THE UNIVERSITY OF MICHIGAN 3648-4-T AFCRL 787 STUDIES IN RADAR CROSS SECTIONS - XLVI THE CONVERGENCE OF LOW FREQUENCY EXPANSIONS IN SCALAR SCATTERING BY SPHEROIDS SCIENTIFIC REPORT NO. 4 CONTRACT AF 19(604)-6655 by T.B.A. Senior 31 August 1961 Report No. 3648-4-T on Contract AF 19(604)-6655 Prepared for ELECTRONICS RESEARCH DIRECTORATE AIR FORCE CAMBRIDGE RESEARCH LABORATORIES OFFICE OF AEROSP.ACE RESEARCH UNITED STATES AIR FORCE B'EDFORD, MASSACHUSETTS

THE UNIVERSITY OF MICHIGAN 3648-4-T Requests for additional copies by Agencies of the Department of Defense, their contractors, and other Government agencies should be directed to: ARMED SERVICES TECHNICAL INFORMATION AGENCY ARLINGTON HALL STATION ARLINGTON 12, VIRGINIA Department of Defense contractors must be established for ASTIA services or have their "need-to-know" certified by the cognizant military agency of their project or contract. All other persons and organizations should apply to: U.S. DEPARTMENT OF COMMERCE OFFICE OF TECHNICAL SERVICES WASHINGTON 25, D. Co:: —-- -. - ii II__________

THE UNIVERSITY OF MICHIGAN --. 3648-4-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. m II "The Zeros of the Associated Legendre Functions Pn (p') of Non-Integral Degree", K. M. Siegel, D. M. Brown, H. E. Hunter, H. A. Alperin and C. W. Quillen (UMM-82, April 1951), W-33(038)-ac-14222. UNCLASSIFIED 20 pgs. III "Scattering by a Cone", 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. Co Orthwein and C. E. Schensted (UMM-92, February 1953), AF-30(602)-9. UNCLASSIFIED. 70 pgs. V "An Examination of Bistatic Early Warning Radars", K. M. Siegel (UMM98, August 1952), W-33(038)-ac-14222. SECRET. 25 pgs. VI "Cross Sections of Corner Reflectors and Other Multiple Scatterers at Microwave Frequencies", 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. 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 (UMM116, October 1953), AF-30(602)-9. UNCLASSIFIED. 38 pgs. ---- _______________________________________- iii _________________________________________________

THE UNIVERSITY OF MICHIGAN 3648-4-T X "Scattering of Electromagnetic Waves by Spheres", H. Weil, Mo L. Barasch and T.A. Kaplan (2255-20-T, July 1956), AF-30(602)-1070. UNCLASSIFIED. 104 pgs. XI "The 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 "Description 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. SECRET. 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", Ho Weil, R.R. Bonkowski, T. Ao Kaplan and M. Leichter (2255-12-T, August 1954), 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", Ko 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. _________~___**____ ~iv _... ____.__ ______

THE UNIVERSITY OF MICHIGAN -, 3648-4-T 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. XXI "Radar Cross Section of a Ballistic Missile IIr', 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 Spheroid", 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 Defense", M.L. Barasch, H. Brysk, J.W. Crispin, Jr., B.A. Harrison, T. B. A. Senior, K. M. Siegel, H. Weil and V. H. Weston (2778-1-F, April 1959), AF-30(602)-1953. SECRET. 362 pgs. XXV "Diffraction by an Imperfectly Conducting Wedge", T. B. A. Senior (2591-2T, 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. UNCLASSIFIED. 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. __ _ _ _ _ __ _ _ _ _ _ __ _ _ _ _ _ _ V I II I V I I I I

THE UNIVERSITY OF MICHIGAN 3648-4-T XXVIII "The Physics of Radio Communication via the Moon", M. Lo Barasch, H. Brysk, B.Ao Harrison, T.B.A. Senior, K. M. Siegel and Ho 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., Bo A. Harrison, Ro 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. XXXI "Diffraction by an Imperfectly Conducting Half-Plane at Oblique Incidence", T. BA. Senior (2778-2-T, February 1959), AF-30(602)-1853. UNCLASSIFIED. 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. 61 pgs. XXXIV "An Infinite Legendre Transform and Its Inverse", P. C. Clemmow (2778-5T, 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. Clemmow (2778-6-T, April 1959), AF-30(602)-1853. UNCLASSIFIED. 39 pgs. XXXVI "Diffraction of a Plane Wave by an Almost Circular Cylinder", P.C. Clemmow and V.H. Weston (2871-3-T, September 1959), AF-19(604)4933. UNCLASSIFIEDo 47 pgs. XXXVII "Enhancement of Radar Cross Sections of Warheads and Satellites by the Plasma Sheath", C. L. Dolph and H. Well (2778-2-F, December 1959), AF-30(602)-1853. SECRET. 42 pgs. -ii- ~i~~~- ~~~i~ii-i- -- iii~iV1 -i iM i- w. ~ ~i ii ~ ~i i

THE UNIVERSITY OF MICHIGAN 3648-4-T XXXVIII "Non-Linear Modeling of Maxwell's Equations", J. E. Belyea, R. D. Low and K. M. Siegel (2871-4-T, December 1959), AF-19(604)-4993, AFCRC-TN-60-106. UNCLASSIFIED. 39 pgs. XXXIX "The Radar Cross Section of the B-70 Aircraft", R. E. Hiatt and T. B. A. Senior (3477-1-F, February 1960), North American Aviation Purchase Order LOXO-XZ-250631. SECRET. 157 pgs. XL "Surface Roughness and Impedance Boundary Conditions", R. E. Hiatt, T. B. A. Senior and V. H. Weston (2500-2-T, July 1960). UNCLASSIFIED. 96 pgs. XLI "Pressure Pulse Received Due to an Explosion in the Atmosphere at an Arbitrary Altitude, Part I", V. H. Weston (2886-1-T, August 1960), AF-19(604)-5470. UNCLASSIFIED. 52 pgs. XLII "On Microwave Bremsstrahlung From a Cool Plasma", M. L. Barasch (2764-3-T, August 1960), DA-36 039 SC-75401. UNCLASSIFIED. 39 pgs. XLIII "Plasma Sheath Surrounding a Conducting Spherical Satellite and the Effect on Radar Cross Section, " K-M Chen (2764-6-T, October 1960), DA-36 039 SC-75041. UNCLASSIFIED. 38 pgs. XLIV Integral Representations of Solutions of the Helmholtz Equation with Application to Diffraction by a Strip", R. E. Kleinman and R. Timman (3648-3-T, February 1961), AF-19(604)-6655. UNCLASSIFIED. 128pgs. XLV "Studies in Non-Linear Modeling-II: Final Report", J. E. Belyea J.W. Crispin,Jr., R.D.Low, D.M.Raybin, R. K.Ritt, O. Ruehr, and F.B.Sleator (2871-6-F, December 1960), AF-19(604)-4993, UNCLASSIFIED. 95 pgs. XLVI "The Convergence of Low Frequency Expansions in Scalar Scattering by Spheroids", T. B. A. Senior (3648-4-T, August 1961), AF19(604)-6655. UNCLASSIFIED. 143 pgs. vii

THE UNIVERSITY OF MICHIGAN 3648-4-T PREFACE This is the forty-sixth 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 boundaryvalue 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 ix I_______ __ ____ ____ __

THE UNIVERSITY OF MICHIGAN 3648-4-T TABLE OF CONTENTS Page SUMMARY xi I INTRODUCTION 1 II THE SPHERE 3 III THE GENERAL SPHEROID 11 IV THE SPHEROID OF SMALL ELLIPTICITY 16 V THE DISC 26 VI THE OBLATE SPHEROID OF ELLIPTICITY NEAR TO UNITY 41 VII THE OBLATE SPHEROID OF INTERMEDIATE ELLIPTICITY 51 VIII PROLATE SPHEROIDS 74 IX DISCUSSION OF RESULTS 101 ACKNOWLEDGEMENTS 118 REFERENCES 119 APPENDIX A Spheroidal Coefficient Expansions 120 APPENDIX B The Rayleigh Series for A Disc 128 ____________ __________________ x.^_____________________________

THE UNIVERSITY OF MICHIGAN 3648-4-T SUMMARY For the scalar problem of the diffraction of a plane wave by a spheroid the exact solution is known, and at low frequencies the expression for the far field amplitude can be expanded in a series of increasing positive powers of ka, where k is the wave number and 2a is the interfocal distance. This is the Rayleigh series, and is convergent for sufficiently small values of ka. In order to determine the range of frequencies for which this expansion is applicable an essential factor is the radius of convergence, and the paper is devoted entirely to the calculation of this quantity. Attention is concentrated on the case in which the plane wave is incident nose-on, and the radius of convergence is obtained as a function of the length-to-width ratio for prolate and oblate spheroids, hard as well as soft. For other angles of incidence it can be shown that the radius is not greater than this, and in most instances it would appear to be the same.,- - - -,,Xt~~,, -xi,

THE UNIVERSITY OF MICHIGAN - 3648-4-T I INTRODUCTION In recent years it has become increasingly apparent that one of the most difficult problems in diffraction theory is the quantitative description of the scattered field at wavelengths which are comparable with the effective dimensions of the body. Even in those cases where an exact solution of the scattering problem exists in the form of a wave function expansion, a large number of terms are necessary to calculate the rapid variation as a function of frequency which is often typical of the resonance region. Moreover, there are as yet no approximate methods specifically designed for treating this frequency range, and for this reason attempts have been made to push the high and low frequency approximations as far as possible in the hope of narrowing the gap. In particular, some success has been achieved in applying high frequency techniques such as the geometrical theory of diffraction even when the wavelength is as large as a typical dimension of the body. At the other end of the frequency spectrum the scattered field can be expanded as a series of increasing positive powers of ka, where k is the wave number and a is a dimension representative of the body. This is the so - called Rayleigh series and for sufficiently small values of ka (that is, for sufficiently low frequencies) the expansion can be shown to be convergent. Nevertheless, for any body the radius of convergence is almost certainly finite and sets an upper limit on the __ _ _ _ _ _ _ _ __ _ _ _ I _ _ _ __ _ _ _ _ _ _ _ _ 1 __ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _! _ _ _ _ _ _ _ _

THE UNIVERSITY OF MICHIGAN 3648-4-T portion of the resonance region to which the Rayleigh series is applicable. It is obviously desirable to know this radius before attempting to predict the resonant behavior using a finite number of terms in the low frequency expansion. The present paper is concerned with scalar scattering by prolate and oblate spheroids with particular reference to the convergence of the Rayleigh series. In ~2 the series for the sphere is considered briefly and this serves to illustrate the methods which are available for assessing the convergence. For the more general problem of the spheroid the expression for the far field amplitude is derived in ~3 and this is followed (~4) by an analysis of the case in which the ellipticity is small (almost spherical bodies). The next two sections are devoted to the problem of an oblate spheroid whose ellipticity is unity (a disc) or only slightly different from unity (almost a disc), and in ~7 the convergence for oblate spheroids of intermediate ellipticity is determined. In ~8 the convergence is calculated for a prolate spheroid whose ellipticity is near to unity (a body which is almost a vanishing rod), and a study of the intermediate ellipticities then completes the discussion of the prolate spheroid. In all cases the bodies considered are either soft or hard (Dirichlet or Neumann boundary condition respectively at the surface), and the resulting values for the radius of convergence are displayed in 69. It must not be assumed, however, that these values would also be applicable if the boundary condition differed from the above. ------ - ---- ---- --- --— ~~~~~~~~~~~~~~~~~~~~- 2 - -- Ill_ I [ -III I I

THE UNIVERSITY OF MICHIGAN 3648-4-T II THE SPHERE A limiting case of both the prolate and oblate spheroids is the sphere and it is convenient to begin by considering the Rayleigh series for this more simple body. In view of the symmetry possessed by the sphere it is sufficient to take a field which is incident in the direction of the negative z axis of a Cartesian coordinate system (x, y, z), and if the field is also assumed to be a plane wave, it can be written as i -ikz V =e (1) where the time factor e has been suppressed. In terms of spherical polar coordinates (R, 0, 0) with x =R sin 0 cos 0, y =R sin 0 sin 0, z =R cos 0, V has the expansion 00 V = (-i) (2n+l) j (kR) P (cos 0) (2) n n n =0 and if the scattered field is written similarly as 00 ao V = (-i) (2n+l) A h (kR) P (cos0), (3) n n n application of the boundary condition at R = a gives _ — -- --- ----- ---- - --- 3 ____________________

THE UNIVERSITY OF MICHIGAN 3648-4-T j jn(P) A (4) n A h (p) with p = ka. A is either unity or a/3p depending on whether the body is soft (Dirichlet boundary condition at the surface) or hard (Neumann condition) respectively. If each Hankel function of argument kR is replaced by the first term in its ikR asymptotic expansion for large kR, the coefficient of e- in the scattered field is kR the far field amplitude, and from equation (3) its expression is seen to be oo f(cos 0) =2i (-1)n+ (n + A P (cos 0). (5) 2 nn n=O This series is absolutely convergent for all (real) values of p. Moreover, within some neighbourhood of the origin in the complex p plane, each A (n = 0, 1, 2.... ) can be expanded in a series of positive (integral) powers of p and n is therefore an analytic function of p within the region. It then follows that if the terms in (5) are re-arranged, an expansion for f(cos 0) is obtained which includes only positive powers of p and is convergent for values of p inside the smallest circle of convergence of the individual A. This is the Rayleigh series, and for n the sphere the terms up to and including p have been given by Senior (1960a). The calculation of the least circle of convergence is a trivial matter. From equation (4) it is apparent that the only singularities of A are poles at the zeros of the spherical Hankel function (or its derivative), and the location of these zeros is 4 ______________

THE UNIVERSITY OF MICHIGAN 3648-4-T such that the singularity nearest to the origin is provided by one of the smaller values of n. Using the expressions for the h (p) it is found that whereas h (p) has a a no zero, both - h (p) and hl(p) have a zero at p = - i, and the zeros of a hl(p) and h2(p) are (-i + 1) and - (-3i + 3) respectively. Accordingly, Al is infinite on the unit circle and since all the other coefficients are regular inside, the entire Rayleigh expansion must converge forl Pl < 1. The fact that a singularity exists for whichi p | = 1 shows that the expansion does not converge outside this region, and in consequence the Rayleigh series for both hard and soft spheres converges only for ka < 1. (6) The above method is based upon the location of the smallest singularities in the complex p plane, and for this purpose it is essential to have available the exact expressions for the individual A. If these are not known, or if their complication is such that the location of the singularities is not practicable, the method is no longer applicable, and it is necessary to employ a more intuitive approach. The one which has proved most valuable involves a comparison of the numerical coefficients in the low frequency expansions for the A, and although the method cannot be regarded as rigorous, it is generally sufficient to indicate the radius of convergence with a reasonable degree of certainty. The new technique will be illustrated with reference to the sphere. ____ ___ ___ ___ ___ ____ ___ ___ _IIIII_ — 5 _ _ I _ I _II_ _ _ _ _ __ _ _ _ _ -

THE UNIVERSITY OF MICHIGAN - 3648-4-T The starting point for the analysis is the expansion of the A in the form r =1 r A = p a{ 1+, p (7) n n where m is some integer and the a are independent of p. In general, the a r r will be complex and for most bodies only a small number of them will be known. For the sphere, on the other hand, it is a straight forward matter to determine as n many of the a as are desired by inserting the series developments of the Bessel functions into equation (4), and in Tables I and II the values are given for r < 11 and n = 0, 1 and 2. If the moduli a are now plotted sequentially (r = 1, 2, 3,....) for each n, the curves shown in Figures I and II are obtained. In reality, each curve only has a meaning when r is an integer, but it is tempting to join the discrete points by the continuous curves shown; an infinity then implies that the corresponding power of p has zero coefficients. These curves confirm in a striking manner the radii of convergence previously found. When n = 1 (soft) and n = 0 (hard) the curves are asymptotic to the value unity, and this represents the smallest radius of convergence (and hence the radius of convergence of the low frequency expansion for f) in accordance with equation (6). The fact that for the soft sphere A has no singularity in the finite portion of the p plane is reflected in the upward trend of the corresponding curve in Figure I.

.. THE UNIVERSITY OF MICHIGAN..... —3648-4-T TABLE I. CONVERGENCE COEFFICIENTS FOR SOFT SPHERE n=0 n=l n=2 (ao)-1' lar| (arl)1 lali (ar2) r |la2| (a 0) (a 1)r __ar _ _r _ 1 1 1 1 0 o O oo 2 2 1.2247 3 1.2910 2.0494 3 5 21 3521 3 1 1.4422 1 1.4422 0 oo 3 3 2 3 4 -21.6548 1.2359 0 oo 15 7 2 2 1 5 - 1.8639 - - 1.2011 2.1411 45 5 45 6 4 2.0703 111.1614 -1.9753 315 27 297 7 _1 l2.2746 711.1376 2 1.9151 315 175 189 2 67 5 8 — 2.4771 1.1193 - 1.9518 2835 165 1053 2 11515 9 — 2.6781 - 1.1054 2.0999 14175 2835 3969 4 8313 2 10 2.8780 1.0943 2.2297 155925 20475 6075 2 29543 2 11 — 3.0767 - -1.0853 1- 1.9239 467775 72765 2673

TUE UNIVERsITy OF 3648-4-T MICHIGAN / /l 0 11ft 0 / II / 00 Y) 0 CI) f 0.!: I;!!/~ ~ ~ ~ ~~In 0! O BO U~~~~~~~~~~~m = o)

THE UNIVERSITY OF MICHIGAN 3648-4-T TABLE II. CONVERGENCE COEFFICIENTS FOR HARD SPHERE n=0 n=1 n=2 o (arO a-r I (ar l-r ar (ar2)-r lar21 1 0 oo 0 oo 0 oo 3 3 25 2 - 1.2910 - 1.8258 - 2.2450 5 10 126 3 -1 1. 4422 1.8171 0 oo 3 6 3 3 5 4 - 1.2359 - 1.7479 - 2.3858 7 28 162 2 1 2 5 - 1.2011 1. 5849 - - 2.3220 5 10 135 6 11 1. 1614 -1.8732 185 2. 1038 27 216 16038 71 29 n 7 - 1.1376 21.7400 2.0829 175 1400 1701 67 89 250 8 67 1.1193 1. 5277 250 2. 0731 165 2640 85293 1151 523 107 9 I- 1 1.1054 - 31.5202 - 7 2.0599 2835 22680 71442 10 8313 1.0943 37 1.6609 2557 2.0242 20475 218400 2952450 29543 4918 793 1 - 543 1.0853 918 1. 6107 - 751 1 9875 72765 931392'1515591'

THE UNIVERSITY MICHIGAN OF 3648-4-T I CHGIGAN N I~j ii I I 0) II I I~~ - --- -^- 10 |~~~~~ t! r') 0 C)0 10O

THE UNIVERSITY OF MICHIGAN 3648-4-T III THE GENERAL SPHEROID Having obtained the radius of convergence of the low frequency expansions for the sphere, we now turn our attention to the more general problem of the spheroid and seek to determine the corresponding limits on the convergence using the methods previously described. For this purpose a necessary preliminary is the derivation of an expression for the scattering function, and since the solutions for prolate and oblate spheroids can be deduced from one another by a trivial change of parameter, the analysis will be given for only the first type of body. Consider, therefore, a prolate spheroid which is defined in terms of the prolate spheroidal co-ordinates (, rj, 0) by the equation e =o. Incident upon the body is a plane wave travelling in the x z plane of a Cartesian co-ordinate system (x, y, z) where 2 /2 2 2 1/2 x =a (1-)(-1)1/ cos, y =a (-r1 ) (- 1) sin 0 z = arA, with 2a as the interfocal distance, and if the direction of incidence makes an angle' with the positive z axis, the field can be written as i ik (x sin ~ + z cos ~ ) (8) V = e The incident field can also be expressed as a sum over angular and radial 1._________________________ 11 I_____________________

THE UNIVERSITY OF MICHIGAN - 3648-4-T spheroidal functions, and by postulating a similar expression for the scattered field V the unknown amplitude factor can be determined from the boundary condition at the surface. The details of the analysis are given in Senior (1960a), and it is there shown that S (c, cos) ( Vs =2 (2- )i1 mn A R (c, )S (c, rT) cos m V 2j om N (c) mn mn mn' mn (9) m=O n=m with A m mn (10) mn AR(3) (c, ) mn o where nowA is unity or aD/D depending on whether the body is soft or hard respectively, and c = ka. The notation is that of Flammer (1957) and the reader is referred to this book for the definitions of the symbols and functions here used. In the far field c.- kR, where R is the distance from the center of the spheroid, and since (3) 1 i (n mn ci c as c~ —oo, the far field amplitude is oo co S (c, cos ) f(r, )=- 2i o(2- ) m () A S (c, ) cos m 0. (11) om N (c) mn mn LI,.. mn m=0 n=m For sufficiently small values of c (that is, for sufficiently low frequencies) all the terms in (11) can be expanded in positive integral powers of c, and by re-arranging 12 _ __ __ _ ____ ____

THE UNIVERSITY OF MICHIGAN 3648-4-T the resulting series the Rayleigh expansion is then obtained. Of the various factors which make up the individual terms in (11), the angular functions are free of singularities, whereas A (see equation 10) has poles mn at the zeros of the denominator. In addition, of course, there is the possibility of singularities arising from the vanishing of the normalization constant N for some mn (complex) c, but if such a singularity were the nearest to the origin of the c plane, the convergence of the Rayleigh series would be independent of the precise shape of the spheroid. This is intuitively unlikely, and it will be shown later that any singularities of N can be discounted as far as the convergence of the series is conmn cerned. The radius of convergence is then specified by the coefficient A whose mn pole has the smallest modulus, and is given by the smallest zero of the radial functions R(3) mn In the particular case r =r (incidence along the negative z axis) the expression for the far field amplitude can be simplified by observing that S (c, - 1)=0 mn unless m = 0. The summation over m now contributes only the first term and accordingly S (c,-1) f(, r)-2i N (c) A S( (c ). 12)' ( on on n= on Using this equation the Rayleigh series for both soft and hard prolate spheroids have been computed up to and including terms in c, and the results are given in Senior (1960a).,__.....__-_________________________ 13

THE UNIVERSITY OF MICHIGAN - 3648-4-T An interesting feature of equation (i2) is that all the functions which can affect the convergence of the low frequency expansion also appear in the expression for f(rn, ),' 7& 7T, and it therefore follows that the radius of convergence when r 7Tf cannot be greater than it is for incidence along the z axis. Since this conclusion also holds for an oblate spheroid (as can be seen by replacing c by - ic and e by it in (11) and (12), it is clear that the convergence of the Rayleigh series is not directly related to the radius of curvature at the'point' at which the incident field strikes the body. This rules out one of the ways in which the convergence could depend on ~ and. Because of the complication involved, a detailed study of the dependence of the radius of convergence on' has not been pursued, and the subsequent analysis is confined to the case = 7r. The study is then aimed at an investigation of the way in which the convergence varies with I, and accordingly the results which are foun o represent only the upper bound on the convergence when ~ /Tr. Nevertheless, preliminary calculations do suggest that this upper bound is in fact the actual radius of convergence, implying that the convergence is independent of the angle P at which the field is incident. When' = 7r the far field amplitude is given by equation(12) and in the following sections the radius of convergence of the expansion for f(r, 7r) is determined for certain selected values of 5. From these results it is possible to infer,,________________________,14, __ _ _ __ __ __ _

THE UNIVERSITY OF MICHIGAN 3648-4-T the convergence for all A, and to this end we start by considering the problem of an almost spherical body. 15 _

THE UNIVERSITY OF MICHIGAN 3648-4-T IV THE SPHEROID OF SMALL ELLIPTICITY The ellipticity of the (prolate) spheroid 0 = 6 is e =1/o and if 0 - oo, c - 0 in such a way that c tends to a finit limit p, the spheroid degenerates into a sphere of radius p/k. The amplitude function defined in equation (12) then reduces to the sphere amplitude shown in equation (5), and this fact is most clearly seen by expanding the radial spheroidal functions in terms of the spherical Bessel functions. From Flammer (equation 4. 1.15) we have 00oo on r-n on () c (c)R() (c, i dr (c) jn(c (13) o on r=0, 1 00 oo' where cn (c) = d (c) and the summation extends over even or odd values of o r r=0, 1 n according as n is even or odd (denoted by a prime attached to the summation sign). Similarly, 0' c n (c)R3), irn dn (c) h (ca) (14) o on r n r=0, 1,, 1 The analogous formula for an oblate spheroid is e = 2 + 1 + 1 --- - --- -- --- - - --- -- — __ ___ _ _,16,, _ _ _ _ _ _ _, _ _

THE UNIVERSITY OF MICHIGAN 3648-4-T and since d on(c)= (r-n)+ 0(c2) r for small c, it follows that as c-o 0 the spheroidal amplitude coefficient A reduces to n A h(cO) n in agreement with the spherical coefficient (4). Morever, 00 oo' S (c,) d on(c) P (cos 0) on r n r=0, 1 --- P (cos O) n and on 2 ^,00 )d (c) N (c) - 2 2 on z1. 2r + 1 r =0, 1 2 2n+' which completes the reduction of equation (12) to equation (5). A consequence of this identification is that in the sphere limit the convergence of the Rayleigh series for the soft body is determined by the (3) smallest zero of R (c, c) as a function of c i, and for the hard body by the ol a (3) smallest zero of Roo (c, ~). This suggests that when the ellipticity is small but not zero a possible method for assessing the convergence is to expand From henceforth we shall omit the suffix'o' from the radial variable specifying the spheroid. __ _ _ _ _ __ _ _ _ _ _ __ _ _ _ _ _ 17., — - - - - - --

THE UNIVERSITY OF MICHIGAN 3648-4-T (3) R (c,) in terms of the spherical Hankel functions, and calculate the perturbaon tion of the zeros of h (ce) by retaining only the leading powers of e and c. Although it would appear that two small but unrelated quantities are now involved, this is in fact not so, and the requirement that the spheroid goes over into a sphere of radius p as c —O implies that e = c/p for all values of c under discussion. Let us therefore begin by considering the soft spheroid and attempt to locate (3) the smallest zero of R( (c, a). From the recurrence relations defining the on on spheroidal coefficients d, we have r don n 2 4 n+2 n+2c + don n + 2 c4 d 1+a c +0(c n n don n 2,4) n-2 n-2 c +( and in general don 0 (c In-r ) r where n (n+l)(n+2) Cn+2 2 (2n+1) (2n+ 3) 18

THE UNIVERSITY OF MICHIGAN 3648-4-T 2-(2n - 1) (2n+ 3) n + - odd a = -- (+ for n n 00 n2 2 even 2(2n - 1)2 (2 n+ 3) and n n(n-l) 2 2(2n-1) (2n+l) on with an-2 0 if n < 2. Bearing in mind that the factor c is cancelled by a like o factor in the expression R) (c, ~) and can therefore play no part in the conon vergence analysis, equation (14) now gives on( R(3) n 2 2 n cO (c) R() (c,) (1+a c )h (ce)- c a h (ce) o on n n n n-2 -c +2 h2 (c) + (c 4) -C a +2 n+2 and since (n+l)(2n-1), 2n-1 h n(cO) -- 1 hn(CO) + h' (c) hn - 2 L (ce)2 I n ce n n+ ) n(2n+3)1 2n+3 n + 2( (ce)2 Jn( ce where the prime attached to the Hankel function denotes differentiation with respect to the whole argument, we have co (c) R (c ) 1+ c (an2 + a )+ (n+l)(2n-l)a o on n-2 n n+2 n-2,,, - -- -- - -- - - ^-^ -- 19 ___ ^ ^ __ -

THE UNIVERSITY OF MICHIGAN 3648-4-T + n(2n+3) a n+2 h (c) - ce (2n-l) an ( 2 l[ 3/ 1 n-2 - (2n+3) a+2 h' (c) +0(c4). n+2 n (3) The zeros of R) (c, 0) are therefore given by the roots of on h (p) =e (2n-1) an - (2n+3) an2 ph (p) + 0(e ), (15) n n-2 nT 2 n and the smallest root in the complex p plane determines the smallest radius of convergence of the expansions for the individual A, and hence the radius of convergence of the Rayleigh series. In passing it should be pointed out that to order e equation (15) is simply hn(p) = 0 (16) and consequently a first order analysis will not reveal any change in the radius of convergence. This is otherwise obvious from the fact that the ratio of the major to 2 -1/2 2 the minor axes is (1 - e ) / =1+0 (e ), which implies that the first order terms in e or c correspond only to a change in the radius of the sphere but not to any deformation of shape. To obtain the solution of equation (15) it is assumed that p =p +7e + O(e) n where p is a solution of (16). If this is inserted into (15) it follows immediately that, _...._.__ 20 _...___,_ _,

THE UNIVERSITY OF MICHIGAN 3648-4-T = pn {(2n-1) n-2 - (2n+3) a n2 and since n n n (2n-l3)a l - (2n+3) a' l n-2 n+2 4 (2n-1)(2n+3) the perturbed root is therefore e 4 P Pn 1+ {1 (2n1)( +0(e 4. (17) It will be observed that the terms in braces are real, so that the effect of the perturbation is merely to change the magnitude of the root whilst leaving its phase unchanged. Moreover, the terms are also positive for all integer n (includin zero) and thus the magnitude of the perturbed root exceeds that of the unperturbed, showing that the effect of the perturbation is to increase the radius of convergence of the expansion for A. For a fixed ellipticity the percentage change increases with increasing n a 1. To determine the smallest radius of convergence it is only necessary to recall that pn also increases with increasing n > 1, and whereas p oo, p = - i. Accordingly, the smallest perturbed root is that corresponding to A l, and is p =- i (1+ - + 0(e4). (18) The modulus of this represents the value of c. specifying the radius of convergence and consequently for a soft prolate spheroid of small ellipticity the Rayleigh series ____ ____ ____ ___ ____ ____ ____ ___ 21 __ __ __ __ ____ _ ___________ _____ __

THE UNIVERSITY OF MICHIGAN 3648-4-T converges for c < 1 + O +0( 4). (19) 5~2 When the spheroid is hard rather than soft a similar treatment is possible based upon the location of the roots of R R() (c,). By expanding this derivative in powers of c we have 1 on(C) a R(3) +n 2 )2 an nh(c) (c, )=( 1 +cv c )h' (c5)-c h (c ) c o D0 on n n n-2n-2 2 n 4 -c 2"n+2h'+2 (c) + (c ) and since hn,(c~) = — 1 - c 2n-1f ( n-2 ({ - (n-2)(2n-1) nn 1 (n-2) n ht W) h I(ccc- h' (cn-2 _ 2 n c_ _ _2 4 nn... hn(Cc h' (c) )= - (2n+3)} h (c) + 2n+3 1 - n(n3) h(), n+2 ()2 n cE ( 2 n the zeros of - R (c, R) are the roots of the equation no~n 2 — --- --- ------ 2p(c2 4)2 n ph' (p) = e (2n-)(n-2)(n+1) n-2 (2n+3 n(n+3) a +2 h (p) + (e4). (20) If it is now assumed that p =p' +' e +O (e4) _ _ _ _ _ _ _ _22 _ _ _ _ __n'

THE UNIVERSITY OF MICHIGAN 3648-4-T where p' is a solution of the equation h' (p) = 0 the second derivative of the Hankel function can be eliminated using the relation h" (Pt- 1 n(n+l)}hn (') n n, 2 Jnn Pn to give,, (2n-1)p - n-2)(n+l) n - (2n+3) p - n(n+3) +21 - n(n ) Pt n I ) n+2' 2 n Ln Moreover, n n n (n+1) (n-2)(n+l) (2n-1) n - n (n+3) (2n+3) an+2 - (2n )(2+3) and hence 2 92 e1 n 4 P =PT e+ n(n+l) - n+ 0( (21) P=Pn [1+ (2n-1)(2n+3) nn'2n (21) Pn - In contrast to equation (17) the terms in braces involve both e and p', and consequently the root differs from p' in phase as well as amplitude unless p' is n n either real or purely imaginary. The final step is to insert the values of p' and select the root of smallest n magnitude. Since Pt -i and P =-i 2e- i 7/4 23

THE UNIVERSITY OF MICHIGAN 3648-4-T with p' increasing as n increases, it is clear that the smallest perturbed root is now produced by A, and the fact that p' is purely imaginary then leads to a perturbed root which differs from the unperturbed root in amplitude alone. The root which therefore specifies the radius of convergence is 2 e 4 p -i(l+ -)+ (e4) (22) (cf equation 18), implying that for a hard prolate spheroid of small ellipticity the Rayleigh series converges for cK< + - +0( ). (23) 3~ Providing ~= oo, this exceeds the radius for the soft body. Using the above results it is a trivial matter to deduce the radii of convergence for the oblate spheroids. The oblate coefficients A differ from the on prolate coefficients only in having c replaced by -ic and 5 by it and since the p n and pn are unaffected by this transformation, the formulae for the perturbed zeros can be obtained from (17) and (21) by changing the sign of e. It now follows that for the soft oblate spheroid of small ellipticity the Rayleigh series converges for cE<l- 2-I +0(E4 ) (24) 5t (cf equation 19), and for the hard oblate spheroid, when.._______________________ 24 __....

THE UNIVERSITY OF MICHIGAN 3648-4-T cg <1- + 0( 4) (25) 32 (cf equation 23). It will be observed that both radii are less than unity and whereas the deformation of a sphere into a prolate spheroid served to increase the range of ct for which the Rayleigh series converge, the reverse is true when the spheroid is oblate. On the other hand, it should be noted that c~ is the semi-major axis only when the spheroid is prolate and since it is more natural to express the convergenc criteria in terms of the maximum dimension of the body, the above limits on the convergence for an oblate spheroid are better written as c(2 +1)1/2 < 1+ +0(-4) (soft) (26) 10E2 2 1/2 1 -4 c( + 1) < 1+- + 0( ) (hard) (27) 6~ When expressed in this manner it is seen that any deformation of a sphere into a spheroid produces an increase in the radius of convergence. _____________________________ 25 _ ______ ____

THE UNIVERSITY OF MICHIGAN 3648-4-T V THE DISC A part from the sphere the only other spheroidal body for which precise results are easily obtained is the disc. This is the limiting case of an oblate spheroid as -- O0 (or e -E 1), and has the advantage that the amplitude coefficients A are such as to permit a direct location of the singularities in the complex c plane. In addition, the disc can be treated by methods other than those involving spheroidal functions, and in recent years several integral equation techniques have become available. These are particularly suited to the derivation of the low frequency expansion and since a significant number of terms can be obtained without undue effort (Bazer and Brown, 1959), the approximate radius of convergence can be inferred using the intuitive argument described in ~ 2. This provides a check upon the conclusions reached from a study of the singularities of the A on In seeking to calculate the radius of convergence it is convenient to begin with the rigorous method, and for this it is necessary to have an exact expression for the A when e = 0. Bearing in mind that oblate spheroidal coordinates are now required, the amplitude coefficient A is defined as AR(1) (-ic, i) A -on (28) on (3) on 26

THE UNIVERSITY OF MICHIGAN 3648-4-T (see equation 10), and by making the transformation c-O-ic and C it in equation (13) an expansion for R) (-ic, i~) is obtained in the form on oo' (1) 1 r-n Ron (-ic,i) = on (-i) (). (29) c (-ic) 0 r=0,1 But in the limit O = 0, r (ce) = S(r) jr(c)= + & (r-l) and hence for even values of n don (-ic) R() (-ic,i0) = (-i)n - (30) on on (ic) c (-ic) a on~o L R( (-ic, i1) =0 (31)?=0 whilst for odd values of n R (-ic, i) = 0 (32) on on a R(1)1( i i) d 1 (-ic) a on c(ic) a ^ 1 on (-3 o-inc. 3 (33) _________________________ 27 ______,,,_____

THE UNIVERSITY OF MICHIGAN 3648-4-T Accordingly, for the soft disc the A are identically zero for all odd values of n, and it is only necessary to consider the form of the coefficients when n is even. And similarly, for the hard disc the expansion for the far field amplitude is confined to odd values of n, so that in this case only the A for odd n have to be conon sidered. For the radial functions of the third kind the expansion in terms of spherica Hankel functions deduced from equation (14) is not appropriate to the determination of the functions in the limit t = 0, and an alternative expansion is desirable. From Flammer (equation 4. 4. 19) we have R( (-ic, i R (-ici) 1 + i Q (-ic) (tan1 - — 2 + i g (-ic,i) (34) on on [ on 2 i on where n! 2 Q (-ic) = -1 -- e n even (35) on c 2 n n on (-i) 2' 2' o 3 (n+1) 2 c L Icn3 (n+l)! (ic) n odd (36) =1__ n odd (36) c 2n n-l n+1 _| ncd on(2 2 1 and g (-icif) B 2r + n even (37) on 2r r=O0 o n 2 =B01 B 2r n odd (38) B2r r=0,_____,__,__________^^ 28 - - -__

THE UNIVERSITY OF MICHIGAN 3648-4-T Since gon(-ic iO) = 0 when n is even, as is the g - derivative when n is odd, an on expression for the B o is not required at this stage, and using the values found for the radial functions of the first kind it follows immediately that for n even (3) (-ic, iO) = R(1) -ic - i (-ic) (39) on on 2 on and for n odd a (3) a = c+ f T.! for all n, with the even values applying to the soft body and the odd values to the hard. The singularities of A are therefore given by the equation Hence i Qon (-ic)=1 (42) on 2 on (41) and substituting the expression for Q (-ic), this becomes on on + n n iT 0 2n n n r2 2. 2 for n even, and on + 3(n+l)! f 44) 2n n-1I n+1l rV2r 2 2 for n odd, where for convenience the variable r has been introduced in place of -ic. ____________________________________ 29 ___________ ________ _

THE UNIVERSITY OF MICHIGAN 3648-4-T The problem of finding the radius of convergence of the Rayleigh series is now equivalent to the solution of equations (43) and (44), with the required radius specified by the root of smallest magnitude. Before attempting the solution it is desirable to give some thought to the values of n which may provide this root. In the first place, we remark that for both the soft and hard discs the values differ from those found for the sphere. Thus, for a soft sphere the amplitude coefficient A has no singularities in the finite portion of the c~ plane and the smallest singularity belongs to Al, whereas for the soft disc n is limited to even values. This suggests that for some particular ellipticity two of the amplitude coefficients must have (smallest) singularities which are equal in magnitude, and the same sort of transition also occurs for the hard bodies. Accordingly, the sphere results give no direct indication of the values of n which must be considered. On the other hand it is relatively easy to determine the roots of (43) and (44) when n is sufficiently large. Taking for example equation (43), the spheroidal coefficient d (r) can be represented by the first term of its expansion in powers of r providing n is large and r < < 4n n, and from the recurrence relations defining the d it can be shown that 0 n-l 2 don _( ) n 1 ( l o 230 - I1 —.

THE UNIVERSITY OF MICHIGAN 3648-4-T Substituting into (43) then gives 1 1i 1| n+ -'n! n —* n +-l - 2 2(45) (2) - 2n nl n-l n-l (45) 2 2' 2 2 and for large values of n the right hand side is asymptotic to ( n+ 1 (2n) 2 e \e Hence 4n f 2im -1 r-e- exp L2n+l (46) (m = 0, 1, 2,...., 2n), and for n sufficiently large the roots are 2n+1 in number and 4n spaced equally around a circle of radius - in the complex r plane. It will be obe served that this radius is proportional to n and such that the assumed representation of d (r) is valid. Since it is the smallest root which is required out of the 0 totality of roots of (43) for even n, it is now apparent that only the lower values have to be examined in detail, with the probability that the lowest value (i. e. n =0) will provide the root of smallest magnitude. For equation (44) the analysis is similar in all respects and the fact that the difference between the right hand sides of (43) and (44) is precisely compensated by a like difference in the formula for d and d means that the asymptotic be0 1 haviour of the roots is also given by (46). Once again it is expected that the lowest value of n will produce the smallest root, but since n is confined to odd values for ______~_~_~___________ ~31 __ ________

THE UNIVERSITY OF MICHIGAN 3648-4-T the hard disc, the appropriate value here is n = 1. In Figure III the roots obtained directly from equation (45) are compared with the asymptotic form for large n, and in interpreting the graph it must be remembered that the even integers refer to equation (43) and the odd integers to equation (44). Unfortunately, for the smaller values of n not even the formula given in equation (45) is sufficiently accurate for our purposes, and whilst both (45) and (46) would suggest that of the 2n+l roots one of them is always real, this is true only in an asymptotic sense, and for all finite n the smallest root of each equation has an imaginary part which cannot be ignored. This is clearly seen by a study of the tabulated values of d (r) and d (r) for real r (see, for example, Flammer), and from these it would appear that when n is even don(r) n o0 21n 1 nnI r 2' 2 and when n is odd on 3(n+1) 1 d (r) #-' # 1 n n-l I n+l rI 2r 2 * 2 as r tends to infinity through real values. Indeed, the accuracy of these representations is such that for r > 6 5, d0 (r) differs from / by less than one unit in o 2r the fifth significant figure. Since dn (r) and d n(r) are non-negative for real r, it 0 1 32,__

THE UNIVERSITY OF MICHIGAN 3648-4-T x 10x 98 x 7 6 _ X / 5 / 43 - \ / - Asymptotic Formula y / x Computed from equation (45) 2 / Exact (smallest) root 0 I I I I 0 I 2 3 4 5 6 7 n FIGURE III. THE SOLUTION OF EQUATION (45) ________________________ 33 ____________

THE UNIVERSITY OF MICHIGAN 3648-4-T now follows that equations (43) and (44) can have no real roots in the finite portion of the r plane, and this immediately rules out the possibility of using tabulated values for the spheroidal coefficients in the solution of the equations. We are on on therefore compelled to rely on the series developments of d (r) and d (r) in o 1 powers of c, with enough terms included to ensure an accurate determination of the roots, and we shall begin by examining the problem of the soft disc, for which n is restricted to even values. When n is even the relevant equation is (43) and taking first the case n = 0, the equation from which to calculate the singularities of A is oo d~ (r) + /. (47) O 2r In Appendix A it is shown that d00,, o 2 o 4 o 6 o 8 o 10 12) d~~ (r) = 1 - a4 r + a2 ra + a5 r 0 (r o 1 2 3 4 5 where 0 -2 o = 5555555556 x 102 1 ao =4-135802469 x 10-3 2 3 = 2- 125500406 x 10-4 0 -6 a = 3 903827192 x 10 4 0 -7 a5 =3.787031007 x 10,.,- -- -, - -- -,- -- -- - --- 34,,,,

THE UNIVERSITY OF MICHIGAN 3648-4-T 10 and writing r = se, (47) splits up into the two real equations o2 o4 06 oo10 1 - a s cos 20 + a s cos 40 - a s cos 60 + c cos 80 + ao s cos 100 1 2 3 4 5 12 + Fs +0(s2) = — cos, (48) 02 o4 06 08 o 10 al s sin 20 - 2 s sin40 + a3 s sin60 - 4 s sin80 - a s sin 100 +0(s2) =- sin 0 (49) in which either the upper or the lower signs must be taken in conjunction with one another. A casual examination of (48) and (49) shows that if a root of these equations exists for some particular value of 0, a further root can be obtained by changing 0 into 27 -0, and consequently it is sufficient to confine attention to the range 0<0< v. In addition, the magnitude of the coefficients is such that for the equations with the upper signs the smallest root will almost certainly occur within the range O<0< / 2, and for the equations with the lower signsthe corresponding range is T /2 < 0 < 7T. After a few trial calculations it is found that the smallest root has 0 approximately 8~ and a magnitude somewhat greater than 3 2. Unfortunately, for,_______________________________,35 ________

THE UNIVERSITY OF MICHIGAN - 3648-4-T 0 values of s as large as this the convergence of the series for d (r) is extremely slow, and even the terms is given in (48) and (49) are insufficient to determine the root with reasonable accuracy. Although the labour involved in deriving 0 additional terms in the expansion for d is such that it seems unprofitable to pursue the matter further, it is possible to use the tabulated values of d (r)for real r o 12 (see, for example, Flammer) to estimate the coefficient of c, and this turns out -8 to be negative and of order 2 7 x 10. With this additional coefficient available, the calculation of the smallest root can be refined to give r = 3.25 exp{i 0 044 7r, and greater accuracy is only possible by including still more terms in the expansion for d00 o For the reasons stated previously it is to be expected that this is the smallest root of equation (43) for all (even) values of n, but in the interests of completeness the equation for n = 2 has also been investigated. We have d (r) = r 1-a2 r r + a3r + 4 r + O(r, o02{ = r 2 2 24 26,28 10 o 45 1 r 2 r+3 r +Q~4 r with 2 -3 a2 =5 668934252 x 103 2 a2 = 2 394192749 x 103 a2 = 5 248224599 x 10-5 a2 = 8 783406594 x 106 4..36 _______________________________ 36 __________ ____ ___ -- ____________

THE UNIVERSITY OF MICHIGAN - 3648-4-T which can be inserted into equation (43) to give 22 24 26 28 1-a s cos20 - a s cos4e s os 60 + s cos 80 1 2 3 4 + (s10 + 45 1 cos 50 +0(s )= - - - cos -, 2 2 2s 2 s 22 24 26 28 a s sin 20 + 2 s sin 40 - a s sin 60 - a s sin 80 10 + 45 1 ~ sin 50 + o(s )= - - i-sin 2 2is 2' and whilst the number of terms in d is not sufficient to permit an actual calculation of the roots, it has been verified that no root exists whose magnitude is less than 3- 5. We are therefore led to the conclusion that the smallest root of equation (43)(and hence, of (42) for n even) is provided by the case n = 0, and is c = 325 exp i (0.5 0.044)T (50) corresponding to a singularity of A. The magnitude of this root represents the smallest radius of convergence of the expansions for the individual A, n even, and accordingly for a soft disc the Rayleigh series converges for c < 3-25. (51) In comparison with the above, the problem of a hard disc is relatively easy. When n is odd the relevant equation is (44), and taking first the case n = 1 the equation with which to calculate the singularities of Aol is _____________________________________ 37

THE UNIVERSITY OF MICHIGAN 3648-4-T ol _ 3 7 d (r) - (52) 1 2 +2r As shown in Appendix A, ol, 1 2 16 1 8 10 d (r) = 1 - r + ar4 - a r+ r + r 0(r ) where 1 -2 = 6 x10 1 = 2 216326531 x 10-3 2 ca3 =4-316150166 x 10-5 a1 = 2 412282939 x 10-7 4 and equation (52) then splits into the two real equations 12 4 0 6 18 1 - a s cos 20 + a 4 cos 40- s cos 60 + cos 80 1 2i 3o COS68-tq23 4 10 + 3 ir 30 + (s ) = - - - cos (53) s 2s 2 1 2 14 1 6 1 8 ac s sin 20- a2 s sin 40 + c s sin 60 - a s sin 80 +O(s ) =+ 3 sn 30 54) + NS,1= - sin (54) S Ts 2 Once again it is sufficient to restrict attention to 0<0<7v, and the magnitude of the a1 are such that the smallest root almost certainly lies within the ranges 0 < 0 < 7/3 or 7/2 < 0 < 2 7r/3 depending on whether the upper or lower signs are chosen. It is now a straight-forward matter to show that,,_.,_,__,,,__,,_, 38 _____,______-__ _,_,_,

THE UNIVERSITY OF MICHIGAN - 3648-4-T r = 2 1255 exp i 0. 62456 7rt and because the modulus is so much smaller than that found in the case n = 0, even the fewer terms shown in (53) and (54) give the root to a high degree of accuracy. For completeness the corresponding equation for n = 3 has also been investigated, and since o3 23 32 34 36 38 10 1 (= -7'r Il+a1 r -a2 r ~3 r + a4r +0(r ) with 3 -3 a3 = 8 888888889 x 103 a2 = 2 504052873 x 104 a3 = 1 111387425 x 10-5 3 3 = 1- 961434362 x 10-7 the equation specifying the singularities of Ao can be written as 32 34 36 38 1 + a s cos 20 - s cos 40 - 3 s cos 60 + a s cos 80 10, + 525 ~s 70 +0(s )= -- 2 cos - 3 S 2s 32 34 36 38 - a1 s sin 20 + a2 s sin 40 + a3 s sin 60 - ac s sin 80 ~1 c2 3 4 10 + 525 2r 70 +0(s )= - 3 sin -. 2s Since the number of terms is insufficient, a precise root has not been obtained, but --- -- - -- --- -- - 39 -

THE UNIVERSITY OF MICHIGAN - 3648-4-T it has been verified that no root exists whose amplitude is less than 4 5. Bearing in mind the asymptotic behaviour of the roots as a function of n, it is now concluded that the smallest root of equation (44) (and hence, of equation (42) for n odd) is provided by the case n = 1, and is c 2- 1255 exp {i (0-5 0 62456), (55) corresponding to a singularity of A. Accordingly, for a hard disc the Rayleigh series converges for c < 2. 1255, (56) which is significantly smaller than the radius of convergence for the soft disc. As a final check, the radii of convergence have also been determined from the actual coefficients in the Rayleigh series using the alternative approach referred to in ~2, and the details are given in Appendix B. Such a check is desirable in view of the fact that in applying the rigorous method there is always the possibility that a dominant singularity may have been overlooked, and it is therefore pleasing to find that the results agree with those obtained above. 40 __,,.____.,_____

THE UNIVERSITY OF MICHIGAN 3648-4-T VI THE OBLATE SPHEROID OF ELLIPTICITY NEAR TO UNITY It is convenient to follow up the discussion of the disc by considering the problem of the oblate spheroid which is almost a disc, and for which the ellipticity is almost unity. This implies that 0< <<1, and by obtaining the expansions of the amplitude coefficients A in terms of A, it is possible to investigate the perturbation of the singularities consequent upon the presence of the non-zero parameter ~. It is then a trivial matter to deduce the changes in the radii of convergence. The expression for the A is shown in equation (28), and taking first the on radial function R) (-ic, it), the Bessel functions in equation (29) can be replaced b on the leading terms in their series expansions for small ce to give o.o -n-2 don (1) n o F (c) 2 2 R (ic(1) = (-i) c (-i, i) - d 0(3 (58) onc 3 (-ic) c d n 0 0 wOne )sevnnon when n is even, and ________ _______ ________ ___ _ — 41

THE UNIVERSITY OF MICHIGAN 3648-4-T on d1 (-ic) R( -ic = ()n- 1 + 0(3 ) (59) on (-ic) R(1) n 1 d1 (-ic) 2 Ra o (n-ic) (-i) - o- + 0(~ ) (60) 0 when n is odd. For the radial functions of the third kind we have from equation (34) Ron (-icei~) -R() (-ic i{l-i 2 Q (-i (12- + 0(3) )+ igon (-iei~) (on on 0 2 on 0' on where g^on^ 0o (-icit)=g~n + 0(53) for n even, and gn(-ic,i) = + B~ n + 20( 4) o0 2 for n odd. Moreover, from Flammer (equations 4. 4. 25 to 4. 4. 27) on (-ic) d on (-ic) 1 on.n 1 B 1 - 1-c Q (-IiC ) 0o c do on on (-ic)J o o for n even; Bon. 1 n cn (-ic) Bon 3in+l 1 o o 2 on c d (-ic) ____ ____ ____ ____ ____ ___, ___ 42 _

THE UNIVERSITY OF MICHIGAN 3648-4-T on c on (-ic) B2 B on c n-1'" B - X (-ic) B - Q (-ic) 2 2 on o 3 on on ic for n odd, where X (-ic) is the eigenvalue, and consequently when n is even 7 - = l i 72r1 Q (-ic) (1 — i + e in lBon - o + 0( ) } | (1) on(-ic) R (-ic,iF) i c (-ic) on 4. 2\ n+ 1 on o (3) (-i, on ^lR (-1c,1^) o /0L don (-ic) on o aR (-ic) on ic (-ic) on ic 2 2 on o R(3 1+- -I Q (-Ic) + Bo R(3) 3 5, o o 3 o don (ic) R (-ic,i) d- / d (-ic) on o o on + 1 + i 1 5 i Q n(-ic) + R(1) (-icid ) don (-Ic) -1 on n-2 1 on n R cc (-ic) 3 on f n-i on a on o o + 0(2 2) 43

THE UNIVERSITY OF MICHIGAN 3648-4-T It now follows that for a soft spheroid of ellipticity - 1 the amplitude coefficient A has the form on on 2 1 A = l-i|-QO (-ic)+i- +F(- (61) on (-ic) ( I*~~~~~ 0 for even values of n, and in the disc limit this reduces to the expression given in equation (41). When n is odd 3 on (-ic) r on -ic)n A - ic 1+d (-i c) 11 Lcn(_ic) 9 T c (-ic) l J on 9 con(-ic) L' 9 {con(i) Qon o 0 + < 2)' (62) and is zero when g = 0, which agrees with the original finding that only even values of n contribute towards the soft disc solution. Similarly, for a hard spheroid of ellipticity near to unity 3 c don(-ic) 2 on 3 01 -ic) 2 on 3 on_ d (i \ 5 -ion-/ 3 onic)1 0 0 0 5 2 3 7 oT:5 o2 on for n even and on \r2 ic- 9iX (-ic) c (-ic) I A 21 - i Q (-i c)+ + 0( 2) (64) c d o(-ic) J 44,____........,__________ 44,,,,___________________

THE UNIVERSITY OF MICHIGAN 3648-4-T for n odd, and here again the limiting values are in agreement with those employed in I5. In order to find the radii of convergence for the individual A, and hence the radius of convergence of the Rayleigh series, it is necessary to locate the smallest singularities of the A in the complex c plane. Taking first the case of on the soft body, the singularities for even values of n are given by the roots of the equation.. T c 0(-ic) 1 - 7 -c) + o + (3 )=0, (65) nd (-ic) 0 and as such can be obtained by a perturbation analysis based on the singularities for a disc. If c = c is a solution of (65) with e = 0, and if n c=c + t ~ + 0(2 ) (66) n n is the corresponding solution of (65) with the term in e retained, substitution of (66) into (65) gives immediately o ~a o n on (.ic) a d (-ic) V =-i 1+c n n on dn (-ic) ) d (-ic) o o n and consequently the roots of (65) are on -1 c = c 1 _ i -~_- 1 +2c n +0(. (67) c n on n (-iC n) L n d -ic) L d (-ic o 0 n

THE UNIVERSITY OF MICHIGAN - 3648-4-T In view of the restriction to small values of A, the smallest root is almost certainly produced by the smallest c, and from the results in 5 it is known that for n even c <c, n=2,4,6,..... O n The value of c is shown in equation (50) and using now the expansions for d~ (t) 00 and c0 (t) /doo0 (see Appendix A) with t replaced by -ic, the perturbation of 0 the singularity can be computed. From equation (67) it is found that c=c { 1- i (1-027 +0-1992 i) + 0(2) (68) and thus the effect of the non-zero parameter is to modify c in both phase and amplitude. Of most importance, however, is the reduction in the amplitude, implying a reduction in the radius of convergence, and from (68) the actual radius is 3- 25 (1 - 1027 ~). (69) This formula is valid as long as terms in 2 are negligible, or until another singularity becomes smaller in magnitude, and bearing in mind that for even values of n >0 the c exceed c by 0- 25 at the very least, it seems reasonable to regard (69) as holding for ~ as large as 0- 1 or even 0. 2. All this is based on the assumption that the coefficients A for odd n have on no singularities which are smaller in magnitude that the one whose expression is ____l _______________________ 46 ________ __

THE UNIVERSITY OF MICHIGAN 3648-4-T given in (68). If such a singularity were dominant, the radius of convergence of the Rayleigh series would change discontinuously in the limit as the spheroid became a disc, and although this is intuitively unlikely, it is necessary to investigate these other singularities to make sure that they do not include the smallest. From equation (63) it is clear that the only singularities of A for odd n on are those which correspond to the vanishing of cn (-ic), and using the expansions derived in Appendix A it has been verified that the smallest root of the equation for n = 1, namely d d (-ic ) + d~1 (-ic ) + (-ic)..... =0, 1 3 5 is approximately 5 2 exp i(0 5 + 0 083) r}. Since this exceeds c in magnitude and the singularities for higher (odd) values of n are even larger, it follows that for the soft body the singularities of the odd coefficients have no effect on the convergence, and accordingly the radius of convergence is as shown in (69). Turning now to the case of the hard body, it is convenient to consider first the coefficients Aon for odd n, the singularities of which are given by the equation i x (-9i c) c(-ic) c (1-i7r (-ic) + on 0 ( 0. (70) even respectively. 47 _

THE UNIVERSITY OF MICHIGAN 3648-4-T Here again a perturbation analysis is applicable, and writing C=C +7Z n +0(2 ) n n where c is defined as before, we have n r ( on ){ 1 - d Aon(-i n n 2 n1 n n = 3iX (-ic o - n cnd -ic I d (-ic )J and consequently the roots of equation (70) are 1+ on n o n i 2 a cn 1 n-ic 2 c Ic don (-ic 3 n don (-ic 1 n n 1 n 1 n n When n is odd the c of smallest magnitude is provided by n = 1 and if the value for cl shown in equation (55) is inserted into the expansions for d1 and c the corresponding singularity of Aol is found to be c=c1 {1- (0-2615 - 0.5719i) +0(2) (72) It will be observed that as a result of the finite ellipticity cl is modified in both phase and amplitude, but since the amplitude is decreased, a deformation of a disc serves to decrease the radius of convergence. Providing terms in 2 are negligible, the actual radius is 2.1255 (1- 0 2615)(73) --- --- --- -- -- - ----- --- ~ ~~ on (-ic - - - - -- - - -

THE UNIVERSITY OF MICHIGAN 3648-4-T and it only remains to verify that no coefficient A for even values of n has a on smaller singularity to assert that (73) is the radius of convergence of the complete Rayleigh series. The singularities of the A when n is even are given by the zeros of on on c (-c), and the smallest root of the equation for n = 0, namely 0 d~ (-ic) + d (-ic) + d (-ic) +..... =0 0 2 4 is approximately 4 1 exp{ i(0. 5 + 0.05) 7r. This is greater than cl in magnitude and since the singularities for higher (even) values of n are still larger, the even coefficients can be discounted as far as the overall convergence is concerned. It follows that for the hard spheroid of ellipticity almost unity the radius of convergence is as shown in (73). In keeping with the form of presentation used in ~4, it is convenient to express the above results in terms of the maximum dimension of the body. For an oblate spheroid the semi-major axis is c( + 1)/2, which differs from c only by terms 0(2 ) for small A, and consequently the radii shown in (69) and (73) are equivalent to the following convergence criteria: c(2+1)1/2 < 3.25(1-1.027E) + 0(02) (soft) (74) c(2 + 1) /2 < 2.1255(1-0.2615~) + 0( 2) (hard) (75) --- --- ---- --- --- --- -II_ 49 _ ___ ____

THE UNIVERSITY OF MICHIGAN 3648-4-T It will be appreciated that these two equations summarize the conclusions of ~5 as well as of the present section, and it is of interest to note that whilst the soft body gives the larger radius of convergence, its convergence decreases more rapidly with increasing C. This is eminently reasonable in view of the fact that both soft and hard spheres have radii of convergence equal to unity. 50

THE UNIVERSITY OF MICHIGAN 3648-4-T VII THE OBLATE SPHEROID OF INTERMEDIATE ELLIPTICITY In order to complete the discussion of the oblate spheroid it is necessary to consider the convergence of the Rayleigh series when the ellipticity is neither small nor near to unity. Unfortunately this is a difficult task and whilst it is (3) possible to obtain several different integral expressions for Ron (-ix, it) on (see, for example, Flammer equation 5.4. 1), no expansions are available by means of which the zeros can be determined analytically. Moreover, the zeros 2 almost certainly correspond to complex values of c, so that any attempt to discover them by purely numerical means (i. e. by computing an integral expression for a variety of c and ~) would be an extremely laborious undertaking. Nevertheless, it is important to have some estimate of the convergence for these ellipticities, and this is particularly true in view of the change in the dominant singularity which takes place somewhere within the range 0 < e < 1. Thus, for a soft spheroid, the coefficient Ao specifies the convergence as long as the ellipticity is small, but by the time that the body has become disc-like (e N 1) the coefficient Aoo has taken over; and with the hard spheroid the behaviour is just the opposite. As a consequence, the'curve' giving the radius of convergence as a function of g may well possess an abrupt change of slope for some value of the ellipticity, and the nature of the'transition' is therefore a problem of some interest __ __ __ __ __ __ __ __ __ __ __ _ ~51,,- - - 5 1 —---—.

THE UNIVERSITY OF MICHIGAN 3648-4-T In the absence of any other method with which to explore this region, it is necessary to rely on a numerical comparison of the coefficients in the expansion of the Aon', with the hope that sufficient terms can be included to give a reliable estimate of the convergence. For the values of g under consideration it is convenient to write the ratio of the radial functions as 00 -1 (1)( ~' on Ro1 = Qon (-ic ) r0,1 dr (-ic)Qr(i) + r=,1 dp/r (-ic)P 1(i) r=0,1 r (76) (see Flammer, equations 4. 2. 3 and 4. 2. 7), where Qon'(-ic) is as defined in equations (35) and (36), and Pr(ii) and Qr(id) are the Legendre functions of the first and second kinds respectively. Attention will be directed only at the cases n = 0 and n = 1, and since the solutions for the hard body can be obtained from those for the soft by differentiating the functions of the radial variable, it is sufficient to write down the expansions for the soft body alone. Taking first the coefficient Ao1, we have T1 3 Q (ol ((-ic)- (77) so that' Unless otherwise stated, the argument of the Legendre functions is it. 52, I

THE UNIVERSITY OF MICHIGAN 3648-4-T 00 dol 0 01 dr r V dp/r r-1 -1 o3 1 3 P L-dlol QI d01 Q1 C3ol 2 ) -, ol2 -1, 1 Q1 r=l - 1 A = 1 = 1 0~ 2L3 9 Ql 9 1 Q- Go d P 9_Q r r dl P 1) 1. 1 9 Q1 Q -9 531 L r = 0 r(7) Bo 1 1 P = 0 B + 3 ( Q32.3 QI I Qi i P1 B3 = - QI 3 Q1 P 1 P 1 P 2 r3 Q3 3 e2 B4 — 2B 32. 5 Q 22. 32 Q1 32. 54 aP Q 1 0 32.5.72 \P Q/ i P1 B5= 3.52 Q1 Ill __ _ _ __ _ _ __ I __ _ _ __ _ _ 53,,__ _ _ _ __ _ _ _ _ _ __ _ _ _ _ _

THE UNIVERSITY OF MICHIGAN 3648-4-T B =1 (B4+ 2 B2) P3 1 B 53 PO 5" 3.^2 / P 3.-5.72 p 2.23 5.77 Ql B6 2 4+ B2 -2.B2 22 2 2 2 2 23 5 3.5 P 3.57 P 2.3.5.7 P 1 1 P____, P 229 /P Q3 2 Q5.7 Q 3.52 Q / 35. 7 Q1 2. 335.7 Q1 3 2357.211 Pi 4 / Q\ 1 Q 4 52 82i P 3.5472 Q1 1 2 229 B8-_ ____) _P B (6+ B -2532 52 3 252 4 237 3 l ~~1 /4 P\ 3 -272 4 (B4++ B2) 3.25.7 5 5213 P1 1 P 11r P 2 P2 1 P + 4 B2 - _ 74 2.5 4 3 22 3. 5.7.11.13 P 2.3.5. 7 Q1 3.5 3.5.7.11 1 P 12542 p Q386 p Q - _ _ --------- 3_ - — + P 5 2.345.7.11 Q1 3.587211.13 P1 Q1 357.13 p1 Q 2 ) Q7 1 ( Q ^. 33537.11.13.17 P Q1 33527.11213.17 p1 Q ___________________ _ ~54,_ _ _ _,________

THE UNIVERSITY OF MICHIGAN 3648-4-T 3121 i P 36567 Q1 Hence 3 _ ol2 P1 A = (dl ((ic) cr (80) 01 9 & 70rQ1 r r=0 where the ar are given by the equation ~r Y s Br-s = 0 (81) r = 1,2, 3,...., with a_ 1. For 0 g r < 9 the cr have been computed for a sequence of values of e spanning the range from a disc (t = 0) to a point at which the convergence shown in equation (26) can be assumed to be applicable. The values of the Legendre functions were obtained from the N. B. S. Tables (1945), reinforced where necessary by direct calculation of the functions from their formulae, and the results are shown in Table III. It was then a trivial matter to determine the convergence coefficients ar defined as -1/r Iarl = rll r r and these are also tabulated together with the values of ( 2 + 1)1/2 larl indicating the convergence measured in terms of the semi-major axis of the spheroid. This -~~______________ _ ~55 ___ _________

THE UNIVERSITY OF MICHIGAN - 3648-4-T TABLE III. CONVERGENCE COEFFICIENTS FOR Aol (SOFT) _ =1.2 = =0.6 = 0. 4 r |a (2+1)/2 a a I1 (2]a +1)/2a1 a a I (2+1)/2 laI = | arl (2 +1 rl | r r I r 2 1.2 0254 0.91190 1.4244 5.23870x10-1 1.3816 1.6112 3.81764x10-1 1.6184 1 7431 3 8. 01696xl0-1 1.0765 1.6815 1.74623x10-1 1.7891 2. 0864 8.48364x10-2 2.2758 2. 4511 4 1.52720 0.89955 1.4051 2. 45103x10-1 1.4212 1.6574 1.15639x10-1 1.7148 1.8469 5 1.83195 0.88597 1.3839 1.62005xl10-1 1.4391 1.6783 5.45946x10-2 1.7888 1.9266 6 2.54840 0.85565 1.3366 1.43436x10-1 1.3822 1.6119 4.11153x10-2 1.7022 1.8333 7 3.38311 0.84020 1.3124 1.12973x10-1 1.3655 1.5924 2. 48936x10-2 1.6948 1.8254 8 4. 58890 0.82658 1.2912 8. 22250x10-2 1.3665 1.5936 1.59641 x10-2 1.6773 1.8065 -2 9 6.12600 0.81770 1.2773 7.51219x10-2 1.3333 1.5549 1.05680x10 1.6579 1.7856 g =0.3 = 0.2 =0. 1 r ar larl 1(2+1)/2lal a a (2+W 2arl larl (2+1/2al I r( 2 1)~2a I cr r 1r l ar 2 3. 24571 x10-1 1. 7553 1. 8326 2. 75740x 1-1 1, 9044 1. 9421 2o 34498 x10-1 2. 0651 2. 0754 3 5.40952x10-2 2.6441 2.7605 3.06378x10-2 3.1958 3.2591 1.30276x10-2 4.2499 4.2711 4 7.71550x10-2 1.8974 1. 9809 5. 05071x10-2 2.1094 2.1512 3.24774x10-2 2.3556 2.3673 5 2.86240x10-2 2.0355 2.1251 1. 32196x10-2 2.3755 2. 4225 4.54659x10-3 2.9408 2.9555 6 2.02689x10-2 1.9151 1.9994 9.28174x10-3 2.1814 2.2246 3.85537x10-3 2.5254 2.5380 7 1.02668x10-2 1.9234 2.0081 3.64290x10-3 2.2303 2.2745 9.34045x10-4 2.7090 2.7225 8 7.43396x10-3 1.8455 1.9268 1.90913x10-3 2.1872 2.2305 4.73578x10-4 2.6036 2.6166 9 3 33953x10-3 1.8843 1.9673 8.67370x10-4 2.1888 2.2321 1.53654x10-4 2.6529 2.6661 __= 0. 05 = 0. 01 = 0. 0 rL r | | arl|, (g2+1)L/2|a | |r la r | (22+)1/21ar| (g2+L/2a 2 2.16460x10-1 2.1494 2. 1521 2. 03171x10-1 2. 2185 2.2186 2. 00000xlO-1 2.2361 2.2361 3 6. 01278 x10-3 5.4993 5.5062 1.12873 x 10-3 9.6044 9.6049 0 o 4 2. 58893 x10-2 2. 4930 2.4961 2. 15442 x102 2.6102 2.103 2.05714x10-2 26405 2.6405 5 1.88152 x10-3 3.5083 3.5127 3.23203 x10-4 4.9901 4.9903oo O 6 2. 37246 x10-3 2.7382 2. 7416 1.56921x10-2 2.9335 2.9336 1.41037x10-3 2.9861 2.9861 7 3.28993 x10-4 3.1444 3.1483 4.92560x10-5 4.1244 4.1246 0 c0 co 8 2.00580x10-4 2.8958 2.9024 8.81922x10-5 3.2123 3.2125 6.99710x05 3.3066 3.3066 9 4.33278 x10-5 3.0535 3.0573 5.28150x10-6 3.8580 3.8582 0 co o 56,,,t,, ___

THE UNIVERSITY OF MICHIGAN 3648-4-T I I I! I I I I I! Io! n I I I I I I I 0 0 -Ic CX,1f c 0 0 d 0 0 d 8 II ii Fi II II II II II II ri CD o

THE UNIVERSITY OF MICHIGAN 3648-4-T last set of data is plotted as a function of (1 + 1/-2)1/2 in Figure IV, and it will be observed that as g decreases the overall level of the curves increases accompanied by a build-up of oscillations. In the main, the maxima and minima occur at odd and even values of r respectively, and when ~ is less than 0. 2 the first few minima for each curve increase in magnitude as r increases, whereas the maxima decrease In other words, the amplitude of oscillations decreases with increasing r. As e becomes smaller, the maxima increase in size, and are infinite in the limiting case of the disc. Since an infinity corresponds to the absence of that power of c, the result implies that no odd powers of c occur in the expansion of Ao1 fcr a disc, a fact which is otherwise obvious when equation (80) is compared with (62). Indeed, when e = 0 the ar are merely the coefficients of (-t)r in the expansion of {cl (t)} and our previous consideration of this function showed it to be regular for Itl < 5. 2 approximately. To find the convergence of the expansion for Aol when ~ is not zero, we have to determine the limiting values of the quantities ( 2 + 1)1/2 Jar 1 as r - oo, | and in practice the most convenient way of doing this is to express them as multiples of the corresponding convergence coefficients for the sphere (see Table I). Taking for example the case ~ = 0. 4, the resulting ratios are r = 4 1.4944 r=5 1.6040 r = 6 1.5785 _58

THE UNIVERSITY OF MICHIGAN 3648-4-T r = 7 1. 6046 r = 8 1. 6140 r=9 1.6153 the limit of which is estimated to be 1. 62. Bearing in mind that for the soft sphere the A1 converges when c(2 + 1)/2 < 1, the radius of convergence for = 0. 4 is therefore 1. 62. This procedure proved effective for all except the smallest values of I, and here the interpretation of the ratios was made easier by the fact that the successive minima on each curve increase with increasing r, whilst the maxima decrease. Since the trend is relatively uniform, it is possible to obtain a reasonable estimate of the convergence even when the mean level of the curve is still increasing at r = 9, and the results are shown in Table IV. TABLE IV. RADIUS OF CONVERGENCE FOR A0o (SOFT) i___ _ 1.2 0.6 0.4 0.3 0.2 0.1 0.05 0.01 0 c(2+1)1/2 1.16 1.42 1.62 1.78 2.01 2.42 2.76 3.38 5.2 It is believed that these are accurate to within lqo for the larger I, but the error could conceivably be as much as 50o for 5 as small as 0. 01 ____~__ ________~_.~ ~59 ___________________

THE UNIVERSITY OF MICHIGAN 3648-4-T For the hard body the expression for the amplitude coefficient Ao1 differs from that in equation (78) in having all functions of g replaced by their first derivatives, and consequently the expansion can be decuded from the above by the simple process of differentiating each Legendre function. In this instance, however, the ar are required only to indicate the convergence for those values of g between the ranges for which either the near-disc formula (75) or the near-sphere formula (derivable from (21) with n = 1) is valid. Even then an accurate determination is unnecessary unless the curve of convergence against (1 + 1/A2)1/2 is found to dip below the one for AO, and this is indeed fortunate in view of the almost random nature of the results. The calculations have been carried out for ~ = 1. 2, 0. 6, 0. 4 and 0. 3, and the ar are given in Table V together with the convergence coefficients larll and (2 + )1/2 l ar deduced therefrom. The last of these represent the convergence measured in terms of the semi-major axis of the spheroid, and are plotted as a function of (1 + 1/ 2)1/2 in Figure V. If anything, the curves are notable only for their lack of uniformity, and any attempt to deduce the ultimate level of each curve as r -- co is largely a matter of guesswork. Nevertheless, by concentrating on the minima and comparing these with the values for the sphere (see Table II), it is possible to come up with some approximate values for the radius of convergence, and these are shown in Table VI. 60 -—,,-,,-,-, —

THE UNIVERSITY OF MICHIGAN 3648-4-T TABLE V. CONVERGENCE COEFFICIENTS FOR A (HARD) = 1.2 -=0.6. a r.lal + lalIar a (2 +1)1/21 l r Ian~ +lIar r 2 4.84690x10-1 1.4364 2. 2437 1.49754x10-1 2.5841 3. 0136 3 -5.47521x10-1 1.2224 1.9094 -1.88580x10-1 1. 7438 2.0336 4 -4.36995x10-1 1.2299 1.9212 -7.88932x10-2 1. 8869 2.2005 5 -4.65054x10-1 1.1655 1.8206 -3.38517x10-2 1. 9683 2.2954 6 2.13025x10-2 1.8993 2.9668 2.67026x10-2 1.8291 2.1331 7 4.09194x10-1 1.1362 1.7748 3.07891x10-2 1. 6442 1.9174 8 5.74060x10-1 1.0718 1.6742 1.50781x1l0-2 1.6893 1.9700 9 3.26687xl0-1 1.1324 1.7689 -2.42565x10-3 1.9524 2.2769 -0.4 ~ =0. 3 r ar la2ll (2+1)1/21 |a a (g+1)1/2la 2 8.15714xl0-2 3.5013 3.7710 5.48205x10-2 4.2710 4.4591 3 -1.31421x10-1 1.9669 2.1184 -1.10656x10-1 2. 0829 2.1746 4 -3.78142x10-2 2.2677 2.4424 -2.43580x10-2 2.5313 2.6428 5 -5.66945x10-3 2.8138 3.0306 1.14628x10-3 3.8738 4. 0444 6 1. 69996x10-2 1.9721 2.1240 1.30692x10-2 2. 0604 2.1511 7 1. 05818x10-2 1.9152 2.1627 5.62521x10-3 2.0961 2.1884 8 1.80478x10-3 2.2027 2.3724 -2.50891x10-4 2.8188 2.9429 9 -2.660133x10-3 1.9373 2.t0865 1. 90734 x10-3 2.0053 2.0936

THE, TE UNIVERSITy OF 368 —T MICHIGAN 3648 -4 -T II I I I I I I il I.I I I o co~~~~~~~~~~vc rr~~~~~~~~~~~~~~~~~~0 I~~LVI I~~~~ 1~~~~~~~~~~~~~~~:18 IIIc co~ 0 0 i c~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ — ~ ~~~~~c- r o~~~~~~~~~~~~~~~~~~~~~~~~~~~C T% ~/~(T +~).4~C~) 62

THE UNIVERSITY OF MICHIGAN - 3648-4-T TABLE VI. RADIUS OF CONVERGENCE FOR Aol (HARD) 1.2 0.6 0.4 0.3 c( + )1/2 1.6 1.8 1.9 1.9(5) The accuracy of the above results is hard to estimate, but to judge from the way in which they agree with the formulae for e - oo and 1 / 0, they cannot be too much in error. On the other hand, it is comforting to find that the convergence of the Rayleigh series for a hard spheroid is not determined by them. Turning now to the case n = 0, the expression for the amplitude coefficient Aoo for the soft body can be obtained from equation (76), and since %Qoo (-ic) = c (i)}2 (82) cdo (-ic 2 we have oo oo -1 r r+ /r P r- d00 / 00 Q Ao =c(dO0)2 Pi c(d~)2 P + r= ro 2 d Q 00 0 Q0 Q0 O 00 dr r 0 0O (83) This can be written as,,,________________ _,63 -

THE UNIVERSITY OF MICHIGAN | 3648-4-T -1 (d )2 p o r 00 0 1 Aoo = - c(d)2 - ) -i)r (84) 0 o _r with B -1 Po Bo=i B2 2 2 QO 32 O 0 i Po 3 32 QO 1 P2 2 P1 1 P3 2 P2 2\ B4 = B2 +) 4 32 2 3.5 Q2 2.3.5 QO 347 Q 1 (P Q4 3. 527 PO QO 23i P0 5 42 5- 34 5 Qo 1 2 P2 1 P4 5 P1 1 P3 P0 6 =32 327 2 P 3.5 7 2 P 2 237 QO 5 Qo 1 P_ 13 / P2 4 +_ Q 2.3.45.7 Q0 3.6527 \ P 335. (7.11 Q4 1 P o Po Qo 1 Po 6 (P6Q6) 335.7.211 P0 Qo 64 -___________

THE UNIVERSITY OF MICHIGAN 3648-4-T 158i PO B7- 6 36 5. 72 Q0 1 2 13 P2 1 4 P Bg 32 (BB)- BB )B) B8 32 6 32 7 4 34527 2 (B4 - 3 B2) 2 P 32 76 3.2 7 PO 3.5. 7 311 P 1 P6 41 Pi 16 P3 + B2 + - + 3.3 5.7.211 P 2.35527 Q0 355211 Q + 4+ 6 3) 5.711.13 Po Q0 3.45.7.13 Q 2233 5.7.11.13 Q 3 85. 7.11 P0 Q + 2498 (P4 Q 2(P - (85) 0 QO r=O 3.6357.11.13 5P 2 Q0 199i p0 B9 84 3.5 Qo and hence oo A00 -c~d00~20 ) (-ic)r a (85) r = 0 o Q0 r=0 in which the ar are related to the Br by equation (81). The coefficient A for the soft body is analogous to the A for the hard oo ol in that each provides the smallest radius of convergence for the appropriate disc 65 -

THE UNIVERSITY OF MICHIGAN 3648-4-T (or near-disc), but neither is important when the spheroid is almost a sphere. Indeed, for the soft sphere the expansion of the coefficient Aoo has an infinite radius of convergence, and consequently any calculations of the ar based on the above formulae are concerned only with indicating the manner in which the radius approaches infinity for values of e greater than those for which the expression in equation (74) is applicable. The calculations have therefore been limited to e = 1. 2, 0. 6, 0. 4 and 0. 3, and the corresponding ar are shown in Table VII. Also listed are the convergence coefficients | a~ and (2 + 1)1/2 ar |, and the latter are plotted as a function of (1 + 1/2)1/2 in Figure VI. It will be observed that none of the curves show any signs of turning over and are effectively straight lines as far out as the largest r considered. This almost certainly implies a radius of convergence in excess of 2. 5, and even the possibility of an infinite value (as in the sphere limit, e = o) cannot be ruled out entirely. Under these circumstances it would be a risky undertaking to try to estimate the convergence, but it is clear that for e > 0. 3 the radius is too large to play any role in the analysis. When the body is hard the coefficient Aoo differs in having all the functions of the radial variable replaced by their first derivatives, but if an attempt is made to differentiate the Legendre functions in equation (83) a difficulty arises owing to the occurrence of a factor PO (ii) - 0 in the denominator, where the prime denotes a/a. 66 66

THE UNIVERSITY OF MICHIGAN 3648-4-T TABLE VII. CONVERGENCE COEFFICIENTS FORAoo (SOFT) =1.2 =0.6 r lr ol |(2+1) a | /201 arOr a 2 1 1.43939 0. 6947 1 0852 9.70516x10-1 1.0304 1. 2016 2 1.49609 0.8176 1.2771 7.47798x10-1 1.1564 1.3486 3 1. 16479 0.9504 1.4846 4.29535x10-1 1.3254 1.5457 4 7.42627x10-1 1.0772 1.6826 2.06231x10 1. 4839 1.7305 -2 5 4.02260x10-1 1.1998 1.8741 8.51713x10 1. 6366 1.9086 6 1.89628x101 1.3193 2.0608 3.67042x102 1. 7870 2.0840 -1 -3 7 7.91543x101 1.4367 2.2442 9.80720x103 1.9361 2.2579 8 2.95953x10-2 1. 5527 2.4254 2.79074x103 2. 0859 2.4326 9 9.99243x103 3 1.6682 2. 6059 7.08031x10 2. 2387 2.6107' = 0.4 = 0. 3 r ar I|ar 01 |(S2+1/2 arol | er |a 01 0(2+ 1)/21 arOl 1 8.40131x10-1 1.1903 1.2820 7.81653x10-1 1.2793 1.3356 2 5.93803x10-1 1.2977 1.3977 5.32816x10-1 1.3700 1.4303 3 3.11415x10-1 1.4753 1.5889 2.68529x10-1 1.5500 1.6182 4 1.37798x10 1.6413 1.7677 1.14727x10-1 1.7182 1.7939 -22 5 5. 27307x102 1.8013 1.9401 4.25182x10-2 1.8806 1.9634 -2 6 1.75955x10 1.9068 2.1199 1.37261x10-2 2. 0437 2.1337 7 5 19818x103 2.1199 2.2832 3.92111x10-3 2.2070 2.3042 -3 -4 8 1.36373 x10 2.2812 2. 4569 9.93377x10 2.3733 2.4778 -4 9 3.16332x10 2.4484 2.6370 2.21463x10-4 2.5473 2.6595,_ _ _ _ _ _ _ _ _ _ _ _ 67_ _

THE UNIVERSITY OF MICHIGAN 3648-4-T Ico 0 _CY E zO O 0O 6 CN~~~~~~CY zC 0o OCO,,,,,,68 ~ II 11 ~ ~ ~

THE UNIVERSITY OF MICHIGAN 3648-4-T P The difficulty is most easily overcome by multiplying through by p- prior PO to the differentiation, and the expression for Aoo in the hard case is then Qtt2 0. odrQ1. d -1 ~~doo o Pt Aoo = c(d0~~)2 2 0 d~ ) o Q which can be written in the form 3 r 00 - A - (d)2 ) [ EB (87) dr=0 B0=1 B1 = 0 2 3 P(t 1 Qt 1 Pt 2- 7 32 52 Pi 7 Q1 2 Qi i 2 B3-32 ^0 3 0 Pt Pt Pt B1~ Pt 1 13 4 4 1 6 4 7 32 52 Pt 3. 5. 7 335 5.11 P' 7.11 P 2 2 2 1 (2 Q 3 + 1 (P + 27 32 52 Q 3.5 Q 2 QwithO i~~~~~~~~~6

THE UNIVERSITY OF MICHIGAN 3648-4-T 3 Q' 0 1 /2 3 P l 1 / 13 4 P 1 P ) pt 375 P' 1 (46 2498 P4 2 P6 1 8 3 3 3.35.7.11 33 5.7.13 P2 5.7 P2 3.5.13 P2 1 / 13 Q2 4 Q4 1 Q _ 1i 1 1 Pt 335.7 735 Q' 5.11 Q' 7.11 Q'0 -^^^-1 51+ + 5) 33 (27 Q. 3.5 Qb 2.3.5.7 Q0 33 (227 -+ —+0 Q 0 23i P 7 6 2 3.5 Q, 0 It will be noted that as a consequence of the differentiation two terms of the series have been'lostt, and the expansions given in Appendix A now serve to determine the Br up to and including B7. From equation (87) it follows immediately that 3 P' c oo 2 2 r ~A ~( ) (ic) a (88) Q0 r=0 _____________________ —------ 70 ----------

THE UNIVERSITY OF MICHIGAN 3648-4-T where the ar are related to the Br by equation (81). These have been computed for e = 1. 2, 0. 6, 0. 4, 0. 3, 0. 2, 0. 1 and 0. 05, and the values are given in Table VIII. Also listed are the convergence coefficients ar~ and (2 +1) /2 a 0, and the latter are plotted as functions of (1+ 1/2)1/2 in Figure VII. The regularity is at once apparent, and by comparing the coefficients with those for a sphere it is possible to estimate the convergence with a reasonable degree of accuracy notwithstanding the smaller number of the ar~ available. The results are shown in Table IX. TABLE IX. RADIUS OF CONVERGENCE FOR Aoo (HARD) 1.2 0.6 0.4 0.3 0.2 0.1 0.05 c(Q2+1)1/2 1.08 1.21 1.32 1.43 1.56 1.82 2.10 This completes the discussion of the oblate spheroid, and in combination with the formulae of ~~4, 5 and 6 the radii of curvature obtained above are sufficient to specify the convergence of the Rayleigh series for both hard and soft spheroids of any ellipticity. Rather than summarize the data here, however, we shall now go on to consider the prolate spheroid and reserve the presentation of the final results for ~ 9. _______________________________ 71 ________________________________

THE UNIVERSITY OF MICHIGAN 3648-4-T TABLE VIII. CONVERGENCE COEFFICIENTS FOR Aoo (HARD) _' = 1.2 s = 0.6 =0. 4 =1.2 =0.6 0.4 r a larOl (l2+1 / la /l ar aar| ( 2+)1/2 arOl ar al (22+1)/2 arO 2 1. 35095 0.8604 1. 3439 6. 65151x10-1 1.2261 1.4299 5.1565410- 1 1.3926 1.4999 3 9.76000x10-1 1.0081 1.5747 2.72000x10-1 1.5434 1.7999 1.54667x10-1 1.8630 2.0065 4 1.95652 0.8453 1.3205 4.12168 x10-1 1.2481 1. 4555 2.22677x10-1 1.4557 1.5678 5 2.52862 0.8307 1.2975 3.31620x10-1 1.2470 1.4542 1.42324x10-1 1.4769 1.5907 6 3.74135 0.8026 1.2537 3.29480x10-1 1.2033 1.4033 1.20057x10- 1.4238 1.5335 7 2.65924 0.7876 1.2303 3.07444x10-1 1.1835 1.3802 9.40407x10-2 1.4018 1.5097 = 0. 3 = 0. 2 = 0.1 r ar |ar ~ (2+1)l/21arO| lar~| (2 + 1/2aO1 a r lar~o (2 +1)L/2Ia o 2 4.52337x10-1 1.4869 1.5524 3.94778x10- 1.5916 1.6231 3.41084x10-1 1.7123 1.7208 3 1. 09000x10-1 2. 0934 2. 1856 6. 93334x10-2 2. 4342 2. 4824 3.36667x10-2 3. 0969 3.1123 4 1.58031x10 1 1.5860 1.6558 1.07294x10-1 1.7473 1.7819 6.69587x10-2 1.9658 1.9756 5 8.64984x10-2 1.6316 1.7034 4. 70389x10-2 1.8429 1.8794 1.92256x10-2 2.2041 2.2151 -2 6 6.63515x10 1.5717 1.6409 3.22539x10-2 1.7724 1.8075 1.31998x10 2 2.1079 2.1184 7 1.70346x10-2 1.5476 1,6157 2.03886x10-2 1.7439 1.7784 6.25585x10- 2.0645 2.1748 =0.05 r a r ar~ (~2 + 1 /21 arO 2 3. 14966x10-1 1.7818 1.7840 3 1.67083x10-2 3.9116 3.9165 4 4.98593x10-2 2.1162 2.1188 5 8.66864x10-3 2.5847 2.5519 6 4.65610x10-3 2.4472 2.4503 7 2.34398x10-3 2.3753 2.3783 72 _

THE UNIVERSITY OF MICHIGAN in I0 r C C0O CI D C o I oR H R Oi T 8 FIGURE VII. CONVERGENCE COEFFICIENTS FOR HARD OBLATE A _ __________________________________ _ 73 _ ___ ______ _____ _____

THE UNIVERSITY OF MICHIGAN 3648-4-T VIII PROLATE SPHEROIDS For a prolate spheroid, e is confined to the range 1 4 f 4 oo and the lower limit (g =1) represents a rod of zero thickness joining the foci of the coordinate system, whilst the upper value (E =oo) again corresponds to a sphere. In spite of the change in the coordinates, some of the analysis for the oblate spheroid is immediately applicable here. In particular, when the ellipticity is small (~ large compared with unity) the convergence of the Rayleigh series has already been determined (~4), and if the body is soft the series converges for 1 -4 c~ <1+ -- +0(-4), (19) 52 whereas for the hard body the criterion is ce <1 + -- + 04 ) (20) 32 where c~ is now the semi-major axis of the spheroid. At the other extreme ('l1) the spheroid approaches a'vanishing' rod and since the volume of this is zero it is not surprising to find that each term in the expansion of the far field amplitude is zero in the limit e = 1, leading to a null Rayleigh series. On the other hand, for all g f 1, no matter how close to unity, the Rayleigh series exists, and it is therefore meaningful to consider the convergence as - + 1. ~~___ _ _ _ __ _ _ _ __ _ _ _ _ ~74 _ _ _ ___ __ __

THE UNIVERSITY OF MICHIGAN, 3648-4-T When e = - 1 is small, the most convenient representation of the radial functions is 1 ~ Ron(1)= ( ) r (on ) (2 l)r (n even) on k(1) (c))(2 1)| on r=0 (89) oo for__ \r con (C) e.2 r (n odd) (1) (c) 2r (90) on r=0 (Flammer, equations 4. 4. la and b), both of which are equivalent to on (1) c (c) R (c,) = +o(e) (91) k (c) on for small e. Here, con (c) 2n n n I o(c 2n 22'on (1 2) 2 — don (c) (n even) k(1) (c) on 2n n-1i n+l1 2 2 on =.. —--- c d (c) (n odd) 3(n+l)! and is finite for all finite c. In addition, (1) 2 i R3 (c,)=R 1) - on log +i (c,) (92) on 2c c (c) on (Flammer, equations 4. 4. 9 and 4. 4. 6) where._____________________,,75 _ __,___ ____

THE UNIVERSITY OF MICHIGAN 3648-4-T oo gon(c )= bo (c) ( -21)r (n even) r=0 00 =Ibo (c)(2-1)r (nodd) and both of these imply r=0 go(c,) = bn (c) + 0(E) (93) o0 when e is small. Accordingly, for a soft spheroid of ellipticity almost equal to unity 2 (c1 ) on kon (c) Fok 1+ o1 on o c ) A on = 1~ 2-T on j log. + ib (c) on + (e) L c (c) c (c) 0 0 (94) and the singularities are given by on(c) on 2 2 on o I o log -- 2cb (c) +2ic +0(e)=0. (95) E 0 k(1) (c) k( c)J on on The first term in this equation is infinite in the limit e = 0, whereas the third term does not contain e and is finite for all finite c. It is therefore apparent that when e = 0 the only possibility of obtaining a finite root is to have the second term become infinite, and such an infinity must then arise from the factor bon (c). o The expression for this gives c (c) X (c) (1) on on o....~~~k^c., (c)

THE UNIVERSITY OF MICHIGAN 3648-4-T where oo r-1 oo 4r-4s-1 on on 2 r (2r- s)(2 s+ 1) dp/2r r=O s=O r=l 00 r 00 = don X 4r-4s +1 ( odd) rZf 2r+l., 2r-s+1 pr-1 (n -odd) r=O s=0 r=l (see Flammer, equations 4. 4.16a and b), and though X (c) is finite, b On (c) is on o infinite at all zeros of c (c). As shown ing6, the zero for which c is smallest is 0 provided by n = 0 and is approximately c =41 expf+ i 0.05w2 which now represents the limiting value of the smallest root of (95). Consequently when c — 0 the radius of convergence of the Rayleigh series for the soft body approaches 4. 1, and the criterion for convergence is therefore c <4.1. (97) It is of interest to note that as in the case of a soft disc (an oblate spheroid of ellipticity e =1) the convergence is dictated by the amplitude coefficient A, and for the next coefficient, Aol the radius is approximately 5 2, corresponding to the smallest zero of c (c). 0 When e is small but not zero, some idea of the way in which the convergence varies as a function of e can be obtained from a perturbation analysis applied to equation (95). For this purpose let ____________________________ __,77 _______

THE UNIVERSITY OF MICHIGAN 3648-4-T c=x + f(c) n be a root of the equation when ~ E 0, where f(E) tends to zero with ~ and x is a zero of c (c). If this is substituted into (95) bearing in mind that the first two 0 terms are the dominant ones, we have approximately X (c)n 1 log- 2 on = f(c) n n log 2 where X (c) =2 ~ on n/ c a/ace (c) J =x 0 n Hence, c=x (1+ (98) log - C for small e, and as required the second term is zero in the limit e = 0. Into (98) we now insert the value of x corresponding to the smallest zero on of c (c). This is 0 x =41 exp f-i 0.05 7T (99) and consequently for e d 0 the radius of convergence for A is -__________________________ 78 _ _ _____ _ ___

THE UNIVERSITY OF MICHIGAN 3648-4-T (100) C~ = (+x + 2 (1+ ) 100) log - If e is sufficiently small this is also the radius of convergence of the Rayleigh series, and under these circumstances equation (100) can be written as c = 4-1(1+ R 1 (101) log 2 e C where R denotes the real part. Unfortunately the expansions in Appendix A do not contain enough terms to enable us to calculate? with the accuracy desired, but using the terms which are available it is found that R t = - 1-5. e o Although it would be unwise to rely on the second figure, the result does give some indication of the convergence for E * 0, and accordingly for a soft spheroid of ellipticity almost equal to unity the convergence criterion is taken as cK < 4*1 {1- 15 (log _-_ ) - (102) It will be observed that the radius is always less than 4. 1 and approaches the -10 limiting value extremely slowly. Indeed, for 1 = 1 + 10 the right hand side of (102) is still only 3.8. 79

THE UNIVERSITY OF MICHIGAN 3648-4-T In the case of the hard body the analysis is similar to the above in most respects. For the radial functions of the first kind we have from (89) and (90) on a (1) c 2 (c) -R( (c,) = - 2 + 0(c) (n even) (103) D) on (1) k (c) on 1y on on (1) (c (c) - 2 c2 (c + O(e) (104) k(1 ) L 2 J on (n odd) and since 1 2 on C2 = n - c )c (c) (n even) 2 4 on o 1 2 on = (k c -2) c (c) (nodd) 4 on o (Flammer, equations 3. 2. 1a and b), where X is the eigenvalue, (103) and (104) on are both equivalent to on (c) a (1) 1 2 co -R (ca,) = - n -c2 + O(E). (105) a on 2 on (1)(c) on Also, a d( on o on n( = b o (c) + 2 bon (c) + O(e) (n even) o n =2b (c) + 0() (n odd) on but here again the differences are compensated by differences in b (c) for n even ons 4.6a and b) and odd, and inse rting the expresions for b (c) (Flammer, equations 4. 4. 6a and b) we have __________________ _ ____ _ 80 _

THE UNIVERSITY OF MICHIGAN 3648-4-T (1) c) c (c) ag go, c' ) = 2(kon c6)jb0"'(c)J _ + 0(e) (106) | 2 for all n. Hence con (c) ic 2 J o on A = —-.on on on (c) i + 1 1 2 + 1 1 c ()2 on 2 (ton-c )log +-(X on-c )- c b0(c) (11 ) - 1+ 0() 21 4 o- 2 0 (c) on the singularities of which are given by on 110 2 c (c) - (X - c2 )log + (X -2) cbon(c) ) 0 -1+ 0(e)=0 (108) E 2 on [ on 0(e) on The first two terms of this are infinite when c=0, but owing to the markedly different rates at which 1/c and log - approach infinity as e —- 0, it is obvious that the only possibility of having a finite root is to have the third term become infinite, and this again leads us to the zeros of cn (c). Since the smallest zero is provided I 00~~~~~~0 by c (c) and has a magnitude 4 1, the radius of convergence of the Rayleigh series for the hard spheroid must tend to 4 ~1 as e — 0, and the convergence criterion (97) therefore applies to both soft and hard bodies. In each case the convergence is dictated by the amplitude coefficient A, and for the coefficient A the radius is oo ol approximately 5. 2. When e is small but not zero, the variation in the radius of convergence as a function of e can be determined in the same manner as for a soft spheroid. Bearing _____________________________ 81 ~_- _ __ _ _ _-.

THE UNIVERSITY OF MICHIGAN 3648-4-T in mind that in equation (108) the dominant terms for small e are the first and third, the root which corresponds to c = xn when e L 0 is c =xn (1+ En) (109) where {Xon (c) - c x (c) V _ I on on c a/ac con (c) c=x o n and the smallest of these is obtained by taking n = 0 with x as shown in equation (99). The radius of convergence for A is therefore c =4.1 (l + R R') (110) e o (cf equation 101), which is also the radius of convergence for the Rayleigh series providing e is sufficiently small. Using the formulae in Appendix A, To has been computed and its real part found to be R ~' =- 4 2 e o Although the accuracy of this leaves much to be desired, the value is as good as can be obtained without including a significantly larger number of terms in the expansions (particularly for the d /), and in consequence for a hard spheroid of ellipticity almost equal to unity the convergence criterion will be taken as._______________ _ ~82 -

THE UNIVERSITY OF MICHIGAN 3648-4-T c~ <4.11-42 (e-1). (111) -3 It will be observed that the radius differs negligibly from 4 1 for c < 10, and as such the result is in marked constrast to that for a soft spheroid. In order to bridge the gap between the ranges of g for which the criteria (19) and (23), (102) and (111) are applicable, it is necessary to resort to a numerical comparison of the actual coefficients in the expansions for the A. This is a simon ple matter for the hard spheroid, and since the amplitude A specifies the convergence of the Rayleigh series as TV oo and ^l1, it is not surprising to find that this is true for all ~. What is more, as ~ decreases from infinity the radius of convergence rapidly assumes the value indicated in (111). For the soft spheroid, on the other hand, the convergence is determined by the amplitude Aol for large ~ but by A in the limit as — > 1. and the value of | at which Ao takes over differs from unity by an extremely small amount. In consequence there is a wide range of r for which no formula is available for calculatin the convergence, and here the numerical approval is indispensable. We shall therefore begin by considering the case of the soft spheroid. For a soft oblate spheroid the expansion of the amplitude coefficient Aol is given in equation (80) and from this the analogous result for a prolate spheroid can be obtained by changing c into ic and ~ into - it. The expansion then proceeds in powers of (-c) and the coefficients have been calculated for a sequence of 8 ranging 83

THE UNIVERSITY OF MICHIGAN 3648-4-T from 1. 7 down to 1. 00001. These are listed in Table X together with the convergence coefficients a | and I ja which can be deduced therefrom, and the r r last-named are also plotted in Figure VIII. The regularity of the curves is at once apparent and it is interesting to compare them with the curves for the oblate coefficient A 1 (see Figure IV). As the oblate e increases from zero the oscillations rapidly die down, and this process continues systematically as passes through infinity and then decreases through prolate values to unity. Such a correspondence between the two sets of curves is typical of the amplitude coefficients A and A1 for both the soft and hard bodies, and may well be true of the A in general. on The radius of convergence for a given ~ is represented by the limit of the curve E a 1 against r as r — o, and notwithstanding the fact that r = 9 is the largest value for which numerical data is available, it is possible to estimate the limit by comparing the e a with the convergence coefficients for the sphere ( = oo). The radii of convergence obtained in this manner are shown in Table II, and it is believed that they are in error by no more than 2 /o. TABLE XI. RADIUS OF CONVERGENCE FOR A01 (SOFT) 1.7 1.5 1.2 1.1 1.05 1.01 1.001 1.0001 1.00001 c] 1.09 1.12 1.24 1.35 1.47 1.72 1.98 2.15 2.31 84

THE UNIVERSITY OF MICHIGAN 3648-4 -T TABLE X. CONVERGENCE COEFFICIENTS FOR Aol (SOFT) = 1.7 =1.5 = 1.2 r r r r la r | lar I 1a rl 2 - 1.35650 0.85860 1.4596 - 9.65819x10-1 1.0175 1.5263 - 4.55854x10-~ 1.4811 1.7773 3 il. 28114 0. 92073 1. 5652 i8 04850x10- 1. 0751 1. 6126 i3 03903 x101 1.4874 1.7849 4 2.41991 0.80177 1.3630 1.27498 0.94107 1.4116 3.27106 x10-l 1.3223 1.5867 5 -i3.62948 0. 77274 1. 3137 -il. 65126 0. 90456 1. 3568 -i3. 13539 x 10-1 1.2611 1. 5133 -1 6 - 5. 74246 0. 74728 1.2704 - 2.23329 0. 87467 1. 3120 - 3.05722 x10 1. 2184 1.4620 7 i8. 98530 0.73077 1.2423 i2.99614 0.85491 1.2824 i2. 97658 x101 1.1890 1.4268 8 1.40864 x 10 0.71846 1.2214 4.02445 0.84026 1.2604 2. 89798 x 10- 1.1675 1.4009 9 -i2.20773x10 0.70904 1.2054 -i5.40489 0.82905 1.2436 -i2. 82153x10 1.1510 1.3811 g = 1.1 g = 1.05 1. o0 r an! ~jan1J la Ija 1i arrJ r ~er | ~larl] | 1all C |arJari I ai] C r arll ] 5arll 2 - 2.96522 x 101 1.8364 2. 0201 - 2.10609 x 10 2. 1790 2.2880 - 1.19178 x 10 2.8967 2.9257 -2 3 il.81208x10-1 1.7672 1.9439 il.22855x10-1 2.0116 2.1121 i6.68718 x102 2.4637 2.4883 4 1. 57735x10-1 1.5870 1,7455 9.13523x10-2 1.8190 1.9099 3.96746 x10-2 2.2406 2.2630 5 -il.29209x10-1 1.5057 1.6563 -i6.64917x10-2 1.7197 1.8057 -i2.39639x102 2,1091 2.1302 -1 -2 -2 6 - 1.06255x10-1 1.4530 1.5983 - 4.82729 x102 1.6572 1.7401 - 1.44379x102 2.0265 2.0468 7 i8.74495x10-2 1.4164 1.5580 i3.50924x10- 1,6137 1,6944 i8.70589x1021 1.9693 1.9890 8 7.19562 x10-2 1.3895 1.5285 2. 55083 x i2 1.5819 1.6609 5.25093 x10-2 19274 1.9467 9 -i5.92080xlCT2 1.3690 1.5059 - 840x 1.5575 1 6354 -i3.16664x102 1.8955 1.9144 g = 1.001 1 = 1.0001 = 1. 00001 ar | arl |'r l arll a r | r r -2 -2 -2 2 - 7.13139x10 3.7447 3.7484 -5.06039x10-2 4.4454 4.4458 - 3.91919x102 5.0513 5.0513 3 i3.96585 x102 2.9324 2.9353 i2.81161 x102 3.2886 3.2890 i2.17735 x102 3.5812 3.5812 4 2.01798x102 2.6532 2.6559 1.32610x10-2 2.9468 2.9471 9.82237x10-3 3.1765 3.1765 -2 -3 -3 5 -il. 04154x102 2.4915 2.4940 -i6.21950x10 2.7622 2.7624 -i4.31950x101 2.9711 2.9711 6 - 5.39444x10-3 2.3879 2.3902 -2.92904x10-3 2.6437 2.6440 - 1.90090x10- 2.8412 2.8412 7 i2.79962 x10 l 2.3158 2.3181 il.38501 x10 2. 5607 2. 5610 i8.40880x10 2.7500 2.7500 -3 -4 -4 8 1.45428x10 2.2629 2.2652 6.56370x10 2.4995 2.4998 3.73397x10 - 2,6821 2,6821 -3-4 -4 9 -i7.55322x10 2. 2227 2. 2249 -i3. 11101 x10 2. 4529 2. 4531 -il. 65941 x10 2. 6303 2. 6303 85 _

THE UNIVERSITY OF MICHIGAN 3648-4-T I I I I I I I I I I I I I I I I I! 1 I 0 O I I I I II I11 O 0 0 0 0 0 LII II II II I II II II = 0 co 0 0 0 0- C F4 C8 PL

THE UNIVERSITY OF MICHIGAN 3648-4-T Even when g is as small as 1- 00001 the radius of convergence is still only 2. 3, and this is far short of the value 5- 2 (corresponding to the smallest zero of c ) which is reached when e = 1. It is also much less than the radius for A o 00 and in order to determine the convergence for Ao out to at least the neighbourhood of the cross-over point there are two possible methods of attack. The first of these would appear to be the most logical and is based on the formula (98). By taking n = 1 and xl = 52 exp + i 0 083 7, the formula gives immediately the decrease in the radius of convergence as a function of g in terms of the factor V1, but unfortunately the expansions in Appendix A are not sufficient to compute X l(c) when I c is as large as 5 2. In addition, there is reason to doubt the validity of -50 (98) unless - 1 is vanishingly small (less than, perhaps, 10 ). We shall have more to say about this in a moment. The second method is merely to extend Table X to smaller values of g and in this connection the computation can be simplified somewhat by analyzing the behaviour of the a as 6 -> 1. From the expressions for the Legendre functions P (a) and Qn(W), we have P () = 1 + 0(E) n n Qn() + -- + O(E) for small e where S= 2/log 2, and hence ___87

THE UNIVERSITY OF MICHIGAN 3648-4-T B + 0(c ) 2 5 B 4 + O(c S) 7 3 37 4 3 5 7 B = $ + O(c S) 5 3. 5 -6484 + o(c E ) 6 3457 82 s+ 0(oE) 3.5.7 B - 312401 B =- + O+0 S 8 3 6 3 3 5 7 11 B9:i 3121 S+ 0(c +). 9 35 667 The coefficients ca are therefore r - 5 682 5 ^=-i -+ 4'o( 3 32 57 5 _ 2 ~-_ 2 2 3.5 35 6484, 10369 a6 4 4-410369 62- 30( 34557 34547 5 _________________________ 88 ______....____

THE UNIVERSITY OF MICHIGAN 3648-4-T 82 +i 116 2 1 3+ 0(E) =-i - --. + i 2 + 0(eS) 354 7 3257 3. 5 312401 + 77083 C2 7372 3 1 4 O( 8 33 567311 3457 34557 54 0() 73 3121 3-i 256502 2 i 5492 3 _i 4 + 0(eS) 36567 3657 357 32 53 from which the convergence coefficients ac can be determined as before. By r comparing the la | for e = 10-5 with the values shown in Table X it is found that the error produced by the terms in e S in less than 50/0 for r> 3 and decreases rapidly with c. Using the above expressions the a have been computed for = 1+ 10 r m = 3, 4(2) 12 (4) 20, and these, together with the [a i, are shown in Table XII. The corresponding radii of convergence are given in Table XIII, and though it is difficult to estimate their accuracy, it is believed that the error is not more than 5 /o at the very most. TABLE XIII. RADIUS OF CONVERGENCE FOR Aol (SOFT) 10-6 10-8 10-12 10-16 10-20 1024 1-32 | -40 LcEl 2.45 2.62 2.90 3.12 3.30 3.44 3.66 3.81 89

THE UNIVERSITY OF MICHIGAN 3648-4-T TABLE XII. CONVERGENCE COEFFICIENTS FOR Aoi (SOFT) - = 10-68 = 10-82 = 10-126 r a |r lar 1 I ar larll ar larl ar lar rr Iar 1 __1_ jr _al r r -2 -2 -3 2 - 2.75697x10 6.0226 - 2. 09273x10- 6,9126 -1.41222x102 8.4149 - 1.06569 x103 9.6869 -2 -2 -2 -3 -3 3 il. 53165 x10 4.0267 il. 16263 x10 4.4142 i7. 84568 x10 5.0326 i5.92048 x10 5.5277 4 6.58912x103 3.5099 4.86257x103 3.7869 3.18528x10-3 4.2093 2.36673x10 - 4.5338 -3 — 3 -3 -4 5 -i2.68253x103 3.2681 -il.88176x103 3.5082 -il.16308x10-3 3.8626 -i8.36645x10-4 4.1256 -4 -4 6 - 1.08140x10-3 3.1213 7.12431 x10 3.3461 - 4.07099 x104 3.6733 - 2.79274x104 3.9114 7 i4.37865 x10- 3.0186 i2.69942 x10 3.2346 il.41159 x104 3.5485 i9.13964x 10 3.7758 -4 -55 8 1.78423 x104 2.9415 1. 03072x10 3 1503 4.92791x105 3.4548 3. 00042 x10 3.6758 5-5 -5 -6 9 -i7.28965x10 2.8820 -i3.95440x105 3.0847 -il.73403xl0 3.3806 -i9.94091 x10 3.5962 e = 10-20 = 10-24 c = 10-32 = 10-40 r ara a a a arI la r L ^r |ll r 1 |r | Ir larll 2 - 855709x10 3 10.810 - 7.14858x10 3 11.828 -5.37809x10 13.636 - 4. 31051 x10 15.231 -3 -3 -3 -3 3 i4 75394x10 5.9472 i3.97143 x103 6.3147 i2. 98783x10-3 6.9430 i2. 39473x10 3 7.4745 -3 -3 -3 -4 4 1.88244x10 4.8009 1.56252x10 1 5.0295 1.16601x101 5.4115 9.29945x10 4 5.7265 5 -i6 51833 x 10 4. 3368 -i5. 33352 x104 4.5144 -i3. 90677 x 10 4.8044 -i3. 08012 x 10 5. 0383 -4 -4 -4 -5 6 - 2.10759x10 4. 0992 - 1.68544x104 4.2548 - 1.19716x104 4. 5044 - 9.32618 x10 4.6958 7 i6.61960x10 3.9539 i5.13286 x 10 4.1003 i3.49108 x105 4.3323 i2. 62118 x105 4.5133 -5 -5 -6 8 2. 07570x10 3.8491 1.55274x10 3.9913 1. 01505 x10 4.2091 7.24215 x 10 4.3906 -6 -6 -6 9 -i6.56738 x10 3.7657 -i4.73362 x106 3.9053 -i2.88338 x 10 4.1264 -i2.00485 x106 4.2963 90 _

THE UNIVERSITY OF MICHIGAN 3648-4-T From a study of these results it is seen that the radius is still increasing even at the smallest value of e, a fact which is otherwise obvious from the form of the expressions for the a. On the other hand, not all of the change in the ar is automatically reflected in an increase in the radius, and for sufficiently small S the changes in the ar for r less than some fixed number have no effect on the convergence. This is most easily seen by dividing the a by6< and examining the convergence indicated by the remaining coefficients. As r increases, the contribution of the higher powers of 5 becomes more important due to their relatively larger coefficients, and even for r < 9 the terms ing may still dominate when e = 10-40 Thus, for r = 9 the ratio of theS2and S contributions is 1. 3 when = 10-40 and does not fall to 0.1 until e = 10-507 Under these circumstances it is questionable whether a formula such as (98) 2 in which terms in 5 are neglected can be expected to hold unless e is extremely -40 small, and it is therefore not too surprising to find that even for e = 10 the rate at which the radius increases is not yet consistent with a formula of the type 5.2 (1- V). Ultimately, however, the radius must assume this dependence on 6, but it may well be necessary for e to be appreciably smaller than the values considered in -500 Tables XII and XIII, and possibly of order 10 To pursue the analysis of the convergence to such values would be a trifle academic. l_________________________ 91 ___l —------

THE UNIVERSITY OF MICHIGAN 3648-4-T In view of the above results it is clear that the radius of convergence for the amplitude coefficient Ao is of no concern until e becomes extremely small oo -70 (less than, perhaps, 10 ) and one may hope that the criterion (102) is then applicable. Nevertheless, in the interests of completeness we have computed the coefficients in the expansion of A in powers of (-ic) for e = 1. 7, 1.5, 1 2 and 1 1, oo and the results are displayed in Table XIV. The convergence coefficients 5| a | are plotted in Figure IX, and the extent to which the curves resemble those for the oblate body (see Figure VI) is quite striking. In the prolate case, however, the curves turn over somewhat sooner and minima occur for r < 9 with the smaller values of ~. Although it is impossible to obtain any reliable estimates of the convergence from these curves, the radius would appear to be of order 4 for = 1- 2 and 1. 1. For the hard body, the problem of finding the radius of convergence of the Rayleigh series is more straight forward. We have already seen that when e is large or near to unity the convergence is determined by the amplitude coefficient A and as — 1 the radius rapidly approaches its limiting value 4 1. Since the 00oo corresponding limit for Aol is 5 2 and is approached at a comparable rate, it is natural to expect that the coefficient A will specify the convergence for all A, and this is indeed the case. 92 _

THE UNIVERSITY OF MICHIGAN 3648-4 -T TABLE XIV. CONVERGENCE COEFFICIENTS FOR A (SOFT) oo =1.7 _ =1.5 r a |a ~l | a ~l a |a ~| |a ~|l r r r r lr 1 -il. 48156 0. 6750 1.1474 -il. 24267 0. 8047 1.2071 -1 2 - 1. 35548 0, 8589 1. 4602 - 9. 22893x10 1. 0409 1,5614 -1 -1 3 i9.28991 x 10 1,0249 1, 7423 i5.12812x10i 1.2493 1.8740 -1 -1 4 4.94757x10 1,1924 2.0270 2.18152 x10 1.4632 2.1948 ~~~~-1 ~-2 5 -i2.11560x10-1 1.3643 2,3193 -i7,25571x10 1.6899 2.5349 -2 -2 6 - 7. 39309x10 1. 5436 2, 6241 - 1. 88449x10 1. 9385 2. 9078 7 i2. 10389 x 10 1.7361 2, 9513 i3. 55228 x10 2.2384 3.3575 -3 -4 8 4. 72423x10 1.9530 3.3202 3.39789x10 2.7139 4.0709 -4 -5 9 -i7. 41712x10 2.2272 3.7862 -i6.30718 x10 2.9288 4. 3932 =1.2 =1.1 r a ar II ar \ar~l 1 -i8.34063x10-1 1.1990 1. 4387 -i6. 56918 x 0-1 1.5223 1.6745 -1 -1 2 - 3.62036x10 1. 6620 1. 9944 - 1.90671x10 2.2901 2. 5191 -1 -2 3 il.16371x10 2.0483 2.4579 i4.00143x10 2.9237 3.2160 -2 -3 4 2. 62887x10 2.4835 2. 9802 4.91704x10 3.7764 4.1540 -3 -4 5 -i3 37091 x103 3.1221 3. 7466 -i3.95666x10 4.7922 5.2714 -5 - 6 - 7. 91241 x10 4. 8263 5. 7916 - 2.60938x10 3.9559 4. 3515 -4 -5 7 i2.04309x10- 3.3659 4. 0391 i8.06818x10 3.8437 4.2280 -5 -5 8 6. 22227x10 3. 3555 4. 0266 1.15107x10 4.1435 4.5578 -6 9 -i9.79192 x10 3.6022 4.3227 -i7.58100x10 4.7866 5.2653 93 _

THE UNIVERSITY OF MICHIGAN 3648-4-T 5. 0 4. 0 3. 0 =1.2 -1.5 2. 0 1 2 3 4 5 6 7 8 9 r FIGURE IX. CONVERGENCE COEFFICIENTS FOR SOFT PROLATE A 00oo 94 _ _ _ _ _ _ _ _ -

THE UNIVERSITY OF MICHIGAN 3648-4-T The expansion of the oblate coefficient A is given in equation (28), and 00 by changing c into ic and g into -it the analogous result for the prolate coefficient is obtained. The expansion then proceeds in powers of (-c) and the coefficients ac r have been computed for ~ =1.7, 1 5, 1.2, 1 1, 1.05 and 1.01. These are listed in Table XV, together with the convergence coefficients aJ and |a0 deduced r r r therefrom. The last named are plotted in Figure X. Once again the similarity of the curves for the prolate and oblate coefficients (Figure VII) is apparent, and by comparing the coefficients with those for the sphere (=oo) the radius of convergence has been estimated as shown in Table XVI. TABLE XVI. RADIUS OF CONVERGENCE FOR A (HARD) 00 1.7 15 1.2 1.1 1.05 1.01 c~ 1-16 1.22 1-48 1.76 2.10 2-95 The error in these values is probably less than 2%o, but as ~ decreases the less regular nature of the curves may increase the error to some extent. In particular, for = 1- 01 the radius could be out by as much as 50/, and for g =1- 001 the j a Iwith r, 7 are too scattered to give any reliable indication of the convergence. __________________________ 95 _

THE UNIVERSITY OF MICHIGAN 3648-4-T -- cD 0 0 CMl CM C CO - LO CD CO 0 CO. C CD C 0 CD CO 0 LO CO 0 CD NO N d ~ CO CD CO CO 0 - 0 C- - co 00 CO UP C CM M T- I u 0 O 0 C 0 0 C 0C ~f CM CM CM' " o 0 CO CO L.O T LO.-' 0 i CO -- o - cO C oc C Q I I I I I C O I o o 0) N CO O I ) N o C 0) Co CO CO O CoC O O Co cO CO O o C II I I I I C CD O CD CD CD CCD CD CD. 0 tC x b C - N tN x Co N C o 0 C C co O r-q Co LO Co Co L: - --- co L O o I O CL 00 -, — CO ]C Cl 00 d d O CA co 0 CD iO CO ci-l COM N ) CO C O IC) C H P- r-H r- r r-H O C O Cl C CLO Cl z O O Co 0) CO C OcO 0 O Cl 0) N CO D LO CD5 O Co C 00 LO Cl oO C CO L O 0 c - U 11 _ -H -H -O O OCO CO0 — 0 0 Cl Cl Cl Cl I I I U~~~~~~~~~~~~~~~~~~~~I I I IC I I F4 O O o C 0 O O CO O P5 (.OO) CD N O O Cl O Cl N Cl I CO C0 COO cL- C) OC'- l N Co o oCo o 0 o CD - N o C C - N C < dCo C o o0) ~ o - o- o CO Co OCl o C-oD -4 CO CO NL NO c CoO CO Clo N CO U? I IC O' C 0O Co Co 0C) 0) CO O4 N^ L 0)D I 0) OO ClO CL - m CO C - 00 0C NO.' I., C C CMc O a t CO 0 0 O O C CO00 -L CD L C9,, f: 5 O I CD C M L O t n 3 O 03 a O n C c) 0 ) Co N C O coOc ~ QUO 0) C C 0 C m O r-I I' OT-HCD I T-lCCo 0) COl I N 0 0 )CO C Cl Cl' -- C CO CO Cl0 0) 0) CO 0) 0) 0 0) N Co Cl CO:t T —(-L E o. c 11 0 0) N 0) Co COc QC U0 0) CO o N co C I IC) N CO'-4 0) CO CD O O 00 CD 0< s< s^ s^ s 5H O O C~ Cr3 kO LN IC S3 Cl I.w I I *I II I i i t II ~~ ~~ ~- 96

THE UNIVERSITY OF MICHIGAN 3648-4-T 4.\ 0 g 1. 01 3.(O =1.05 2. 0- 1.1 --— ~ —-~~ —-— =1. 2 1. 0 —. I i I I I I I 1 2 3 4 5 6 7 r FIGURE X. CONVERGENCE COEFFICIENTS FOR HARD PROLATE A0o 97

THE UNIVERSITY OF MICHIGAN 3648-4-T To complete the discussion of the hard spheroid it is only necessary to consider the convergence of the expansion for Aol and verify that the radius is nowhere less than the radius for A. This is known to be true for S large or near to unity, and for the intermediate range the analysis in ~7 can be used to compute the coefficients a in the expansion of Aol in powers of (-c). The results are given in Table XVII and the convergence coefficients e a1 are plotted in Figure XI. As in the case of the oblate spheroid (cf Figure V), the curves are characterized by an irregular set of peaks which make difficult any accurate estimate of the convergence, but by comparing the levels of the minima with those for the sphere curve, the values shown in Table XVIII have been deduced. TABLE XVIII. RADIUS OF CONVERGENCE FOR A (HARD) ol S 1-7 1.5 1.2 1.1 c5 1.56 1.62 1.85 2-09 Although the errors associated with these results are impossible to assess, the radii are in good agreement with the formula for large A, and their trend is not inconsistent with the general formula for the convergence when 9 is close to unity. 98 ___________

- THE UNIVERSITY OF MICHIGAN 3648-4-T TABLE XVII. CONVERGENCE COEFFICIENTS FOR A01 (HARD) = 1.7 = 1.5 \a1 fla?1! a 11 r r l r rl Il I |r | r 2 -8.01283xlO-1 1.1171 1.8991 -6.07163 x1-1 1.2834 1 9250 3 -i4.94909 x10-1 1.2642 2.1492 -i2.81094 x10-1 1. 5266 2. 2899 4 -5. 27131 x10- 1.1736 1. 9951 -2. 61935x10-1 1. 3978 2. 0967 5 i8. 52514x10-1 1. 0324 1. 7551 i3. 75071 x10-1 1.2167 1. 8250 6 4. 96922xl01 1.1236 1.9102 2. 09175xl0-1 1.2979 1.9469 7 i 1. 04854 x 10-1 1.3801 2. 3462 i 4.13844 x 10-4 3. 0431 4.5646 8 7.49680x10-1 1.0367 1.7623 1.73763xl0-1 1.2445 1.8668 -i11.04910 0.9947 1.6910 -i2.31610x10-1 1.1765 1.7647 ~ =1.2 2 1.1 r r ar1ll a 1 Iaj _l glar'l 2 -3. 56897x10-1 1.6739 2.0087 -2.81934x10-1 1.8833 2.0717 3 -i7.27010x10-2 2.3960 2.8752 -i2.99021 xl-2 3.2218 3.5440 4 -5.33162 x10-2 2.0811 2.4973 -2.06402x10-2 2.6383 2.9021 5 i6. 06176x10-2 1.7518 2.1021 i2. 04491 x10-2 2.1770 2.3947 6 3. 19757xl0-2 1.7750 2.1300 1.06186xl10-2 2.1330 2.3463 7 -i8.31925x10-3 1.9821 2.3785 -i3.40594x10-3 2.2518 2.4770 8 4.15955x10-3 1.9844 2.3812 -1.41363x10-4 3.0284 3.3312 9 -i7.17694x10-3 1.7307 2.0769 i7.26718x10-4 2.2322 2.4554., ----- --- --- ---- ----- -— _ _,,99

THE UNIVERSITY OF MICHIGAN 3648-4-T I I I I 1 II II I- I IoT~~~c o Iq — - —.. — --- - - 100 1_~. __ _. r,. Q,. C.i.]. - ---—. — — _ _ _ _ _ _ _ _ _ _ _ 1 0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

THE UNIVERSITY OF MICHIGAN 3648-4-T IX DISCUSSION OF RESULTS The analysis in the previous sections is sufficient to specify the radius of convergence of the Rayleigh series for a spheriod when a plane wave is incident nose-on, and we shall now gather together the results of the various calculations. Before doing so, however, a few words are necessary about the style of presentation to be adopted. The quantity of interest is, of course, the radius of convergence and this is inversely proportional to the wavelength. Since it is also dimensionless, it must be proportional to c, and the factor of proportionality is then at most a function of the spheroid's shape. From the physical point of view a convenient choice of factor is one which associates the radius with the semi-major axis of the spheroid and this has the advantage of being non-zero for all ellipticities. The radius of convergence is therefore defined as the limiting value of either ce or c(2+ 1)1/2 for which convergence exists, with the first and second applying to prolate and oblate spheroids respectively, and in any graphical presentation it is natural to choose this as the ordinate. Having taken the frequency dependence into account via the choice of ordinate, it is desirable to have the abscissa independent of k (and hence c) and a function only of the variable e specifying the spheroid. In addition, the abscissae for prolate __ _ _ _ _ __ _ _ _ _ _ __ _ _ _ ~101

THE UNIVERSITY OF MICHIGAN - 3648-4-T and oblate bodies should correspond to one another as much as possible, but because of the different ranges of the variable it is apparent that the mathematical description of the two scales will almost certainly differ. The formal analogy between the two types of body is one in which 2 is replaced by _2, suggesting that the horizontal scale should be a function of ~2 rather than of A, and ultimately log (1 7 1/52)1/2 was chosen as the function, where the upper and lower signs refer to prolate and oblate respectively. This satisfies all the required conditions, and since (1 1/2)1/2 = w where w and A are the maximum dimensions perpendicular and parallel to the direction of the incident field, the abscissa is continuous through the transition from prolate to oblate bodies. Turning now to the actual results, it will be recalled that for both types of spheriod, hard as well as soft, the radius of convergence of the Rayleigh series is specified by the convergence of the expansions for one or other of the amplitude coefficients Aoo and Aol. In Figure XII the radii of these expansions for a soft oblate spheriod are presented. When 2 is much greater than unity, the convergence of the Aol expansion can be obtained from the formula (26), and it will be observed that the curve goes smoothly into the values deduced from a numerical comparison of the coefficients in the expansion and listed in Table IV. The resulting graph is almost a straight line on the logarithmic plot, but ultimately a small amount of curvature becomes apparent (E < 0. 1 approx) and the radius finally ________________________ 102 _ ____-__ _____

THE UNIVERSITY OF MICHIGAN 3648 -4 -T CO 0 ".k~~~~~~~~~~~~~~~~~~~~~~ aX~~~~~~-''. + FZ eo H'I 0', - 0'. (T b 0 H k.I o 1-..~~ - 2~o 0 I ~' z, \1~. 0 \ *. X \ _ 1 0 ____ \~~~~~~~~(t'. t \ ~.. "S 0~~~ \ ** u~~10

THE UNIVERSITY OF MICHIGAN 3648-4-T ol01 reaches (at e = 0) the value 5. 2 corresponding to the smallest zero of c. For the amplitude coefficient Aoo the radius of convergence is infinite when g = oo (the sphere), and though it is presumably finite for all other A, the curves in Figure VI suggest that the radius is still in excess of 2. 6 by the time g has decreased to 0. 4 (w = 2. 69 e). If e is still smaller, say e < 0. 2, the formula given in (74) becomes applicable, and this indicates a radius of convergence which increases slowly towards a maximum value of 3. 25 at e = 0. It would therefore appear that the radius has a minimum for e somewhere near 0. 3, but since the level is in excess of the corresponding radius for Ao1, it does not affect the overall convergence of the Rayleigh series. This last can be found by selecting the smaller of the radii for Aoo and Ao,1 and accordingly the convergence of the Rayleigh series is determined by the convergence of AO1 out to the point at which the two curves cross. This occurs when e = 0. 017 (i. e. w = 59J), and thereafter the radius for Aoo is the dominant one. The final curve showing the convergence of the Rayleigh series is given in Figure XVII, and the kink at e = 0. 017 is at once apparent. It is not known whether this has any significance beyond the obvious one implied by the mathematics. When the oblate spheroid is hard the results are even more detailed and are presented in FigureXIII. Taking first the amplitude coefficient Aoo, the radius of convergence is specified by (27) for 52 > > 1 and is in excellent agreement with.__________________-.- 104 ___________

THE UNIVERSITY OF MICHIGAN 3648-4-T O 0 0 So I S S Q o ~ _1 C4- C) I-1 o 0 0 ~'d~~~~~~ Q z 0 so 0 eQ 0 0 \', I/ (+~ *

THE UNIVERSITY OF MICHIGAN'- 3648-4-T the radii for smaller i deduced in ~8 and listed in Table IX. The curve is concave w upwards at least as far as - = 20, but the curvature is quite small and since the 0 limiting radius for g = 0 is only 4.1 (corresponding to the zero of co ) it must reverse itself somewhere beyond the point ~ = 0. 05. For the coefficient Ao1 the radius is ~2 when = co and providing 2 > 1 the variation as a function of e can be obtained from equation (21) with n = 1. The resulting formula is c(E2+ 1)1/2 = f (1 + 1 )+ O( -4 (112) 5~2 and the curve is shown in Figure XIII. This can be assumed to cater for values of i greater than (about) 2 and for smaller e it is necessary to rely on the calculations in ~8. Notwithstanding the difficulty in deducing radii from the convergence coefficients in Figure V, the estimates appear remarkably accurate and these bridge the gap between the regions for which the formulae for large and small e are applicable. When e << 1, (75) gives a radius of convergence which increases slowly with decreasing i, attaining a value 2.1255 at ~ = 0. The crossover point at which the radii for Aoo and AO1 are equal is 0. 053 (w = 19Z), and by selecting the smaller of the two at each value of A, the radius of convergence of the Rayleigh series is obtained. This is shown in Figure XVII and, like the curve for the soft oblate, is characterised by a kink at the point where the dominant coefficient changes. For 106 _ —__ —_____

THE UNIVERSITY OF MICHIGAN 3648-4-T all ellipticities, the radius for the hard body is less than that for the soft, and whereas with the former the dominant coefficient changes from AoO to Ao1 as g decreases, the reverse is true for the latter. In the case of the prolate spheroid the results are not quite so complete, and this is particularly true when e is close to unity. From the physical standpoint it is not surprising to find that difficulties arise as e -+ 1 since the entire Rayleigh series vanishes in the limit, but more important mathematically is the fact that the values of I c corresponding to the radii of convergence are of order 4 or greater, and for such large values the spheroidal function coefficients are extremely hard to compute. In addition, the rate at which the radii approach their values for e = 1 is so slow that any uncertainties in computation are reflected in errors over a wide range of I, and under these circumstances the numerical approach described in ~2 is indispensable. For the soft body the radii of convergence of A and A are plotted as functions of (1 - 1/ 2)1/2 in Figures XIV and XV with the latter providing an abscissa which runs through 20 orders of magnitude. If 2 is large compared with unity, the convergence of the expansion for A01 is given in (19), and when e is such that the formula no longer holds (say, C < 2) the convergence can be found from Table XI. It will be observed that the values deduced by the numerical technique are in excellent agreement with (19). As ~ decreases, the radius continues to 107

THE UNIVERSITY OF MICHIGAN 3648 -4-T I 11 0 CQ l' 0o 0 * C ) L E o o o ] * 0: 0 0 I U.' =: h0 > M 0*8 \ o o __ 1 ~_____ ~ __ T ~ - 0~~~~~~~C ~~~~~~~~~~F o o o o~~~~~~ cj 6; r;~~~~" 108~~~~~'

THE UNIVERSITY OF MICHIGAN 3648-4-T 0 o. Cc co I| ~ o ~ ~'o I c*o -0 otf I H H cC ~C3c T - Q:*~ ~~~co 0.., I

THE UNIVERSITY OF MICHIGAN 3648-4-T increase but at a slower and slower rate (see Table XIII), and even when e - 1 has fallen to 10-40 (corresponding to w/L = 10-2 in Figure XV) the radius is still far below the limiting value of 5. 2 appropriate to the'vanishing' rod (t = 1), and does not yet have the logarithmic dependence on e which equation (98) possesses. Ultimately, this dependence must obtain, but it probably does not do so before the radius of convergence of the amplitude coefficient Aoo takes over. For the coefficient Aoo the radius is infinite when ~ = oo and of order 4 when = 1. 2 and 1.1. As e -^ 1 the radius approaches its limiting value 4.1 according to formula (102), but here again e - 1 may have to be extremely small before (102) can be assumed to be applicable. Nevertheless, it is almost certain that the radius of convergence exceeds that for Ao1 until the latter crosses the 4.1 level, and some preliminary calculations suggest that this occurs when w/9 is approximately 10-37. For all practical purposes, therefore, the convergence for the soft body is determined by the amplitude coefficient Ao1 alone, and the resulting radius is given in Figure XVII. When the prolate spheroid is hard some of the above difficulties do not occur, but there is now a range of e which cannot be treated adequately by the numerical technique, and for which no formula for the convergence is applicable. Fortunately, however, the same amplitude coefficient specifies the radius at both ends of this range, and the end points of the two curves can be joined up withouttoo much possibility of error. __________________ _ ~110

THE UNIVERSITY OF MICHIGAN - 3648-4-T The radii of convergence for Aoo and Ao1 are shown plotted as functions of (1 - 1/2)1/2 in Figure XVI, and taking first the amplitude coefficient Aoo, the radius is unity when e = co. Its value increases with decreasing e and is initially given by (23), but this no longer holds if 2 is not large compared with unity, and thereafter the radius must be obtained by the numerical technique. It will be observed that the formula (23) goes over smoothly into the values listed in Table XVI. When e is near to unity the radius is given by (111), and although it would appear that this should be applicable for ~ - 1 as large as 10-2 (i. e. w/ = 0. 14), the curve does not then join up with the tabulated values. The discrepancy, however, can probably be attributed to the computation of Re o'o The number of terms used in the expansion for XOO (c) is insufficient to give a reliable determination of To' for Id as large as 4.1, and from a consideration of the signs of the subsequent terms it can be shown that to include them would increase the magnitude of Zo', possibly by as much as a factor 2. Such an increase would restore the agreement between the results of the two methods for calculating the convergence, and it is on this basis that the continuation of the curve shown in Figure XVI has been arrived at. For values of e less than (about) 104 (i. e. w/A < 10-2) the change in Re't has no significant effect on the convergence, and with further decrease of the radius remains constant and equal to 4.1. For the amplitude coefficient Ao the limiting value of the radius is 5. 2 (corresponding to the smallest zero of c~l) and is approached with a rapidity _____________________ _ 111 ------------ ---- --

THE UNIVERSITY OF MICHIGAN 3648-4-T 4.0 / / / / / o deduced from series for n/ I o 2. ~ ~' postulated continuation o deduced from series for n =1 /' * 1.0 - c l*I I 1 10-1 10-2 10-3 (1 -/2)1/ 2 FIGURE XVI. RADIUS OF CONVERGENCE FOR HARD PROLATE SPHEROID 112

THE UNIVERSITY OF MICHIGAN 3648-4-T comparable to that for Aoo as -4 1. When ~ is large compared with unity the radius can be found from equation (21) with n = 1 (or, alternatively, from (112) by deducing c and changing the sign of ~ ), and is consistent with the values for smaller e listed in Table XVIII, but for e lying between 10-1 and (say) 10-2 no information is available. Nevertheless, it is unlikely that the radius is anywhere less than the radius for Ao (if it were, there would be two hard prolate bodies for each of which Aoo and Ao1 had the same convergence), and it seems probable that the curve for A 0o is more or less parallel to that for Ao1. The convergence of the Rayleigh series is then specified by the coefficient Aoo for all ellipticities. The final results are shown in Figure XVII in which the radius of convergence for the soft body is represented by the solid line, and the radius for the hard body by the dashed line. All bodies, prolate as well as oblate, are encompassed by this graph and since the horizontal scale is logarithmic, the mid-point corresponds to the sphere (w =1), with the prolate bodies occupying the portion to the left (w < M), and the oblate bodies the portion to the right (w >A ). Thus, for the oblate spheroids the radius of convergence for the soft body everywhere exceeds that for the hard, and with the prolate spheroids the reverse is true except in the limit of a'vanishing' rod, where the two radii are equal. With this exception, the only case in which the two radii are equal is the transitional body, the sphere. In Figure XVII the ordinate is k times the semi-major axis, and is therefore c or c(52 + 1)1/2 depending on whether the spheroid is prolate or oblate,,____ _ _ _ _ _ _ _ _ _ 113. —

THE UNIVERSITY OF MICHIGAN 3648-4-T 4,o-~~~~~ M4I 2 I *D rb C' ~~~~~~~Ia CQ I 0 J000' =000' V ~ ~~\ o cl^/ C) 0 0 0 0 0 eaueDJeAuoD jo snipet _________________ 114 _

- THE UNIVERSITY OF MICHIGAN 3648-4-T respectively. The discontinuity in slope at the mid-point occurs entirely as a consequence of this change in ordinate, and if it is taken as cC (for example) throughout, the curves for both the hard and soft bodies are continuous in all their derivatives at w = 2. On the other hand, for the oblate bodies the quantity which is plotted would then be zero in the limit of a disc, and would no longer provide a meaningful measure of the convergence of the Rayleigh series. The radius of convergence is essentially the upper bound on the frequency for which the low frequency approximations are valid, and it will be seen from Figure XVII that of all the spheroidal bodies the sphere has the least radius. For a thin prolate spheroid, however, the radius can be as large as 4. 1, and this is sufficient to include the first two minima in the pattern for the backscattering cross section as a function of c~ (see Siegel et al, 1956). With this body, therefore, it is possible to penetrate the'resonance region' to a significant extent by using' low frequency techniques, whereas for a sphere the radius of convergence corresponds only to the first maximum in the pattern. It may be desirable to end with a word of warning. All of the above analysis has been carried out for bodies which are hard or soft in the sense that a Newmann or Dirichlet boundary condition respectively is applicable at the surface, and if the However, as ct approaches the radius of convergence, the number of terms which must be included in the low frequency expansion to get a reliable estimate for the field may become impossibly large. [ _______ - 115. __ __ __ ___

THE UNIVERSITY OF MICHIGAN 3648-4-T boundary condition is other than one of these, the radii of convergence which have been found no longer apply. This is easily seen by considering a partially reflecting sphere whose boundary condition is such that* (1 +iQ a)(Vi+VS)= 0 (113) ap at the surface. For a sphere which is predominantly hard or soft, 2 > > 1 or << 1 respectively, and under the condition (112) the amplitude coefficients A are given on by the formula jn(p) + i a jn(p) on hn(p)+ iQ - hn(p) ap When n = 0 the denominator is simply ik p (1 + iQ - -) P P which has a zero at p = po where P= Po 1 + iQ and the modulus of this is less than unity for all Q, whether real or complex, This is the analogue of an impedance boundary condition in electromagnetic theory (see, for example, Senior 1960b). 116

THE UNIVERSITY OF MICHIGAN 3648-4-T providing Im 2 < 1/2. Indeed, po - 0 as Q2 -e 0 indicating a radius of convergence for the exapnsion of Aoo which approaches zero with Q. In the limit Q = 0, however, the above zero disappears and the radius for the entire Rayleigh series reverts to the value found in ~2. Such discontinuities are a direct consequence of the fact that under a mixed boundary condition the coefficients Aon have no expansions which are uniform in Q. This is equally true for bodies other than the sphere, and it is to be expected that for a spheroid a similar behaviour will obtain. In general, the singularity provided by the'joining' parameter Q will be the dominant one, thereby producing a reduction in the radius of convergence. The convergence may then bear little resemblance to that for the corresponding'perfect' body. __ __ __ __ __ __ _ __ __ __ __ _ _ 117, - - - - - -- -—, -

THE UNIVERSITY OF MICHIGAN 3648-4-T ACKNOWLEDGEMENTS It is a pleasure to acknowledge the assistance of Mr. H. E. Hunter and Mrs. P. A. Marsh who painstakingly carried out much of the tedious computation involved in this study. The work described herein was supported by the Air Force Cambridge Research Laboratories under Contract AF 19(604)-6655.,,_,,,,, _. 118 - -

THE UNIVERSITY OF MICHIGAN 3648-4-T REFERENCES [13 Bazer, J. and Brown, A., IRE Trans. on Antennas and Prop. AP-7 S12 (1959). L2] Flammer, C., Spheroidal Wave Functions, Stanford Univ. Press, (1957). [3] Jones, D. S., Commun. Pure Appl. Math. 9 713 (1956). 4] NBS Math. Tables Proj., Tables of Associated Legendre Functions, Columbia Univ. Press, (1945). [5] Senior, T. B. A., Can. J. Phys. 38 1632 (1960). [6] Senior, T.B. A., Appl. Sci. Res. 8(B) 418 (1960). i7] Siegel, K. M., Schultz, F. V., et al., IRE Trans. on Antennas and Prop. AP-4 266 (1956). ~_________________ _ ~119 -

THE UNIVERSITY OF MICHIGAN 3648-4-T APPENDIX A SPHEROIDAL COEFFICIENT EXPANSIONS Apart from the eigenvalues XOn, the most fundamental quantities are the spheroidal coefficients dOn, and their expansions are involved in much of the prer ceding work. The derivation of the expansions is, in itself, not a trivial task, but if the corresponding terms in the expansion for XOn are known, the task is at least straightforward. Unfortunately, not all of these terms are known to the required accuracy, but by assuming in advance the form of the expansions for the dOn it is possible to derive simultaneously the expansions for the spheroidal coefficients and the appropriate eigenvalue. The process will be illustrated in reference to the case n = 0. The prolate spheroidal coefficients dOn(t) are defined by the recurrence r relation (r + 2)(r + 1) 2 dOn + 2r(r+l)-i 21 dyn -t2 2 Onn + r(r+l) - A + t (2r + 3)(2r + 5) r+2 On (2r-1)(2r+3) r(r-1) 2 On (2r- 3)(2r -1) together with the normalizing conditions 120

THE UNIVERSITY OF MICHIGAN 3648-4-T - l)r)/2 r! ) — do1 n (A. 2) r r r r n n I n r-0= 2 2 2 2. for n even, and r-1 n-1 0,, (-1) 2 (r+l)! O (-1) 2 ( (n+l)J,r r-1 11 ~ n 2l!n2l|d (A. 3) 2r r- r+1 j r n n - i n+1 r=f 2 2 2 2 2 for n odd. From these equations it is apparent that all the expansions proceed in even powers of t and, in addition, On r r r r r o with Do 1 andD' Dot, Dt.. O... The coefficientsD, Dr. D.....ITeofi arel independent of t. If (A. 4) is substituted into (A. 1) with r = O., we have immediately that 15 t t 4 -jj (X 00(A. 5) D2 +t D + D2"+.... ( (A. 5) and hence t2 x00- -+o(t4). 121

THE UNIVERSITY OF MICHIGAN 3648-4-T Knowing this one term we can now derive all the Dr and Drt by merely inserting the expansions for d and d00 into the recurrence relation and equating the coeffir r-2 cients of tr and t r - 2. It is found that r -1 D=- D (A. 6) r (r+1)(2r -3)(2r-1) r-2 from which all the D can be calculated, and r-1 1 2r(r+1)-1 1 Dr' =D - 2D -2. -. Dr (r+ 1)(2r - 3)(2r- 1) r(r+ 1) (2r -1)(2r+ 3) 3 J which can be shown to imply r Dr' = Dr (A. 7) 9(2r+ 3) thereby specifying the Dr' r It will be observed that a knowledge of D2 and D2' specifies X00 through t6. Moreover, from the recurrence relation with r = 4, 21 11 D4+ t D4'+t D4" +..... =6 + t2 ) (D + D, 4t4 L 00 21 4 2 ~| + t D2" +..... )+ (A. 8) and since D4 and D4' are known, D2" and D'" can be calculated, which in turn 122 -

THE UNIVERSITY OF MICHIGAN 3648-4-T specify X00through t 0. Similarly, from the recurrence relation with r = 6 we obtain D4" and D4'" and hence, from (A. 8), D2i and D2V, which then give k00through t14. To calculate the expansion for the d00 correct to 0(t ) it is only necessary r to carry this process one stage further (in the course of which the terms in 00 are determined through t 18), and the resulting expansions are 2r 2 13 46 85648 d00 2 6 d 1 3 - t4+ t6- t8 2 32 327 3 5. 7 3.5. 7.11 3.6547311.13 +0(tl0) d~0, t 4 ct0 - " 4 2498 2608 n 4 d00= —-- - - t2-31- t4+ t6+-0(t8)Jd~~ 0d 4=1 2 2 { 2 14 t t4 + ot6) d 6 3.35.211 L 3.25 3.55.217 00 { t2 + +0(t } d0 d 1t + t) + 02 C 6 0J 81 35. 27.11.213 17. 19 L 00 2 00 d 12 + o(t2),d with d00 = 0(t12) d00 for r > 12. Substitution into the normalizing condition (A. 2) ____________________________________ 123 ______ _______

THE UNIVERSITY OF MICHIGAN 3648-4-T with n = 0 then gives 1 67 3037 100403 dO00 1 __ 2_ t4 -2 4 - t 78 ~ 232 233 452 2 436527 2 2 738547 89075591701 + --- 4 — 2 — t 10 + O (t 12). 2.83105. 7.11 13, 17. 19 (A. 9) Before leaving the case n = 0 it is convenient to gather together the other expansions which are required in the course of the analysis. The coefficients dOn for r > 0 are defined in Flammer (p. 27), and when n = 0 p/r 5 41 283 t2 2 2 4. 6 _ P/2 2 3.5 2.3 3. 5 3. 7 2.3 511 + O(t lod0~ t4 64 512 0 00 2 4 6 8)~0 d =1 — t + 4- t4- - t + 0 (t ) do p/6 2.3 5.7 13 223.5213 d~~ = 8 {7,113 - 31t 2 + O(t4)} d0, P/8 233 5.711.13 1 [____ _ _ _ _ _ _ __ _ _ 124

THE UNIVERSITY OF MICHIGAN 3648-4-T 10 oo l l ~~~~700 1.t 1 + 0(t2) d00 - 2.34537. 11.13.17 00 The function c~ (t) is 00 00 00 00t d2 4 6 - co~(t )=d 1 + + +- + 0 0t ):d00 d00 0 0 0 and has the expansion o 1 11 2 6 571 c0(t) = d 1 t-t- + - t t 0 ~ L 2 4 6 3 3 5 3P5 7 3 378 1924952 1 12 + 10 t + ( (A.110) S ( i125 3 2 and finally 00 2 00 2 00 2 1 8872 01 N (t) 002 1(d 1 1. 00 00 0 1 4 4 6 47 8 305 365.7 38547 872 1587878 t 12 + 0(t14 (A. 11) 00.*7.11 3 5. 7.11. 13 125

THE UNIVERSITY OF MICHIGAN 3648-4-T From an examination of this last result it would appear that No0(t) can have no zeros for which it l < 4. When n = 1 the appropriate expansions can be obtained in a similar manner to the above, and because the coefficients now fall off more rapidly with increasing powers of t, a smaller number of terms proves to be sufficient. We have O t2 { 2 9 229 4 9125420 1 01 F2 _1 2 2229 (t d =- {1+ - t - 9 t 4 t6- +, 0 3 3552 3 537211 35.67.211.13 9 33527t11(13 17 4 () 1 01 t3 4 42386 n 189210 d = 1 2 + t - 6 t6 + O(t6)d 5 325.72 20 1 3_ —- --- --- 126751 3 13 1 dol - 1 d~+ 0t)2+(t Jd1 9 335.2711213 17 +(t d01= 10 0( with d01 - 0(t) dl for r > 9, giving r 3 2 543 4 42827 6 18922109 8 10 d0 1- t t t t + 0(t 2.52 230472 243245672 273 577 11 (A. 12) ______________________,126,__, _......,,__.__

THE UNIVERSITY OF MICHIGAN 3648-4-T Also, 01 t2 2 2 53 11 6 d - 1- t2 + t4 - 5 t6 (t8 01 p/1 5 1 p/ 2. 3 3. 5 2.3. 5 7 5.7 1 01 t4 2 8 4 6 d01 d = -- 12 1 -— t + 2 3 t + (t d /3 2.232 5.7 30 5 6 - t8 d 01 1+0(t 2 ] d 01 P/5 2.335.1 1 001 ( 1 F t 32 226 218725910 c d01 W4 7 1 — +{+4 - t6+ - t8+ O 10 0 1 L 52 52 345572 34587 4112 J (A. 13) 2d012 N (t) = 2 (d1) {1 + + t 6 - 114 01 3 1 7 3.567 3.5874 69336 10 0 t12) 30 4 4) 5 07.411.13 and from this last result it is obvious that N01(t) can have no zeros for which It > 5. -, 127

THE UNIVERSITY OF MICHIGAN 3648-4-T APPENDIX B THE RAYLEIGH SERIES FOR A DISC As noted in~ 5, the problem of diffraction by a circular disc (and the related problem of diffraction by an aperture) can be solved by methods other than those involving spheroidal coordinates, and this is particularly true at the low frequency end of the spectrum. Thus, for example, using an integral equation approach first proposed by Jones (1956), Bazer and Brown (1959) have derived a sequence of terms in the low frequency expansion for a circular hole and from this the solution for diffraction by a disc can be found by Babinet's principle. For a plane wave normally incident on a hard disc the expression for the far field amplitude is 1 f(r, 7) = - - sinh (ct cos 0) gl(t) dt (B. 1) (see Bazer and Brown, 1959) where gl(t) is given by the integral equation tg (t) =t sinh ct + i — in h [c(t-s) cosh ct sin h cs} (s) d 7T 1 t-s S gl() ds, -1 3 and since the kernel is of order c for small c, it is a straight-forward matter to obtain the expansion 2 3 4 5 2 s M )=ctl +2 t2 2ic3 1 4 t 2ic5 4 +t2 g(t)=ct 1c - + -c + -- + 225 L__I 9 120 ---

THE UNIVERSITY OF MICHIGAN 3648-4-T + 6 t6 4 2i c 71 t2 t4 c6(5040" 8 17.27 + 35 52+ 2 + 2520) 8/ t 28 2 2\ 2 c9 /8 4 + _ _ t 2e + 8-_ +c 362880 -202 4052 2 + 27575 729 31 2 t4 t6 10 1 t 2 + + 0(. + 198450 + 14175 +136080) j (B.3) If f(rj, 7) is now written in the form f(rTr) -2i A P (cos 0) (B.4) n=0 the integral relation ect Cos 0 P (cos 0) d(cos 0) = 2 (-i) jn(ict) (B. 5) -1 n n can be combined with equation (B. 1) to give ni r 1 A =- (2n+l)(-i)n+ i -(-1) j n(ict) g(t)dt, n 2n 1 0 implying that A =0 n for n even, and A n ( n+)-i)n j (ict) g1(t) dt (B. 6) for n odd. oreover, for n odd. Moreover, ________________________ 129 ________________________

THE UNIVERSITY OF MICHIGAN 3648-4-T ct (ct) 2 (ct)4 (t)8 10 10, 3ct) = 1 10 + 280 +15120 + 1330560 +(c ) and hence, by sub(ctstitution of (B. 3) into (B. 6), J3(it) —i 2i35 2 4 6 8 ++0i A (ic - i 31 1+ 825 792 61776 7413120 5 t2 4 \ 6 87 05(ic) ci 1 + + c 25 - + O(c ) 5,(ict) =i 10395 26 1560 159120 24186240 and hence, by substitution of (B. 3) into (B. 6), C 4 2 2i 3 3 4 2i 5 7 A -1 - 1 c + c + -26c + - c 1 8 4- 6 2 i 7 403 81 2 + c 7858620 84 c + c /9 1 (1824 + 0(c 10 4018225 \ 104104 2 729 61226225 2 c. c J 10 2 2i 3 28 4 2i 5 61 A i6615 - 1- l+-c + c + c +-c 3- 757r 63 9 2673 2025 +1( 360 4 >6 2i 7 31583 81 11011 2 - 7858620 88 225 8 + 2i 9 1 1321171 4 10 + 18225 104104 C 729 2 2 +( 7r 729 226225 C 56 2 2i 3 7 4 2i 5 29 A - i 1+ + C c + c S 6615w1 35-1 9 71-5 w 975 130

THE UNIVERSITY OF MICHIGAN 3648-4-T +1 ( 396 4 6 2ic7 5087 + 81 14365 2 + 1289925 14 1001 4 8 2i c9 1 [ 47938 4 \ 10 l 14 1001 42ic _ 2925 \ 610470 2J 7r 729 193375 ~2J lc ) The corresponding convergence coefficients ja n defined in the manner of equation (7) are shown in Table XIX and plotted in Figure XVIII. It will be observed that for n = 1, 3 and 5 the curves are almost identical. This is also true for larger values of n, and for large r it would appear that the common asymptotic limit lies somewhere between 2. 0 and 2.3. It is therefore concluded that all the A for odd n have the same radius of convergence for their low frequency expann sions, and accordingly the radius of convergence of the Rayleigh series for the hard disc lies within the range 2. 0 to 2 3. The significance of this result in terms of the amplitude coefficients Ao on is most easily seen by observing that A =,drS (-ic,-1) I or or A d o- A (B.7) n / n N (-ic) or ror as a consequence of equations (12) and (B. 4). Since S is free of singularities in the complex c plane, whilst Nor has no zeros which can affect the discussion, the radius of convergence of the low frequency expansion for A is equal to the sma n lest radius of convergence for the individual A (r = 1, 3, 5,....), and the fact that 131

THE UNIVERSITY OF MICHIGAN 3648-4-T TABLE XIX CONVERGENCE COEFFICIENTS FOR A HARD DISC rr r r 1 Co oo oo 2 2- 500 2 512 2.504 3 2 418 2.418 2 418 4 3-006 3 126 3. 179 5 2.191 2.205 2.211 6 2 466 2.452 2 447 7 2. 317 2.346 2. 352 8 2.200 2.178 2 184 9 2.802 2.714 2. 686 132

THE UNIVERSITY OF MICHIGAN 3648-4 -T \I TO I^~~~I _ - (D, - Z/ I, I II IC) _..... Q II0.'~~ u (Da)~~~~ rOW)~~~~~~ do~ r) Cz O K ADI _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 3 3

THE UNIVERSITY OF MICHIGAN 3648-4-T this is true for all (odd) n accounts for the similarity of the various curves shown in Figure XVIII. It follows immediately that one of the A must have a radius of convergence between 2* 0 and 2. 3, which is in agreement with the results of the more rigorous analysis of ~5. Turning now to the problem of a soft disc, the far field amplitude for a plane wave which is normally incident is 2cr1 f(r 7r) = 2c cosh (ct cos 0) g2(t) dt (B. 8) 0 where g2(t) is given by the integral equation g2(t) = cosh ct + sinh c(t-s)] () ds. (B. 9) 7i -1 The integrand is here of order c for small c, so that a solution by iteration is again possible, and though it proves desirable to calculate a somewhat larger num ber of terms than Bazer and Brown (1959) have provided, it is a straight forward matter to show that 2 2 3 2 g2(t)=l1 — + c... + c 4t v2 + 414 ___ 2ic5T t4 ( 2 45 37 7T 3 3 37 16 16 5 6 t6 t4 2 8 34 + -4 - 2+ + C'+ t7r 97 \30 7 3 1357 a 134.......

THE UNIVERSITY OF MICHIGAN 3648-4-T (64 80 508 2i c _4 1 - 6 9 4 2025 7r 2 30 7r2 4 + -8 44 + 11 1_64 32 4 4 4 8 _ 2 +1575/ 6 3 74, 2 -2205 \37r 1357r 7r / ^ 37r 9 71 / J 8 Jt8 t6 4/ 2 5 2(32 8 + c — +t -- t 6 1260 7r 15 7r 378 ) - 37 57r 746 \ 256 448 1856 1112 A 14 75 8 6 4 14175r 7r fr 9 7r 675 4 33075 7r 2 i c9,^ 6t( 1 -2 1 2 2 16 23 2i7019{ 2 1701o2+ 5670 1 \I 9! 1260 7r 4 r 945 r 567 t2 32 256 50 11 \ 256 512 k G 6 13564 2 19845 8 6 \3 7 135 r 567 / \ 7r 97' 7936 2864 2 \ 10j t10 t +-+ + -10+ 2025 74 357212 25515 10! 90720 2l +6/ 1 11 \ t4 8 26 407 4v - 2) t 6 4 21 315 7 34020 - \ 15 7 315 4 141750 T t2 128 1184 22448 3844 37r8 135 6 42525 4 496125 2 1024 256 14272 78224 2852 0c 11 10 8 6 4 7r. r 675 7 1275757 626875 7rJ 135 _

THE UNIVERSITY OF MICHIGAN 3648-4-T If the far field amplitude is written in the form (B. 4), the coefficients A n for the soft disc are given by the equation An = 2(2 n + 1) (-i)n+ 1 + (-1)j jn(ict) g2(t) dt 0 and hence 1 A =n (2 + 1) (i)n+1 j(ict) g(t) dt (B. 11) for n even, and A =0 n for n odd. Since t (ct)2 (ct)4 (ct)6 (ct)8 (ct)1 ( c 12 (ict) 6 120 5040 362880 39916800 + O(c ) (ct) 2+ (ct)2 (ct)4 (ct)6 (ct)8 j2 (lct) =15 1 { 14 + 504 33264 3459456 + 518918400 + O(c) (i (ct)4J 1+(ct)2 + (ct)4 + (ct) + (ct)8 j4 (ict) 945- ( 22 1144 102960 14002560 (ct)10 12 2660486400 + c 0 c2 136

THE UNIVERSITY OF MICHIGAN 3648-4-T A =c _ i 1 2ic +4 2 2 2ic(4 1) 1 2ic A - 1 -..- -- - --- - + 16 16 2+ 4 2i c5/16 20 127\ - c. ir4 9 2 75w 4 9 2 202 64 32 4 \ 6 2i c7 64 _ _.e+, -- 6 3 74 972 2205, 7T 3r 112 464 278\ 3/256 512 9 4 6752 33075/ 8 9 6 9wr 675 / w 9w +7936 2864 + 2 8 4+ - 2551 2025 w4 35721 2 251 2i c / 256 64 3568 19556 713 \ -- w 6 4 4+ 82687 w 8 6 675 74 127575 82687 1024 _ 2560 18496 36928 254684 8 6 4 7.10 97r 675 7 35721 7r 2232562572 - 175175 c10 + 0(c ) 1715175 3 3 i - 2ic (4 12 2 2i 41 23 2 9w w 6 97r7 4/1 40 5 4 2ic 16 148 38 42 17 w 4 2 55 21wr 63/w 137

THE UNIVERSITY OF MICHIGAN 3648-4-T 64 704 7048 8 6 2i c7 64 \ 63 r 4 14175 2 4455 7T 7 6 637 14175 / 7 272 132 18377 \ /256 3712 20048 + _..2 - 1819125/ 4 21 7r 1752 181912 / 63 76 4725 r4 33472 2 8 2ic 9 256 4160 2 + - 7 8 6 36382572 33033 8 63 6 16064 48392 60638 + 5959453 2835w4 28066572 /1024 2048 137728 11255488 295857592 \ T10 7 78 4725 76 9823275 7r 22347950625 r2 78 c10 + o(c 11 558967 ic 2 ic5c 30 2 2i c3 /4 37 4 525rT c 2 99J \7r /16 64 140\ 4 2ic5 16 236 5627\ 7 33 7r,,\7 99 7r 6 4 0 4228 4719 2i c 64 46 299 7 18427 52 4719 7r 6 144 20872 256 256 5888 117 270 2 2425 8 99 6 138

THE UNIVERSITY OF MICHIGAN 3648-4-T +2925856 449536 1 8 675675 7r 47297257r 2011 2i c9 256 6592 11693936 906316 r 7r8 99 76 2027025 4 51081037r2 35155667 \ /1024 9728 2223808 33432534135 10 33 8 75075 ~ 33 78 75075w 150100864 + 742415224 2 N 10 127702575 42735943752 196222 54273594375 + 0(c11)}. The corresponding convergence coefficients | ar defined in the manner of equation (7) are shown in Table XX and plotted in Figure XIX. For n = 0, 2 and 4 the curves are far less regular than the analogous ones for a hard disc, and even when terms in c are included the curves still remain apart. It is therefore difficult to estimate the radius of convergence with any certainty, but bearing in mind that for sufficiently large r the curves must all approach a common limit, the more regular curve for A4 can be used to indicate the limit. It would appear from this that the radius of convergence of the Rayleigh series lies somewhere between 3. 0 and 3. 4. 139

THE UNIVERSITY OF MICHIGAN 3648-4-T TABLE XX CONVERGENCE COEFFICIENTS FOR A SOFT DISC 0 2 4 r la [1aa a a 1 1 571 1.571 1 571 2 2. 336 4. 003 7.863 3 2.795 3 393 3 679 4 3 106 3.710 3 973 5 3. 870 4.081 3.681 6 3. 903 3. 634 3. 499 7 4 280 4 138 3 836 8 4 458 4.223 4.258 9 4.033 3.631 3.521 10 4- 230 3. 563 3. 459 140 __ —

THE UNIVERSITY OF MICHIGAN 3648 -4-T o w q, I 2P -t.,) /. /'I / 0 vair~~~~~- 0 0,,. 0 / z // /' H i~o' u..0 14~. / z.... ~' / \.o' c - \J

THE UNIVERSITY OF MICHIGAN 3648-4-T The relationship between the coefficients An and Aon is given by.... ~ S (-ic,-1) A or or A (B.12) n n N (-ic) or or r=0 (cf equation B. 7) and by the same argument as before it follows that the limiting value of thej an for large r represents the smallest radius of convergence of the expansions for the individual A or It will be observed that this is in good agreement with the radius of convergence of A found by the more rigorous analysis of ~ 5, oo and this confirms that the singularity of A is the one which dictates the radius of oo convergence of the Rayleigh series for a soft disc. Unfortunately, for the particular problem of the soft disc difficulties are experienced with both the available methods for estimating the convergence, and it comes as no surprise to find that these have a common origin. In seeking the solution of equation (47) it was found that a large number of terms must be included in the expansion for d0 in order to determine the root, and even when terms as high 12 as c are taken into account it is still not possible to find the solution with an error of less than about 1 /o. This is due to the fact that for values of c I of the same order of magnitude as the root the leading terms in d00 do not decrease rapidly and, 0 indeed, the convergence coefficients for d0 behave in a similar manner to those 0 shown in Figure XIX. Such a behaviour is reflected in all the d00 and hence, through r equation (B. 12), in all the A for n = 0, 2, 4..., which then leads to difficulties in n __________________________________ 142 ___ —------------------

THE UNIVERSITY OF MICHIGAN 3648-4-T applying the intuitive method. This is in marked contrast to the case of the hard disc. 143 ___

UNIVERSITY OF MICHIGAN 1113 9015 I2 2009