The actual profile used was n = cos, sal 2 n = ql {1 n - a ) - a 2 c (o - C2) 2 2 1 - J + 2 22 (- a) (1 ( - ) sin 2 ' 2 < jlal (see Figure 4-6) for which the impedance and its first derivative are continuous, 0 < jaj <_, but due (apparently) to the contributions of the concave regions, no combination of a2 and n2 was found to reduce a/x below its value for the uniform impedance n = 1. We now reverted to the uniform impedance and added tips to produce the profile ~~n = 1 I0( 1 al < 2 n = n2 + (1 - n2) sin ' 2 ) a2 < < Starting with a2 = 0.18786 corresponding to a tip length of 1 inch, the cross sections were determined for a sequence of increasing n2 > 1 to determine 45

01 5224-1 -T n / U2 0*, F - -6 ~-2 c2 Figure 4-6: Impedance profiles for broadside incidence. 46

015224-1-T the minimum S/A. The results are listed below and in each case the minimum occurs for l2 = 1.115: a2 = 0.18786, /X = -18.65 dB 0.22483 = -18.77 0.25441 = -18.89 0.26106 = -18.89 where the tip lengths are 1, 0.5, 0.1 and 0.01 inches respectively. Although it is doubtful whether the full effect would be realized if the length was less than about x/4, it is evident that by raising the impedance at the edges to 0.115 we can reduce the average cross section 3 dB below that for the uniform impedance, thereby achieving more than 31 dB reduction in the cross section of the bare body over a wide frequency range. In an aspect region about edge-on (0 = fr) the analytical approximation is significantly in error because of the omission of surface wave effects, and it is then necessary to resort to the computer program RAMD. To illustrate this we consider the problem of minimizing the backscattering cross section over an angular range of + 30 degrees about edge-on. The problem is one of practical importance and similar to that previously investigated by Senior and Liepa (1977). Although the analytical approximation is no longer applicable, the underlying concepts are still valid and, in particular, the edges of the cylinder continue to play a dominant role in the scattering. This is evident 47

015224-1-T from computed data for the backscattering cross section at edge-on incidence. The results for the uniform impedances n = 0 and 1 at f = 4.5(0.1)5.5 GHz are plotted in Figure 4-7, and since the frequency variation is close to sinusoidal, we can determine the magnitudes of the two contributors to the scattering from the levels of the maxima and minima. Thus, for n = 0, IP1[ = 0.077, IP21 = 0.046 whereas for n = 1 IP11 = 0.135, IP21 = 0.010 For a 15 degree half-angle wedge, P = iO.078 and -iO.137 for n = 0 and 1 respectively, and these now serve to identify P1 with the scattering from the front edge of the ogival cylinder. Presumably, therefore, P2 is due to the rear edge whose contribution appears as a result of surface waves on the cylinder. This process can be repeated at other aspects throughout the angular region. As 0 decreases it is clear that at least one of the contributors is no longer frequency independent, but if we attribute the entire frequency variation to the rear edge contribution we can compensate for it by using the differing levels of two adjacent maxima or minima to deduce the values appropriate to the 5 GHz frequency. The magnitudes of P1 and P2 obtained in this manner are shown for n = 0 in Figure 4-8. The close agreement between the lower set of values and the theoretical curve for the scattering from a wedge clearly identifies P1 as the front edge contribution, and the minor departures are no more than to be expected in view of the deductive analysis employed, The 48

01 5224-1 -T 0.015 l- \ \/\ / \ 0.010 0.005 4.5 5.0 5.5 frequency (GHz) Figure 4-7: Edge-on backscattering cross sections of the 15~ half angle ogival cylinder with impedance n = 0 ( --- —) and n = 1 (- ). 49

C) C) -C) C 0) N) co - ^c — ~O ^- ^ CD C 00) -h CT 0 D C^ 0 0 -* C) CD D 3 C- 0 CD =3 C It CLN o 0 Q DC C C) CDt (D) r+t — h, e 0 I CD OC> c-t- 1, CL 0) r+ I (j D N1 CD Q. I C CD O>< 0- C) (D O rtD c+ (A 0 C~L t/ V ^^ (D 0 XI c3 - XI C-t- —. X= CL -J. — J C D) C-+ —. \ t =r -'* Q_ V-a- CD - -S- OCD C 0 CD \ rt (T>\ CQ -S CL x^0 (00 Kh n> Y CD CDL to c-I-CD K * 0) 0 U) I C C.+

015224-1-T other points then define the rear edge contribution P2 whose magnitudes exceed those of the front edge for 0 < 1750 and reach a local maximum at 0 = 153~. This is almost precisely the location expected for the traveling wave peak according to the formula 0 = 180 - 49.9, degrees (Senior and Knott, 1968). As 0 decreases below 153 degrees, IP21 also decreases and then increases again in accordance with the theoretical expression for (direct) scattering from a wedge coincident with the rear edge of the cylinder. The above process for finding the magnitudes of the two contributors is a very tedious one and a number of other methods have been tried to reduce the labor. One that is reasonably effective is to draw cumulative plots of a/x versus 0 at all of the frequencies and to sketch in the bounding curves. The latter are then assumed to represent IP + p 2 2 7T 1 - 21 from which IP11 and IP21 can be found. The cumulative plots for n = 0 are shown in Figure 4-9, and the accuracy that would result from using the bounding curves can be gauged from the crosses showing the extrema deduced from Figure 4-8. The process is certainly much simpler but its accuracy is also less than that desired, particularly for those aspects and/or impedances where P2 has a significant frequency dependence, and we have therefore used the original method in all our analyses. This requires the construction of individual plots of a/A versus frequency, 4.4 < f < 5.6 GHz at each of the angles 0 = 150(1)1800 for each impedance variation considered, but the results seem worth the labor. 51

015224-1-T 0 -10 -30 -40 180 170 160 150 c) (degrees) Figure 4-9: Cumulative plots of the backscattering cross section for r = 0. The crosses show 2/" (IPl1l: IP21)2 with |IP1 and IP21 taken from Figure 4-8. 52

015224-1-T The values of IP I and IP21 obtained appear accurate and meaningful and the only difficulties experienced were in seeking to deduce the frequency variation of IP21 from two adjacent and low minima. In such cases it was found desirable to extend the frequency range for which data is computed to include a second maximum and to use two maxima instead. IP11 and IP21 for the four uniform impedances n = 0, 0.5, 1 and 2 are plotted as functions of 0, 150 < 0 < 1800, in Figures 4-10 and 4-11. The front edge contributions are in excellent agreement with the theoretical values (shown as the dashed curves in Figure 4-10) for a 15 degree half-angle wedge, and, as expected, the rear edge contribution decreases with increasing n, the decrease being most dramatic for 0 X 160~. To obtain a broadband reduction in the backscattering cross section it is necessary to reduce both sources individually, and in the case of P2 it is obvious that n should be as large as possible. If, for example, n = 1, Figure 4-11 shows that IP21 is reduced by an amount ranging from 14 dB at 180 degrees to 18 dB at smaller angles. To reduce IP1I, however, requires a value of n in the vicinity of 0.3, and based on the simulation of the leading edge by an infinite wedge we would expect IP11 to be zero at edge-on incidence if n = 0.25882. To test this conjecture, data were now computed for the ogival cylinder with the uniform impedance n = 0.25882. The values of IP2I are included in Figure 4-11 and are comparable to estimates made by interpolating the results for n = 0 and 0.5. Figure 4-12 shows a plot of the corresponding front edge contribution along with the theoretical curve for a wedge. It is seen that IP1I decreases from 0.0038 at 0 = 1800 to a minimum of 0.0028 at 0 = 174.5~ and then increases rather rapidly. From the consistency of the data points for adjacent values of 0 it is believed that the effects are real, and the 53

015224-1-T 1 -n = 2 10-12 410: D mn = 0.5 180 170 160 150 q (degrees) Figure 4-10: Deduced magnitudes of the front edge contribution for n = 0, 0.5, 1 and 2. The broken lines are the theoretical wedge results. 54

01 5224-1 -T 1 -r:O /1~~ ~~r n= 0.25882 lO ] / "'/I n = 0.5 10- / -2 1 -^ y^ = 21 180 170 160 150 q (degrees) Figure 4-11: Deduced magnitudes of the rear edge contribution for n = 0, 0.5, 1, 2 and 0.25882. 55

-rD *@ x 0(D x N( CL \ 00 (D \1 rM Q \ x C\ I5 0 \:: c. c:n -::C ('"-' x " C x CD.-,.. o( \ X,1i- C\ X CD S. \ OC(D \ rD 0 C) C -h \ _0 D -! \ % ~ 0C _ \ ci n o\ X L0 X < 0 ) 0~\ X 1J IS\ X c, - 3. -. c- x ^^s. x T*~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0- S.

015224-1-T rapid increase for 0 < 1700 is compatible with the decreasing angle of the ogival cylinder compared with a wedge. For 0 > 1700, however, no convincing explanation for the behavior has been found. The values of IP1| are consistent with those for a wedge having an impedance n = Inleia with Inlj 0.264 and a = 2.50, but for scattering at these low levels it is not unlikely that the curvature of the surface in the vicinity of the edge is also a contributing factor. We must also be careful not to lose perspective, and over the range 170 < 0 < 1800 the computed front edge contribution in Figure 4-12 is more than 25 dB below the already-low return for the perfectly conducting edge. To reduce both the front and rear edge contributions we obviously require an impedance which is as large as possible over most of the surface (including the rear portion) to decrease the rear edge return, but which is also close to 0.25 in the vicinity of the front edge. In addition,the impedance variation should be smooth with no discontinuities in at least an/as to avoid the introduction of new sources of scattering, and a general variation which meets these criteria is 1 1 - n = 2 (m + nl) - (m - n1)cos rs/s5l, ~ s 2 sl (55) n = nm < s1 < s where nl is the impedance at the front and s is the surface distance measured from the front to the rear (s = Sm). As an illustration, suppose that the maximum achievable n(= m) is 1 and that the taper extends over the entire front half of the body, i.e., s1 = sm/2. For nl = 0.25882 the resulting front edge contribution is plotted in Figure 4-13, 57

015224-1-T where we have included for comparison the data for the uniform impedance n = 0.25882 and the theoretical wedge return. IP11 now has a very shallow minimum at 0 = 1780, and is greater than it is for the uniform impedance. The impedance taper has therefore increased the front edge contribution, showing that IP1I is not entirely determined by the impedance at the edge. IP21 is plotted in Figure 4-14 with the corresponding curve for the uniform impedance n = 1. As expected, the rear edge contribution is greater than for the uniform impedance since the taper has removed some of the absorber from the surface. The average cross section a obviously increases with decreasing 0 and is dominated by the front edge contribution for 0 < 160~. To reduce a it is therefore necessary to reduce IPl1 for 0 a 160~, and this we can do by increasing the value of n at the front edge, even though this may serve to increase |P1| for 0 close to 180~. With this objective in mind, results were obtained for the same sl and nm as before, but with n1 = 0.26287, 0.27497, 0.29498 and 0.39973. For a uniform impedance wedge these values are such as to produce zeros in the backscattering from the edge at 0 = 1750, 1700, 1650 and 1500 respectively. Not surprisingly, the effect on |P21 is relatively small. In all four cases the values lie between those for the uniform impedance n = 1 and for the tapered impedance commencing with nl = 0.25882, and to increase nl from 0.25882 to 0.26287 reduces IP2I by less than one percent. For the largest n1 (= 0.39973) the data for IP21 are included in Figure 4-14. With IP1I, however, n1 has a considerable effect and the data for the four values of n1 are plotted in Figures 4-15 through 4-18, along with the corresponding curves for wedges and the uniform impedance n1. As nl increases, IP1(180~)I initially decreases, and then increases for nl > 0.28. The minimum as a function of 0 also becomes more pronounced, occurring at an angle somewhat greater than predicted by wedge theory. It is deepest for n1 = 0.29498 but then rises rapidly with increasing n1, suggesting that the net return from 58

_. * -- * * ~ w -c0 I x tl O CD \ I X 0: o -. 0 CD Q\ I - U) C- O 5 Q) CD\ D w D L 0'\ ' ( DD 3 \ I X c x\ \ -- - -0\ X -J - X \ C-+ C+ - X - -h\\ x S-+ - _- \\ X CD- — J 0 CD (D cL C _ _=E:. \\ X -n C c\ \ X C03):\ \:* 0(D\ \ X "- c'- \ =r QC c -^ \ iD CD = -h \L \ C D \ \ X c 3 c- i \ \ =3 - S = \ X C- I 01 \D {D II I'D 0 I -+h /) \N x CD -' N O I ~ 0 \ -S I O - \ l1 E0. O\ \ X P m00\ \' 0D + N) \ \ X0 N0 VC \ X D 0 C+ I =3 x (D - CD (D C CD C\ * <* CO S \ s 00 3 ~ I3X 0 I, n — '. -.. U"( u'_ X -_ 0 CC~-_C)

09 L = u aouepadWL uuo.Lun aq:; Jo ( —) fLnsea eqa; B1M paeedwmo '(***) CL66~'0 = Li '(xxx) Z88SZ'0 = 1u pue Z/ s = 1s 'L = u W 4LM (99) aL.toad aouepadwi aq,; oj; uoLnqLt;uoo a6pa aeaj aq jo; sapn;.u6ew paonpaa:L-= ajn6.j (saaj6ap) 4 OS 091 OLL 081 0 -- O'O IZdj x*x. ~ ~0 X 1- L-xbZZ LO x^f 0 0 x x x -x0 '0 KJS * * K ~~X KK K ~ ~ f ~ ~~~~~0'0 / ~ ~ ~ ~ ~~~~IO I- 1-t~~~~~~~~S 10~/

015224-1 -T 0.08 x x 0.06 x x 0.04 x x x IP llI/ X 0.02 " / Kx x K x x x x x 180 170 I (degrees) 160 150 Figure 4-15: Deduced (xxx) magnitudes of the front edge contribution for the impedance profile (55) with n1 = 0.26287, nm = 1 and sl = sm/2, compared with the theoretical wedge result for the uniform impedance n = 0.26287. 61

015224-1-T 0.08 0.06- x x x 0.04 1 I^~~~~~ x~~ ~~~~~~~ / x IPi I x X y/ 0.02 x xX X X X/ x t 1'80 170 5(degrees) 1 1 Figure 4-16: Deduced (xxx) magnitudes of the front edge contribution for the impedance profile (55) with nl = 0.27497, ram = 1 and s = sm/2, compared with the theoretical wedge result for the uniform impedance n = 0.27497. 62

015224-1-T 0.08 0.06 -x x x x x p x x 0.02 -xK xK x 0.02- x X X X: 'xx x xx x / 180 170 c (degrees) 160 150 Figure 4-17: Deduced (xxx) magnitudes of the front edge contribution for the impedance profile (55) with nl = 0.29498, rn = 1 and sl = sm/2, compared with the theoretical wedge result for the uniform impedance r = 0.29498. 63

015224-1-T 0.08 0.06 0.04 1~x_ x x x \ x,x X X x \x 0.02 x x X x K 180 170 (degrees) 160 150 Figure 4-18: Deduced (xxx) magnitudes of the front edge contribution for the impedance profile (55) with n1 = 0.39973, nm = 1 and sl = sm/2, compared with the theoretical wedge result for the uniform impedance n = 0.39973. 64

015224-1-T the front edge is made up of a contribution associated with the actual impedance at the edge and one due to the impedance variation. For cross section reduction purposes the impedance profile that is most desirable depends on the particular objective, and in Figure 4-19 the average cross section a/X is plotted as a function of 0 for the above five values of nl. The edge-on backscattering is least if nl = 0.26287 or 0.27497, but when the cross section is averaged over the entire range 1800 > 0 > 1500, the latter impedance is slightly superior, and reduces the average bare body cross section by 21.1 dB. However, the averaged (over angle) cross section is actually least for the impedance n1 = 0.39973, which yields a reduction of 22.0 dB in the bare body return. It also gives the greatest reduction in the traveling wave lobe, and the resulting cross section is now almost constant over the entire aspect range. We have also examined the effect of different impedance tapers using the same values of nl (= 0.25882) and nm (= 1) as before, and, in particular, data have been computed for the cosinusoidal taper (55) with sl = sm/4 and sm/8. These correspond to taper lengths of only 0.75X and 0.375X respectively at 5 GHz, and not surporisingly even the front edge contribution now shows some frequency dependence, especially for sl = sm/8. However, by using adjacent maxima and minima in the individual plots versus frequency, values of IP1l and IP21 have been deduced appropriate to 5 GHz, and the results for IP21 are shown in Figure 4-20. For the most part the data points lie between the curves for the uniform cylinder with n = 1 and the cylinder whose tapered impedance has s1 = sm/2, and the variability near 0 = 1550 is no more than expected in view of the uncertainties in the deductive analyses. For the front edge contribution, however, the reduced values of sl substantially increase the scattering. As seen from Figure 4-21, the results for s /4 and s /8 are comparable to those 65

99 ' ( — ) Lnsae (0 = u) &poq aeq aq; q4LM pajedwoo '(.) ~t660'O '( ---) 86162'0 '( ---) 6LZ'O '(- -) Z8Z9Z'0 '(- -— )Z88SZ'0 = puB Z/ws - Ls 'L u q LM (SS) aLi4o.d aouepadwL aq JAOJ suoOL;as ssoJO 6u.La^;;e~sq eq a6eJaAV:6L-q ajn6lj OSL 09L (saaO6ap) L OZL 08L,......'..................... —.............................................. 1.01 o0 -J000*0 1-L-tZ9 10 l-l-bZZ0lO f~ ~ ~ ~ ~~~z

m-o C~ o o o ~- ~~.* ~. roCN. * *) '~ -- o oo CD -- h n n\ n \.. 0 C \ \ CD -- 11 3 m CS -- D\ -J. Co\ c C) \ N)-* \ \:: ---C CD\ ' QQ \ 3fD 11B OL\X N, 00 0 r r (D~ = * 1c11 0 " GO \Xo \ O' CLD _-_ O X /) -C D \ (D CD o V) 20 -, c-IUo I - - II 0 0 2. o —_ -- Co -o - 0

89 u,8 * '( -) L = avuepadwL LuLoLOun!(eee) Z/lu, =s pue L = u 'Z88SZ'O =ulLi q;iLM (99) aL[iold t(...) 8/Ws '(xxx) / Ws ( ---)z/ls = Ls pue = u '88SZ'O = Lu 4.LM (SS) aLLo.jd:jio uoLlnqLu uoo a96pa,uoj aq4 lo sapn:Lu6ew paDnpoa:LZ-V ajnbLj OSL 09L (saae6ap) OZLL 08 L /O // // /! 0 x 0 /! /0 L / 0 / x / Cx 0 / / /. 0'0 / x / / / x. * L. S. x LO

015224-1-T for uniform impedance wedges (see Figure 3-8) with n = 0.198 and 0.017 respectively, and the large impedance slope now has more effect than the actual impedance at the edge. As a further illustration of the effects of a rapid impedance variation, data have been obtained for the impedance profile n = n + (nm ' n-) sin 21S 0 < s < s ( 1) sin 2s_ -1 (56) n m 1 s1 s < sm with n1 = 0.25882, nm = 1 and s, = Sm/2, and the resulting values of IP21 and IP11 are included in Figures 4-20 and 4-21. |P21 is quite in accordance with expectations and is bracketted by the values for the uniform wedge and the wedge with the taper (55) extending over half the body, i.e., sl = s /2. IP1I, however, is much greater than for the latter wedge, and the minimum centered on 0: 1660 is suggestive of a complex (equivalent) edge impedance having Inl -0.3. For 0 < 1700~ IPl is also less than for the analogous profile (55), implying a larger reduction in a in this aspect range, but if this is the objective, Figures 4-17 through 4-19 show that an even better performance can be achieved using the profile (55) with n1: 0.3; and since the front edge contribution is virtually independent of frequency if sl = sm/2, the results obtained are then meaningful for all frequencies f > 5 GHz. In seeking to realize the above impedances using a coating, one of the main difficulties is the requirement that the impedance be real. As previously remarked, much of the desired reduction in the front edge contribution is lost if the impedance is complex and to demonstrate this fact computations were performed for the impedance profile (55) with sl = s /2, Re. n = 0.25882, Re. nm = 1 and arg n 300 instead of 0. The results are shown in Figures 4-22 69

015224-1-T and 4-23. Over most of the aspect range the values of IP I are substantially larger than for the case rn = 0, and are reasonably approximated by the data for the corresponding uniform wedge. For the larger 0, IP21 is similar to but somewhat smaller than for e = 0, suggesting that Inl rather than Re.n is the determining factor in this aspect range, but for 0 < 1650 the curve shows hardly any evidence of the local maximum that previously characterized the rear edge return. Because of the changed shape of the curve, it would appear that IP21 is influenced by the value of the impedance at the rear as well as by the attenuation of the field on the surface. This is certainly not unreasonable physically and it is unfortunate that there is no easy way to separate the two effects. 70

-J. rN _ -- ' —= aCD 0Z C) 1x IN |~~~~~~~ x D 0 CD 'D:u 3: \ - CD CD - CD C x C-+ - — + X r+ (D \ -'h ) ^ 0e \ X o n c- \ I — h CD -- c+ 0CD \. — h:; — h \ x COCD ' X -_. -O \ 'r r~ c+.\ \ -- + ^0 t) — I X \ -(0 -n t- \N - ) = x\ 0D C). 00 C Y X 0. 0O - C) C, 3 D I r',CXD \ O 0 Q:0 \ 0 0 * c+ II:: II "' ' +C) OC- c - I C 00 0 C. O C, D ro --- O0 I\ D I 0. - - 0 O~

01 5224-1-T 0.04 x x x x x 0.03 // xX 0.02 I/"/,/ / 0.02 /x /xx //. 0.01 -180 170, (degrees) 160 15 Figure 4-23: Deduced (xxx) magnitudes of the rear edge contribution for the impedance profile (55) with Re. n1 = 0.25882, sl = sm/2 and arg n: 30~, compared with the results for arg n = 0 ( ---) and the theoretical wedge result ( ---) for the uniform impedance n = 0.25882 (1 + i//3). 72

015224-1-T 5. ACKNOWLEDGEMENT The author is indebted to Mrs. Austra Liepa for the computer programming involved in this study. 6. REFERENCES 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. Bucci, O.M. (1974), "On a function occurring in the theory of scattering from an impedance half plane", Laboratorio di Onde Elettromagnetiche, Napoli, Italy. Kaminetzky, L. and J.B. Keller (1972), "Diffraction coefficients for higher order edges and vertices", SIAM J. Appl. Math. 22(1), 109-134. Kay, A.F. (1957), "Scattering of a surface wave by a discontinuity in normal reactance with applications to antenna problems", TRG Report AFCRC-TN-56-778. Knott, E.F. and T.B.A. Senior (1973a),"Equivalent currents for a ring discontinuity", IEEE Trans. AP-21, 693-695. Knott, E.F. and T.B.A. Senior (1973b),"Non-specular radar cross section study", The University of Michigan Radiation Laboratory Report No. 011062-1-T. 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-F. Maliuzhinets, G.D. (1959), "Excitation, reflection and emission of surface waves from a wedge with given face impedances", Sov. Phys. -Dokl. 3(4), 752-755. Senior, T.B.A. (1960), "Impedance boundary conditions for imperfectly conducting surfaces", Appl. Sci. Res. 8(B), 418-436. 73

015224-1-T Senior, T.B.A. (1962), "A note on impedance boundary conditions", Can. J. Phys. 40, 663-665. Senior, T.B.A. (1972), "The diffraction matrix for a discontinuity in curvature", IEEE Trans. AP-20(3), 326-333. Senior, T.B.A. (1975), "Half plane edge diffraction", Radio Sci. 10 (6), 645-650. Senior, T.B.A. and V.V. Liepa (1977), "Non-specular RAM", The University of Michigan Radiation Laboratory Report No. 014518-1-F. Ufimtsev, P.Ya. (1962), "Method of edge waves in the physical theory of diffraction", Izd-Vo Sov. Radio, Moscow. 74

015224-1-T APPENDIX A. EDGE DIFFRACTION COEFFICIENT A program has been written to compute the edge diffraction coefficient for backscattering from a 15~ half angle wedge using the expression for P (0, 0, n) given in (38). The required inputs are the real and imaginary parts of the angle x satisfying (12), and a program listing and specimen output are as follows: COMPLEX XF17,F17SQARGF5,F5SQSINEQwSINSQP15,FALFA DATA PI/3 1415927/,P11/2.8797933/,Sl1/.5454545/, SPI2/1 5707963/,RAD/.0174533/,DEG/57 2957795/ ARG-CMPLX((17.*PI)/12. O.) F17=FALFA(ARG) F17SQ=F17*F17 ARG=CMPLX(-(5.*PI)/12o,0.) F5=FALFA(ARG) F5SQ=F5*F5 ROOT2=SQRT(2o) 100 READ(5,500) X 500 FORMAT(2F10.5) WRITE(6,501) 501 FORMAT(1HI,' EDGE SCATTERING COEFFICIENT (15 DEGREES)') XR=REAL(X) XI=AIMAG(X) WRITE(6,510) XRXI 510 FORMAT(1HO,5X,'X - (' F8.5 ','r F8o5,' )'//,4X, &'PHI'rl2X,'MOD',11XX' ARG '/) SINE=CSIN(S11*X) SINSQ=SINE*SINE PHIP=180. PHI= 0O DO 200 I=1,34 FPHIR=RAD*PHI Q=F17SQ/(FALFA(X-PHIR+P11)*FALFA(X+PHIR+P11)) Q=Q*FSSQ/(FALFA(X-PHIR-P11)*FALFA(X+PHIR-P11)) COS6=COS(S llPHIR) COS2=COS(Sllt(PHIR-PI2))*COS(Sll*(PHIR+PI2)) P15=0*23632*COS6*COS6*(COS2-SINSQ)*Q*Q/COS2 AMP=CABS(P15) PHASE=DEG*ATAN2(AIMAG(P15) REAL(P15)) WRITE(6,530) PHIFPAMPPHASE 530 FORMAT(1H,2XF5.1,6XE12.5,5XF7*2) PHIPF=PHIPF-5 200 PHI=F'H I +5 GO TO 100 END 75

015224-1 -T COMPLEX FUNCTION FAIFA (A) COM4PLEX A,COSA DAA PI/3. 1415927/,PI33/.(C951998/,P11/. 2855 93/ COSA=CCCS (A/ (11.,0.)) F ALFA= (COS +COS ( 4. * PI33) ) / (CCS.+CO S (PI33 ) * 6 (CCSA+COS ( 2.*PI33))/(CCSA+COS( 5.*P33)) * 6 (CCSA+COS ( 8.*Pi33) /(CCSA+CC (11. *PI 33) ) * & (CCSA+COS (14.*PI33))/ (CCSA+CS (17.*?I33))* & (CCA+ COS (20.*PI33))/ (COSA+COS(z3.*PI33))* & (COSA+COS (26.* PI'33))/ (CCCS(A/ (22.,.)))*. (CC + CO S (PI/12. ) ) * & (CCSA+COS (3.*P11))/(COSA+COS (2. *P11))* & (COSA+COS (5.*P 1 1))/(COSA+COS(4.*P1 1) ) BETURN END 76

015224-1-T EDGE SCATTERING COEFFICIENT (15 DEGREES) X = ( 1.34707,-0.13234 ) PHI MOD ARG 180*0 0*35281E-01 100.93 175.0 0o35763E-01 102*58 170.0 0.37411E-01 107.33 165.0 0.40797E-01 114*54 160.0 0o46763E-01 123*06 155.0 0.56298E-01 131.60 150*0 0.70495E-01 139.20 145.0 0.90746E-01 145.48 140*0 0*11914E+00 150.45 135.0 0.15921E+00 154.31 130.0 0.21739E+00 157*29 125.0 0.30669E+00 159*58 120.0 0*45759E+00 161.35 115.0 0*76166E+00 162*70 110.0 0*16768E+01 163*73 105o0 0*68786E+05 164.50 100*0 0.19867E+01 -14.95 95*0 0.10699E+01 -14*59 90.0 0*76313E+00 -14.40 85*0 0.60831E+00 -14.36 80.0 0*51376E+00 -14.47 75.0 0*44882E+00 -14*73 70.0 0.40024E+00 -15.16 65.0 0.36124E+00 -15*76 60*0 0.32790E+00 -16.57 55*0 0.29762E+00 -17*62 50*0 0.26850E+00 -18*99 45*0 0o23889E+00 -20.76 40.0 0o20722E+00 -23.07 35.0 0.17179E+00 -26.15 30.0 0.13087E+00 -30.37 25*0 0.83354E-01 -36.40 20*0 0*32150E-01 -45.51 15.0 0*23747E-11 -60*39 77

015224-1-T EDGE SCATTERING COEFFICIENT (15 DEGREES) X = ( 1.44423P-0.22407 ) PHI MOD ARG 180.0 0o71446E-01 112.27 175.0 0.72400E-01 113.03 170.0 0.75389E-01 11522 165.0 0,80796E-01 118.61 160.0 0*89249E-01 122.83 155.0 0.10165E+00 127 47 150.0 0o11922E+00 132.13 145.0 0.14372E+00 136.51 140.0 0.17783E+00 140.43 135*0 0.22596E+00 143.81 130.0 0.29606E+00 146.66 125.0 0.40406E+00 148.99 120.0 0 58720E+00 150*88 115.0 0*95725E+00 152 37 110.0 0.20734E+01 153.51 105,0 0,83987E+05 154.34 100.0 0.24025E+01 -25.11 95.0 0.12847E+01 -24.82 90.0 0.91180E+00 -24.77 85 0 0.72461E+00 -24.96 80.0 0O61117E+00 -25*41 75.0 0*53408E+00 -26.12 70.0 0.47714E+00 -27.12 65.0 0.43208E+00 -28.46 60.0 0*39410E+00 -30.21 55*0 0.36000E+00 -32.46 50.0 0.32734E+00 -35.36 45.0 0.29392E+00 -39.11 40.0 0 25748E+00 -44.04 35.0 0.21539E+00 -50.65 30.0 0.16478E+00 -59.76 25.0 0.10384E+00 -72.70 20.0 0.38029E-01 -91.68 15.0 0.23997E-11 -119.71 78

015224-1-T EDGE SCATTERING COEFFICIENT (15 DEGREES) X = ( 1.57080,-0.25601 ) PHI MOD ARG 180.0 0.10898E+00 125.22 175.0 0.11045E+00 125*65 170.0 0*11495E+00 126.88 165*0 0.12286E+00 128*80 160.0 0.13482E+00 131.21 155*0 0*15178E+00 133.90 150.0 0*17517E+00 136.66 145.0 0o20716E+00 139.35 140.0 0*25112E+00 141.83 135.0 0.31271E+00 144.04 130.0 0o40211E+00 145.95 125.0 0.53975E+00 147.54 120.0 0*77333E+00 148.83 115*0 0~12460E+01 149.82 110.0 0.26739E+01 150*54 105.0 0*10757E+06 150.98 100.0 0.30626E+01 -28.83 95.0 0.16334E+01 -28*91 90.0 0.11587E+01 -29.25 85*0 0.92223E+00 -29*87 80.0 0*78068E+00 -30*81 75.0 0.68616E+00 -32*09 70.0 0*61800E+00 -33*77 65*0 0*56564E+00 -35*96 60.0 0.52293E+00 -38.76 55*0 0.48566E+00 -42.35 50*0 0.45048E+00 -47*02 45*0 0o41398E+00 -53.14 40.0 0.37197E+00 -61+34 35*0 0.31872E+00 -72.58 30.0 0*24684E+00 -88*33 25*0 0*15164E+00 -110*60 20.0 0*49443E-01 -141*09 15*0 0,23954E-11 -178*31 79

015224-1-T APPENDIX B. ANALYTICAL APPROXIMATION For a 15~ half angle ogival cylinder subject to an impedance boundary condition and illuminated by an H polarized plane wave incident in a plane perpendicular to the generators, an analytical approximation to the backscattering cross section can be developed by adding expressions for the edge waves to the physical optics result. The method is equivalent to that of Ufimtsev (1962) or, alternatively, to the equivalent current method of Knott and Senior (1973). B.1 Physical Optics If the incident field is the plane wave having H = zH. with H, = eik i * the impedance analogue of the physical optics approximation is to assume 2n * i ik i H:.. e n * i - nT over the lit portion of the surface and zero in the shadow. On substituting this into (6), the physical optics expression for the far field amplitude P becomes n' p ik(i Ppp * o =k | -- n' ~ i ek( - P I) P ds, lit n' i - and in the particular case of backscattering such that i = - p pP 0 ~ k { [n- P nl} n * P e-2ikp * p ds'. (B.1).lit n' * p + nJ At the specular point n' * p = 1. 80

015224-1-T Let us now apply this to a length a of a wedge of uniform impedance n and interior half angle Q. If the incidence is such that only the upper surface is illuminated, i.e., < 0 < -, then n' * p = sin (0 - a) and p ~ p'= X' COS (0 - giving PP ' O(0i) = in n sin (0 - ) e-2ikx'cos (0 - d) dx' 0 0 Jo Hence p ' ~0(0) 4 sin i - )- q tan (0 -) {1 - e-2ikcO s (- ") 4 sin 0 2)- s + r for 0 7 T/2 + n, reducing to pp ~ o0() = k 1 - n 2 1 +Tn if 0 r /2 + n, and the contribution of the front edge is therefore pP (0)= 4 sin 0i: a;n tan (0- ). (b.2) 4 sin 0 -0 + rn When 0 -?/2 + Q the expression reduces to that in (17), implying a singularity identical to that displayed by the exact edge diffraction coefficient (16). If 0 > T - Q the lower face of the wedge is also illuminated and its contribution to the edge diffraction is 4 sin (0 + Q)+ (3 PP (0~ ) = 4 s in: + 3; 0 tan (0 + Q) (B.3) which is infinite when 0 = 3ir/2 -, as expected. 81

015224-1-T In the case of an ogival cylinder of half angle Q, it is convenient to specify the point of integration using the angle a subtended at the center of curvature. On the upper surface n' = x sin a + y cos a p' = R {x sin a + y (cos cos cos )} giving n' * p = sin (0 + a) p p = R sin (0 + a) - R sin 0 cos Q. From (B.1) the contribution of the upper surface is pp o ) = kR e2ikRsin0cos sin(a + 0) - na) sin(a + O)e2ikRsin(a + )da u 2 J sin + + 0 ce Q-n^~~~~ ~~(B.4) for o < 0 < i where: = min. (rr - 0, ). If 0 > o - Q, however, a portion of the lower surface is also illuminated, and its contribution is -(T-0), i (a-2ikRsin(- O)da pP O) kR -2ikRsin0cosQ - 0 i sin(a - )e2ikRsi 0 pP ~ 0(0) -2 e S - + ()sin(c -a 0 + (B5)e) AR~ 2L~ -S~_ (B.5) The physical optics expression for P is therefore the sum of (B.4) and (B.5), i.e., R -i ksnc2 J-sin(a + 0)- s )+ d -2 -+ kR -2ikRsin0cosQ | sksin(c - 0)e da +-2- sin(- +. -0 (B.6) valid for Q < <._. 82

015224-1-T B.2 Inclusion of Edge Diffraction The contributions of the front and rear edges to the backscattered field can be obtained from the edge diffraction coefficient for a wedge of half angle Q, the exact expression for which is given in (16), reducing to (38) in the particular case Q = 15~. Were we to merely add them to (B.6), however, we would be guilty of tallying the edges twice, since the physical optics approximation already contains some edge contributions, albeit erroneous ones. The latter can be deduced from (B.2) and (B.3), and on subtracting them from (B.6) with account taken of the change in phase center, the scattering from the (unterminated) surfaces of the ogival cylinder is found to be P ()- PP (0) + - sin( tan(0 - s)e surf( ). i s-n p P;1 nl 2i kRcos0si n i sin(o + Q) - n tan( + 2) 2ikRcos0sin ( + 4 sin(0 + Q) + l tan(0 +.)e i sin(O + Q) - n2 + ) ikRcos0sinQ 4 sin(0 +=Q) + n2 tan(O + )e (B.7 where U is the unit step function and n1, n2 are the surface impedances at the front and rear edges respectively. On the assumption that the surface impedance and all its derivatives are continuous (in practice it is sufficient if just n and an/as are continuous), an approximation to the far field amplitude is then obtained by adding the wedge diffraction coefficients to Psurf The result is P(0) = pw( 0,, 1)e2ikRcos0sins + pW( 0, w - 0, N2e-2ikRcos0sinQ U( - 0 - n) + Purf(0) (B.8) 83

015224-1-T where pW(0, 0, n) is given in (16) and psurf(0) in (B.7). Some immediate deductions can be made. If Q < 0 < r - Q and n is everywhere such that n(a) = sin(a + 0) (B.9) implying n - n(- Q) = sin(0 - Q) n2 - n(Q) = sin(0 + n), then pP ~* (0) = psurf() = 0 (B.10) and the scattering in the direction 0 is entirely attributable to edge diffraction. In particular, for broadside incidence (B.9) requires n(a) = cos a, in which case P(7/2) = 2PW(f/2, 7/2, cos a). (B.11) On the other hand, if n1 and n2 are chosen to suppress the edge diffraction in accordance with (19), we are left with only the surface contribution and P(0) = surf(). (B.12) It would appear that the condition for this conflicts with (B.9) for all ogival cylinders and for all angles of incidence except for a strip (: = O) at broadside incidence (0 = ir/2) where the requirement is n = 1. To minimize the backscattering cross section in any given direction is now an optimization process. 84

015224-1-T When 0 = rr/2 + Q the wedge diffraction coefficient for the front edge is infinite unless n = 1. The physical optics estimate is likewise infinite, and since the two infinities are identical (and cancel), it is convenient for numerical purposes to regroup the terms in (B.8). On doing so we have P(0) = Pf(0, 0, nl)e2ikRcos0sinQ + pr(, 0, i2)e-2ikRcos0sin0 U(r - 0 ) + PP ' (0) (B.13) where the front edge contribution is fi sin( _ ) - "l t -I ) = PW(,, ) + 4 sin(0 ) r tan (0 - Q) i sin(0 + n) - n 4 sin(0 + ) + T1 tan(0 + Q)U(0 + Q - r), (B.14) 4 sin(~ + Q + the rear contribution is r sin(0 + n) +n P (0' ~, n2) = P (r - 0, T - 0, q2) - 4 sin(o + ta) + n2), (Bad) and PP ' (0) is given in (B.6). All three terms in (B.13) are finite for all finite n, but the regrouping has changed somewhat the conditions for suppressing the individual terms. Although PP ' ~(0) still vanishes if (B.9) is satisfied, (19) is no longer relevant to the suppression of the remaining terms in (B.13). Thus if 0 = r/2 with Q = 15~, Pf'r(/2, T/2, n) = - i 0.1013411 + 7.02667 sin2 6xJsec2 6 - 0.93301 tan x - ) tan x + 1 (see 18) and vanishes if n = cos x = 1.390, compared with the value n = 1.237 for which PW(7/2, r/2, n) = 0. 85

015224-1-T Programs have been written to compute P(0) for Q = 15~ and any desired impedance using both of the representations (B.8) and (B.13). The results are, of course, identical, and though (B.13) is more convenient numerically, the computation of the individual terms in (B.8) is helpful in seeking to minimize the scattering. Because we have not included any traveling wave effects per se, the accuracy of our formulae diminishes as end-on incidence (0 = r) is approached, particular for surface impedances where these effects are significant. 86

UNIVERSITY OF MICHIGAN oIlOJIENHIIHWMMU I 3 9015 03524 4568