388967-1 -F DIFFRACTION BY A THICK IMPEDANCE EDGE AND AN IMPEDANCE STEP PROTRUSION Final Report by John L. Volakis and Mark Ricoy Radiation Laboratory Department of Electrical Engineering and Computer Science The University of Michigan Ann Arbor, Michigan 48109 for Rockwell International North American Aircraft P.O. Box 92098 Los Angeles, CA 90009 388967-1 -F = RL-2555 September 1985

TABLE OF CONTENTS Page List of Illustrations ii 1. Introduction 1 2. Analysis via the Angular Spectrum Method 1 3. Analysis via the Geometrical Theory of Diffraction 5 4. Comparison of Calculations with Measured Data 5 5. Family and Design Curves 8 References 10 -i

LIST OF ILLUSTRATIONS Figure Page 1 Geometry of a step protrusion with a surface impedance n as shown. 11 1 2 Geometry of a pair of parallel half-planes with a surface impedance n on their outer surfaces. 12 1 3 Two half-planes with an impedance stub inserted. d = 0 constitutes a thick half-plane. 13 4 Illustration of the coupling, reflection and launching coefficients. 14 5 First order diffraction mechanisms from a thick edge. 15 6 Higher order diffraction mechanisms from a thick edge. (a) Direct wave. (b) Image Wave. 16 7 Measurement model of a thick edge. 17 8 Measurement model of a step protrusion. 18 9(a) Comparison of measured and calculated backscatter patterns from a perfectly conducting thick edge with 2z = 0.614 inch. (a) H incidence. 19 9(b) Comparison of measured and calculated backscatter patterns from a perfectly conducting thick edge with 29 = 0.614 inch. (b) E incidence. 20 10 Echowidth of a thick perfectly conducting edge as a function of its thickness 2z/x. 11(a) Comparison of measured and calculated backscatter patterns from a perfectly conducting thick edge with 2t = 1 inch. (a) H incidence. 22 11(b) Comparison of measured and calculated backscatter patterns from a perfectly conducting thick edge with 29 = 1 inch. (b) E incidence. 23 12(a) Comparison of measured and calculated patterns from an imperfect edge with n = 0.689+j.812 and thickness 29. = 0.614 inch. (a) H incidence. 24 -ii

Figure Page 12(b) Comparison of measured and calculated patterns from an imperfect edge with n = 0.689+j.812 and thickness 2z = 0.614 inch. (b) E incidence. 25 13(a) Comparison of measured and calculated patterns for a perfectly conducting step protrusion with a = 0.307 inch. (a) H incidence. 26 13(b) Comparison of measured and calculated patterns for a perfectly conducting step protrusion with a = 0.307 inch. (b) E incidence. 27 14(a) Comparison of measured and calculated patterns from an imperfect step protrusion n = 0.689+j.812 and a = 0.307 inch. (a) H incidence. 28 z 14(b) Comparison of measured and calculated patterns from an imperfect step protrusion n = 0.689+j.812 and z = 0.307 inch. (b) E incidence. 29 z 15(a) Echowidth of a thick perfectly conducting edge for various values of the edge thickness, 2g. (a) H incidence. 30 15(b) Echowidth of a thick perfectly conducting edge for various values of the edge thickness, 2z. (b) E incidence. 31 16(a) Echowidth of a perfectly conducting step protrusion on a ground plane for various values of the step heighth, z. (a) H incidence. 32 Z 16(b) Echowidth of a perfectly conducting step protrusion on a ground plane for various values of the step heighth, a. (b) E incidence. 33 Z 17(a) Echowidth of an imperfect edge (n = 0.689+j.812) for various values of the edge thickness, 2k. (a) H incidence. 34 z 17(b) Echowidth of an imperfect edge (n = 0.689+j.812) for various values of the edge thickness, 2k. (b) Ez incidence. 35 18(a) Echowidth of an imperfect step protrusion (n = 0.689 +j.812) for various values of the step heighth, a. (a) H incidence. z 18(b) Echowidth of an imperfect step protrusion (n = 0.689 +j.812) for various values of the step heighth, a. (b) E incidence. Z -iii

I. Introduction An analytical study was performed for calculating the scattered field by a step protrusion shown in Fig. 1. Measured data were also collected and were found to be in good agreement with the analytical data. Two independent methods were used to obtain analytical expressions: the angular spectrum method [1] and the geometrical theory of diffraction (GTD). The first becomes cumbersome for large t(z > x/4) and is thus suitable for protrusions of small thickness. In contrast, the GTD which is a high frequency method, becomes accurate and efficient for the analysis of thick (z > X/4) protrusions. The accuracy of each technique in their respective regions of applicability was clearly demonstrated by the obtained measured data. 2. Analysis via the Angular Spectrum Method Before proceeding with the application of this method, the protrusion geometry in Fig. 1 was broken down to a series of sub-geometries. First the diffraction by a pair of half planes (see Fig. 2) was obtained by imposing the appropriate impedance boundary conditions on the outer faces of the half planes. The details of this analysis are described in [2]. It was found that the diffracted field is given by sa I elkr Esa = D1(p ';Tc )A(q, e (1) z S 1 vr for the E (plane wave) incidence. In the above D (Iq0) is the soft z c f diffraction coefficient for the impedance half plane and is given by

-2 - I -j J/4 1 - 2n cos 4/2 cos % /2 D c1, ) = 1 cos +U (cos,;n )U (cos; ). S '0 Cos ( + Cos 0 3 1 3 0 1 (2) A(,5o;k) is defined by -jk sin As L (cos )o) 2 A(,-(;) = ek s (cos o(k sin ) + j U (cos ) sin(kz sin). 1 2 (3) In the case of H -incidence we obtain via duality that sa D(;,) e-jkr H D (h o;n)A((,o) e-, (4) where Dh(,;n ) = D (q o;1/n) (5) The split functions Ui(cos )) and Li(cos p) are defined in Appendices I and II of [2] and are in non-integral form. They may, however, appear cumbersome in form but do not present any computational difficulty. It is further noted that (1) and (4) will reduce to the known result for the perfectly conducting n - 0. 1 The next geometry in progression is created by adding a stub with an impedance surface between the half planes as shown in Fig. 3. Subsequently, by letting d -+ 0 one can obtain the diffraction by a thick impedance edge. In turn, image theory can be invoked for the analysis of the original step protrusion shown in Fig. 1. Specifically, we have that

ESP( ) - ESHP( ) - ESHP( 2k - )e sin (6) and HS (p o - P HSHP (p ) -H (,27o - p)e-2jk sin. (7) z 0 z 0 z sP sP In the above E and H denote the scattered fields in the presence of z z the ground plane while EzHP and HzHP correspond to the scattered fields by the impedance half plane of thickness twice that of the step. When the ground plane is replaced by an impedance plane, (6) and (7) are not valid and must be modified as discussed in [2]. The analysis of the geometry in Fig. 3 involves the coupling of the incident field to waveguide modes, the reflection of these modes from the stub, the subsequent re-radiation of the modes as well as their multiple interactions between the waveguide opening and the stub. Since the final geometry of interest is the thick impedance edge in Fig. 2, we may assume in this analysis that the inner walls of the parallel plate waveguide are perfectly conducting without loss of generality. Further, by restricting z < x/2, all modes other than the TEM mode are attenuating and can therefore be neglected for a first order analysis. The TEM mode exists only in the case of H -incidence and thus the results in (1) for the Ez-incidence is a good approximation for the thick impedance edge when z < x/4. Inclusion of the TEM mode (and all of its multiple interactions) for the H incidence results to the following expression for the z diffracted field from a thick impedance edge: Hs Hsa + e-j-/4 e-jkr Co ( o)roLo( (8) z z vA2k Vk 1 - r R 0 0

-4 - where L2+ (cos o) Co(o) = ( (9) L(.) 2j-jkL sin Lt L2(1) o cos (/2 U2(cos ) sin(k sin )(10 L2+ (1) 2+(11) ~ = U2+(1) n - 1 = 1 (12) 1 L (cos () = 2F/k sin (/2 L2+(cos p) (13a) and U (cos ( ) = 2A cos (/2 U2+(cos ((). (13b) 2 According to Fig. 4, Co is referred to as the coupling coefficient, L~ as the launching coefficient, Ro as the reflection coefficient and ro is the plane wave reflection coefficient from the impedance stub. It is noted that the inclusion of the TEM mode was done as if the two parallel half planes were perfectly conducting [3]. However, such an approximation is not expected to compromise the accuracy of the results [4] since the coupling effect is dominant when ( is near 180 degrees where our assumption is valid. The accuracy of the above analysis can be improved (especially for the E -incidence) by the inclusion of additional modes. Expressions for the coupling, launchingin and reflection coefficient have already been obtained via the angular spectrum method and we are currently in the process of including them in the computer program.

-5 - 3. Analysis via the Geometrical Theory of Diffraction As mentioned earlier, this analysis assumes that 2z is large (> x/2). Using GTD one can obtain the scattering from a thick edge (see Fig. 5) in a direct manner. Assuming that the surface coating is lossy, one needs to only consider the first order mechanisms illustrated in Fig. 5. The individual contribution of each of these mechanisms requires the use of the diffraction coefficient for a right angled impedance wedge which is given in [5]. The diffracted field by a step protrusion is obtained via image theory according to Eqs. (6) and '(7). However, in case of a perfectly conducting thick edge or step (n = 0) one should also include the contribution of the higher order 1 riiechanisms illustrated in Fig. 6. These need only be included in the H -incidence case since they give zero field in case of E -incidence. Analytical expressions for the higher order fields were obtained in a closed form via the self-consistent GTD approach. 4. Comparison of Calculations with Measured Data An extensive number of backscatter measurements were collected in order to verify the analysis discussed earlier and in [2]. The major problem in collecting measured data for theory verification purposes is the construction of appropriate test models. In this case our test model must isolate the backscattering by a single thick edge. One such test model is shown in Fig. 7. Provided all measurements are performed in the xy-plane (see Figs. 1 and 5), the only backscatter mechanisms will be from the front edge located at x = 0. Any scattering by the rear tip and the side edges will be negligible. In order to further reduce the scattering due to interactions between the side edges,

-6 - magnetic absorber strips were initially placed around the side edges when performing measurements with the perfectly conducting thick edge. This is shown in Fig. 7. However, it was found that such a precaution was not necessary. An appropriate test model for a step protrusion on the gound plane is illustrated in Fig. 8. Based on the ground plane length in front of the step, it is concluded that the ground plane effect will be present to within five degrees of the edge-on incidence (q = 180 degrees). The analysis discussed earlier was restricted to two-dimensional geometries (see Figs. 1 through 6). However, since the test models are of finite extent along the z-direction, a relationship is required between the measured and calculated results. Using the equivalent current concept it is found that the radar cross section of a finite length edge is given by JZG (14) 3-D k d2-D (14) where Zd is the length of the edge, IES 2 2 D = 2Tr (15) lim r + o HS 2 is the echo width of the edge and Es with Hs denote the total scattered Z Z field. For the models in Figs. 7 and 8, Zd = 1.5 ft.

-7 - Measurements were performed at 9 GHz on two thick edge models corresponding to thicknesses 29 = 0.614 inch and 29 = 1.0 inch. Both of these were measured without coating, however, only the first was tested with a 0.03 inch material coating. This coating was characterized with a relative permittivity of er = 20-jl and pr = 1.4-jl.5 corresponding to n = 0.689 + j.812. Measurements were also performed on the step protrusion model shown in Fig. 8 with A = 0.307, i.e., half the thickness of the first thick edge model. The material coating was placed on the step protrusion model as illustrated in Fig. 1. Figure 9 presents a comparison of the measured and calculated backscatter patterns from a perfectly conducting edge with 2Z = 0.614 inch. As seen for the H -incidence the angular spectrum method is in excellent agreement with the measured data. A good agreement is also observed for the EZ-incidence. However, as indicated earlier, in case of the E -incidence the angular spectrum method needs to be improved by the inclusion of additional modes. It is further noted that the ripple in the measured patterns is due to the interactions between the front edge and the rear tip. We also observe in Fig. 9 that the GTD patterns are in remarkable agreement with the measured patterns for either the E or H cases. This clearly indicates that the simple GTD analysis is applicable to thicker edges (29 > x/2). Such a statement is verified in the edge on echowidth plot of Fig. 10. It is observed that the ideal changeover point from the angular spectrum method to the GTD is at 2z9 x/2. As a result, in the patterns of Fig. 11 which correspond to a perfectly conducting edge with 2z = 1.0 inch, the GTD patterns nearly overlay the measured patterns corresponding to the model in Fig. 7.

-8 - When the aforementioned coating is placed over the thick edge with 2a = 0.614 inch, the results for the H and E incidences are shown z Z in Figs. 12(a) and 12(b), respectively. The agreement between measured and calculated patterns is not as good as that observed for the perfectly conducting case. This is probably due to the test model accuracy in representing the computer model. The patterns given in Figs. 13 and 14 constituted the primary objective of this research. Clearly a good agreement is observed between measured and calculated data for both the perfectly (Fig. 13, H and Ez incidences) and imperfectly (Fig. 14, Hz and Ez incidences) step protrusion on a ground plane. A comparison of the H and E patterns z Z indicates the high cross section associated with the H case when at edge-on (q = 180 degrees). The cross sections for the E case vanishes at edge-on incidence. 5. Family and Design Curves The presentation of the measured data in the previous section served as a verification of the accuracy of our analytical models. Now that the degree of this accuracy has been established we may proceed to obtain a series of patterns for edges and step protrusions as a function of the thickness 2k. The following collection of patterns refer to the geometries in Figs. 1 and 5. Figures 15 through 18 each contain ten backscatter patterns corresponding to a specific value of 2z/x. Among those patterns 22 ranges from 0.01x to x. Note that the case of 22 = 0.468x corresponds to the thickness of the test model. The patterns corresponding to 2. < 0.5x were computed via the angular spectrum method whereas the

-9 -patterns with 2t > 0.5x were computed via the GTD (see Fig. 10). Family curves of the echowidth for both polarizations are included. Figures 15 and 17 refer to a perfectly conducting and imperfect thick edge geometry shown in Fig. 5. In addition, Figs. 16 and 18 refer to a perfectly conducting and imperfect step protrusion on a ground plane as shown in Fig. 1.

-10 - References 1. P. C. Clemmow, "A method for the exact solution of a class of two-dimensional diffraction problems," Proc. Royal Soc. A (205), pp. 286-308, 1951. 2. J. L. Volakis, "Diffraction by a pair of impedance half-planes and an impedance half-plane on a ground plane", The University of Michigan, Radiation Laboratory Report No. 388967-1-T, May 1985. 3. S-W Lee and R. Mittra, "Diffraction by a thick conducting half plane and a dielectric-loaded waveguide," IEEE Trans. Antennas and Propagat., AP-16, No. 4, pp. 454-461, July 1967. 4. T.B.A. Senior and J. L. Volakis, "Scattering by an imperfect right angled wedge," submitted to IEEE Trans. on Antennas and Propagat. 5. G. D. Maliuzhinets, "Excitation, reflection and emission of surface waves from a wedge with given face impedances," Sov. Phys. Dokl., pp. 752-755, 1958.

-11 - A y A X Fig. 1: Geometry of a step protrusion with a surface impedance n as shown.

-12 - A y A I X, I Xk v / li-~ Fig. 2: Geometry of a pair of parallel half-planes with a surface impedance n on their outer surfaces. 1

-13 - d L -' I I/ -- lO1 2 n1 Fig. 3: Two half-planes with an impedance stub inserted. d = 0 constitutes a thick half-plane.

-14 - A 00 0 Ao*o^= I A AC^ 00 L4 l _I I ----- AoLo(0) 00_ Fig. 4: Illustration of the coupling, reflection and launching coefficients.

-15 - Fig. 5: First order diffraction mechanisms from a thick edge.

-16 - ( r n r $1 7 ->1 l l - u I - \a — Fig. 6: Higher order diffraction mechanisms from a thick edge. (a) Direct wave. (b) Image wave.

2f1 am -I 1.5' - - 1.5 = Zd BACK VIEW TOP VIEW FRONT VIEW Fig. 7: Measurement Model of a Thick Edge

SIDE VIEW A x - l K- 1/10" PLATE ff m FRONT VIEW TOP VIEW r U 21 1.5' 00 I - 1.5' 1' - 1.5' - Fig. 8: Measurement Model of a Step Protrusion.

5 BACKSCATTERING FROM THICK HALF PLANE =' IMEASURED (MODEL) Ami n 4~ a. -&4.. ^. u t- *"'~'* — a bHz / IW UA SeTi* 1 ~4~. = 9 GH ANrG-ULAR SPECTRUM (TEM MODE ONLY) Zd = 46 cm GTD V' CIO 77 -7 0 ~,. 3C -5 // r^ ^ '\ ^.Ij / -10 -2 0.-r L, -0 50 —~-~ - ^-~-~ —~-L ----~ ---~ --- —-1 — -— 1 —~ ---~L-^ — 090 0 1508 SCATTERING- ANGLE (DEGREES) 50 Fig_ 9(a): Comparison of measured and cal. p edge With -~" 0....4 inc.*,'. (a) - conducing thick z nidne I — _ I

BACKSCATTERING FROM THICK HALF PLANE 5 --- MEASURED (MODEL) -- ANGULAR SPECTRUM (NO MODES) -.- GTD U) m a z 0 () U) U) Uf) 0 0r C) 0 -5 -10 -15 I I -20 0 30 60 90 120 SCATTERING ANGLE (DEGREES) 150 180 Fig. 9(b): Comparison of measured and calculated backscatter patterns from a perfectly conducting thick edge with 2z = 0.614 inch. (b) E incidence.

-21 - BACKSCATTERING FROM THICK HALF PLANE HZ - INCIDENCE INC. ANG. = 179.9 SCAT. ANG. = 179.9 Normalized to fT(2k)2 20. 10. r< A r3 c4-).r0 w{.> Angular Spectrum 0. -10. GTD -20. 0.00 0.50 1.00 1.50 2.00 Thickness, 2s in wavelengths Fig. 10: Echowidth of a thick perfectly conducting edge as a function of its thickness 2z/x.

BACKSCATTERING FROM THICK HALF PLANE 5 ' Hz incidence J - MEASURED (MODEL) 2, = 2.67 cm j J --- GTD freq. = 9 GHz i0 Zd =46cm (1). '/=0 0 30 60 90 120 150 180 z -l U) -20 -. / Fig. 11(a): Comparison of measured and calculated backscatter patterns from a perfectly conducting thick edge with 2z = 1 inch. (a) Hz incidence. I r~ l I3

BACKSCATTERING FROM THICK HALF PLANE 15 10 0 -v, U) c, 0I v> 0, FC) 5 0 I I -5 -10 -15 -20 0 30 60 90 120 SCATTERING ANGLE (DEGREES) 150 180 Fig. 11(b): Comparison of measured thick edge with 29 = 1 and calculated backscatter patterns from a perfectly conducting inch. (b) Ez incidence.

BACKSCATTERING FROM THICK HALF PLANE 0 -5 --- MEASURED (MODEL) --- ANGULAR SPECTRUM (TEM MODE ONLY) --- GTD -10 U) cn 0 —.,. C) -- o LL C/) rU) CD -15 -20 -25 I rN 4 -I -30 -35 0 30 60 SCATTERING 90 120 ANGLE (DEGREES) 150 180 Fig. 12(a): Comparison of measured and calculated patterns from an imperfect edge with n = 0.689+j.812 and thickness 29 = 0.614 inch. (a) Hz incidence.

-5 - /- MEASURED (MODEL) E incidence I --- ANGULAR SPECTRUM (NO MODES) 2t = 1.56 cm I! G\\TD freq. = 8.930 GHz\. - Zd = 46 cm -= (0.689, 0.812) // / /V &. -15. /X, 1' / 'I -305, // 3 -15 -/ -/ -20 // - / LLf // /. U -25 0 1 // O -/ I / / I / / / I 30 I f,_ _ // I I - I I I -35 - 0 30 60 90 120 150 180 SCATTERING ANGLE (DEGREES) Fig. 12(b): Comparison of measured and calculated patterns from an imperfect edge with n = 0.689+j.812 and thickness 22 = 0.614 inch. (b) Ez incidence.

BACKSCATTERING BY A STEP PROTRUSION ON A G.P 15 - i l H incidence -- MEASURED (MODEL) 2z = 1.56 cm I!.. ANGULAR SPECTRUM (TEM MODE ONLY) freq. = 9 GHz - GTD Zd = 46 cm 77-~ ~o - i I Cn 5 -I/ i ~ 'I.. / LZi / \ \ / ~ -10 0 30 60 90 120 150 180 SCATTERING ANGLE (DEGREES) Fig. 13(a): Comparison of measured and calculated patterns for a perfectly conducting step protrusion r). -15 F) with a = 0.307 inch. (a) Hz incidence.

BACKSCATTERING BY A STEP PROTRUSION ON A G. P 5 0 - MEASURED (MODEL) --- ANGULAR SPECTRUM (NO MODES).. GTD Or) m cn CO vz 0 I — Cr) LJ U) 0 Q: U) -5 -10 -15 -20 I ^i -25 -30 0 30 60 90 120 150 180 SCATTERING ANGLE (DEGREES).., Fig. 13(b): Comparison of measured and calculated patterns for a perfectly conducting step protrusion with z = 0.307 inch. (b) Ez incidence.

BACKSCATTERING BY A STEP PROTRUSION ON A G.P 5 0 MEASURED (MODEL) --- ANGULAR SPECTRUM (TEM MODE ONLY) —..- GTD.-% 0U) m Q Io 0 LU U) 0 (r -5 -10 -15 -20 -25 -30 0 30 60 90 120 SCATTERING ANGLE (DEGREES) 150 180 Fig. 14(a): Comparison of measured and calculated patterns from an imperfect step protrusion n = 0.689+j.812 and z = 0.307 inch. (a) Hz incidence.

BACKSCATTERING BY A STEP PROTRUSION ON A G. P i d. L.... %, -5 "! 'Y,, E incidence z / 2z = 1.56 cm / \ I freq. = 8.930 GHz / / Zd = 46 cm \ 7, = (0.689, 0.812) / / \ 5 - / /o 0 30 60 90 120 150 180 F\\ SCATTERING ANGLE (DEGREES) Fig. 14(b): Comparison of measured and calculated patterns from an imperfect step protrusion -20 o \r \ ~ — MEASURED (MODEL) I I --- ANGULAR SPECTRUM (NO MODES) -35 I I! -40 I 0 30 60 90 120 150 180 SCATTERING ANGLE (DEGREES) Fig. 14(b): Comparison of measured and calculated patterns from an imperfect step protrusion n = 0.689+j.812 and Q = 0.307 inch. (b) Ez incidence.

-30 - BACKSCATTERING FROM THICK HALF PLANE HZ - INCIDENCE ETA = ( 0.001, 0.000) 20. 10. p — 1 '),4 r: ti 0 a 0. -10. j h g f e d c b -20. -30. 0. 30. 60. 90. 120. 150. 180. SCATTERING ANGLE(DEGREES) Fig. 15(a): Echowidth of a thick perfectly conducting edge for various values of the edge thickness, 2z. (a) Hz incidence.

-31 - BACKSCATTERING EZ - INCIDrENCE FROM THICK HALF PLANE ETA = ( 0.001, 0.000) 20. 10. a b c d e f 9 h i J 2~/X.01.05.1.25.3.4.468.55.75 1 i, — -^ CfL 0 a-U P4 0. -10. e d - c b a -20. -30. 0. 30. 60. 90. 120. 150. 180. SCATTERING ANGLE(DEGREES) Fig. 15(b): Echowidth of a thick perfectly conducting edge for various values of the edge thickness, 2z. (b) Ez incidence.

-32 - BACKSCATTERING BY HZ - INCIDENCE STEP PROTRUSION ON G.P. ETA = ( 0.001, 0.000) 15. 5. Q:l 0 -(9 -5. -15. J i h 9 f e d c b -25. -35. Fig. 16(a): 0. 30. 60. 90. 120. 150. 180. SCATTERING ANGLE(DEGREES) Echowidth of a perfectly conducting step protrusion on a ground plane for various values of the step heighth, a. (a) H incidence. z

-33 - BACKSCATTERING BY EZ-INCIDENCE STEP PROTRUSION ON G.P. ETA = ( 0.001, 0.000) 15. 5. 0-. C)q 0lla u, -5. -15. -25. -35. 0. 30. 60. 90. 120. 150. SCATTERING ANGLE(DEGREES) 180. Fig. 16(b): Echowidth of a perfectly conducting step protrusion on a ground plane for various values of the step heighth, a. (b) E incidence.

-34 - BACKSCATTERING FROM THICK HALF PLANE - T - Tf ' ENT ETA = (.0E9, a.812) 15. -1 I 1 5. - I 2z/ a -.01 b -.05 c -.1 d -.25 e -.3 f-.4 g -.468 h -.55 i -.75 j-1 --,, —. to P4 j - h f -e g 9 -5. -15. a C b -25. -35. 0. 30. 60. 90. 120. 150. SCATTERING ANGLE(DEGREES) 180. Fig. 17(a): Echowidth of an imperfect edge (n = 0.689+j.812) for various values of the edge thickness, 2z. (a) HZ incidence.

-35 - BACKSCATTERING FROM THICK HALF PLANE EZ - INCIDENCE ETA = ( 0.689, 0.812) 15. - 5. - a b c d e f g h i j 2S/x.01.05.1.25.3.4.468.55.75 1,-4 H-.. -5. -15. j h 9 f e d - c - b a -25. -35. 0. 30. 60. 90. 120. 150. 180. SCATTERING ANGLE(DEGREES) Echowidth of an imperfect edge (n = 0.689+j.812) for various values of the edge thickness, 2z. (b) Ez incidence. Fig. 17(b):

-36 - BACKSCATTERING BY STEP PROTRUSION ON G.P. HZ - INCIDENCE ETA = ( 0.689, 0.812) 15. 5. (f) Q1 p E0-4 PL -5. -15. -25. -35. 0. 30. 60. 90. 120. 150. 180. SCATTERING ANGLE(DEGREES) Fig. 18(a): Echowidth of an imperfect step protrusion (n = 0.689+j.812) for various values of the step heighth, Z. (a) Hz incidence.

-37 - BACKSCATTERING BY STEP PROTRUSION ON G.P. EZ-INCIDENCE ETA = ( 0.689, 0.812) 15. 5. F-q Cl9 0 U k9 0D 44 -5. -15. -25. -35. 0. 30. 60. 90. 120. 150. SCATTERING ANGLE(DEGREES) Echowidth of an imperfect step protrusion (n = 0.689+j.812) 180. Fig. 18(b): for various values of the step heighth, Q. (b) Ez incidence.