AFXVrL TB"R- 7 5-912 012643-42-.T 'VMiClw Ji/: IT OF GA 00-TWA r 1 ", %I VI 17 IV I,-% Ml' I -IL,;i i I Xa -o —icn LcLoractory FIELD PENETRATION INTO A CYLINDRICAL CAVITY I I j T. B. A. &miC-,nior tiluarv 1975 Prepared fir T h I. Dikew',.,-nood Corporation.Albuqu~,rquo, New Mlexico 12643-2-T = RL-2259 Ann Arbo~r, Michigan

FIELD PENETRATION INTO A CYLINDRICAL CAVITY 0 T.B.A. Senior Radiation Laboratory Department of Electrical and Computer Engineering The University of Michigan Ann Arbor, Michigan 48104 January 1975 Abstract To test the efficacy of a direct integral equation approach to the study of cavity-aperture interactions, the problem of an E-polarized plane electromagnetic wave incident on a thin, perfectly conducting cylindrical shell with a slit aperture is considered. A computer program is constructed for the solution of the appropriate E field integral equation. Data are presented showing the behavior of the field inside the cavitv and in the aperture for a variety of aperture and cavity dimensions, and some information is obtained about the SEM singularities and their dependence on aperture size. This study was performed under subcontract to The Dikewood Corporation 1009 Bradbury Drive, S.E. University Research Park Albuquerque, New Mexico 87106

, PREFACE The author is indebted to Mr. E. F. Knott for developing the computer program used in this study.

'CONTENTS Section Page I INTRODUCTION 1 II MATHEMATICAL FORMULATION 2 III NUMERICAL RESULTS 6 IV CONCLUSIONS 9 APPENDIX 29 REFERENCES 40 i

ILLUSTRATIONS Figure Page 1 The geometry 2 2 Shell current amplitudes for 0 = 100, a = 0 and various values of ka. 10 3 Shell current amplitudes for 0 = 10, a = 0 and various values of ka. 11 4 Shell current amplitudes for 0 = 30, a = 0 and various values of.ka. 12 5 Shell current amplitudes for = 30, a = 0 and various values of ka. 13 6 Aperture field amplitudes for 0 = 100, a = 0 and various values of ka. 14 7 Aperture field amplitudes for 0 = 100, = 0 and various values of ka. 15 8 Aperture field amplitudes for 0 = 300, a = 0 and various O values of ka. 16 0 9 Aoprt.ure field amplitudes for ~ = 30~ a = 0 and various values of ka. 17 10 Field amplitude at the center of the aperture for 0 = 100 and 30~, and a = 0. 18 11 Aperture field amplitudes for 0 = 10, ka = 2.5 and a = 0(45~)180~. 19 12 Aperture field amplitudes for 0 = 30, ka = 2. 5 and a = 0(450)180~. 0 20 13 Interior field amplitudes for 0 = 10~, a = 0 and various values of ka. 21 14 Interior field amplitudes for 0 = 10, a = O and various values of ka. 22 15 Interior field amplitudes for 0 = 30, a = 0 and various values of ka. 23 16 Interior field amplitudes for 0 = 300, a = 0 and various values of ka. 24 17 Interior field amplitudes for 0 = 30, ka = 2. 5 and c =0(45~)180. 25 ii

Illustrations, continued 18 Field amplitude at the center of the cavity for 0 = 10~ and 30~, and a = 0. 0 26 19 Field amplitude at the center of the cavity for 0 = 100 and 300, and a = 0 27 20 First interior resonance for 0 < 0 < 40~. 28 0 o - iii

* SECTION I INTRODUCTION The present study was motivated by some recent investigations of the coupling of an electromagnetic field into a spherical cavity. In the particular case when the cavity is bounded by a thin, perfectly conducting shell having a circular aperture, it is not unnatural to expand the interior and exterior fields in spherical modes, and as shown in ref. 1, it is then possible to calculate the fields inside the cavity. Unfortunately, there are difficulties, most of which are attributable to the poor convergence of the interior mode expansions in the vicinity of the boundary. These have been discussed by Senior (ref. 2), who has proposed instead an alternative formulation based on the E field integral equation for the total current induced in the shell. The resulting coupled integral equations for the tangential components of the current appear quite amenable to solution by the moment method, and are also convenient for the numerical determination of the complex frequency singularities of the singularity expansion method (SEM). In order to demonstrate the efficacy of this approach, we here consider the simpler two-dimensional problem of a plane wave incident on a thin, perfectly conducting, cylindrical shell having a slit aperture. In most respects, this problem is physically and mathematically akin to the spherical one, and our original interest was to pursue the solution only far enough to verify that there are no difficulties involved. Nevertheless, the problem does have interest in its own right, and data have been obtained for the currents, aperture and interior fields for a variety of aperture angles and ka in the range 0. 25 K ka < 4. 0 where ka is the electrical circumference of the (closed) cylinder. Selected data are presented in Section II, along with information about the first few complex singularities and their dependence on aperture size. A description and printout of the computer program are included as an appendix. 1

SECTION II MATHEMATICAL FORMULATION A thin, perfectly conducting cylindrical shell of radius a having a slit aperture of half angle o is illuminated by an E-polarized plane wave incident in a plane perpendicular to the z axis of the cylinder. In terms of the polar coordinates (p, 0, z) the equation of the shell is p = a, o <0< 27 -, and the incident electric field is taken to be Ei ^ -ikpcos( -a) (1) - z e -iwt (see Figure 1), where a time factor e has been assumed and suppressed. Figure 1. The Geometry Since the shell is infinitesimally thin, it can be represented by an electric current sheet of strength J (s'), where J (s') = z J (s') is the total current borne by the shell. The scattered electric field at a point having the position vector is then the shell. The scattered electric field at a point having the position vector e is then 2

kZ ( ES-() = - t) H (kR) ds (2) C where Z = 1/Y is the intrinsic impedance of free space, H( is the Hankel function of the first kind and R is the distance between the integration and observation points. The integration is along the shell (lone sidet only) in the plane perpendicular to the z axis containing the observation point, and consequently s' = a ' with 0 <''-< 27rThe total field is obtained by adding (1) and (2), and if we now allow the observation point to lie on the shell and use the boundary condition at a perfect conductor, the following integral equation results: 27r-0 -ikacos(0 -a) = ka ) H(1)(kR) d. (3) "0 This is mere Ir a sPnecr i! case of an intra>o! eI i o n+r for resis t+ir es h -c+Qgootcs npr;O tr>viou, lconsidered by Knott et al (ref. 3), and a computer program for its solution is described in the Appendix to this report. Having found the current J, the field at any point can Z be calculated from eqs. (1) and (2). We note in passing that for an H-polarized incident plane wave, the integral equation for the total current induced in the shell is more complicated than (3), but a rather general computer program with which its solution can be obtained is available (ref. 4). In contrast to the mode-matching method used by Senior and Desjardins (ref. 1) in the solution of the corresponding problem for a sphere, the direct integral equation approach is convenient for the calculation of the complex frequency (SEM) singularities, and the determination of their dependence on aperture size. For the interior region p < a, the cylindrical mode expansion has the form a J (kp) e 3

and if the cavity were closed, the singularities would be the real resonant frequencies corresponding to the zeros of J (ka), i. e. mn where u mn few are is the mth zero of J (u). In order of increasing magnitude, the first n w = 2. 405 c/a w = 3. 832 c/a w = 5. 136 c/a w = 5. 520 c/a (m=l, n=0) (m=l, n=l) (m=1, n=2) (m=2, n=0). Each has its counterpart in the case of a spherical cavity, and the first two are even similar in magnitude to those for a sphere. If the slit is now opened, the complex frequencies must take on a negative imaginary part associated with the radiation damping of the modes, and in addition it is expected that the real parts will decrease with increasing aperture size. The exterior region p > a is rather different. The cylindrical mode expansion here is b H() (kp) e n n n = -oo and for a complete perfectly conducting cylinder the complex frequencies are = v c/a mn 4

where v is the m zero of H(v). The nature of these zeros is discussed in mn n, ref. 5 (see also ref. 6), and the zero with the smallest imaginary part is 10 =-2.404 - i0. 3405. In addition, there is a branch point at w = 0, which has no counterpart in the case of a finite body. 5

SECTION II NUMERICAL RESULTS With the aid of the computer program described in the appendix, the integral equation (2) has been solved to give data for the fields inside the cavity and in the aperture, as well as some information about the complex frequency singularities. No difficulties have been experienced in any of the more than 100 individual runs that have been made so far. Although many of these runs have been directed at the complex frequency singularities, with the program interrupted following the computation of the determinant, it is evident that only a small selection of the data can be presented here. For the data which follow, the number of sampling points used was increased almost linearly with ka, from a minimum of 12 for ka < 1. 0 to a maximum of 48 for ka > 3.5. The largest ka considered was 4. 0, and most attention was directed at cavities having aperture half-angles 0 = 10 and 300 with the plane wave at symmetrical incidence, 0 i. e. a = 0 The amplitudes of the total currents J (0) induced in the shell are illustrated in Figures 2 through 5. Since a = 0, the currents are symmetrical about 0 = 180~, and only the ranges 0 _ 0 < 180 are displayed. As expected, the currents are infinite at = 0. The curves become increasingly complex as ka increases; note the enhanced values of J for 0 = 30 when ka = 2.3, i.e., close to the first z o resonant frequency of the cavity. The amplitudes of the corresponding aperture fields are shown in Figures 6 through 9. The fields are zero at 0 = _+ 0 in accordance with the edge condition, and in contrast to the shell currents, the aperture fields are rather simply behaved. This is in line with the observation in ref. 1, though we note that even for 0 = 300 and ka = 4. 0, the aperture is still only 0. 67k in width The fields are a maximum at the first resonant frequency of the cavity, but very small at the next (ka = 3. 83), and the variation with frequency is brought out in Figure 10, where the 6

amplitude at the center of the aperture is plotted as a function of ka for 0 = 10~ and 30 0 = 30. Not surprisingly the fields are larger for the larger aperture, but the general behavior is remarkably close to that found in ref. 1 for a circular aperture into a spherical cavity. The effect of oblique incidence is illustrated in Figures 11 and 12. The program was also designed to compute the fields inside the cavity at sampled points along the line 0 = 0 from the center of the cavity to the aperture, and some data for the amplitudes as a function of position are presented in Figures 13 through 16. The curves are all for a = 0, and the strong excitation when =30 and ka = 2. 3 is very clear. Increasing a decreases the excitation, and this is shown in Figure 17 in which the amplitudes for a = 0(45) 1800 and ka = 2. 5 are plotted. To bring out the resonance effect, the field amplitude at the center of the cavity is plotted as a function of ka for 0 = 100 and 0 = 30~ in Figure 18. As expected, opening up the aperture O0 0O detunes the cavity and shifts its resonance to a lower frequency. This is more evident at the first resonance than at the second, and because of the sharpness of the first resonance. particulariv for the smaller aperture, we nave repiotted te cidaa uf Figure 18 on a logarithmic scale in Figure 19. In line with our original objective of investigating the integral equation approach in all phases of its operation, we have also given some attention to the complex frequency singularities of SEM. To locate a singularity, the program was run at each of a set of complex frequencies surrounding the expected value, with the program (in general) interrupted once the determinant had been computed. From an examination of the results, a new set of frequencies was selected, and so on until the zero of the determinant was found to the accuracy desired. No attempt was made to mechanize the procedure. A plot of the first interior resonance as a function of the aperture half-angle 0 is shown in Figure 20. Although any opening of the cavity must shift the frequency into the lower half of the complex w plane and decrease its real part, the effect is very small for a 10 angle, but increases rapidly with increasing o. The computed values Z: I ~- I~-15 L — J ~ ~YILIV DCII 7

are listed in Table 1, along with isolated data for other resonances. For the second Table 1 SEM SINGULARITIES wc/a Interior Exterior de first second first 0 2.405 3.832 -2.404-i0.341 10 2. 400 - i 0.001 20 2. 359 - i 0.015 30 2. 302 - i 0.067 3.827 - i 0.003 -2.38 - i 0.39 40 2.25/- i 0. 160 interior resonance, even 0 = 300 gives only a small shift in frequency comparable 0 to that produced ov 0 = i0 at the first resonance. The first exLerioi' resonance proved more difficult to locate: the initial (sparse) sampling of the complex frequency plane pointed inexorably to the logarithmic singularity at w = 0, and only after a more detailed search was the zero found. As regards the determinants considered, it is believed that the data for the interior resonances given in Table 1 have at most an uncertainty of unity in the third decimal, but no statement of absolute accuracy is possible without a more detailed investigation of the effect that the number N of sampling points has. 8

* SECTION IV CONCLUSIONS The results obtained leave little doubt that the E-field integral equation is an effective approach to the analysis of the thin-shell cavity problem, and provide encouragement for a study of a spherical cavity using the formulation given in ref. 2. However, the data for a circular cylinder are also of interest themselves, and it would seem desirable to pursue the present calculations further to locate more of the SEM singularities, including their complete dependence on J, and their excitation coefficients. 9

4.0' 3.0 -zo [Jz(0)i 2.0 -1.0-.5 2.3 0 0...f I a I I I I I 10 30 60 90 120 150 0, degrees Figure 2. Shell current amplitudes for- = 100, a = 0 and various values of ka. 10 180

4.0L 3.0 zJJ (0)l 42.0-1 \ V \0 5 83 83 3.5.0 10 30 60 90 12J0 1o 180 0. degrees Figure 3. Shell current amplitudes for 0 = 10o, a = 0 and various values of ka. 11

8.0 6.0 zo IJZ(P I 4.0 -2.0 - 2.3 2. v1 1.3 30 60 90 120 150 180 4, degrees Figure 4. Shell current amplitudes for o = 30~, a = 0 and various values of ka. 12

4.0 - 3.0 -Z IJZ()1I - 2.o 2.0 - 2.5 01 3( I -I I I ) 60 90 120 150 180 0, degrees Figure 5. Shell current amplitudes for 0 = 300, a = 0 and various values of ka. 13

0. 8 2.3 0. IEz() I/ 0.4 -0.2 -On 2 0 -10 I I i 1[ 0, degrees Figure 6. Aperture field amplitudes for 0 = 10~, a = 0 and various values of ka. 14

3.5 0.6 -IEz(0) I 0.4 -O. 2 -0.2 -0 4.0 2.5 a I -10 I 0 0, degrees I 1 10 Figure 7. Aperture field amplitudes for 0 = 10, a values of ka. 15 = 0 and various

4.0 3.0 -IEz(0)I - 2.0 -1.0 -0 1.0 0.5!I I -30 0 0, degrees I I 30 Figure 8. Aperture field amplitudes for 0 = 30, a = 0 and various values of ka. 16

4.0 - 3.0 3.5 3.8 - 2. -3 4. 0 2.5 3.83 -30 0 30 0, degrees Figure 9. Aperture field amplitudes for 0 = 30~, a = 0 and various values of ka. 17

4.0 - 3.0- - 30 2.0-: 30~ 1.0"0 10 ~ 0/ 0 ~ ~ j --- —---— ^ --- —--— ^ --- —--------- --------- ka Figure 10. Field amplitude at the center of the aperture ~)for = 10~ and 300, and c = 0. o

0.4 - 0.3 -E z(0) 0.2 - 90~ 1 8 0 135~ 0.1 - 0 -1 a I I.... i........~.... 0 I 6 0, degrees I i6 Figure 11. Aperture field amplitudes for 0 0 = 10~, ka = 2.5 and a = 0(45~)180~. 19

1.0 00 0.75 45~ IE() // 1/ \ \ 0.5 0,25 1350 0 -30 0 30 5, degrees Figure 12. Aperture field amplitudes for 0 = 30~, ka = 2.5 and a = 0(450)180~. 20

ft 0. 0. IEz(P) I 2.3 2.0 0. 0.5 0. 0 0 1.0 p/a Interior field amplitudes for 0 = 10 a and various values of ka. 21 Figure 13.

2.4 -1.8 - 4.0 EZ(PI /I / 3.5 3.0 - 1.0 0.5 p/a Figure 14: Interior field amplitudes for 0 = 10, = 0 and various values of ka. 22

8.01 -6.0 2.3 4.0 2.0 2.0 0 1.0 p/a 0~ a and various values of ka. Figure 15. Interior field amplitudes for 0 = 30, 23

8.0 6.0 IE (P)I 4.0 3.5 424 2.0 2. 5 3.0 1.0 0.5 0 p/a Figure 16. Interior field amplitudes for A = 30~, a 0 and various values of ka. 0 24

4.0 3.0 JEZ(P)j - z. Ui 1. 00 1.00 900 1350 0 1.0 1 00 0. 5 P/a 0 Figure 17. Interior field amplitudes for 300 00 9rr c

0 0 CIO a- Q- 1 -— ft -..m..w - -- - - - - - - - - - - - - - - - - --- - - - - - - - - - - - - -. - - - -- -ft. N 0 4 0 1 t cl r-4 CD C).f-4 P4 0 0 CY') I 06 C) C) tc;...ON w 26

I I II II II 10 1 -1 10 -2 I rI II 30~ I I N) 10~ / / / 0 \10 N% %4 N N41.N / / / / / / / / / / / / / 0 ka Figure 19. Field amplitude at the center a = 0. of the cavity for 0 =10 and 30, and 27

0~ 0 20 30~ -U. i Im. wa/c o -0. 2 -0. 3 I I I I I 2.2 2.3 2.4 Re. wa/c Figure 20. First interior resonance for 0 < 0 < 40. O - 28

APPENDIX SLOTTED CYLINDER COMPUTER PROGRAM INTRODUCTION We here describe the essential features of the computer program that solves the two-dimensional integral equation (3) for the total current induced in the wall of a perfectly conducting cylindrical shell having a longitudinal slot. The electrical size of the cylinder and the angular size of the slot are the only body variables subject to control by the user of the program, but there is no limit to the number of angles of incidence that may be specified. The program is named RAMP and is written in FORTRAN for MTS (Michigan Terminal Service) at the University of Michigan, where the basic machine is an IBM 360 computer. Because some features of the MTS system differ from those of other computer facilities, minor alterations would have to be made if RAMP were run elsewhere. The program is patterned after others developed at the Radiation Laboratory that solve similar two-dimensional oroblems but involving morpe ennr.qa nrnfi!pv However, little attempt has been made to exploit the specific symmetry of the slotted circular cylinder, so that the program is not as efficient as it might be. The matrix for this geometry is, for example, symmetric, and the running time could be reduced somewhat were this symmetry to be used. MATHEMATICAL FORMULATION The program solves the E field integral equation (3) for any complex number k whose real part is the free space propagation constant. The contour C extends from one edge of the slot to the other along the cylindrical shell, and is subdivided into M cells, where M is an integer specified on input. Thus, each cell has an angular width A 0=2( - 0)/M 29

where 0 is the half angle subtended by the slot. The width chosen reflects the judgement of the user, since small widths tend to improve accuracy but also to increase machine running time. Much recent data generated by RAMP were obtained using cell widths of X/12 or less, implying M> 12 ka. The linearlization of eq. (3) produces M simultaneous equations for the unknown currents J, assumed constant over each cell. Program RAMP creates an M x M matrix of complex numbers associated with this system of equations, inverts the matrix and then multiplies the resultant matrix by the incident field to obtain the surface currents. The currents, now being known, may be used to calculate the scattered electric field at any point in space from eq. (2). In the actual computation, of course, the program approximates the integral by a discrete sum of M contributions. Aithouh eq. (2) aillows the fTelds to be calculated anywhere, the nrnrgram is very specific and computes them only over the slot aperture (at constant radius a) and along a radius that terminates at the midpoint of the slot. Moreover, it is the total field that is computed, so that once the integration required by eq. (2) is performed, the program then adds the incident field (1) to obtain the total field. PROGRAM DESCRIPTION Program RAMP is an outgrowth of a previous, more efficient version in which the wavenumber k was a pure real number. In general, however, the interior resonances occur at complex wavenumbers, especially for aperture half angles greater than about 10 degrees, hence the generalization to complex k was necessary. In order to achieve this, two modifications were made: a) the provision of a complex number (KFAC in the program) which is chosen by the user, specified on input and subsequently multiplied by the (real) free space wavenumber to generate complex k, and b) the development of a subroutine to generate Hankel functions for complex arguments. It is mainly this latter provision that makes RAMP less efficient than its predecessor. 30

RAMP consists of a MAIN program and two subroutines, FLIP and HANK, MAIN reads all input, prints all output, fills the matrix elements, computes the field distributions from the current distribution and indexes through the desired range of incidence angles. FLIP performs two functions; it inverts the original matrix and multiplies the inverted matrix by the incident field to obtain the surface currents. The bulk of FLIP is virtually a copy of IBM's matrix inversion routine, but modified for complex elements, with a handful of statements added at the end to perform the matrix multiplication'operation. The MAIN program prudently calls for the inversion from FLIP only once, and thereafter expects FLIP to merely supply new current distributions for new angles of incidence. Subroutine HANK is based on the ascending series representation for the Bessel functions of the first and second kinds of order zero. The program decides how many terms of the series to use from the criterion n= 6 + 1. 35 IWI where n is the number of terms used and W is the complex argument. Although the -5 criterion was selected so as to provide absolute precision of better than 10, internal round-off errors tend to be worse than this for complex arguments having large -5 imaginary parts. For real W the accuracy is better than 10 for arguments as large as 10. 0. Since HANK uses double precision arithmic on the IBM 360, it would have to be modified for, say, the CDC-6600 system. Briefly, the program RAMP operates as follows: input data for a single frequency and aperture half angle are read from two consecutive cards from the input stream, as described below. The first entry on the first card is the integer M (the number of sampling ints on the cylinder profile) and is also used as a key to shut down the program; i.e., if M = O - which can be synthesized with a blank card - the program terminates. The MAIN program then fills the matrix, calling on subroutine HANK for 31

the appropriate Hankel functions. The incident field structure is computed for the first angle of incidence, and MAIN then calls FLIP to invert the matrix and supply the current at each cell on the profile. The currents are weighted and summed according to eq. (2) to obtain the total electric field distribution at N discrete points over the aperture and at L discrete points along a radius. N and L are controlled by the user as input data. Since the weighting function is a Hankel function, much of the machine time is spent carrying out operations in subroutine HANK. After the fields over'the aperture and along a radius have been computed and printed on the output record, MAIN indexes to the next angle of incidence by adding an increment (specified as input data). Subroutine FLIP is called, but since the matrix has already been inverted, FLIP merely supplies new values for the surface currents, which MAIN then uses to compute new field distributions. If the new angle of incidence exceeds the limit specified by the user, the program returns to the input stream and reads the first of a pair of cards required to specify a nw geome+try..As mentioned earlier, a blank card will shut down the program. INPUT DATA FORMAT The two input data cards required for a single geometry should contain the following information: Card 1: FORMAT (315) M the number of sampling points on the profile N the number of field points in the aperture L the number of field points along the radius Card 2: FORMAT (8F10. 5) A the radius of the cylinder in inches HANG the aperture half-angle in degrees WAVE the incident wavelength in inches KFAC a complex number whose real part (always 1.0) and imaginary part constitute the fourth and fifth entries on the card, 32

FIRST the first angle of incidence in degrees LAST the last angle of incidence in degrees INK the increment to be used in indexing through the incidence angles In hindsight, the variables A, WAVE and KFAC could have been incorporated in a single complex variable (say KA) making the input structure a little less complicated, but at the time the program was being prepared the variations to be studied were not known precisely. This is because the previous version of RAMP (for real ka only) was modified by the inclusion of the factor KFAC to permit complex values for ka. Consequently the user must decide beforehand what value of (complex) ka is of interest, then calculate A, WAVE and KFAC to be read in as input in order to generate this value. PROGRAM LISTING A listing of the main program and the two subroutines are given on pages 34 through 37. The entire program occupies 30584 32-bit bytes of storage as listed; the required stora~ce wiii change, of course, if the arrays are re-dimensioned so as to accommodate other cylinders (i.e., different surface field sampling points). The program is presently limited to 50 sampling points. OUTPUT SAMPLE Pages 38 and 39 contain a sample of output to illustrate the format The first page lists the current distribution over the cylinder surface and the second lists the total field distributions over the aperture and along a radius. 33

IMiPLICIT COMPLEX(K) C OM PLEX AA (50 r51)rKJ (5(0) rPINK(5) SUMB REAL LASTIINiK DI MENS IO0N F E i-A(50 ) F iAT-(5 C) D-ATA RE-DrDIG/ j.017'4532c,,57.29578/ 10 READ 130, MN,,TL IF (M.EQ.G) GO TO 95 READ 2CCr,f AHANGWAVEKFACFISTLASTINK K=6.2831 85*K-FAC/WAVF KA=K*A KTA=2. ';*KA D FEEF 1, E=2. C'* (1 83. 0-hANG) /M DEv.AT=RFi*DEEFJ'. E ADFFFA*DEFEATI/WAVWE KDAO. 25*KA'*DE-E.AT DO 1 5 I= 1, F EE( I) H AN G +D7E F EAE* (I - 05) 15 F EA T(I)=PA7-1*FE El(I) DO 35 I=1,M1 DO 35 JTh1,M I F (I.B-'jQ.J) GO0 TO 2 5 A NG =A BS ( F EA T ( I), -EAT(3)T) *0.5 SANG=SIN (ANG,-) K RK TA S AN'G C A LL I iA ((RK 1) AA (I,,J)=KDA*KH GO TO7 3 5 35b CO NT INI>; DEL=2.?2Ki"'ANG/1N TET-A=lF:?T- INK 40 TETATl-LTA+I-N'K I F (TF~ T- T. LA ST) GO0 TO 10 T!.HE=RFD* TETA DO '45 I=1,M K AN G=KA'iCO S(THE - F A T(I))*C MPL X (9.3-1. 45 P IF;K(I) =C EX P(KAe N G) I F (T ETAi. EQ0. F IRI ST) O TO 50 C ALL F-L I P (A A. "PINK,,\rK JrSUMfA NG,2) GO TO 62 50 CALL FLIP(7kAA,MPTNK,KJB,DMAG,1) 55 PRT111 '400,r KA,,HANGTFlT-ArDMAG PRINT 3CC DO 6-0 I=1,M AMP=CA135(KJ (I)) F ASE=DIG,*ATA N2(Al Fl~CG(KJ (I)) REAL (KJ (I)) 60 P RIN T 5 00,i I rFE 7"(I1),A MPFAS ErI 62 PRINT 730,n, KA,F{ANG,TrETrA PRINT 900 -DO 75 J=1,N S UM=CM.PL X (.0C,0. 0) AliGLE=DEL* (3-0. 5} -HANG ANG"=RED*A NGLF DO 7 0 = 1,fM SANG=SIN(Q.5*'ABS(ANGi-FEAT(I))) 34

K R=KTA*S ANG CALL HANK (KR,KH) 70 SUM=-SU'-KJ (I) *KH S UM=KDA* SUi KANG=KA*COS (THE-ANG) *CMPLX (0.C,-1.0) S UM=SUI+CEX P (KA N G) AMP=CABS (STJ'") FASE=DIG*ATAN2 (AIMAG (SU!), REAL(SUM)) 75 PRINT 6%0, AItGL, SUMIA'P,FASE IF (L+MN.LT.47) GO TO 82 PRINT 700, KA,HANG,TETA 82 PRINT 800 DIP=A/L LL=L+1 DO 90 J=1,LL R=DIP* (J- 1) RA=R/A S UM=C-PLX (0.,0.0) DO 85 I=1,M KR=K A*SQRT( 1. 0+ RA* (RA-2. C*COS (FEAT (I) )) ) CALL I.ANK(KRKH) 85 S U=SUI-KJ (IT) *Kr T S UM= KDAS U M 1V f'4t — *r It - f- f') *- ' -- f' r% f II I V f t A f r I )> Uri = ulrl + C LX (x K \ *i: ) AMP=CABS (SUM) FASE=DIG*ATAN2 (A!MAG (SUM),REAL(SUM)) 90 PRINT 6CO, RSUM^AMPFASE GO TO 4C 95 CALL SYSTEM 100 FORM AT (315) 200 FORMAT (8F10.5) 300 FORMAT (' CELL NUMBER ANGLE CURRENT AMPLITUDE ' &'CURRENT PHASE CELL NUMBEF'/) - - 400 FORMAT ('1' 27X,'SLOTTED CYLINDER /20X 'KA' 14X 2F8.3/2 X, ' APERTURE HALF- ANGLE',F13. 3/2rX, 'INCIDENT FIELD DIRECTION' &F8.3/20X, 'DEERMINANTI,E21.5//).. -.. 500 FORMAT (I7, F1 5. 2, F17. 5, F 16. 2, I1 2) 600 FORMAT (F9. 2,F17. 5F12.5 F16. 5 F14. 2) 700 FORMAT ('1',27X 'SLCTTED CYLINDERj/2X, KA, 14X,2F8.3/2CX, '#APFRTIJRE HALF- ANGLE',F13.3/2CX,'INCIDENT FIELD DIRECTION F. 3/) 800 FORMAT ('0 DISTANCE',8X,'FIELDS ALONG A RADIUS',7X,'AMPLITUD.E-. &8X,' PHASE'/) 900 FORMAT ('0 ANGLE',12X, 'APERTURE FIELDS',1 X, AMPLIT"UDE', &8X ' PHASE '/) END 35

SUBPROUTINE 'FLIP (ANXYD,DMAG, TAT) COMPLEX A(50,51),X(50j),Ys(53),D,BIGA,HOLD DIME NSIO N L (50), M ( C) IF (IA T. GT. 1) G O TO 1 50 D=CM PLX (1.0,0. 0) DO 80 Ki1,N L (K) =K M (K)=K BIGAA (KOK) DO 20 J=KN DO 20 T=KIN 10 IF (CAB3S(BIGA).GE.CABS (A (I,))) GO TO 2D B IGA =A (I, J) L (K) =I M (K) =J 20 CON72INUE J=L(K) IF (J.LFr.K) GO TO 35 DO 30 11,N HCLD=-A (K, I) A (K, I) =A (JI) 30 A (J, I) =H1OLD 35 I=(K) IF (I. LE.K) GO TO 45 DO 4D J= 1,rN H OL D A (J, K) A (3, K) =A (J,:) 140 A (JrI) = lOTT) 45 IF (CA Ei(BiUA F) GO TO D=C r.PLX,Ij.~ P ET1,UIP 50 DO 55 r=1,N IF (I.EQ. K) GO TO 955 A (I, K) =- A (I,K) /B I'GA 55 CONTINUE DO 65 I=1,N DO 65 J=1,tl IF (I.EQ.K.CO.J..EQ.K) GO TO 65 A (1, J) =A (I, K) *A (KJ) +A (, J) 65 CCNTINU'r DO 75 3=1,N IF (J.EQ.,K) GO TO 75 A (K,rJ) =A (KJ) /BIGA 75 CONTINUE D=D*BIGA 80 A(KK)=1.O/BIGA BN=FLOAT (N) DMAGCABS (D) * (2.0 **BN) K=N 100 K=K-1 IF (K.LE.O) GO TO 150 I =L (K) IF (I.LE.K) GO TO 120 DO 110 J31,N HOLD=A (JK) 36

A (J, K) =A (34I) 110 A(JI)=HOLD 120 J=M(K) IF (J.LE.K) GO TO 100 DO 130 I=lfN HOLD=A (K,I) A (KI I) =-A (34f ) 130 A (JrI) = HOLD GO TO 100 150 DC 200O I=lfN Y (I) =C'qP LX(J0 ) DO 2CC J=1, N 200 Y (I) =A (TJ)*X(J) +Y(I) R ET UJRN END SUBROUTTNE flANK (ZfB) COrMPLEX zIll CONPLELX*16 JfYfZ"7,DUM REAL*8 ABrF A=DB LB ---, (IBL (Z/j) ) B=D BL E (A T!" -. AG7 ( Z)) ZZ=-J).2'')5*DC..'PLX (A*A~-B*Br2,2`*A*f3) F=CDAt3S ('Z Z) IXSiZ..G L (F) I"F=1. 0 DO 10 K=2,NPM B1l1. J/DFL CAT.' (K) 10 F=F+B B11.0~/DFLCAT (K+ 1) A=DFLOAT (K*K) DUM 11 Z Z/A J =1. +DUIJ Y=FA+ (F+B) *D{JNI DO 390 I=1 iM7,~ K=MM-I+ 1 B=1. /DFrLCAT (K) A=DFLOATA (K*K) DUM=ZZ/A F=F- B Y=F+DUJN*Y 30 J=1.0+DU~1I*J Y =(0.5 77 ~21 566 4901 93 +C LOG (0. 5*Z)) *J - Y Y=C. 6366 198*Y A=DREAL (J)-DIMAG 3(Y) B=DIMAG (J) +DPAL (Y) H=CMPLX (SNGL (A),SNGL (B)) R ET URN END I37 i;,.;A

SLOTTED CYLINDER KA 2. 300 - 0.200 APERTURE HALF-ANGLE 30.000 INCIDENT FIELD DIRECTION 0.0 DETERMINANT 0.47255E-14 CELL NUMBER 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ANGLE 35.C'00 45.CO 55.CO 65.C0 75.00 85.C0 95.00 105.CO 115.C00 125.C 135.C0 145.00 155.00 165.00 175.f D 185.COi 195.CC 205.CO) 215.,O 225 5.C 235.0, 245.CC 255. C 265.Ci 275.CO 285.00 295.00 3C5.CO 315.C 325.CO CURRENT AMPLITUDE CURRENT PHASE 8.97951 3.05661 1.77597 0.85976 0. 49555 0. 93305 1.41335 1.74348 1.89528 1.88554 1.76117 1.58776 1.43204 1.33522 1. 29643 1. 296414 1.33520 1. 432C 5 1.58773 1.76118 1.88556 1. 95 25 1.74348 1.41337 0. 933C 6 0. 49554 C.85976 1.77595 3.05662 8.97952 -122.16 -114.21 -114.87 -123.67 178.16 143.93 144.67 152. 30 161.23 169.21 174.67 176. 5 174.62 170.74 167.79 167.78 170.74 174.62 176.50 174. 67 169.21 161.23 152.31 144.67 143.93 178.16 -123.67 -114.87 -114.21 -122.16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 CELL NU BER 38

SLOTTED CYLINDER KA 2.300 APERTURE HALF-ANGLE INCIDENT FIELD DIRECTION -0.200 30. C C 0.0 ANGLE -28.00 -24. C00 -20.00 -16. 03 -12. 00 -8.0 C -4.00 0.0 4.00 8. 00 12. 00 16.00 20.0C 24.00 28.00 APERTURE FIELDS -0.98 021 -1.36907 - 1.66180 -1.88785 -2.05832 -2. 17806 -*2.24 926 -2.27291 -2.24927 -2.17806 -2.05832 -1.88784 - 1.66179 -1.36 9C6 -0. 9802C 0.64451 0.94723 1. 17919 1. 36086 1.49933 1.59731 1.65587 1.67536 1.65587 1.59731 1. 49932 1.36835 1.17918 0.94721 0. 6445 A PLITUDE 1.17312 1.66481 2.C3766 2.32721 2.54650 2. 70C99 2.793 4 2.82364 2.793C-5 2.70099 2.54649 2.32720 2.03765 1.66479 1. 17310 AMPLITUDE PHASE 1 46.67 145.32 144. 64 144.21 143.93 143.74 143.64 143.61 143.64 143.74 143.93 144.21 1 44. 64 145. 32 1 46.67 PHASE 83. 7 86. 15 89.13 92.82 97. 42 103.11 113.33 118.07 126.81 135.54 143.61 DISTANC FIELDS ALONG A RADIUS C.0 i. 0t 2.00 3. 00 4. 00 5.00 6. CO 7.00 8.00 9. 00 10.00 0. 32772 P '. C n,I 0. 4 669 -0. 14866 -0.37864 -C.64 144 -0.93482 - 1.25537 -1.59489 -1.94 090 -2.27291 2. 9R781 -.J.,): 3. ' 6825 3.- 1551 2. 9076 1 2.75355 2. 56475 2. 35369 2. 13142 1.9 484 1.67536 3. 0' 572 3. C 663'4 3. r 6 iO 1 3. C 917 2.93216 2. 82727 2. 72981 2.6674 1 2. 66207 2.71947 2.82364 39

REFERENCES 1. T.B.A. Senior and G.A. Desjardins, Field Penetration into a Spherical Cavity, Interaction Note No. 142, Air Force Weapons Laboratory, August 1973. See also IEEE Trans. EMC-16, 205-208, 1974. 2. T.B.A. Senior, The Spherical Cavity Problem, Interaction Note No. 220, Air Force VWeaspons Laboratory, January 1975. 3. E.F. Knott, V.V. Liepa and T.B.A. Senior, Non-Specular Radar Cross Section Study, 011062-1-F (AFAL-TR-73-70), The University of Michigan Radiation Laboratory, Ann Arbor, Michigan, 1973. 4. V.V. Liea, E. F. Knott and T. B.A. Senior, Scattering from Two-Dimensional Bodies with Absorber Sheets, 011764-2-T, The University of Michigan Radiation Laboratory, Ann Arbor, Michigan, 1974. 5. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, U. S. Department of Commerce, 1964, pp. 359, 373. 6. E. Jahnke and E. Emde, Tables of Functions, Dover Pub., New York, 1945. 40