TH E U N I V E R S I TY OF MI CHI GAN RADC TN-59-164 ASTIA AD-215 487 STUDIES IN RADAR CROSS SECTIONS XXXV ON THE SCALAR THEORY OF THE DIFFRACTION OF A PLANE WAVE BY A LARGE SPHERE by P. C. Clemmow April 1959 Report No. 2778-6-T on CONTRACT AF30(602)-1853 TASK hi577-3-... - j. _'korJ ROMEAI/IR -PEVELOP NT CENTER AIR ]RESEARCHO Ai' 1fEtlOPENT COMMAND UNITED STATES AIR FORCE GRIFFISS AIR FORCE BASE, NEW YORK

T HE U N I VE R S IT Y OF M I C H IG AN 2778-6-T PATENT NOTICE: When Government drawings, specifications, or other data are used for any purpose other than in connection with a definitely related Government procurement operation, the United States Government thereby incurs no responsibility nor any obligation whatsoever and the fact that the Government may have formulated, furnished, or in any way supplied the said drawings, specifications or other data is not to be regarded by implication or otherwise as in any manner licensing the holder or any other person or corporation, or conveying any rights or permission to manufacture, use, or sell any patented invention that may in any way be related thereto. Qualified requestors may obtain copies of this report from the ASTIA Arlington Hall Station, Arlington 12, Virginia. ASTIA Services for the Department of Defense contractors are available through the "Field of Interest Register"? on a?tneed-to-knowt' certified by the cognizant military agency of their project or contract. ii

THE UNIVERSIT Y OF MIC HIGAN 2778-6-T STUDIES IN RADAR CROSS SECTIONS I "Scattering by a Prolate Spheroid", F. V. Schultz (UMM-42, March 1950), W-33(038)-ac-14222. UNCLASSIFIED. 65 pgs. II "The Zeros of the Associated Legendre Functions pm(,ct) of Non-Integral Degree", K. M. Siegel, D. M. Brown, H. E. Hunter, nH. A. Alperin and C. W. Quillen (UMM-82, April 1951), W-33(038)-ac-14222. UNCLASSIFIED. 20 pgs. III "Scattering by a Conett, K. M. Siegel and H. A. Alperin (UMM-87, January 1952), AF-30(602)-9. UNCLASSIFIED. 56 pgs. IV "Comparison between Theory and Experiment of the Cross Section of a Cone", K. M. Siegel, H. A. Alperin, J. W. Crispin, Jr., H. E. Hunter, R. E. Kleinman, W. C. Orthwein and C. E. Schensted (UIM-92, February 1953), AF-30(602)-9. UNCLASSIFIED. 70 pgs. V "An Examination of -Bistatic Early Warning Radars", K. M. Siegel (UlMM-98, August 1952), W-33(038)-ac-14222. SECRET. 25 pgs. VI "Cross Sections of Corner Reflectors and Other Multiple Scatterers at Microwave Frequencies"t, R. R. Bonkowski, C. R. Lubitz and C. E. Schensted (UMM-106, October 1953), AF-30(602)-9. SECRET - Unclassified when Appendix is removed. 63 pgs. (S). VII "Summary of Radar Cross Section Studies under Project Wizard", K. M. Siegel, J. W. Crispin, Jr. and R. E. Kleinman (UMM-108, November 1952), W-33(038)-ac-14222. SECRET. 75 pgs. VIII "Theoretical Cross Section as a Function of Separation Angle between Transmitter and Receiver at Small Wavelengths", K. M. Siegel, H. A. Alperin, R. R. Bonkowski, J. W. Crispin, Jr., A. L. Maffett, C. E. Schensted and I. V. Schensted (UMM-115, October 1953), W-33(038)-ac-14222. UNCLASSIFIED. 84 pgs. IX "Electromagnetic Scattering by an Oblate Spheroid", L. M. Rauch (UMM-116, October 1953), AF-30(602)-9. UNCLASSIFIED. 38 pgs. X "Scattering of Electromagnetic Waves by Spheres", H. Well, M. L. Barasch and T. A. Kaplan (2255-20-T, July 1956), AF-30(602)-1070. UNCLASSIFIED. 104 pgs. iii

T HE UNI VE R S IT Y OF MICHIGAN -2778-6-T XI "'he Numerical Determination of the Radar Cross Section of a Prolate Spheroid", K. M. Siegel, B. H. Gere, I. Marx and F. B. Sleator (UMM-126, December 1953), AF-30(602)-9. UNCLASSIFIED. 75~ pgs. XII "Summary of Radar Cross Section Studies under Project MIRO", K. M. Siegel, M. E. Anderson, R. R. Bonkowski and W. C. Orthwein (UMM-127, December 1953), AF-30(602)-9. SECRET. 90 pgs. XIII "tDescription of a Dynamic Measurement Program", K. M. Siegel and J. M. Wolf (UMM-128, May 1954), W-33(038)-ac-14222. CONFIDENTIAL. 152 pgs. XIV "Radar Cross Section of a Ballistic Missile", K. M. Siegel, M. L. Barasch, J. W. Crispin, Jr., W. C. Orthwein, I. V. Schensted and H. Weil (UMM-134, September 1954), W-33(038)-ac-14222. SECREiT. 270 pgs. XV "Radar Cross Sections of B-47 and B-52 Aircraft", C. E. Schensted, J. W. Crispin, Jr. and K. M. Siegel (2260-1-T, August 1954), AF-33(616)-2531. CONFIDENTIAL. 155 pgs. XVI "Microwave Reflection Characteristics of Buildings", H. Weil, R. R. Bonkowski, T. A. Kaplan and M. Leichter (2255-12-T, May 1955), AF-30(602)-1070. SECRET. 148 pgs. XVII "Complete Scattering Matrices and Circular Polarization Cross Sections for the B-47 Aircraft at S-band", A. L. Maffett, M. L. Barasch, W. E. Burdick, R. F. Goodrich, W. C. Orthwein, C. E. Schensted and K. M. Siegel (2260-6-T, June 1955), AF-33(616)-2531. CONFIDENTIAL. 157 pgs. XVIII "Airborne Passive Measures and Countermeasures", K. M. Siegel, M. L. Barasch, J. W. Crispin, Jr., R. F. Goodrich, A. H. Halpin, A. L. Maffett, W. C. Orthwein, C. E. Schensted and C. J. Titus (2260-29-F, January 1956), AF-33(616)-2531. SECRET. 177 pgs. XIX "Radar Cross Section of a Ballistic Missile II", K. M. Siegel, M. L. Barasch, H. Brysk, J. W. Crispin, Jr., T. B. Curtz and T. A. Kaplan (2428-3-T, January 1956), AF-04(645)-33. SECRET. 189 pgs. XX "Radar Cross Section of Aircraft and Missiles", K. M. Siegel, W. E. Burdick, J. W. Crispin, Jr. and S. Chapman (WR-31-J, March 1956). SECRET. 151 pgs. iv

THE UNIVERSITY OF MICHIGAN 2778-6-T XXI "Radar Cross Section of a Ballistic Missile IIItt", K. M. Siegel, H. Brysk, J. W. Crispin, Jr. and R. E. Kleinman (2428-19-T, October 1956), AF-04(645)-33. SECRET. 125 pgs. XXII "Elementary Slot Radiators", R. F. Goodrich, A. L. Maffett, N. E. Reitlinger, C. E. Schensted and K. M. Siegel (2472-13-T, November 1956), AF-33(038)-28634, HAC-PO L-265165-F31. UNCLASSIFIED. 100 pgs. XXIII "A Variational Solution to the Problem of Scalar Scattering by a Prolate Spheroidtt, F. B. Sleator (2591-1-T, March 1957), AF-19(604)1949, AFCRC-TN-57-586, AD 133631. UNCLASSIFIED. 67 pgs. XXIV "Radar Cross Section of a Ballistic Missile - IV The Problem of Defensett, M. L. Barasch, H. Brysk, J. W. Crispin, Jr., B. A. Harrison, T. B. A. Senior, K. M. Siegel and V. H. Weston (2778-1-F, April 1959), AF-30(602)-1853. SECRET. XXV "Diffraction by an Imperfectly Conducting Wedge", T. B. A. Senior (2591-2-T, October 1957), AF-19(604)-1949, AFCRC-TN-57-591, AD 133746. UNCLASSIFIED. 71 pgs. XXVI "Fock Theory", R. F. Goodrich (2591-3-T, July 1958), AF-19(604)-1949, AFCRC-TN-58-350, AD 160790. UNCALSSIFIED. 73 pgs. XXVII "Calculated Far Field Patterns from Slot Arrays on Conical Shapes", R. E. Doll, R. F. Goodrich, R. E. Kleinman, A. L. Maffett, C. E. Schensted and K. M. Siegel (2713-1-F, February 1958), AF-33(038)28634 and 33(600)-36192; HAC-POs L-265165-F47, 4-500469-FC-47-D and 4-526406-FC-89-3. UNCLASSIFIED. 115 pgs. XXVIII'"The Physics of Radio Communication via the Moon", M. L. Barasch, H. Brysk, B. A. Harrison, T. B. A. Senior, K. M. Siegel and H. Weil (2673-1-F, March 1958), AF-30(602)-1725. UNCLASSIFIED. 86 pgs. XXIX "The Determination of Spin, Tumbling Rates and Sizes of Satellites and Missiles", M. L. Barasch, W. E. Burdick, J. W. Crispin, Jr., B. A. Harrison, R. E. Kleinman, R. J. Leite, D. M. Raybin, T. B. A. Senior, K. M. Siegel and H. Weil (2758-1-T, April 1959), AF-33(600)-36793. SECRET. 180 pgs. XXX "The Theory of Scalar Diffraction with Application to the Prolate Spheroid", R. K. Ritt (with Appendix by N. D. Kazarinoff), (2591-4-T, August 1958), AF-19(604)-1949, AFCRC-TN-58-531, AD 160791. UNCLASSIFIED. 66 pgs. v

T HE U N I VE R S IT Y OF M I C H I G AN 2778-6-T XXXI "Diffraction by an Imperfectly Conducting Half-Plane at Oblique Incidence", T. B. A. Senior (2778-2-T, February 1959), AF-30(602)1853. UT4CLASSIFIED. 35 pgs. XXXII "On the Theory of the Diffraction of a Plane Wave by a Large Perfectly Conducting Circular Cylinder", P. C. Clemmow (2778-3-T, February 1959), AF-30(602)-1853. UNCLASSIFIED. 29 pgs. XXXIII "Exact Near-Field and Far-Field Solution for the Back Scattering of a Pulse from a Perfectly Conducting Sphere", V. H. Weston (2778-4-T, April 1959), AF-30(602)-1853. UNCLASSIFIED. 55 pgs. XXXIV "An Infinite Legendre Integral Transform and Its Inverse", P. C. Clemmow (2778-5-T, March 1959), AF-30(602)-1853. UNCLASSIFIED. 35 pgs. XXXV "On the Scalar Theory of the Diffraction of a Plane Wave by a Large Sphere", P. C. Clemnow (277-6-T, April 1959), AF-30(602)-1853. UNCLASSIFIED. 39 pgs. vi

T HE U N I VE R S IT Y OF MIC HIGAN 2778-6-T PREAACE This is the thirty-fifth in a series of reports growing out of the study of radar cross sections at The Radiation Laboratory of The University of Michigan. Titles of the reports already published or presently in process of publication are listed on the preceding pages. When the study was first begun, the primary aim was to show that radar cross sections can be determined theoretically, the results being in good agreement with experiment. It is believed that by and large this aim has been achieved. In continuing this study, the objective is to determine means for computing the radar cross section of objects in a variety of different environments. This has led to an extension of the investigation to include not only the standard boundary-value problems, but also such topics as the emission and propagation of electromagnetic and acoustic waves, and phenomena connected with ionized media. Associated with the theoretical work is an experimental program which embraces (a) measurement of antennas and radar scatterers in order to verify data determined theoretically; (b) investigation of antenna behavior and cross section problems not amenable to theoretical solution; (c) problems associated with the design and development of microwave absorbers; and (d) low and high density ionization phenomena. K. M. Siegel vii

T HE U N I V E R S I T Y OF M IC H I G A N 2778-6-T Table of Contents Page 1. Introduction 1 2. The General Nature of the Solution 3 3. The Radiation Field 6 4. The Field at an Arbitrary Distance From the Sphere 20 5. The Neumann Problem 26 Appendix A 28 Appendix B 32 Appendix C 36 References 39 ix

T HE UNI V E R S I T Y OF M I C H I G A N 2778-6-T SUMMARY The problem, with boundary condition either U = O or a U/ r=O, is solved by a method different from those already given by other authors. The fundamental idea is analogous to that introduced by the present writer to treat diffraction by a circular cylinder, and is carried out by means of an infinite Legendre integral transform and its inverse. Useful integral representations for the field, previously obtained by other means, are rederived. First, a discussion of the radiation part of the scattered field is given; this includes the two special cases of forward and back scattering, the former of which yields the total scattering cross-section. Subsequently, the field at a finite distance is considered. xi

T HE UNIVERSIT Y OF MI C H I G AN 2778-6-T 1. Introduction The purpose of this paper is to treat the problem of diffraction by a large sphere by a method analogous to that recently used by the present writer (Clemmow 1959 a, hereinafter referred to as I) to solve the problem of diffraction by a large circular cylinder. For the cylinder, an infinite Fourier integral transform and its inverse were used; for the sphere, an infinite Legendre integral transform and its inverse (Clemmow 1959 b, hereinafter referred to as II) are appropriate. Otherwise, however, the technique in the latter case is closely parallel to that in the former. The present paper, therefore, follows much the same lines as I, and may conveniently be read in conjunction with it. In the cylinder problem, the use of infinite Fourier transform analysis seems first to have been exploited by Friedlander (1954), though there is a hint of it in an early work by Debye (1908). On the other hand, an infinite Legendre transform analysis for the sphere problem appears to be new>, unless it be traced back in the history of the theory of radio propagation around the surface of the earth to papers by March (1912) and Rybczynski (1913). Admittedly, March introduced an integral representation involving Legendre functions, but the connection, if any, with the present theory is sufficiently disguised to make a pursuit of the comparison certainly 1 * Though there is some similarity to the techniques used by Felsen; see, for example, Felsen (1957).

THE UNIVERSITY OF MICHIGAN 2778-6-T tedious and probably unprofitable. It may, however, be remarked in passing that the justification of the dismissal of Marchts work by Love (1915) and Watson (1918) is perhaps questionable. For the criticism was chieflylevelled at the fact that March's representation was invalid at all points on the radial line from the center of the earth pointing directly away from the transmitter; and yet this in itself, as the present treatment, for example, bears witness, is no evidence of the unsoundness of the theory. There is no need here, then, to add further to the discussion given in I of the general background to the problem and of the relevant literature. References in the body of the paper draw attention to previous work mainly for the sake of avoiding duplication of specific calculations. The incident field is taken to be a plane wave, and for the most part the development is presented in terms of the Dirichlet problem, in which the wave function vanishes on the surface of the sphere. In P 2 the fundamental idea is stated and expressed mathematically by means of an infinite Legendre integral transform and its inverse. In 43, the radiation field is considered; the discussion includes both that in the forward direction, from which the total scattering cross-section is obtained, and that in the backward direction, which happens to be a special case as regards the mathematical treatment. In ~ 4, attention is turned to the field at any finite distance from the center of the sphere. In ~ 5, the 2

T HE UN IVERS IT Y OF MI CHI GAN 2778-6-T closely analogous Neumann problems in which the normal derivative of the wave function vanishes on the surface of the sphere, is briefly discussed. 2. The General Nature of the Solution The problem considered is that of the scalar plane wave specified b;y ui -ikr cos 9 U e (1) falling on the sphere r a a, where (r, e, A) are spherical polar coordinates (see Fig. 1).. The convention adopted is that the physical i U Figure 1 The configuration space is embraced by the range of values of e between 0 and Tr, and the analysis is evidently independent of 0. The Dirichlet boundary condition UTO on rh a is chosen, where U is the wave function of the total field.'The suppressed time factor is exp(iz t). 3

T HE U N I VE R S IT Y OF M IC HI GAN 2778-6-T Following the argument of I, the aim is to obtain a convenient form of the solution by expressing it in terms of functions which are not individually periodic in E. This is easily done with the use of the infinite Legendre integral transform defined and discussed in II. In the notation of II, the function 0Q L_ 5 p (eI, e2; ) cot (X) E _ ~(e) dy, (2) -00 where 2 11 r- -ika cos e e E for e < < c -ika cos 8 -ik a for e = e', e = e2, (4) 0 for e< 8l e e 2 for arbitrary values of 8, e2 (e > e ). Two supplementary remarks have 2 2 1 to be made. First, (2) is meaningless when e has one of the values -0, ~ T, 2 IT, + 3X,..., because E (8) is then logarithmically

T HE UNIVERSITY OF MICHIGAN 2778-6-T singular. Secondly, the path of integration in (2) avoids the poles at v = + 1, + 2, ~ 3 ~... by running parallel to the real V axis; whether it lies entirely above or entirely below the real axis is immaterial, since the residue of the pole at vL n cancels that of the pole at v_ -n. Then, likewise, the function co A 1- - 2; Ha (kr) cot (lrL)E (&)div (5) -r (2) - (2) -o HU (ka) 2 has the value minus (4) on r = a. It is thus an outgoing field which on the surface of the sphere cancels the incident field for e between 81 and e2, and is zero there for e less than 81 or greater than e2. Clearly, then, an exact representation of the scattered field is given by the superposition of all functions (5) corresponding to non-overlapping ranges [. e2j which together span the full range -oo to oo. Again following the argument in I, it may be said in general terms that for ka>>l the scattered field is given to a good approximation by the expression (5) with the range e1, e)2] roughly spanning that of physical space, namely O3, Tr];and that the further the range [8e, e2v is from some such "primary" range, the smaller, at a given point in physical space, the expression (5) becomes. Furthermore, for n=0,-l+ + 2,+-3,..., 5,

T HE U N I V E R S IT Y OF MI CHI GAN 2778-6-T E () + 2nrr) = i e E (-) i (6) (see, equation ()), and it follows from (3) that (see II. eqluation (8)) and it follows from (3) that n 2irnv, ) (7) P(81 - 2n r, - 2n r; v ) = ()n e29n7 2 i 2 A decrease of 2n lr in both 8 and 0 is therefore equivalent in (5) to 1 2 an increase of 2n'r in e, so that an interpretation in terms of rays travelling around the sphere becomes evident if the solution is built up by taking successive [8e1 82] ranges to be of width 2 nr. 3. The Radiation Field 3.1. The Case 8 # 0 In this section the radiation field, that is, the part of the scattered field of order l/(kr) for kr >> 1, is considered for all directions other than that in which the incident wave is travelling. For any fixed V, as kr — oo'-ikr (2) (kr) (2 ) ir i~r e (8) V(kr) 6

THE U N I VERSIT Y OF M ICHIGAN 2778-6-T and (5) is consequently asymptotic to 00oo -ikr /(2 ka) i iur e-ikr P(e1 e2; v) i r -I(2 ) e -\- ----— e —-- ----- e cot (rrv)E (8)dv. 7r'/r kr (2) -oo H, (ka) (9) The justification for substituting (8) into the integrand in (5), where the variable of integration v runs to infinity, is essentially that given for the corresponding step in I. The behaviour in the complex v-plane of the integrand in (9) is now considered. As noted in II, the only simgularities of EBAs (@) are simple poles at' = 1/2, 3/2, 5/2,...; and it is evident, from (3), that p(81, 2; v ) can have no others. But the integrand in (9) contains the factor cos( Ir ), so that its only singularities in addition to the irrelevant poles at v = O, + 1, 2, ~ 3,... are simple poles arising from the zeros of H(2) (ka). Furthermore, for any fixed value of e (sin e H 0), and any fixed value of arg v in the range -i1 < arg v <rr as 1 Wtoo i'e E (8)~ ~/( -i)r e I i7r e B (e4.'e i _ _ _ _, (10) -~ V 2 /( v sin 8) where l/L} has a positive real part, and i/(sin 9) is positive for real values of e between O and Ir. 7

T HE UNIVERSIT Y OF MIC HI GAN 2778-6-T It follows that the path of integration in (9) may be replaced, if 82 <- 1r/2, by one enclosing the poles of 1/H(2)(ka) in the upper half-plane, and if @ > 3 r-/2, by one enclosing the corresponding poles in the lower half-plane. For any range 91, 823 outside [- r/2, 3 r/23, the integral (9) can therefore be evaluated by the calculation of residues. The main contribution to the scattered field is contained in the expression (9) with @ =- 7r/2, e2 =3 7r/2. In this case the appropriate method of evaluation is that of steepest descents. In order to apply this method, p(-7r/2, 3 ir/2; v) is expressed as the sum of terms whose individual behaviour can be represented, for the most part, by exponential functions with comparatively simple exponents. As a first step in achieving this (cf. I) the l/ path of integration in (3) is distorted in the way indicated in Figure 2, where the regions of convergence at infinity are shown shaded. The contribution to p(-lr/2, 3 ir/2; u ) of the curved portions of the new t path of integration i~s _ /(u e e i [Hr (ka) - H( (ka) (1 __ I2ka, I I I ll, I,- In essence, this result is to be found in Watson (1944) p. 175. But since the integration is now in the complex y-plane, in contrast to the more standard complex cos ty -plane used by Watson, it seems worthwhile to give a direct derivation. This is done in Appendix A.

T HE UNIVERSITY OF MIC HIGAN 2778-6-T - r/2( 31,/2 -Tr I r20 2wr Figure 2 Distortion of the PY path of integration in p(- lr/2, 3 Tr/2; P) defined by (3) Furthermore, the contributions from the straight portions of the path are conveniently taken in conjunction with the functions p(-1r/2+2nir, 3T/2+2nTr; v ), n =l, 2, 3,.... In effect, then, these functions can be replaced respectively by (_)n l/(lr) tiri + 2irrnv -~isvr [H (1I)(ka)H (12)(k)] (12) The radiation part of the scattered field now appears in the form -ikr U s p(e) e (13) kr 9

T HE U N I VE R S ITY OF MIC H IGAN 2778-6-T where (1) n H, (ka) Equation (14) should be cot ( )equatio E () dof I. In both -00 H1) (ka) x P H(2) (ka) 1 C'-n 5 I oot (irta v /) -2ir)ni 2irn d)E ()d'. (15) Equation (14) should be compared to equation (12) of I. In both cases complete rigour has been sacrificed because the integrals involved are not strictly convergent. However, there is no fear of error if a procedure analogous to that in I is adopted, and equation (14) is replaced by oo H(1)(ka) P (lo) W ~ - cot (r v) E (9) d V (2) (ka) n~(,~ I H~ (ka) (o (8) +e )n) dv O H (2)(ka) - l0

T HE U N I V.E R S IT Y OF M I C HI G AN 2778-6-T As far as the stages in its derivation are concerned, equation (15) corresponds to equation (13) of I. In contrast to the latter, however, equation (15) must undergo a further slight transformation before it can be evaluated by standard techniques, because the factor cot (T- v) in the integrand of the first integral spoils an immediate application of the method of steepest descents. The difficulty is overcome by writing in that integrand cot (Tr )-i + [l- i tan (rv)] cot (Try') )n -2iir n v =i - 2 cot (irA) L (-) e for Im <O; (16) n=l so that, if the path of integration is now definitely specified to run below the real axis, the result is ()(ka) P(e) 1 = H,/(ka) E (e) dv H (2)(ka) 00oo 00 H(1) (ka) _ 2 i. nvie_2iw (17) i Z. 11cot. (r ) e n=O H(2)(ka)' -OO ~ ~ ~ 1

T HE U N I V E R S IT Y OF M I C H I G A N 2778-6-.T The integral in the first term of (17) can be evaluated by the method of steepest descents. If (10) is used, together with the Debye asymptotic form for the Hankel functions, the saddle-point is seen to be located at v = ka cos (8/2). The steepest descents path is somewhat as shown in Figure 3, being asymptotic at infinity to the lines of zeros of H(l)(ka); and it should be observed that the poles of E (8), which #ka cos (k C) Figure 3 The steepest descents path in the complex i-plane for the first integral in equation (17) lie along the negative real axis, do not interfere with the path distortion. The resulting contribution to P(s) is (see Appendix B) 1 ka e2ikasin() 1 + (ka () 2 ka sin3(~8) 12

T HE U NIV E R S IT Y OF M I CHI G AN 2778-6-T Derivations of (18) by different methods have been given by Franz and Depperman (1954) and by Keller, Lewis and Seckler (1956). The integrals in the sum in (17) can be evaluated by closing the contours around the poles in the lower half-plane, these being at the zeros, x = v say, s- =1, 2, 3,..., of H(2)(ka). If it is noted that s (cf. equations (9) and (11) of II) -2irirvTr; e E (8) -E 1(e) = -ilre tan( Tr;)P (-cos e), (19) X}-2 L2 2 the resulting contribution to P(e) is seen to be nW oo (1)(ka) X-2 - ) 1s i.H. (ka).-i.r(2n+l) s p l(-cose). n=0 s0H=l (2)(ka) ( (20) Expression (20) agrees with the analogous one given for a point source by Franz (1954). It is worth making a further remark in connection with the particular case e = T' (back scattering). As it stands, the first integral in (17) is meaningless at 9 = 71T, because of the logarithmic singularity there of E (e). Furthermore, in the steepest descents evaluation described, the 13

T HE U NIV E R S I T Y OF M I C H I G A N 2778-6-T asymptotic representation of E 1(8) requires I sin ej to be much greater than unity at the saddle-point v = ka cos (8/2). It would seem, therefore, that as E approaches 7F, the calculation is only valid for increasingly large values of ka. Nevertheless, the result (18) is well behaved at 8-ir, and on physical grounds no startling variations of field-strength would be expected in the vicinity of the back scattering direction. It may therefore be plausibly anticipated that (18) does, in fact, hold for all values of 8, other than those too close to zero. To establish this by a rigorous mathematical treatment of the first term of (17) would appear to be quite difficult. On the other hand it is possible to rewrite this term in a form which remains valid at e='r, and it is then a relatively straightforward matter to evaluate it for this particular case. For if the path of integration, shown in Figure 3, is displaced so that it runs "symmetricallytt through the origin, a sign change of the variable of integration v gives the form H 1 (ka) 2irr _ I 1.... -- [ E (e) - e E (i) dv, (21) H 21' (2) (ka) dV, (21) which from (19) is (1) ii~-i- H (ka) V tan (Tr e) e _P (-cos 8) d. (22) (2) H (ka) 14

T HE U N I VERSITY OF MICHIGAN 2778-6-T And at E8 1T the expression (22) is H(1)( (k1) E iV tan (I-V) e d V. (23) H (2) H 2)(ka) A direct steepest descents evaluation of (23) is spoilt by the presence of the factor tan(irV ) in the integrand. But a modified steepest descent treatment can be applied, and the expression (18) with 8 — r recovered (see Appendix C). This gives added support to the contention that the result of a direct evaluation of the first term of (17) holds uniformly up to e = -. 3.2. The Case e 0= The analysis of the previous section is now supplemented by a consideration of the radiation field in the forward direction. This gives the total scattering cross-section, osay, through the relation -=- 4 ~Im P(O). (24) k2 The required modification to the treatment in ~3.1 is analogous to that made at the corresponding stage in I. The symmetry now present in the problem is preserved by taking the original primary range for J1, ] as 15

T HE U N I V E R S IT Y OF M I C HIG A N 2778-6-T [-3 7r/2, 3 Tr/2] rather than [- T7/2, 37T/2], and the (/ path of integration in the integral (3) for p (-3 1T-/2, 3 7r/2; V ) is distorted as shown in Figure 4. The result is (2) 00 l ~ CFH (ka) -2i Tr nV + * -_ -oo HV (ka) -2i[r Hv (ka) 2iTr n cot(TrjL')E (e) d. (25) In order to put -- 0, (25) must be transformed so that the only Legendre functions appearing are P (cos e). The procedure is similar u-~ to that which converts (15) into the sum of (22) and (20). Since (equation (12) of II) 16

T HE UNIVERSIT Y OF MICHIGAN 2778-6-T -2 ir -3 ir/2 IF / i O rr Figure 4 Distortion of the V path of integration in p-(-3 7r/2, 3 1T/2; v) defined by (3) E ((e) - E (e) =- rtan (-Tr )P (cos e), (26) 2 -21 -2 the aim must be to make the factors multiplying E (8) in the integrands 1)-2 )1 odd functions of ). This is achieved by using the identity (16) in that part of the integrand of the first integral in (25) which does not have this character, namely the part not involving the Hankel functions. If, then, the paths of integration are taken to run below the real axis, (25) can be written 17

T HE UNIVERSITY OF MI CH I G AN 2778-6-T (1) __rFH (ka) P(e) e - - ). cot( ) E (e) d. profte ir a1-e o (2) (ka) 2) -_O 2i0 co H( (h a)2i -2i T n l (2o )(2) H (ka) Consider, first, the summation part of (27). A sign change of V in part of the irntegrands, and use of (26), shows that it is - (- ) L 1+ H(2[(ka) e P, (cose) d); (28) (2) -ao U -~ and (28) in turn can be evaluated by closing the path of integration around the poles V j, s = 1, 2, 3,..., of l/H( (ka) in the lower half-plane, with

THE UNIVERSIT Y OF MICHIGAN 2778-6-T H( )(ka) as the resulting contribution to P(c)1 -21 e s, -.. (co e) The isolated integral in (27) is handled as follows. It is noted that, by changing the sign of ), it can equally well be written with E 1(e) replaced by E 1(e) provided the path of integration runs above, instead of below, the real axis. If, now, half the sum of the two integrals is forned, each path of integration can be taken to run "'symmetrically" through the origin from the third to the first quadrant in the complex p -plane; for only the poles of cot (iTr ) are thereby crossed, and the contributions of the residues of those at V+ tn are easily seen to cancel. Again appealing to (26), the resulting contribution to P(@) can be written iJ 5 H)(k) 1[i tan (Tr) o (1e e p (cos) dV. H(2) (ka)J V (30) 19

T HE U N IV E R S I T Y OF M IC H I G A N 27786rr The combination of (30) and (29)'with 0 0 gives P(0) =~i e$ Ftni t ( ( ) (- d L(2) H ((ka a) ns -2'-iW n P (1) > ( ~2i~n ) eS nl1 a' H)(ka)JI (3 ) This'i just th@e for~ obtained by Franz and 3Beckmann (1957)9 who derie from the int egral seera terma of the asyptotie approxdrinaeton in'inverse powers of kao 4 The Fied at an Arbitrary Dis tance From the Sphire In this seetion the problem is considered for unrestrcted viuaes oi kr 0so that (5) cannot be approximated by (9)o Two methods are given fo. obt aining epressions for the field9 corresponding r espectively to those designated I and III in ~4 of Reference Io 20

T HE UNIVERS IT Y OF MIC HIG AN 2778-6-T In the first method the scattered field is deduced directly from the final expression obtained for the radiation part in i 3.1. This expression is given by (13) and (17), where the path of integration in the first integral in (17) is that shown in Figure 3, and the summation part of (17) is replaced by (20). Loosely stated, the procedure is to write 1iT7 e, -ii7rv (2) /(kr) (kr) (32) J(kr) with the appropriate value of 2', in place of exp (-ikr)/(kr), which is the asymptotic approximation to (32) as r -oo. In the region where the result converges, it is an outgoing solution of the wave equation free of singularities for r> a, whose asymptotic form for kr-*oo is precisely that of U5. It must therefore be U, which thus appears in the form -liar 1 1J L' H())-iioE d(1) -4 H( )(ka) (2) 4 i k ~,~ ___ H~ (2) U — (kr)-e E()d /(27r) V(kr) H(2)(ka) 1r/(k1r) ni=O l (ka) 3H(2)3(ka)() /21(ka) H (kr) o0V e-~7. SH 1)s H(2(-)ose). sl )(ka) (33) 21

T HE U N I V E R S I T Y OF M I C H I G A N 2778-6-T The path of integration in the integral is that shown in Figure 3, and the integral is only convergent for 7~/2 < 8 < 3 -'/2. Hence, (33) gives the scattered field in the half-space on the illuminated side of the plane through the center of the sphere perpendicular to the direction of propagation of the incident wave. The half-space 8 <7T/2 includes the shadow region, where the total field can become very small. This suggests that a consideration of the total field, rather than the scattered field alone, might yield an expression which is valid for points of observation not catered for by (33). The inclusion of the incident field, in a suitable representation is the basis of the second method. The representation is obtained merely by writing r for a in (2), (3) and (4). If 8 is allowed to lie in the range 0 to T, the primary range for El, 82 is L- /2, 3-71/2], and the corresponding contribution to the total field is 1r ILJpr' /2, 7r/z; )p(-7r/2,37r/2;v) H(2) (34) where 3 lr/2 p (- 7/2, 3 7r/2;) = 3- kr co E (V)sin (35) 22

T H E U N I V E R S I T Y OF MIC HI GAN 2778-6-T The analysis can now proceed in a manner closely parallel to that of ~3.1. The $1 path of integration in p (-7/2, 3 7r/2; P ) and p (- r/2, 3 T7/2; v ) is distorted as in Figure 2, and the contributions r to (34) associated respectively with the curved and straight portions of the ypath can be treated separately without giving divergent integrals. For the former, p(- 71/2, 377/2; V ) in (35) is replaced by (11), and p (- Tr/2, 31Tr/2; L) by (11) with r written for a. The latter is taken in conjunction with the contributions arising from ranges [El, 82e outside the primary range. The total field thus appears in the form e__>, i' 7r [IH(i,,) ((ka) (2) _~i rVH(1)(kr) (k (kr)] cot(' -)E (e)d =/(27r) /(kr) ) e (2) (ka)c(2 n ~~ e (kH, ( ka) -00.oo oo (l) 1+: 7/- f n-i" e [(1) H (ka) (2) (- 2 H kH (kr) J(2ir) /(kr) HL,(r)- (2) H((r) n=1 00 HI (ka) -2i'r n j 2i r n x cot(ir V)(e + e )E (8)d. (36) respectively if the point of observation is outside or inside the shadow region. 23

THE UNIVERSITY OF MICHI GAN 2778-6-T For points outside the shadow region, the technique is essentially the same as that employed in ~ 3.1 for the evaluation of (15). The paths of integration are taken to run below the real axis, and the identity (16) is used for the factor cot( rI ) in the integrand of the first integral of (36). The result is -oo H (ka) U- Ve1 (kr H (ka) )() E (2 1) W-~(2f) k(kr) n 2)(37) For r/2 3 /2, the path of integration in the integr)l in (37) agrees with (33) pro02ided 1 ()n- \ VeH (1) (2) (kr) E =j s U_ _e ~_ 1 VSIYS -[()Tr ( V/(2s ) I/(kr) e (kr) (38) -e~ H (kH r) E d) = e (2-ir) (kr)a1 For~~~~~~2

T HE U N I V E R S I T Y OF MI CHI GAN 2778-6-T Equation (38) is the analogue of equation (33) of I. It can be shown to hold for those values of 8 for which the integral converges, namely 7/2 to 3 1T/2.' For 0 < e < Tr/2 (but still outside the shadow region), the path of integration in the integral in (37) can be deformed so that both terminations at infinity are in the lower half of the complex L) -plane. The corresponding contribution to U is then conveniently written:____ H (ka) H)( /e2~' /() l;; >T/ (ka) (2)(kr) e-~ lr E (e) dv, /(27r) V(kr) IV (2)ka) H (ka) (39) where the path C (Figure 5) traverses two saddle-points; one near p _ ka cos (e/2) associated with the reflected wave, and one near v = kr sin & associated with the incident wave (Franz and Beckmann, 1956). For points inside the shadow region the evaluation of (36) is more straightforward. The sign of p is changed in the parts of the integrands of the integrals in the sum which contain the factor exp (2i7rrn v ), and the identity (26) then leads immediately to the well known result (Bi7~(1) (2) U=-7~r/(2 r)e4 1 0n c e-1i1 VSH(l)(ka)H V (kr)- 2i1rn (o VC7 1/ (kr) - n s 2) n=O s=l {ii [H )(ka)]} (40) *A proof of (38) can be constructed with the help of the infinite Legendre integral transform and its inverse. 25

T HE UN I VE RS IT Y OF M I C H I G AN 2778-6-T saddle-points CN \ Figure 5 The path C in the complex v-plane for the integral in (39) 5. The Neumann Problem The foregoing analysis needs only slight modification for the case when the boundary condition is aU/ a r -- on r=a. The contribution to Us analogous to (5) is r 1 (k aq (e1,e2; v ) (2) ) — V.... H. (kr) cot (Tr J)E (8) d,', (41) where the dash denotes differentiation with respect to the argument of the Hankel function, and 2 q(e13 e2; j) 1- S cos eika cosYE ( )sin Vdv. (42) 26

T HE U N I V E R S I T Y OF M I C H I G AN 2778-6-T For it is clear that on r a the part of a U / a r corresponding to (41) cancels the normal derivative of the incident field for 81<9<8 e s and is zero for 6 <81, 8, 2. Furthermore, from (3) and (42), q(8 1 " 92; d9 (ka) [ (t 1)2 2; * (43) Evidently, then, the analysis for the Dirichlet problem applies also to the Neumann problem if H(l)(ka)//(ka), H(2)(ka)//(ka) are throughout replaced respectively by H(1) (ka)/V/(ka)] H(2)(ka)//(ka) 27

T HE U N I V E R S IT Y OF M I C H I G A N 2778-6-T Appendix A The object of this Appendix is to show that (1) _-i COi3 e7 ~ ~~ e_2 ~ (2)(z), (^.1) X Ce E ( )sind q — /( e H((Al) (-1) (2) = i e E- 1S( e)sinyd Hr (z),e (A.2) (1) where the paths of integration run between certain regions of convergence at infinity in the complex $U-plane. For real positive values of z the situation is depicted in Figure (A-I), where the regions of convergence at infinity are shown shaded. It is tacitly understood that the paths of integration do not intersect the branch cuts of E 1(B) which start from the branch points at 0, + r, 21r2, 3 T... and travel to infinity along straight lines parallel to the negative imaginary axis. Now an expression for E (Lp) valid in the upper half of the complex -plane is (see equation (7 q) -plane is (see equation (7) of II). ((o.... e F(~, J+. I.l; + e ) _=ei....g......( e. *(A.3) n=O n ( I - n)I

T HE U N I V E R S I T Y OF M I C H I G A N 2778-6-T (-2) (1) -2'rr I 2' O 2 rr (-1) (2) Figure A-1 Paths of integration and regions of convergence at infinity in the complex l)-plane Also (1) r ip? -iz cos ~ e e sin 9p d'j (-2) (1) =z.!. eiu c d -iz cos W ) d V iz di (-2) ~' i -is z cos r e e d (-2) -- 2-r- e J (z). (A.4) 29

T HE U N I VE R S IT Y OF M I C HI G A N 2778-6-T Hence, from (A.3) and (A.4), (1) -iz cos ) 5 e _ ( Y))sin/ d (-2) 2e~i? e-~iTrf ( )n (n-~)! ( +n-~).;J =. 2e, -.(U 2n+~) J (z) B zu ( )n! (v+n)! wt+e2n + n=O 2 -~i'n'"V (2 r') e J (z4: (A-5) where the last identity is a special case of the Gegenbauer addition theorem (Watson (1944), p. 370, equation (9)). But (A.5) can be written (-:) (1) i TrS t, e \'lE _(Y) sin d (-2) (-1) - i'rr u2 - /( —) e ir e H(1 (3) (A.6) 3o

T HE UNIVERSIT Y OF MICHIGAN 2778-6-T Now allow z to be temporarily complex, z =\ zI exp (i; ) say, where 0 < <17 <. Then it is not difficult to see that the path of integration of the first integral on the left hand side of (A.6) can be chosen so that, for all values of Y on it, exp (-iz cos YV) becomes exponentially small as I z/ s oo (it is simplest to consider W = Tr/2). The first term on the left hand side of (A.6) is thereby identified with the first term on the right hand side. Thus (-2) _ e~ B1, 2 Coir e (2) e E s( ) sin cy d H) Ir e e. (Z). (A.8) Y-r 21 J, (-1) Finally, (A.1) follows from (A.8) by changing the sign of v; and (A.2) follows from (A.7) by changing the sign of L and changing the variable of integration from q to + 2r-. 31

T HE U NI V E R S I T Y OF M I C H I G A N 2778-6-T Appendix B The object of this Appendix is to show that, for ka>l, (1) H (ka) E (e) dv Hk (ka) 2ik sin ( ef2 2 ka sin3(I e) (B.1) where the path of integration is that shown in Figure 3. If a new variable of integration oC is introduced via the relation J = ka cos o, (B.2) the first two terms of the Debye asymptotic forms for the Hankel functions give (1) HV (ka) r i(5 - 2 sin ) 12ika (sin o - co s ) R ) -,2-i 1- - e H(2)(ka) 12 ka sin3 o (B.3) and the first two terms of the relation (see equation (6) of II) 2 e i EEV( ) iT (V _F(~, ~; +l; -2 ine ) (B.4) -2.I J(sin 0) 2sin3 32

T HE UNIVERSITY OF MICHIGAN 2778-6-T give E - lil 1 i cot 8 ]eika8 coso E 1(e-i.,/(je e e V-2 2 IV(ka sin 9 cos o ) 8 ka cos (B.5) When (B.3) and (B.5) are substituted into the left hand side of (B.1), the exponential part of the integrand is exp [ika f(o), where f(o ) = 2(sino(- occosok) + e cos. (B.6) The saddle-point, then, is evidently at oc e /2, and with the notation = o - e, (B.7) s = sin (~ 8), c = cos (~ 8), (B.8) it is adequate to write ika f(O4) 2ikas ika 2 3 4 2 k2a2c28 6 ikas (B +i =lka (3f -s 9k) 3 9 2- 3s2 10 92 sin /(cosos)= sc [1+ 13 il B.10)2 2sc 8c 0.) sin co)eikaf(o ) 2ikas 10-9s 2 (2 2- 4 6 8c2 a k a c4 ~ (B.ll) 33

T H E U N I VE R S IT Y OF MI C H I G AN 2778-6-T The procedure is now standard. Expressions (B.3), (B.5) and (B.11) are substituted into the integral of (B.1), the path of integration for A is taken from -oo exp (ilr/4) to oo exp (itr/4), and the integration is performed term by term. To the accuracy required, the second term in the square brackets in (B.3) and (B.5) need only be taken in conjunction with the first term (unity) in the square bracket of (B.ll). The result appears explicitly in the form 2 2 21 kae2ikas. i 1O 29s 2 f5 2s 5-2s 2k 2 4kas 3s 2kas2 k12kas2 ka 4c2s 3s3 (B.12) which simplifies to the right hand side of (B.1), the order term in (B.1) being evident from inspection. The algebra in this derivation has been written out at some length for two reasons. First, to show how the terms which would individually blow up 2 at @ = Fr, namely those in (B.12) with c in the denominator (which originate in (B.5) and (B.10)), in combination remain finite there. Secondly, to give an indication of the labour involved. Since a simple answer is obtained as the sum of a considerable number of terms, it might be thought that a better technique would be available, and it is interesting to question whether the completely different approach of Keller, Lewis and Seckler (1956) has here 34

T HE U N I V E R S I T Y OF M I C H I G A N 2778-6-T an advantage. The author understands, however, from private conversation with J. B. Keller, that their Luneberg-Kline method, which obtains the asymptotic form for ka >> 1 without recourse to an exact solution, also becomes laborious if higher order terms are sought.

T HE UNI V E R S ITY OF M I C H I G A N 2778-6-T Appendix C The object of this Appendix is to show that HXD) (k a) e2ika ~ Fi H(2) 2ka H (ka) (C.1) where the path of integration is that shown in Figure 3, but with e= IrT so that it runs through the origin. The method adopted is bated on an argument of the steepest descents type, but direct application of the standard technique in the manner of Appendix B is spoilt by the factor tan (7r u) in the integrand, However, in view of the symmetry in i of the integrand, only one half of the path of integration need be considered. If this is taken as the portion for which lm VL <O the identity taen..)ne (Ca2) n=1 can be used, and the left hand side of (C.1) written as tha i t rns truhheo(C.3) n=l

where i 7r Hl (ka) I - e d (C 0.4) 0~ J~ HZ(2)(k) i-7rV1H (ka) -2i1Tnn I ne ()2) 2 e dD, na 2i2,3,..., (C.5) n H (ka) the lower limits of integration being at infinity in the third quadrant of the complex V-plane. To evaluate Io and I, the new variable of integration oa is introduced as in Appendix (B) (equation (B.2)), and the asymptotic form (B.3) used for the quotient of the Hankel functions. The evaluation of I then follows closely the pattern of Appendix B, the calculation being, in fact, considerably less laborious since, effectively, s =1, c = O, and the function E (9) is absent. The result is 2ika 5i I ~ - ka e (1 - ). (C.6) 0 12 ka For I, n = 1,2,3,..., the same type of argument gives, in the first n place, liT 2 n ika 2 ikal 2i8rnka18 I ru(-) 2ie (ka) e dS (C27) Here, =4o<- 1Tr/2 (cf. equation (B.7)), and the factor exp (-2i1r-n - ), which distinguishes the integrand of I from that of I, has been replaced 37

T HE U N I VE R S ITY OF M I C H I G AN 2778-6-T by exp (2irrn ka# ). It can be seen, though hardly without some further working, that the inclusion of the next approximation to exp (-2itrn LJ ), 3 which would introduce the further factor 1-i7r n kap /3, would only yield terms negligible to the present order of approximation. Now in (C.7) write - = ~t-rr n, (C.8) to obt ain 21.2 I nr (_)n 2i ika (ka2 -ika Tr n i ( 2- ) ika i 2 n~n IT n n.l _ _ i 2ika (-). (C.9) 2 r2n Since oo ()n+ 2. = _ (Co10) n=l n 12 the substitution of (C.6) and (C.9) into (C.3) gives -2 ka e (1 - ), (C.ll) 2ka which is in agreement with (C.1).

T HE U N I V E R S IT Y OF MI C HI G A N 2778-6-T Refe rences Beckmanns P. and Franz, W. (1957) Zeitschrift fur Naturforschung 12, 533. Clemmow, P. C. (1959a) The University of Michigan, Radiation Laboratory Report 2778-3-T (Reference I). Clemmow, P. C. (1959b) The University of Michigan, Radiation Laboratory Report 277E-5-T (Reference II). Debye, P. (1908) Physikalische Zeitschrift 9, 775. Felsen, L. B. (1957) University of Michigan Research Report R-574-57, PIB- 502. Franz, W. and Depperman, K. (1954) Ann. Phys. 1_, 253. Franz, W. (1954) Zeitschrift fu{r Naturforschung 9, 705. Franz, W. and Beckmann, P. (1956) I.R.E. Trans. AP-4, 203. Friedlander, F. G. (1954) Comm. Pure Appl1 Maths. D, 705. Keller, J. B., Lewis, R. M. and Seckler, B. D. (1956) Comm. Pure Appl. M1"aths. 9, 207. Love, A. E. H. (1915) Phil. Trans. Roy. Soc. A 21_, 105. March, H. W. (1912) Ann. Phys. _, 29. Rybczynski, W. v. (1913) Ann. Phys. 41, 191. Watson, G. N. (1918) Proc. Roy. Soc. A 2_, 83. Watson, G. N. (1944) Theory of Bessel Functions (Cambridge University Press). 39

UNIVERSITY OF MICHIGAN 3 901 5 02827 60801111111 3 9015 02827 6080