AFCRL-66-182 71_ —'_-T 7133-3-T = RL-2148 Copy THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL ENGINEERING Radiation Laboratory Studies in Radar Cross Sections L - Diffraction and Scattering by Regular Bodies IV: The Circular Cylinder O. EINARSSON R. E. KLEINMAN P. LAURIN P.L.E. USLENGHI February, 1966 Scientific Report No. 3 Contract No. AF 19(628)-4328 Project 5635 Task 563502 Contract With: Air Force Cambridge Research Laboratories Office of Aerospace Research United States Air Force Bedford, Massachusetts Administered through: OFFICE OF RESEARCH ADMINISTRATION. ANN ARBOR

THE UNIVERSITY OF MICHIGAN 7133-3-T AFCRL-66-182 STUDIES IN RADAR CROSS SECTIONS L - DIFFRACTION AND SCATTERING BY REGULAR BODIES IV: THE CIRCULAR CYLINDER by 0. Einarsson R.E. Kleinman P. Laurin P.L.E. Uslenghi Scientific Report No. 3 Contract AF 19(628)-4328 Project 5635 Task 563502 February 1966 Prepared for AIR FORCE CAMBRIDGE RESEARCH LABORATORIES OFFICE OF AEROSPACE RESEARCH UNITED STATES AIR FORCE BEDFORD, MASSACHUSETTS

Unclassified Security Classification - -- DOCUMENT CONTROL DATA - R&D (Security claesiflcation of title, body of abstract and indexing annotation must be entered when the overall eport ia clasified) 1. ORIGINATING ACTIVITY (Corporate author) -a. REPORT SECURITY C LASSIFICATION The University of Michigan Unclassified Department of Electrical Engineering 2b GROUP The Radiation Laboratory 3. REPORT TITLE Studies in Radar Cross Sections L - Diffraction and Scattering by Regular Bodies IV: The Circular Cylinder 4. DESCRIPTIVE NOTES (Type of report and inclusive date) Scientific Report No. 3 5. AUTHOR(S) (Lest nrme, first name, tnitil) Einarsson, Olov Laurin, Pushpamala Kleinman, Ralph E. Uslenghi, Piergiorgio L.E. 6. REPORT DATg 7. TOTAL NO. OF PAGES 7b. NO. OF REFS February 1966 323 168 Se. CONTRACT OR GRANT NO. *4. ORIGINATOR'I REPORT NUMBER(S) AF 19(628)-4328 7133-3-T b. PROJECT NO 5635. skb. OT 5ER R PORT N O($S) (Any other nmrbere that may be aseigned Task 563502 dai rpot. d. AFCRL-66-182 10. A VA IL ABILITY/-IMITATION NOTICES Distribution of this document is unlimited. 11. SUPPI-EMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY Air Force Cambridge Research Laboratories Office of Aerospace Research, USAF ____Bedford. Massachusetts 13. ABSTRACT A survey of electromagnetic and acoustical scattering by a circular cylinder is performed. Theoretical methods and results for infinite and semiinfinite cylinders, and experimental ones for finite cylinders, are included. Only time-harmonic fields are considered, and dielectric cylinders are not taken into account. DI DD I.AIN 64 1473 Unclassified Security Classification

Unclassified Security Classification 14. LINK A LINK B LINK C KEY WORDS - ROLE WT ROLE WT ROLE WT Electromagnetic Acoustical Scattering Circular cylinders Theoretical Experimental I ii IIIII INSTRUCTIONS 1. ORIGINATING ACTIVITY: Enter the name and address of the contractor, subcontractor, grantee, Department of Defense activity or other organization (corporate author) issuing the report. 2a. REPORT SECURITY CLASSIFICATION: Enter the overall security classification of the report. Indicate whether "Restricted Data" is included. Marking is to be in accordance with appropriate security regulations. 2b. GROUP: Automatic downgrading is specified in DoD Directive 5200. 10 and Armed Forces Industrial Manual. Enter the group number. Also, when applicable, show that optional markings have been used for Group 3 and Group 4 as authorized. 3. REPORT TITLE: Enter the complete report title in all capital letters. Titles in all cases should be unclassified. If a meaningful title cannot be selected without classification, show title classification in all capitals in parenthesis immediately following the title. 4. DESCRIPTIVE NOTES: If appropriate, enter the type of report, e.g., interim, progress, summary, annual, or final. Give the inclusive dates when a specific reporting period is covered. 5. AUTHOR(S): Enter the name(s) of author(s) as shown on or in the report. Enter last name, first name, middle initial. If military, show rank and branch of service. The name of the principal author is an absolute minimum requirement. 6. REPORT DATE: Enter the date of the report as day, month, year, or month, year. If more than one date appears on the report, use date of publication. 7a. TOTAL NUMBER OF PAGES: The total page count should follow normal pagination procedures, ie., enter the number of pages containing information. 7b. NUMBER OF REFERENCES: Enter the total number of references cited in the report. 8a. CONTRACT OR GRANT NUMBER: If appropriate, ente the applicable number of the contract or grant under which the report was written. 8b, 8c, & 8d. PROJECT NUMBER: Enter the appropriate military department identification, such as project number, subproject number, system numbers, task number, etc. 9a. ORIGINATOR'S REPORT NUMBER(S): Enter the official report number by which the document will be identified and controlled by the originating activity. This number must be unique to this report. 9b. OTHER REPORT NU!MBER(S): If the report has been assigned any other report fIumbers (either by the originator or by the sponsor), also enter this number(s). 10. AVAILABILITY/LIMITATION NOTICES: Enter any limitations on further dissemination of the report, other than those imposed by security classification, using standard statements such as: (1) "Qualified requesters may obtain copies of this report from DDC." (2) "Foreign announcement and dissemination of this report by DDC is not authorized." (3) "U. S. Government agencies may obtain copies of this report directly from DDC. Other qualified DDC users shall request through,! (4) "U. S. military agencies may obtain copies of this report directly from I1DC. Other qualified users shall request through (5) "All distribution of this report is controlled. Qualified DDC users shall request through, If the report has been furnished to the Office of Technical Services, Department of Commerce, for sale to the public, indicate this fact and enter the price, if known. 1L SUPPLEMENTARY NOTES: Use for additional explanatory notes. 12. SPONSORING MILITARY ACTIVITY: Enter the name of the departmental project office or laboratory sponsoring (paying for) the research and development. Include address. 13. ABSTRACT: Enter an abstract giving a brief and factual summary of the document indicative of the report, even though it may also appear elsewhere in the body of the technical report. If additional space is required, a continuation sheet shall be attached. It is highly desirable that the abstract of classified reports be unclassified. Each paragraph of the abstract shall end with an indication of the military security classification of the information in the paragraph, represented as (TS), (S), (C). or (U). There is no limitation on the length of the abstract. However, the suggested length is from 150 to 225 words. 14. KEY WORDS: Key words are technically meaningful terms or short phrases that characterize a report and may be used as index entries for cataloging the report. Key words must be selected so that no security classification is required. Identifiers, such as equipment model designation, trade name, military project code name, geographic location, may be used as key words but will be followed by an indication of technical context. The assignment of links, rules, and weights is optional. I pI Unclassified Security Classification

l i THE UNIVERSITY OF MICHIGAN - 7133-3-T ABSTRACT A survey of electromagnetic and acoustical scattering by a circular cylinder is performed. Theoretical methods and results for infinite and semi-infinite cylinders, and experimental ones for finite cylinders, are included. Only time-harmonic fields are considered, and dielectric cylinders are not taken into account. iii

9MISSIN

I THE UNIVERSITY OF MICHIGAN 7133-3-T TABLE OF CONTENTS ABSTRACT iii I. INTRODUCTION 1 1.1 Preliminary Remarks 1 1.2 Brief Historical Survey 2 1.3 Scattering from Finite Cylinders 4 II. EXACT SOLUTIONS FOR AN INFINITE CYLINDER 7 2.1 Precise Formulation 7 2.2 Plane Wave, Spherical Wave, and Line Source Incidence 21 2.3 Dipole Sources 56 2.4 Scattering of Evanescent Waves 87 ml. LOW FREQUENCY APPROXIMATIONS FOR AN INFINITE CYLINDER 92 IV. HIGH FREQUENCY APPROXIMATIONS FOR AN INFINITE CYLINDER 98 4.1 Geometrical and Physical Optics Approximations 98 4.2 Geometrical Theory of Diffraction 107 4.3 Asymptotic Expansions of Exact SOlutions 115 4.4 Impedance Boundary Conditions 133 4.5 Radar Cross Sections 139 V. SCATTERING FROM A SEMI-INFINITE CYLINDER 145 5.1 Electromagnetic Scattering from a Perfectly Conducting SemiInfinite Solid Cylinder 146 5.2 Scattering of a Scalar Plane Wave by a Semi-Infinite Cylinder 162 5.2.1 The Boundary Condition au/ p = 0 when the Angle of Incidence is Neither 0 Nor 7r 162 5.2.2 The Boundary Condition au/ap when the Angle of Incidence is 0 165 5.2.3 The Boundary Condition u = 0 166 5.2.4 The Far Field 168 5.2.5 Numerical Computations for the Boundary Condition au/ap = 0 171 5.3 Radiation of Sound from a Source Inside a Semi-Infinite ThinWalled Tube 176 5.3.1 General Solution 176 5.3.2 Field Inside the Tube 178 5.3.3 The Far Field 185 5.3.4 Cylindrical Resonators with an Open End 196 m v

THE UNIVERSITY OF MICHIGAN 7133-3-T Table of Contents (cont'd) 5.4 Scattering of a Plane Electromagnetic Wave from a SemiInfinite Thin-Walled Tube 197 5.4.1 General Solution 197 5.4.2 Field Inside the Tube 202 5.4.3 The Far Field 203 5.4.4 Axial Incidence 206 5.5 Electromagnetic Radiation from a Source Inside a Semi-Infinite Thin-Walled Tube 209 5.5.1 General Solution 209 5.5.2 Field Inside the Tube 214 5.5.3 The Far Field 224 5.6 The Wiener-Hopf Factorization 235 5.6.1 Explicit Expressions 235 5.6.2 Low Frequency Approximations 243 5.6.3 High Frequency Approximations 245 5.6.4 Numerical Computations 254 VI. EXPERIMENTAL DATA 264 REFERENCES 312 vi

THE UNIVERSITY OF MICHIGAN 7133-3- T I INTRODUCTION 1.1 Preliminary Remarks This is the fourth in a series of reports on electromagnetic and acoustical scattering by selected bodies of simple shape. The previous reports dealt with the sphere (Goodrich et al, 1961), the cone (Kleinman and Senior, 1963) and the prolate spheroid (Sleator, 1964). The choice of the circular cylinder was dictated by various considerations. Firstly, the large number of theoretical and experimental results which have been published on this shape during the last one hundred years is in itself sufficient to justify the writing of the present report. Secondly, the circular cylinder has often been used for the development and testing of approximation methods of general applicability, in both the low and high frequency limits. Finally, it is a shape of considerable interest in practical applications such as scattering by the central part of a missile and radiation and scattering by cylindrical antennas. In this report, the emphasis is placed on scattering rather than on radiation problems, i.e. the source is usually located off the surface of the cylinder. Radiating slots and gaps in the cylinder surface are not considered, and the interested reader should consult the various excellent monographs on this subject, such as the books by King (1956) and by Wait (1959). Although the case of an electric dipole on the surface of the cylinder is examined, the problem of the equivalence of dipoles and slots is not discussed. Only the case of time harmonic fields is considered explicitly. This choice is justified by the fact that an arbitrary field can always be decomposed into the sum f monochromatic waves by Fourier analysis, and that most of the literature does ineed consider time harmonic fields only. However, there exist a few works on difraction of pulses by circular cylinders, for example Friedlander's (1954) and Baraat's (1965) in which a Laplace transform method is adopted. The propagation of I 1

THE UNIVERSITY OF MICHIGAN 7133-3-T acoustic pulses from a circular cylinder has been investigated by Barakat (1961). Useful information is also contained in a book by Friedlander (1958). Chapters II, m and IV are devoted to exact solutions, low and high frequency approximations for a cylinder of infinite length. Special emphasis has been given the equivalence between acoustical and electromagnetic boundary value problems. Wherever possible, the case of an impedance boundary condition has been consid dered, and the results for soft, hard and perfectly conducting cylinders have been derived as limiting cases. In Chapter V, the scattering by a semi-infinite circular cylinder is investigated. Chapter VI is devoted to a brief survey of measurement techniques and experimental results. It seemed reasonable to include such measurements on (necessarily) finite cylinders, even though a theoretical chapter on finite cylinders does not appear in this report. The principal reason for the exclusion of such a chapter is that a satisfactory theory for a cylinder of finite length does not yet exist; most works on this subject deal with the two limiting cases of a thin long cylinder (a wire) and of a short fat cylinder (a disc), and a comprehensive treatment of either one of these two cases would require a separate book. A brief outline of the main existing works on scattering by a finite cylinder is given in section 1.3. A considerable effort has been made to take into account all relevant contributions to the subject of this report and to give due credit to bibliographical sources. However, a complete bibliographical listing has not been attempted. The authors are indebted to their colleagues of the Radiation Laboratory for criticism and comment. This report has been typewritten by Miss C. Rader, and the figures have been drawn by Mr. A. Antones. 1.2 Brief Historical Survey Some of the following historical remarks are taken from a recent paper by Logan (1965). They illustrate the principal studies on scattering by circular cylinders until the beginning of World War II. The numerous contributions which have 2

THE UNIVERSITY OF MICHIGAN 7133-3-T appeared in the technical literature during the last twenty-five years are adequately described in the following chapters of this report, and therefore will not be mentioned in this section. The first important study on the scattering of waves by a circular cylinder is contained in section 343 of Rayleigh's Theory of Sound (Strutt, 1945), which was completed in the Spring of 1877. In that section, Rayleigh showed how to separate the wave equation in circular cylindrical coordinates, and carried out the analysis explicitly for the case of a plane sound wave normally incident on a cylinder of gas of given density and compressibility and with radius small compared to the wavelength. Four years later, Rayleigh published a paper (Strutt, 1881) in which, on the basis of Maxwell's theory of electromagnetism, he solved exactly the problem of the scattering of electromagnetic waves by a dielectric cylinder; he reconsidered this problem almost thirty years later (Strutt, 1918). Rayleigh's solution is valid for normal incidence (the case of oblique incidence has been investigated only recently by Wait (1955)), and it is known that for this case the acoustical and electromagnetic problems are essentially equivalent, as Rayleigh showed in 1897. A few years earlier, the first study of the scattering of plane waves by a perfectly conducting infinite circular cylinder had been published by J. J. Thomson in his Recent Researches in Electricity and Magnetism (1893). In 1905 and 1906, Seitz published two papers on diffraction by a metal cylinder, which contain various numerical results. In those same years, Debye (1908) succeeded in proving that the exact solution for the circular cylinder leads to results consistent with the predictions of geometrical optics. Also, Nicholson (1912) published an interesting work on the pressure exerted on a perfectly conducting cylinder by an incident electromagnetic wave, and Bromwich (1919) discussed the separabili of Maxwell's equations in orthogonal curvilinear coordinates, of which the circular cylindrical coordinates are a special case. Bromwich's paper is especially inter esting because he does not restrict himself to harmonic time dependence. 3

THE UNIVERSITY OF MICHIGAN 7133-g-T It was not until the advent of the radar that a new series of studies on scattering by cylinders began. In 1941, the well-known book by Stratton appeared, and in that same year a theoretical report by Moullin and Reynolds was distributed, in which the case of plane waves normally incident on infinite circular cylinders was considered, and the numerical data obtained from the exact solution were displayed in a number of graphs. 1.3 Scattering from Finite Cylinders The purpose of this section is to provide the reader with some bibliographical references on scattering and radiation by finite cylinders. Although these two problems are closely related, the latter has received much more attention in the literature owing to its importance in antenna applications. Hallen (1938) was the first to obtain an approximate solution for a thin cylindrical antenna, i. e. for values of the cylinder radius much smaller than the wavelength. He derived an integral equation for the unknown current distribution on the surface of the cylinder, and solved it approximately by an iteration technique. Since Hallen's attempt, several authors have introduced modifications in the integral equation: Van Vleck, Bloch and Hammermesh (1947), King and Middleton (1946), Gray (1944), Duncan and Hinchey (1960), Kapitsa, Fock and Wainshtein (1960), among others. Van Vleck, Bloch and Hammermesh (1947) presented two independent methods for deriving approximations to radar back scattering from thin cylinders. In solving for the current induced on the thin cylinder, they assumed that this current consists of four trigonometric functions, two of which correspond to forced terms, that is, to the voltage impressed by the incident wave, whereas the remaining two were attributed to resonant parts, i.e. to the current present on the cylinder at the resonant frequencies. In the first of their two methods, the end-condition is imposed for choosing two of the parameters in the expression for the current, viz., the current vanishes at the ends of the cylinder. The other two parameters are determined by imposing the conservation of energy. In the second 4

THE UNIVERSITY OF MICHIGAN 7133-3-T method, two of the parameters are found by equating the terms in Hallen's equations corresponding to the voltage impressed by the incident wave and the other two are determined using the end-condition with an iterative procedure. The reader is referred to the original paper for details and for a discussion of the advantages of both methods. Storer (1951) and Tai (1951) have independently applied variational methods for calculating the scattering cross section. Tai expressed the back scattering cross section as a function of the unknown current on the cylinder. This function is transformed with the help of Hallen's integral equation into one which is stationary in the unknown current function. By substituting various trial functions for the current into the stationary functional and then determining the free parameters by the Rayleigh-Ritz method, the back scattering cross section is estimated. The trial function used by Tai is a linear combination of the currents on the cylinder at the first and second resonant frequencies, and it had been previously adopted by Van Vleck et al (1947). Williams (1956) has used an extension of the Wiener-Hopf technique for calculating the scattering of a plane sound wave by a finite cylinder. Williams' method is parallel to the method of Jones (1952); it involves the Laplace transformation of the differential equation before applying the boundary conditions and the reduction of the problem to the solution of two complex integral equations. Although these equations cannot be solved exactly, an approximate solution has been obtained under the assumption kl >> 1 (i = length of the cylinder). Williams has also obtained explicit expressions for the end-condition by taking into consideration the resonance of the system. Wilcox (1955) has conducted a detailed study of scattering of electromagnetic radiation by finite cylindrical shells. He has used the integral equation method and has obtained approximations to the scattering cross sections in terms of the tangential electric field on the axial extension of the cylinder surface in a form which I I 5

THE UNIVERSITY OF MICHIGAN 7133-3-T is stationary with respect to variations about the correct values. The tangential component of the electric field due to scattering by a semi-infinite cylinder is used as a trial function in the stationary expression. Wilcox has also derived variational expressions for the far field in the thin cylinder approximation. A high frequency asymptotic solution of scattering by a solid conducting finite cylinder is given by Kieburtz (1962). However, his results are incorrect except for the first order term, because they are based on an erroneous assumption concerning * the locations of the singularities of the Fourier transforms of the field components Numerical methods are available for calculating the current distribution from integral equations. Govorun (1962) has obtained numerical results for the symmetric part of the surface current on a solid cylinder excited by a plane wave at broadside incidence. The length to radius ratio varies from 6 to 65536 and the length from X/8 to 7X/4. The paper also contains results for a cylindrical antenna with a circumferential gap of finite width. His solution converges quite rapidly for thin cylinders. Williams (1956) has included a few numerical results in his previously-mentioned paper. Much of the work done in the area of finite cylinders has been devoted to thin cylinders in consideration of the practical applications to antennas. King (1956) has written a book which gives an extensive treatment of this subject Kieburtz reports in a private communication that the method for construction of an asymptotic series expansion used in his recent paper (Kieburtz, 1965) can be applied to the cylinder problem. 6

THE UNIVERSITY OF MICHIGAN 7133-3-T II EXACT SOLUTIONS FOR AN INFINITE CYLINDER In this chapter, the boundary value problems of scattering of electromagnetic energy by an infinitely long circular cylinder are formulated and solved exactly. The relationship between vector and scalar problems is examined, and various types of sources are considered: plane waves, cylindrical waves, dipoles, and evanescent waves. It is assumed that the cylinder is made of a perfectly conducting material. In some instances, however, the more general case is considered in which an impednce boundary condition may be applied at the surface of the cylinder. The rationalized MKS system of units is employed throughout. * 2.1 Precise Formulation This section deals with the problem of finding the electromagnetic field external to an infinitely long (perfectly conducting) circular cylinder embedded in a uniform, homogeneous and isotropic medium of electric permittivity E, magnetic permeability 1p and zero conductivity, which medium may be taken as free space. The homogeneous Maxwell's equations 3E VAH = E -a-t (2.1) aH VAE = -, (2.2) govern the behavior of the electric field E and of the magnetic field H at all ordinary points in space, but do not describe the fields at the source points. By taking The following vector notation is used: vectors of arbitrary magnitude will be underlined, e. g. E; unit vectors will be denoted by carets, e. g. A; scalar products indicated by dots, e. g.. * E; and vector products by wedges, e. g. VA E. 7

THE UNIVERSITY OF MICHIGAN 7133-3-T the divergence of both sides of equations (2.1) and (2.2), and with the convention that at some time the fields may become solenoidal, which is certainly the case if, for example, E = H = 0, one finds the auxiliary equations e -t-oo — t-oo V-H = V E = 0. (2.3) Equations (2.1), (2.2) and (2.3) are satisfied by the incident or primary fields E_ and Hi, by the total (incident plus scattered) or diffracted field E and H, and therefore also by the secondary or scattered fields Es and Hs, which represent the disturbance introduced in the primary fields by the infinite cylinder. The presence of the cylinder is accounted for by requiring that on its surface the total electric and magnetic fields satisfy the impedance boundary condition: E- (E )= Zp$,, (2.4) where Z is the surface impedance and, a unit vector perpendicular to the surface of the cylinder and directed from the surface into the surrounding medium. The case of perfect conductivity corresponds to Z = 0. If the sources of the primary fields are specified, the surface impedance Z is given, and a radiation condition (which is necessary to ensure uniqueness) is assumed, then the boundary value problem is well set and may be formulated directly in terms of either the electric or the magnetic field. However, it is often advantageous to reformulate the problem in terms of auxiliary functions from which the field quantities may be derived through simple operations of differentiation. Such auxiliary functions, or potentials, may be chosen in a variety of ways (see, for example, Stratton, 1941). Following the procedure adopted by Kleinman and Senior (1963), we define E and H in terms of a vector potential A through the relations: E = VAVAA, (2.5) I 8

THE UNIVERSITY OF MICHIGAN 7133-3-T H = e -VAA. (2.6) at Maxwell's equations are satisfied by (2.5) and (2.6), provided that a2A VAVAVAA + 2cEVA' =0, (2.7) At that is, (VAVA+ El 2- = Vf, (2.8) at where f is an arbitrary scalar function of position and time. Formula (2.8) may be rewritten as (v2-e: 12)A = Vf, (2.9) at2 where V2 operates on the Cartesian orthogonal components of A. Any electromagnetic field can be derived from such a vector potential; in particular, there exists a potential which gives the field exterior to a conducting cylinder. Instead of obtaining the fields E and H from (2. 5) and (2. 6), we may use the E = -I at VAA, (2.10) H = VAVAA, (2.11) where A is still a solution of (2.9). It is then possible to express any electromagnetic field in the form: -A ^J

THE UNIVERSITY OF MICHIGAN 7133-3-T a E = VAVAA- a VAA2 -1 t 2' (2.12) a H = VAVAA + e V/\A ' -2 at 1 with A1 and A solutions of (2.9). Expressions (2.12) are obviously redundant, in the sense that a great freedom of choice is left for Al and A2. Such freedom can be used to construct the vector potentials from scalar potentials. Let us set Al = (2.13) 2 2 where c is a constant vector. It was proved by Whittaker (see, for example, Nisbet, 1955) that any electromagnetic field can be derived from (2.12) with the vector potentials restricted to the form (2.13), provided that the scalar functions 01 and 2 are two independent solutions of the wave equation (V - e 2 = 0. (2.14) The potentials A and A2 so determined are usually called electric and magnetic Hertz vectors, and denoted by Te and!T respectively. The electric Hertz vector )e originates a field which is characterized by the absence of a component of H in the direction of c (transverse magnetic (TM) field), whereas the magnetic Hertz vector TT originates a field for which E c = 0 (trans-m verse electric (TE) field). Except when otherwise stated, in the following we shall consider the particular case of monochromatic radiation. The propagation constant k in the medium surrounding the cylinder is then given by 2 k = w = 2, (2.15). I\ 1U

, THE UNIVERSITY OF MICHIGAN 7133-3-T where w is the angular frequency and X the wavelength. In the preceding relations, the operator 8/at is replaced by the multiplicative factor -iw. The time dependence factor e is suppressed throughout. If rectangular Cartesian coordinates (x, y, z) and cylindrical polar coordinates (p, O, z) connected by the relations x = p cos p, y = p sin p, z = z are introduced (Fig. 2-1) so that c = i is parallel to the axis z of the cylinder, then the field components defined by (2.12) are, in cylindrical coordinates: E = al+ j ao2 p 8paz p ' = 1 a 21 W. 2 p = p aaz- ap p 12 l z* 1 k2l Ez a2 k 2 az a2 WE H -8 2- pe p apaz p 8^ I 1 a22 ai Hp = p '5z + aiw a' a2o 2 H = - + k2 8 z2 2, (2.16) Observe that if V1 -0 then E - 0 (TE case), while if 2 =- 0 then H - 0 (TM case). The boundary conditions (2.4) at the surface p = a of the cylinder now be come: E = ZH,, z 0 E = -ZH, at p = a, or also, by (2.16): a~1 a2~1 a+ k2i a2 - a 0az 2 a20 aZ a0az Z A2 + 2 + k22 Ml ap az at p = a. 11

I x A i z THE UNIVERSITY 7133-3-T OF MICHIGAN (P1. 1, Z 1) \R / \\R r ((p,, z) I I - z I y P -- I /'._- _ — - - / / R =r-rI FIG. 2-1: GEOMETRY OF THE CIRCULAR CYLINDER 12

THE UNIVERSITY OF MICHIGAN 7133-3-T In order to satisfy these relations, in the case of perfect conductivity (Z = 0) it is sufficient to require that either k2 b 0 and - 0, 1 ap p=a p=a or 2- 0, and = 0 p=a whereas in the case in which Z f 0 both TE and TM fields must, in general, be present. However, if 01 and 02 are independent of z (two-dimensional problems, a/az 0), then the boundary conditions can be satisfied by requiring that either p=a ~1- 0, and (- Z i) 1 =0 0 - p=a The scalar wave functions 1 and b2 can be represented by linear combinations of the elementary wave functions, n h1) 2 h2 in +ihz h (1) (v\k -h ein Z (2.17) n, h (1) n where n is a real integer (n = 0, +1, t2, etc), and h a parameter which is, in general, complex and whose values may cover a discrete set as well as a continuum spectrum. If h assumes only one value, as it happens, for example, when the primary source consists of a plane wave, then ~1 and q2 are proportional to E and H, respectively. 13

THE UNIVERSITY OF MICHIGAN 7133-3-T In the case in which all primary sources are located within a finite distance from the origin r = 0 of the coordinate system, the fields E and H are required to satisfy the Silver-Muller radiation condition lic -r (VA) + ikr = 0 uniformly in r, (2.18) r — ) oo H where r = rr = x +yi +zi, and the angle e of Fig. 2-1 is restricted to the x y z range 0< 6 < 0 < ir- 6, with 6 arbitrarily small and constant. From this condition and from Maxwell's equations it follows that the fields are of the form ikr f(r) as r —o. r If the primary source is a line source parallel to the cylinder axis, then the problem is two-dimensional and condition (2.18) is no longer valid. In such cases it is sufficient to require that lim p1/OD - ik = 0 uniformly in ~, (2.19) p -4' ca \-p1 = 0 2 that is, L1 and '2 are of the form ikp f() E as p —o. Finally, in the case of plane wave incidence it is necessary to separate incis S dent from scattered fields; the scalar wave functions s1 and l2 which originate the 1 2 scattered fields satisfy the two-dimensional radiation condition (2.19). Two kinds of primary fields are of practical interest, and will be examined in detail in the following two sections of this report: plane waves, and sources located at a finite distance from the scatterer. The first kind is important in radar 14

THE UNIVERSITY OF MICHIGAN 7133-3-T scattering where the target is usually assumed to be illuminated by an incident plane wave. The second type is relevant to the case of an antenna mounted on the scatterer. The antenna may be taken to be a dipole, since more complicated sources can be considered as distributions of dipoles. On the other end, a plane wave is the limiting case of a dipole going to infinity in a direction perpendicular to its moment. From the strictly mathematical viewpoint, it is therefore sufficient to consider the simple case of a dipole source at a finite distance from the cylinder; the solution of the scattering problem for any given source distribution can subsequently be obtained by superposition. In order to arrive at the definition of dipole, let us consider the scalar point source defined by ikR e 0o R where R = jr-_1J = {(pcos0-p cosf ) +(psinm-p sin1)2 + (z-z 1) is the distance between the source point (p1, ~1 Zl) and the field or observation point (p,,z). An electromagnetic source at the point (p1, 0,, z1) can be derived from b in may ways. For instance, we may take either ^ = and 21 = 0 or =0 and 02 =- in equations (2.16). In the first case the primary field components are 0ikRA those of an electric Hertz vector I = -~ iz whereas in the second case the R ikR fields are originated from a magnetic Hertz vector - = eR. In the notation of Stratton (1941), Be and Em represent electric and magnetic dipoles of moment = 47rei and m = 47riz, respectively. z -^ 15

THE UNIVERSITY OF MICHIGAN 7133-3-T The components of the fields of an electric dipole (with moment p( )= 47re ) Z are: E= (z-z) [P-P cos(0- - P 1 1 R3 R R ) i k2 3ik 3 ikR = R3 -4 + 0 2 _ 3ik kR i ikR He (i-iwE si B, a w 1 fIIP kP -1 3ik\R +.... z H =. - z pole by a perfectly conducting infinite cylinder can be described by a scalar wave function, which is single-valued and twice-differentiable in each of the quantities 1' P, 0 ZP1. 01 z1 satisfies the wave equation (V +k2 ) = 0 and the boundary condition 16

THE UNIVERSITY OF MICHIGAN 7133-3-T / ikR \ p=a and originates a diffracted electromagnetic field given by equations (2.16) with ikR 1 R '1 ' 2 0 which satisfies the radiation condition (2.18). Similarly, the scattering of the electromagnetic field of an axially oriented magnetic dipole by a perfectly conducting cylinder can be described by a scalar wave function L2, single-valued and twice differentiable in p,, P1, 0, Zl' such that 2 2 ikR (V+k) =0, a 0 P p=a and that the diffracted field obtained by putting 1 = 0 and 02 = e /R+04 in (2.16) satisfies condition (2.18). If the surface of the cylinder has a nonzero impedance, then two independent scalar wave functions are needed to describe the scattered field produced by an axially oriented dipole source. In the general case of a dipole source arbitrarily oriented with respect to the cylinder axis, it is still possible to derive the solution of the scattering problem from two scalar wave functions 01 and 02, representing series of electric and mag netic multipoles oriented along the axis. An alternative but entirely equivalent approach consists in finding explicitly the Green's functions for the cylinder; such Green's functions are customarily grouped together into a dyadic Green's function, whose derivation for the case of a perfectly conducting cylinder is briefly outlined in section 2.3 of this report. The use of dyadic Green's functions is noteworthy I 17

THE UNIVERSITY OF MICHIGAN 7133-3-T for its elegance, and has been particularly advocated by Schwinger (1943, 1950), Morse and Feshbach (1946, 1953), Tai (1953, 1954a, b) and van Bladel (1964), among others. When the boundary value problem is two-dimensional, i is possible to formulate simultaneously both scalar and vector scattering problems. We shall limit our considerations to the case of a plane wave incident perpendicularly to the cylinder axis. A scalar or vector plane wave can be considered as a limiting case of a scalar point source or of a dipole removed to infinity. If we let z = 0 and 01 = r, then R ~ pl+pcos 0 when p1 becomes very large, and ikR ikp1 e e ikx R P1 as Pi -* oo, ikp1 so that if the source strength is renormalized by neglecting the factor e /p 1 a A 1 scalar plane wave propagating in the i direction is obtained. In the two-dimensional case, a/az - o, and equations (2.16) simplify, becoming J 2 E = I P P a ' 2 Ep = -im ap E = k 2 z 1 H =1-L p p aP a1 H0= iwe ap Hz = k 2 z 2 * (2.21) ikx By taking b1 = e and 2 = 0 in relations (2.21), one finds that Ei = k2 ikp cos z i E= P E = Hi = 0 z=0 — w 18

THE UNIVERSITY OF MICHIGAN 7133-3-T Hi = -\lesinpE, H = - 7cos0E i which are the components of an electromagnetic plane wave propagating in the direction of the positive x-axis with E = k e i and H - e k e i. Alterikx z y natively, by making 1= 0 and 2 = e in (2.21), one obtains that which are the components of a plane wave propagating in the i direction with E =; k2e ik and Hi= kiX i y z If a function ks (p, 0) which is single-valued and twice-differentiable in both variables p and O, can be found such that (V +k = 0 (2.22) (1 k (e i =0 (2.23) p=a lim 1/2 /a S \ p li p/2O -ik )L1 = 0 uniformly in p, (2.24) then: (i) the acoustic velocity potential of the field scattered by a cylinder with i ikx s r =?k/(6) in the presence of the plane wave b = e is given by b1; here 6 is the density (mass per unit volume) of the medium surrounding the cylinder and r is the normal acoustic impedance, i.e. the ratio of pressure to the normal component of the velocity at the surface of the cylinder (in particular, C = 77 = 0 in the case of a perfectly soft cylinder); 19

THE UNIVERSITY OF MICHIGAN 7133-3-T (ii) the electromagnetic field diffracted by a cylinder with relative surface impedance rl = Z VE/l in the presence of the plane wave E = - /V;; H = k ei ikx s is given by relations (2.21) with 1 = e +1, 2 =. If a function 02(p, 0) can be found which satisfies all requirements imposed on 51 except (2.23) which is replaced by ( + ikn)(e ikx+) = 0, (2.25) then: (i) the acoustic velocity potential of the field scattered by a cylinder with i ikx s 7 = wj/(Nk) in the presence of the plane wave i = e is given by b (in particu-1 lar, = = = 0 in the case of a perfectly rigid cylinder); (ii) the electromagnetic field diffracted by a cylinder with? = Z ~/ in the presence of the plane wave H = e7 E = k e is given by relations (2.21) with ikx s '1= o, 2= e +. l In closing this section, we point out that instead of dealing with the scattering problem from the point of view of differential equations, we could approach the problem from the equivalent viewpoint of integral equations. In the vector case, this approach would involve the use of dyadic Green's functions as previously mentioned. The introduction of dyadic Green's functions is avoided in the case in which one wants to determine only the electromagnetic field at the surface of the scatterer (Maue, 1949). Once the surface fields are known, the fields at any point in space may be obtained through an integration over the surface of the scattering body (Stratton, 1941). The use of Maue's integral equation in place of the differential (wave) equation usually represents a complication of the problem, which may be counterbalanced by two simplifications: (1) the number of independent variables is reduced by one and the introduction of a special space coordinate system is unneces sary, and (2) the integral equation is the only requirement imposed on the unknown - --- 20 _

THE UNIVERSITY OF MICHIGAN 7133-3-T function, that is, the boundary conditions are automatically satisfied. Maue's integral equation formulation is therefore particularly useful when the scattering problem involves boundaries which are not coordinate surfaces in a system of coordinates for which the wave equation is separable, but obviously loses most of its interest in the simple case in which the scatterer is an infinite circular cylinder. 2.2 Plane Wave, Spherical Wave, and Line Source Incidence In this section, the scattering of a plane electromagnetic wave by an infinite circular cylinder of radius a with relative surface impedance rn = Z ~/ is considered. The formulas which give the diffracted field components as infinite series of eigenfunctions are derived for the case of oblique incidence, and they are subsequently specialized to the case in which the incident wave propagates in a direction perpendicular to the axis of the cylinder. The transformation of the infinite series solutions into contour integrals in the complex plane is discussed. Particular attention is devoted to the case of normal incidence on a perfectly conducting cylinder. The behavior of both near fields and far fields is investigated in detail, for the cases in which either the electric or the magnetic incident field is parallel to the cylinder axis. Finally, the scattering of cylindrical and spherical waves is also examined. Let us consider the incident plane electromagnetic wave ~! o. o. n xik(x sina +z cos a) E = (-cosacos4 +sinI3 +sinacos3i )e (, E- x y z (2.26) iT * e ik(x sina+ z cosa) H= ie/.u (-cosasinm i -cospi +sina singi )e, - V x y Z which propagates in the direction of the unit vector i i A EAH A A k = = sinai +cosai, (O<a<7r) IEiAHiI x 21 -1

THE UNIVERSITY OF MICHIGAN 7133-3-T The incident electric field E forms the angle P with the (x, z) plane of Fig. 2-2. It is easily verified that, according to formulas (2.16), the scalar functions E1 i _ z = cos eikp sina cos 0+ ikz cosa 1 2 2 2 e k sin a k sine (2.27) Hi i z sin ikp sina cos 0+ ikz cosa k sin a k sina generate the incident fields (2.26). The scattered fields E and H may be derived from two scalar functions ~1 and O2 through formulas (2.16). The functions 1 and;2 are linear combinations of the elementary wave functions (2.17). Since the cylinder is assumed to be of infinite length, the dependence of the scattered fields on the coordinate z must be the same as for the incident fields, that is, both incident and scattered fields vary according to the factor e. Hence h = kcosa, and only the plus sign is considered in the exponential of (2.17). Furthermore, the wave functions containing J do not satisfy the radiation condition (2.19), and must therefore be disregarded, so that we finally have: OD s = H(1) in+ikzcosa aH (kpsina) e 1 = ' n n n=-oo (2.28) s = ()p in + ikzcos a E bnHn (kp sina) e n=-oo and therefore the scalar wave functions which generate the diffracted fields are given by: 22

I - THE UNIVERSITY OF 7133-3-T A z MICHIGAN k incident wavefront 1 X I 2a. FIG. 2-2: GEOMETRY FOR PLANE WAVE INCIDENCE. 23

----- THE UNIVERSITY OF MICHIGAN 7133-3-T ikz cosa OD '1 r =e 2cosBe i+ 1 A H (kp si nn a)e 2.- n n k sina L n=(2.29) i 0 i 0 + ikz cosa iina cos i2 = 2 = \/sina c e + 2k sina =-00 + i i B H(1)(kpsina)e 0}, n=-co where we set, for convenience,.n a = A, n 2. n k sina.n n k si2n n k sina (2.30) The coefficients A and B are determined by imposing the boundary conditions at the surface of the cylinder. One finds that (Levy and Keller, 1959): 1 A = - n A cosp3(J sina-inJ)(l7H(1)sina- iH () + nJ H(1)(ns + n n n n n n..na J 2n!n Cosa + sin3 -2 7r (ka) sin c (2.31) B n 1 A s in sina - iJ' )(H1)sina- ir H(1) si n n n n + rJ H() n n Q ncosa kasina J 7c 3 2n cosa ] r(ka) sina where 24

THE UNIVERSITY OF MICHIGAN 7133-3-T I (1) (1) (1) I )(1) I ncos r H(1)\2 A = (H sina - irH )(n H (1sina - i H )+ rn csin H H) n n n n \kasina n / (2.32 and the primes indicate the derivatives of the Bessel and Hankel functions with respect to the argument ka sina. In deriving (2. 31), the well known formula ) ikp sina cos 0 e o00 = i Jn(kp sina) eino n=-oo has been used. In particular, in the case of a perfectly conducting cylinder, one has that (A ) n r=O J (kasina) = - - cosj, H (ka sina) n (B ) n =0O J' (ka sina) (1= sin3. (2.33) H(1) (ka sina) n The components of the scattered fields are given by: oD s ikzcos.n in (1)-. E e ie ' iA cosaH - n n n-o L nB H(1 kp sina n J = ikzcosa ne nin(o nf H(1) + iB H(1)'} 0 n- kpsia n n nJ ni=-oo I 1 00 E = e sina i A H(l)ein~ z n n n=-oo (2.34) OD e1 e H1'nA Hs ~ ikzcos a ~. n (1) (1) I kHcs ee e H+iB cosaH, P E/ kp sin n n n n=-oD w 25

THE UNIVERSITY OF MICHIGAN 7133-3-T s r- ikz cos a n ino (1), nB^O~a m(l H = Do e 1 i e 'S{iA - Hkp He 3 _ n n - kp sina n CD s f ikz coso a i (1) ino Hz= I e sina iB H(1)e i n=-ao where the argument of the Bessel and Hankel functions and of their derivatives is kpsina, and A and B are given by (2.31). n n It is seen that if the cylinder is not perfectly conducting, then the z-component of the scattered electric (magnetic) field is different from zero, even if the z-component of the incident electric (magnetic) field is zero (see also Wait, 1955)*. Formulas (2. 34) become less complicated when the incident wave propagates in a direction perpendicular to the axis of the cylinder (normal incidence, a = 7r/2). 5 5 In such case, the function i1 generates a TM field and the function if a TE field, 1 2 whose components are obtained through relations (2.21): tln connection with this remark, it should be noted that the method given by Stickel (1962) for deriving the diffracted electromagnetic field corresponding to oblique incidence from the field corresponding to normal incidence is valid only for perfectly conducting cylinders. 26

THE UNIVERSITY OF MICHIGAN 7133-3-T E = cos 3 z = — n-oD.n -f(1) ino i A H( li(kp) e n n CD p -lp kp - n n n=-oo H8 = Iicosj3 inA H() (kp)ei, Hio no n n=-0o -TMfield (2.35) Hz = / sing l i BH(k) e zn n=n=-w *I OD ES sin8 I n (1) ino n=-oo.9 TE field where we set: J (ka) - iJ(ka) n,n n (k1) r1a) H (ka)- irH (ka) n n B n J (ka) - ir J (ka) n n H(1)(ka)- 1 H(1)(ka) n n (2.36) I The scattered electromagnetic field for normal incidence is thus given by the superposition of a TM field proportional to cos, and a TE field proportional to singJ; we can therefore limit our considerations to the particular cases 3=0 (E parallel to the cylinder axis) and B = r /2 (H parallel to the axis). 27

THE UNIVERSITY OF MICHIGAN 7133-3-T From formulas (2.35), one derives the following asymptotic expressions for the components of the far scattered field: -i ikp Es ^ r e cos A EA cos non0 [ + (l/kp), z n e co p7TOD H5" a f7iEs5+O(1/kpk Hs. 4 r-00 * HA e cosnc +O(1/kp TM field for kp -; *7ro s 2 4 ikp *D- eikp H e sinf3 ~E B cosng7 EI+O(1/kp] E - 2/r e sinf nen B cosn0. 3/2 +0(1/kp =0 J(kp ES H7 [l+0(l/kp, TE field for kp -+ co The infinite series solutions which have been obtained are of practical value as they stand only when the radius of the cylinder is not large compared to the wavelength. The number of terms of each series required to give a good approximation to the infinite sum is of the order of 2ka, so that numerical results are easily obtained only when ka is not very large compared to unity. The procedure commonly adopted in the case of large values of ka consists in replacing the infinite series I 28

THE UNIVERSITY OF MICHIGAN 7133-3-T by contour integrals, which may be evaluated asymptoticallyf. As an example, let us consider the case of a perfectly conducting cylinder (r7 = 0) with E parallel to the axis. From formulas (2.35) and (2.36) it follows that ZO J (ka) n n (1) E s =- 1 (1) H )(kp)cosn, (2.37) n=0 H (1)(ka) n n where E = 1 and E = 2 for n >1. Treating the summation over n as a residue O n series, the summation is replaced by a contour integral C in the complex v plane (Fig. 2-3) taken in the clockwise direction through the origin and around the poles at v = 1,2,..., giving: J (ka) H((kp)cos v i -iv E = - ve dv. (2.38) z H (i) (ka) sin vi Similarly, in the case of a perfectly conducting cylinder with H parallel to the axis, one has that D J'(ka) (1) H_ =- 7____1 -- H (kp)cosn, (2.39) n=0 H (ka) n which becomes: s J(ka)H (kp) cosv0 -(iv H -iFe/ \ I -e dv. (2.40) vH (ka) sin rnv "The contour integral solutions may also be introduced directly, without making use of the infinite series solutions (see, for example, Clemmow, 1959a). 29

THE UNIVERSITY OF MICHIGAN 7133-3-T Im v D E O& 0 F G Re v 0 IA B FIG. 2-3: THE COMPLEX v-PLANE m 30

1 THEH UNIVERSITY OF M~lICHHIGAN I 7133-3-T The current density J at the surface of the conducting cylinder is given by: J = PAHIp=a so that when E is parallel to the cylinder axis J = J i = i(H+H )pa az z z 0 0 and when H is parallel to the axis A 2 z 7rka OD Vg7; je.n cos nO n=O H (ka) n (2.41) J =B j 0 =- (H + H p=a = ka cosn(1) ra n=O n H()(ka) n (2.42) In general, for a perfectly conducting cylinder and arbitrary direction of incidence: J = 2 ikzcosa n cos.n z 2rkasina EFT e - -^ n.-1 -. % L 00D 2 + cotgasinm3. n=l 1 sin no H (ka sina)J n (2. 41a) 00 2i ikz cosa cos i n ka A s eik Z cosi e ) (2. 42a) -7rka n0 n H (kasina) n The surface current densities J of formula (2.41) and JB of formula (2.42) may be z rewritten in the form:'" * The signs of the right hand sides of formula (13.4) in the book by King and Wu (1959) and of formulas (4) and (5) in the English translation of Goriainov's paper (1958) appear to be incorrect. 31

THE UNIVERSITY OF MICHIGAN 7133-3-T 2i ___ _cosy0 2 2= C 0 e dv (2.43) -iv 2 os rW 2 H H (ka)sin vr The path C of integration that appears in formulas (2.38), (2.39), (2.43) and (2.44) may be deformed in various ways in the complex v plane, to give contour integrals which either can be evaluated asymptotically in a direct manner (e.g. by saddle point technique), or can be converted into a rapidly convergent residue series. Results of these approximation techniques are given in Section IV. For example, in the case of formula (2.38) the contour C may be deformed as indicated in Fig. 2-3, so that S AB CD E F SG C A B D E F where the points A, B, D, E, F, G lie on a large semicircle of radius M with center at the origin v = 0, and the contour EF surrounds the zeros v (I = 1, 2,...) of H (ka) which lie in the first quadrant, and which are first order poles of the integrand function. When the radius M is increased to infinity, the contour integral along the semicircle vanishes: SB E G A D F (M=oo) (M=oo) (M=oo) The remaining two integrals can be manipulated to give (for details see Imai, 1954): 32 - m

THE UNIVERSITY OF MICHIGAN - 7133-3-T s 1 D H(2)(ka), iv(2 -0) E, ^(ka) H l(kp)e dv+ B H (ka) (M==o) (ka) (1) e +e + iv(2- ) -iv H () e dv (2.45) +. (j_ i r - ivr' E H (ka) e - e (M=Oco) Both integral representations (2.38) and (2.45) are as exact as the infinite series (2.37). If a similar calculation is performed on formula (2.40), one obtains: D (2)' 7r H (ka) iv( - ) z 2 = ( a yjlL 1), ( )H (kp)e d~+ (M=Oo) F1 JI(ka) e ivo+ eiv(2r- ) -iv2 + ~f4 \ 1~ ` Hv (kp) e dv, E H (ka) e -e (M=oo) (2.46) (1)' where now the contour EF encloses the zeros v (s = 1 2,...) of H (ka) which s 1/ lie in the first quadrant. The scattered fields may be expressed as integrals over the surface S1 of the cylinder, by using the vector analogue of Green's theorem (Stratton, 1941). If the cylinder is perfectly conducting, one has that s 1 H (r) = J(r) A dS, (2.47) where r and rI are the radius vectors for observation point and source point respectively, J is the surface current density, dS = a dodz. is an element of the ikR cylinder surface, the function Pi = e /R with R= Ir - r was previously intro0 - 33

THE UNIVERSITY OF MICHIGAN 7133-3-T duced in Section 2.1, and the gradient operates on the coordinates of the source point. Formula (2.47) gives the scattered magnetic field as a function of the surface current density. If J is known, the determination of Hs at every point in space depends only on the evaluation of a surface integral. In the particular case of normal incidence, the double integral is easily reduced to a single integral by observing that J is independent of z1 and that dz i7rH( )(kR1) 00 1 with R1 = { +a 2ap cos(0 - 01 (2.48) One then finds (Riblet, 1952): 7f H (p, ) = - i ( H()(kR1)d, (2.49) 4 - 1 ' J-7 where R1 = (apl -2)/R1. In particular, if the incident magnetic field H is parallel to the cylinder axis, one has that Hs(p,) = ika C ( ) a(s)(kR )( )1 do (2.50) -7T with Jo given by formula (2.42), whereas for the other polarization (E1 parallel to the z axis) it can easily be proven that -7r 34

THE UNIVERSITY OF MICHIGAN 7133-3-T with J given by formula (2.41). The results (2.50) and (2.51) can also be obz tained by observing that we are essentially dealing with scalar problems with Neumann and Dirichlet boundary conditions, respectively, and by applying Green's theorem with - H (kR ) as Green's function. Formulas which give the scattered field as a function of the current density on the surface of the cylinder, such as (2.50) and (2.51), are especially useful when ka is large compared to unity. An integral equation for the current density J is obtained by adding the incident magnetic field H' to both sides of equation (2.50) and by choosing the obserz vation point on the surface of the cylinder. Since the integrand of (2. 50) has a singularity at 0= 01 when p -> a, particular care must be taken in evaluating the limit p -a (Maue, 1949; Riblet, 1952). The final result is: ). - ika coa () 1. 1 -2 -A e2 J(01)H1 2kasin 2 sin 2 -if ' (2.52) If E is parallel to the z axis, the surface current J (0) satisfies the integral equation: * Jz() = -2 ~/t Ccos ei+ + 2 J 1)Hl (1 kasin 2 sin l d 2 1 2 2 (2.53) If a Fourier series expansion is assumed for the surface current, the unknown coefficients of the expansion may be determined either by direct substitution of the series in the integral equation for the current, or by means of a variational principle (Papas, 1950). Of course one finds again the series (2.41) and (2.42), which were previously derived from the wave equation by separation of variables. * Formula (28) of Riblet (1952) contains some misprints. 35

THE UNIVERSITY OF MICHIGAN 7133-3-T In order to present a qualitative discussion of the behavior of the current on the surface of a conducting cylinder a diagram of both amplitude and phase of the i z surface current density J = J e Z, computed by means of formula (2.41), is z z shown in Fig. 2-4 for the case in which ka = 3. 1 (King and Wu, 1959). The curves are obviously symmetric with respect to the plane of incidence. The phase velocity v of the surface current is given by the formula: z v /v = -ka/(d4 /do), (2.54) Z 0 Z where v = (eu) 1/2is the phase velocity of the incident wave, and db /d0 is the 0 z slope of the phase curve in Fig. 2-4. It is then seen that the phase velocity of the current around the cylinder surface is greater than v in the illuminated region 0 about T = t, it decreases to a value less than v at the shadow boundary 0= 7r/2, and maintains a nearly constant value 0.7v from 0 = 7r/2 almost to 0= 0. This means that a given phase of the surface current density creeps around the cylinder from the shadow boundary into the umbra region as a traveling wave whose velocity is sensibly constant and less than the phase velocity of the incident wave. An identical traveling wave of current exists on the other side of the cylinder. The amplitude of the surface current decreases rapidly with 0 except near 0= 0 and 0 = iT, where both amplitude and phase of J are stationary. The behavior at 0 = Itr is z easily understood by observing that the element of cylinder surface is there parallel to the incident wavefront. The behavior near 0 = 0 is a consequence of the interference of the two traveling waves which propagate in opposite directions around the cylinder and produce a standing wave. The standing wave is clearly observed only around A = 0, that is in the region where the amplitudes of the two interfering waves are of the same order of magnitude. Analogous considerations apply to the case in which the incident magnetic field is parallel to the cylinder axis. However, the current is now in the 0-direction so that the waves propagating around the cylinder may be regarded as longitudinal, whereas in the case of E parallel to the axis, the waves are transverse to the 36

[ THE UNIVERSITY OF MICHIGAN 7133-3-T iz FIG. 2-4: CURRENT DENSITY J = IJ e ON A CONDUCTING CYLINi z DER, WHEN E IS PARALLEL TO THE AXIS AND ka = 3.1 (King and Wu, 1959). J iQ FIG. 2-5: CURRENT DENSITY J0= Je i ON A CONDUCTING CYLINDER, WHEN H1 IS PARALLEL TO THE AXIS AND ka = 3.1 (King and Wu, 1959). M 37

THE UNIVERSITY OF MICHIGAN 7133-3-T direction of propagation. Whenever a standing wave is produced by the interference of two longitudinal waves, regions of concentration of electric charges exist on the surface of the cylinder. Such concentrations do not occur when two transverse.i waves interfere. Phase and amplitude of the surface current density Jp = JI e for ka =3.1 are given by the continuous lines of Fig. 2-5 (King and Wu, 1959). From Figs. 2-4 and 2-5 it is seen that the amplitude IJ[ decreases more slowly than I J i as I 0 decreases from ir to zero. This explains why the standing wave pattern is much more evident in Fig. 2-5 than in Fig. 2-4. Also represented in Fig. 2-4 is the difference FE - (V between the phase i V E of the total electric field EZ= EZ| e at a point 0 on the surface of the cylinder and the phase (1 = ka cos 0 of the incident field at the same point. The broken lines of Fig. 2-5 represent amplitude and phase of the traveling waves of surface current density, as given by Fock's high frequency approximation (see chapter IV). A detailed graphical representation and a discussion of the properties of the total electric (magnetic) field in the vicinity of the cylinder when the incident electric (magnetic) field is parallel to the axis were given by King and Wu (1957, 1959). The numerical results necessary for such discussions may be obtained easily through formulas (2.37) and (2.39), when ka is not large compared to unity. The main features of the diffraction phenomenon as well as surface current distribution for the case of E parallel to the cylinder axis and a = 0.16 X are illustrated in Fig. 2-6 (Carter, 1943). The amplitude of the total electric field in the back scattering direction 101 = Ti (Fig. 2-6a) resembles the amplitude 2| sink(x - a)| I of the total field originated by the reflection of a plane wave incident perpendicularly on an infinite conducting plane at x =-a. In both cases, the amplitudes exhibit standing wave patterns whose maxima and minima are practically located at the same points along the negative x-axis. However, in the case of the plane (ka = co), the amplitude of the standing wave is a periodic function of x with period X/2, whereas in the case of the cylinder, the oscillations of the amplitude pattern about the constant 38

THE UNIVERSITY 7133-3-T OF MICHIGAN 1u -I m"C E' Wove Direction IrN r Primary Field - 1.U i l..If.7,. Total Electric Intensity (R.M.S.) I 1' V Y v Total Electric Intensity (R.M.S.) I,,,,... a. _ I I I I 2.0 1.0 P/X 0 P/A (a) I I I 1.0 0 (b) FIG. 2-6a, b: DIFFRACTION OF A PLANE WAVE BY A CONDUCTING CYLINDER, WHEN E1 IS PARALLEL TO THE AXIS AND a = 0.16X (Carter, 1943). 39

11.1 Ila ngle (relativ at 0 - incident "Wft urren sity wove I I.*- 0-3 X -4mi I I J* I 100 4 N 1 2 3 Phase gl r mplit de 4h. 0 1 N O%% uftft so*.000 I - ~11 -z C,, 0 -01 0 zU FIG. 2-6c: CURRENT DISTRIBUTION ON THE CYLINDER SURFACE (Carter, 1943).

THE UNIVERSITY OF MICHIGAN 7133-3-T amplitude characteristic of the incident field decrease as Ixl increases, and vanish at x = -oo. The diffracted electric field in the back scattering direction may therefore be considered as produced by the interference of the incident wave with a cylindrical wave propagating radially outward from the cylinder (scattered wave). Since the diffracted field very near the cylinder along the negative x-axis is very similar to the field near a conducting plane, one may try to describe the field in the illuminated region around 101 = 7r in terms of the plane tangent to the cylinder surface at the azimuth 0 and of the plane wave arriving at an incidence angle 7T - 1 1. For a given value of 0, one still finds a standing wave pattern in the radial direction, which is more and more spread out as 1 01 decreases from 180 0; the distance between two adjacent maxima (or minima) is equal to X/(21cos 01). The spreading of the pattern is evident in Figs. 2-6a and 2-6b, where the cases I 1 = T, 7r/2, and zero are illustrated. The agreement between this simple interpretation and the exact result (2.37) is good whenever cos 0 is bounded away from zero. Thus we find that in the illuminated region I| 0 17r, the total electric field may be interpreted in terms of a standing wave in the radial direction and of a traveling wave which moves along the surface of the cylinder away from the negative zaxis with a phase velocity greater than v. Similar results are valid for the case in which the incident magnetic field is parallel to the axis of the cylinder (King and Wu, 1959). The behavior of the scattered electric field near the cylinder when E is parallel to the axis is illustrated in Fig. 2-7, for 0 = 0, r/2, and r and for various values of ka (Adey, 1958). It is seen that the back scattered field amplitude never exceeds the incident field amplitude, and, for a fixed kp, increases with the cylinder radius (Fig. 2-7c). On the contrary, the amplitude of the forward scattered field sometimes exceeds the amplitude of the incident field and oscillates about that value (Fig. 2-7a), so that the scattered field in the shadow region and in the vicinity of the cylinder surface does not behave like a divergent wave. If a plane wave is incident on an infinite cylinder perpendicularly to its gen erators and at an angle o0 with the x-axis, then the far scattered field components 41

o 0.8 10 ~- *_5.97 0.6 0o.6 E E t'1 E 0.4 ---.4 0.2 -: (a) Z(b) 0 5 10 15 20 0 5 10 15 20 15 30 35 40 45 kp kp m 1.2- - j I - -- 2000 ~ —, 2 1.E 0 3 \.0 k: =2 -- 090 -0.2 --- —--- ________ 4 __________ T 0 5 40 15 20 25 30 35 40 45 0 5 10 5- 20 - ~, o kp kp C) FIG. 2-7: SCATTERED FIELD ES = jES e PRODUCED BY A PLANE WAVE INCIDENT ON Z A CONDUCTING CYLINDER, WHEN E1 IS PARALLEL TO THE AIS. AMPLITUDE |E| AT (a) = 0, (b) = 7r/2, (c) 0 =; (d) Phase e = S-kp at = 0 (Adey, 1958). s S >. FIG.2-7 SCTTEED FELDE Ee PODUCD B A LAN WAV INIDET2O

THE UNIVERSITY OF MICHIGAN 7133-3-T in the electromagnetic case, or the far scattered pressure and velocity potential in the acoustical case, may be expressed in the form i(kp - 7r) f(,o) ji e, as p --. (2.55) The far field amplitude function f(0, 0 ) has the following well known properties (see, for example, Karp, 1961): f(0, ) = f(0 +r, 0+r), (2.56) 27r f(, )I 2d = -27r Re f(, ) (2.57) U0 Equation (2.56) may be obtained as a limiting form of the reciprocity theorem for Green's function; relation (2.57) constitutes the so-called forward amplitude theorem. In the case in which E is parallel to the generators of a metal cylinder (acoustically soft cylinder), it can be proven that f(0, 0 ) is a function of the difference (0- p ) if and only if the cylinder has a circular cross section (Karp, 1961). The far scattered field amplitude pattern that a plane wave at normal incidence on a circular cylinder produces in the azimuthal plane depends upon the value of ka; a particular case is shown in Fig. 2-8 (Faran, 1951). For small values of ka the pattern is nearly independent of 0 (Fig 2-8a), but lobes develop as ka increases (Fig. 2-8a,b and c). The phase of the far scattered field is a complicated function of both 0 and ka. In the case of electromagnetic scattering with E parallel to the axis of the metal cylinder, the far scattered electric field easily follows from relation (2.37): Eikp -i ePi n Jn(ka) pa k -. 4 H. cos n. (2.59) n=O H(1)(ka) n 43

THE UNIVERSITY OF MICHIGAN 7133-3-T (a) ka = 1.7 (c) ka = 5.0 (b) ka = 3.4 FIG. 2-8: AMPLITUDE PATTERN OF THE SCATTERED PRESSURE pS PRODUCED BY A PLANE WAVE WITH PRESSURE pi NORMALLY INCIDENT ON A PERFECTLY RIGID CYLINDER. The scales show the quantity lim 1/2 Jps/pi'(7rkp/2)l/2 as afunction of 0. (Faran, 1951). 44

THE UNIVERSITY 7133-3-T OF MICHIGAN 220r c, c <: ka FIG. 2-9: DIFFERENCE BETWEEN THE PHASES OF THE FAR FORWARD SCATTERED FIELD AND OF THE INCIDENT FIELD, WHEN Ei IS PARALLEL TO THE AXIS OF THE PERFECTLY CONDUCTING CYLINDER. (Adey, 1958). 45

THE UNIVERSITY OF MICHIGAN - --- 7133-3-T The difference e between the phase of the far scattered field in the forward direction 0 = 0 and the phase of the incident field, both evaluated at the same point, is therefore given by: OD J (ka)1 argn (1) 4 (2.60) n=0 H (ka) n and decreases monotonically from the value 5?r/4 at ka =0 to the limit value 37r/4 for very large cylinders, as is shown in Fig. 2-9 (Adey, 1958). In the case of E1 parallel to the axis of the metal cylinder, the back scattering cross section aE per unit length, defined as lirm 2 z lim 2r 1 c0E = 27rp 7 == O 27p -, aE p —, 1 p --- 2 i f2~ — z 1 is easily derived from relation (2.59). One finds that 4 E= A(ka), (2.61) 'E k where A(ka) = ) n (2.62) n=O H (ka) n This is also the cross section per unit length for an acoustically soft cylinder. Similarly, in the case of H parallel to the axis of the conducting cylinder, the back scattering cross section aH per unit length defined as H lirm z lim = p - -> O 27p - -- oo i. z 2 46

THE UNIVERSITY OF MICHIGAN 7133-3-T may be obtained from relation (2.39). One finds that a= B(ka), (2.63) where D J (ka) 2 B(ka) = E ~ (k 1)n a ) (2.64) n=O n H(1(ka) n This is also the cross section per unit length for an acoustically rigid cylinder. The quantities A(ka) and B(ka) are plotted in Figs. 2-10 and 2-11 for ka 10 (Senior and Boynton, 1964). The broken lines show the geometrical optics approximation W ka (see section IV). It is seen that aE is always larger than the geometrical optics approximation a = 7ra and increases monotonically with ka, whereas oH oscillates about the geometrical optics value as ka increases. The ratios a /a and a l/ are plotted in Fig. 2-12. E g.o. H g.o. The total scattering cross section a total per unit length is defined by the ratio of the time averaged total scattered power per unit length of cylinder to the time averaged incident Poynting vector. In the case when E is parallel to the axis of the perfectly conducting cylinder, one has that 7r s 1 sE(P 0) tota k Re i E (a, 1 ad (2.65) - T P1=a where the bar above Es indicates the complex conjugate. Observing that z ES(a, 01) = -E (a, 01), and that z Jz 7r i- eci boe p1(p: the total scattering cross section becomes: 47

6. 4 2 - G0 0 2 3 4 5 6 7 8 9 10 ka FIG. 2-10: CROSS SECTION uE PER UNIT LENGTH; A(ka)= kuE/4 (Senior and Boynton, 1964) (- -- geometrical optics approximation). ( ---goeria pisaprxmto) H z M C! 0-4 r) — 4 0 v I I O Z4

of- - - 7.. I ko 0 t 2 3 4 5 6 7 8 9 10 FIG. 2-11: CROSS SECTION aH PER UNIT LENGTH; B(ka) =kuH/4 (Senior and Boynton, 1964) ( --- geometrical optics approximation).

0va v q:y k1. Og. o0. C.A3 0.5 0 1 2 3 4 5 6 7 8 9 10 ka FIG. 2-12: CROSS SECTIONS F (- - -) AND o (-) PER UNIT LENGTH, NORMALIZED TO THEIR GEOMETRICAL OPTICS ~PPROXIMATION a (Senior and Boynton, 1964). g. o. 0-3 z C) C) z ttl CA

THE UNIVERSITY OF MICHIGAN 7133-3-T total kRe d1l} = ar -ikaRe J()edos 1 1 = a /C Rel \ J z(p)e do1 J. -JT V'1 (2.66) According to notation (2.55), the scattered field Es given by (2.51) may be rewritz ten in the form: E( i(kp- 7) E (p,0) - f(,0)rkp e 4 as p- oo, (2.67) where _ -ika cos(ol- p) f(p,0) = - / \ Jz(l) 1 4 J- 7r (2.68) From formulas (2.66) and (2.68) the important result follows (Papas, 1950): 4 = - - Re f(O, 0). 'total k (2.69) The total scattering cross section per unit length is therefore proportional to the real part of the far field amplitude function, evaluated in the forward direction. Since the phase of f(0, 0) is given by (8 + ir/4), where 0 is the angle plotted in Fig. 2-9, the real part of f(0, 0) cannot be positive, and therefore one has the obvious result that atota is never negative and becomes zero for ka = 0. From formula (2.59) it follows that J (ka) f0, 0) = - En ( no H(ka) n (2.70) hence 51

I THE UNIVERSITY OF MICHIGAN 7133-3-T 4 - (2.71) n=0 J (ka) + N (ka) n n where H(1)(ka) = J (ka) + iN (ka). n n n In the case in which a finite and nonzero impedance exists at the surface of the cylinder, the calculations of far fields and cross sections become rather complicated. From the computational viewpoint it is then advantageous to use the so-called phase shift analysis. In this method the incident plane wave is expressed as a sum of modes, each of which is characterized by a certain angular dependence. The perturbation that the scattering body introduces in the field of the primary wave at large distances from the scatterer manifests itself by a shift 'y in the phase of the radial dependence associated with the nth angular dependence. The phase shifts y are entirely determined by the boundary conditions and are, in general, complex quantities. Their knowledge permits the determination of the amount of scattered radiation and of its angular distribution. The phase shift analysis is outlined in the following; further details may be found in the literature (Lowan et al, 1946; Lax and Feshbach, 1948). i ikx It was previously found that when a plane wave & = e propagates perpendicularly to the axis of a cylinder on whose surface an impedance boundary condition holds, then the scattered field is given by: J (ka)+i(C-iD)J (ka) s 7_ n n n (1) E =I (kp)cosn, (2.72) n=0 H(1) (ka)+ i(C - iD)H(1) (ka)n n n i 5 where C and D are real quantities. In the acoustical case, b and b are velocity potentials and C - iD = u6/(?k) is the relative (or specific) acoustic admittance of the surface of the cylinder; the density 6 and the normal acoustic impedance | were defined in section 2.1. In the electromagnetic case, if E is parallel to the cylinder axis then x = EZ, I = E and C- iD = Z is the relative surface I m,I 52

THE UNIVERSITY OF MICHIGAN 7133-3-T admittance, whereas for Hi parallel to the axis, one has that i = Hi, b = H, and C - iD = Z \/P is the relative surface impedance. Formula (2.72) may be rewritten in the form: co =- E= inle n sinYs H (kp)cosn, (2.73) n n n n=0 where the phase shifts 7y are given by: n= 6 2 + arctan QUn- i(C-iD)Tn}, (2.74) with U =tan(6'-6 +2) T = 2 Jn(ka) + Nka) n n n 2 n 2 n n (2.75) J (ka) / J'(ka) 6 = arctan N (ka)) 6 = arctan n N (ka) n N' (ka) \ n / \ n / Similarly, the diffracted field is given by = i'+ = = e n {cos7nJn(kp) + siny nN (kp, (2.76) n-O n n n n J and the total scattering cross section per unit length by: c1 2Imy 4Imny 6total k En {1-2e cos(2Ren)+e n. (2.77) n=0 i In particular, when E is parallel to the axis of a conducting cylinder one finds that tanyn = tan6 n, so that relation (2.77) reduces to the form (2.71). The quantities 6, U and T which appear in (2.74) can be computed without specifying the impedance at the surface of the cylinder. Lax and Feshbach (1948) published tables of T and U for the parameter values n = 0(1)20 and n n ---------------------— 53 --- —-----— I

THE UNIVERSITY OF MICHIGAN 7133-3-T ka= 0(0.1)10, whereas 6, 6' and f{j2(ka)+N2(ka) had been previously tabulated by Lowan and associates (1946) for the same values of n and ka. Tables of { (ka)+N (ka)} for n=0, 1 and ka = 0(0.2)16 may be found in the book by Watson (1922). The phase shift analysis is obviously not limited to cylindrical scatterers. For example, Lowan (1946) and Lax and Feshbach (1948) also carried out extensive computations for absorption and scattering by spheres. Finally, the method can be applied to the more general case in which the impedance on the surface of the scatterer varies with the direction of incidence, by associating with the nth angular dependence a quantity (C - iD ) which is a function of the summation index n. n n This section ends with a few remarks on the scattering of cylindrical and spherical waves. If a line source parallel to the cylinder axis is located at (p = b, = 7), then the incident cylindrical wave is given by (Stratton, 1941): O0 A E (-1)n H()(kb) J (kp)cos n, p<b n=0 n = (2.78) co A E (-1) J (kb)H(1)(kp)cosn, p >b n n n n = n=0 2 2 1/2 where R = (p +b +2bp cos 0) /. For a comparison with the case of plane wave i ikx (1) incidence in which i = e, it is useful to choose A= 1/H (kb), so that in both cases i is equal to unity along the cylinder axis. The scattered wave may be written in the form: * Formulas (26) and (27) of Adey (1958) contain some misprints. 54

THE UNIVERSITY OF MICHIGAN 7133-3-T CDJ nJ (ka) ( = -A en (-1) n1 H( (kb)H (kp)cosn, (2.79) n=0 n H (ka) n where = = 1 for = 0, = for = 0 p=a p=a and the coefficients have been determined by imposing the boundary condition on the surface p = a of the cylinder. However, these calculations are sometimes unnecessary, since many results concerning the scattering of a cylindrical wave can be derived from the known results for the scattering of a plane wave. Thus, if the field at the point (p = b, 0 = 00 (c C00 )) due to a plane wave propagating in the direction 0 = 00 is known, then, or the basis of the reciprocity theorem, the far field in the direction 0 = 180 Jue to a line source at (p = b, 0 = 0~ (or 1800)) is also known (Adey, 1958). Furthermore, it is shown by Kodis (1950, 1952) that if the line source is sufficiently far from the cylinder and this is not too large, namely if a2/(2b2) < 1 then the scattered field is essentially equal to that produced by an incident plane ikz ik(R - b) wave = e, whereas the incident field may be taken as equal to e. A few patterns of the far field amplitude of the pressure wave scattered by a rigid cylinder were computed by Faran (1953) for different values of kb. Faran concluded t' ore should expect little change from the plane wave scattering pattern, provided that a/, < 0.1. Zitron and Davis (1963) computed a quantity proportional to the amplitude of the far scattered field as a function of 0, for ka = 1.0 with kb = 2, 5, 10, 20; ka = 3.4 with kb = 6.8, 13.6, 17, 68; and ka = 10.0 with kb = 100, 200, 500, for both Dirichlet and Neumann boundary conditions; they also pointed out that Faran's curve for kb = 6. 8 is incorrect. In the case of the hard cylinder, other results 55

THE UNIVERSITY OF MICHIGAN 7133-3-T were published by Shenderov (1961), who plotted the amplitudes of both scattered and total far fields as functions of 0 for ka = 2, 6, 10 and b = 1.2a, and compared numerical and experimental diagrams of the amplitude of the total far field for ka = 6, 10 and b = 5.2a. Finally, since the entire following section is devoted to the case of dipole sources, we here limit ourselves to pointing out that useful relationships exist between the current distributions on perfectly conducting cylinders of arbitrary cross b-1 section (in particular, circular), illuminated by plane or spherical waves. These relationships were established by Brick (1961), who utilized integral equations for the electromagnetic field put into an appropriate form by means of a Fourier integral operation, and obtained the leading terms of series expansions in powers of (kR ) by the method of steepest descents. Brick's results are thus valid for kR >> 1, where R is the distance between the source point and that point of the cylinder axis which belongs to the azimuthal plane containing the observation point. 2.3 Dipole Sources In this section, we consider scattering from infinite cylinders when the excitation is an elementary source, i. e., an infinitesimal dipole, located at a finite distance from the cylinder. Although some results are included for the case when the dipole is on the surface of the cylinder, the subject of radiating slots, and the equivalence of slots and dipoles, is not treated. The reader is referred to the exhaustive treatment of Wait (1959) which also has an extensive bibliography. Our concern here is with sources off the cylinder and the limiting case is touched on only briefly for comparison. In 1943 there appeared two independent treatments of the problem of scattering of dipole fields by cylinders, Oberhettinger's and Carter's. Oberhettinger derived the exact field scattered by a cylinder with the electric dipole oriented parallel to the cylinder axis and also presented far field asymptotic approximations. Carter used the reciprocity theorem and the results for plane wave incidence to cal culate the scattered far field for dipoles and arrays of dipoles. 5 56 -

THE UNIVERSITY OF MICHIGAN 7133-3-T There are are a variety of methods for deriving the expressions for the field scattered by an arbitrarily oriented dipole. Although differing in detail, they share two essential features, a representation of the field in terms of electric and magnetic Hertz vectors and, ultimately, satisfying boundary conditions on the surface by suitably matching the scattered part of the Hertz vectors with the incident field through the expansion ikR OD Oc~ c- O.D -ih(z-z - e = 2 E cosn(-io ) - dh Jn(kp n (X> o n=0 n n -J J-ao where p< = min (p, p), p> =max (p, p), _2 2 The process is completely analogous to that employed earlier in the case of plane wave incidence (cf. Eq. 2.28) though more cumbersome due to the integration over h. Here, as before, some special cases offer inviting simplification, e.g. when the dipole is parallel to the axis of the cylinder and one Hertz vector suffices to represent the entire field (this is the case treated by Oberhettinger (1943); see also Wait (1959) and Harrington (1961)). It is also possible to construct the field due to an arbitrarily oriented dipole by suitably operating on the field scattered by an arbitrarily incident plane wave (Senior, 1953). The resulting expressions, regardless of the method used to obtain them, are cumbersome and the elegance exhibited by any particular method is usually compensated by an atrocious calculation equivalent to matching coefficients in expansions of the incident and scattered fields. 57

THE UNIVERSITY OF MICHIGAN 7133-3-T In treating scattering problems when the source is an arbitrarily oriented dipole or distribution of dipoles (e.g. current), it is now fashionable to employ the dyadic Green's function, e.g. Morse and Feshbach (1953), Van Bladel (1964) and Tai (1953, 1954a, b). This procedure enables one to formally discuss the solutions of scattering problems without actually calling into play the inevitable complicated representation of solutions of particular problems. Of course, if it is these particular solutions which are sought, then the tensor or dyadic Green's function, as a labor saving device, loses much of its value. Still it does provide a systematic way of presenting the results and a brief discussion will be given followed by the explicit representation of the dyadic Green's function for a cylinder with a number of special cases. If the time harmonic Maxwell's equations for a current source J at a finite distance from a perfectly conducting scatterer S, are written VA E = i H, (2.80) VAH = -iEE+J then the electric field satisfies the inhomogeneous vector wave equation VAVAE-k 2E = ir, the boundary condition nAE1 =, S and the radiation condition r AVAE+ikE = o(l/r). (2.81) If V denotes the volume exterior to S, then the solution of the problem may be written as 58

THE UNIVERSITY OF MICHIGAN 7133-3-T E(r) = G(r,) * J(r )dr (2.82) where G(r, r )' is the dyadic Green's function and satisfies VAVAG-k G = 6(r-r )I I is the identity dyad, n = S rAVAG+ikG = o(1/r) The nine rectangular Cartesian components of the Green's tensor are nothing else than the total electric field components for electric dipole sources oriented along the coordinate axes. That is, if iwtJ is chosen to be i 6(r - o) then equation (2.82) yields E(r,r) = G(r,r ). - =- 0 X If we use a superscript x to denote the orientation of the dipole source, i. e. E (r) = G(r,.r ) -0 X then clearly Ey() = G(r, r )L, - — O y E (r) = G(r,ro). =- 0 Z Thus we see that it is possible to construct the components of the dyadic Green's * A double underlining will be used to denote dyads. 59

- THE UNIVERSITY OF MICHIGAN 7133-3-T function if we know the response of an arbitrarily oriented dipole source. In fact, these are entirely equivalent pieces of information. For a perfectly conducting infite cylinder the dyadic Green's function is given by Tai (1954b) as i G(r r) = 8 — '-1 8ir __Mr) )+ahMj-hr + (3)(h, r) Mn (-h )+aM(3) +~en n '1 n-en (-h, r )+ M(3)(-h,rl+ )n -1 n on (-h,r1) + 3N (-h, r + )(-h1 r+ N(3 (-h,141, for p >p1, (2.83) where 2 k2 2 X = k -h Men(h,r) 0 N (h,r) e o ihz n sin a cos '1 n - Zn(kp) no (X n =n cos p z- p) sin ih= (XP) sin0 +- Z (X) np + 2 C ( os A +X Z (Xp) s n zi, n smin' J'(Xa) n n J (Xa) n n The superscript (1) on M and N signifies that Z is J, the Bessel function of 8 —n -n n n the first kind. The superscript (3 means Z is the Bessel function of the third kind (1) n or Hankel function H. When p1 > p, r and r1 must be interchanged in (2. 83). n - -- I 60

THE UNIVERSITY OF MICHIGAN 7133-3-T Equations (2.82) and (2.83) have been employed by Tai (1964) to obtain explicit far field results for two electric dipole orientations, longitudinal and transverse (also called vertical and horizontal dipoles). The integration in (2.82) is easily accomplished since J is always a delta function. The integration in (2.83) is carried out by the method of saddle points (e.g. Wait (1959)). In the results both cylindrical and spherical coordinates are employed, so that the familiar factor e /r is exhibited. Recall that /2 2 r = Vp +z VP + Z sin = p a) Longitudinal Dipole The current is given by 6(p-po) 6(0 -o) 6(z) ^ J =-ire A z — -, (- P where A is a constant (A = p( /e, where p(1) is the dipole moment). The total far electric field is (compare with (2.29)): ikr 2 -ikp sinecos(0-0) z > J (kasin e) E = -0 e k sineA {e ~ ~ - E (-i)f - -- X -47rr nL =n H(1)(kasin0) n XH(1)(kp sin )cosn(0j-. (2.84) n o0 The far field of a longitudinal dipole in an azimuthal plane (constant 0, 0 < 0 < 27r, see Fig. 2-14) is proportional to the far field of a line source (radiating at a longer wavelength) parallel to the cylinder through (po, 0 ) (see equations (2. 78) and (2. 79) where p = b, 0 = 7r). If the dipole radiates with wave number k, the equivalent 0 61

THE UNIVER x S IT Y 7133-3-T OF MICHIGAN tr, 6,0) Iy FIG. 2 -13 FIG. 2 -14 ly x Dipole FIG. 2-15 62

THE UNIVERSITY OF MICHIGAN 7133-3-T line source radiates with wave number ksin. In the plane of the dipole (e = 7r/2) the wave numbers are equal. Explicitly, the far field of an electric line source parallel to the cylinder axis (E = i AH( (kp+p - 2pp cos(0- ) ) is: z o 0 0 iir ikp- 4 e 4 -ikpcos(- 0o) ( 0 n H( (kp )cosn(0-0 ( (2.84a) H(1) ka ) n o n The field of an array of longitudinal dipoles of strength A, at points (p., 1), all in J J (kasin) ) -Z en (1)(l sd Hn (psin cos n( - n=O H (kasin) n n (2.85) Note that to this order in 1/r only Ee is nonzero. b) Transverse Dipole 6(p-po )6(0-0) 6(z) 0 0 J = -iuEA x The total far field in the plane of the dipole (6 = 7r/2, z = 0) is j 63

THE UNIVERSITY OF MICHIGAN 7133-3-T ikr -ikpocos(0- o) J'(ka) Ek2 sine ~ (-in+1 n X 47rr 2 n=0 H (1) (ka) n X [H(l(kpo) sin[n - (n- 1)0 + Hn+l(kp) sin[n0- (n+ 1)01] (2.86) The field due to an array of such dipoles of varying strengths, A, is Jikr (ka) eikr 2 -ikp cos(- 0) 1 J(ka) E: k2 I A, sin01e (- 1) n - ~nL-2 0 nl H(1), (ka) n A H() (kp) sin [n-n- + H()(kp )sinn-(n+ ) (2.87) In this case only Ep is nonzero. p Note that in equations (2. 84) to (2. 87) the constants A or A, may be complex, A = IA, le x, thus governing both amplitude and phase of the source. Carter (1943) has carried out a large number of calculations of far field patterns from arrays of longitudinal and transverse dipoles. The problem with which Carter was concerned was that of achieving omnidirectional azimuthal radiation patterns for arrays of dipoles around a cylinder (transmitting antenna on the Chrysler building). His approach was to calculate the radiation patterns of various symmetri arrays of dipoles with a variety of related phase differences between dipoles and the observe which pattern most closely achieved the desired shape. While this may not now be considered the most direct way of treating this problem, the patterns calculated by Carter may prove useful for other purposes. Carter employed the reciprocity theorem and the known results for plane wave incidence to arrive at the expressions (2.84) to (2.87). This avoided the asymptotic evaluation of the integral in (2.83), since the asymptotics are in a sense already carried out for plane wave incidence. For plane wave incidence Carter has 64

THE UNIVERSITY OF MICHIGAN 7133-3-T computed the magnitude of the electric field as a function of distance from the cylinder for = 0, 7/2, r, 37r/2 and ka=l. He also calculated surface current for this case and this has already been given in Fig. 2-6c. In the case of dipole excitation, Carter has made numerous calculations. For a single longitudinal dipole the magnitude of the electric field in the far zone is given as a function of azimuthal angle in Fig. 2-16 for a dipole.24X from the cylinder axis and various cylinder radii (-a/X =.0016,.0318,.08,.16, o.24 ). Also included in this figure is the pattern of a transverse dipole located.24X from a cylinder of radius. 16X. The nearly circular pattern for the smallest radius is quite distorted, without any definite relationship between the geometrical shadow and the shape of the pattern. However, for large cylinders the pattern is quite different from the patterns of small cylinders as seen in Fig. 2-17, where the far field is plotted for a longitudinal dipole.878X from the axis of a cylinder of radius. 383X. Carter attributes this to resonance effects. Carter's results for arrays of dipoles include longitudinal (vertical) dipoles (1,2, and 4 fed in phase) located.878X from the axis of a cylinder of radius.383X (see Fig. 2-17), and 4 transverse (horizontal) dipoles fed in phase, phase rotation, and pairs in phase, other pairs out of phase (see Figs. 2-18 to 2-20). The transverse dipoles are located.796X from the axis of a cylinder of radius.637X. The patterns given here are in the plane containing the dipoles; additional patterns in other planes are contained in Carter's original paper. A word of caution is called for regarding the use of Carter's analytic expressions of the far field. Numerous errors were found in these formulae, e.g. using Carter's equation numbers, terms in equation (48) should alternate in sign, equation (50) is completely incorrect, a factor j is missing from equation (51), terms in equation (55) should alternate in sign, etc. A spot check of the computed patterns, however, indicates that the correct formulas were used and the errors apparently are typographical. A complete recalculation of all the patterns was not undertaken. It is recommended that any quantitative application of Carter's results be accompanied by a suitable verification. I 65 1

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 2-16: FAR-ZONE ELECTRIC FIELD PATTERN IN THE AZIMUTHAL PLANE; All curves except one are for dipoles parallel to the cylinder axis (Carter, 1943). 66

H z c m -1 0 FIG. 2-17: FAIL-ZONE ELECTRIC FIELD PATTERN IN THE AZIMUTHAL PLANE, FOR DIPOLES PARALLEL TO THE CYLINDER AXIS (Carter, 1943).

3H 0.796 X 2:z 1H 0 CA '-4 0 0 FIG. 2-18: FAR-ZONE ELECTRIC FIELD PATTERN IN THE AZIMUTHAL PLANE, FOR FOUR TRANSVERSE DIPOLES FED IN PHASE (Carter, 1943). \~cct~ co FIG.2-~: FA-ZOE EECTRC FELD ATTRN N TH AZMUTHL PAN! FOR OUR RANVERS DIPLESFED N PHSE Cartr, co3)

Xo c. FIG. 2-19: FAR-ZONE ELECTRIC FIELD PATTERN IN THE AZIMUTHAL PLANE, FOR FOUR TRANSVERSE DIPOLES; DIAMETRICALLY OPPOSITE DIPOLES FED IN PHASE, PAIRS IN QUARTER PHASE RELATION (Carter, 1943). H -q m Pd C0 H cn I0 z P-4 Q)

THE UNIVERSITY OF MICHIGAN 7133-3-T 6 0 -4 0 CNI 70

THE UNIVERSITY OF MICHIGAN 7133-3-T Lucke (1951) derived the Green's function for a dipole in the presence of a cylinder (both elliptic and circular). The result is contained in (2.83) and shall not be repeated. An earlier version of Lucke's work (1949) contained an error in the expression for the field but this was removed and agreement with Carter's result was obtained. Lucke calculated the scattering pattern (see Fig. 2-21) for a longitudinal dipole located.238X from the axis of a cylinder of radius.076X. This is almost identical with one of the cases calculated by Carter and the patterns do coincide. (Lucke's scale is a factor ka =.5 smaller than Carter's and this must be taken into account to obtain agreement). Sinclair (1951), as Lucke, obtained some results for antennas in the presence of circular cylinders while investigating the more general case of elliptic cylinders. His results agree with Carter and are easily obtained from the expression for the Green's function given in the beginning of this section. Wait and Okashimo (1956) have calculated radiation patterns of a radial dipole and pairs of diametrically opposite in-phase radial dipoles, located on the surface of a cylinder for cylinder radii a/X =. 0315,.125,.335,.915, 1.54. The patterns were computed from a theoretical result equivalent to (2.86) and (2.87) for special values of 0p (0 =0 for single dipole and 0 =0 01 = T for the pair). The patterns were compared, with excellent agreement, with the experimental results of Bain (1953) and are presented in Figs. 2-22 and 2-23. Oberhettinger (1943) computed the far field pattern for an electric dipole parallel to the axis of a cylinder of radius a = X. The dipole locations were 5/4X, 3/2 X, 7/4 X, 2X from the cylinder axis and the patterns are all in the horizontal plane (0 = 7r/2, see Fig. 2-24). The complexity of the pattern is seen to increase with distance of the dipole from the cylinder. Levis (1959, 1960) has made extensive calculations of a radial dipole on a cylinder. This corresponds to a transverse dipole as defined in (2.86) when the dipole is normal to the surface of the cylinder as well as its axis, i.e., 0 = 0. He has computed the real and imaginary parts of E and E as well as their magnitude 0 0 71

THE UNIVERSITY 7133-3-T OF MICHIGAN.238X 0.238X I --- I FIG. 2-21: FAR-ZONE ELECTRIC FIELD PATTERN IN THE AZIMUTHAL PLANE, FOR A DIPOLE PARALLEL TO THE CYLINDER AXIS (Lucke, 1951). 72

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 2-22a: RADIATION PATTERN (PROPORTIONAL TO THE AMPLITUDE OF THE FAR-ZONE ELECTRIC FIELD) IN THE AZIMUTHAL PLANE, FOR A DIPOLE ON THE CYLINDER AND PERPENDICULAR TO THE CYLINDER AXIS. Case a/X = 0.0315 (Wait and Okashimo, 1956). 73

THE UNIVERSITY 7133-3-T OF MICHIGAN FIG. 2-22b: Case a/k = 0.125 (Wait and Okashimo, 1956); - - - experimental (Bain, 1953). 74

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 2-22c: Case a/X = 0.335 (Wait and Okashimo, 1956); --- experimental (Bain, 1953). 75

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 2-22d: Case a/X = 0.915 (Wait and Okashimo, 1956); --- experimental (Bain, 1953). 76

THE UNIVERSITYOFMCGA OF MICHIGAN 7133-3-T I I I4 1% 11'4 FIG. 2-22e: Case ~~~~~a/X=15 (atad ksio,15) - - exprimntal(Ban, 153) 77

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 2-23a: RADIATION PATTERN (PROPORTIONAL TO IE1) IN THE AZIMUTHAL PLANE, FOR TWO RADIAL DIPOLES DIAMETRICALLY OPPOSITE ON THE CYLINDER SURFACE AND FED IN PHASE. Case a/X = 0.0315 (Wait and Okashimo, 1956). 78

THE UNIVERSITY 7133-3-T OF MICHIGAN FIG. 2-23b: Case a/X = 0. 125 (Wait and Okashimo, 1956). 79

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 2-23c: Case a/X = 0. 335 (Wait and Okashimo, 1956) 80

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 2-23d: Case a/X = 0.915 (Wait and Okashimo, 1956). 81

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 2-23e: Case a/X = 1.54 (Wait and Okashimo, 1956). 82

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 2-24a: RADIATION PATTERN (PROPORTIONAL TO [EeI) IN THE AZIMUTHAL PLANE OF A LONGITUDINAL DIPOLE LOCATED AT A DISTANCE d FROM THE AXIS OF A CYLINDER OF RADIUS a = X. Case d = 5 X (Oberhettinger, 1943). 4 83

THE UNIVERSITY 7133-3-T OF MICHIGAN FIG. 2-24b: Case d = X (Oberhettinger, 1943). 84

THE UNIVERSITY 7133-3-T OF MICHIGAN 7 FIG. 2-24c: Case d = X (Oberhettinger, 1943). 4 85

THE UNIVERSITY 7133-3-T OF MICHIGAN FIG. 2-24d: Case d = 2X (Oberhettinger, 1943). 86

-- THE UNIVERSITY OF MICHIGAN 7133-3-T for X varying from.05 to.5 in increments of.05. The field quantities are tabulated to five figures for the full angular range in increments of 5. It must be noted that Levis has oriented his cylinder along the x-axis, thus in order to compare his formulae and calculations with Carter, Wait and Okashimo, etc., one must rotate coordinates using the relations given explicitly by Levis. 2.4 Scattering of Evanescent Waves In relation to certain physical phenomena, such as the Smith-Purcell effect, it is of interest to investigate theoretically the scattering of an evanescent (or surface) wave by an infinite metal cylinder. This study has been performed by Ronchi et al (1961), and is summarized in the following. Another work on this subject has recently been published by Levine (1965). A concrete physical situation, in which this problem arises, is illustrated in Fig. 2-25. A plane electromagnetic wave is totally reflected at the plane interface A-A which separates two media with different refractive indexes, giving origin to a surface wave in the medium with smaller index of refraction. The surface wave, whose amplitude decreases exponentially as the distance from A-A linearly increases, and whose planes of constant phase are perpendicular to A-A, is scattered by the infinite metal cylinder C. It is assumed that the distance d between interface A-A and cylinder axis is so large compared to both cylinder radius and wavelength of the incident radiation, that multiple scattering effects may be neglected. The notation of Fig. 2-2 is used, but it is now assumed that the plane of incidence (k, i^) forms the angle 7y with the positive x-axis. According to formulas z (2.16), the components of the electromagnetic field incident on the cylinder are generated by the scalar functions: = iexp [ik(xcos ysina + ysin7ysina + zcosaJ, (S A 1,2), where 87

THE UNIVER incident wave A __ - SITY 7133-3-T OF MICHIGAN reflected wave \/ A A2 o FIG. 2-25: A SURFACE WAVE IS ORIGINATED BY TOTAL REFLECTION AT A-A, AND IS SCATTERED BY THE CYLINDER C. 88

THE UNIVERSITY OF MICHIGAN 7133-3-T i cos B ol k2s k sincr i sin 0o2 = -— sin k sina In order that the 'is represent an evanescent wave attenuated in the i direction, the quantities cos a and sinysina must be real, whereas cosysina = iQ with Q real and positive. These conditions may be satisfied in two different ways: Case 1: a real, Case 2: 7r X. 7 = 2 - i ', a = -1, 7 = - iY", y' real positive; a real (e.g. positive), O 7" real positive. Case 1 is termed the case of "small attenuation", and Case 2 the case of "large attenuation" (Ronchi et al, 1961). The incident wave functions become, for small attenuation, I = io exp(-kxsinh' sina) exp [k(ycoshy'sina + zcos ) ( = 1,2), whereas for large attenuation i = 0i exp(-kx cosh7" sinha ) exp ik(ysinhy"sinha + zcosha I) (I = 1,2). Let us call 6 the angle between the z-axis and the direction of propagation of the phase, which is parallel to the (y, z) plane, Then, in Case 1 tan6 = tana coshy', and in Case 2 [I tan 6 = tanha sinhy. 89

THE UNIVERSITY OF MICHIGAN 7133-3-T The parameters of the evanescent wave, which have an immediate physical meaning, are the angle 6 and the attenuation constant h, which equals ksinhy' sina in Case 1 and kcoshy" sinha in Case 2. One can easily express a and y', or a and y", in terms of and h: terms of 6 and h: r sin 2 cos2) sinf = in 6 - 2 cos 6 k2 Case 1 l sinhy = sih 2 2 sinha = \|(- cos 6 -sin 6) - Case 2 coshy" = 1 2 k2 2 cos 6 - sm 6i h2 It is then easily seen that Case 1 is valid for h/k< Itan6J and Case 2 for h/k > Itan6I. In particular, Case 1 describes the condition of normal incidence 6 = 7r/2. In the case of small attenuation, one has that 00 i i ikz cos \a n y i-nn+ = = i e. J (kp sina) e ' n=-wo (2.88) (I = 1, 2). 90

THE UNIVERSITY OF MICHIGAN 7133-3-T It is then found that the scattered field components are given by (2.16) with s,i ikzcos a E a ()(. + ine = e a H (kp sin) e n=-oD ( = 1,2), (2.89) where al J (kasina) n H (ka sina) n J? (ka sina) n n2 H (kasina) n The corresponding results for large attenuation are obtained by replacing a with (-ia ) and y' with (y"+i ) in (2.88) and (2. 89). From the expressions of the o 2 field components, it can be proven that in Case 2 the radiated field vanishes. This is related to the fact that the wave number kcosha in the direction of the cylinder axis is larger than the wave number k in free space. Thus the scattering cross section per unit length of the cylinder, which shows some sort of resonance for increasing attenuation (Ronchi et al, 1961), vanishes for all h >k|tand. 91

THE UNIVERSITY OF MICHIGAN 7133-3-T m LOW FREQUENCY APPROXIMATIONS FOR AN INFINITE CYLINDER In the case of an infinite circular cylinder, the low frequency or long wavelength limit (ka << 1) can be derived easily from the exact power series solution. The exact solutions for normal incidence are given in Eqs. (2.35) and (2.36). As particular cases, we shall consider the parallel (3 = 0) and perpendicular (13 = 7r/2) polarizations of the incident electric field. The scattered field due to an incident wave with E parallel to the axis of a perfectly conducting circular cylinder is given by (2.37). Since ka << 1, we may limit our considerations to the first term of series (2.37); thus we have, in the far field: 2 J (ka) i(kp-) (3.1) E s -x -k- e (3D1) z kp H(1)(e 0 which is independent of the azimuthal angle 0. According to (2.61) and (2. 62), the back scattering cross section per unit length is given by J2(ka JE (a (3.2) E k J2(ka) + N2(ka) o o and since ka -- 0: 2 a --. (3.3) k(logka)2 In the approximation (3.1), the total scattering cross section is also given by (3.2) and (3.3): cTE - (E)total. (3.4) This result easily follows from Eq. (2.71). m I 92 ---

THE UNIVERSITY OF MICHIGAN 7133-3-T In the case in which H is parallel to the cylinder axis, the bistatic cross section per unit length is given by J'(ka) 2J (ka) 2 o 1 H k (1)' ( + (1) cos (3.5) H (ka) H (ka) o 1 Two terms in the exact series solution must now be considered, since they are of the same order in the limit ka -*0. In particular, the back scattering cross section corresponds to 0 = iT, and is given by: UH (ka). (3.6) H 4k The total scattering cross section is obtained by integrating (3.5) over all values of 0 (Panofsky and Phillips, 1956): (a ) 31- r (ka) a. (3.7) H total 4 Lord Rayleigh was the first to use potential theory solutions to construct long wavelength acoustic and electromagnetic approximations for both two-dimensional and three-dimensional problems (Strutt, 1897). The advantages of Rayleigh's method are not very evident in the case of a circular cylinder, because the exact solution is well known. Rayleigh first determined the fields in the region a < p < X by using potential theory approximations. He then introduced these intermediate fields in Green's integral in order to derive the far field form. In particular, he showed that the potential of a scatterer in a uniform field is the near field limit of the corresponding scattering problem solution, and that this yields the first term of the far field expansion. Let a plane sound wave, whose velocity potential is given by fi e-ikp cosp 38 = er whos (3.8) be incident on a circular cylinder whose radius is small compared to the wavelength. 93

THE UNIVERSITY OF MICHIGAN 7133-3-T By expanding in series of Bessel functions (Strutt, 1945; sections 341, 343): e-p os = J (kp)-2i().+2(-i)n (kp) cos n+... (3.9) For small ka, i 1 22 i = 1- k 2a2 -ikacos +... (3.10) 4 p=a t iL 1 2 -ka-ikcos9+... (3.11) dr 2 p=a The velocity potential of the wave diverging from the cylindrical obstacle is given by Ws = S D (kp)+S1Dl(kp)+... (3.12) 0 o 1 1 where So, S1,... are trigonometric functions, / rk2 2 22 3 4 4 Do(kp) = (y+log 2i)1 -. + 2 2 2 dD (kp) D(kp) = d(kp) kp L 22 2 2- 2 k +... (3.13) 2 2.4 and y = 0.577215... is Euler's constant. Suppose that the material of the cylinder has density 6' and compressibility m', and let 6 and m be the corresponding quantities for the surrounding medium. All special cases can easily be obtained bygiving appropriate values to a' and m'. Inside the cylinder (Strutt, 1945; section 339):... --- 94

THE UNIVERSITY OF MICHIGAN 7133-3-T 2 A2 14k' i 2 2 2 2 /A i - k'- + + A - 2. Cos inside 0 2 2 22. 2 4 2 4 2 2.4 2.~4 8 314 (3. 14) where k' is the internal value of the wave number. Outside the cylinder (Strutt, 1945; section 341): tsi B B + log ) + B cos (3.15) outside 2i 1 kp The conditions to be satisfied at the surface of the cylinder lead to the following equations: -A k2a2 = -k2a2+ 2B (3.16) o o -A (1 k'a = 1-ka +B) (3.17) o 4 2i 3k.22\ B A ( )= -ik- 2 (3.18) ka a' k'` Ba Ala -ika + (3.19) Solving the above equations, we get 1,221 k' 61| 1 2 2 m'-m B ka2 1- - k2a 2 k am (3.20) o 2 2 6( 20MI 2 2 6'-6 B1= -ik a 6 (3.21) We can now write the velocity potential of the scattered wave at a large distance from the cylinder as I I 95

I THE UNIVERSITY OF MICHIGAN 7133-3-T,s 22ka/22 I m-m 6'-6 S -k2a2e (i)kp 2m + 67+6 cos. (3.22) The conditions m' -- o, 6' = co correspond to the case of a hard cylinder, s 2ir2a2 F+1 lcO 2 vt- 1 (3.23) ~ 1/2X3/2 L2 Poo - 8- ] ~ The case m' = 0 gives rise to an extreme case when the zero order in circular harmonics becomes infinite and the first order term is relatively negligible. This iTs i corresponds to a boundary condition of evanescence of (0 + O ) in which i/ = 1 (often referred to as the soft cylinder boundary condition). Therefore: S (Q+log )+l = 0. (3.24) For large kp: ~ Q i2 (3.25) + log' 57 The problem of electromagnetic wave scattering is analytically identical to the problem of scattering of acoustic waves in two dimensions by small cylindrical obstacles. One can identify pi and Os with E or H where E is the electroz z z motive intensity parallel to z and H the magnetic force parallel to z. We also z replace 6 by the electrical conductivity oa and 1/m by a, the permeability, and 6? and l/m' by a' and M', the values of Ca and a inside the cylinder. The expressions for E and H are identical with (3. 22) and (3.25). z z Lamb (1924, sectiacluded a section 304) has included a section on plane wave scattering by a cylindrical obstacle in his book on hydrodynamics. He has made reference to Rayleigh's contribution; the method used in Lamb's book is identical to Rayleigh's 96

THE UNIVERSITY OF MICHIGAN 7133-3-T method, for the case of normal incidence of a plane sound wave scattered by a rigid circular cylinder. He has also given the expression for the total scattering cross section for the above case in the long wavelength limit, and this agrees with our equation (3. 7). The expressions for scattering cross section in cases of E and H-polarized waves for arbitrary cylinders in two dimensions are derived by Van Bladel (1963). He has applied the low frequency limit for the Green's function in the integral equation and his results for the special case of circular cylinders agree with equations (3.3) and (3.7). It is clear that the scattering cross sections depend markedly on the polarization of the incident waves, and this has been discussed by Kerr (1951). As ka increases, E/ag decreases monotonically and aH/g o oscillates, departing markedly from the fourth power law (see Fig. 2-12). Some considerations on the low-frequency total scattering cross section for a cylinder with impedance boundary conditions are developed by Lax and Feshbach (1948). 97

I THE UNIVERSITY OF MICHIGAN 7133-3-T IV HIGH FREQUENCY APPROXIMATIONS FOR AN INFINITE CYLINDER In this chapter, we consider only the cases of cylindrical and plane waves at normal incidence for both polarizations of the electric field (E parallel or perpendicular to the axis of the cylinder). The results obtained for a perfectly conducting cylinder by means of geometrical and physical optics approximations, of the geometrical theory of diffraction, and of asymptotic expansions of exact solutions are presented in Sections 4.1, 4.2 and 4.3, respectively. The case of a cylinder with a nonzero surface impedance is briefly examined in Section 4. 4, and all results on radar cross sections are collected in Section 4.5. 4.1 Geometrical and Physical Optics Approximations Let us assume that the wavelength X of the incident radiation is very small compared to the radius a of the cylinder. Under this assumption, the scattered field in the illuminated region may be approximately determined by a simple raytracing technique. Consider a plane incident wave propagating along the x-axis, perpendicularly to the axis z of the cylinder; since the problem is essentially twodimensional, we may restrict our considerations to the azimuthal plane of Fig. 4-1. A thin beam AA' of rays impinges on the cylinder surface at BB' where it is reflected and scattered in the angular range PP'; we want to determine amplitude and phase of the scattered field Vs at the point P(p, 0), located at a large distance from the cylinder (p - oo, o- ). The energy per unit time carried by the incident beam is proportional to 1 12 (AA = | a| a9' sin2 whereas the energy that the scattered wave carries through PP' per unit time is proportional to |s(,) 2(PP') = = s( p, 22pBOB; II 98

THE UNIVERSITY OF MICHIGAN 7133-3-T Pt Al FIG. 4-1: GEOMETRICAL OPTICS APPROXIMATION 99

THE UNIVERSITY OF MICHIGAN 7133-3-T since no absorption occurs at BB', these two expressions must be equal by conservation of energy, that is WS(P., Psin^ i 2 2 The phase of the scattered wave is easily determined by observing that the difference between the geometrical paths A OP and ABP of Fig. 4-1 is given by 2asin. 0 2 Thus arg~s5(p,) = k(p - 2a sin ) +, where 6o is a constant angle which depends on the boundary conditions. For an o acoustically hard cylinder (H parallel to axis of conducting cylinder ) o = 0, while (< = 180~ for a soft cylinder (Ei parallel to axis of conducting cylinder). Therefore, a plane electromagnetic wave normally incident on a perfectly conducting cylinder with i A ikx E = i e Hi= ' - ikp-2asin ) z g.o. P whereas a plane electromagnetic wave with the other polarization, i. e. 1 \i ikx produces a geometrical optics far scattered field (Hs) = / s e (4.2) z g.o. 2 100 -

THE UNIVERSITY OF MICHIGAN 7133-3-T In particular, the geometrical optics back scattered fields are: (E b.) = - ik(P-2a) (4. la) z g.o. V2p for E parallel to the axis, and (Hb- )g) a e ik(P-2a) (4.2a) z go. 4E/ 2p= for H parallel to the axis. In Section 4.3 it is shown that the geometrical optics fields given by (4. 1) and (4.2) are the leading terms of asymptotic expansions of the exact fields. The physical optics approximation represents a refinement of geometrical optics. It is recalled that the physical optics approximation method consists of two steps. Firstly one obtains the total electromagnetic field on the illuminated portion of the surface of the scatterer by assuming that at every point the incident field is reflected as though an infinite plane wave were incident on the infinite tangent plane, and the field on the shadow portion of the scatterer is assumed to be zero. Then, an integration over the illuminated surface of the body gives the scattered field. The physical optics current density J on the surface of a perfectly conducting cylinder has been given by Riblet (1952) for both polarizations of the incident plane wave. If i ^ ikx E i e Z then from (2. 41): J ( -\ iA 2 H'. ikzcos0 37T J = J0)i 2 =-i 2 sine, for < 0< p=a (4.3) '0x, for | < 2 2 101

THE UNIVERSITY OF MICHIGAN 7133-3-T whereas if A t- ikx HI = i\E//X e then from (2.42): _ = J~ ~ - 2H = - 2 IE/pe for z 0 ~ 2 p=a (4.4) 0, for I|1<j If the expressions (4.3) and (4.4) are substituted into formulas (2.51) and (2.50) respectively and an approximate evaluation of the integrals is carried out, the physica optics approximations to the scattered fields ES and H are obtained. z z Riblet (1952) considered (4.3) and (4.4) as the leading terms of asymptotic expansions of the current in inverse powers of ka. By substitution into Maxwell's equations and imposition of the boundary conditions, he was able to determine a first order correction to (4.3) and (4.4). The absolute value of the ratio between the correction terms and the leading term is equal to [ka sin3 (/2)1. These corrected physical optics currents represent an improvement with respect to the approximations (4.3) and (4.4) only in the angular range 27r/3 < 0 < 47r/3. Finally, we shall give a brief account of the Luneberg-Kline method (see, for example, Keller et al, 1956) for obtaining the high frequency expansion of the field reflected by an arbitrary obstacle, and shall state the explicit results for a circular cylinder. Assume that the wave function O., (V+ k2)~1 = 0, has an asymptotic expansion of the form 00 eik~ v (x,y,z) - e, -k)n as k —oo. (4.5) n=O (ik)n Inserting (4.5) into the wave equation and equating to zero the coefficient of each E 102

THE UNIVERSITY OF MICHIGAN 7133-3-T power of k, we find: (V )2 = 1, (4.6) 2Vv V + v V2 = -v2v (4.7) (n=0,1,...; v 1= 0) The eiconal equation (4.6) determines the phase function A, whereas the v 's are n obtained from (4.7) by iteration. If s denotes the arc length along an optical ray (i.e. a curve orthogonal to the wavefronts e = constant), then the solution of (4.7) can be written in the form: V8'1/2 -W [. v (s) = v(S.) F G(s) 2 1G(s 1/2 rG(t) 11/2 V2v (t)dt (4.8) n n o LG 2 n- 1 o 0 where G(s) denotes the Gaussian curvature or, in two dimensions, the ordinary curvature, of the wavefront ( = constant at the point s on a ray. In particular, it is easily seen that v0 varies along a ray as the inverse of the square root of the cross sectional area of a narrow tube of rays, as was previously found by energy conservation. This method has been applied by Keller et al (1956) to a variety of problems, among which is the reflection of a plane wave by a large circular cylinder. For a i ikx soft cylinder (11 = 0) and = e, the reflected field at a point P(p, 0) is (see Fig. 4-1): pa 103

THE UNIVERSITY OF MICHIGAN 7133-3-T 1 ( a. 3 *^~ ( refl. 2 es 2sin 2 x oD 3n n I 0 2 h- 2 X -_ahn 16 ika sin ) (a/2) sin- 2 (4. 9) n=O h=O =0 where the angle 0 is shown in Fig. 4-1. The coefficients ahn satisfy the following recursion relations for h $ 0: ahi = h1 (2h+4i+2n-3)(6h-4 -2n-1)ah, 1 n + + (2h- 41 - 2n+ 5)(2h- 41 - 2n+3)ahl, 1, n- + +[24(h- 1h- 2 - n)- 6 ah_2 n-1 + + 12(1 - h)(2h- 4- 2n+3)ah-2, -, n- + +9(2h-5)(1-2h)ah3 n1 + 9(2h-5)(2h- 1)ah3.1-1 n-1} while for h = 0 we have: 3n a p, a = -2. aofn 1 hln 'ooo particular, t he first few te s of he series (4. 9) are: In particular, the first few terms of the series (4.9) are: (4.10) (4.11) 104

THE UNIVERSITY OF MICHIGAN 7133-3-T ('1srefl. - exp k (s- a sin 2)] + 16ka 2s sm smn sm sin 2 2 2 + )2 - 3 9sin+o2s (s)3 (15sin 2 15). sm 2 (4.12) In a plane z = constant, the family of wavefronts is a family of parallel curves with the optical rays as common normals; these rays do in general possess an envelope which is called the caustic of the wavefronts; the distance s which appears in (4. 9) and (4.12) is measured along a ray from the caustic to the observation point P(p, 0). With reference to Fig. 4-1, we have that a 0o s = BP+ 2sinm- (4.13) In particular, in the far field (p -A ow, 0 ~ 0): a 0 s *P- sin 2 (4.14) so that formula (4.12) becomes: sp ' i2' ik(p- 2a sin2) siref. s e (p — >co), (4.15) where (Kellereft al, 1956): where (Keller et al, 1956): 105

THE UNIVERSITY OF MICHIGAN 7133-3-T A - 1+ - 1 (3477 7218 + 38174.15a) (16ka) \in 2 sm s2 sin 2 and, in particular, A ~ -1 as ka becomes very large. i ikx For a hard cylinder (82/8pI = 0) and 2 = e the reflected field at a p=a point P(p, 0) is still given by (4. 9) and (4.10), but (4.11) must be replaced with the following (Keller et al, 1956): 3n ao - ah + 16(21 +n- l)ahl n-i + 16(4- 2 - n-2h)ah, l 1 n-1] own Ri-dLnh h-l,jQn-l h-,-l, n-lj | n=l (4.16) a =+2. 000 Explicitly, the first few terms of the expansion are: (02 refl. - smi- exp k 2(s-asin2)X j' -16 [ 80 + 3p + 2a ( - -; + sin 2 a sin (rf asin- e Si2 (4.17) 2 a(3 a3 o 6 2 _0 where s is given by (4.13) and is indicated in Fig. 4-1. In the far field, approximation (4.14) applies and result (4.17) becomes: (s)refl. B -sin e ( - n), (4.18) I 106

THE UNIVERSITY OF MICHIGAN - 7133-3-T where (Keller et al, 1956): IBI " 1 + ( 3477 6642 + 3049 + (4.18a) (16ka) sin 2 sin 2 sin 2 and, in particular, B '~ 1 as ka becomes very large. The far fields (4.15) and (4.18) include the terms 0(k ) in the asymptotic expansion (4.5) and they coincide to 0(k ) with the results that Imai (1954) obrD tained by saddle-point evaluation of the in the exact expressions (2. 45) and -.2 (2.46), respectively. Imai did not carry his computations through 0(k ). The leading terms in (4.15) and (4.18) are the geometrical optics fields (4. 1) and (4.2), respectively. Keller et al (1956) plotted the amplitude of the back scattered fields (0 = r) vs. ka, for both polarizations and for 1 < ka < 4, and the amplitude of the far scattered field vs. 0, for both polarizations and for ka = 4, 40 and infinity; their diagrams are based on formulas (4.15) and (4.18). The Luneberg-Kline method does not take into account the diffraction effects, but considers only the reflected part of the scattered field; the remaining part is the so-called creeping wave contribution, which is described in the following sections. 4.2 Geometrical Theory of Diffraction The geometrical optics approximation does not account for the presence of nonzero scattered fields in the region of geometrical shadow, and often represents an insufficiently accurate approximation in the illuminated region. A better approximation is represented by the so-called geometrical theory of diffraction of Keller, which is an extension of geometrical optics. For a description of this theory, the reader is referred to a paper by Keller (1956), in which the extension of the laws of optics is presented in two equivalent forms. In the first form, the different situations in which diffracted rays are produced and the different kinds of diffracted rays which occur in each case are explicitly described. The second formulation is based on an extension of Fermat's principle. The equivalence of the two formulations 107

THE UNIVERSITY OF MICHIGAN 7133-3-T follows from considerations of the calculus of variations. Keller's theory assigns a field value, which includes a phase, an amplitude and, in the electromagnetic case, a polarization to each point on a ray. The total field at a point is postulated to be the sum of the fields on all rays which pass through the point. Keller's theory has been developed for both scalar and vector fields and for objects of various shape and type (e.g., acoustically hard and soft bodies, perfect conductors, dielectrics); the results depend upon the nature of the object in an essential way. For example, a very detailed application to the diffraction of a scalar or vector wave by a smooth convex opaque object of any shape has been made by Levy and Keller (1959). From its similarity to geometrical optics, Keller's method can be expected to yield good results when the wavelength is small compared to the obstacle dimensions. However, it has been found that in most cases the results are useful even for wavelengths as large as the relevant dimensions of the scatterer. An important advantage of the method is that it does not depend on separation of variables or any similar procedure, and it is therefore especially useful for shapes more complicated than a circular cylinder. In fact, in the case of a circular cylinder, the solution obtained by the geometrical theory of diffraction coincides with the leading terms in the asymptotic expansion of the exact solution for large ka (Levy and Keller, 1959). If R is the distance between the observation point (p, 0, z) located off the cylinder surface (p > a) and a line source parallel to the cylinder axis and located at (p = p >a, =0) and if i i1 2 4 i =i -1 k e 4 (4.19) is the incident field, then the scattered field bs may be written as s g.Os +o,' (4.20) g.o. d 108

THE UNIVERSITY OF MICHIGAN 7133-3-T where rg is the geometrical optics field (which, in particular, is zero in the g.o. geometrical shadow), and d is the diffracted field which is given by (Levy and Keller, 1959) S(p.0) = (tk)12 [(p2 a(p- a2) - a2 X rexp2ik 2 /2 2 21/2] i7r X exp [ika-aa)0 + exp ika -aa )(27r - D 1-exp [2r(ika-acer), X exp - (ika- aa) (arc cos - + arc cos ). (4.21) P P The diffraction coefficients D1 and the decay exponents aI are determined by comparing (4.21) with the leading terms of the asymptotic expansions for exact solutions -1 a, = ik-ivla, (4.22) (7 [ (2) 1/2 i /4 OH (ka) DI = e k a O (1) k 4 (1) =. (4.23) I=V The expressions on the right-hand sides of Equations (4.22) and (4.23) are defined below, in Section 4.3. Values of ca and D based on these equations, and the operator S are given in Table I for the three types of boundary conditions considered in this report (Levy and Keller, 1959). In this table, A(q) is the special representation for the Airy integral which has been employed by Keller and Franz, and which is related to the integral Ai of Miller (1946) by the equation: 109

THE UNIVERSITY OF MICHIGAN 7133-3-T 1-4 E5 I 110

THE UNIVERSITY OF MICHIGAN 7133-3-T SO A(q) = S cos(t3- qt). (4.24) 0 For a scatterer of general shape, the diffracted field is given by formulas of which (4.21) is a particular case (see, for example, formula (11) of part one in the paper by Levy and Keller (1959)). These formulas involve the incident field, various geometrical quantities, the diffraction coefficients Dl and the decay exponents a. If we assume that the leading terms in De and a, depend only upon the radius of curvature of the scatterer's surface in the normal plane tangent to the optical ray and on no other geometrical property of the scatterer, then DI and aI can be determined from the field diffracted by any object of simple shape, e.g. a circular cylinder. Thus, the geometrical theory of diffraction is of no help in determining the high frequency behavior of the field scattered by a circulary cylinder; on the contrary, it is the knowledge of this behavior (achieved by asymptotic expansion of the exact solution) which allows us to determine in the easiest way the geometricall diffracted field for any smooth convex opaque object. Of course, small correction terms to D1 and a, do involve other geometrical properties of the scattering surface, and in order to determine these additional terms it is necessary to consider the particular shape of the scatterer. If the line source is removed to infinity and we take into account only the first term ( = 0) in the series (4.21), and if the electric field of the incident plane i ikx wave is E = i e,then the far back scattered field is given by (Levy and Keller, " Z 1959): Eb.s. - ik( - 2a)Jk (4.25) where (425) where 111

THE UNIVERSITY OF MICHIGAN 7133-3-T J(ka) = 1-4T3264/3 [A6(q?)] (ka)F16 exp [-7rq 61/(ka)/3 sm X Xexp - + a(r+2)+i7rqol6 /(ka) /3cos 3, (4.26) q' = 3.372134 is the smallest zero of the Airy function A(q), and Ai(q') = -1.059053. 0 0 The first term on the right-hand side of (4.26) represents the geometrical optics contribution, in agreement with (4. la). The second term in (4.26) represents the diffracted field, and may be neglected to the level of accuracy of formula (4.25). Thus IJ(ka)l = 1, as is shown by the broken line in Fig. 4-2. i A ikx For the other polarization (rigid cylinder), such that H = i t/h eik the - z far back scattered field is approximately given by (Levy and Keller, 1959): Hb'S Av eik(p-2 H(ka), (4.27) z 2p where H(ka) = 1+2r 3/261/3 1 (q (ka)1/6 exprq61/3(ka)1/3 sinr i -1/ /33 X exp [2+ ika(7r+2)+irqo 6 /3(ka) /3cosj], (4.28) q = 1.469354 is the smallest zero of the derivative A'(q) of the Airy function, and A(q ) = 1.6680. The first term in (4.28) represents the geometrical optics contribution, in agreement with (4.2a), whereas the second term represents the diffracte field contribution. The quantity IH(ka)| is plotted in Fig. 4-3. If the geometrical optics portion of (4.25) and (4.27) is replaced by the asymptotic expansion of the reflected field in inverse powers of ka (Keller, Lewis and Seckler, 1956), then higher order corrections to the diffracted field must also be introduced. The most important of these corrections deals with the decay exponent a, and for cylinder and sphere it was found from an analysis of the asymptoti 112 I

THE UNIVERSITY OF MICHIGAN 7133-3-T 1. 1 1 1 0. 1 2 3 4 5 6 7 8 9 10 FIG. 4-2: AMPLITUDE OF FAR BACK SCATTERED FIELD FOR A SOFT CYLINDER, NORMALIZED TO THE GEOMETRICAL OPTICS VALUE a/2p; THE DIFFRACTED RAYS HAVE A NEGLIGIBLE EFFECT (Levy and Keller, 1959) 113

THE UNIVERSITY OF MICHIGAN 7133-3-T 1 2 3 4 5 6 7 8 9 10 FIG. 4-3: AMPLITUDE OF FAR BACK SCATTERED FIELD FOR A RIGID CYLINDER, NORMALIZED TO THE GEOMETRICAL OPTICS VALUE {a/2p. (Levy and Keller, 1959). 114

THE UNIVERSITY OF MICHIGAN 7133-3-T expansion of the exact solution by Franz (1954). With these modifications, the quantities J(ka) and H(ka) in Eqs. (4.25) and (4.27) are replaced by J(ka) and H(ka), respectively, where (Levy and Keller, 1959): 5i 127 J(ka) = 1+ + 2 - 16ka 512(ka)2 -2 expi7T ( +2ka+12) 4r3/264/3 [A(q)] (ka-l/6 e -(2 12) (4.29) 3/2 1/3 1 r [ -/2 1/6 exp e (v1 +2ka+ i2) +2i 3/26- /3q A1(qj (ka)1/ 1ep(12i (4.30) with:..* 2 + a 1/3 3 1/ 2 e 1 20 _ 180 1/3 i a1/3 -i - q21 v ka\+ q ) / e (i Oq 3o) 1 (4.32) The quantities IJ(ka)J and I|H(ka)l are plotted in Figs. 4-2 and 4-3, respectively. 4.3 Asymptotic Expansions of Exact Solutions In this section, we shall review the main results obtained by Franz and his collaborators (Franz and Deppermann, 1952; Franz, 1954; Franz and Galle, 1955), Imai (1954), Wetzel (1957), and by the school of Fock (Fock, 1945, 1946; Goriainov, 1958). It is oDviously impossible to give an account of the many papers written on M 115

THE UNIVERSITY OF MICHIGAN 7133-3-T high frequency scattering by circular cylinders, and therefore only those works which contain new important information are explicitly mentioned, whereas many others are simply listed in the bibliography at the end of this report. We shall first consider the case of line sources parallel to the cylinder axis, and then discuss in some detail the case of plane waves at normal incidence, for both polarizations. Let us consider a line source located at (p = p, =0) parallel to the axis of the cylinder, and such that = H((kR), (4.33) 4 o where, as usual, R is the distance of the observation point from the line source. Then the total field;b must satisfy the equation -1 (V2+k2) = -p 6(p-p )6(- ), (p>a) the boundary conditions at p = a and the radiation condition (2.19) at infinity. One has that (see Eq. (2.79) where, however, the source is at 0 = 1800):. S 2J (ka) (1) ~ (1) E L - (k p H- (k H (kp>)cosn (p a), (4.34) n=O n-n OH (1)(ka) n where p< and p> are, respectively, the larger and smaller of p and p, and the operator 2 is given in Table I of Section 4.2 for the three types of boundary conditions considered. The sum (4.34) may be written as eiv(0 - ) H )(kp>) 8 sin(r ) (1)(ka) X LH1) (ka) J(kp-<)-Jv(ka) H (1k)( j dv (4.35) V V V J 116

-- THE UNIVERSITY OF MICHIGAN --- — 7133-3-T where the contour C1 encircles the entire real axis in the clockwise direction, or also, replacing v with -v on that part of C for which Im v < 0, as coS L(-7r)] H (kpa) -4 = -ssin(7 H())(k 2 v x [H)(1)J (k)J (k - J (ka)H(1)(kp<)]d, (4.36) where the contour C2 is in the upper half-plane, just above the real axis and running parallel to it. Both results (4.35) and (4.36) are as exact as the series solution (4.34). The contour integral (4.36) has been asymptotically evaluated by Franz (1954) If the observation point (p, 0) lies in the geometrical shadow, the contour C can be 2 closed in the upper half-plane and the integral evaluated by computing the residues at the zeros v of OH (ka): i-/ OH o2- H(ka) = T z clv l H1(kp )H 1(kp). (4.37) 4 sin(7rv) 1a0 H(1) () V V This residue series converges rapidly in the shadow region (source and observation point geometrically invisible to each other) and very slowly in the illuminated region. Therefore, if the field point belongs to the illuminated region, it is convenient to split the integral (4. 36) into the sum of two integrals by means of the relation cos L(0 - 7r)] = eivr cos(vo) - ie iv sin(v7r), (4.38) and then convert one of the two integrals into a residue series, obtaining (Franz, 1954): F i 117

THE UNIVERSITY OF MICHIGAN 7133-3-T ' cos(l',)1 iVW OH{2 (ka) (1) (1) = - ' e FI H (1)(kp )H (kp) + 4 sin(vr) H(1ka) Vi 0 I9) + X ej (1 ) H(L (ka)H 2)(kp) - H 2)(ka)H (kp d. (4.39) 8 OH(1)(ka) v v v 2 V The residue series in (4.39) converges everywhere except in the forward direction 0 = r; however, (4. 39) is of interest only in the illuminated region, since in the shadow we may profitably use the simpler representation (4.37). The contour integral in (4.39) may be asymptotically evaluated by saddle point technique, and represents the sum of the primary field and the reflected part of the scattered field, whereas the residue series represents the creeping wave contribution to the scattered field. The quantities q, and (2) OH ()ka) K-; (ka) v (J1 are given by the second and third columns of Table I in Section 4.2, and ve is related to q- by: 1/3 if v-ka+( ) e q1 +... (4.40) In order to compute t from (4.37) or (4.39), it is necessary to know the positions of source and field point, so that the appropriate asymptotic expansions of the Hankel functions of arguments kp and kp may be used. For example, if both line source o and observation point are very far from the cylinder (p - co, p — co), then (4.37) becomes: 118

THE UNIVERSITY OF MICHIGAN 7133-3-T expk(P a /+p(- a2)1/ ^-} exp(iuv)+exp iV(2 7r -0i 2ik (p2- a2)(p2- a21/4 1 1 -exp(27riv) O (2) a(ka) exp (i arc cos a + arc cos-)] L[a (1) (4.41) v _ v=v= In the remaining part of this section we consider in some detail the case of a plane wave incident perpendicularly to the axis of the cylinder. The first rigorous treatment of this problem in the high frequency region is due to Franz and Deppermann (1952), who introduced the concept of creeping waves; they based their derivations on Maue's integral equation (1949). The same problem has been treated by Imai (1954) in a different way; he starts from the solution for the scattered field in the form of an infinite series and transforms it into contour integrals (see formulas (2.45) and (2.46)), which he evaluates by the saddle point method and the residue theorem obtaining the scattered field at a large distance from the cylinder. Specifically, a saddle point evaluation of the two integrals \ in formulas (2.45) and (2.46) gives the reflected part of the scattered field, whereas the other integrals F are transformed into residue series whose terms correspond to the creeping waves of Franz and Deppermann. The results of Imai are not given here, because more accurate approximations were derived by Franz and Galle (1955)*. The convergence of the asymptotic series obtained by Franz and Galle (1955) is rather poor over certain regions of the azimuthal angle 0. This inconvenience Imai (1954) pointed out some errors in the derivations of Franz and Deppermann (1952); for example, he showed that their contributions due to reflections from the shadow side do not exist. In turn, Franz (1954) remarked that the numerical values which appear in Imai's formulas (7.8), (7.18), (7.22) and (8.16) are incorrect. I 119

THE UNIVERSITY OF MICHIGAN 7133-3-T has been avoided by Goriainov (1958), who has derived asymptotic expansions for both surface current and far field and for the whole range 0 < 0 < 27r by applying a technique developed by Fock (1945, 1946)*. Goriainov's results for surface current density and far field differ by at most a few percent from the exact results for ka= 5 and, of course, become more accurate as ka increases. In the following, besides the azimuth 0, 0 < 0 < 2ir, we shall introduce the angle C, -r < r <7r, defined as? = 0, for 0 <, = p-27r, for > r. Also, we shall make systematic use of the symbol m, defined as: m = (ka/2)1/3. Let us consider a plane electromagnetic wave incident in the direction of the negative x-axis, such that Ei A -ikx Hi A ' e-ikx E ie, H=i E/.e. (4.42) z - y The current density on the illuminated portion of the surface (p = a, I j< 77/2) is given by (Franz and Galle, 1955): For a history of what is presently known as the Fock method see the first volume by Logan (1959), in which a detailed discussion of the notations employed by various authors is also given. A brief exposition of Fock's theory has been given by Goodrich (1959), and an excellent treatment of high frequency diffraction methods in general may be found, for example, in a paper by Logan and Yee (1962). 120

I THE UNIVERSITY OF MICHIGAN 7133-3-T = (+IH) ^ 2 \cos0e 01+ -- +3si0 + z 0 0p=a L 2ka cos30 2(ka cos30)2,J. 71 I- 13 _-1 + i e/ e m D n n exp vn(2-)] + expivn (+ 2 1-exp(i27rv ) n (4.43) where X.r 2 3 3 -1 3 n 1 v r ka + e ma -e m n n 60 7Oka (1 - 3 -5 1 / ( 281 4 + e m I 12600a - a +. 12600 n360n / (4.44) 1 D ~ 1 13 -2 D -1+e m n Ai'(-a ) andthe a 's (n=1,2,...) are (with a change in sign): 2 3 a -i 2 59a 2 223a3 n 3 -4 n +...1 2 2(37 3- )+ - 7-e --- +-... 10 12600 3150(ka)2 18 (4.45) the zeros of the Airy integral in Miller's notation Ai(-a ) = 0. n (4.46) The first few values of a and Ai'(-a ) are given in Table I (Logan and Yee, 1962). n n The first group of terms in (4.43) represents the optics contribution to the surface current, and the first term itself is the geometrical optics current; this development is numerically useful only if kacos >> 1. The summation over n represents the creeping wave contribution to the surface field. m 121

THE UNIVERSITY OF MICHIGAN 7133-3-T TABLE II n a Ai(-a ) n n 1 2.33810 74104 59767 +0.70121 08227 20691 2 4.08794 94441 30971 -0.80311 13696 54864 3 5.52055 98280 95551 +0.86520 40258 94152 4 6.78670 80900 71759 -0.91085 07370 49602 5 7.94413 35871 20853 +0.94733 57094 41568 The current density on the shadowed portion of the surface (p = a, < 7r/2) is given by (Franz and Galle, 1955): 1I - 1 J = (-H H'Hp z 0pa - 3 -1 iE/ e m x ~ iue/e 3m X X D n n exp[w(0- 2)+ exp[n (f2 1- exp(i2irv ) n (4.47) where v and D are given by (4.44) and (4.45). The creeping wave series (4.47) n n is no longer useful for computational purposes when one approaches the shadow boundary = t+r/2. An alternative representation of the surface current, which is especially useful in those angular regions where (4.43) and (4.47) fail to converge rapidly, has been derived by Goriainov (1958). An expansion which may be profitably employed in the transition region about the shadow boundary II-Il7-<,, m' (4.48) is the following: 122

THE UNIVERSITY OF MICHIGAN 7133-3-T Jz = (H +H)pa i/ -1 [ )(mn)e lka+ f(0)(m) ika z 0 0 p~a I =0 I — (4.49) where r1 = 0- + 27r, The Fock afcton function () is a defined as (see, for example, -r1 = +27,. (4.50) particular case of the more general function f (e), Logan (1959), vol. 1): f(>)()= - Ir eit I wl(t)d 1 (-OD < < OD), (4.51) where wl(x) = 7 [Bi(x) + iAi(xj 2 (4.52) with Ai and Bi Airy integrals in Miller's notation (1946), and r is a contour which starts at infinity in the sector 7r/3 < argt < 7r, passes between the origin and the pole of the integrand nearest the origin, and then ends at infinity in the sector -7r/3 < argt < 7r/3. For e > 0, one can write f(i)() as a residue series: 57r *.-...Ir - A,, 6 (e) = e 6 n en n Ai(-a ) e n n (4.53) Values of f( )() are given by Logan (1959, vol. 2) for I =-1(1)5 and g = 0. 5(0.1)4. 0. * The values tabulated by Logan correspond to our Eq. (4.53); in the headings of his tables, a factor (-1) is missing. 123

THE UNIVERSITY OF MICHIGAN 7133-3-T If, in the first approximation, we limit ourselves to the first term (S = O0) in the series (4.49), we find that after some manipulation (see Goriainov, 1958) we may write: JZ i m m-1 0)(o ) ea o + Fi(mkacos(7r - 0)) j~ i ( /co ~-~ Z7 OO (O 0 r / 2) (4.54) where 3 F(9) = f(O)() e 3 (4.55) Formula (4.54) provides a smooth transition between (4.49) and the geometrical optics result. At large distances from the cylinder surface (p > a), the incident wave (4. 42 produces the scattered field (Franz and Galle, 1955): - ik(p - 2a cos E O \ cos e z 2 2 3i 16kacos 2 i + 15 33 5 2kacos3 512(kacos-) 32(kacos2 )2 4(kacos ) 2 2 2 2 + m(2kp) /2exp p + i)J n C n -1 / 4 2 1 \ X ^+i 8kp + 8kp + exp [vn(r~~l~~E - n to -exp(i2rvn) nI (4.56) (4.56) f where 124

THE UNIVERSITY OF MICHIGAN 7133-3-T 13 -23 -i -4 n C - Ai(- a 1+e m - +e m - + n n30 1400 281 a2 +_ 1 K29- n)+...j, (4.57) 3150(ka)2 9 and v is given by (4.44)*. The first group of terms in (4.56) represents the optics n contribution to the far field, and the first term itself is the geometrical optics field; this development is numerically useful only if kacos (r/2)>> 1. The summation ove n represents the creeping wave contribution to the far scattered field, and is prac-1 tically applicable only for 0 - Ir I > m. In the far field (p - co) and in the back scattering direction ( =0), equation (4.56) becomes (see (2.67), in which f(0, 0) is now PE): Es ikp -i E,PL PJ e 4 (4.58) z E Vrkp where 1 4 - i2ka1 5i 127 + 1sn Crn P ~ V-r kae + + +. me E 2 6ka 512(ka)22 n (7 (4.59) In the far field (p -> oo) and in the angular region 0 - r << m, the dominant contribution to the scattered field may be written in the form (4.58) with PE given by (Goriainov, 1958): * - The quantity C1 of Franz and Galle (1955), given in their formula (17a), contains an error: the factor 3/x6 in the denominator must be replaced by 61/3 125

THE UNIVERSITY OF MICHIGAN 7133-3-T PE --- -im i;p(m(-7r)) e (0 )+p m(r- -)) e p, (4.60) where 1 1 v(t) P(9) = expt) t (4. 61) r with wl given by (4.52) and 1 2 1/2 i (t) v() = e exp(it) dt (4.62) VW =oe xp( wI(t) -0o The reflection coefficient function p(.) is tabulated in Logan (1959, vol. 2) for F =-1.60(0.01)1.60, and for = -3. 0(0. 1)2. 0. In the particular case of forward scattering (0 = r), a more refined approximation has been derived by Wu (1956); the forward scattered field is still given by (4.58) with PE'-ka-M m —Mm +1 i 121 +-MIm3 1 + 1 -1.T (l+ ojM2)m - 1 K9M~ 281+ E o 30 m 140 26003 1 361M 73769 7 +5821200 ( 36 0M4 m +... (4.63) where 7r.27r 3 3 M = 1.2550 7437 e M1 = 0.5322 5036 e 0 '. T 3 M2 = 0.0935216, M = 0.772793 e, 2 3 2-r. 21r M4 = 1.0992 e. (4.64) 126

THE UNIVERSITY OF MICHIGAN 7133-3-T It is easily seen that for p = iT, formula (4.60) coincides with the first two terms of the expansion (4.63). The total scattering cross section follows immediately from (4.63) and from the forward scattering theorem (2. 69); the explicit result is given in Section 4.5. The previous treatment from (4.42) to (4.64) contains the most relevant results for the case of E-polarization. We shall now investigate the case of H-polarization (hard cylinder), namely, of a plane electromagnetic wave incident in the direction of the negative x-axis, such that E = -1i V e-i Hi A -ikx - i e — Z; (4.65) (notice that all results from (4.65) to the end of this section are normalized with respect to the incident magnetic field). The current density on the illuminated portion of the surface (p = a, | 1 < 7 /2) is given by (Franz and Galle, 1955): J = - (H + Hs) p z z p=a -ikacosp 0 i 1+ 3 sin. - - 3 32 +... + 2ka cosL (kacos 0) + D n n exp lvn(- - ) + exp iv (. + )J 1 - exp(i2rn ) n (4.66) where * 7r 1i 3 v s ka+e m3 -e n n -1 0 10m 1 (3 +4+) - (oon3 7 n i -5/ 6113 28113n\ 3 -5 1 (-5 2 + n n -e m 2000 — + + + 2000 n 2 63 2268 3~n (4.67) 127

THE UNIVERSITY OF MICHIGAN 7133-3-T.7r 7.T D Ie 3 m-2 (1 + -n +e 3m l -4 21 (-4 +e m \1-6 \) + 3 1+ 1 3 353 n ^ 6(ka)2 5003 7875 +300 (4. 68) and the 3 's (n = 1, 2,...) are the zeros of the derivative of the Airy integral in Miller's notation (with a change in sign): Ai'(-j ) = 0. (4.69) n The first few values of:n and Ai(-3 ) are given in Table II (Logan and Yee, 1962). TABLE I n n Ai(-Bn) 1 1.01879 29716 47471 +0.53565 66560 15700 2 3.24819 75821 79837 -0.41901 54780 32564 3 4.82009 92111 78736 +0.38040 64686 28153 4 6.16330 73556 39487 -0.35790 79437 12292 5 7.37217 72550 47770 +0.34230 12444 11624 I The first group of terms in (4.66) represents the optics contribution to the surface current, and the first term itself is the geometrical optics current; this development is numerically useful only if kacos >> 1. The summation over n represents the creeping wave contribution to the surface field. The current density on the shadowed portion of the surface (p = a, I0- 7r < 7/2) is given by (Franz and Galle, 1955): expfiV I + expif f VT = -(H + H D (4.70) z zp=a n - exp(i2?r n n le m 128

THE UNIVERSITY OF MICHIGAN 7133-3-T where v and D are given by (4.67) and (4.68). The creeping wave series (4.70) is n n no longer useful for computational purposes when one approaches the shadow boundary = + 7r/2. An alternative representation of the surface current, which is especially useful in those angular regions where (4.66) and (4.70) fail to converge rapidly, has been derived by Goriainov (1958). An expansion which may be profitably employed in the transition region about the shadow boundary |l-F 2 | m, (4.71) is the following: I(0)m ikal ( - ika - (0) Jf = -(H + Hs ) - E (~)(-gm ) e k S + (4.72) where rn and rl are given by equations (4.50). The Fock function g )() is a particular case of the more general function g (e), defined as (see, for example, Logan (1959), vol. 1): r 1 1 () where w;(t) is the derivative of w (t) given by (4.52), and r is the contour previously defined for (4.51). For e > 0, one can write g (5) as a residue series: n 1-1 - 9-( ) ( )e- Ai(-j e (4.74) g (= ) e. Ai(-n) e n n Values of g (e) are tabulated by Logan (1959, vol. 2) for I = -5(1)5 and = 0.5(0.1)8. 0.* * The values tabulated by Logan correspond to our equation (4. 74); in the headings of the tables, a factor (-1) is missing. 129

THE UNIVERSITY OF MICHIGAN 7133-3-T If, in the first approximation, we limit ourselves to the first term (1 = 0) in the series (4.72), we find that after some manipulation (see Goriainov, 1958) we may write: J )(m( )e ) O-G (mcos(7r- )ei'kcos(7r), (0, 7r/2), (4.75) where 3 1 3 G(9) = g( ({) e 3 (4.76) Formula (4.75) provides a smooth transition between (4. 72) and the geometrical optics result. At large distances from the cylinder surface (p > a), the incident wave (4.65) produces the scattered field (Franz and Galle, 1955): Is a ik( p-2acos(2 ) sin 2 - 1 3i z 2 8kp 16ka cos-2 2 i 15 33 7 - + 3{ j 2 2 + 2 - '3{\ 2+"':53 512(kacos2) 32(kacos 512 os2 32 os )2 4(kacos 3) ] -1/2 e (,-i.- expiv (7r+) +expiv, (7r- l) m(2kp -1/2 expk l (+)J p 5 - n 1 - exp(i2.n Pn' n n -21 \ Xw +i +ee (477) where I. -- 130

THE UNIVERSITY OF MICHIGAN 7133-3-T n) -2...F;;2) 2 30n 1 -i 3 -4 25(a)2 ( 6 53 1260 11340) (4.78) 25(ka)153 0 n n and v is given by equation (4.67). The first group of terms in (4.77) represents the n optics contribution to the far field, and the first term itself is the geometrical optics field; this development is numerically useful only if ka cos (r/2) 1. The summation over n represents the creeping wave contribution to the far scattered field, and is practically applicable only for | - r >m. In the far field (p - co) and inthe back scattering direction ( = 0), equation (4.77) becomes: s v P ~ ikp- i H2 ~ 4hi:2 -sPe, (4.79) z H rkp where 7r 5ri - i2ka 1 i 1 1 12a 1i 353 1 n Prv 7rkae l k-+a 2+ ' I. H 512(ka) n sin(7rv ) n (4.80) In the far field (p -- o) and in the angular region - T | m, the dominant contribution to the scattered field may be written in the form (4.79) with PH given by (Goriainov, 1958): si ~a( —)! imV q(m(0or)) eika(- r)+qt m p)e(7r-) e' (4.81) where 131

I I THE UNIVERSITY OF MICHIGAN 7133-3-T _1___ + 1v'(t) 48 q(5) = 2 exp(it) v(t) dt, (4.82) ith v and wl given by (4.62) and (4.52), respectively. The reflection coefficient function q(9) is tabulated in Logan (1959, vol. 2) for = -2.00(0.01)2.00, and for =-3.0(0.1)3.0. In the particular case of forward scattering (0 = Tr), a more refined approximation has been derived by Wu (1956); the forward scattered field is still given by (4.79) with m 1- 1 I (t r-4 3 3 3 PH -ka-M m- (M + + 1 3 3 MM - 100 1340 M3 1260 o 15M-3 4 M 400 \5239080 M M+-M — ^M -,m +.. where 3 3 M5 =-1.088874119 e, M =-0.93486491 e - 13 M2 = -0.1070199, M = -0.757663 e, 2 3 27r i - -i -i3 M4 -1.1574 e M2 -3.70409389 e 27r M = 0.41682138 e M = 3.17579652 7r 27r -i -i2 — 3 3 M_5 = 2.55965945 + 3.12247506 e M_6 = 2.06575721e -5 -6 -i3 M = -1.36515171 - 2.94764528 e 3. (4.84) 132 132

THE UNIVERSITY OF MICHIGAN 7133-3-T It is easily seen that for p = ir, formula (4.81) coincides with the first two terms of the expansion (4.83)'P. The total scattering cross section follows at once from (4.83) and from the forward scattering theorem; the explicit result is given in Section 4.5. 4.4 Impedance Boundary Conditions Some considerations on the case of impedance boundary conditions have already been developed in Sections 4.2 and 4.3 (see, for example, Table I in Section 4.2 and the discussion on line sources in Section 4.3). Most authors limit their considerations to the scattering cross section (Lax and Feshbach, 1948; Rubinow and Keller, 1961; Sharples, 1962). An asymptotic evaluation of the reflected field for plane wave incidence can be found in Keller et al (1956). The far back scattered field, produced by a plane wave at normal incidence with the electric field parallel to the cylinder axis, may be obtained as a particular case of the results given by Uslenghi (1964). If the incident field is such that i ^' ikx E= i e, (4.85) z and the impedance boundary condition (2.4) is valid, where Z = rg /7e is the surface impedance, then the far back scattered field may be written in the form b.s ikp- i 4 n E' 7kp e 4A-+2 (l)nAn (4.86) with the coefficient An (n = 0, 1, 2,... ) given by the first of relations (2.36). Treating the summation over n as a residue series, the summation is replaced by a contour integral C in the complex v plane taken in the clockwise direction around the poles at v = 1, 2,...; following a Watson transformation, the contour C is then deformed to include the poles of the integrand which lie in the first quadrant (see Fig. 2-3). Thus, the far back scattered field is obtained as a sum of two contributions: The diagram of g(C) in Fig. 3 of Goriainov (1958) appears to be incorrect. 133

THE UNIVERSITY OF MICHIGAN 7133-3-T b.s. bs bs.. E (E f ) + (Eb ); (4.87) z z refl. z cr.w. the reflected field arises from an asymptotic evaluation of the term containing A in (4.86) and from a saddle point evaluation of the line integral, whereas the creeping wave field is represented by the residue series due to the complex poles of the integrand. One finds that The case in which the relative surface impedance is close to unity is of considerable interest in applications to absorbers; for rT = 1, formula (4.88) becomes: b(E..) Fl -i ikp - i2k a i (_3i b.s 88ka (Ez g.o. (4.89) rn=1 n= The creeping wave contribution is given by (E ) 212 4 e Ee sin(-r v)w (t cn)( + 2)] (4.90) where w is given by (4. 52), w (t) t, m = n n w{(t ) (t imr) (4. 92) 134

THE UNIVERSITY OF MICHIGAN 7133-3-T which may be obtained from the values of wl(t)/w (t) that were computed by Logan and Yee (1962) when t lies in the first quadrant. A similar analysis of the far back scattered field may be carried out for the H-polarization. An approximate expression of the forward scattered field (and of the total scattering cross section) has been found by Sharples (1962) who used an extended form of the Kirchhoff-Fresnel theory of diffraction, and arrived at numerical results for values of the relative surface impedance either large or small compared to unity. The method of Sharples is an extension of a previous work on soft cylinders by Jones and Whitham (1957), and leads to more accurate results than the variational technique developed by Kodis (1958). The quantities v of (4.91), and the corresponding quantities for the H-polarization, are the roots of the equation -1 )r = rn, for E-polarization, H (ka) + ig H (ka) = 0, P (4.93) V L = r, for H-polarization, which have a positive imaginary part; in the particular case rl = 0, the v 's are n given by the asymptotic expansions (4.44) and (4.67) of Section 4.3. The roots of equation (4. 93) have been studied in detail by Streifer (1964), for the two cases in which e = 0(m ) and e >0(1). If we indicate with a and 3 the opposites of the n n roots of Ai and Ai', which are given in Tables II and m of Section 4.3, then (Streifer, 1964):.T.7 2 3 3 -1 n 1 1 v, ka+e ma -e m -1 - - - i n n 60 70ka 10 i -i 7 i 3 e -2 -2 1 -2 ( —2 i 31 13 4 i Sn 2ka 5( 2 1 2 -4 -5 2m +0(m 5) for >0(1) xn (4.94) 135

THE UNIVERSITY OF MICHIGAN 7133-3-T whereas: n -i 37r2 13 e 1 v, ka+e 3 m- e + +/-1 1 n mn 10 \n 6 25ka -i1 2 n - 4 -3 10 -n3 + i~e On m- 4 n ka- 1+0) 3 + 43 28 4n 7r + i3 e 3-2 n + 1 4 nJ m - 25 n 2 i 37. 7f 10 ~11 - 1e 20 \ 2 5pn X (1+ 45n 15j4 n 7r m+3 e ( 7 5 5 25 4 3e 4 287)m n n n 41n 3 -2 i 3 i 3 - - m + m e X 126 05 2 1 \ 2.5(1 21 7 \ 6+O(m -5) 23'7 5/3 20/P 839 n n n n for = O(m-2). for ~=O(m ). (4.95) If the radius of the cylinder is very large compared to the wavelength, then only the first creeping wave, corresponding to that root v1 of either (4.94) or (4.95) which has the smallest imaginary part (hereafter called the "first root"), gives a sizeable contribution to the scattered field. The position of the forst root vl in the complex v-plane is indicated in Fig. 4-4 for = 0, 1, and infinity, and for various values of ka. The position of v1 for two fixed values of ka and for e varying from zero to infinity is plotted in Fig. 4-5. Finally, values of v1 for different values of ka and = 1 are given in Table IV (Streifer, 1964); these values are in good agreement with those obtained by Weston (1963). 136

THE UNIVERSITY OF 7133-3- T MICHIGAN 7 5 Im v 3 1 m4 T T 120 Re v 40 60 80 100 FIG. 4-4: THE FIRST ROOT v OF (4.93) FOR THREE VALUES OF? AND VARIOUS VALUES OF x = ka (Streifer, 1964). 137

00 — 1 ka = 10 5 ka = 20 3 OD 5 3 2 -3-4 2 1.5 1.5 Imv 1 Imv 1.0 3.0 - _/__ Im0v.7 0.75 3 C~ 0.50 30.50 c. 00 0.35 2 -0.3 0.2.20.10 20.10 2 c11 Re v 12 21 Re v 23 (a) (b) FIG. 4-5: THE FIRST ROOT v1 OF (4.93) AS A FUNCTION OF ~, FOR (a) ka = 10 AND (b) ka = 20. (Streifer, 1964) H c! Z rn 1 --' I a - 3 0 o ~

THE UNIVERSITY OF MICHIGAN 7133-3-T TABLE IV First Root l of H(1) (ka) + iH()(ka) 1 v v The = 0. (Streifer, 1964). ka - ka)m ka v - ka)m ]..- = - 1 -=1 4 1.051 + il.300 14 1.118 + il.542 5 1.064 +il.355 16 1.123 + il.561 6 1.075 +il.394 18 1.128 +il.578 7 1.084+ il.425 20 1.131 + i1.592 8 1.092 +il.450 30 1.142 + il.643 9 1.098 + il.471 40 1.148 + i1.675 10 1.103 + il.489 50 1.151 + il.699 12 1.112 + i1.518_ 4.5 Radar Cross Sections In this section, we shall state the principal results on high frequency back scattering and total scattering cross sections for a perfectly conducting cylinder, and mention briefly the various techniques which have been used in the case of impedance boundary conditions. The geometrical optics approximation to the back scattering cross section per unit length of the cylinder is given by 0o. g.o. = ra, (4.96) and is the same for both polarizations. The agreement of (4.96) with the exact results is excellent even for relatively small ka in the case of E-polarization (see Fig. 2-10), whereas it is unsatisfactory for H-polarization (see Fig. 2-11). A more refined approximation to the back scattering cross section is obtained by computing the far back scattered field with the aid of the formulas given in Sections 4.3 and 4.4. The formulas of Section 4.3 may be used to compute the bistatic cross section. 139

. THE UNIVERSITY OF MICHIGAN 7133-3-T For many practical purposes, it is sufficient to determine only certain average characteristics of the scatterer, such as, for example, the total scattering cross section. According to the forward scattering theorem of formula (2.69), which holds for both polarizations, atota can be easily derived from the forward scattered field. Thus, in the case of the E-polarization, it follows from (4.63) that (Wu, 1956): (at tal)E ~ 4a +0.49807659(ka)2/3 - 0.01117656(ka)4/3- 0.01468652(ka)-2 + + 0.00488945(ka)8/3 +0.00179345(ka)10/3 +.. (4.97) whereas for the H-polarization, it follows from (4.83) that (Wu, 1956): (atotal)H 4a i- 0.43211998(ka) 2/3- 0.21371236(ka)-4/3 + 0.05573255(ka)-2 - 0.0005534(ka)8/3 + 0.002324932(ka)10/3 +.. (4.98) In particular, the geometrical optics atotal is given by (a ) =4a, (4. 99) (total)g.o. 4a99 for both polarizations. The total scattering cross section, normalized to its geometrical optics value (4.99), is shown in Fig. 4-6 for E-polarization and in Fig. 4-7 for H-polarization. In both figures, the exact value computed from the exact series solution (such as (2.71) for E-polarization) is shown in full line; the approximate values given by the first few terms of (4.97) and (4.98) are shown in broken lines. It is seen that the first three terms of (4.97) give an excellent approximation to the exact value of ata for all ka > 1, whereas in the case of (4.98), the first three terms represent a good approximation for all ka 4. The technique employed by Wu (1956) to arrive at (4.97) and (4.98) allows us to find any finite number of terms in the asymptotic series. It consists in solving 140

THE UNIVERSITY OF MICHIGAN 7133-3-T 1.5 1.4 1.3 1.2 1.1 1.0 0-% 0 b v 0 4 8 ka 12 16 20 FIG. 4-6: NORMALIZED TOTAL SCATTERING CROSS SECTION u /(4a) AS A FUNCTION OF ka, FOR ELECTRIC FIELD PARAL9DL TO AXIS; (I) GEOMETRICAL OPTICS WITH ONE CORRECTION TERM, (H) GEOMETRICAL OPTICS WITH TWO CORRECTION TERMS. (King and Wu, 1959) 141

THE UNIVERSITY OF MICHIGAN 7133-3-T 1.0 0. 9 0.8 0.7 0.6 0.5 -$ 0 4r. X -r ""& 0 4 8 ka 12 16 20 FIG. 4-7: NORMALIZED TOTAL SCATTERING CROSS SECTION a /(4a) AS A FUNCTION OF kg, FOR MAGNETIC LD PARALLEL TO AXIS; (I) GEOMETRICAL OPTICS WITH ONE CORRECTION TERM, (II) GEOMETRICAL OPTICS WITH TWO CORRECTION TERMS. (King and Wu, 1959) 142

- THE UNIVERSITY OF MICHIGAN 7133-3-T the reduced wave equation in the region outside the cylinder by considering this region as a Riemann surface with infinitely many sheets; this procedure is essentially different from that given by Franz and Deppermann (1952). A different approach to obtain (4.97) and (4.98) has been developed by Beckmann and Franz (1957). Before the 1956 paper by Wu, various attempts were made to obtain a high frequency expansion for atotal following essentially two different ways. Wu and Rubinow (1955) performed very extensive transformations on the exact series solution for the forward scattered field, and succeeded in determining the first correction term to geometrical optics for both polarizations; their method was, however, too cumbersome to permit the determination of higher-order terms. An entirely different approach was adopted by Papas (1950), who used the variational method of Levine and Schwinger. For example, for the E-polarization, Papas finds that -1/2 total, 4a k -( ); (4.100) although the leading term of this formula has the correct value 4a, the higher order terms are incorrect. Subsequent works by Wetzel (1957) and Kodis (1958) proved that it is very difficult for the variational method to provide even the first correction term to geometrical optics. Kodis, for example, finds that (total)E 4a +0.746(ka 2/. (4.101 and it is seen by comparison with (4.97) that the numerical coefficient of the second term of (4.101) is in error by about 30 percent. Finally, we mention a few works on the determination of total for a cylinder with impedance boundary conditions. The phase shift analysis procedure which was described in Section 2.2 permits to calculate the approximate high frequency cross section; however, Lax and Feshbach (1948) give explicit results only for the sphere. The determination of the scattering cross section (and of the shift of the shadow 143 J, L W

THE UNIVERSITY OF MICHIGAN 7133-3-T boundary) for a cylinder with impedance boundary conditions was performed by Rubinow and Keller (1961), who also extended their results to any smooth two- or three-dimensional object. A different approximation method was developed by Sharples in 1962 (see remarks in Section 4.4). 144

THE UNIVERSITY OF MICHIGAN 7133-3-T V SCATTERING FROM A SEMI-INFINITE CYLINDER This section is devoted to the scattering of electromagnetic and acoustic waves by a semi-infinite cylinder of circular cross section. Both a thin-walled tube and a solid cylinder are considered. The boundary conditions are -a = 0 (rigid ap cylinder) or u = 0 (soft cylinder) for the scalar case and it is assumed that the cylinder is perfectly conducting in the electromagnetic case. When the scattering body is a thin-walled semi-infinite tube the sources can be located either inside or outside the tube. In the former case we assume the solution of the corresponding infinite waveguide problem to be known. That is, the amplitude and phase of all modes are known at the point corresponding to the end of the tube. The semi-infinite cylinder problems are solved by employing the method of Weiner and Hopf (1931) for treatment of integral equations in the interval (0, w). However, the calculations will be somewhat more straightforward if one does not formulate the problem as an integral equation but instead takes the Fourier transform of all quantities before applying the boundary conditions. This approach has been used by Wainstein (1949) and Jones (1952), among others. To illustrate the method we will treat the problem of electromagnetic scattering from a semi-infinite rod (i. e. a solid cylinder with a plane end surface). The corresponding scalar problem for plane wave incidence has been treated by Jones (1955) and as in that case the final expressions contain the solution for the semiinfinite thin-walled tube plus additional terms which make the solution fulfill the boun dary condition on the end surface. These additional terms are not expressed explicitly but only given as the solution of an infinite system of linear equations. Contra to the infinite cylinder case, the solution of the electromagnetic scattering problem for the solid or tube-shaped semi-infinite cylinder cannot be constructed from the scalar problems with boundary conditions u = 0 and E- = 0 respectively by taking the incident scalar waves as the component along the cylinder axis of the incident electri - 145

THE UNIVERSITY OF MICHIGAN 7133-3-T and magnetic field. The scattering field due to an incident TE field, for example, consists of both a TE and a TM part. 5.1 Electromagnetic Scattering from a Perfectly Conducting Semi-Infinite Solid Cylinder Let (p, p, z) be cylindrical coordinates as in Fig. 2-1 and let the semi-infinite circular cylinder occupy the space pS a, z >0. As before, the time dependence e will be suppressed throughout. We write the total electromagnetic fields as E Ei+ Es (5.1) H =Hi+ H where Ei and Hi denote the incident field (the field obtained if the rod were absent). The scattered fields Es and HS satisfy the Helmholtz equations (V2+ k2)ES =0 (5.3) (V2+ k2)H = 0 (5.4) outside the rod (k = w Ei = 27r/X) and the following additional conditions: (i) Es Ei Es = -E p = a, z >O (i) E; =-E p=a, z>0 zp=a, z E = -Ei E = E i p<a, z = p p (ii) E, Hs satisfy a radiation condition at infinity (iii) Es(a,' z).(z-1/3) E(a,,z) Oz 2/3) Hs(a, z ) - z-1/3) H(a, ) 0(1) as z -+-0 where 00 is an arbitrary fixed angle. 146

THE UNIVERSITY OF MICHIGAN 7133-3-T Condition (iii) is the edge condition, necessary to ensure uniqueness of the solution (Bouwkamp, 1946; Meixner, 1949; Heins and Silver, 1955; Van Bladel, 1964). We assume temporarily that k = k + ik. (k > 0, k. > 0) and allow k. -- 0 in r 1 r 1 1 the final results. This assumption is equivalent to introducing losses in the surrounding medium and consequently the Fourier transform of all field quantities related to the (outgoing) scattered wave will exist in the ordinary sense. We expand all field components in a Fourier series with respect to p and take the Fourier transform with respect to z. Thus, for example 27r oD zn(,) = 7 E (p ze- ddz (5.5) 0 -OD from which the original field is obtained as E (p,, Z) = i(p,a)e da. (5.6) n=-oo If we associate an imaginary part to k it follows that the fields are exponentially decreasing as |z J-.ox and that their Fourier transforms are analytic functions of a in the strip -k. < Ima < k.. 1 1 Let F(z) be a function, exponentially decreasing as |z | -o, and 1(a) its Fourier transform. We introduce the following notations 'F(z), z > 0 F F(z) = { t0, z<0 ro, z >0 F (z) = F(z), z < 0 Denoting the Fourier transform of F (z) and Fh(z) by 7 (a) and 3(a) respectively, we have 147

THE UNIVERSITY OF MICHIGAN 7133-3-T T - - d2i-t (5.7) 7 (a) = - ()i (5.8) i jv - a where the path of integration passes above the pole y = a in (5. 7) and below it in (5.8). Equations (5. 7) and (5.8) are easily obtained from application of the Cauchy integral formula to T(a) in its strip of analyticity. +7 (a) is analytic in the lower half-plane (Ima < 0) and 7 (a) in the upper one. The z-component of (5.3) in cylindrical coordinates reads aE QE2 E aE a 2E a1 1 z z s - + ---- + k2E =0 (5. 9) s 2 z and the same equation is valid for H. The corresponding equation for 6 and Xs z zn zn is the radiation condition is ( k(Pk a) ( a = Z (aa) - p>a= (5.11 pzn H1)(a = fk2- ) >a H ^r -a ) 5 s n n(p.a) (aa) pH1 >a (5.11 H(1) k2 a2 "Sn(P,a,) = _s (a,a), n >a (5. 12 zn zn H(1) k a n 148 I

THE UNIVERSITY OF MICHIGAN 7133-3-T The solution for the region p < a, z < 0 satisfying the boundary conditions on the plane end surface can be obtained by use of images. We use the superscript I to denote quantities related to the field in this region and define rI- a 2 I- I-(T1 k (.(p,a)+t (p,) = - (a,a)+ (a,-a) 2 (5.13) n/ 1J^ k - ) p<a (p,a)- (p,-a) = (a,a)- (a,-a) a) (5.14) n p<a The fields obtained by inserting these n and n in (5.6) are the total fields in the zn zn region p < a, z < 0 if there is no source of the incident fields in p < a. If there are sources for p < a we have to add the incident field plus its reflection by a perfectly conditing infinite plane at z = 0 to obtain the total field. Thus, for p < a, Source at po< a I-(Pa) t= s-(pa)i+(p-a) (5.15 o zn zn zn XI(p,a) =X,-(p,a)+. (p -o) (5. 16 zn zn zn Source at p >a, zn(p, ) = es(pa)+ )p a) (5.17 (p^a) = s(p, a)+ (p,a) (5.18 zn zn zn where the + and - superscripts denote a division according to (5.7) and (5.8). That the fields obtained from (5.13) and (5.14) satisfy the boundary condition for z = 0 follows from the fact that the pertinent E0 and E are both odd continuous functions of z and thus vanish at z = 0. 149

THE UNIVERSITY OF MICHIGAN - 7133-3-T If E and H are known for p = a, -co < z < o, they can be obtained everyz z where from equations (5.11) - (5.14). The remaining field components can then be derived from Maxwell's equations. As a special case we obtain the following equations between the field components at p = a: i H(1)(ar) iH (aK);pn(a, a) = - [K26 (a, a)+ 6s (a,a) (5.19) zA.ucH ) n ' a zn +J n iwE H (aK) s an sn 5 ) ( a,) a= - (a,+ t (a (5.20) 2 zn (1) zn ax KH (aK) n iJ (aK) n(aa)- (a, —a) = K J(arc) K O n(a,) — (a,-a)) + h n znOn = a)(aK L + sI(a, )+ ((a, -al (5.21) a \ zn zn w /T J' (aK),a)+\ a, - a n a - a) + n +a (a, 2n, +,;na n K.J (aK)na a aK~ n X 2 aa - (a, E)+ (a,'-a (5.22 (5.22 where K = k -a (the branch whose real part is positive when a = 0) and H(1)' n and J' denote the derivative with respect to the argument. We define n 22 e (a) = c a 2 (a, a)+ana~ (a,a) (5.23 n on zn h (a) = 2a2, (a, ) + ana. (a, a). (5.24 n On(' n 150

- THE UNIVERSITY OF MICHIGAN 7133-3-T For the incident field we have iWK a2H(1)'(aK) i H(1) (ax) i (a,) = n H(1)(aK) n i e (a) n f zn(a, a) zn (5.25) (5.26) if the source is at p < a, and iJ (aK) i (aa) = n e (a) -zn (a,) n =uca J-(aK) n 2 iwEK a J'(ac) (a) = n (a,a) n J (ax) zn n (5.27) (5.28) if the same source is at zn (a,a) = -- n(aa) - 0 zn On p > a. Combining (5.19) - (5.28) and using the fact that (total field) we obtain (total field) we obtain naa X (a,) + (-a) = zn zn i H(1) (aic) i a2 n n J (aKc) n J' (aK) n (e-(a)- ei(a)) n n J (ax) + n J' (ac) n (5.29) + 2 h (a)-h (-a) = iWeKa n n n J (aKc) n - (a, a) - (a, a)\zn zn J' (ac) n n (a -a) (5.30) J (aK) zn(a ' n i i where all quantities except e (a) and i (a) are related to the total fields. Equations n zn 151

THE UNIVERSITY OF MICHIGAN 7133-3-T (5.29) and (5.30) are valid if the source of the incident field is at p > a. If the i source is at p < a the corresponding relations are obtained by replacing e (a) by e (a)-e (-a) in (5.29) and 6i (a) by ~ (a)+. (-a) in (5.30). n n zn zn zn Employing the Wronskian we can write H(1)(aK) J (aK) n n -2i (5.31) (1) JP(aK)(1) H (aK) n 7 raXJ (a) (aK) n n n n - 'n 2i (5.32) H(1) J n(ax)?r ax J (aK)H(1)(aK) n n n We now perform a factorization such that L (a)L (-a) = riJ n(aK)Hn )(aK) (5.33) n n n n M (a)M (-a) = -riJ' (aK)H) (aK) (5.34) n n n n where, for Ima > -k., L (a) and M (a) are analytic, have no zeros and behave as 1 n n 0(1/fC) as Ia-o. We define L (a) - L (a) (5.35) M (a) - M (a) (5.36) -n n M (a) - Ll(a). (5.37 Further details about the functions L and M are given in Section 5.6. Equations n nas (5.29) and (5.30) can now be rewritten as 152

am THE UNIVERSITY OF MICHIGAN - 7133-3-T + +e (k) (-ita)( z(a,a)+ Jea(a M (-)- F (a)+fn(a)- 2.a KM (a) a k(k-a)M (k) n n n h(a)-h (-) h (-k)-h (k) i 2n L(t)-G(a) +g (a)-i n L (k) w((k+a)a WE(k+a)a2 2' (a,a) h (-k)-h (k) Zn. n n _ + G-(a)-f-(a) a(k+a)L () i L((k)+Gn(a)-gn(a) (5.39 n WE(k+a)a where J (aK)M (-a) F (a) = n n e (-) (5.40 n caJ'(ac) n n KcJ'(aK)L (-a) G(a) = n n (a,-a) (5.41) n (k+a)J (a) zn -4 n and i e (a), source at p a ) (a2 2 (5.43 n e a) - e (-a), source at p < a F) n n - t. (a, a) source at p >a n (5.40 '(a,a)+ (a,-a), source at p <a n zn o m 153 a-. 153 I

THE UNIVERSITY OF MICHIGAN 7133-3-T The left hand side in (5.38) and (5.39) is analytic wrien Ima < k. and the right hand side is analytic when Ima > -k.. They consequently represent a function analytic in the whole a-plane. The behavior of the Fourier transforms of the field components when (a - oo is given by the edge conditions and L (a) and M (a) are 0(1 /) as la j — a. From this it can be concluded that both sides tend to zero at infinity and the common analytic function is consequently identically zero. The constants e (k) and n n h (-k) - h (k) in (5.38) and (5.39) can be determined by putting a = -k in (5.38) and a = k in (5.39) and using the relations e (k) = kna (a, k) and n zn h+(-k)- hn(k) = -kna[+ (a,-k)+ n(a, k) obtained from (5.23) and (5.24). Pern n zn forming this we end up with the following expressions: e(a) = a(k+a)M (a).2k a M n(k) -g(k) -G(k n 4k2a2M24k ) - n n2 L2(k) n n -nL (k) (f-k) - F(- -k) + (f(a) - F-(a) n \n n 2 n y (5.44 (a,a) = L (a) n 22 nL(k) (g(k)- G(k)) - L4k a2M (k)-n L (k) n n n -2M (k) (-k)-F:(-k))] + a(k+a) (g nn 2 -n n (5.45 Employing equations (5.8), (5.40) and (5.41) we obtain nC( k 1 )Mn(-Y) e(-'Y) Fa J, =ak n2 dy (5.46 a a 'y - ('y - -a) -OD n3 II J 154

THE UNIVERSITY OF MICHIGAN 7133-3-T O2' Ln(-Y)6 (a, —y) n n zn G(a) = 1 5 - d-y (5.47) (k+-y) n ( k2 - (7 - a) where the path of integration passes below the pole y=a. Thus, equations (5.44) and (5.45) are in fact a system of integral equations. The only singularities of the integrands in (5.46) and (5.47) in the lower half-plane are simple poles and by completing the contour of integration by a large semi-circle in the lower half-plane F (a) and G (a) can be calculated by means of residues. We n n write OD A nm F-a) += (5.48) m=1 nm a) xA B Gn-u) (5.49) m=O nm where _ Mn ( )e-(a' )j12 A - 1 n nm n nm nm (5.50) a a? (n-p ) nm nm 2 L(a )j. (a,a ) B n nm nm zn nm nm m > a a (k- a ) nm nm (5.51) B n L (k)Cz (a,k) no a n zn and Jn (j ) on o j, <nl <. n nm no n' 155

THE UNIVERSITY OF MICHIGAN 7133-3-T a - n- Ima' > 0 nm 2' nm nm) = 0 nrm 0 = jno <jnl <.. no ni. Ima >0 nm (we define j = j = 0 although they are not zeros of J and J. ) 0 1 Inserting a = a' and a = a respectively in (5.44) and (5.45) we obtain the folnlm nlm lowing infinite system of linear equations for the coefficients A and B ~ n m nm aa I(n2-j1 ) n A Mf(a- )j 2(k+a I) - A - O akn L k)E L k -(fZ(-k) - 2ak Mn(k)(g(k) - k+a) nL2(k) (n+ nm_2a2k2M2(k) - 4a2k2M2k n n n a(k- a'/) D Am - 2 ) t -f (at) 2=1 n+ nm / n = 0,1,2,... Q = 1,2,3,... (5.52) 156

a a~a (k-a ).L2(a )j2 n nl n) nim akn L (k) [1k (W z nk El nk~ L M__ rim -k gk)(+ 2a2k2M2(k) + 2M Wk)(Z a'-k Hhi nL2 nL k (r+2a km M( - 4a 2k2M 2(k) El a(k+a 1i) + 2 (aln OD Ea g(k~ BN rim!+aI n rim/ ni = 0,1,p2... I = 1,92,93,... z4 0 - PcJ1 2ik L2(k) M (k l El Ea2k2M (k) I k+a M=O 'ix. rim B= nio nL 2(k) ~r+2a2k~M2(k)) El \n - 4a km M (k) El 0 -0 (5.53) Wheri A arid B are calculated, the solutiori to the problem is obtairied by irisertirig in (5.44) arid Z rim rim (5.45) F (a) from (5.48) arid G-(a) from (5.49) arid I

HzI 2a~k2M(k) L(k) nL()n n n L jal - k m1nm OD +f,(-k)) -2a k2 M(k) m+I1 t F~ (-k) =2 n nL 2(k) n z4 m1 nm C.A n = 1$2 3,p 40 -i a? -kH;d (1) P-4 H 0.4 I.-A 00 1t0. J-11 0 0 z

THE UNIVERSITY OF MICHIGAN 7133-3-T The functions F (a) and G (a) affect only the boundary values on the plane n n end surface. E (a, z) and E (a, z) vanish for z > 0 for every choice of F and G. z 0 n n In particular, by taking G (a) = F (a) = 0, (5.44) and (5.45) express the solution for n n a semi-infinite thin-walled tube. In this case, the f (a) and g (a) are defined by the n n upper alternatives in (5.42) and (5.43) regardless of the position of the source of the incident field. The field components at an arbitrary point can now be calculated as a summation over n = -cx to o and an integration over a from -a to co of expressions containing the quantities en(a) and zn(a, a) of (5.44) and (5.45). We write X(p, ) = xi(p,, ) + ein (a)H (pK+B H) (pKi e da or (5.55 p >a, -o < z < oo where X stands for an arbitrary field component. The path of integration r is as indicated in Fig. 5-1 after the imaginary part of k is put to zero. When p < a and Ima zeros of Jn(aV77) or J (ak- a) bah Read branch cut FIG. 5-1: PATH OF INTEGRATION FOR THE INVERSE FOURIER TRANSFORM I m 159

THE UNIVERSITY OF MICHIGAN 7133-3-T the source of the incident field is located at p > a we have o X(p,, z) += i A(a)J (p )J(p e (5.56) n=-wJ X(p,n, z)=Xi(P,,z)+ Xr(p,,z)+ c ) + B (a)J(p ez d n=-oD p<a, -o <z <0 (5.57) where Xr is the incident field reflected by a perfectly conducting infinite plane at z = 0. The expressions for A(a) and B(a) in (5.55) - (5.57) are given in Table V. The radiation far field can be obtained from (5.55) by estimating the integral by the method of steepest descent. If we introduce spherical coordinates (r, 0, 6) such that z = r cos 0, p = r sin we get ikr OD X(r,e,i, ~ x+e ikr 1. ein_(-i)nEiA(kcose)+B(kcos0) (5.58) r nr n=-wo when kr -a is krsin2 >> 1. Equation (5.58) is obtained under the assumption that the source of the incident field is located at a finite distance. For plane wave incidence, the scattered field in the region 0 < e, where 0 is the angle between the direction of propagatio of the plane wave and the positive z-axis, contains an additional part equal to the field reflected by an infinite cylinder. A and B in (5.58) are still given by Table V with b(a) and c(a) belonging to p > a. The terms containing p in the denominator then contribute only to higher order terms and should be disregarded. 160

THE UNIVERSITY OF MICHIGAN 7133-3-T TABLE V Relation Between the Field Components and the Quantity X of Equations (5. 55) - (5. 58) l I x A(a) B(a) II i E P E z H P H H z pK a en n2 c(a) 2 2n c(a) pK 3 2b() pwxc a PKa i 2 b(a) wulc a ia- c(a) K 1 2 b(a) 22 K a 0 - b(a) 22 wLcK a - c(a) K 0 e (a)-e (a),n. n b(a) = n n H(1 (aK) n where e (a)-e (-a) b(a) = n n J' (aK) n;n,(a, a)- zn(a, a) c(a) = zn zn H(1)(aK) n.(a, a) + (a. — a) c() = n zn J (aK) n when p > a when p <a 2 2 2 K = k -a. 161

THE UNIVERSITY OF MICHIGAN 7133-3-T 5.2 Scattering of a Scalar Plane Wave by a Semi-Infinite Cylinder The case when the cylinder is solid, i.e. a rod of circular cross section, has been treated by Jones (1955) (cf. also Matsui, 1960) for both the boundary condition a- = 0 and u =0. His results also contain in principle the solution for a thin-walled dp tube and therefore we will not treat that case separately. Since the problem under consideration can be treated in a way quite analogous to that given in the previous section, the results will be given without derivation. As noted before, if the solution to the electromagnetic scattering problem is known, u1 = H and u2 = E do not yield the solution to the scalar problems with boundary conditions 3u /ap = 0 and u2 = 0 i i i i2 for the incident waves u1 = Hz and u2 = Ez respectively. The reason is that u1 and u2 so constructed do not satisfy the correct edge condition at the open end of the tube. This section contains essentially the results given in Jones' paper, written in conformity with our earlier notations (time dependence e- ). Let the cylinder occupy the space p < a, z > 0 and the incident plane wave be given by u = exp(ikp sin.i cos 0+ ikz cos0.) (5.59) i.e., the angle between the direction of propagation and the positive z-axis is 0.. 5.2.1 The Boundary Condition au/ap = 0 when the Angle of Incidence is Neither 0 Nor 7T Let the total field be given by (0) (1) u +u +u(p,,z) in p>a u(p,,z) in p <a, z <0 0 in p<a, z>0 where I I 162

THE UNIVERSITY OF MICHIGAN 7133-3-T I (1) u ikzcose. co i 1 ' n=-oe rl=-cD I 2 = -e J' (ka sin 0.) (n ) H(1H (kpsine.)eino H (kasin.) n 1 n 1 (5. 60) is the field reflected by an infinite cylinder (cf. Eq. 2.34). We have u(p,, z) = 1 21T 1 27r -o n=-oo n=-oD P (a) n H(1) iaaz in (1;),( H (Kp) e doa, rn n I einS p>a (5.61) P (a) n J J c p)+ ia2 L n m=0 iaz e d, p <a where 2ij'2 p (a ) nm n nm f nm, m>l 2 2 aa' (jt2 -n )J (j ) nm nm n nm 2i6 P (k) = on o fno no ak (6 = 0, nm; = 1, n=m) mn and as before 2,2 2 K = k -a Jt(jt ) = n nm 2 n Ca' = k2 -nm a The path of integration 0 = j < j <... (we define j = 0 although no nl 31o, it is not a zero of J1) 1 positive real or imaginary is given by Fig. 5-1 and passes above all poles of P (a). n The equation for P (a) is n 163

I I THE UNIVERSITY OF MICHIGAN 7133-3-T I 00 riP (a) G 6 Ga G a n n on o n nm (a+k)M (a) a-kcos8. a+k — 1a+ n 1 nm (5.62) where G = n i ( - cos9.)M (-kcos0.) i n i sin. H (ka sine.) 1n 1 and M (a) is the split function defined by (5.34). The constants a n nm by the following equations: 6 a - a 1 on oo nm a' a nr nr a' - kcos. at + k - ' a + a' nr i nr m=1 nr nm are determined r = 0,1,... (5.63) where a' nr 2c' (j,2 -n2) nr nr f' (a' +k)M (a' )n2 Lnr nr n nr at = - 1 00 2kL2(k) The edge condition requires that a. m / as m - co and we must also Inm have OD 6aa 6 a +m= a on oo +anm m=l = 1. (5.64) 1 The solution for a semi-infinite thin-walled tube is obtained by putting f = 0 in nm (5. 61) and a = 0 in (5.62) for all n and m. Inside the tube, i.e., p < a, z > 0, nm the integral in equation (5.61) can be calculated by residues. Thus, II 164

THE UNIVERSITY OF MICHIGAN 7133-3-T iP -i J P (a )J (Of ) ia' z iP o(k) ikz. in0 nm n nm n nm a nm u(p = kae +, e 2, 2... e n=-oo m=1 aa' (j2 -n )J (f nm nm n nm p <a, z >O (5.65) 5.2.2 The Boundary Condition au/ap = 0 when the Angle of Incidence is 0 ikz Here we take the incident wave to be e. The total field is assumed to be ikz e +u in p <a ikz -ikz e +e +u in p<a, z<0 0 in p<a, z>O. The result is ' H(1)( rS) eia da p > a 2 HKH (Ka) u(p,,z) = (5.66) F h J Om 1 1 P(a) + 1 mo a iaz 27 T | 0 Kf(Ka) J 2 2 rn nm where 2iP(a' ) m a J (j ) m$0 nm o om h 2iP(k) o ak The equation for P(a) is iriP(a) 1 -m gf3(a+k)L (a) a+k z f a+ a' (5. 67) 0 1 om where j3 = -7rakL (k). 0 1 165

THE UNIVERSITY OF 7133-3-T MICHIGAN The equations for the constants A3 are 2o~'In t +k)L ]' or 1 or 1-9i a' +k or o om m=l or om r = 0,1,... $ (5.68) 5.2.3 The Boundary Condition u = 0 We assume the angle of incidence to be neither 0 nor 7 and write the total field as (0)+ (2)+, Z) u +u1 +u(p,~,z) in p a u(p,,z) in p< a, z < in p a, z 0 where u(0) is the incident wave given by (5.59) and () ikz cos e. O u2) =-e I n=-oD. 7 me e J (kasine.) n 1 (1) \ n] (1) H (kpsin0 )e H (kasin.) n 1 n 1 (5.69) is the field reflected by an infinite cylinder (cf Eq. 2.29). We obtain u(p,,z) = 1 27r 1 27r R (a) e (K1)(5 H( )(cp) e da, n p>a (5.70) n=-wD jp J p\nm) n a le Jn(Kp)- 2 2)nm e m=l a - a nm iaz -.% da, p<a where 166

I THE UNIVERSITY OF MICHIGAN 7133-3-T 2j R (a ) nm n nmn gnm = 2 a Jn+l(Jnm) and as before 2,2 2 K = k -a J(j )=0 n nm o < nl <Jn2 < positive real or imaginary. n nm The equation for R (a) becomes n 7rR (a) n L (a) n n-k a - k cos e. 1 OD -m= m1 n nm a+ a mn (5.71) where (i)nL (-kcos.) n 1i ^, (1 -k sin 8.H (ka sin.) 1 n i and L (a) is defined by (5.33). The 7nm are determined from n n 2 -2a a 7 2 a2 j L(a ) nr n nr 1 a -kcos. nr I OD = +nm m=l nr nm r = 1,2,... (5.72) We also have = 0(m-7/6) nrm as m — OD 167

THE UNIVERSITY OF MICHIGAN 7133-3-T and 'ym =1 (5.73) m=1 The solution for a semi-infinite thin-walled tube can be obtained by putting g = 0 in (5. 70) and m = 0 in (5. 71) for all n and m. Inside the tube the field can be expressed as (,.. i innm n J nm nm nnm a nm (5.74) U(p,z; = -i e 2 ' ' --- — e n=-oo m=l a a J (j nm n n m p <a, z > 0 5.2.4 The Far Field For points at large distances from the origin the integrals in equations (5. 61), (5. 66) and (5.70) may be evaluated by the method of steepest descent. The integrand has a pole at a = kcos i. which for certain points of observation will be close to the saddle point. To overcome this difficulty a method by Vander Waerden (1951) is used. We introduce spherical coordinates (r, 0, 0) p = rsine, z = rcos 0 and following Bowman (1963b) we write the field in the far zone kr sin2 >> 1 as ikr ikrcos(8- 8.) uU (, ) r + H( -i)A(H) e s ikr 2 sgn(0- 0i)A(0)T(r, 0-) k (5.75) where 1 D.. p (kcos0). U(0,) = _1 (_i)P n, co.) ein- (5.76) n=-o k sin 0 IH (ka sin 0) n 168

THE UNIVERSITY OF MICHIGAN 7133-3-T - -i- J' (ka sinm0.) A() e1 e4 n(5.77) 7r n=-ow HWm7 H (kasine.) i i n when the boundary condition is a = 0 and ap 1 —,..n D R (kcos0). U(0, ) 1 (i)nl n i (5.78) n=-o H (ka sin ) n I -i J (kasinm.) A(0) = eH E (1)2i e (5.79) 'n=-o Dsin0. H '(kasine.) ' i n i when the boundary condition is u = 0. The total field is obtained as before by adding the incident and reflected waves to u. The functions H(0 -.) and sgn(O- 0) are the Heaviside step function and the signum step function defined by 1, x>O 1, x>O H(x) = sgnx = (5.80) LO x<0 -1, x x<0 The function T(r, 0- i) is given by * T 1 -i2w2 _ 4 T(r,0- 0) = e 2 erfc l -i)w - (5.81) 0-0. where w= (kIr sin 2 and the error function is defined by e2 2 t2 erfc z = e dt (5.82) JZ 169

THE UNIVERSITY OF MICHIGAN 7133-3-T The second term in (5. 75) removes the reflected wave u(1) and u(2) respecively in the region 9 > 0.. The last term can be interpreted as a transition contribuion which assures a continuous field across the shadow boundary = 0.. Thus it compensates for both the jump in the reflected field and the singularity of the first erm at 0 = 0.. For fixed 0 90. the transition term is asymptotically smaller than the other two; in practice it can usually be neglected when w > 4. For ka << 1 and the boundary condition u= 0 the only term in (5. 70) which must be considered is that for n = 0. For values of a such that |aaI << 1 we can write the equation for R (a), (5.71), as o R R(a) T1 i om o f 1 ome L (a) oa-kcos0. - a+a I o 1 m=l om 7 a -kos 1+ i(a - k cos 0.) a- kcos0. ex (0)) (a-kcos0.) 7rR (a) e 1 (5.83) 7oLo(a) where ( = ia > and R( )(a) is the value of R (p) when 7 =0 (all m), i.e. m=l om the case of a tube-shaped cylinder. Near the saddle point we have [aa << 1 and in estimating the far field we can consequently replace R (a) in the integral of (5. 70) by the expression given in (5. 83). This integral is then just that which occurs in the diffraction by a semi-infinite thin-walled tube at the positions p = a, - r < z oo. Thus, the far field for the semi-infinite rod is the same as that produced by a semi-infinite thin-walled tube of the same diameter, but longer by C, subject to the same incident field. Under the assumption that 7 =0, m > 2, Jones calculated r to be 0.087 a and he estimates the correct value to be close to 0.1 a. There is no corresponding result for the boundary condition - = 0. 170p ~ 170

THE UNIVERSITY OF MICHIGAN 7133-3-T 5.2.5 Numerical Computations for the Boundary Condition au/ap = 0 Jones shows that when the angle of incidence is zero, 2 2 \ t(u+2) dt = 2(1-3 ) (5.84) 2 Sz=0 o a 0 The modulus of the left hand side of (5.84) represents the average pressure amplitude on the end of the rod if u stands for the velocity potential of a small-amplitude sound wave. The constant f3 is obtained from (5. 68). Jones solves this equation approximately by assuming successively (i) m= 0, m>0 (ii) 3r=0, m>l1 (iii) 3 =0O, m>2. m m m The average pressure amplitude as a function of ka for the range 0 <: ka< 10 is given in Fig. 5-2. The plotted curves correspond to the third and seventh (f3 = 0, m > 6) approximation, the latter calculated by Matsui (1960) for ka < 3. When the incident wave is propagating along the axis of the cylinder it satisfies the boundary condition along the cylindrical surface p = a. The scattered energy is consequently finite and can be obtained by integration over the end surface only. The scattering coefficient c is related to the constant (3 by c = 1-2Re{f3J (5.85) 0 1 quotient of the scattered energy and the incident energy per unit area. The scattering coefficient is plotted against ka in Fig. 5-3. For small ka (ka < 2) we can use the approximation (ka)2. (5.86) 4 171 w

- 3rd approximation 2.0 c o p4 1.5 - q) 1.0 I 0 1 2 3 4 ka 5 6 7 8 9 FIG. 5-2: THE AVERAGE PRESSURE AMPLITUDE ON THE END OF THE ROD WHEN THE ANGLE OF INCIDENC IS ZERO. (- Jones, 1955; --- Matsui, 1960). -q ti c! C z <?0 3 -0 O - az YE

3rd approximation 1.0 0.5 I t | IAI I - I - I - I- -- I 0 1 2 3 4 ka 5 6 7 8 9 10 FIG. 5-3: THE SCATTERING COEFFICIENT WHEN THE ANGLE OF INCIDENCE IS ZERO. (-Jones, 1955; --- Matsui, 1960) 0-i z I (T ~-4 C);o~ -. 1 EdA w) C4Z

THE UNIVERSITY OF MICHIGAN 7133-3-T When the angle of incidence is not 0, Jones shows that, to a first approximation (which is better the smaller ka), the average pressure amplitude on the end of the cylinder is the product of the average pressure amplitude when the angle of incidence is 0 and the amplitude of the symmetric wave that is produced in a hollow semi-infinite cylinder occupying the same position as the rod. The results given by Jones are reproduced in Fig. 5-4. Jones also calculates the pressure on the end of the rod due to a pressure pulse when the angle of incidence is zero. Let the pressure of the incident sound pulse be given by rsoo+i -ik(v t-z) e 0 p (t,z) = H ) k dk (5.87) )dk (5 87) a2r i k J ~-oD+ iE where v is the speed of sound, t is the time and H(x) is the Heaviside step function defined by (5.80). According to (5.84) the average pressure ikz (i.e. total pressure/end area) due to the incident wave e is 2(1-3). The average pressure due to po(t, z) is consequently o+ +iE -ikv t _10 p(t) = k [ -1 k)] dk (5.88) J-o + is The integrand has a simple pole at k=0 with residue 1/2 since 0 = 1/2 when ka = 0. Thus the above integral may be written as O 1-203 (k) -ikv t ' 1 0 e dk -oD k = + l [(F(k) - l) sin(kv t)+ G(k) cos(kv t)] (5.89) J-0o 174

THE UNIVERSITY OF MICHIGAN 7133-3-T 2.0 1.5 ' 0S 10 0 bD 0.5 0 1 ka 2 3 FIG. 5-4: THE AVERAGE PRESSURE AMPLITUDE ON THE END OF THE ROD FOR VARIOUS ANGLES OF INCIDENCE (Jones, 1955). 175

THE UNIVERSITY OF MICHIGAN 7133-3-T where 2 [ - (k)] = F(k)+ iG(k) and we used the fact that Bo(-k) is the complex conjugate of 8 (k). Jones computes the integral in (5.89) from the earlier values of | (k) (third approximation) by replacing F and G by parabolic approximation over o the intervals (0, 1/2), (1/2, ),.... The result is shown in Fig. 5-5. The curve given there is within 1 percent of 12~}l/2 0. 915+0. 745 t - ]/ Jones also constructs expressions for the distant fields and for 1o whose first variations are zero for small variations of a, 7y and O3 about their correct values. nm nm m However, he does not use these expressions in the numerical computations. 5.3 Radiation of Sound from a Source Inside a Semi-Infinite Thin-Walled Tube 5.3. 1 General Solution The problem of scalar diffraction when the source of the incident field is located inside the tube has been treated by Levine and Schwinger (1948) and indepenau dently by Wainstein (1949) for the boundary condition - = 0 (rigid tube). op We assume that the tube is located at p = a, z > 0 and that a single arbitrary mode is propagated in the negative z-direction. The velocity potential of this incident mode can be written as (0) n nm a nm u =Am cosn0e (5.90) n nm m /0 when n= 0, m >1 when n 1 where J'(j' )=0, =j <j' <... and n nm no nl.2 t 2 ]nm a' = k nn 2 a a is positive or positive imaginary. The field inside an infinite tube for arbitrary ex citation can always be written as an infinite sum of cylindrical modes. The solution 176

THE UNIVERSITY 7133-3-T OF MICHIGAN 2.0 1.8 1.6 Q) bD ) a Q> (1 — C 1.4 1.2 1.0 0.8 0 1 2 v 3 4 a a 5 FIG. 5-5: THE AVERAGE PRESSURE ON THE END OF THE ROD DUE TO AN INCIDENT UNIT PRESSURE PULSE. v IS THE SPEED OF SOUND (Jones, 1955) 0 177

THE UNIVERSITY OF MICHIGAN 7133-3-T of the corresponding diffractionproblem for the same excitation inside a semi-infinite tube is then obtained by summing up the contributions from all modes. The total field due to the incident mode of (5.90) is 1 F (a)F (a ) 1, Fn(Q>) )(1) iaz - con I —H (1ip)e da, p>a u(p,,z) = r n (5.91) (0) 1.\ _n___ iaz u + cosn0 K J' (Pa) J(p)e dc, < a 2 2 2 where, as before, K = k -a and the path of integration is given in Fig. 5-1, passing above all real poles of the integrand when Rea < 0 and below them when Rea > 0. We have (k+a)M (a) F (a) = -A (k+a' )M (a ) n-' (5.92) n 2 nm nm n nm ai+a nm where the split function M (a), as before, is defined by (5.35). 5.3.2 Field Inside the Tube When p < a, z > 0, evaluation of the integral in (5. 91) by means of residues yields Jn(j ~) -ic' z Jn(jf P) ic~zl n nin a nm n a u(p,. z) = A ncosn(jm) e. - e eLJ (it (5.93) where 4(j, ~(k+an )M (a' 1) R -(5.94) mm 4(ja -n2) nm is the reflection coefficient of the incident mode and 178

THE UNIVERSITY OF MICHIGAN 7133-3-T ),2 (k+a' )(k+a'l )M (a' )M (a' ) (n) n. nm nl n nm n nf ^W 2 2R a (^ + ) (5.95) 2(j - n ) nm m~. can be called the conversion coefficient of mode A into mode A nm ni We write the coefficients R as mDl ie(n) (n) (n) m RI'Rl e m (5.96) The moduli and the phases of these coefficients for a symmetric incident wave (A, A 1) as functions of ka in the range 0 < ka < jo2 (jo2 = 7. 016) are given in Figs. 5-6 and 5-7. The modulus of the reflection coefficient of mode All for ij <ka <j12 (jl = 1.841, j2 = 5.331) is shown in Fig. 5-8. If we introduce some auxiliary functions connected with the split,function, the quantities in Figs. 5-6 through 5-8 can be expressed as |(0) kaRe P1 (k) R = e, 0 ka< j =3.832; (5.97) and R(O k+all kaRe P(k) I (0 k+a l aallRe Pl(all) 11 11 k ReP (k)+a ReP (a ](0) = 2k 2t i 11 1 1)l ol k- ll R() 2 k-a ea'kReP (k)+a Re P (5.98) fore (ja< 98) for Jol..<ka<Jo2; 179

.61 IR \ IR()I.5 0ool I11.4 ((.3 '2- R(~) 00 10.1-.09- R(I I I r II I I.08.07.06.05.04.0 0 1 2 3 4 5 6 7 FIG. 5-6: MODULI OF REFLECTION AND CONVERSION COEFFICIENTS FOR SYMMETRIC MODES A0, Aol (Wainstein, 1949; IR(O)I for ka < 3.832 is also given by Levine and Schwinger, 1948). H - z -I ci)?d 0 0 z

o(0) mn a/a 0.6' 0.5 120~ 1000 0. op..s IC I 1j H C) m dl a 0. 0. 0.1 0 1 2 3 ka 4 5 6 7 8 FIG. 5-7: PHASES OF REFLECTION AND CONVERSION COEFFICIENTS FOR SYMMETRIC MODES A00 A AND END CORRECTION FOR MODE A00. (Wainstein, 1949; I/a is also given by Levine and Schwinger, 1948).

THE UNIVERSITY OF MICHIGAN 7133-3-T 1.0.9.8.7.6.5.4.3.2.1.09.08.07.06 6 FIG. 5-8: MODULUS OF THE REFLECTION COEFFICIENT FOR MODE A (Wainstein, 1949). 182

I THE UNIVERSITY OF MICHIGAN 7133-3-T 1 (1)1 k+a' aa{ Re Sl(a ) k-al for Jll ka<jl I i i - 12'J (5.99) 0(1) = ka Im P (k), 00 1 e(= aallIm Pl(a 1) 11 11 111' e(O)+e(0) =(O) = e(o)= oo- 11 lo ol 2 (5. 100; where the functions P1 and S1 are defined by equations (5.222) - (5.223). Using the approximate formulas (5.233) and (5.252) we find (ka)2 -4 (0) e 2 (ka)4og 1 19) I 0 L, 6 7 ka +12 -OO1, ka<l (5.101; where logyl = 0.5772 in Euler's constant, and l(0 ) e V3 - k(k a ),oo 3 2 (k) 2 1 1 <ka<j' =3.832. (5.102) At ka= 1, (5.101) and (5.102) yield values larger and smaller respectively than the correct one by about 3 percent. If the incident mode is A and the frequency is so low that all the higher modes are exponentially damped, the field inside the tube for large z is given by u=A r-ikz R(O) ikzl u = A e +R e 00 00 (5. 103) Equation (5.103) represents a standing wave of amplitude ul = A 1 + JRO) -2 R cos(2kz + e()) (5.104) The first node is consequently located at z = -i, where [m m 2k) = e(0) 00 -- 183 (5. 105)

THE UNIVERSITY OF MICHIGAN 7133-3-T The length I is called the end correction and it determines the resonant frequencies of cylindrical resonantors open at one end. When ka - 0 the end correction tends to the limit lim r ka-*0 a -1 arctan 1 dx 1 7 2 ( 1 (x. n i 0.f 3. pOD 1 2 M log 2I lK l dx = 0. 6128 21 (x)5 1 6 (5.106) The quantity /a is plotted in Fig. 5-7, and some numerical values, together with those of — n IR(), are given in Table VI. ka oo TABLE VI End Correction and a Function Related to the Absolute Value of the Reflection Coefficient for Mode A I oo Ba(O ++ ka kaR() 00 ka ka i/a ka I/a IIII ~I I II 0.05.10.15.20.25.30.35.40.45.50.613.612.611.608.604.610.598.594.590.586.581 0.0245.0485.0719.0948.117.139.160.180.200.219.55.60.65.70.75.80.85.90.95 1.00.576.571.565.560.554.549.544.538.533.527.235.251.266.281.290.311.325.333.351.364 Wainstein (1949) The phase of R) is given by 0) = 2ka -. 00 00 a When ka > jm i.e. above cut-off, the power transported by the mode A nm nm of (5.90) is * This value was obtained by Brooker and Turing as reported by Jones (1955) and independently by Matsui (1961). Levine and Schwinger (1948) give 0.6133 and Wainstein (1949) 0.613. 184

THE UNIVERSITY OF MICHIGAN 7133-3-T nm 2 o nm 2 nm n nm where p is the density of the surrounding medium, vo the velocity of sound and e =1, ' =E2 =...= 2 are the Neumann numbers. Thus, the fraction of the power of the incident wave Anm which is converted into the mode A and propagated towards z = ao is r2 (n) = ( i' 2 IR(n) (5.108) atm..n 2 ) | E2 2 (n) (n(n) r = r (5.109) which is a consequence of reciprocity. The total power reflection coefficient for the mode A is ^nmn r(n) (n) r - me (5.110) m =0 where the summation is taken over all propagated modes. In Fig. 5-9 the power reflewcti aon ad nversion coefficients ogether with the total power reflection coefficienr t are given for the symmetric modes A, A 1 5.13.3 Te Far Field If we introduce spherical coordinates (r, e, 0) such that z = r cos 0, p = r sin0, and evaluate the upper integral in (5. 91) by the method of steepest descent, we obtain * 185

THE UNIVERSITY OF 7133-3-T MICHIGAN 1.0.5.1.05.01.005.001 0 1 2 3 ka 4 5 6 7 FIG. 5-9: POWER REFLECTION AND CONVERSION COEFFICIENTS OF SYMMETRIC MODES A AND Ao (Wainstein, 1949). 00oo ol 186

THE UNIVERSITY OF MICHIGAN 7133-3-T the total far field' ikr.n a(k+cr' )(l+cos0)M (a' )M (kcos0) 1 (_-1) nm c nm n u(r, 0,0).~ A cos n e... n. n —m sin O(kcos +c' )H (ka sin 0) nn n (5.111 The power radiated per unit solid angle about the direction (0, 0) is given by 0= 2 u p(9) = i pvko u 2 6( 0 (5. 112 This quantity divided by the power of the incident A nmatt. Using (5. 111) and (5.107) we get pattern f (6, 0. Using (5. 111) and (5.107) we get mode is called the power f (e, ) nrm E k(k+a ) 2|M(a' ) 2(1 + cos 0)2M (kcos 0)12 n nmn n nm n 3 n 2 21 (1)' 2 47rat (11 — sin 8(kcose+a' )2 H() (kasin) ) nm,2 nm 2 cos no (5.113, ) ) ) ) where as before, E =1, el =e =. = 2 and ka >j. The relation between o 1 2 nm f (0, ) and the power gain function, i. e. the radiated power related to an isotronmpically radiating source pically radiating source, is f (e, I) G (8) = 42 nm nm' P a/47r (n) rad 1 -r m (5.114: where r(n) is the total power reflection coefficient given by (5.110). If we insert in (5.113) the expression given by (5.215) for M (a) Ll(a), we obtain As pointed out by Noble (1958), the sign for the far field in the directions O =0 and = 7r when n=m=0 as given by Levine and Schwinger (1948) (Eq. III, 12, 13) and by Morse and Feshbach (1953) (Eq. 11.4.33) seems to be in error. m ] 187

THE UNIVERSITY OF MICHIGAN 7133-3-T J (ka sin 0) f (e) exp ka (Re P (ka) + cos0 Re P (k cos 0l 00 2. 20 (1)(k 1 1 i Hr sin | (a0)|. (5.115) when 0<ka<j'l =3.832, and ol f = -Lm m 1 oo00 2 ' \ k-a kcos0- 2|(1) 7r m=l lm 1 sin21 0H( (ka sin e) X exp ka(RePl(k)+ cos Re P(k cos (5.116) when j' ka <j'( +1 For the lowest mode with p-dependence cos no (also om o(m +1)' 0 including A ) the corresponding expression is ~nanl /tan 2 Tk sin e) En a' tan ((ka sin e) t^ -e, o I n. I2 T X 2 n exp laReS (a' )+kcos0ReS (kcosJ 2p n nl n n 2 X 2e 2 cos no cos 0-cos e0 nl (5.117) when j' < ka < j where 0n is defined by ni Jn2 nl kcos = -a' (5.118) nm nmn The power patterns of the mode A for different values of ka in the range oo O < ka < 4.023 is given in Fig. 5-10 and for modes Ao and A 1 for ka =4.023 in oo ol Fig. 5-11. The power gain function in the forward direction ( = 7r) for mode A 00oo is t nl \2 The factor tan — /tan- is missing from the corresponding formula (Eq. 81) in Wainstein (1949). 188

THE UNIVERSITY 7133-3-T OF MICHIGAN watts 4) 0 0. So 0 To i 0.4 0.2 190- 500 1800 FIG. 5-10: RADIATED POW PER UNIT SOD ANGLE FOR INCIDENT MOD A00 CARRYING TE POstei 1 WATT. (Levine and Schwinger 1948 (ka C 3.832) and Wainstein, 1949 (ka = 4.023)) 189

watts 0.12 0.10 ka=4. 023 0.08 0.0 0 cd 0.06 P4f01(e0) ~ 0.04 0 003060900410 10010 FI.51"RDAE0OE PRUI OI NL O NCDN OE 0ADA0 FIG.-11:RADIATE POWER POER UNIAT SOLI kANL FOR02 (WINCtIDNT MODE4A9 ND H z 0-4 C.e4 H -4 l w0.0 I

THE UNIVERSITY OF MICHIGAN 7133-3-T Gr) = (a)2 (5.119) oo 00 1- R(0) 2 00 This function is plotted against ka for ka <j' = 3.832 in Fig. 5-12. The exact power patterns of Figs. 5-9 and 5-10 can be compared with the Kirchhoff approximation, in which the radiation field is calculated from the incident field and its normal derivative at the open end. The radiated power per unit solid angle for the incident mode A carrying 1 watt of power obtained from the Kirchhoff approximation is E k sin 9 J' (ka sin9) 2 of(5.13)wfi a ((5.120) fK nm2 0) n n Cos 2no (5.120 nm 2 cos 2 -cos cos. nm 2. nm where 0 is defined by (5.118). If we compare (5.120) with the exact expression nm of (5. 113) we find that nmm n' nmnm' 47r.,2 n 2o n m (5.121) The Kirchhoff approximation therefore gives the correct value for the radiated power in the direction 9 = (7r/2 <0 7 < r). It also gives correctly the directions of nm nm zero radiation if 0 > 7r/2. From (5. 113) we obtain the special cases (ka) 3 (0) f (0) R m=0,,2... om 47Raa MO om (5. 122} f (0,0) = 0 n>l nm 191

THE UNIVERSITY OF MICHIGAN 7133-3-T %00 0 0 0 0 p 04 PL4 11 0 9.0 7.0 5.0 3.0 O.0 0 1.0 2.0 3.0 ka FIG. 5-12: THE POWER GAIN IN THE FORWARD DIRECTION (8 = r) (Levine and Schwinger, 1948) 4.0 192

THE UNIVERSITY OF MICHIGAN --- 7133-3-T E k J' (ka) f (r/2, ) = 2(n) cos2n (5.123) nm 2a' H(1)(ka) mm nm n (ka)2 f (r ) = ( 00 47r (5.124) f (7r,) = 0 n>O, m l> nm The corresponding values for the Kirchhoff approximation for the mode A are J2(ka) k2 fK () = O. K 2 fK-(7) (ka) (5.125) 00oo 00oo 47r oo 4 f (0), f (7T/2) and fK (r/2) are plotted against ka in Fig. 5-13. 00 00 00 For low frequencies the Kirchhoff approximation can be improved if we add the reflected A mode to the incident field. This modified Kirchhoff formula for oo incident A mode reads 00 00 sinoo (J1(kasin0) 2 2 f(e) = si n ) 1- L os0)1 o (l+cos0) - 2sin 9Re { (5.126) A comparison with (5.122) shows that this expression yields the correct value not only for 0 = r but also for 0 = 0. A comparison of the exact and approximate power patterns for ka = 1.0 is given in Fig. 5-14. As our problem is self-adjoint, there is a reciprocity relation between the results of this section with those obtained for scattering of a scalar plane wave in Section 5.2. The principle of reciprocity can be stated as follows. To the incident eikr mode A of equation (5. 90) we relate the far field cos n - g () and from an nm r n incident plane wave vegiven by (5.59) (i.e. propagating in the direction (0b, 0)) we assume the excitation inside the tube to be 193

watts 0.05 - 0.04 - - 9 = -- If 0.03 ~ / \ 0 1 2 3 4 5 6 CARRYING 1 WATT OF POWER W 1949). 0 1 2 3 4 5 6 FIG. 5-13: RADIATED POWER PER UNIT SOLID ANGLE IN THE DIRECTIONS 0 = 0 AND 0 = 7r/2 FOR INCIDENT MODE A^- CARRYING 1 WATT OF POWER (Wainstein. 1949). H a 7 -m w H X < O z 7

watts.07 ka = 1.0.06- f00(e) 2.05 ~.04 Exact 03K- irchhoff CD 03 Modified 02 0 Kirchhoff Q).01 0 0~ 30~ 60~ 900 ~ 120~ 150~ 180~ FIG. 5-14: COMPARISON OF EXACT AND APPROXIMATE EXPRESSIONS FOR RADIATED POWER PER UNIT SOLID ANGLE FOR INCIDENT MODE A CARRYING THE 1 WATT OF POWER AT ka = 1.0 (Levine and Schwinger, 19R). -q z < PI tri - - C) -4 0 -c3 C) 0 TZ

THE UNIVERSITY OF MICHIGAN 7133-3-T u(p,,z) = _ Z2 B n n(j ) cosn enm n=0 m=l n anm Then 2E ij'2 A B = n n ( 27r -.) (5.127) nm nm, n2 n2)a2 nm nm It is readily checked that (5.127) conforms to (5.65) and (5.111). It may be noted that if we want to use the principle of reciprocity to calculate the field at an arbitrary point inside the tube due to an incident plane wave, we must know the far field excited also be all evanescent modes. Equation (5.111) is still valid for a mode below cut-off, but the quantity f (0, 0) of (5.113) has lost its physical meaning as there is no energy transported by the incident mode in that case. For ka < 1. 841 the absorption cross section for an incident plane wave, defined as the ratio of the power transmitted into the tube to the power incident per unit area, is related to the power pattern of mode A by a (e.) = X2 Or-0.) (5.128) a i 00 1 where X = 27r/k is the free-space wavelength. 5.3.4 Cylindrical Resonators with an Open End We assume the resonator to consist of a tube of length L closed at one end by a rigid wall and open at the other end (Fig. 5-15). The resonator is excited by a plane wave propagating along the positive z-axis u( = Ae. (5.129) If we neglect all higher modes, the field inside the tube is represented by the mode A alone, reflected with reflection coefficient 1 at the closed end and with reflec00 (0) tion coefficient R() of (5. 94) at the open end. Summing up all these traveling waves 00oo 196

THE UNIVERSITY OF MICHIGAN 7133-3-T incident plane wave -0 T 2a I~ open end closed end z z=0 z=L FIG. 5-15: CYLINDRICAL RESONATOR WITH AN OPEN END. we obtain the velocity potential ikz ik(2L- z) u(Z) Ae +e u(z) = A i2kL 1 -Re (5.130) As a measure of resonance we take the quantity |= u(L) 0 2A (5.131) which is equal to the ratio of the pressure amplitude at the closed end to the pressure at an infinite plane screen located at z = L. Introducing the end correction I defined by (5.105) we obtain 1 g =,,.,.. (5.132) 1 + R I + 2|R| cos 2k(L+I) Fig. 5-16 shows g as a function of ka for a resonator with L/a = 7.82. 5.4 Scattering of a Plane Electromagnetic Wave from a Semi-Infite Thin-Walled Tube 5.4.1 General Solution Let the semi-infinite tube occupy the space p = a, z >0 and the incident plane wave be given by 197

THE UNIVERSITY 7133-3-T OF MICHIGAN 's 0 o a) P-4 CO 0) 34 m pq 50 10. 5 1.5 L/a = 7.82 0 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ka FIG. 5-16: RESONANCE CURVE FOR A CYLINDRICAL RESONATOR WITH LENGTH TO RADIUS RATIO 7.82 (Wainstein, 1949) 198

THE UNIVERSITY OF MICHIGAN 7133-3-T ik(x sin.+ z cos 0.) E = (-cos. cos i +sin i +sine.cos3 )e -1 x y 1 z E ik(x sin. + z cos 0.) H1= -cs (osos +sine.sin-c )e - 1 1 X y 1 z 0 (5. 133 ) I I i.e., its direction of propagation is parallel to the (x, z)-plane and makes an angle 0. 1 with the (y, z)-plane. The configuration is given by Fig. 2-2 with 9. exchanged for a, The problem has been treated by Pearson (1953) for the special case B = 0 (incident TM-wave) and by Bowman (1963a, b, c, d) in a series of unpublished memoranda. Its solution is, in principle, given in Section 5.1 as soon as the functions fn(a) and g (a) of (5. 42) and (5.43) are determined from the incident field. Inserting x =p cos p in (5. 133) and using ikpsin. cos e e 00 = inJn(kp sini) ein n=-oD (5.1341 we obtain ikzcoso. O E (a,, z) = sin. cos B e i in J(kasine.) ein z 1 n i n=-oo ikzcos90. OD-cos0. cos3 E (a, p, z) = -e i nJ (kasine.)+ n =-o L- kasin. n 1 n+isL1 J'(kasin-.)+CD + i sin3 J' (ka sin e ineino n I (5.135: ) I From the definitions of f (a) and gn(a) in (5.42) and (5.43) we get The sign after 1 in the quantity Nn(ik) defined in connection with equation (2) in Pearson's paper seems to be incorrect. I m 199

THE UNIVERSITY OF MICHIGAN 7133-3-T 2i sin J' (ka sin.) n 1 f (a) = n f+(-k) = n M (kcos 0.)(a - kcos 0.) n 1 1 2insin J' (kasin 0.) n i M (kcos 0.) k(1 + cos 0.) n i 1 (5. 136) gn(a) = n n+l 2i sinO cos 3J (ka sin8.) 1lcsi n ) 1i ka(l+cos0.)L (kcos.)X( -kcos.) i n i 1 (5.137) g(k) - 2i+ cos J (ka sin i.) n 1 2 k asin0.L (kcos.) 1 n i where M (a) and L (a) are the split functions defined by (5.33) and (5.34). Inserting (5.136) and (5.137) in (5.44) and (5.45) and putting Fn(a)= G (a)= 0 yields the solution to the problem. In accordance with (5.55) and (5.56) we write tion to the problem. In accordance with (5.55) and (5.56) we write oD X(p,,z) = X (p,,z )+ e n=-o S r A(a)H1(p)+ B(a)H (1)+(p. eiazda n j (5.138) p > a, -ao < z < ao X(p,0,z) = X (p,, )+ 2 em n=-oo Jr A()p iaz A(a) J (p) + B(a) J n(p) e da, n t n _1 (5.139) p <a, -o <z <oo where as before #c = Vk -a and the path of integration r is as indicated in Fig. 5-1 with the addition requirement that it passes below the pole a = k cos. X stands i 1 for an arbitrary component of the total field, and X is the corresponding component of the incident field. The functions A(a) and B(a) are given in Table V if we take - or nn 6VUU

THE UNIVERSITY OF MICHIGAN 7133-3-T e (a) b(a) = n H (aK) n Czn(a,a) c(a) ='"H(1)(aK) n n(a, a) c(a) = J n(aec) when p > a when p < a (5.140) e (a) b(a) = n 9 where, from (5.44) and (5.45) and (5.136) and (5.137), r 2aknL (k) iacosMk)J (kasin0.) e(a) =i a(k+a)M(a) n n n n 2 M2(k) sin (kcos.) nsinjL (k) J(kasin 0.) k(l + cos 0.)M (k cos e.)J i n I1 asin, J' (ka sin Oi)(k- a)~ M (k cos O.)(a - k cos O.) n 1 1 (5.141) n(a, a) = r 2aknL (k) nL na} ) -n n L4k a M (k)- n L (k) n n in cos 3 L (k)J (kasin 0i.) x ak sine. L (k cos e.) i n 1 2sinI3M (k)J'(kasinO.) n n i1 k(l+cos9.)M (kcos.)/ i n i icos sini.J (kasin.)(k+a) 1 + i.. - k(+ cos.)L L(kcos 0)(a-kcosO.) (5.142) As a special case of (5.138) and (5.139), the curreng flowing in the wall of the tube is given by -j(0, z) = H (a +0, 0, z)- Hz(a- 0,,z) y z ZZ i 0 - 7 9 0 n=-co inp en(a) iaz e a3 k2 e c(A a krc M (a)M (-a) n n (5.143: 201

THE UNIVERSITY OF MICHIGAN 7133-3-T j (, z) = Hp(a+0,, z)- H(a-0,, z) * -O 1i I0 71 n=-a 7T,; - 0 fln-Wo / nare (a) k (a,a) n Zn iaz d 44 2 e (da.) kK a M (a)M (-a) aK L (a)L (-a) n n n n (5.144) 5.4.2 Field Inside the Tube For the interior of the tube, i.e. p < a, z > 0, the integral in (5.139) can be calculated by means of residues. The contribution from the pole a = k cos e. cancels the incident field and we obtain the H and E components of the total field expressed as te sums of TE- and TM-modes respectively: as infinite sums of TE- and TM-modes respectively: OD ODI J (j I -) ia' z H (p, Z) = 7 A ei n nma e nm H(P.,,z) nm j J(j )e n=-oD m=l n nm C DJ (j n-) ia z E(p,,z) = j ~.B ein ae n n=- o m=l n nm (5.145) (5.146) where as before J_(ji )= 0, < < <j <.. n nm nl n2 L 2 J2n 2 Inm Jn(j ) =0 0' <jnl< <.. and a' = k -- n nm Jnl n2 nm 2 i a are positive or positive imaginary. The constants A nm IT e(a' ) n | nnm nm \I ^ / 2 and B are given by (5.147) (5. 147) 202

THE UNIVERSITY OF MICHIGAN 7133-3-T and j ~- (a,a ) mnn.. nm B =-i 2 n (5.148) nm 2 a a nm The remaining field components are readily obtained from (5.139) in the same manner, or by Maxwell's equations, from the knowledge of H and E. Specifically z z the surface currents on the inside of the wall of the tube are 0(,z) = Hz(a,,z = eE ne A 'e n n=-oo m=l (5.149) 00 Oa iat z i io /n anm nmz jz(,z) = -H(a,,z e m(z = r os, the integral in (5. 138) can be estimated by the method of steepest desce When 0 > i we have to cross over the pole a = k cos i to deform the contour of integration into the path of steepest descent. The contribution from the integral along the path of steepest descent when 0 * 0. is given by (5.58) and for 0 >. the residue at a = kcos. yields an additional term which removes the reflected wave X. As in the scalar case we use the method of Vander Waerden (1951) to obtain an expression continuous at =0. (cf. Section 5.2.4). We write the E and H components of the total fields in the far zone kr sin 0 ~ 1 as II 203

THE UNIVERSITY OF MICHIGAN 7133-3-T E Eir ikrcos(O- 0.) 7. e " Ii E, H(' r H -)AEH(0) -krsin0 ZJ z ikr + sgn(i- )AE H(0) T(r, - 8) (5 s150) where c i)n + 1 (z(a, kcos0) ) UE(9 ) = I (-i) (1 e 0 (5. 151) n=-oo H ((ak sin ) n Tf -7 sini.J (kasine.) n ( -i4 n i i 0 A(h '- u e is (5.152) f1 E V" n=-oD ( H (ka sin.) n i and OD 1 0c ^e (kcos0) ) U (e, ()n n o en (5.153) H7r 2 (1)' o n=-oo (ak) sin 8 H (ak sin 0) n - rsin- J' (ka sin 0.) n A 4 ) = sinio A = j IT J'e 2sin ( ) Ie. (5.154) n=-oo H (ka sin 0.) n 1 The functions H(.-90) and sgn(9. -9) are the Heaviside step function and the signum function defined by (5.80) and T(r, 8-9.) is given by (5.81). If 0 > 8i, i.e. the observation angle is outside the domain of cylindrical waves, the scattered field is a spherical wave given by 204

THE UNIVERSITY OF MICHIGAN 7133-3-T EES UE(,0) ikr E ( ro 0s ^*0 e 0 E J sine - sine r (5.155) nrH nr Hs f Ur uH(eO, ) ikr o 5 FgUH(,g) esir (r,,) - 9 = sin sine r 0 V 0 In the negative z-direction, i.e., 0 = r, we can still use (5. 155) in spite of the fact that the condition kr sin >> 1 is not satisfied. In the sums for U and U only the terms n = + 1 will contribute and we get: s J eik|z I 2ia cos B Ll(k) J1(ka sin.) E (,0,z) = - H 2 v 2_ (5.156) IO Y i ] ka M (k)-L ki sinoLl(kcos0i) E ikz| I 4ika sinJ3M (k) J(kasinO.) Es(_Oz)= Hs ~-1 1 Y - ' r22 2 22 o 1 k a M 1(k)- L(k)J (1+ cos i)Ml(kcos i) (5.157) as as z ---oo. Along the outer surface of the tube, p = a+0, the fields when z- co can be estimated from the integrals of (5.138) by deforming the path of integration in the upper half-plane into a U-shaped contour around a branch cut from k to k+ioo. The symmetric term n =0 is dominant, and due to the boundary condition the components E, E and H of the total field are identically zero. For the remaining components of the total field, representing surface charges and surface current, the results are: b.1 r 205

THE UNIVERSITY OF MICHIGAN 7133-3-T ikzcosi. 00 icosjcos9. E (a+O, z) ~ -- e ' i ( H( + Vn=-o sin. H sin 0.) i n i n si2icos 3J (kasin i)L (k) ikz + 2 2 0 1 in i 2 (1 ka sin. L (k cos 0.) 2z r (kasin0.) H (kasin.) i o i In 2- +ii n 1 2 2 Yka (5.157) ikzcos0. ( "o. 2i ci n ei in C os _ B -H (a+0, 0, z) - e e i e - + 0 n=-Co kasine.H (kasin.) 1i n nsinl3cos.i ikz n sin, cos \i 2icos 3 J(ka sin0 i)L (k) ikz (kasinO.) H (kao i nO +i. 1 n i 1 2 1Yka 1[ ikz cos e. OD —. o 2i I.n in sin/ H (a+ z) e i e ()' 0 n=-o ka H (kasin0.) ln 1 _(k) sing Jo(ka sin.) - -ikz 0 1_____o______._ -i _ ae ( +ke-(k)ei'+ke (k)e L (kcos80.)(1- cos8.) 1 -1 2 i kz as z -oo, where In = 0.5772156649... is Euler's constant. The terms containing the summation over n from -oo to oo are the fields pertaining to an infinite cylinder (cf. equations 2.41 and 2.42). 5.4.4 Axial Incidence The case of axial incidence, i. e. 0. = 0, is of some interest because in that case the sum over n only contains the terms for n = +1. Thus, we need to know only the split functions M (a) and L (a) (- M (a)) to solve the problem. 1 1 o 206

THE UNIVERSITY OF MICHIGAN 7133-3-T The solution is obtained as a special case of the previous results if we put 0. =0o. It is no restriction to assume the incident field to be polarized in the x-direc — I tion, i. e. the incident field is j ikz E =ie x (5.158) i A.ikz H = Ef le The components of the total field are then m E (p,0, Z) = E1 +~ Cos 0 p p 7T H (1) (icp) 1 H(1) ra 1 e1(a) H~1 (K) iN 2 3 (1)'I e da a pK H 1(ecp)/ E (p 0, ) =E I ' ino E (p,0, Z) = Hf (,cp) a C (a. a) H (1 iaz) 1Zi' 1ia (1) 2 1) e da H(1 (ica) PIK2 H (i)(ca)/ 11 - ela z da (5.159a) F(Kca) 6-(a. H (1) a e (a) H (1), t,a)H1 (icp) 1 2 (1) 2 2 (1)' ~PK H 1(ica) a kic H 1(Kcay H (p,0, z) = Hi + p p 'IO 71T X iaz d 0 0 E.(, )H(1)? I (1) (p _ (Icp) a e (a) H (,p Cos 0S e1(y A 0 r KH(')(ca) pa2kKi3 H(1 (eca)/ 1 1 Xe izda 207

THE UNIVERSITY OF MICHIGAN 7133-3-T.0 Sinm0 e 1(a) H1 110 iarz H (p, 0,z) = - o i e da (5.159b) Z 7r k H (Ka) when p > a, where el(a) and 6~ (a,a) are obtained from (5.141) and (5.142) by putting n=1, e.=0 and 3= r. Thus, 1 2a4k2M (k) e (a) = -2 2 2 2- (k+ca)M (a) 1 4k a M(k) L(k) 1 (5.160) aLl(k) - (a, a) 2 L(a) Z 4k a M (k)-L (k) 1 1 The field components for p < a are given by expressions identical to those of (5.159) with H and H exchanged for J and J' respectively. n n n n The H and E components of the field inside the tube (p < a, z > 0) are z z H (p,0,z) = -2i sin0 i 2 i i e z o m=l 3 i2 -1) 1 l mf nm The far-zone scattered field is s s 2 eikr (am cos9) k mo 7r r sinO H(aksin) (5.162) E (p = -2ikr e (kcosi) Ez = - -H2 (1) Ei acr (amksi, ai 208

THE UNIVERSITY OF MICHIGAN 7133-3-T when krsin0 >> 1. In the back scattering direction 0 = r the polarization is the same as that of the incident field, and from (5.156) we obtain (OO) eikz ia 2k E (O, 0, z). z — + -oo (5.163) x Iz 4k2a M2(k)-L2(k) Along the outer surface of the tube the asymptotic forms of the nonvanishing components of the total field are 2 4k a M (ika2 k + E (a+ O, 0, ) H (a+,e0 (i+a2 4-a- (os+ L2(k p l Z 22 2 2 H o 4k a M1 (k)- L (k)/ (5.164) ce 8ia k M(k) ikz ro 1 e H (a+ O0,, z) -sin 22 2 2 (a o )4k2a2M2(k)- L (k) z as z - oo. 5.5 Electromagnetic Radiation from a Source Inside a Semi-Infinite Thin-Walled Tube 5.5. 1 General Solution We assume that the tube is located at p = a, z > 0 and that a single waveguide mode is propagated in the negative z-direction. The incoming mode can be either a TM mode with the axial component of the electrical field strength J (j P) -iac z E = E n nfi cosne (5.165) n nm or a TE mode with the H -component J j' ) -i' z n nm n > 0, m >1 -- 209

THE UNIVERSITY OF MICHIGAN 7133-3-T where Jn(jnm) 0, O< nl < n2< n(j m)=0 ' <nl <... and.2 12 2 nm 2 'nm a \ k a k nm 2 nm 2 a a are positive or positive imaginary. For arbitrary excitation inside the tube we have to calculate the field in a corresponding infinite waveguide at z = 0 expressed as an infinite sum of waveguide modes and then sum up the results from each mode. The problem has been studied in detail by L.A. Wainstein (1948c; 1950a, b). We can obtain its solution from the results of Section 5. 1 by proper choice of the functions f (a) and g (a) in (5.44) and (5.45). For that purpose it is suitable to den n compose the total field into the field obtained from the surface current at p = a, 0 < z < co, connected with the incoming waveguide mode, plus the scattering of this field from the semi-infinite cylinder. For the scattering problem so formulated the expressions of Section 5.1 are directly applicable and 23 o/w Kx a el(a) H ~ M (a)M (-a) n nm 2E (a+a' ) n n n nm (5.167) f.22 Ka2a +j2 a 1 K a urnm nm (a,a) = -— L (na)L (-a) E a +nH I zn 2E n n nm jn(a+a ) nm 2 '(+ a' ) o nm nm i i where c =1, =E.. =2. The only singularities of e (a) and t (a) are at o 1 2 n zn a = +k. Inserting the expressions of (5.167) into (5.42) and (5.43) and separating the result according to (5.7) and (5.8) we get 210

THE UNIVERSITY OF MICHIGAN, 7133-3-T e f((a) = H n n nm wu aM (a' ) o nnm (a+ a ) nm i e f(-k) = H nn nm En gn(a) = - En nfn nm I uo a M (a' )-M (k o n nm n k- a' nm (5.168) ia(k+a )L (a ) nkL (k) nm n nm n (a+a ) nm awe (k+a)(k-c' ).nm nm o nm iaL (a ) nL (k) n. n nm Eg(k)=-E -H " ng n nm j nm 2awE (k- a' ) nm o nm The quantities e-(a) and i- (a, a), related to the Fourier transforms of the tangenn zn tial components of the total electric field strength at p = a, -oo < z < 0, are obtained from (5.44) and (5.45) on putting Gn(a) = F (a) - 0. We write an arbitrary component of the total field as X(p, 0, z) = -, (p) e da, (a,)H )(pK)+ B(a, )H() z a p >a, -oD <z < Co (5. 169) X(p,0,z) = XI(p,0z)+ 2 [A(a, )J (p)+B(. )J (p e da, (5.170) p >a, -oo< z <oo where as before K =k -a and the path of integration r is as indicated in Fig. 5-1. X xtands for a component of the fields of the incoming waveguide mode and the functions A(a, 0) and B(a, 0) are given in Table VII if we define 211

THE UNIVERSITY 4 7133-3-T OF MICHIGAN 2k3a5nM (k)L (k)L (a ) P (a) = n n n nmfl (k+a)M (a) nm 0 PH( ) C a kM (a' ) n nm ( k-a 2(a+a' ) am n2kL2(k) \ (k- a' )NI nm 7 (k+ a)M (a) n (5. 171) 2 Q (a) = -L (a ) E i n nm nm ((k+a )(k+ a) nm + 2(a+a ) \ nm n2k L2(k) L(c) QH(a) = - H v (~ 2a2k2nL (k)M (k)M (a' ) n n n nm (k-a' )N nm L (a) n where N = 4k2a2M2(k)- n2L2(k). n n As a special case of (5.169) and (5.170), the current flowing in the wall of tne tube is given as -j(0, z) = H (a+0, 0, z) - H(a -00, z) = -H cosn0e nm -ia' z nm + + 0 0 i 7ra k r H nmPH(a) cos n0+E PE (a)sin n iaz 2- e da K M (a)M (-a) (5. 172) (5.172) 212

I THE UNIVERSITY OF MICHIGAN 7133-3-T TABLE VII Relation Between the Field Components and the Quantity X of Equations (5.169) and (5.170) X A(a, 0) B(a, 0),, E p E z H P -n2 b(a) cos no - bH(a) sin no 32 E plC a 2 cH (a) cos n0+ cE(a) sin n pK -i E(a) cos - cH(a) sinn A n - -- [H(a) cos nO+ CE() sin nO PK -32 bE()cosn - bH(a)sinn] pGu K a o 2 bH(a) cos n+ bE(a) sin no 0 a cE(a) cos n - CH(a) sin n 1-. [b (a)cosnO+b ()sinnj K a 0 — 2 2bH(a) cos n+ bE(a) sin n w2 2 K a o a c E(c) cos n0- cH(C) sinno K 0 0 H z I I A where E PnmE(a) bE (a) = H()'(a) H P (a) b (a) nm= H 1 H (aK) for p >a Enm QE(a) CE(a) = -(1) H (aK) n H nmQH(a) c(a) = () H H() (ac) n E PE(a) bE(a) = J'(ac) n H P (a) bH(a) = n... H Jn(aic) n E nQ (a) c (a) = E J (aK) n H QH(a) c H(a) = — J-(a H ) J (aK) n for p < a _J 213 --

I I THE UNIVERSITY OF MICHIGAN 7133-3-T j (0,z) = H (a+0,0,z)-H(a- 0,,z) e -Ca z naca' -at z I o ka c nm nm nm =- --- E cosnpe + H sinnpe +, I ij nm nij2 nm 1o nm + o1 p0 7ra 1~ 34 a kK M (a)M ()-a) n n k mQH (a) sin n - En QE(a) cos nl irz +. e da. K L (a)L (-a) I (5.173) n n 55.52 Field Inside the Tube When p < a, z > 0 the integral in (5. 170) can be evaluated by means of residues. Thus, the E and H components of the field can be written as z z I rJn ) - Ez(p,, z)= E cos n n nm a e n nm -ia z > nm + I =0 nE Rm mS. n e +o + H sin no, TnH nm m = J (j.) n ni a Jn(jn) n n ia z IJ e (5.17, Hz(p, p, z) = Hnm cos n ^ nm 1 Jn(j, ) J (j,) n nm aI n rnu -ia' z nm e 0o = nH0 f=0 Jn (j P) n n a ii iI ia' z1 00 +E sin no TnE nm Q= ml l:o Jn(j' P) i j (j ) e n nia ia[ z ni (5. where 214

THE UNIVERSITY OF MICHIGAN 7133-3-T nE Rml RnE nE - -R m e mli i 2 QE(an a ao iOnH nH _nH ml R = - R e,2 - P (a' ) 3ka 2 H n lu ~ ^ r i (5.176) nH _a _i T = 2 QH(anl) a an 12 TnE = - nE Tm I o \ akat (J 2 - n2) ) For symmetric modes (n =0), T = TnH = 0 and the reflection and conversion niFor symmetric modes (n=), T coefficients are easily expressible in terms of the auxiliary functions P (a) and S (a), n n connected with the split functions, of equations (5. 220) and (5. 225). Thus, for the reflection coefficients we have I k+ aol aaol ReP (a o R11 k- a e ol E = aolImPool) 11 OH aa ll RePl(all) IRH =ae 11 1( 11) oH Ol = acra1 ImPl(all) jol < ka < jo2 jll < ka < j12 (5.177) (5.178) 215

THE UNIVERSITY OF MICHIGAN 7133-3-T Approximate expressions for the reflection and conversion coefficients of arbitrary E or H modes can be obtained using the function U(s, g) defined by (5. 258). nmn nm The absolute value of reflection and conversion coefficients of the symmetric modes H l, Ho2 and of the reflection coefficient of mode H11 as functions of ka in the ol' o2 11 range j <ka < jo (j1 = 1.841, j =10.173) are given in Fig. 5-17 and the corresponding phase functions of symmetric modes Hol H 2 are shown in Fig. 5-18. The absolute value and phase of reflection and conversion coefficients of symmetric modes E E for jo < ka < o3 (jo = 2.405, j3 = 8.654) are shown in Figs. 5-19 and 5-20. When the incident mode is Eol and jo < ka < jo2, or when it is Hn and jl < ka < jo2 n= 0, or jl< ka < jnl' n >1, the only undamped mode traveling in the positive z-direction is the reflected incident mode. For large z the z-dependence of all components of the field is given by Z(z) = e-ihz + R eihz (5.179) where h is the wave number of the incident mode and R is the reflection coefficient given by (5.176). Equation (5.179) represents a standing wave of amplitude Z(z) = l1+|RI| 2|R I cos(2hz + ). (5.180) Thus the first node or antinode is located at z = -S where 2 ao = 0oE (5.181) for an incident E 1 mode and ol 2at = el (5.182) for an incident Hn mode. As in the scalar case we call the length I the end correction related to the pertinent mode. In Figs. 5-18 and 5-20 the quantity l/a is plotte for incident modes Hol and El respectively. ol olI Il 216 I

THE UNIVERSITY OF MICHIGAN 7133-3-T 1.0 0.1 0.01 0.001 FIG. 5-17: ABSOLUTE VALUE OF REFLECTION AND CONVERSION COEFFICIENTS OF SYMMETRIC MODES H, H AND OF THE REFLECTION COEFFICIENT OF THE MODE H.' (Wainstein, 1948c; 1950a). 217

THE UNIX TERSITY OF MICHIGAN 7133-3-T This page is blank 218

0.30 120~ O. 25 - - 1000 OH 11 l/a 0.20- 800 _ o. 5- oH 600 CD u 0.10 -40 0.05- -20~ OH 22 I ' i 1' 3 4 5 6 ka 7 8 9 10 FIG. 5-18: PHASE OF REFLECTION AND CONVERSION COEFFICIENTS OF SYMMETRIC MODES H01, H02 AND END CORRECTION. (Wainstein, 1948c) 01' 02^ H z trn co m P< 00 o ^ ~-A I 0 O ~rl O Z

1.0 0.5 0.1 0 -' CD3 I C3 H z tC H C) z O) C) 0.05 0.01 2 3 4 ka 6 7 8 2 3 4 5 6 7 8 9 FIG. 5-19: ABSOLUTE VALUE OF REFLECTION AND CONVERSION COEFFICIENTS OF SYMMETRIC MODES E 0. E02. (Wainstein, 1948c)

0.6 -120~ 0.4- 80~: 0.3- /-60~; 0 U0.2 40~ TM modes 2 3 4 5 6 7 8 FIG. 5-20: PHASE OF REFLECTION AND CONVERSION COEFFICIENTS OF SYMMETRIC MODES E0, E2 AND END CORRECTION. (Wainstein, 1948c) 01' 02 H z 0-h -t 0 0 q) z 0 0 0-( 0 z

THE UNIVERSITY OF MICHIGAN 7133-3-T There exist some symmetry relations for the coefficients R, RnH T n ml' mi' in'' TH. From (5.171) and (5.176) we obtain nm nE I n nE R - R.2 om.2 Rm Jnm Jnf a (.,2 na2 ) 2 a? (2j- ni nm nH- ni _ Rn nm4 m (5.183) 2 4 mm I 4 m 'm nm nQ U a (j -n12 ) n2 nmTnH -_ - ru TnE 2 )m E 4 ml Jnm 0n & Furthermore, for symmetric modes (n =0) and ka j > j, om of.Q o o oE mmoE E O= O = (5.184) il) 2 oH The same relation is true for 0 if ka >j >j ml om 3ol The power flow connected with the modes of (5.165) and (5.166) if ka >j or nm ka >j' is nm 4raka E o nm a 12 nm 2.2 nm n nm (5.185) H: 7ra4kal (j n2.n2 H n o in nm 2 p = 0nm 2~j4 nm n nmn If ka > jn, j the fraction of the power of an incoming undamped E mode conerted ins nand Hn modes respectively is verted into the E and H modes respectively is f ns 222

THE UNIVERSITY OF MICHIGAN 7133-3-T.2 2 a 2 nE _nmant p nE rP R rm/.2 m Jnlanm (5.186).2 a' (j' -n 2 2 tnE g~o nam ns nnsE t ms E 4 ms o iTaE 'is nm Similarly, the fraction of the power of an incoming undamped H mode converted into the H and E modes respectively if ka > j' I is nf ns ni ns ri nm 'nm (5.187) 4 E j' a 2 nH 0 ro __nm ns nH ms A.2 a Ij2 2 ms ns nm nm nE nE Im rmi rnH= r1m (5.188) nE.nH t =t tnm ml The total power reflection coefficients of the modes E and H are nm nm m 223

THE UNIVERSITY OF MICHIGAN 7133-3-T I s o 0 n E 1 E nH I=l s=l (5. 189) I s 0 0 nH nH + tnH r -Zr ~Z+ r t m= rmi ms m =1 m s=1 where the summation is to be taken over all undamped modes. 5.5.3 The Far Field We introduce spherical coordinates r, 0 such that p = r sin and z =r cos 9 2 2 and assume that kr sin >> 1. The integral in (5.169) can then be estimated by the method of steepest descent. According to (5.58) the result is E9(r,9,) - _ H = E (5.190) E (r, ) =, o H sn te Ec sine (-i) n H-r ------- E Q (kcos e) cos no+- S QH(k cos e)sin n0 e 2 (1)' F Pkcsnn+ r 7(kasin) H (kasin) nm E n Equation (5. 190) is still valid in the negative z-direction (9= =r) in spite of the fact that the condition kr sin 9 ~ 1 is not satisfied. It is readily checked that the far field of order 1/r in that direction vanishes except when n = 1 in which case the result is w 224

THE UNIVERSITY OF MICHIGAN - 7133-3-T 1 E (0,0, z) = x H (O,0,z) = x ka Ll(a )L (k) iklz -H y lm 2E k22 2 2-L I E i0m 1j a f1T-m1I E k a3M (a )M (k) ikizi E -, H M1 im 1 e \ Y lm (k-a )[4k2a2M2(k) L2(k1 1m l1 1 - J ) (5. 191) When z tends to infinity along the outer surface of the tube (p = a+ 0) the integral in (5.169) can be estimated by deforming the path of integration into a U-shaped contour around a branch cut from k to k+ ioo. The asymptotic behavior of the nonvanishing components of fields representing surface charge and current so obtained are, for n=0, p ikaL (a )L (k) E (a+0,z) =- H -E om om o ke p om(. 2z 2 om ka '1 (5.192; H (a+O,z),- -H z nm ka L(a' )L(k) ikz aL1 om 1 e 2(k+a' ) 2 om z as z — o, where AIn1 = 0.5772156649... is Euler's constant. For n >1 we have I E (a+0, z) = H -' 2-n(i)n- a 2n+ +2L (k)M (k) n - n n (n- 1): L4k2a2M2(k)-n2L (k n n ikz e n z X E 2ka M (k)L (a ) p n n nm coIn + o "n..... cos no + n H nm o nM (a' )L (k) n nm n sinnl k-a' J nm (5. 193a] 225

THE UNIVERSITY OF MICHIGAN 7133-3-T 22-n(_ in-1a 22n+2L (k)M (k) ikz 2 (-i) a k L (k)M (k) ikz H (a+O0,,z) 2~ X z (n-1)! [4k -n L (kL z n n nM n () n o 2ka2M (k)L (a ) n nm n on n n nm Hn k-a'- cosn0 - - E Dnnnm sin n nm k- n ~' \^ n L nm o nm (5.193b) as z-oo. By a well-known formula, the field outside the tube can be expressed as a surface integral of the field components over the opening and the outer cylindrical surface. In the Kirchhoff approximation the field over tne opening is assumed to be that of an infinite waveguide and the field on the outer cylindrical surface to be zero. It is readily shown that the field obtained from this approximation (in Kottler's formulation, cf. Stratton (1941), p. 468) is identical with the field calculated from a surface current in the wall of the semi-infinite tube equal to that of an infinite waveguide. Thus, the field of the Kirchhoff approximation is what we called the "incident field" in our formulation of the problem as a scattering problem. The components of this field are obtained from (5.169), (5.170) and Table VII if we put P (a) = 0 and define PH(a), Q (a) and QH(a) such that H PH(a) = nee(r) nm H nn (5.194) H nQK()- iE Q(a) = E i (a,a) nm H ni E n zn where e (a) and q (a,a) are given by (5. 167). Thus the far field according to (5. 190) is (5.190) is 226

THE UNIVERSITY 7133-3-T OF MICHIGAN (-i)n- J(kasin ) E 6-2 sin O [E L nm 2 2s 2 a k sin 0 J (kcos0+a )Cosn nm nm + — H n E nm a ka' sin2 O+j' cos0 nm anm sinno ikr e r j2 (kcose+a' ) nm nm ikr H cos n -- nm r (5. 195) EK&r~e,6,) (-i)n-1 kaJ'(kasin ) o n E 2(kcos0+a' ) ~ nm Comparing (5.195) with the exact expression in (5.190) we find Ee(r, 0,) E nm' - E (r, 9?0) m EK(r, ',0) (i)n-a k3k J' (j ) nm n nm 2j nm ikr E cos no e nm r (5.196) o (-i)n+la3ka, (j 2 0 nm nm -n2)J (j' ) n nm ikr X H cos n e nm r where 0 and 0' are defined by nm nm kcosO = -a nm nm kcos i' = -at nm nm (5.197) The power radiated into a unit solid angle about the direction (0, p) is p(,0) = E2+ IEI2 )r2. (5.198) 227

THE UNIVERSITY OF MICHIGAN 7133-3-T We call this quantity divided by the power of the incident E or H mode the power pattern fE (0, 0) and fH (0, 0) respectively. Thus, using (5.185) and (5. 190) nm nm we get 2 2 E e j ( QE(kcose) 2 fE ( 4.)nm cos2n + nr a 4ka sinH( )(ka sin8) nlM n PE(kcos0) 2 + E sin2n 2 (1)',(k sin )i f (kasin 0) H (ka sin )) n (5.199) E E j Q (k cos) 2) (, 0 nm H sin2n + mo r a ka' (j,2 -n sineH((kasi0) nm nm n PH(kcos) 2 ) +,H- - cos2no (ka sin )2 H(1) (ka sine) n From (5.196) we see that E ka fE (O m,) = fE(0, 0) = n2j ka -j2 nm( nm nm, 47r n anm nm (5.200),2 2 E ka 2 -n nm am n m nm 47r,2 nnm nm nm where as before E =1, E =E = =2. Thus for Eam modes the Kirchhoff o 1 2 nm approximation gives the correct value of the radiated power in the direction (0, 0) where 0, 7r/2 < < 7, is given by (5.197). If another mode E can propagate am nm undamped in the waveguide the power pattern vanishes in the direction (.,9 0) which also is correctly given by the Kirchhoff approximation. The same is true for H modes in the directions (e',0) and (0', 0) respectively. nm ni 228

THE UNIVERSITY OF MICHIGAN 7133-3-T The relation between f (0, 0) and the power gain function, i.e. the radiated nm power related to an isotropically radiating source, is f (e, 0) G (0,) -= -p( = 4 nm (5.201) nm ' /47r n rad 1 - r m where r is the total power reflection coefficient given by (5.189). The fraction of m the incident power which is radiated into the space outside the tube for H mode is oH oH 2 o 1- r = p- 1R11 when j <ka <j This quantity is shown in Fig. 5-21 as a 0o2' function of ka/j1I compared with the result obtained from the Kirchhoff approximation. The power pattern for modes Hol, Eol and H1l are shown in Fig. 5-22, together with the results from the Kirchhoff approximation. When ka >> 1 we can use the function U(s, g) of (5.238) to obtain an approximate expression of the far field. For example, inserting (5.257) and (5.261) into (5.190) yields, for H and E modes respectively om om eikr exp U(s, q )+ U(Som qH E e(r,) H e ) \E om r i4r (kcos ' +a ) H (ka sin 0) sin-m cos 2 1 2 2 X (5.202) 27rakJ (kasin0)sin 1 f, 2r/2 < < sin o 2 where s= \2ka os, =-); o mOm H 4 8ka/ 229

C3 00 O co Cd k a) 0 a) c4. CD3 l — 03 H H Pd 0n - M C) Z 0 z 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 ka/j11 2.0 = 3.83171...) FIG. 5-21: RADIATED POWER FOR INCIDENT H1 MODE CARRYING (Wainstein, 1948c). 1 WATT OF POWER (ill

THE UNIVERSITY OF MICHIGAN 7133-3-T watts Hll mode, ka = 2.0 watts watts 0.3r 0.2 0. 1 0 180~ FIG. 5-22: RADIATED POWER PER UNIT SOLID ANGLE FOR INCIDENT MODES EACH CARRYING 1 WATT OF POWER (Wainstein, 1948c, 1950a). (The solid curves for Hll mode are renormalized by using the fact that the Kirchhoff approximation gives the exact value at 01l (113. 0~). Consequently their absolute magnitude is rather uncertain. ) 231

THE UNIVERSITY OF MICHIGAN 7133-3-T Eikr iakexp[(s,qE)+U(sO,qj Ee(r,e) N' E E o Om r j(kcos 0+a ) om om 9 l om sin 2 (1) e, 0<e<7r/2 21r H (ka sin ) sin X 2 (5.203) 0o 0 akJ (ka sin 8) sin —co 7r/2<< r 0 2 2 where s = 2k acos0, s = qE = a+- ) om k om E 4 8ka and the angle 8 and 8' are defined by (5.197). Wainstein (1948c) reports that om om for Hol and El modes and ka equal to that of Fig. 5-22 (higher than the cut-off frequency by 6 percent and 5 percent respectively), the difference between the exact power patterns and the approximate ones obtained from (5.202) and (5. 203) does not exceed 2-3 percent. This indicates that for practical purposes the condition ka >> 1 is fulfilled as soon as the incident mode is above cut-off. For general H and E nm nm modes, somewhat more complicated approximate expressions are obtainable in the same manner. As j > j > n, when n > 1, those formulas should also be useful nm rn as soon as the incident mode is above cut-off. As pointed out in Section 5.6, the approximate expressions of Mn (a) and L (a) obtained by using the function U(s, g) have a small jump at a = 0. Consequently the approximate radiation patterns display a small jump at 0= r/2, the amount of which is a measure of the accuracy of the approximation. When ka sin >> 1 is satisfied along with the condition ka >> 1, we can replace the Bessel and Hankel functions of (5.202) and (5.203) by their asymptotic forms. For an E mode with 0 < < r/2, the result is om 232

THE UNIVERSITY OF MICHIGAN 7133-3-T Om 0 iinak sm 0cos8 exl(ikr-kasin9+-)+U(s.q E (r,, o)E 22 F(r, E) Om 2 V i om cose - cos em Vk^ om om (5.204) where F(r, ) = rsi exp (s, qE The factor before the function F(r, 8) is the field of an incident plane wave propagating in the direction o0 scattered by a half-plane tangent to the tube. If we express the incident waveguide mode as a superposition of plane waves repeatedly reflected at the wall of the tube, the direction of propagation of these plane waves is also given by 8. Thus the first factor of (5.204) can be interpreted as the geometomI rical optics contribution to the far field. The factor rsi "expands" the cylindrical half-plane waves into spherical waves as the distance from the edge increases. When 7r/2 < < r, substitution of the asymptotic form for J in (5.203) yields the expressions of (5.204) plus an identical wave originating from the opposite edge of the wall of the tube. The function U(s,qE) tends to zero as ka->co for every fixed 0 f 7r/2. At 8 = 7r/2 it is discontinuous in such a manner that it compensates for the jump in the geometrical optics approximation. This also means that the term U(s, q) is approximately zero for ka so large that the incident mode is not close to cut-off. As in the scalar case the principle of reciprocity can be used to relate the results of this section to those obtained in Section 5.4 for the scattering of a plane electromagnetic wave. We assume that the plane wave is that of (5. 133) (i. e., is propagating in the direction (., 0)) and write the field inside the tube due to this wave as m 1 233

THE UNIVERSITY OF MICHIGAN 7133-3-T I 0) co J (j' ~) ia' z Hz =L Z(A cosno+B sin n) n nm a enm n=0 nm nm m n nm n=0 m=l n nm (5.205) co 0Jn (Jn -) ia z n nm nm J n=0 m=1 n nm To an H mode incident from z = oD given by (5. 166) we relate the radiated far field In H ikr E ' Fnm(0)sinnp - pH ikr E G (0) cosn ea nm r In the same manner the far field pertinent to an E nm mode (5. 165) is taken as ikr E ^v FM (e)cos n - anm r ikr E^/a GE sinnr 0 nm r We then have A H = (-1)n o nm nm 'O i2E4 24njnm 2GH (r-0) sing3 ka 4a (j'2 n2) nm 1 nm nm J B H nm nm = (-1)n+l C 4ij'4 o Jnm Ao k 2a4 (j' -n) nm nm FH (Or-0.) cos 3 nm I C E = (-1)1+l nm n m i2E j n nm FE (Or-0.)cosj3 k2a4 nm 1 k a a am K - m 234

THE UNIVERSITY OF MICHIGAN 7133-3-T i4j2 D E (-1)n nm GE (7r-0.)sin nm nm 2a2 nm i k a an nm (5.2 2t) where =, =1 =... =2. It is readily checked that equations (5.206) conform to equations (5. 145), (5. 146) and (5. 190). From (5.206) the absorption cross section, defined as the ratio of the power transmitted into the tube to the power incident per unit area, can be related to the power patterns of the modes above cut-off. Thus, for jl < ka < j (jl1 = 1.841, j = 2.405) where only the Hll mode is above cutoff, we have 2 4 a(i)= X (j' H(-e, 0)sin2 +f f(r-0 i9,7r/2)cos2Ii (5.207) a i \3 1 11 i where X = 27r/k is the free-space wavelength. The absorption cross section for ka = 2 can consequently be constructed from Fig. 5-22. 5.6 The Wiener-Hopf Factorization 5.6.1 Explicit Expressions The fundamental step in the Wiener-Hopf technique is to find the split functions L (a) and M (a) analytic in the upper half-plane and such that L (a)L (-a) = iiJ (aK)H()(aK) (5.208=5.33) n n n n (1)' M (a)M (-a) = iJ (a) (ac)a) (5.209=5.34) n n n n where K=k - and where Ln(a) and Mn(a) behave as O(1/V-) as' a — o.. (See Wiener and Hopf, 1931; and Paley and Wiener, 1934.) These conditions determine the split functions completely except for a factor + 1 but all physical quantities are independent of the choice of this sign. As we have defined L (a) = L (a), -n n M (a) = M (a) and M (a) = Ll(a), it is only necessary to determine L (a) for -n n o 1 n II 1 235 --

-THE UNIVERSITY OF MICHIGAN 7133-3-T n > 0 and M (a) for n > 1. In the following we will give formulas only for L (a) in n n those cases where the corresponding expression for M n(a) is obtained by just replacing the Bessel functions by their derivatives. An explicit expression for the function log L (a) is obtained by applying n (5.8) to log iJn(aK)H(1)(H. Thus, n n. { f log (> i J ( -2 H( ) ( y2 L (a) = exp 1 pog.. k2 2)H)ak2 dj (5.210) where the path of integration passes below the pole y = a and where P designates the Cauchy principal value as y ]-+ oo. Wainstein (1948c; 1949; 1950a, b) factorizes the functions 7TKJ (aK)H()(ax) in which case the integral corresponding to (5.210) n n exists in the ordinary sense. The split function, analytic in the upper half-plane, is then instead 1|-ia(k+ a) L (a). Multiplying both numerator and denominator of the integrand of (5.210) by y+ a and observing that the logarithm is an even function of y we get La a logiJ 22 H (a T dlyj L (a) =expa (di.. (...1 (5.211 where the path of integration is indicated in Fig. 5-23a for the case when a is real and I\a < k. To determine the field quantities everywhere, it is sufficient to know L (a) and M (a) for positive real values of a such that 0 <a < k and for all positive n n imaginary values. L (a) and M (a) are continuous and different from zero at those n n points. The integrand in (5.211) is an even function of y and we can therefore integrate only over the interval (0, ao). Change of the variable of integration to v define by v = ak- for 0 <k and v=a - k for k<7 yields 236

THE UNIVERSITY OF MICHIGAN 7133-3-T (1) L (a) = \iJ(aC)H (aK) X n n n Y (v) ) vka v +arc tan J (v) -iog Jn(V)H( v) aoa n -d X exp a P 2 22 2r 2 dv + X (v2 a22) ja2k2 2 1 (v -a K )\a k -v OD v log 2I (v)K (v)] + i 2a dv (5.212) +hr S/(a22 v2) aa2k+v2 H va1 (aK +vv 0 M (a) = 7riJ'(aK )H (aK) x n n n (v)1)' eva v + arc tan n ilog ( |v)H ' l(v) + v (v) n n aa n Xexp dv + 7r 2 2- ) 2 a -v (a K +v Va k +v v2 _n 7T n (1) where -7r/2 <arctan 7r/2 and I (v)=i nJ (iz), K (v)= H '(iz) are modin n n 2 n fied Bessel functions. These expressions are valid if a is real and -k <a < k. However, L (a,) nnl and Mn (a ) take the form 0 'oD for real an and a' respectively, where as before nm 0, Inm 0 Jno j < j 0 =ij < j' <.. nnm nnm = no inl n O' (we define j =j =0 although they are not zeros of J and J') and 00 lo 0 1 m 237

THE UNIVERSIT! 7133 - Ak -a -anl -an2 kY OF MICHIGAN,3-T 1 m-7 brancr cut Al a I ean2 ani+ it) brachcut I -.=-*~- l Re oy an a an +k branch cut (I A2 v tf branch cut (b) Im~y cuts A31 Ak Re 7 an2 a ank branch cut (C) FIG. 5-23: PATHS OF INTEGRATION FOR THE SPUIT FUNCTIONS 238

THE UNIVERSITY OF MICHIGAN 7133-3-T.2 2 2 nm, 2 run nl 2 nm 2 na a are real and positive real or imaginary. Equations (5.202) and (5.213) are also valid for an arbitrary positive imaginary value of a if the square root factor is substituted by 1. In that case the use of principal value is no longer necessary. L1(k) can also be obtained from (5. 212) by putting K = 0. Thus ~r Y i(v ) ka a +arc tan - ilog JV)H(1) L (k)= exp.- 2 dv 00 log 2I (v)Kl(V ) + i ( 1 dv (5.214) va k2+v J A different expression for L (a) and M (a) is obtained if the contour of inten n gration is deformed according to Fig. 5-23b. This scheme has been employed by Wainstein (1948c; 1949; 1950a, b) and what follows is a generalization of his work. We write a + a -P (a) (1) rum 2n (5.215 L(a) = )(af) o (a) = riJ (ai)H (a) m e (5.215 o at +a S (a) M-a) (ax (ae) e (5.216 n n n at -a m=l nm where m and m' are the smallest non-negative integers such that j > Ka, o +o n(m+1) j' ka, the product should be disregarded. I mm I w 239

THE UNIVERSITY OF MICHIGAN 7133-3-T If we combine the integrals along both sides of the branch cut in Fig. 5-23b, observing that the small indention around y = k gives no contribution if a 4 k, we get k P (a) = 2 (P)\ n ra \ Jo n ny(a?7) +arctan n 2 2 Yn(a1?|7) 2 2 +arc tan. 2^ T icO -2 (P)o 7r a Jo 2 2 7y - a I * ( Yn(ax)" T dy- +arctan n) 1 aa m a +a log nm m=l nm (5.217) The functions 7r 11 + arc tan 2 J (x) n and Y' (x) + arc tan 2 J'(x) n are discontinuous and jump by ir at the zeros of J (x) and J (x) respectively. We n n introduce instead the continuous functions Y (x) 7 (1) n i n (x) = + arg H(x) = arc tan Y-x + + miT 2 n nJ (x n (5.218 if jn <x < j(m+l) nm n(m+l) Y'(x) n(x) = + argH( )(x) = arctanJ" + + +mr n 2 n J(x) 2 n if jn <x<jt, nm n(m+l) (5.219: 240

THE UNIVERSITY OF MICHIGAN 7133-3-T Thus, we have n (jn ) = 7r I > 0, n > 0 1>1, n >1 Q2'(0) = i', n ~2'(j' ) = Jr nfl) Insertion of n (x) into (5.217) gives a divergent integral which is compensated by a similarly divergent series. By subtracting the divergent part from both the integral and the series, the result can be written as 2 r vfQ (v)-aKQ (aic) ReP (a) = 2 \ dv, n rT -) a -v (a2 )a'2v v (5.220) Q (aic) 2 r ImP (a) = n-, +_ +log - n aa 7 kaL mm+ a aa arc tan - + 0 \a 2 2 m m=m +1 \ -a k 0 J nm +(m +1)+ v 2 - a222 + a.rccos 21 (v2- a2c(2).v -2 (5.221) for a real, 0 <a < k. Here m is zero or an integer such that j < ka < jn(m + 1 1 1 nm +1 and ~-(m+ 1) = 1+- + +... + -- - (y = 0.577216... ) is the logarithmic derivative of the gamma function. When a is positive imaginary, the expressions take the form ka a) v~(v) n ax ReP (in) = 2 ( 2 2)_]a k2- dv- n(a) n 7j (a2K 2- v2)J -v 0 (5.222) 241

THE UNIVERSITY OF MICHIGAN 7133-3-T ImP (in) = +logk - [2 (aK)-ac log +r +V(m +1)ImP (i~) -- 2 \ la +r(!n ka n m) - n(a) - mlog /+ I mm) + 0 nm On v (v)t-h-aKe (arh)-afo - a Ii + l d- v- a klo( 1 (5.223) 2 2 2 - 2 (v - a a K2k 2a for 0 < r < co. Here m iz zero or an integer such that jnm < Ka < in(m,r where o nm 2 2 as before Ka = ak +rl In the same manner, the expressions for ReS (a) and ImS (a) are obtained from (5.220) and (5.223) by substitution of 12' (x) and j' for 2 (x) and j respecn nrM n nm tively". When a — k the integral in (5.220) defining Re P (a) and Re S (a), n > 1, o n diverges, but this is compensated by a contribution from the indentation around 'y=k in Fig. 5.23b. Thus, we have o a +k b ka 4 (v) \ L (k) = T om exp o a -k 2v log v m oma k - k 0 v( ka 1 ka2 ka 'lr - dv+ - log log g +i — ImP (k (5.224; 7r va 2 2 b 2 ' Wainstein (1949) indicates that ' (x) should be defined as 0' (x) =arg H (x)1 - - which seems to be incorrect. 242

THE UNIVERSITY OF MICHIGAN 7133-3-T where b <1 if ka >1 and b=ka if ka < 1, a +k L(k) = nm exp -ka P(k) n>l (5.225) I m0' m +k vka '(V)-T. i-. M(k)=- n exp- 2 -2dv+ [alm=m (k)+ n ak a -k 7r 2 n m1 nm 0 v n 1 (5.226) As before, m or m' is an integer or zero such that j < ka < j + and o o knm n(mo+1) inm <ka < in(,) respectively, and the product should be put equal to 1 if m = 0 or m' =0. o o 5.6.2 Low Frequency Approximations A low frequency expansion of L (a) and M (a) can be obtained from (5.220) through (5.226) by using the power series expansions of J and Y or J' and Y' n n n n respectively. Since 2 (v) behaves as - 7r 2(y+ log) when v - 0, the expansion for L (a) contains a series of inverse powers of log(ka) and consequently it can be expected to give accurate results only when log ka > 1. The expression for L (a) is L (a) o L (k) -{-A-1 log(l-)+A- 2 /2(e) - o o 2 w2 - A-3 [ log(1 - - 0 u 2(1 u) ]} (5.227) where II I 243

THE UNIVERSITY OF MICHIGAN 7133-3-T A = -2 log ak- 2y+ i7, k- a 2k r ' 2k ' = 0. 577216... (Euler's constant) and rx J2(x) =g - 1~-t)dt (the Dilogarithm) - ~-\ 2^ 2 \ L (k) ~ - A - 2?(3)A 00 =L (- 2(3)A ) co (3) = ~ 3 = 1.2020569... n=l n (5.228) (5.229) (5.230) Equations (5.227) and (5.229) are given by Hallen (1961) in connection with his treatment of cylindrical antennas. We also give only the results for L (a) and Ml(a) which are the ones most easily obtained. ReP1(a) ~ RePl(ir) " a(k2- a2) k+a ka 4a k- 4a 2 2 2 a(k +l ) a k ka arc tan- 2 2r r) 2 for a real, 0 <a <k (5.231) for 0 < r < o Re Sl(a) ~ ReSl(in) ~ 22 2 a (k - a )-4 k+a ka a2(k2 -- 4log k. + - for a real, 0 < a <k 4aa k- a 2 2 22 a (k + r2)- 4 k ka arc tan- + - for 0 < rl < co 2at7 ri 2 (5.232) (k) 2 ak2 19 ILl(k)| exp - - + (T- - + logka) 1 4 1 (5.233) 244

THE UNIVERSITY OF MICHIGAN 7133-3-T |M(k) exp k 13 a2 k (+ + logka) (5.234) The expression for jLl(k)| is given by Levine and Schwinger (1948) and they report that it gives values in excess of the correct value by less than 3 percent if ka < 1. 5.6.3 High Frequency Approximations A high frequency expansion of Im P (a) and Im S (a) is obtained by employing instead the asymptotic series n 1 v+- + -1)( /1-25) + ( -1)(P2-114+10 73) nv)V v-(-< -- + 2(v) n 2(4v) 6(4v)3 5(4v 3 2OD + (p - 1)(53 - 1535/? +54703 - 375733) (5235) 14(4v)7, (v) - n 3 + 46. -63 3 +185/?_ -20532 +1899. n 2(4v) 6(4v)3 5(4v)5 ( )(4v) 5(4v) v -4. 00 (5.236) where =4n. Hence I m 245

THE UNIVERSITY OF MICHIGAN 7133-3-T IM (.,. 2 -+1+-,,, are tanr a + n aa 7r aka 1m+l. a2 -a2 m) ka- oo V nm (n 1\ 2 a K a j7 (4 - 1)arccosk + (m +1) + -arccos -- - 0 a 2aa 8a2 / a a arc cos - F arc cos(+ - 1)(~ - 25) k. 1 +.+A - 384 a( a)3 (Ka 2k) - aa(Ka) 1-3 2 v(K-2-1 (ka)2(a1 1 3.. (2v+1) I2 -I for a real, 0 <a <k, where A is the coefficient of v in (5.235). 2 27r /r log..... j 1 -a k i P (i) k a.l +,' -(m+1)fl7 \ 22 o K.log(+.. (5.238) 7 k na 8na 2 for a = irg, n > 0. The corresponding expressions for Im S (a) are obtained by substitution of the coefficients of (5.235) against those of (5.236). To obtain an approximate expression for Re P (a) and Re S (a) valid for larg values of ka, we deform the contour of integration in (5.211) according to Fig. 5-23c Apart from the integrals along A3 and Ao we get contributions from the pole at = a and from the first quadrant of the large circle used to complete the contour. The integral along AI is readily performed. It only contributes to the imaginary parts of P (a) and S (a) and is essentially equal to the series term of (5.222) or 246

THE UNIVERSITY OF MICHIGAN 7133-3-T (5.223). The real parts of P (a) and S (a) are given by n n K (u) XT n 22o - arc tan - (u),.~n 2 7TI (u) ReP (ar) = -1 - 2 2 udu (5.239) (u + a2) a k2 +u -K' (u) Un Soo 5- arctan n '-n 2 7rI' (u) ReS (a) = -1+ (1) n udu (5.240) n r 2 + 2 2 (u + a K Va2k + u where a is real and 0 < a < k. Consider the formula 2 U - u 5; 2+132lo ~1eUKn(U) J7r 2 +2 lo U n- i n -oD =- rilog ia e g H (5 241) where 3 is real and positive and the line of integration lies below the branch cuts. The quantities 7Y, 72,...M - are the complex zeros of K (u) in the third quadrant, and we put 7y = 0. The number of zeros is M 2 - 41 - (-_1) and they are dis2 4 -tributed close to the curve indicated in Fig. 5-24. Equation (5.241) is a generalization of a formula given by Jones (1955). By changing the sign of the variable of integration in the range (-ao, 0), and separating out the term containing the product over M which can be easily integrated, we get 247

THE UNIVERSITY 7133-3-T OF MICHIGAN - Im u -na Reu -in 2 a = t2-1 = 0.66274, where t is the positive root of cosht = t. o LOCATION OF THE ZEROS OF K (u) AND THIRD QUADRANT. FIG. 5-24: K'(u) IN THE n Jo 0 / e2U K (u) \ u log du U2 2 TrI (u)-i(- K (u)/ n n = -Irilog ^ 7 i e 4 (1) 2n M 2 m 1 2+ 2 mT^J (5.242) Taking the imaginary part of both sides, we have n10 K (u) u n (-)n arctan -du 22 2rI (u) u2 ++ Ii 2 [2 (J n( fl n/Jy21 M -2 log m=0 2 ( 248

THE UNIVERSITY 7133-3-T OF MICHIGAN and in particular, by letting 3 -- oo, 50 rK (u) (-1)n+1 uarctan -du = 7r rI (u) du = n By an identical procedure we obtain 4n -1 16 Re 2} i (5.244) ( K' (u) u n arc tan du 2 7r' (u) n =r 2 2 J[(J()+ y2(p M' -z m=0 245) (_1)n+ uarctan n du = - 7In (u) n 2 + Rem=Ol 4n 22 16 + Z3 (5.246) where y' are the complex zeros of K' (u) in the third quadrant and ' = 0, m n 0 Mt'= n+1-(-1)l]. We have We have K (u) arc tan _ = O(e ) as u — oo n (5.247: and the same is true for -K'(u) arc tan -( 7r (u) n We therrefe obtain an asymptotic expression as ka —oo from (5.239) and (5.240) by replacing (a2k2 + u2- 1/2 b by replacing (a k +u) by 249

THE UNIVERSITY OF MICHIGAN 7133-3-T 2 1 _2 ak 3k3 2a k and thu S 2 1+ — 2 r n 2ak 2 n-n ka — oo 42 M + 1 4. _ Re 2 2 a 33 \ - Re 2a k m=0 L 1\2+ 2 2 1^ -^m - -(-1)n f() (5.248) 2 1+- 2 Mt 4r - 2k?r (2 2 ReS (a) -1 - k log -a- 2 (aK)+Y,2(ax n L2ak n n o 2 2 -m=0oo m ka -— oo - ml m 3k3 (4) f(a) ( -4n2-3 Re + -1)n- 4 f(a) 2a3 k, 16 m=0: (5.249) where K =k- 2 and -log O<K< aa k- a f(a) = - a arc tan k < K alal k k (a real) (5.250) (a imaginary) The divergence of the integrals in (5.239) and (5.240) when a -tk is compensated by a contribution from the indentation around a = k of the path of integration, so that L (k) v o ka 1lo ka 1 2 4 log- 64a 2k2 (5.251) k- a om ka-Xoo 250

THE UNIVERSITY OF MICHIGAN 7133-3-T / k+a En-l)'.2 m k Om ka Rn[!2n] 1L l ka m L______og ILn'I 11 k-a 2 4 2 r ka o om M M 2^ 0 1 4n -*2 log(2ka) + 2Zlog 2 ( -1 m=1 4a k m=016 n>1 (5.252) 0 / k+ at 1 nd-l)2 M (k) 1 k- m exp -2 1 log 2- + ak k- a 2 4 27 kaa —oo _ )nlog(2ka)+ 1 E log|8 | 2 2 3 I6 R {m}2 (- )n 2 m=l 4a k 1 m=O n >1 (5.253) where, as before, m or m' is an integer or zero such that j < ka < jn(+1 or 0 o nio n(mQ+1; j < ka < n(m+1). Jones (1955) reports that (5.248) and (5.252) for RePl(al) Jnm mm + / 1 u (S >) and IL1(k1 respectively yield values differing fromt he exact ones by less than one percent when ka > 2. Wainstein (1948c) has derived an asymptotic approximation to L (a) and L1(a) in terms of a universal function U(s, g). Bowman (1963c) shows that both L n(a) and M (a) can be represented asymptotically in terms of the same function for all values of n. The function U(s, g) was introduced by Wainstein in connection with the problem of diffraction by two parallel half-planes. We start from the formula log 7ra\k -7 Jn 7 )H aVk - i _ n 2 log (-ia(k+ a) L (a)) = 35 i - y-aAY J-(2.254) (2.254) 251

THE UNIVERSITY OF MICHIGAN 7133-3-T where the path of integration passes below the pole y = a. integration in (5.254) by y =ksinT and obtain _____ - k log vakcosT J (a log ia(k ) Ln(a)) - cos~ kIin C We change the variable of k cos)H(1) (ak cos T) n d - a (5.255) The path C goes from T +io =to - ico and below the point T = arc sin a. t2 2 If we introduce into (5.255) the asymptotic approximation (1) r 2i2n (x)- + 2_ 1 'rxJ (x)H( (x) n-, - e 2- +.. n n L 8x2 x — > co (5.256) where 2 (x) is given by (5.235), we can deform the contour C into a path of steepest descent C. From (5.256) and (5.235) it follows that C goes through a saddle point o o at = 0 at an angle -7r/4 with the real axis. Thus, we may replace the exact integral in (5.255) by the usual steepest descent approximation and we can write i 7- + U(s, qL) L (a) V 1 4 L n (a(k+ a) ka->co (5.257) if a is real and 0 <a < k or if a is positive imaginary, where U(s,g) = 21 log(l-e dt t-ie 4 -t1 - t-se (5.258) and where s = a, qL= Q(ak). We have U(0+, q) = - log(l+e2). (5.259) 252

THE UNIVERSITY OF MICHIGAN 7133-3-T By comparing the corresponding value of L (0) with the exact value n L (0) = 7riJ (ak)H (ak), (5.260) n n n and indication of the error in (5.257) can be obtained. In an analogous manner we have i + U(s ) (5.261) Mn( \a(k+a) e ka —+ ao if a is real and 0 <a k or if a is positive imaginary where a s = a" qM = n(ak) It should be noted that all approximate formulas for ka ->oo given here are obtained from the ordinary asymptotic expansion for the Bessel functions, i.e. the order is kept fixed as the argument tends to infinity. This means that we have to require ka > n, when n > 1 in order to apply the formulas. Consequently, a high frequency approximation for scattering of a plane wave at nonaxial incidence cannot be obtained because in that case functions of order 2 or 3 ka are needed to obtain sufficient accuracy. We have given formulas for L (a) and M (a) valid if a is real and 0 <a < k, or if a is positive imaginary. As we have seen in the preceeding sections of this chapter, the physical quantities of interest are given as inverse Fourier transforms of functions involving L (a) and M (a). When z > 0 we can deform the path of integration for the inverse Fourier transform (Fig. 5-1) into A of Fig. 5-23b, and when z < 0, into a corresponding contour around a branch cut from -k to -iao. As L (a) and M (a) are regular in the upper half-plane they take the same values on n n both sides of the branch cut in Fig. 5-23b. When z < 0 and the branch cut goes from -k to -ioo we get the values on the upper and right side of the branch cut a+ and on 253

THE UNIVERSITY OF MICHIGAN - 7133-3-T the left and lower side a - from 7riJ (arc)H(1)(aK) L (a+) = (5.262) n L (-a) n ri J (aK)H(2)(aK) L (a-) =- n (5.263) n L (-a) n \~2 2 where aK = a -, as before, is the branch that is positive when a =0. The corresponding formulas for M (a+) and Mn(a-) are obtained by replacing the Bessel functions by their derivatives. 5.6.4 Numerical Computations Numerical computations of Ll(a r), I = 0, 1... 6 have been performed by Matsui (1960) for 0 4 ka < 3 using (5.212) and (5.214). Jones (1955) reports numerical computations of Ll(a l), - = 0, 1, 2, for 0 4 ka < 10 by Brooker and Turing. They use a formula obtained by deforming the path of integration into a of Fig. 5-23 (cf. equation 5.239). The numerical values are shown in Table VmI*. For ka < 0. 5, using a formula similar to (5.226), Hallen (1956) has computed the end admittance of a tube-shaped antenna, which admittance is equal to the complex conjugate of the quantity 2 4ir 0 LL1 ] His results, converted into a graph of L (k), are shown in Fig. 5-25. iwt * Brooker, Turing and Matsui use the time factor e. Their split function is there fore the complex conjugate of Ll(a). 254

H z m 3.(.en Cen 0 2.0 - 1.0 FIG. 5-25: REAL AND IMAGINARY PARTS OF L (k) AS A FUNCTION OF ka. (Halleu, 1956) conv(The diagram is crsion frostructed from Eq. 5. 229 for small values f ka and due to the conversion from Hall~n's curve, the accuracy for large values of ka is rather poor.) z rll (-4 m?0 CA PW4 H 04 co GO I~ —! l l w GO I 00 0 1T7 0 -4 0 z C)

THE UNIVERSITY OF 7133-3-T TABLE VIII Real and Imaginary Parts of MICHIGAN Llal~) ka.0.1.2.25.3.4.5.6.7.75.8.9 1.0 1.1 1.2 1.25 1.3 1.4 1.5 1.6 1.7 1.75 1.8 1.9 2.0 2.1 2.2 2.25 2.3 2.4 2.5 2.6 2.7 2.75 2.8 2.9 Re L1(k)} Im 1(k M J M J 1 1.0000.9957.9831.9633.9375.9070.8730.8364.7983.7595.7205.6818.6440.6073.5719.5382.5060.4757.4471.4204.3955.3724.3510.3313.3133.2969.2820.2686.2566.2459 1.0000.9747.9078.8180.7206.6252.5375.4601.3942.3394.2952.2606.0000.0609.1198.1752.2259.2716.3117.3464.3757.4000.4196.4348.4460.4536.4580.4595.4586.4554.4504.4437.4356.4264.4161.4050.3932.3808.3680.3549.3414.3276.2747.3654.0000.1501.4235.4541.4634.4568.4391.4139.3839.3510 * Matsui (1960), Jones (1955) M = Matsui J = Jones 256

THE UNIVERSITY OF MICHIGAN 7133-3-T Table VII (cont'd) Re {ZL1(k} Im L1()} Imka M J M 3.0 3.25 3.5 3.75 4.0 4.25 4.5 4.75 5.0 5.25 5.5 5.75 6.0 6.25 6.5 6.75 7.0 7.25 7.5 7.75 8.0 8.25 8.5 8.75 9.0 9.25 9.5 9.75 10.0.2366.2348.2173.2086.2145.2867.2900.2790.2625.2439.2253.2078.1922.1789.1684.1612.1588.1705.2068.2054.1982.1884.1776.1668.1565.3136.3163.2799.2405.1888.1903.2208.2391.2493.2532.2518.2464.2316.2260.2119.1954.1750.1388.1640.1801.1901.1957.1978,1970.1937.1882.1809,1717.1604, 1452.1473.1395.1334.1298.1305 a10 - k ad 11 2 11 2 a I >1 Is real aud positive or positive imagiary. 257

THE UNIVERSITY OF 7133-3-T MICHIGAN Table VI (cot'd) ka J 'm Ll1I M J.0.1.2.25.3.4.5.6.7.75.8.9 1.0 1.1 1.2 1.25 1.3 1.4 1.5 1.6 1.7 1.75 1.8 1.9 2.0 2.1 2.2 2.25 2.3. 2.4 2.5 2.6 2.7 2.75 2.8 2.9.4533.4537.4547.4564.4588.4618.4653.4693.4737.4785.4836.4890.4947.5006.5065.5125.5184.5243.5298.5350.5397.5436.5467.5487.5494.5483.5453.5398.5313.5193.4533.4556.4618.4715.4836.4976.5124.5270.5395.5476.5480.5356.0000.0000.0002.0005.0011.0022.0036.0055.0000.0003.0021.0066.0080.0111.0148.01M3.0245.0305.0373.0451.0539.0637.0746.0866.0999.1145.1304.1476.1662.1862.2075.2301.2537.2782.0147.0272.0449.0687.0996.1385.1858.2413 258

THE UNIVERSITY 7133-3-T OF MICHIGAN Table VIII (cont'd) ka 3.0 3.25 3.5 3.75 4.0 4.25 4.5 4.75 5.0 5.25 5.5 5.75 6.0 6.25 6.5 6.75 7.0 7.25 7.5 7.75 8.0 8.25 8.5 8.75 9.0 9.25 9.5 9.75 10.0 all M J M J Re {Llll)} l.5031.5028.4378.3206.1101.3440.3761.3553.3223.2880.2562.2286.2055.1878.1733.1644.1616.1749.2202.2194.2111.1997.1872.1748.1631.3031.3026.3624.3982.3080.1148.2052.2531.2768.2851.2831.2742.2606.2437.2244.2026.1770.1328.1614.1809,1930.1998.2024.2015.1977.1915.1832.1730,1604.1437 " lam ll}.1527.1439.1372.1332.1338 where j 1= 3.83171... 259

THE UNIVERSITY OF 7133-3-T MICHIGAN Table Vm (cont'd) Re ZIf 12} Im{L1a12 ka M J M J.0.1.2.25.3.4.5.6.7.75.8.9 1.0 1.1 1.2 1.25 1.3 1.4 1.5 1.6 1.7 1.75 1.8 1.9 2.0 2.1 2.2 2.25 2.3 2.4 2:5 2.6 2.7 2.75 2.8 2.9.3526.3527.3532.3539.3549.3562.3576.3593.3594.3604.3631.3671.3611.3631.3652.3674.3697.3720.3744.3768.3791.3815.3721.3778.3837.3894.0000.0000.0001.0002.0005.0009.0015.0023.0033.0046.0060.0078.0098.0121.0146.0175.0207.0242.0280.0321.0366.0414.0466.0521.0579.0641.0706.0773.0844.0917.0000.0001.0009.0028.0061.0110.0177.0264.3837.3859.3879.3897.3913.3926.3937.3943.3945.3942.3934.3919.3946.3986.0371.0500.4008.4001.0649.0819 I 260

THE UNIVERSITY OF 7133-3-T MICHIGAN Table VIm (cont'd) {ReL 12 Im {L1(12)} ka M J M 3.0 3.25 3.5 3.75 4.0 4.25 4.5 4.75 5.0 5.25 5.5 5.75 6.0 6.25 6.5 6.75 7.0 7.25 7.5 7.75 8.0 8.25 8.5 8.75 9.0 9.25 9.5 9.75.3896.3954.3850.3657.3275.3145.3331.3488.3628.0991.1004.1196.1376.1483.0196.0884.0902.0989.1137.1344.1608.1929.2294.2676.2997.3030.3748.3837.3878.3846.3705.3399.2842.1886.0167.2805.2921.2750.2507.2258.2024.1831.1197,1131.1750.2088.2256.2313.2295.2225.1666.1536.1443.1391.2119.1987.1832.1654 10.0.1398.1426 a12 = where j12 = 7.01559... 261

THE UNIVERSITY OF 7133-3-T MICHIGAN Table VIII (cont'd) ka.0.1.2.3.4.5.6.7.8.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 Re L of13)} M.2988.2989.2992.2996.3002.3009.3018.3027.3038.3049.3061.3074.3087.3101.3114.3128.3142.3155.3168.3181.3193.3204.3213.3222.3229.3234.3237.3238.3236.3230.3221 M.0000.0000.0000.0001.0003. 0005.0009.0013.0019.0026.0035.0045.0056.0069.0083.0099.0117.0136.0158.0180.0205.0231.0259.0289.0321.0354.0389.0425.0463.0502.0543 (ReB Llo)} M.2640.2641.2643.2645.2649.2654.2660.2666.2674.2681.2689.2698.2707.2716.2725.2734.2743.2752.2761.2769.2777.2785.2792.2798.2803.2807.2809.2811.2810.2807.2803 Im {L^)}14 M.0000.0000.0000.0001.0002.0004.0006.0009.0013.0018.0023.0030.0037.0046.0056.0066.0078.0091.0104.0119.0135.0153.0171.0190.0211.0232.0255.0279.0303.0329.0355 a13 where 13= 10.17347... a14= lo 14= 2 14 - a where j14 13.3236 262

THE UNIVERSITY OF 7133-3-T MICHIGAN Table VIII (cont'd) ka.0.1.2.3.4.5.6.7.8.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 Re Li(a'15) M.2391.2391.2393.2395.2398.2401.2406.2410.2415.2421.2427.2433.2440.2446.2453.2460.2466.2473.2479.2486.2491.2497.2502.2506.2510.2513.2516.2517.2517.2515.2512 M.0000.0000.0000.0001.0001.0003.0004.0007.0009.0013.0017.0022.0027.0033.0040.0048.0057.0066.0076.0086.0098.0110.0124.0137.0152.0168.0184.0201.0219.0237.0256 Re Lli(f1)i M.2201.2202.2203.2204.2207.2209.2213.2216.2220.2225.2229.2234.2239.2244.2249.2254.2259.2265.2270.2274.2279.2283.2287.2291.2294.2296.2298.2299.2299.2298.2296 M. 0000.0000.0000.0000.0001.0002.0003.0005.0007.0010.0013.0017.0021.0026.0031.0037.0043.0050.0058.0066.0075.0085.0095.0105.0116.0128.0141.0154.0167.0181.0196 15 I 15 2 a where j = 16.47063... whr 15 a l6 '16 - a where a J16 19.61586... 263

THE UNIVERSITY OF MICHIGAN 7133-3-T VI EXPERIMENTAL DATA This chapter contains the experimental data for scattering by circular cylinders. Most of the material is relevant to scattering by infinite cylinders, but some results for finite cylinders are included. Scattering of a plane acoustic wave by an infinite cylinder is presented in view of the direct correspondence to scattering of an electromagnetic wave. The details of the measuring techniques will not be discussed here, but appropriate reference are given in each case. The infinite cylinder is considered first, followed by the finite cylinders. The excitation is by a plane wave or point source. The notation used is given in the following diagram: t --- --— Receiver I I I I Source A 1 I 0I O II I,,I ength 3. 28 cm. Figures 6-1 through 6-3 are plots of the amplitudes and phases of the back I o -I -- -- 6.1 Scattering of Plane Waves by an Infinite Circular Cylinder Using the parallel plate technique, Adey (1955) measured the amplitude and phase of the diffracted electric field for perfectly conducting circular cylinders with ka = 2, 3.4 and 5.97. The incident radiation was a plane wave propagating perpendicular to the cylinder axis with the electric vector in the axial direction and wavelength 3.28 cm. Figures 6-1 through 6-3 are plots of the amplitudes and phases of the back and forward scattered fields. 264

THE UNIVERSITY OF MICHIGAN 7133-3-T Current distribution measurements on conducting cylinders for a plane wave i incident perpendicular to the axis of the cylinder with the magnetic vector parallel have been performed by Wetzel and Brick (1955). The image plane technique was used, and the normalized amplitude and phase of the surface current, as functions of the angle measured from the center of the illuminated side, are presented in Figs. 6-4 and 6-5. The cylinder was perfectly conducting with ka= 12 (cf. Eq. 2.42). Cook and Chrzanowski (1946) studied the absorption and scattering of a plane sound wave by a simulated infinitely long circular cylinder whose axis was perpendicular to the direction of incidence. Figure 6-6(a) shows the absorption cross section for a Fiberglas cylinder of radius 2.88 in. over the frequency range 100-5000 Hz. Figures 6-6(b) and (c) correspond to the cylinders having one and two layers respectively of cattle felt of thickness 7/8 in. wrapped around them. Theoretical curves are given for comparison purposes. Acoustic scattering by circular cylinders of infinite length immersed in a liquid medium has been treated by Faran (1951). He measured the amplitudes of waves scattered by metal cylinders in water. Figures 6-7(a) and (b) show the scattering amplitude patterns for brass and steel cylinders respectively with ka= 1.7. The direction of the incident plane wave is indicated by the arrow in each diagram. Figures 6-8(a), (b) and 6-9(a) give the scattering patterns for brass, copper and stee cylinders, respectively with ka = 3.4 and Figs. 6-9(b), 6-10(a),(b) show the corresponding quantities for ka = 5.0. 6.2 Scattering of a Point Source Field by an Infinite Circular Cylinder Kodis (1950) used the image plane technique to measure the scattering of elecromagnetic waves by conducting and dielectric cylinders. The electric field was directed along the cylinder axis and the source was a horn antenna operating at 24 GHz (X = 1.25cm). Figures 6-11 through 6-22 show the amplitude and phase of the diffracted field for brass cylinders with ka =3.1, 6.3 and 10 respectively. The heoretical values for point source excitation are related to those for a line source parallel to the axis. 265

-- THE UNIVERSITY OF MICHIGAN 7133-3-T Wiles and McLay (1954) employed a technique similar to that of Kodis (1950) to measure the diffracted electromagnetic field amplitudes for brass cylinders of infinite length. The incident cylindrical wave had a wavelength of 3.2 cm and electric vector parallel to the cylinder axis. Figure 6-23 shows the relative intensity of the axial component of total diffracted electric field for ka = 2.494, measured in the range 5 < ky < 25. Bauer, Tamarkin and Lindsay (1948) used ultrasonic waves at 1145 kHz (X = 1.3 mm) to measure the scattering by steel cylinders in water. Relative pressure distributions at different points at right angles to the direction of propagation, for various distances of the obstacle and the receiver from the source are plotted in Figs. 6-24 through 6-31. The models used were 1/4 and 1/2 in. steel rods and 5/8 in. polystyrene tubes. 6.3 Finite Cylinders Measurements of scattered pressure for finite cylinders using acoustic waves have been made by Wiener (1947). The wooden cylinders had length and diameter equal, and the results were compared with theoretical values for an infinite cylinder. The scattering of electromagnetic waves (X = 3.13 cm) by brass cylinders of length-to-width ratios 4.44, 8.89 and 13.32 has been determined by Giese and Siedenthopf(1962). The measurements were confined to the far fields as a function of the scattering angle (angle between the radius vector to the point of observation and the cylinder axis). Meyer, Kuttruff and Severin (1959) used the Doppler method to measure the electromagnetic back scattering cross sections of finite cylinders at a wavelength of X =3.2 cm. Figures 6-32 through 6-39 give plots of the differential cross section as a function of the scattering angle 0. The maximum back scattering cross section (in the plane 0 = 90 ) is presented as a function of i/X in Fig. 6-40, and the corresponding quantity for 0 =0 0 or 1800 is plotted in Fig. 6-41. 266

THE UNIVERSITY OF MICHIGAN 7133-3-T 6.4 Thin Cylinders Figure 6-42 shows a set of measurements reported by Van Vleck, Bloch and Hammermesh (1947) for a thin cylinder (I = 900 a) at broadside incidence. The normalized back scattering cross section is compared with theoretical values calculated by Lindroth (1955). Figure 6-43 shows the results of measurements made by Liepa and Chang (1965) on the back scattering cross section of a silver-plated stainless steel cylinder 1/16 in. in diameter. The frequency of the incident electromagnetic wave was maintained at 2. 370 GHz (corresponding to ka = 0.0394), and the length of the cylinder was varied from 30 in. to 1.5 in. (Note that this curve is not directly comparable to Fig. 6-42 because 1/a is kept constant in the latter, whereas ka is constant in the former.) A similar set of data for a silver plated stainless steel cylinder with ka =.0222 is given by King and Wu (1959) in Fig. 6-44. The values are compared ith data for a cylinder with ka =. 0202 (Liepa, 1964) and are in good agreement. 267

kp kp f0) -c! q H I-q z Z4 m -4 cn 0 z kp kp FIG. 6-1: MEASURED (- * ) AND THEORETICAL ( —) AMPLITUDE AND PHASE OF TOTAL ELECTRIC FIELD IN THE FORWARD DIRECTION (0 =0) FOR A METALLIC CYLINDER WITH Ei PARALLEL TO THE AXIS (Adey, 1955).

THE UNIVERSITY OF 7133-3-T MICHIGAN Q) PC -le4 kp FIG. 6-2: MEASURED (- * -) AND THEORETICAL (-) AMPLITUDE OF TOTAL ELECTRIC FIELD IN THE BACKWARD DIRECTION (0 = r) FOR A METALLIC CYLINDER WITH E1 PARALLEL TO THE AXIS (Adey, 1955). 269

THE UNIVERSITY OF 7133-3-T MICHIGAN III I I I 201 - 0 -10 -20' -30' ka = 2 I I S 0 28 32 FIG. 6-3: kp MEASURED (* * *) AND THEORETICAL ( —) RELATIVE PHASE OF TOTAL ELECTRIC FIELD IN THE BACKWARD DIRECTION (0 = 7r) FOR A METALLIC CYLINDER WITH Ei PARALLEL TO THE AXIS (Adey, 1955). 9'7

. P. 0 t) -1 1-A 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 -1 z m frf 0-4 w-3 c < r 3 O 0 -0 z 0 20 40 60 80 100 120 140 160 T- 0 in degrees 180 FIG. 6-4: AMPLITUDE OF CURRENT DENSITY J ON A METALLIC CYLINDER FOR Hi PARALLEL TO THE AXIS (Wetzel ahd Brick, 1955). ka= 12 WITH

p4 -4 k bf) (1) 4) 2000 1800 1600 1400 1200 1000 800 600 exact theory -4 w3 -c — A CA3 C.3 C.3 I!0! - c! z -e C) 0-4 0) z) 400 200 0 20 40 60 80 100 120 140 160 r- 0 in degrees 180 FIG. 6-5: RELATIVE PHASE OF CURRENT DENSITY Jo ON A METALLIC CYLINDER WITH Hi PARALLEL TO THE AXIS (Wetzel and Brick, 1955). FOR ka= 12 z

I IA0 0 U C o -. I.4 ----- -.0 -I ).6 --,., _ - - /'i. -A/:// I. I. I. 0. 0. 4 / 1 1 1. 4 --- — I0 — 2 - ---- --- J -- - - 6 - - -.- - —! 4...,___. H z I.-4 PU <( 0'I H 0 0 V.a l1 Po I I I i.2.3.4.5 1 2 3 45.1.2.3.4.5 1 2 3 45 Frequency in kHz (b) (a) (c) FIG. 6-6: MEASURED (. *.) AND THEORETICAL (o o o — ) ABSORPTION CROSS SECTION FOR A FIBERGLAS CYLINDER: (a) BARE, (b) WITH ONE LAYER OF HAIR FELT, (c) WITH TWO LAYERS OF HAIR FELT (Cook and Chrzanowski, 1948). C) 0

tri C trl (a) (b) O FIG. 6-7: MEASURED (e * ) AND THEORETICAL (-) SCATTERED PRESSURE AMPLITUDE l FOR PLANE WAVE INCIDENCE: (a) BRASS CYLINDER, ka = 1.7; (b) STEEL CYLINDER, ka = 1.7 (Faran, 1951). 0

(a) (b) FIG. 6-8: MEASURED (* * *) AND THEORETICAL ( —) SCATTERED PRESSURE AMPLITUDE FOR PLANE WAVE INCIDENCE: (a) BRASS CYLINDER, ka = 3.4; (b) COPPER CYLINDER, ka = 3.4 (Faran, 1951). C3 O C) -4 z OT s4

C-3 C-3 003 H C, C) c -4 -D1;o P-4 0 1, 01, (a) (b) FIG. 6-9: MEASURED (I ' I) AND THEORETICAL ( -) SCATTERED PRESSURE AMPLITUDE FOR PLANE WAVE INCIDENCE: (a) STEEL CYLINDER, ka = 3.4; (b) BRASS CYLINDER, ka = 5.0 (Faran, 1951).

-A w C!3 CA3 I q H C! z-4 c) cP (a) (b) FIG. 6-10: MEASURED (* * *) AND THEORETICAL ( —) SCATTERED PRESSURE AMPLITUDE FOR PLANE WAVE INCIDENCE: (a) STEEL CYLINDER, ka = 5.0; (b) ALUMINUM CYLINDER, ka = 5.0 (Faran, 1951).

1. 6r 1. J 1.4 0000 plane line source 0 H Q).4-4 9-4 a1) 0. 0. 0.1 incident field 0 0 C,31 H zi (-4 Hlt a C) 0 z7 0.2 0l 0 5 10 15 29) 25 k 30 35 40 45 50 FIG. 6-1 1: MEASURED (o o a) AND THEORETICAL (.-) kx = 47r (Kodis., 1950). AMPLITUDE IE zI FOR ka = 3.1 AND

160 140 - 0 120 100 9 80 CD 0 incident fields -20 -40 - 0 5 10 15 20 25 30 35 40 45 50 ky FIG. 6-12: MEASURED (o oo) AND THEORETICAL (-) PHASE OF E FOR ka = 3.1 AND kx = 47r (Kodis, 1950). H co 0 zo I - K

1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 0 plane wave line source incident field H z CA4 0-< CA) P —~ I 0 hI CO 0 a) 4.) a) P4 0 5 10 15 20 25 30 35 40 45 50 ky C) 0 z FIG. 6-13: MEASURED (o o kx = 4ir (Kodis, o) AND 1950). THEORETICAL (-) AMPLITUDE IEZ| FOR ka = 6.3 AND

160 140 120 E 100 E 80 a) 80.a 5 60 P. 40 a) 0 00 I. -4 I 00 H trl z (ir -4 a 0 0 Z iO line source incident field 20 O -20 - -40 0 FIG. 6-14: 5 10 15 ky MEASURED (o o o) AND THEORETICAL (-) PHASE OF E FOR ka = 6.3 AND kx = 47r (Kodis, 1950). z

1.6 1.4 1.2 1.0 0.8 0. 6 plane line source H incident - field --- a) ra.9.4P. a) 0 00 z H0 C) W 0 — z 0.4 0.2 0 FIG. 6-15: D 5 10 15 20 25 30 35 40 45 50 ky (o o o) AND THEORETICAL ( —) AMPLITUDE IEI FOR ka = 10 AND kx = 47r MEASURED, (Kodis, 1950' 1).

1300 0 H ii 200 14 b) 100 w~ 0o Cw a) a1) -4 a) 0 0 incident field C/3 C3A W line source 40 z CO) 0 P-4 0 z 20 0 -20 -40 0 5 10 15 20 25 30 35 40 45 50 ky FIG. 6-16: MEASURED (o 0 a) AND THEORETICAL ( —) kx = 47r (Kodis, 1950). PHASE OF E FOR ka z = 10 AND)

1. 1. 1 1 H M! plane wave line source incident field _ ~0 384 000 z rri C,) C-3 10.0 '1 0 0 0.1 0.1 0. C)4 0 z 0 5 10 15 20 25 k 30 35 40 45 50 FIG. 6-17: MEASURED (@00o) AND THEORETICAL (-~) AMPLITUDE IE I FOR ka (Kodis,. 1950).Z - 3. 1 AND kx = 2Oir

THE UNIVERSITY OF MICH 7133-3-T I C)( I - \ \ I I (D I C.) I I, ~~~ ~ ~I, I, I, I, I IGAN ~ LO LO IV 0 11 AI C — Z44 C to C 0 o l o -- 04 LO T-0 1 I I 1. I a, o 0 0 o CD R Cal 0 T-< T4 V-4 T-4 0 0 0 0 00 CD CNI o 0 c 't soaaiuT asu'cld OATIVIIJ 285

1.1 plane wave line source 1.2 in0 incident a) 0.8 ~06 0.4 0.2 0 \/ I I i 40 45 50 FOR ka = 6.3 AND kx = 2Oir (/3 $-A P-3 I Hz ci H 0-4 01 041 z 0 5 10 15 20 25 30 35 ky FIG. 6-19: MEASURED (o o o) AND THEORETICAL ( —) AMPLITUDE IEI (Kodis., 1950).Z

160 140 120 100 0 00 0 0 p4 0 -0 80 60 40 20 line O/0 incident field H z I-eH C) CAD z -0 0 -o =20F -40 o FIG. 6-20: 5 10 15 20 25 30 35 40 45 50 a ky MEASURED (,aoo) AND THEORETICAL (-.-) PHASE OF E FOR ka = 6.3 AND kx = 2Oir (Kodis, 1950).Z

1.6 1.4 1.2 plane wave line source incident field 4) Po 0 tE.rP 04 0) pc; 1.0 0.8 0.6 0 o 0 to 00 00 I45 50 z w -q Z 0.4 0.2 0 0 5 10 15 20 25 30 35 40 ky FIG. 6-21: MEASURED (o o o) AND THEORETICAL (-) AMPLITUDE IE | FOR ka = (Kodis, 1950). 10 AND kx = 207r

(0 I p4 af) a) 160 140 120 100 80 60 40 20 line s ource 6.mo'0 0 0 Hz M *il 0 P-4 4C) z incident field 60 I mopool- MP.-O 0 -.IfuLg -20 piano wave I4 I 1 I1 I I I I -40 5 10 15 20 25 30 35 40 4y FIG. 6-22: MEASURED (@0 oo) AND THEORETICAL (-) PHASE OF E kx = 207r (Kodis, 1950). Z 45 50 FOR ka = 1OAND

THE UNIVERSITY OF 7133-3-T MICHIGAN 3: 1I 3 * h *h~ ko " ' ky FIG. 6-23: MEASURED (o o, x x) AND THEORETICAL ( —) RELATIVE INTENSITY OF THE TOTAL DIFFRACTED FIELD E FOR A BRASS CYLINDER; ka = 2.494 (L - left of the cylinder, R - right of the cylinder) (Wiles and McLay, 1954). 290

THE UNIVERSITY OF MICHIGAN 7133-3-T 20 15 a) ~i a) a1) a) I0 5 Uv 0 a m -I - I A - I -k - - - - - - I OR 0 200 400 ky 600 800 1000 FIG. 6-24: RELATIVE SCATTERED PRESSURE AMPLITUDE FOR A 1/41IN. STEEL ROD RITH ka = 14.4, x0= 12. 7cm AND x =6. 6cm (Bauer et al, 1948).0 291

15 a) q.1 rio P4 0 5 o o 0 200 400 ky 600 800 1000 FIG. 6-25: RELATIVE SCATTERED PRESSURE AMPLITUDE FOR A 1/2 IN. STEEL ROD WITH ka = 28.8, x = 12.7cm ANDx = 6.6cm (Bauer et al, 1948). 0 Z m,4 co CA) 0 (C z ~zr ~^

THE UNIVERSITY OF MICHIGAN 7133-3-T 20 r 0) Po I 4) 0 0> a) $h 15 - 10o 5S 0 I I I I I 0 200 400 600 800 1000 ky FIG. 6-26: RELATIVE SCATTERED PRESSURE AMPLITUDE FOR A 1/4 IN. STEEL ROD WITH ka = 14.4, x = 12. 7cm AND x = 46.6 cm (Bauer et al, 1948). O 293

10 P4. *P4_ 0 I Jq\vJf 0 2060 400 ky 600 S00 1000 FIG. 6-27: RELATIVE SCATTERED PRESSURE AMPLITUDE FOR A 1/2"1 IN. STEEL ROD WITH ka. = 28.8, x0= 12. 7cm AND x = 46. 6 cm (Bauer et al. 1948). Hz 0 -0li z3(1

THE UNIVERSITY OF MICHIGAN 7133-3-T 20 r 15 a) qz P0 4~) $< a) a) k P4 10 5 F rslru 0 I I I I I 0 200 400 ky 600 800 1000 FIG. 6-28: RELATIVE SCATTERED PRESSURE AMPLITUDE FOR A 1/4 IN. STEEL ROD WITH ka =14.4, x = 54.3 cm AND x= 10cm (Bauer et al, 1948). o 295

THE UNIVERSITY OF MICHIGAN 7133-3-T 20 15 a) "0 a) 10 5 0 0 200 400 ky 600 800 1000 FIG. 6-29: RELATIVE SCATTERED PRESSURE AMPLITUDE FOR A 1/2 IN. STEEL ROD WITH ka = 28.8, x = 54.3cm AND x = 10cm (Bauer et al, 1948). 0 296

THE UNIVERSITY OF MICHIGAN 7133-3-T 15r a) ~0 *P4 ra a) A4 a) a) 10 5 F 0 1 I 0 200 400 ky 600 800 1000 FIG. 6-30: RELATIVE SCATTERED PRESSURE AMPLITUDE FOR A 1/4 IN. STEEL ROD WITH ka = 14.4, x = 54.3cm AND x = 40cm (Bauer et al, 1948). o 297

THE UNIVERSITY OF MICHIGAN 7133-3-T 15 0> '0 ig 4, 0 m k P4 4) or-I vr 0c P4 10 5 0 0 200 600 800 1000 ky FIG. 6-31: RELATIVE SCATTERED PRESSURE AMPLITUDE FOR A 1/2 IN. STEEL ROD WITH ka = 28.8, x = 54.3cm AND x = 40cm (Bauer et al, 1948). 298

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 6-32: BACK SCATTERING CROSS SECTION (IN db/cm 2) FOR INCIDENCE AT ANGLE e TO CYLINDER AXIS. UPPER HALF: a, (E1 PARALLEL TO AXIS); LOWER HALF: acr (E1 PERPENDICULAR TO AXIS); I = 6 cm, X = 3.2cm, ka = 0.588 (Meyer et al, 1959). 299

THE UNIVERSITY 7133-3-T OF MICHIGAN FIG. 6-33: BACK SCATTERING CROSS SECTION (IN db/cm ) FOR INCIDENCE AT ANGLE e TO CYLINDER AXIS. UPPER HALF: a,, (E' PARALLEL TO AXIS); LOWER HALF: ca (Ei PERPENDICULAR TO AXIS); I = 9cm, X = 3.2cm, ka =.882 (Meyer et al, 1959). 300

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 6-34: BACK SCATTERING CROSS SECTION (IN db/cm ) FOR INCIDENCE AT ANGLE e TO CYLINDER AXIS. UPPER HALF: a,, (Ei PARALLEL TO AXIS); LOWER HALF: ca (Ei PERPENDICULAR TO AXIS; i = 14cm, X = 3.2cm, ka = 1.375 (Meyer et al, 1959). 301

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 6-35: BACK SCATTERING CROSS SECTION (IN db/cm ) FOR INCIDENCE AT ANGLE 0 TO CYLINDER AXIS. UPPER HALF: a,, (E1 PARALLEL TO AXIS); LOWER HALF: ao (E1 PERPENDICULAR TO AXIS); I = 18cm, X = 3.2cm AND ka = 1.766 (Meyer et al, 1959). 302

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 6-36: BACK SCATTERING CROSS SECTION (IN db/cm2) FOR INCIDENCE AT ANGLE 0 TO CYLINDER AXIS. UPPER HALF: a,, (El PARALLEL TO AXIS; LOWER HALF: ac (E' PERPENDICULAR TO AXIS); I = 22 cm, X = 3.2cm, ka = 2.16 (Meyer et al, 1959). 303

THE UNIVERSITY 7133-3-T OF MICHIGAN FIG. 6-37: BACK SCATTERING CROSS SECTION (IN db/cm ) FOR INCIDENCE AT ANGLE 0 TO CYLINDER AXIS. UPPER HALF: a,, (El PARALLEL TO AXIS; LOWER HALF: C( (E1 PERPENDICULAR TO AXIS); S. = 28 cm, X = 3.2cm, ka = 2.74 (Meyer et al, 1959). 304

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 6-38: BACK SCATTERING CROSS SECTION (IN db/cm ) FOR INCIDENCE AT ANGLE e TO CYLINDER AXIS. UPPER HALF: al (Ei PARALLEL TO AXIS); LOWER HALF: aL (E' PERPENDICULAR TO AXIS); I = 32 cm, X = 3.2 cm, ka = 3.14 (Meyer et al, 1959). 305

THE UNIVERSITY OF MICHIGAN 7133-3-T FIG. 6-39: BACK SCATTERING CROSS SECTION (IN db/cm ) FOR INCIDENCE AT ANGLE e TO CYLINDER AXIS. UPPER HALF: o,, (E1 PARALLEL TO AXIS); LOWER HALF: ua (Ei PERPENDICULAR TO AXIS); I = 40cm, X = 3.2 cm, ka = 3.92 (Meyer et al, 1959). 306

THE UNIVERSITY OF MICHIGAN 7133-3-T I730. 0 / x 70 " 7 2 5 70,o JO FIG. 6-40: MAXIMUM BACK SCATTERING CROSS SECTION (9 = 90') OF A CYLINDER AS A FUNCTION OF LENGTH. MEASURED (o o o) AND THEORETICAL (-) aI; MEASURED (x X ) AND THEORETICAL ( —) al (Meyer et al, 1959). 307

THE UNIVERSITY OF MICHIGAN 7133-3-T 100 10 t b 0.1 0.1 1 2 5 10 20 Ilk --- 50 FIG. 6-41: MEASURED (o o o) AND PHYSICAL OPTICS (- - -) BACK SCATTERING CROSS SECTION IN THE AXIAL DIRECTION (0 = 0~ or 180~) (Meyer et al, 1959). 308

THE UNIVERSITY OF MICHIGAN 7133-3-T 0.5 0.4 0.3 cb b 0.2 0.1 0 kl FIG. 6-42: BACK SCATTERING CROSS SECTION OF THIN WIRE AT BROADSIDE INCIDENCE FOR i/a = 900; EXPERIMENTAL (o o o) (Van Vleck et al, 1947); THEORETICAL ( —) (Lindroth, 1955). 309

THE UNIVERSITY OF MICHIGAN 7133-3-T 2.4 2.0 1.6 1.2 Can b 0.8 0.4 0 0 0 4 8 12 16 ke 20 FIG. 6-43: MEASURED BACK SCATTERING CROSS SECTION OF STAINLESS STEEL CYLINDER AT BROADSIDE INCIDENCE; ka = 0. 0395 (Liepa and Chang, 1965). 310

THE UNIVERSITY OF 7133-3-T MICHIGAN 3.0 2.5 2.0 ^11 IT W E -, A I I a -| 1-1 I I I I I I I I L rt — 4-F ifrL^ b 1.0 0.5 0 0 2 4 6 kJ 8 10 12 FIG. 6-44: MEASURED BACK SCATTERING CROSS SECTION OF SILVER PLATED STEEL CYLINDER AT BROADSIDE INCIDENCE; o o aFOR ka = 0.022 (King and Wu, 1959); x xFOR ka = 0.0202 (Liepa, 1964). 311

THE UNIVERSITY OF MICHIGAN 7133-3-T REFERENCES Adey, A. W. (1955) "Diffraction of Microwaves by Long Metal Cylinders, " Can. J. Phys., 33, 407-419. Adey, A. W. (1958) "Scattering of Electromagnetic Waves by Long Cylinders, " Electronic and Radio Engineer, 149-158. Bain, J. D. (1953) "Radiation Pattern Measurements of Stub and Slot Antennas on Spheres and Cylinders, " Stanford Research Institute Tech. Rep. No. 42, Stanford, California. Barakat, R. G. (1961) "Propagation of Acoustic Pulses from a Circular Cylinder, " J. Acous. Soc. Am. 33, 1759-1764. Barakat, R. G. (1965) "Diffraction of a Plane Step Pulse by a Perfectly Conducting Circular Cylinder, " J. Opt. Soc. Am. 55, 998-1002. Bauer, L., P. Tamarkin and R. B. Lindsay (1948) "The Scattering of Ultrasonic Waves in Water by Cylindrical Obstacles, " J. Acous. Soc. Am. 20, 858 -868. Beckmann, P. and W. Franz (1957) "Computation of the Scattering Cross Sections of the Sphere and Cylinder by Application of a Modified Watson Transformation, " Z. fur Naturforschung 12a, 533-537 (in German). Blacksmith, P., R. E. Hiatt and R. B. Mack (1965) "Introduction to Radar Cross Section Measurements, " Proc. IEEE 53, 901-919. Bouwkamp, C. J. (1946) "A Note on Singularities Occurring at Sharp Edges in Electromagnetic Diffraction Theory, " Physica 12, 467-474. Bouwkamp, C.J. (1954) "Diffraction Theory, " Reports on Progress in Physics 17, 35-100. Bowman, J. J. (1963a) "Scattering of Plane Electromagnetic Waves from a SemiInfinite Hollow Circular Pipe (Axial IncidenceY'Internal Memorandum No. D 0620-125-M41-10, Conductron Corporation, Ann Arbor, Michigan. Bowman, J.J. (1963b) "Scattering of Plane Electromagnetic Waves from a SemiInfinite Hollow Circular Pipe (Off-Axis Incidence) Internal Memorandum No. D 0620-127-M41-13, Conductron Corporation, Ann Arbor, Michigan. Bowman, J.J. (1963c) "On the Vainshtein Factorization Functions," Internal Memorandum No. D 0620-135-M41-20, Conductron Corporation, Ann Arbor, Michigan. 312

I THE UNIVERSITY OF MICHIGAN 7133-3-T Bowman, J.J. (1963d) "Far Scattered Fields Due to Thin Semi-Infinite Cones and Cylinders," Internal Memorandum No. D 0620-162-M, P0047-23-M, Conductron Corporation, Ann Arbor, Michigan. Brick, D.B. (1961) "Current Distributions on Cylinders Excited by Spherical Electromagnetic Waves," IRE Trans. AP-9, 315-317. Bromwich, T.J. I'A. (1919) "Electromagnetic Waves," Phil. Mag. 38, 143-163. Brysk, H. (1960) "The Radar Cross Section of a Semi-Infinite Body, " Can. J. Phys. 38, 48-56. Carter, P.S. (1943) "Antenna Arrays Around Cylinders, " Proc. IRE 31, 671-693. Clemmow, P.C. (1959a) "Infinite Integral Transforms in Diffraction Theory," IRE Trans. AP-7, S7-S11. Clemmow, P. C. (1959b) "On the Theory of the Diffraction of a Plane Wave by a Large Perfectly Conducting Circular Cylinder, " The University of Michigan Radiation Laboratory Rep. No. 2778-3-T, Ann Arbor, Michigan. Cook, R. K. and P. Chrzanowski (1946) "Absorption and Scattering by Sound Absorbent Cylinders," J. Acous. Soc. Am. 17, 315-325. Debye, P. (1908) "Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens, " Physik. Z. 9, 775-778. Duncan, R.H. and F.A. Hinchey (1960) "Cylindrical Antenna Theory," J. Res. NBS 64D, 569-584. Faran, J.J. (1951) "Sound Scattering by Solid Cylinders and Spheres, " J. Acous. Soc. Am. 23, 405-418. Faran, J.J. (1953) "Scattering of Cylindrical Waves by a Cylinder, " J. Acous. Soc. Am. 25, 155-156. Fock, V. (1945) "Diffraction of Radio Waves Around the Earth's Surface, " J. Phys. USSR 9, 255-266. Fock, V. (1946) "The Distribution of Currents Induced by a Plane Wave on the Surface of a Conductor, " J. Phys. USSR 10, 130-136. Franz, W. (1954) "On the Green's Functions of the Cylinder and the Sphere, " Z. fir Naturforschung 9a, 705-716 (in German). Franz, W. and K. Deppermann (1952) "Theory of Diffraction by a Cylinder as Affected by the Surface Wave, " Ann. der Physik 10, 361-373 (in German). I 313

THE UNIVERSITY OF MICHIGAN 7133-3-T Franz, W. and R. Galle (1955) "Semiasymptotic Series for the Diffraction of a Plane Wave by a Cylinder, " Z. fuir Naturforschung 10a, 374-378 (in German). Friedlander, F. G. (1954) "Diffraction of Pulses by a Circular Cylinder, " Comm. Pure Appl. Math. 7, 705-732 Friedlander, F.G. (1958) Sound Pulses (Cambridge University Press, Cambridge, England). Giese, R. H. and H. Siedentopf (1962) "Ein Modellversuch zur Bestimmung der Streufunktionen Nicht Kugelf6rmiger Teilchen mit 3 cm Wellen, " Zeit. fuir Naturforschung 17a, 817-819. Goodrich, R. F. (1958) "Fock Theory,? The University of Michigan Radiation Laboratory Rep. No. 2591-3-T, Ann Arbor, Michigan. Goodrich, R. F. (1959) "Fock Theory-An Appraisal and Exposition, IRE Trans. AP-7, S28-S36. Goodrich, R.F., B.A. Harrison, R.E. Kleinman and T.B.A. Senior (1961) "Studies in Radar Cross Sections XLVII - Diffraction and Scattering by Regular Bodies I: The Sphere, " The University of Michigan Radiation Laboratory Rep. No. 3648-1-T, Ann Arbor, Michigan. Goriainov, A.S. (1956) "Diffraction of Plane Electromagnetic Waves on a Conducting Cylinder, t Doklady, AN SSSR 109, 477-480 (English translation by M. D. Friedman, ASTIA Document No. AD 110165). Goriainov, A. S. (1958) "An Asymptotic Solution of the Problem of Diffraction of a Plane Electromagnetic Wave by a Conducting Cylinder,? Radio Engr. and Electr. Phys. 3, 23-39 (English translation of Radiotekhnica i Elektronica 3). Govorun, N. N. (1962) "The Numerical Solution of an Integral Equation of the First Kind for the Current Density in an Antenna Body of Revolution, " USSR Computational Math. and Math. Phys. 3, 779-799. Gray, M. C. (1944) "A Modification of Hallen's Solution of the Antenna Problem, " J. Appl. Phys. 15, 61-65. Grib, A.A. (1955) "On a Particular Solution of the Equations of Plane, Cylindrical and Spherical Waves, " Doklady, AN SSSR 102, 225-228. Hallen, E. (1938) "Theoretical Investigations into the Transmitting and Receiving Qualities of Antennas, " Nova Acta Regiae Societatis Scientiarum Upsaliensis, Ser. IV, II, No. 4. 314

THE UNIVERSITY OF MICHIGAN 7133-3-T Hallen, E. (1956) "Exact Treatment of Antenna Current Wave Reflection at the End of a Tube-Shaped Cylindrical Antenna, " IRE Trans. AP-4, 479-491. Hallen, E. (1961) "Exact Solution of the Antenna Equation, " Trans. Roy. Inst. Techn. 183. Harrington, R.F. (1961) Time-Harmonic Electromagnetic Fields (McGraw-Hill Book Co., Inc., New York). Heins, A.E. and S. Silver (1955) "The Edge Conditions and Field Representation Theorems in the Theory of Electromagnetic Diffraction, " Proc. Cambridge Phil. Soc. 51, 149-161. Hochstadt, H. (1959) "Some Diffraction by Convex Bodies, " Arch. Rational Mech. Anal. 3, 422-438. Imai, I. (1954) "The Diffraction of Electromagnetic Waves by a Circular Cylinder, " Zeit. Physik 137, 31-48 (in German). Jones, D. S. (1952) "A Simplifying Technique in the Solution of a Class of Diffraction Problems," Quart. J. Math. 3, 189-196. Jones, D. S. (1955) "The Scattering of a Scalar Wave by a Semi-Infinite Rod of Circular Cross Section," Phil. Trans. Roy. Soc. London, Ser. A 247 p. 499. Jones, D.S. (1957) "High-Frequency Scattering of Electromagnetic Waves," Proc. Roy. Soc. 240A, 206-213. Jones, D.S. and G.B. Whitham (1957) "An Approximate Treatment of High-Frequenc Scattering," Proc. Cambridge Phil. Soc. 53, 691-701. Junger, M. C. (1953) "The Physical Interpretation of the Expression for an Outgoing Wave in Cylindrical Coordinates, " J. Acous. Soc. Am. 25, 40-47. Kapitsa, P. L., V.A. Fok and L.A. Vainshtein (1960) "Symmetric Electric Oscillation of an Ideally Conducting Hollow Cylinder of Finite Length," Soviet Phys. 4, 1089-1105. Karp, S. N. (1961) "Far Field Amplitudes and Inverse Diffraction Theory, " in Electromagnetic Waves, ed. R.E. Langer (University of Wisconsin Press, Madison, Wisconsin) 291-300. Keller, J.B. (1955) "Diffraction by a Convex Cylinder," IRE Trans. AP-4, 312-321. eller, J.B. (1956) "A Geometrical Theory of Diffraction, " in Calculus of Variations and Its Applications, ed. L.W. Graves (McGraw-Hill Book Co., Inc., New York, 1958) 27-52. eller, J.B., R.M. Lewis and B.D. Seckler (1956) "Asymptotic Solution of Some Diffraction Problems, " Comm. Pure Appl. Math. 9, 207-265. I- -315

THE UNIVERSITY OF MICHIGAN 7133-3-T Kerr, D. E. (1951) Propagation of Short Radio Waves (McGraw-Hill Book Co., Inc., New York). Kieburtz, R. (1963) "Scattering by a Finite Cylinder, " in Electromagnetic Theory and Antennas, ed. E.C. Jordan (Pergamon Press, New York) Part I, 145-156. Kieburtz, R. (1965) "Construction of Asymptotic Solutions to Scattering Problems in the Fourier Transform Representation," Appl. Sci. Res. 12B, 221-234. King, R. W. P. (1956) The Theory of Linear Antennas (Harvard University Press, Cambridge, Massachusetts). King, R. W. P. and D. Middleton (1946) "The Cylindrical Antenna; Current and Impedance," Quart. Appl. Math. 3, 302-335. King, R. W. P. and T. T. Wu (1957) "The Reflection of Electromagnetic Waves from Surfaces of Complex Shape - II Theoretical Studies, " Cruft Laboratory Sci. Rep. 13, Harvard University, Cambridge, Massachusetts. King, R. W. P. and T. T. Wu (1959) The Scattering and Diffraction of Waves, (Harvard University Press, Cambridge, Massachusetts). Kleinman, R.E. and T.B.A. Senior (1963) "Studies in Radar Cross Sections XLVII - Diffraction and Scattering by Regular Bodies II: The Cone, " The University of Michigan Radiation Laboratory Rep. No. 3648-2-T, Ann Arbor, Michigan Kline, M. (1955) "Asymptotic Solutions of Maxwell's Equations Involving Fractional Powers of the Frequency," Comm. Pure Appl. Math. 8, 595-614. Kodis, R. D. (1950) "An Experimental Investigation on Microwave Diffraction," Cruft Laboratory Tech. Rep. No. 105, Harvard University, Cambridge, Massachusetts. Kodis, R.D. (1952) "Diffraction Measurements at 1.25 Centimeters," J. Appl. Phys. 23, 249-255. Kodis, R. D. (1957) "On the Green's Function for a Cylinder, " Scientific Rep. No. 1391/8, Brown University, Providence, Rhode Island. Kodis, R. D. (1958) "Variational Principles in High-Frequency Scattering, " Proc. Camb. Phil. Soc. 54, 512-529. Lamb, H. (1924) Hydrodynamics (Cambridge University Press, Cambridge, England) Lax, M.(1950) "On a Well-Known Cross-Section Theorem," Phys. Rev. 78, 306-307. Lax, M. and H. Feshbach (1948) "Absorption and Scattering for Impedance Boundary Conditions on Spheres and Circular Cylinders," J. Acous. Soc. Am. 20, 108-124. I 316

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