UNIVERSITY OF MICHIGAN WRRC-24-J 16 February 1955 THE TNUTEHICAL DETERtINATION OF THE RADAR CROSS-SECTION OF A PROLATE SPHEROID K. M. Siegel, B. H. Gere, F. B. Sleator Willow Run Research Center of the Engineering Research Institute UNIVERSITY OF ICHTIGAN The exact curve is found for the nose-on radar crosssection of a perfectly conducting prolate spheroid whose ratio of major to minor axis is 10/1, for vaites of T times the major axis divided by the wavelength less than three. The exact acoustical cross-section is also found for the same range of parameterso Introduction Analysis of the numerical work required to obtain exact scattering cross-section curves by the method of Mie or Hansen, even when the scatterer is a coordinate surface of a separable system, quickly suggests that there should be a better method. Much work has been done in the effort to find better methods'2 and an approximate method for the spheroid problem has been developed *This article is a shortened version of "Studies in Radar Cross-Sections XIThe Numerical Determination of the Radar Cross-Section of a Prolate Spheroid" K. Y. Siegel, B. H. Gere, I. Marx and F. B. Sleator (U1-126, December 1953). Now Chairman, Department of Mathematics, Hamilton College, Clinton, N. Y. 1. W. Franz and K. Deppermann, "The Creeping Wave in the Theory of Diffraction", McGill Symposium on MIicrowave Optics, June 1953. 2. M. Kline, "Asymptotic Solutions of Linear Partial Diffraction Equations and the WKB Method", ibid.

y F_ ED\ F.... -U N UNIVERSITY OF MICHIGAN by Stevenson, but up to the nresent the results of References 1 and 2 havc been applied only to cases where the exact solutions were already known. Thus, although it wa knomwn a nriori that the magnitude of the computations involved would be enormous, the classical solution for the spheroid problem was carried out, and with the aid of the Mark III Digital Computer and its excellent staff at the'raval Proving Ground at Dahlgren, Va., the numerical results presented here were obtained. This problem required roughly five times the capacity of the Mark III. To obtain 22 values of the radar cross-section it was necessary to run the computer for ten weeks, part of the time on a 2h hour seven-day-a-week basis. Since it was expected that for ratios of semi-major axis a to semiminor axis b near unity, the curve of nose-on radar crosr-section C s. wavelength A would approximate that of a sphere, it was decided that more information would be obtained by examining a case with a larger value of a/b. Thus a value of 10 was chosen for this ratio. Furthermore, analysis revealed that for a fixed amount of machine time the number of values of cross-section of a given body which could be computed to a given accuracy decreased sharply as the wavelength decreased. 3. A. F, Stevenson, "Solution of Electromagnetic Scattering Problems as Porer Series in the Ratio (Dimension of Scatterer) / wavelength, Applic.tion to Scattcring by an Ellipsoid" Journal of Applied Physics, Vol. 24, p. 1141, (1953). The present Daper contains some of the numerical work referred to in Stevenson's "note added in proof". LUJrBIILASSL F1E

F CLAgSS FDE UNIVERSITY OF MICHIGAN Since the Rayleigh solution was immediately available for the region of large wavelength, it was felt that the present computations would most profitably be concentrated in the region of the first maximum. Previous analysis of the sphere problem indicated by analogy that the first maximum for the prolate spheroid should occur at a larger wavelength than that predicted by physical optics, and also that the R^yleigh solution would form an upper bound on the curve in its region of validity. Furthermore it was felt that the 10/1 spheroid should behave electromagnetically rather like a thin wrire, and accordingly the abscissa5 of the successive maxima as predicted by the thin wire theory of Van Vieck, Bloch, and Hamermesh were computed. On the basis of this information a set of values of 2na/A was chosen to span the region in which the first maximum might occur. In addition to commuting the radar cross-sections, the Mark ITII recorded enough intermediate information so that the exact acoustical answers were easily obtainable by hand comoutation, h. J. H. Van Vleck, i. Eloch, and Ti. Hamermesh, "Theory of Radar Reflection from >Wires or Thin metallic Strips" Journal of Applied Physics, Vol. 18, D. 274, (19h7). -U —---— 3 -fJ1NLCL/A\5S11FIEE

U N~LL~~\~ FKELO U N UNI VE R S ITY OF MICHIGAN _ 40 - ----— r- _ -I I Abscissac of mLfria Q IJ_________ ___ _____ ________(hin Wire Theor)__ "tI 20 - -ht t t _ _ _ _ _ _' I__ _! 4o1. 0 - - -_ — 1 —_ —-_ _ __I / _ _I_ _- 1" 1 I't.11- _ —----- -_I -............. - ____ _____.ci, — _/__-t -/ ---,, / / I i i I 1 X I I' rl / i * 1' -i --....I >OiQa-t- -1/ — ~ — Vector (Electromzagntic )|i!ark III t — t)(v$ / _ t — x-Sca3ar J Results — ctor. 0t 7T.....____ Rayleih Law. 001-, / ___ -.- - aar... — O K ~ __/ Results of Spence & Granger,) C /- -—,-Physi cal Optics. OOC- ---- --,,_.. --- -'.1.2.3..7 1.0 2 3 5 7 10 2na/A iCg. 1 - Back-Scattering Fi;-on a Prolate Spheroid JLLC-/A-S5 [F L-L-J

FFL UNIVE R S I TY OF MICHIGAN G A N Results The results shown in Fig. 1 were obtained from the following formulas: The geometric optics solution is true2 The physical optics solution is R 0,S L u - - - I, a Cos hal -) ] h | Ue note that 1 O^ = (JG.O. (k^vI-/A) k oC Vte also note that 1/- o x + [/-]. The Rayleigh electromagnetic answer is 3 aI/ <- b~ I rL k -ay. N,(rte where T = 41Tab2/3 1 and M T- r / - af _ ^ 6 ) ON- L ( -- U OK For,a = /?ox <7 ^- ( ka ) | The Rayleigh acoustical answer is cr y.,T - d Wal1. S 1 1 -L

LUJ C 4/A55 F FIO UNIVERSITY OF MICHIGAN where / - ab6 f- a b a /a a 1. - (L a.- _ i'a, J For / = /O G ~ /6 b7(ka)f Q. 9y. c Ox Ray. For ka Z 1, 2, 3, one ray use the work of Snence and qranger 5 to obtain the nose-on acoustical cross sectiono As stated nreviously, if one considers a 10/1 prolate spheroid as a thin wire one could use the work of Ref. 4 to predict the abscissas of the successive maxima. The appropriate formula is ^rr f i k Ac /, 87 t,^ot kc for odd maxima, (-taen for even maxima. 5. R. D. Spence and S. Granger,'Scattering of Sound from a Prolate Spheroid" Journal of the Acoustical Society of America, Vol. 23,:~o. 6, p. 701, ~(1' 9 ). Wl 1NCLL 1/A I5E F

U N I VE R S I TY OF M I C H IGAN When a/b = 10, the left hand side of this expressions reduces to TT 4.12 - 1.5 loge ka o On the basis of similar analyses for the sphere it was felt that four terms in the field expansions would be sufficient to guarantee accuracy of two sirnificant figures in the results for ka 3 3. CDnsequently the exact curve is drawn out to ka 3, Pnd beyond this the conmuted Doints are plotted on the chance that the fourth order results for slightly higher values of ka might be of sane value. At several values of ka both third and fourth order results were obtained in order to give some idea of the magnitude of the errors involved in truncating the infinite deterninants. The differences between third and fourth order solutions are small for ka < 3. In order to show that the truncation error wns small at ka * 3 and that a fourth order solution was sufficiently accurate, the acoustical cross-section was comruted for all orders up to 9. The results are tabulated here: n 1 2 6 7 8 Tb/^a17.7.22 7.95 2.0 1.88 1.88 1.88 1.88 netails of the nume rical analysis used in obtaining the exact answers are given in Apnendix A. The exact cross-section formulas are nresented in,pprndix B, while a list of the quantities which were tabulated by the Msrk uII is given in Appendix C. UiJcNCl/A\5 FE1DJ

.-.. U UNIVERSITY OF MICHIGAN Conclusions The abscissa of the first maximum of the nose-on cross-section curve for a thin prolate srheroid can be obtained quite accurately from thin wire theory. The ordinate of the first maximum for a 10/1 prolate snheroid is only slightly higher than that for a sphere. This suggests that for all prolate spheroids such that l<a/b1O0, the ordinate of the first maximum could probably be predicted to two significant figures by linear interpolation between the sphere solution and that of the 10/1 spheroid. The striking difference between the sphere and the prolate spheroid solutions in both the electromagnetic and acoustical cases is that for the snheroid the second maximum has a larger ordinate than the first. This is the first such nroblem solved in either electromagnetic or acoustical theory in which the first maximum is not the greatest. Among the many scientists who worked on the numerical aspects of obtaining the exact solution on the Mark IIT Electronic Calculator, the authors wish to single out J,. Bauer, R. Beach, D. Y. Brown, D. F. Eliezer,. L'. Fleishman, G, H.. leissner, H. E. Hunter, K. Kozarsky, R. A. Niemann, L. M. Rauch, and I. lryman for special acknowledgement. U NiCL/kA\i55 IFE

U iJiCl/A\ F FEE J.-.-UNU NIVERSI TY OF MICHIGAN Appendix A Computations The essential mathematical formulation of the problem has been given by 3chultz in the oreceding article, and the expressions appearing there were used in the computations with no appreciable modification. For the range of parameters used in the oresent nroblem, however, it was necessary to compute the spheroidal coefficients d which express the spheroidal wave functions in terms of spherical ones, and which Schultz assuned known. This was accomplished in the manner specified by Flammer6 A three-term recurrence relation is obtained by substituting the expansion in associated Legendre functionsof the angular spheroidal (I) function S^ (?) into its differential equation and then applying the differential equation and recurrence relations for the Legendre functionso This equation may be written nm <n mn rA m rn; dkt; t \k \k + %, \;, s o (1) khere E (r k +.t),(6 ) k (2) oF "srot Sphrd vti o t e(i+ kRe) C. Flamer, C, t Wk- am k ) 60 C. Flamer, "Prolate Spheroidal Wave Functions", Technical Report No. 16, Stanford Research Institute, February 1951. -J —L- L/A 5 F —-E 11E --

.-...U N UNIVERSITY ()F MICH I G AN _ k - +;^2k -o*989 | and A his the separation constant for the radial and angular spheroidal functions. Expansions of A in positive or negative powers ma of c are given in Ref. 6. The number of coefficients given there, however, proved insufficient to give the necessary accuracy in most cases, with the result that when the recilrrence relations for the spheriodal coefficients were used repeatedly, the errors built up indefinitely. Consequently an iteration scheme was used to refine the values of the A. One such orocedure is described by Bouwkamp7 m, but this anpeared unsuitable for programning on a digital machine, and a simpler though less direct modification of his technique was accordingly emoloyed, the details are given below. This iteration procedure and the tabulation of the spheroidal coefficients occupied a sizable fraction of the total computation time. The remainder of the operations were straightforward and presented no serious difficulties. However, as stated earlier, the volume of numbers was such that although the Mark TTI was the largest digital computer in use at the time, it was necessary to divide the problem into five successive runs, as illustrated in the schematic diagram shown in Figure 2. 7. C. J. Eouwkarcp, "Cn Spheroidal Wave Functions of Order Zero", Journal of V'atheratics and Physics, Vol. 26, p. 79, (19h7). _______ 10 ________ 1U1NCL/A\5511F1DEJ

UN IVERSITY OF M ICHIGAN I G A ASSUMED RUNi 1 RUN 2 RUT 3 RUN 4 RUN 5 i I Amn r ---- i ~ (Exoct) i * Ilh 1(1 0- D A ik;->_ (4) Ii" a o b.Ak i j i * IAn —---— ~ I ^j'22w~n ( —- ^'!.._-.V! FTG. 2 LOGICAL AND SEQJENTIAL STRUtCTURE OF COMPUATIONS UJCLL/A1^ FE1J- ) J U NC/A55FED

UC LA\SFEO ----- UNIVERSITY OF MICHIGAN As indicated here, the first machine run included, in addition to the approximate values of the separation constants, computation of certain Bessel and Legendre functions, made necessary by the limitations of all previously existing tables. These were obtained from standard power series formulas. The second run included the iteration procedure for the refinement of the A^n, which may be outlined as follows, for each set of values of c, m, n: (1) The anproximate value of An is substituted in eouation (3) to give an approximate value for. and the other two coefficients and Gk are computed exactly, as given by equations (2) and (4), (2) The quantity CQ - G l /n6 is computed, on the assumption that dm /d / is negligibly small. (3) Values of K dmn /ddk are computed k kk+2 k for k - 12, 10,...., using equation (1) in the form 1/K2 - - ( + E+2 C ) / G-2 (4) The quantity Gm2 / K - -Fon - E2 onn is computed. Since the exact value of C m2 is zero, the computed value The routine is,riven here for n even. The obicous modifications are apolied to deal with odd n. 12 ITLCL/A\5 F 1 DEJ

UNIVERS ITY OF MICHIGAN ___ G of.2 / K-2 is a measure of the error in the approximate value of A used. If this does not exceed a certain empiric all established tolerance, the values of Kk obtained in step (3) are used to compute the required spheroidal mn coefficients dk (5) If the value obtained in (4) exceeds the tolerance, it is substituted back in equation (1) which then yields a second approximation to Pm. An average of the first and second approxinations yay be used to repeat the procedure from step (1). The remaining quantities appearing in Figure 2 have been discussed in the preceding article. The specific formula for the back-scattering cross-section, which is not given there, appears in Appendix B. Formulas used in computing the acoustical cross-section are also presented in this appendix. 13 _y NCLL/A\5 5 F EO~~~~~~~~~~~~~~~~~i iiii

UN I VE RS I TY OF MICHIGAN Appendix B 1. Formula for LIadar cack-Scattering Cross-Section If the value = 1 is substituted into the expression for the scattering cross-section derived in the preceding naper (equation 79), the 0 dependence disappears and the resulting formula for the backscattering cross-section can be written "6 - "l4a2 jn / o on 2 nuo n =o dk. 2. Acoustical Cross-Section The problem of acoustical scattering by a prolate spheroid was solved by Spence and Granger. In the present terminology the expression they derived for the nose-on cross-section 0' may be written 4aa2 do n+l m (1) 1n+l #. T7a. r?+1 An n( So) s () ) An((l lo)Sl) c 3 o where ( / dR() (o) _ on (', ) dn( / d A 2 (1) (1) /N n on / on (1) ( 12 d INn on on ( 1 *7 and An is the complex conjugate of An. 1 4

U JJ CASS F J E UNIVERSITY OF MICHIGAN Appendix C Quantities Tabulated in Machine Output For each point shown on the graph of Figure 1 the followinr Quantities were recorded in the course of the comoutations: 1. Associated Legendre functions P n l ( a4rQ i, ^,n ( ) and their derivatives with respect to SO for m 0, 1 rl: - - 2, -2..... -16. 2. Bessel functions Jn (,C ).r' <. 3. Separation constants A, m for m O, 1 n= O, 1, 2, 3 4. Spheroidal coefficients d (kO), and d/p (k ). for m m 0,1 n- 0, 1, 2, 3 k = all required values in the range -16 to +16. 5. Radial soheroidal functi ns ^ )- n~ Tn () and their derivatives with resoect to g. for S 1.005 m = 0, 1 n= o, 1, 2, 3. UJCL/. 15

{LJ JLKA4SS 1F F3 U NIVERSITY OF MICHIGAN 6. 3oundary integrals Ian k for k a 1, 2, 5, 6 all combinations of N and F in the range N O 0, 1, 2, 3 n 0o, 1, 2, 3. (I3 and were not recorded.) 7o Deterrinantal elements Pn, Cn, ^, UT VN, W, for all combinations of N and n in the range N= 0, 1, 2, 3, n =, 1, 2, 3. 8. Radar cross-section 0"'. With the exception ofa ", all quantities are given to 15 significant figures. The values of O are rounded off to 5 significant figures. Particular values of any of these quantities may be obtained on request. 16 UN1Fl0L/ES IFII[Cll