*8-AIBI 998 AIR FORCE INST OF TECH WRIGH4T-PATTERSON AFB ON F/6 21/BELECTROMANETIC SCATTERING BY OPEN CIRCULAR WAVEUIDES.(U)1980 T W JOHN4SON

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( ELECTROMAGNETIC SCATTERING BY OPEN CIRCULAR WAVEGUIDES

By

Thomas Wesley Johnson, Ph.D.The Ohio State University, 1980

Professor David L. Moffatt, Adviser4Open circular waveguides are used to model jet engine inlets.

The exact Wiener-Hopf solution for scattering by a semi-infinitecylinder is studied in the resonance region, where the cylinderdiameter is of the order of a wavelength. In particular, the Wiener-Hopf factorization functions are calculated by numerical integrationand compared to various approximations, to define regions of val-idity. Scattering from the rim is studied as a function of frequency,incidence angle, and time. A ray-optic model for rim backscatteris discussed. The relative power absorption of the five lowest orderwaveguide modes is evaluated. Coupling of incident plane waves towaveguide modes, and radiation by these modes are shown to be relatedby reciprocity.

The waveguide termination model for a jet engine assumes anincident waveguide mode strikes an axially symmetric cone on a flatplate. The various techniques for evaluating scattering by thisstructure are discussed, and the problem is solved for a few cases.

Accession For

.U ('A&I

u :r-nnioqnced E]I : tif icatioDI . ......

P. X ,s ~ilty Codes,Aveil and/or

cpec lal

iJV t I

A-

80-69D

AFIT RESEARCH ASSESSMENT

The purpose of this questionnaire is to ascertain the value and/or contribution of researchaccomplished by students or faculty of the Air Force Institute of Technology (ATC). It would begreatly appreciated if you would complete the following questionnaire and return it to:

AFIT/NRWright-Patterson AFB OH 45433

RESEARCH TITLE: Electromagnetic Scattering by Open Circular Waveguides

AUTHOR: Thomas Wesley Johnson

RESEARCH ASSESSMENT QUESTIONS:

1. Did this research contribute to a current Air Force project?

() a. YES ( ) b. NO

2. Do you believe this research topic is significant enough that it would have been researched(or contracted) by your organization or another agency if AFIT had not?

() a. YES ( ) b. NO

3. The benefits of AFIT research can often be expressed by the equivalent value that youragency achieved/received by virtue of AFIT performing the research. Can you estimate what thisresearch would have cost if it had been accomplished under contract or if it had been done in-housein terms of manpower and/or dollars?

a. MAN-YEARS ( ) b. $

4. Often it is not possible to attach equivalent dollar values to research, although theresults of the research may, in fact, be important. Whether or not you were able to establish anequivalent value for this research (3. above), what ii your estimate of its significance?

a. HIGHLY ( ) b. SIGNIFICANT ( ) c. SLIGHTLY ( ) d. OF NOSIGNIFICANT SIGNIFICANT SIGNIFICANCE

5. AFIT welcomes any further comments you may have on the above questions, or any additionaldetails concerning the current application, future potential, or other value of this research.Please use the bottom part of this questionnaire for your statement(s).

NAME GRADE POSITION

ORGANIZATION TOCATION

STATEMENT(s):

ELECTROMAGNETIC SCATTERING BY OPEN CIRCULAR WAVEGUIDES

ABSTRACT OFDISSERTATION

Presented in Partial Fulfillment of the Requirements forthe Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Thomas Wesley Johnson, B.S.E.E., M.Sc.

The Ohio State University1980

Reading Committee Approved by

Professor David L. Moffatt

Professor Jack H. Richmond ______.__

z Adviser kProfessor Leon Peters, Jr. Department of Electrical

Engineering

L4

ELECTROMAGNETIC SCATTERING BY OPEN CIRCULAR WAVEGUIDES

DISSERTATION

Presented in Partial Fulfillment of the Requirements forthe Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By

Thomas Wesley Johnson, B.S.E.E., M.Sc.

The Ohio State University1980

Reading Committee Approved by

Professor David L. Moffatt

Professor Jack H. RichmondAdviser

Professor Leon Peters, Jr. Department of ElectricalEngineering

ELECTROMAGNETIC SCATrERING BY OPEN CIRCULAR WAVEGUIDES

By

Thomas Wesley Johnson; Ph.D.The Ohio State University, 1980

Professor David L. Moffatt, Adviser

Open circular waveguides are used to model jet engine inlets.The exact Wiener-Hopf solution for scattering by a semi-infinitecylinder is studied in the resonance region, where the cylinderdiameter is of the order of a wavelength. In particular, the Wiener-Hopf factorization functions are calculated by numerical integrationand compared to various approximations, to define regions of val-idity. Scattering from the rim is ;tudied as a function of frequency,incidence angle, and time. A ray-optic model for rim backscatteris discussed. The relative power absorption of the five lowest orderwaveguide modes is evaluated. Coupling of incident plane waves towaveguide modes, and radiation by these modes are shown to be relatedby reciprocity.

The waveguide termination model for a jet engine assumes anincident waveguide mode strikes an axially symmetric cone on a flatplate. The various techniques for evaluating scattering by thisstructure are discussed, and the problem is solved for a few cases.

*11.IJ

ACKNOWLEDGMENTS

The author is deeply grateful for the guidance and encouragementof Professor D. L. Moffatt. The author also benefited from dis-cussions with Professors Leon Peters, Jr., and Jack H. Richmond,who served as members of the dissertation reading committee, andProfessors Robert G. Kouyoumjian, and H. D. Colson, and Dr. PrabhakerH. Pathak, who offered their advice freely. A discussion with Dr.Arthur D. Yaghjian was also exteeely helpful at a critical stageof the research. The dissertation was capably typed by Mrs. LaVerneWemmer. A special tribute belongs to my wife, Catherine, who borewith me during the trying years of Graduate School.

The work in this dissertation was performed by the author whilehe was attending Ohio State University under the auspices of theAir Force Institute of Technology Civilian Institution Program.Funds for computation were provided by the Department of ElectricalEngineering, Ohio State University. Text preparation was supportedunder the Joint Services Electronic Program.

iii

i ii

VITA

April 12, 1951 ............... Born - Detroit, Michigan

1973 ......................... B.S. Electrical EngineeringB.S. Mechanical EngineeringMassachusetts Institute of Technology

1974 ......................... M.S. Electrical Engineering M.I.T.

1974-1980.................... Extended active duty with U.S.Air Force

1978-1980 .................... Graduate StudentOhio State University,Department of Electrical Engineering

PUBLICATIONS

Johnson, T.W. and James R. Melcher, "Electromechanics of Electro-fluidized Beds," I & EC Fundamentals, 14(3), 146-53.

Johnson, T.W., "Design of a Digital Flight Control System Jsing AreaMultiplexing," Proceedings NAECON 76, 403-10.

Johnson, T.W., "A Qualitative Analysis of Redundant AsynchronousOperation," Proceedings NAECON 78, 83-90.

FIELDS OF STUDY

Major Field: Electrical Engineering

Studies in Electromagnetic Theory: Professor Robert G. KouyoumjianProfessor David L. Moffatt

Studies in Communication Theory: Professor C.E. Warren

Studies in Mathematics: Professor H.D. Colson

Studies in Structure of Matter: Professor Richard Boyd

iii

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS ............................................ ii

VITA ...................................................... iii

LIST OF FIGURES............................................ v

LI T O A L S. . . .. . . . . . . . . . .. . . . . . . . . v i

LIST OF TABLS............................................vix

j Chapter

*I INTRODUCTION..................................... 1

II DISCUSSION OF WIENER-HOPF SOLUTION ................. 18

A. On-axis Results 21B. Off-axis Beh~avior 34

C.Reciprocity Considerations 45

III SCATTERING BY A REPRESENTATIVE ENGINE-LIKE

OBSTACLE IN A WAVEGUIDE.......................... 56

IV CONCLUSIONS AND DISCUSSIONS ....................... 71_

REFERENCES ................................................. 77

Appendix

A EVALUATION OF WIENER-.HOPF FACTORIZATION

FUNCTIONS....................................... 80

B SUMMARY OF WIENER-HOPF COUPLING ANDSCATTERING COEFFICIENTS.......................... 110

C ELEMENTS OF DYADIC GREEN'S FUNCTION FORA CIRCULAR WAVEGUIDE............................. 117

iv

LIST OF FIGURES

Figure Page

1-1 GSMT elements for modeling engine inlet ............... 2

1-2 Coordinate system for modeling engine inlet ........... 6

1-3 Electric field lines for TE modes incircular waveguide .................................... 7

1-4 Planar engine model used by Moll and Seacamp .......... 15

2-1 Coordinate system for Wiener-Hopf solutionto semi-infinite cylinder ............................. 19

2-2 Coordinate system looking at origin frompositive Y-axis ....................................... 20

2-3 Coordinate system looking at origin frompositive X-axis ....................................... 20

2-4 On-axis cross section for semi-infinitecylinder exact (from numerical integration) ........... 24

2-5 On-axis cross section for semi-infinitecylinder based on simple asymptoticapproximations ........................................ 26

2-6 Inverse Fourier transform of "exact" on-axisbackscatter with weighting to reduceGibbs-type ringing .................................... 27

2-7 Inverse Fourier transform of asymptoticon-axis backscatter ................................... 29

2-8 Coupling of on-axis incident plane wave tovarious waveguide modes - normalizedpower flow ............................................ 33

2-9 Radiation pattern for TEll mode withka=4 (D/X=1.273) ...................................... 35

2-10 Radiation pattern for TMll mode withka=4 (D/X=l.273) ...................................... 36

v

Figure Page

2-11 Radiation pattern for TEll mode withka=12.77 (D/X=4.065) .................................. 37

2-12 Radiation pattern for TM01 mode withka=12.77 (D/X=4.065) .................................. 38

2-13 "Cavity cross-section" for six lowestorder waveguide modes ................................. 39

2-14 Angle for beam maximum for sixlowest-order waveguide modes .......................... 40

2-15 Rim backscatter cross-section for semi-infinite cylinder with ka=7.261 (D/X=2.311) ........... 43

2-16 Rim backscatter cross-section for semi-infinite cylinder with ka=14.4 (D/X=4.584) ............ 44

2-17 Sources for incident "plane-wave" ..................... 46

2-18 Sources for reciprocity theorem ....................... 49

3-1 Basic geometry for engine scattering model ............ 57

3-2 Coordinates system and dimensions forscattering from an axial conducting cone .............. 57

3-3 Current pulses for moment method solutionof scattering by cone ................................. 63

3-4 Geometry for cone scattering using images ............. 65

3-5 Currents induced on cone in circular waveguidewith incident TE1 l mode, L/a=2.0, b/a=0.5,ka=2.0 solution !y dyadic magnetic Green-sfunction .............................................. 66

3-6 Eigenvalue of TE 1 mode for coaxialwaveguide as a fuiction of innerconductor radius ...................................... 67

3-7 Solution for axial variation of fields forTEll mode in circular waveguide with cone.L/a=2.0, b/a=0.5, ka=2.0 .............................. 69

3-8 Currents induced on cone in circular waveguidewith incident TEll mode, L/a=2.0, b/a=0.5, ka=2.0,solution by coaxial approximation with slowlyvarying center conductor .............................. 70

vi

4-1 Geometry for self-consistent scatteringproblem using flat plate .............................. 73

4-2 Normalized cross-section for axial incidenceon semi-infinite circular waveguide withconducting plate 10 radii down the guide,varying waveguide diameter ............................ 74

4-3 Cross section of semi-infinite circularwaveguide for axial incidence, D/X=.0with flat plate at two to ten radii down guide ........ 75

A-1 Factorization functions L+, M+, $M+ forn=l, a=k. Arrows indicate increasing k ............... 92

A-2 Contour of integration in the complex z-plane ......... 97

A-3 Contour of integration in the complex T-plane ......... 98

A-4 Factorization function M+ for n=l, cz=kcalculated by numerical integration andasymptotic approximation based on two andthree term expressions for phase of Besselfunction .............................................. 103

A-5 Factorization function L+ for n=l. Solidcurve gives a=k, varying k. Dashed curvesvary ct/k holding D/X fixed to the valuesspecified by arrows ................................... 104

vii

-

LIST OF TABLES

Table Page

1-1 SUMMARY OF LOW-ORDER CIRCULAR WAVEGUIDEMODES AND CUTOFF FREQUENCIES ........................ 4

2-1 SUMMARY OF ON-AXIS RCS PEAKS ANDMODE ACTIVITY ....................................... 23

A-1 VALUES OF FACTORIZATION FUNCTIONS L+ AND M+COMPUTED BY NUMERICAL INTEGRATION ................... 83

A-2 SUMMARY OF ASYMPTOTIC FUNCTIONS ATSADDLE POINT IN T-PLANE ............................. 106

viii

LIST OF SYMBOLS

Fourier transform variable for spatial transformwith respect to z

anm,nm Waveguide mode wavenumber for axial propagationof TM and TE modes, respectively

'Ynm 'ynm Waveguide mode decay constant for axial decay ofevanescent TM and TE modes, respectively

En Neumann epsilon1 n=O2 n=1,2,3...

o Polar angle in spherical coordinates (see Figure

1-2)

Unit vector for polar angle in spherical coordinates

eies Polar angle for incident and scattered fields,respectively

Angle of asymptotic approximations of Besselfunctions

xFree space wavelength

p Radial distance in polar coordinates

p Unit vector in radial direction in polar coordinates

a Radar cross section

Azimuthal angle in both spherical and polarcoordinates

Unit vector for azimuthal angle

Azimuthal angle for incident and scattered fields,respectively

n Phase angle of asymptotic approximation of deriva-tives of Bessel function

ix

0i Characteristic impedance of free space0

I- Characteristic admittance of free space

a Radius of waveguideAnm A Coupling coefficients for incident plane wave to

nm TM waveguide modes

Bnm Bnm Coupling coefficients for incident plane wave toTE waveguide modes

c Speed of light in vacuum, free-spaceSC C Radiation coefficients of waveguide modes;

0,d denote orientation of radiated field; E, H denote

TM or TE mode, respectively

D Diameter of waveguide = 2a

Enm Magnitude of TM waveguide mode n,m

Ei,E s Incident and scattered electric field, respectively

E ,EEz,Ee Radial, azimuthal, axial (in polar coordinates),polar (in spherical coordinates) component ofelectric field

(x) Transition function defined by Kouyoumjian andPathak (1974)

fn Ratio of two Wiener-Hopf factorization functions

H,H ,Hz Radial, azimuthal, axial (in polar coordinates)4' Z components of magnetic field

H1 )(),H 1 )'() Cylindrical Hankel function of the first kind(cylindrical Bessel function of the third kind)or order n, and its derivative

i Square root of minus one

In() Modified Bessel function of order n

J Current density

S(),J;() Cylindrical Bessel function of the first kind oforder n, and its derivative

x

Jnm m-th zero of Jn(x), i.e., Jn(Jnm)O

j ' m-th zero of Jn(x), i.e., Jn(jnm)=O

k Free space wavenumberK n() Modified Bessel function of order n

L+() Wiener-Hopf factorization function

m Index of radial variation for a given azimuthalvariation, e.g. the zeroes of JI(x) are jlm,m=l,2 ,3

Jll,jl2Jl 3M+() Wiener-Hopf factorization function

n Index of azimuthal variation

P Power flow

r Radial distance from the origin in sphericalcoordinates

-OT Generalized position vectors locating a point in3-dimensional coordinate system

SS Scattering coefficients for rim of semi-infiniteeeSo0S'SOe'S 4 cylinder

x,y,z Rectangular coordinates

, ,2 Unit vectors appropriate to rectangular coordinates

xi

CHAPTER IINTRODUCTION

The radar cross section (RCS) of jet intakes has been extens-ively studied. There are several reasons that it is of interest.One is simply that it is a major element in the radar cross sectionof aircraft and must be accurately evaluated to estimate total air-craft RCS. Another reason for study is that potentially the RCSof the aircraft could possibly be reduced if the scattering mech-anisms are well understood. A third (though certainly not final)reason is that many aircraft identification or classification tech-niques propose to use modulation of the radar return imposed by theaircraft engine as a significant identifiable feature. It wouldbe a questionable approach to establish such a system on an effectwhich is not well understood, particularly in terms of establishingthe system's susceptibility to intentional confusion or camouflage.

This study is somewhat limited to the region for which DXX;the asymptotic forms developed for higher frequencies may fail inthis region, and the very low frequency techniques (for which littleor no energy penetrates the intake) are invalid.

This study develops two of the practical problems involvedin calculating the RCS. The coupling coefficients at the mouth ofthe intake are known in princple, but difficult to compute in prac-tice. The effect of the engine structure is known only generally,and has been very loosely approximated in past studies.

The jet intake can generally be modeled as an open ended wave-guide with an obstacle (the engine) some distance down the waveguide.Figure 1-1 illustrates the geometry. The problem can then, inprinciple, be solved by the generalized scattering matrix technique.The scattering matrices of the significant scatterers, if known,can be self-consistently manipulated to produce the backscatteredfield. Let

ui represent the incident field,

u bs = the backscattered field,

[Sil] = a matrix, representing (in some sense) the directbackscatter of the open waveguide

C.4-

C

.00

:4-

[ S12] = radiation characteristics of waveguide modes

$S21] = coupling matrix of incident field to waveguide modes

[ S22] = reflection of waveguide modes from the open end[Sb ] = reflection of waveguide modes from the obstacle

IT2J = transmission down waveguide

Tb2 = transmission back from waveguide to aperture.

Note that off-diagonal terms in each transmission and reflectionmatrix will represent mode conversion from one mode to another.

From Figure 1-1 we can easily show that:

ubs = S i + S12r2 (1-1)

= S2 1u1 + S22r2 (1-2)

r 2 = Tb2 Sb T2b rI (1-3)

r = Tb2 Sb T2b (S21ui + S22r2) (1-4)

[I-Tb2SbT2bS22]r2 = Tb2SbT2bS2l1u (1-5)

[(Tb2SbT2b )-' - 22 = S21u (1-6)

r 2 = L(Tb2SbT2b) -1 - $2 2] S2 1u1 (1-7)u S + [TiT-1 i (1-8)

ubs =Sll+ S12 [(Tb2 SbTb) - - 2 2]- S2

The matrices S l S .,., and aS2 have been solved for by the Wiener-1Hopf technique. Thle r ults ar quite complicated algebraically,

The heart of the problem, however, is calculation of the Wiener-Hopffactorization functions. Considerable effort has been given to cal-culating these functions, which is presented in Appendix A. Thisstudy has emphasized circular waveguides, because much prior workhas been done in the area, and the symmetry of the geometry simplifiesthe problem somewhat. Also, since engine geometries are circular,non-circular inlets require an additional model of the mode conver-sion as the energy travels down a waveguide of varying cross-section.Thus our matrices T and T2 are diagonal, having only the relevantphase delay for eac hfmode. his study does not address furtherdevelopment of this effect for non-uniform waveguides.

3

Summary of Circular Waveguide Modes

Propagation in a circular waveguide is limited to a discreteset of modes, which can propagate only for waveguide diameters largerthan a certain minimum (cutoff) diameter. The notation used in thisstudy is consistent with Harrington: TEl7 refers to a mode withelectric field transverse to the axis of propagation. In the Russianliterature (Weinstein) this is referred to as a magnetic mode, sinceit has a z-directed magnetic field, and all other field componentscan be simply derived from it.

The number of modes that can propagate in a circular waveguidegoes roughly as the square of the diameter. Table 1-1 lists the first

TABLE 1-1

SUMMARY OF LOW-ORDER CIRCULAR WAVEGUIDE____________MODES AND CUTOFF FREQUENCIES

Mode cutoff cutoff ka

TEll .5861 1.8412TMOl .7655 2.4048TE21 .9722 3.0542TMll 1.2197 3.8317TEOl 1.2197 3.8317TE31 1.3373 4.2012TM21 1.6347 5.1356TE41 1.6926 5.3176TE12 1.6970 5.3314TM02 1.7571 5.5201TM31 2.0309 6.3802TE51 2.0421 6.4156TE22 2.1346 6.7061TE02 2.2331 7.0156TM12 2.2331 7.0156TE61 2.3877 7.5013TM41 2.4154 7.5883TE32 2.5513 8.0152TM22 2.6793 8.4172TE13 2.7172 8.5363TE71 2.7304 8.5778TM03 2.7546 8.6537TM51 2.7921 8.7715TE42 2.9547 9.2824TE81 3.0709 9.6474TM32 3.1070 9.7610TM61 3.1628 9.9361TE23 3.1734 9.9695

4

28 modes in order of increasing cutoff diameter. The cutoff is cal-culated by noting the Jn(Jnm )=O or J (j'm)=O and then ka=jnm or

j6. Since =i = 1 the cutoff in terms of ka can be simplyx 2derived. It seems more intuitive to discuss D/X. Figure 1-3 presentselectric field line pictures for the first few modes.

This study employs the et time convention; the choice isnecessary for consistency with the vast majority of Weiner-Hopfliterature (Einarsson et al). The coordinate system chosen willbe standard spherical and/or cylindrical coordinates, with theorigin at the center of the waveguide mouth. See Figure 1-2. Thepolar angle is e, the azimuthal angle €. The radius will be desig-nated by r, in spherical coordinates, and p in cylindrical coordin-ates. The waveguide radius is a, and the waveguide extends fromz=O to z=-- at p=a.

Since a large part of the literature is concerned with theWeiner-Hopf solution to scattering by a semi-infinite circularcylinder, an heuristic description will be presented here. Tutorialdiscussions of the Weiner-Hopf technique can be found in [Noble (1958)]and [Morse and Feshback (1953)].

The Weiner-Hopf technique is based on the fact that the fouriertransform of a causal function is entire in a half plane. For example,the function

x(t) =fet t > 0

0 t < 0 (1-9)

has the Fourier transform

X(W) = f eiwt eat u(t) dt = e(ia)t dt-O 0

e-1 - " e1 = 1 (1-10)(i .- a) Le ej

[Recall that we are using e iWt time dependence]. Hence X(w) hasa single pole at w = - ia and is entire (has no poles) for Im(w)>O>-ia.In the Weiner-Hopf technique, the fourier transform of the fieldcomponent is taken with respect to the coordinate along one axisof the problem (the axis parallel to the semi-infinite object).The incident and scattered field are then related by means of the

5

rd

I

Figure 1-2. Coordinate system for modeling engine inlet.

6

TEO, I TE 0,2 TEO,3

TE 1,1 TE 1,2 TE 1,3

4

S2

TE 2,1 TE 2,2 TE 2,3

Figure 1-3. Electric field lines for TE modes

in circular waveguide.

7

TM 0,1 TM 0,2

2iTM I,( TM 1,2

4,4

TM 2,1 TM 2,2

Figure 1-3 (continued). Electric field lines forTM modes in circular waveguide.

physics of the problem. Noble cites three different techniquesfor doing this: field matching, Green's function integral equation,and dual integral equations. The resultant equation is manipulatedso that the left-hand side must be entire in one half-plane and theright-hand side in the other half-plane, with a region of overlap.These are then equated to a third function, which must be entireover the whole a-plane. Because of Liouville's Theorem, this newfunction ends up being a constant, equal to zero. This separatelysets each side of the equation to zero, which leads to the solution(by taking the inverse fourier transform).

For the cylinder scattering problem, the incident field isdecomposed into cylinder waves by the Bessel function additiontheorem. The transform is taken with respect to the z-axis:

(,r : E(z,r, )eilaz dz (1-11)

For each cylinder wave, the incident and scattered fields are relatedby imposing the boundary conditions that the tangential electricfield must vanish at the cylinder walls, and the surface currentmust vanish in free space. These are used to manipulate the probleminto a Weiner-Hopf form. This process is described by [Weinstein(19%9) and Einarsson et al (1966)].

The analysis of interactions at the mouth of an open waveguidegoes back to [Chu (1940)]. By using the Kirchoff approximation theradiation fields for the lowest order propagating modes for a cir-cular, and a rectangular waveguide are calculated. In terms of theGSMT, this would enable us to compute elements of [S1 2].

[Levine and Schwinger (1948)]evaluate the acoustic radiationand reflection characteristics of a hollow circular pipe. Anintegral equation for the velocity potential is solved via the Weiner-Hopf technique. Their study is confined to symmetrical modes incidenton the open end of the pipe. In a circular acoustic waveguide,the symmetrical modes are given by

(p, ) = Jo(jo,mp/a) or Jo(jo p/a) (1-12)

where

Jom are the zeroes of Jo(x)

jom are the zeroes of J (x)

U9

[Pearson (1953)] was the first to apply the Weiner-Hopf techniqueto the electromagnetic problem. He considers a transverse magneticplane wave incident on the open end, and obtains equations for theLaplace transforms of the axial and azimuthal currents on the waveguidewalls. These equations are then solved by the Weiner-Hopf techniques,and the currents can be obtained by inverse Laplace transform. Thefields far from the mouth down the pipe are then evaluated asymp-totically and found to result from propagating modes, thus givingthe coupling coefficients. The backscattered fields are not evalu-ated, nor is the behavior near the mouth studied.

[Jones (1955)] analyzes the scattering of sound waves by asolid semi-infinite cylinder. He considers both hard and soft boundaryconditions although numerical results are evaluated for only thehard (du/dn=O) case. An approximation for high frequency (largediameter) is developed, and a low frequency expression is presented.A variational expression is developed to establish limits of erroron the approximations used in evaluating the exact expressions.Both the pressure field on the cylinder, and the scattered far fieldare evaluated. The end cap pressure time response due to an incidentunit step is evaluated by taking the inverse Laplace transform ofthe frequency domain response.

[Noble (1958)] treats the scalar problem for both radiation ofthe lowest order mode, and coupling of an incident plane wave tothe lowest order mode in a circular waveguide. His discussion istutorial in nature, drawing somewhat from Jones' work. There isa more complete treatment of the general technique used to solveWeiner-Hopf problems than is found in most other references.

[Jones (1964)] applies the Weiner-Hopf technique to radiationfrom a semi-infinite hollow pipe. This study is somewhat tutorialin nature, being part of a textbook, and considers only a few ofthe lowest order modes. Jones notes that TE modes radiate moreefficiently than TM modes. TM modes have higher reflection coef-ficients at the open end of the pipe.

[Einarsson et al (1966)] give an exhaustive study of diffractionby both the infinite and semi-infinite circular cylinder. The back-scatter for a plane electromagnetic wave incident on a solid, semi-infinite, perfectly conducting rod is given. The scalar (sound)scattering from a semi-infinite (both solid and thin walled) tubeis evaluated. The radiation and reflection for a propagating scalarwave incident on an open end are evaluated (reproducing the resultsof Levine and Schuinger and also Weinstein). There is a brief dis-cussion of finite cylindrical resonators with one end open, one endclosed (rigid, Dirchlet boundary condition).

10

The general solution for scattering of a plane electromagneticwave from a semi-infinite thin walled, perfectly conducting tubeoccupies about half the report. This includes both the backscatterfar-field [Sll] and coupling coefficients, for the general planewave (neither TE nor TM). The special case of axial incidence isconsidered. Radiation from a source inside the tube [S2 1] and re-flection from an open end [S2], are evaluated, largely copied fromWeinstein. Asymptotic radiation of the far-fields is compared tothe Kirchoff approximation.

A substantial part of this study concerns evaulation of theWeiner-Hopf factorization functions. A number of forms of integralexpressions which exactly define them are developed. Power seriesapproximations to the low frequency are developed. Unfortunatelythese expressions are still quite complex.

In addition, only the magnitude of the functions is approxi-mated; the phase is not. A large effort to develop high frequencyapproximation yields some useful simplifications, but the resultsare expressed in terms of another unknown function, albeit muchsimpler. Numerical data is given for varying values of ka with the aparameter fixed. This,is inconvenient since, generally, a=kcose,and we are primarily interested in fixing ka and varying a. Experi-mental data are presented for finite cylinders. No experimentaldata are presented for a semi-infinite cylinder.

[Witt and Price (1968)] analyze the problem of a finite tubewithout recourse to Weiner-Hopf techniques. Instead, the directbackscatter from the rim, and the coupling coefficients to waveguidemodes are calculated by what amounts to the Kirchoff approximation.The incident field tangential to the aperture is expanded as a sumof waveguide modes, and a waveguide admittance for each mode is com-puted. The reradiation is also calculated via the Kirchoff method,using the Stratton-Chu integral. The termination is modeled as animpedance, which results in a simple (scalar) reflection coefficient.The reflection coefficient is then transformed to its equivalentimpedance as seen at the mouth of the waveguide. The total waveguideadmittance for that mode is then calculated, and the scattered fieldis expressed as a sum over the waveguide modes, times the incidentfield projection on that mode, times the generalized admittance.

For backscatter with vertical polarization (TE) (which corres-ponds to our * polarization) they present the formula

jka 2 cosO -jkrE (x'O,z') r ey r

(1-13)fvJ1 (v+vo) + I E (v)E (v )(+ (I

Vo(vVo )+v + n=O m=l nm nm o rnm) + Ynm

11

where

vo = kasinO0 (incident)

v = kasinO (scattered)

E nm = weighting coefficient for nm-th mode

ynm = propagation coefficient for nm-th mode

rnm = reflection coefficient seen at aperture for nm-th mode.

The work of Weinstein* in many cases predates the works reviewedhere. However, most of Weinsteins's papers were published in Russianjournals. His book [Weinstein (1969)] contains virtually all ofhis earlier work, and is readily available. It should simply benoted that Weinstein's work did frequently predate work in the West.

[Weinstein (1969)] has collected all of his earlier work on theWiener-Hopf technique into a single volume. The book is tutorialin nature, beginning with the simplest problem, diffraction andradiation by a plane parallel plate waveguide. Having developedthe basic Wiener-Hopf arguments, he proceeds to analyze circularwaveguides, first considering only symmetrical modes (to eliminateazimuthal dependence from the problem). Acoustical problems arethen solved, followed by the general problem for electromagneticwaves scattered and radiated by a semi-infinite circular cylinder,including azimuthal dependence. Comparisons with answers obtainedvia the Kirchoff method are frequent, showing those circumstancesunder which the Kirchoff method works and those under which it fails.There is a substantial discussion of the relative accuracy of Huygensprinciple, compared with edge diffraction; a substantial point ismade that edge diffraction yields more reliable answers.

[Kao (1970)] presents a completely novel approach to scatteringfrom cylinders. He determines the currents on finite cylinders byusing point matching and then calculates radiation patterns fromthe currents. For the semi-infinite cylinder [1970b] he sets uptwo sets of points with slightly different interpoint spacing. Byvarious manipulations of these, he determines the magnitude of thetraveling wave launched on the cylinder by the incident plane wave,as well as the current in the vicinity of the aperturg. However,his analysis is confined to broadside incidence (0=90 ).

[Bowman (1970)] develops ray-optical diffraction expressionsfor the scattering from the end (aperture) of the semi-infinite wave-guide, and compares these with an asymptotic approximation to the

*Also transtated as Vajnshteijn, Wainstein, Vainshteir.

12

exact Weiner-Hopf solutions. For the asymptotic form of the Wiener-Hopf solution, he obtains, for direct backscatter (e=O)

EBS " a eikr /+ ei m mi2mka (-14)E / i m eik}

whereas by ray optical methods, he obtains

+ eiI/4 m i2mkaEBS xa eikr + .e

E/~ m 2- e7 (1-15). = m JFor bistatic scattering with axial incidence he obtains from approxi-

mating the Wiener-Hopf solution letting his e go to r-e and his €to 21-4

eikr 2a /2 cos(kasine-r/4)(sin )ES + --- T F_ / co @

rine cos(0/2)

A + 21]!4 2 c+ e cos (012) imm-3/2 ei2mka (1-16)

From the ray optical approximation he obtains

s + eikr ( 2a 1/2 cos(kasine-n/4r ES ~+r 1--in- cose2 sin

iTT/4 cos2(e/2) im ei2mka+ .o 61) i (1-17)

A 1 cosO m=l 2m- 1-7

In both cases it is seen that the term of O(ka " 1 2 ) is identical

up to the first two terms in the summation

iei2ka i2e4ika + i3e i6ka (1-18a)23/2 + 33/2

vs

iei2ka + i2ei4ka + i3ei6ka (-18b)

+ 272 4 (-8

4! 13

He attributes this difference to the ray-optical approach, for whichhe first considered scattering by plane parallel plates; multiplescattering was effected by assuming a cylinder wave was generatedfrom each edge after scattering. This result was then specializedto having a single point participate in the scattering on each edgebut not modified to take account of the fact that a cylindrical waveis no longer being emanated from each edge. Beyond this comment,Bowman does not analyze further, since the exact result from theWeiner-Hopf solution instructs us how to modify the ray-optic con-tribution.

A comparison of Bowman's results with those of Witt and Priceis illuminating. If we restrict ourselves to on-axis backscatterand disregard terms due to reflection from the termination, theformula presented by Witt and Price reduces to

Es= jka 2 ejkr + EmV 2

Ee ir~+ E (v)2r n= m=l nm

- e jkr + co m 2 (1-19)

n=O m= 1

We observe that, besides the opposite time convention, there is anadditional factor of jka present in the form presented by Witt andPrice. They remark that this is recognizable as the physical opticsapproximation for scattering from a flat conducting disk. Sincethis is not in agreement with the high frequency behavior of the

-J Weiner-Hopf solution, we conclude that the disk is not a good highfrequency model for this problem. In fact, it will be seen laterthat the disk does adequately model the on-axis scattering of acylinder terminated by a perfectly conducting plate.

[Moll and Seecamp (1970)] present a more realistic model ofthe engine geometry. The approach of Witt and Price is used to modelthe scattering, coupling, and radiation at the duct inlet. The studyis confined to TE modes. The termination, however is modeled bytwo sets of blades, each as shown in Figure 1-4, to simulate thefirst stage of a compressor. The blades are modeled as being planar(normal to the z axis). The two sets of blades had different numbersof blades and blade widths and assumed varying relative orientations(stator to rotor). The scattering at the termination is modeledby a similar procedure to that at the inlet. The backscattered fieldis expressed as a sum of modes traveling toward the mouth. The totaltangential electric fields must match at points where there areblades. For each incident mode, integrals were taken over the areacovered by either set of blades, forcing the fields to vanish.The equations thus obtained are used to solve for the scatteringcoefficients for modes generated at the termination. Then, the

14

Figure 1-4. Planar engine model used by Moll and Seacaip.

15

radiation from each mode is computed by matching the radiated fieldplus the incident field to the internal field at the duct aperture,where the internal field is the sum of the modes traveling down thewaveguide plus those scattered back by the termination. The RCSis computed for various orientation of the blade structure and variousrelative orientations of the two bladed structures, giving a rangeof modulation of the RCS caused by the rotor motion. They used 31and 37 blades on the respective blade structures. There is a briefdiscussion of a non-planar cap in the duct termination, but the ideais not developed.

[Lee et al (1973)] are primarily concerned with the measurementerrors made when field strength is measured with a sensor boom.The effect of the presence of the boom is analyzed via the Wiener-Hopf technique, and the relative distortion thus introduced is cal-culated. The relevant part of this paper deals with the calculationof the Wiener-Hopf factorization functions. Although the generalfactorization expression is developed (as an infinite product offactors), the general expression is clearly too complex to be useful.A low frequency form is developed for the n=l case for both L+ andM+ functions. The n=l case is relevant to low frequencies becausethe TEll mode has the lowest cutoff frequency, hence is also theslowest decaying evanescent mode when all modes are cut off. Un-fortunately the low frequency expression presented do not appearto fit very well with data computed in this dissertation by numericalintegration of the exact integral defining the functions.(Lee et al) indicate a constant phase for the low frequency, butit appears that the phase varies rather rapidly as the frequencyincreases from D.C. The formulas do work out to the correct "DC"form.

[Mittra et al (1974)] present a detailed report which includesreconciliation of the numerical solution with experimental data.The complete Wiener-Hopf solution for the semi-infinite cylinderis presented. There appear to be some errors in the results presented,since the scattering coefficients for direct backscatter (Equation(3-86)) could not be reconciled with those presented by [Bowman (1970)];also the two direct polarization solutions for See and S€ do notreduce to the same answer on axis, nor do they reduce to the answergiven by [Einarsson et al(1966)] for on axis backscatter (Equation(5-63)). The solution for large pipes is simplified by an asymp-totic approximation for the L+ and M+ functions; these are expressedas functions of the series

I m-3/2 eim(2kan r+Tr/2) (1-20)m=l

and the same summation, including either even terms or odd terms.

16

The Generalized Scattering Matrix Technique (GSMT) is usedto formulate the problem. Considerable effort is expended expressingthe individual elements of the matrix in terms of the Wiener-HoDfsolution.

Solutions for other geometries are developed by the ray-opticalmethod. The geometries considered are an ellipsoid, elliptical plate,and semi-infinite elliptical cylinder. The semi-infinite ellipticalcylinder is analyzed in terms of diffraction by the edge of thecylinder.

These solutions are then combined to estimate the total RCSfor an aircraft. Numerical results were computed for selected fixedfrequencies for two aircraft. For calculations involving a terminationin the jet intake, this was modeled earlier as a perfect conductor,or as a dielectric plug of infinite depth, or as a dielectric plugof finite depth.

[Chuang et al (1975)] present an extension of the report by[Mittra et al (1973)]. Starting with the exact solution for bistaticscattering from a semi-infinite pipe, they develop approximate highfrequency expressions for the factorization function, based on anasymptotic evaluation of the integral defining these functions.Both factorization functions can be rather simply expressed in termsof a third function, called a modified Lerch function of order 3/2,

L(x,v) = m v ei 2 mx (1-21)

m=

They then derive via the Mellin transformation a twelve-term seriesrepresentation with complex coefficients, which enables the compu-tation of this function for v = 3/2 to be mechanized trivially.The results thus obtained show excellent agreement with results ob-"tained by direct numerical integration of the defining integral,down to the cutoff frequency of the lowest mode of that order. Finallythere is a simplification of the infinite sum which is present inthe scattering calculation mode by use of the asymptotic form withone of the Bessel function addition theorems.

There appear to be a few errors in this paper. These are summar-ized in Appendix B.

(James and Greene (1978)] indicate that both theoretical andexperimental results show substantial sensitivity to wall thickness.They indicate that the exact Wiener-Hopf solution based on infinitelythin walls breaks down when the wall thickness is larger than .OIX.The results of their summary seems to be that the radiation patternsof thick walled pipes are narrower than given by the Wiener-Hopfsolution. Beyond noting this effect, we will not further discussthis problem, since it would require an entirely separate study.

17

CHAPTER IIDISCUSSION OF WIENER-HOPF SOLUTION

The exact Wiener-Hopf solution to the diffraction by a semi-infinite circular waveguide can be found in [Einarsson et al (1966)].The significant formulas from this report are summarized in Appendix

4B. The major numerical problem -- that of calculating the factori-zatior functions -- is discussed at length in Appendix A. This chapterdiscusses the physical significance of some of these formulas. Wereiterate that the e-lwL time dependence is assumed and suppressed.

The coordinate system is a substantial stumbling block sincethe incident and scattered field are defined with respect to 6 andunit vectors which are themselves functions of angle. Further com-plicating matters, [Einarsson et al (1966)] and [Mittra et al (1974)]use different coordinate systems. This study uses the coordinatesystem illustrated in Figure 2-1, which is the same as (Mittra etal (1974)], since it forms a self-consistant reference for scatteringcalculations.

The incident field is assumed to come from the angle i=r.Looking in along the y-axis toward the origin, we see Figure 2-2.Looking in along the x-axis toward the origin we see Figure 2-3.Hence we obtain the following unit vectors.

= - x cose i - z sine i (2-la)

A - (2-1b)

For th,- scattered field, we are not necessarily constrained to s=l'for bistatic scattering). Hence, we obtain general formulas forthe unit vectors.

e x cose s cos~s + y cose s sines - z sine s (2-2a)

f - sines + Y cosos . (2-2b)

In our coordinate system, the incident and scattered fields are in4Ithe same coordinate system. In [Einarsson et al (1966)], the directionof the z-axis is reversed, but ei and the unit vectors for the inci-

18

13!

2A

10-

Figure 2-1. Coordinate system for Wiener-Hopf solutionto semi-infinite cylinder.

19

A

x -

Figure 2-2. Coordinate system looking at origin from

positive Y-axis.

A AI 9 PROJ. ON z-AX IS)

Figure 2-3. Coordinate system looking at origin frompositive X-axis.

20

dent field are defined in such a way that they physically point ini ithe same direction as in our system. Thus cosei, sine i, E E are

all the same in Einarsson's and our coordinate systems. (See Figure

2-2 in Einarsson; El occurs for 6=0, E for 6 - ' with a being re-et

placed by ei). In the scattered coordinate systems, § and $ arereversed in sign (hence also E and E ), and cose has the oppositesign, but sine is unchanged. In addition 4 is replaced by - ssince the x-axis is the same in the two systims.

A. On-axis Results

The most elementary case of the Wiener-Hopf solution can befound by taking the limit as eO0. In taking this limit we willconsider both the backscatter and coupling of energy into the wave-guide.

Taking the limit of Equation (B-3) (which gives the generalbistatic scattering from the rim for 0 incident and scatteredfields) with s=7T, ei=es=S (in the limit 30) we obtain

Se2- ~n [Jn(ka)] 2 [ 2 f2

see 1 im n0 n(-l) 2[L+(k)]2 6 l-f1 (2-3)

where

n Neumann epsilon = l;n=O

= 2;n0

Jn() is the cylindrical bessel function of the first kind

k is the wavenumber = 27

a is the pipe radius.

L+, M+ are the factorization functions; these are functionsof (n,a,k); k is always understood and n is usuallyobvious from the context. Hence the only argument thatis explicitly given is a. L+(k) means L+(n,a=k,k)

fn= nL+(k)/2kaM+(k)

= lim -2i 2 2( a ) 2n (2-4a)S +0 k6 n=l 22n L+(k) " 24

2i 2(-l)(ka)2 n (2-4b)2i _4[L+k)] 2 1_f2-

21

i2

- 4k2a2M+(k)2_ L+(k)2 (n=l understood) (2-4c)

since fo is identically 0.

When combined with Equation (B-2), we obtain

Es eikr -ia2k= 4k2 2 2 2 (n=l understood) (2-5)E 1 4k 2a2 M+(k) -_L+(k)2

This can be seen to be the same as Equation (5-163) of [Einarsson(1966)] with r-*z; the same procedure applied to Sp (which givesthe general bistatic scattering from the rim for S-incident and scat-tered fields) leads to exactly the same result.

S = lim 2i I 1)n n-2 -S +0 W n=O nM+fk)] 2

S2i 2(-I) +W (2-6b)4k (M+(k))2 L1 y7j 2-

-i

4kM+(k)2(l-f ) (2-6c)

-ia2k4k2a2M+(k) 2_L+(k) 2 (n=l understood) (2-6d)

We see immediately from Equations (B-5) and (B-6) (which givethe cross-polarized backscatter) that sin nw=O so no cross-polarizedfield is generated.

The radar cross-section is defined by

a = lim 4nr2 I-r (2-7)IE'l

Applying Equation (2-5), we obtain

22

I : im 47r 2 (2-3krr (4ka2M+(k )2 -L+(k) 2 2-a

2 ak (2-8b)r.wr(4k a M+(k) -+k

: 4T1 4k2a2M+(k)2_L+(k)2I(-b2

_ 2ak (2-9)

7a2 : 4k2a2M+(k)2 _L+(k)2

The results of Equation (2-9) are evaluated and plotted in Figure

2-4. It is of interest to Pote that the normalized cross-section

peaks at D/X=.55, with o/ra =10.4 dB, and then decreases very rapidly.The lowest order propagating mode (TEll) is enabled at D/X=.586;hence this peak occurs just below cutoff for this mode, and as themode is able to transport energy, the cross section rapidly decreases.The first three peaks and corresponding modes are summarized in Table2-1.

TABLE 2-1SUMMARY OF ON-AXIS RCS PEAKS AND MODE ACTIVITY

Location of Height of Cutoff for Modepeak (D/x) peak (dB) mode (D/X)

.55 10.4 .586 TEll

1.68 3.5 1.697 TEl2

2.69 2.5 2.717 TEl3

The rapid variation of cross section in these regions is relatedto the strong coupling of the normally incident plane wave to theTElm modes, as discussed below.

rBowman (1970)] and [Chuang et al (1975)] use their asymptoticforms for L+ and M+ to obtain a simple form for the on-axis crosssection.

see - 1 + e imm-l.5ei2mka (2-10)

This leads to

23

4 -)

0

S-

Hx

0n c

C"~

., 0

T '

24-

lOiD/4 t2

-a le em-l'5ei2mka (2-11)ira/nF MlEquation (2-11) is plotted in Figure 2-5 and yields good resultsfor D/X>l.

Taking the inverse Fourier transform of Equation (2-5) with

the spatial dependence suppressed, we obtain the time domain response,shown in Figure 2-6. This represents the backscattered field as a

time function resulting from a normally incident plane wave, impulsivein time. This result was generated via the discrete Fourier trans-form of the backscattered formula for 0<D/X<3.2. Naturally the timedomain response is dominated by the peak at D/X=.55, which resultsin damped oscillations. The time domain response can be related

(perhaps dubiously at low frequencies) to a ray optic model of scat-tering at the mouth. The optic model was developed by [Bowman (1970)].For example, if we take the simplest asymptotic approximation forthe L+ and M+ functions, given by Equations (A-27), (A-29), (A-31),(A-58) and (A-59), and substitute them into Equation (2-5), suppres-sing the spatial decay and propagation,

E__s 1 -4l ia2k2 4Tar T (2-12)

E1 : 4k a2M+(k)2-L+(k)2

retaining only terms of 0(1) and O(ka-I 2 ), with a unit incidentfield

E a F'e i-a 4+ mei2m(ka-7r/4)Es -- l [+e i (-l 2 (2-13)

r m=l m

[1 + e c/2(m+/2)e (2-14)Es ~ T a' ml VMm 3 / 2 (-4

Taking the first few terms of the summation explicitly

* 31r i2aw 5r i4aw i 7T i6aw

ES -a + elc- + e Te C+e e

rW 242- 4 3 /- 1w_

(2-15)

25

IC).

0

-

IC)I

100

C -

a*0

(A

C)

C.

CD 00D

'1 ZDJLC-)

260

0.11

0.10

0.09

0.08-

0.07

0.06-C-20.05- loE'

0.04

0.03

0.02

0.01

0

-0.01- ct/a

-0.03 -

-0.04-

-0.05-

Figure 2-6. Inverse Fourier transform of "exact" on-axisbackscatter with weighting to reduce

Gibbs-type ringing.

1 27

i 2 amw

Note that e results in a time delay of L- . We employ the Fourierc

transform pair

S_ 4 u(t) (2-16a)

and its Hilbert transform

i -u(-t) (2-16b)

vG

Taking the transform of the frequency domain expression

S2a i4a .6aw

ES .r-a +v/a ((-l+i) eC + (-1-i) ec (1-i)e\_ 4ra 2v /w- 3V3 yr J

(2-17)

leads to

T 2a

u(t- c + - ctF

(2-18)

This is plotted in Figure 2-7.

We note in Figure 2-6 that the sharp peaks at t=m 2 becomeC

almost negligible after m=3, and the response shown in the discreteFourier transform results seems to be dominated by the ringing as-sociated with the lowest frequency resonance. Since this resonanceis not well predicted by the asymptotic forms (compare Figures 2-4and 2-5) it is not surprising that the long-time or steady-stateresponse is not well predicted. This resonance appears to be relatedto the currents excited at the mouth, and to some extent, associatedwith an exterior natural resonance of the cylinder, since all thewaveguide modes are evanescent in this region.

28

HuJ4

04 -)

0.

E0

'4--

C

S-

>

soo

c; cl.

JU

4m W; c; -

290

NA I...... ..

The coupling of an axially incident plane wave to waveguidemodes can be determined from a special case of equations (B-ll) thru(B-14) (which give the general coupling coefficients for incidentplane waves). Taking ei.O we see immediately that only n=l modesare excited. 12

ar e 4ijnmL+(Otlm) f - 2iJlmL+(lm) f1Alm a1 maL+(k) 1-f2

(2-19a)

4ijnL+(lm) f 2iJlmL+(aIm) f2I Am 2 JnL7 m " t maL+ ck ) 2-

- kolma 2M+(k) 1 = 1l )

d (2-19b)

* B 4i(k+aIm) M+(aIm) 1 f,

h im 077T L+jk2) 1-f213Jm

i(k+ajm)M+(ajm) 1ac lm4l- 1 )M+(k) 1 (2-19c)

lm

4i(k+jm)M+(ajm) [Lf kaIm]

BIm kcz a(l - )2M+(k) 1_ I

k m : 1m 1 m

i(k+ajm)M+(ajm) (2-19d)

k al -- , 1 )M+(k) 1-f 2

Jim

In both TE and TM cases, rotation of the incident field from e to $corresponds to an equal rotation of the azimuthal dependence of thewaveguide mode; otherwise the coupling coefficients are the same.

30

t ~m

For purposes of comparing the relative importance of thesecoupling coefficients, the power flow associated with each of themodes can be computed. For TE modes, with an incident plane waveof unit amplitude based on [Collin (1960), p. 179] the power flowis computed as

P = m k m ) ffH (p, ) 2 da (2-20a)

k' a2 21ra n ma cos2ncB nm2pdpd

.2._ nf~~l P o Uo

o n nm)

(2-20b)

c ko,_a 2Bn 12 72

: " 0 nrm n m7 a0 OJ n/ 2a) 2pdp (2-20c)

aT

Jn(Jnm )2 o

P_ 20knm-a Bn0Jnm-1 2 a n(nm (2-20d)(. 7_aj- nim

2 ~ mIr FO (ka)(alma)a

2 n2P j_ 7IB ° 2Jm 2J ~ m (2-20e)

P0 jnm)2n m

We assumed a unit incident field, which has an incident poweri i 2 / codensity of 2 E .IZ o so the normalized power flow becomes(normalized to the average power flow and the area of the waveguidemouth)

P 1 (ka)( ,a)( n2 2 (2-21)

( a2) :7 (j m)2 - nm 2 2-

) 31

Similarly, for TM modes, we obtain

1 Fcno m nm da (2-22a)P 1 = k nm 2 J

-ok.(am)2 2 a Jn(Jnmp/a) 2 27 _ nm(~ -j f 2

O0 OIA'pdpd4

(2-22b)

1,-o kFc nm j m) IAnm I a4 1o_ 2 f Jn(nmP/a)2pdp (2-22c)

0 Jnn(Jnm ) n

_ (.a2o 211[#o knm(Jnm I n m I2P (J nm 2 [a .j.(J nm) 2 (2-22d)

_______ _______ 2 2 ,2

P 1 (ka)(anma) 2 (2-23)/I I = 7 j2 JnmI 2-3v OO ( 2 ) nm

The power flow coefficients are plotted in Figure 2-8 for thefirst three TEim and TMIm modes, for axially incident plane waves.

The reason for the behavior of the on-axis cross section becomessomewhat clearer, since we see that the TE modes all couple muchmore strongly to the axially incident planimwave than the TM modes.Hence, the region in which they are enabled exhibits a much Aredramatic variation due to the size of the change of power absorbed.[Weinstein (1969), p. 151] discusses this behavior and proves thatasymptotically the optical cross section equals the sum of the ab-sorption cross sections for the TElm modes. Figure 2-8 can be com-pared with Weinstein's Figure 53.

32

4J

IC

0

'4-C

00

00 0N

LL.)

(sp 300"

33o

B. Off-axis Behavior

The radiation patterns of the various waveguide modes havebeen produced in the literature; all that is produced here are afew curves for comparison. Figures 2-9 and 2-10 with ka=4 can becompared with (Weinstein (1969)] Figures 46 and 48. Figure 2-11and 2-12 with ka=12.77 can be compared with [Narasimhan (1979)].

Of more practical importance is the relationship of theseradiation patterns to radar cross-section. As is discussed in SectionII-C, the radiation pattern of a single waveguide mode in a certaindirection can be seen, by reciprocity, to be directly proportionalto the coupling of an incident plane wave from that direction tothe appropriate waveguide mode. If, for the moment, we assume thatthe cylinder is terminated with a perfectly conducting flat wall,each propagating mode will be reflected back to the waveguide mouthunattenuated. For a monostatic radar system, this will mean thatat some angle, which happens to couple well to a particular waveguidemode, that waveguide mode will radiate equally well in that direction.Therefore, it becomes of considerable importance to know the directionsand relative strengths of the coupling coefficients of the variouswaveguide modes. Obviously, for 6=O, we can see from Figure 2-8that the TE11 mode dominates all others. From Table 1-1 we see that,for increasing waveguide diameters, the TMoI. TE21 , TM11 , TEo0 arethe next modes to propagate. In a manner analogous to Figure 2-8,the relative importance to radar cross-section for these modes isshown in Figure 2-13. However, Figure 2-13 differs in that thedirection is varied for each mode so that the incident field isassumed to come from the optimum direction. This direction is plottedin Figure 2-14. For example, for D/X= 1.5, the TE01 has a maximumfor E at 300, the TMoI and TE21 modes also have maxima at 300, butfor Ee. However, the relative magnitudes are 0.7, -6.8, and 1.0dB, respectively. Figure 2-13 might be termed the cavity cross-section, since it is the contribution to the RCS which coupling toand radiation from a single mode would produce, assuming that onlythat mode contributed to backscatter. Naturally, this is not true,as other modes will contribute to some extent, as well as the directbackscatter from the rim. We see that for on-axis scattering, the"cavity cross-section" produced by the TEll mode is remarkably similarto that of the disk.

For an incident plane wave, we have from Equations (B-9) and(B-10) (which give the coupling of incident plane waves to the axialwaveguide field)fnm (O4I'I)f(P,@'z)

/ =(2-24){ vo Hz/Ei j nm (Oi'4i)g(Pcz)2

34

oWcc c%

4J-

00

35,

-0

0~I

0 -

LII

4-

-0 -

00

"C,

la~ 0 In 0 lcu m,

36-

o 0

0.

ww-0 -v

E

w I-J

w w

w 4-00

00

1-

0 0 0 0 0

37

0

L)

LII

o(0 -

-

11

4-)4-)

.4-J

-0 a

-2-

S.-

_0 0 0 0 0 0 0 UN It

38

95-

90-

-. 4-

001 -- 5 0

I 0

bD

-5-

-90,

-15-

Figure 2-13. "Cavity cross-section" for six lowest orderwaveguide modes.

39

90-

so-

60-W44

W50-im 40

W 30-

4 20- 1L

10 -TE 2 -E

L TE 11-1ONAXIS I II00 0.5 1.0 1.5 2.0 2.5 3.0

D/

Figure 2-14. Angle for beam maximum for six lowest-orderwaveguide modes.

40

From Equation (B-17), (which give the radiation patterns for thewaveguide modes) we obtain scattered fields of the form[ : COE('sos) CeH(es 4) Enm ikr

Conside _COEes,os) COH (e5,s I) i L i-Hnm e r (-5!0

Consider first the topline of Equatimn (B-9).

E J n(JU p/a) -icn Zz Anm(eioi)cos no n -y-- e nm (2-26)E n

This gives us Enm ) in Equation (B-15) except for the

direction of propagation. If we assume perfect reflection, andignore the sign change, (since we are interested only in magnitude)this leads to

s ikre = (s,s)Ae ( i)e (2-27)EI =OE( n o r

The RCS becomes

E22

= i E41Er(0 )1J E-(- 2 (2-28b)

Since we chose oi=7, we likewise set os=7 and vary e=ei=es to its

optimum value. Note that for computations involving Anm and Bnm

there is a sin no dependence in the modal fields. To adjust thisto conform with Equations (B-15) and (B-16), it is necessary to rotatethe coordinate system. This results in radiation patterns evaluated

at os = Tr. Hence, we obtain, for the remaining three cases:

os = 7/29 oi = i ' es =ei = e (2-29)

41

oo = icE(es s)1 nm(Oi i~)2

¢s = n/2, €i = 7 ', = 8i = 6 (2-30)

a= 41T 1CH(es1s)12 Bnm(i, ) 12

¢s = € = ' es = ei = 0 (2-31)

The backscatter directly from the rim is discussed by [Chaunget al (1975)]. They used a pulse radar and extracted the RCS of therim based on the first return pulse. The agreement between theoryand experiment is better in some cases than others, but fails to matchthe detailed pattern. The normalized RCS can be derived from Equations(B-1) through (B-4) as follows.

E = See E e ikr (2-32)2

alim 4i r 2 IE. (2-33a)

- 4 1soo12 (2-33b)

2ia = l eel 41T i e (2-33)

Similarly

ar _ 2 (2-35)

Comparison of Figures 2-15 and 2-16 with Figures 2 through5 of [Chuang et al (1975)] reveals that despite use of the complexscattering form, and more accurate computation of the factorizationfunctions, some substantial discrepancies still exist betweentheory and experiment. Possihly the exolanation is thatany real cylinder must, of necessity, have a finite wall thickness.[James and Greene (1978)] showed that this leads to substantially dif-ferent results than obtained with the infinitely thin "knife-edge".Of course, the accuracy of the measured results is a possible sourceof discrepancy.

42

mon _ .

0

0)

U CU

4-1

-0 0)

L4f)

04 U)w' /) 0w

cr.C) C

(0 0 CIO

U',-

LO LO

434

I* c

* )

4-

E

cu,

0 L

44

C. Reciprocity ConsiderationsIt is evident from physical considerations that there must

be some relationship between the coupling of an incident plane wave

to a waveguide mode, and the waveguide radiation pattern of thatmode. This will be shown by the reciprocity theorem.

From [Harrington (1969), p. 116-117], if the current ja produces

fields (Ea,Ha), and current Jb produces fields (Eb,Hb), then

- ;(Ea x H- E x Ha).ds = fff(Ea.j b - EJa)dv (2-36)s v

where the surface and volume are of finite extent. Generally speaking,reciprocity is applied to sources and matter of finite extent. Inthis case, however, the matter is of semi-infinite extent. Usually,it is shown that the surface integral vanishes far from all matterand sources. In this case, we will assume that it vanishes externalto the waveguide, and thus need only evaluate the 'power flow'

(E x H ) inside the waveguide.

First, we apply reciprocity to determine what source will producea plane wave incident on the waveguide. We assume that there existsa current dipole, impulsive in space, of either e or 4 orientation,located by R - (ee, 4=6 , r=R), where Ro >> a R X X, as shown

in Figure 2-17. This dipole produces a far-field a? the origin of

Ex= I1 i e k '01 (-cose (2-37a)

E ikI, e ikIr-oI 0 (2-37b)o : 471 F- oI-o kl ik I Rol I'-sineo

E z 0 =k e 0(2-37c)o47 I F-T ol

for e or $ source unit vectors, respectively. Hence, to producea 'plane wave' of unit amplitude in the vicinity of the origin, itis necessary to set

45

RI

Figure 2-17. Sources for incident "plane-wave"t.

46

F = -j kI T e' (2-38a)0 0

fE7 i4rR -ikRIF _ T e (2-38b)

110

For source a we choose

a T 6(a-) 0 (2-39)

I. Iwhere

6(0-e0 ) 6(o-) 6(r-R0 )( = r-r -ITi)n=e- (2-40)

Let us first consider TE modes. These have fields given by

Co J n(3nmP/a) cos no +" i L

H o'Jpa) -sin n-

E ikna 2 Jn(JmP/a) sin no e + iOa m z (2-41b)p enm2 Pjn(Jm) Cos no

E -ika Jn(Jnm p/a) ,cos nO (2-41c)o 3 nm Jn(Jn ) (sin no)

Ez = 0 (2-41d)

H- =e e - anmz (2-41e)= Tnm Jn sin n

47

nminnm z

+ o inma 2n Jn(JnmP/a) -sin e (2-41f)+ 0 JnJnM) \cos n4

4-ji 0' Znmn

where e means the fields propagate toward the waveguide mouth-icn'mze means the fields propagate toward.-

Let the b source be located within the guide at z = -L, as shownin Figure 2-18.

= 2 nm e nmL Jn(jnmp/a) cos 0

o n--nm Jn(Jnm (sin nD

- na Jn(JnmP/a) {-sin n' 6(z+L) (2-42)

-n'm Pdn Jnm \cos n

This current source will generate the appropriate TE mode of unitamplitude propagating toward the waveguide mouth. It will generatethe same mode, of equal amplitude, propagating toward the infiniterecesses of the waveguide. Note that since the leading sign on Hpand He changes with direction of propagation, the discontinuity intangential H exactly matches the surface current. However, Ep Eand Hz are all properly continuous.

Next, according to Equation (B-lO), an incident plane wave

with () polarization produces a field inside the guide.

o oo nmooaHa = fBn~mCO n m e nm (2-43)

o n Bm cos n ninm)

According to Equation (B-17), the TEnT mode propagating toward the

waveguide mouth produces a far-field 'Iattern

b .1eikr /oE = e r Hnm . (2-44)b 04E C

48

- -

N7

//

4 /I

IsI

//

/7.

N

II

Figure 2-18. Sources for reciprocity theorem.

49

Plugging the currents and fields into the reciprocity equation,taking the surface and volume indicated in Figure 2-18, we obtainthe following result. For 6 incidence Ha has sin n@ dependence,so we take the bottom line of Equations ( -41) and (2-42).

0 2 a -ict' Zfff EaJ dv = I f f Bn ,--ev n m -co o 0 nm

J nrImp/a) J'j' p/a)na n n(nmP- dnT n 1. cos n¢ n(n ), sin n

' iL"a ia Jn (n' p/a)2 0 ianm nm an nm(~nm ~cos

o nm m nm

31an(jnmP/a)j n(Jnm ) sin n 6(z+L)pdpddz (2-45)

We invoke the orthogonality of modes to assert that the sum-

mation reduces to a single term, as this integral vanishes exceptwhen the mode and current have the same radial wavenumbers and azi-muthal dependence. When this is satisfied,

2Tr 27tf cos 2nod: f sin 2ndo = T (2-46)-

0 0

If; y 8 ika nm 2 0 inma "inm.jbdv B B rrnm -,j-- e

v nm V i nm

a 2 (Jn(J 'p(jnnmP/a) 2f (nn n (jnm ) pdp (2-47a)0 Tn- p jn(inm ) " n(Jm

2- Bmkainma a (n2a 2 Jn(Jn p/a)2 2

n m Knn +m 2)~

0+ jn(JmP/a) dp

(2-47b)

50

2lBnmkaam a a 2- pan(Jnm /a)2dp (2-47c)

o o

Jn (Jnm)2j nm2

'n8 ka ' a C., nm 2 2 o (2-47d)

i nm 0

bSince E is generated by a field with a sin n dependence, this willrotate the radiated field generated by COH by w/2. Hence, we evalu-ate COH with a cos n4s dependence.

C0H eikr E0 i4rR0 e-ikRfff E J dv fff r CkHv "v -TC 0 - - g ) d

(2-48a)

e ikRo i47R0 -ikRo (2-48b)=F7 COH TRo k

k - C (2-48c)

This surface integral within the waveguide becomes

x H ).ds

271

= f f (Eax Hb - x Ha) • pdpd as z -c (2-49a)0 0

211 a a "ab_ a b b a b a)= f fLT o -(E pH E - (EpH -E Hp)] pdpd (2-49b)

0 0 1

H b

We observe that H a B nmHz

51

Then, since each of E E H and He are based on Hz, the factthat they are scalar Multiples forces the integral to vanish identi-cally. Thus we are left with

Ikaaaa 2(j'm2-n2) €e B 4l i rim e (2-50a)

3Iim4 Bnm =T o CeH

:cH - i ka2-' a a2 (jn e! (Hnm jnm (2-5b)

A (j,, nm

which can be seen to be satisfied by some simple algebraic manipu-lations.

* For a $ incident plane wave, we note that Ha has a cos n depen-dence, so we take the top lines of Equations (2-41) and (2-42) forboth the current and waveguide fields. This results in exactly the

same integrals for both Ea. b and Eb.ja, with 0-€ so that

A - i2 (j 'm n2 -n2 )

CH 2nma B (2-51)

which is satisfied identically.

For TM modes, we begin with

Ez = (jnp/a) rcos 01 e-tanmz (2-52a)n(Jnm) Lsin n

E +inma Jn(Jnm p / a ) os n0 +"i m

E +ia nm Jn(nm) sin (2-52b)

E + inm na2 J n(Jnmp/a) f-sin n ltz (2-52c)_ n- m 2 pjn(Jnm) tcos n e

52

Hp - n- m Pnm/a sin n e (2-52d)cnm on'- cos n

ika 'o J'znm +"

H = i A e (2-52e)i nm P'o Jnim s in np

H :0z

+i nmz

e n propagates toward mouth (2-52f)

-ia nmZe nm propagates toward -

Let

b Eo ikae L na Jn(Jnmp/a) -sin n412/bi= 2_oiae n na

0V o JnmJnJnm Jnm L osn, J

- Jn(jnmp/a) i 6(z+L) (2-53)n ~sin n4

This will generate the appropriate field traveling toward the wave-'guide mouth. However, to generate the discontinuity in tangentialH, the leading sign on all terms must be reversed for the modetraveling toward the infinite recesses of the tube. However, asbefore, this is of little importance since it contributes nothingto the reciprocity integral.

The incident plane wave generates fields

a fAe cos n¢ dn(Jn p/a) -ianmzE = nm . e (2-54)

Azm sin n4 n(jnm)

The modal fields radiate

53

r6b fOE E ek (2-55)E b C nm

0 = E n r )For the ^e incident field we obtain

-lcIiz

ff 0 0i a im n m

p/a n) 2 ianmL

+2~l~ Ae nm 0 ika einm p p 0 inm ,Jn

nmp a) N~ J~sim'a2+ n '((i 2 / a pdp (z-56)inm p nn

0 nm nm pjn.2jnm /a2 d(-6)

am 2 n a2 0 nimpa

f - AnmP /'a) 2 pdp~) (2-56b)

im i

4

2 _ A (2-56e)

Jnm

^a=~ (~e E eikr -i4IR ° e ikR°

ff E J dv =- 0-C- r e 6(-Ro)dv

(2-57a)

._[En 47rI

= T_ CE (2-57b)

Equating these leads to

I T, knma 4 -- A C 4i (2-58a)n Am n 2 417 OE T

0 imnm a4

CE -ik2 tnm a4 A n (2-58b)OE 4jr nm

nm

Finally, for $ incidence, the field generated has sin n depen-

dence, forcing us to the bottom line of the current and field distri-butions (Equations (2-52) and (2-53)). The integrals all end up thesame, resulting in

ik 2cnma 4

C 0(2-59)CE- 4J nm2 nm

Therefore, we have demonstrated the relationship between couplingof incident plane waves to waveguide modes and the radiation patternsof waveguide modes.

55

CHAPTER IIISCATTERING BY A REPRESENTATIVE ENGINE-LIKE

OBSTACLE IN A WAVEGUIDE

This chapter is concerned with scattering by a simplified modelrepresenting an engine geometry in the intake duct. The model consistsof an infinite circular waveguide housing an axially symmetric conecentered on a flat plate, both perfectly conducting, as shown inFigure 3-1. This model is based to some extent on observation ofthe Pratt-Whitney J-57 turbofan jet engine, on display at the AirForce Museum, Wright-Patterson AFB, Ohio. In this study, it isassumed that the TE11 mode is incident; similar procedures couldbe followed for any other incident mode. The coordinate system iscentered at the base of the cone, as shown in Figure 3-2.

The calculation of fields scattered by the cone in situ wouldbe known completely if one knew the exact currents on the obstacle.

-The scattered fields can be obtained straightforwardly from the dyadicelectric Green's function and the dyadic magnetic Green's function,based on the integrals given by [Tai (1973), p. 9].

T(R) = iw 0 ff1 'el(RR').J(') dv' (3-1)

FR(R) = [ m , ' dv' . (3-2)

Although the source singular term for the dyadic electric Green'sfunction is still being discussed in the literature by[Yaghjian (1980)]and others, the Green's functions at points not near the source canbe obtained unambiguously from residue series expansions. Theseexpansions consist of a double summation over n(azimuthal index)and m(radial index) of terms representing TE and TM fields, bothpropagating and evanescent. Hence the coupling to a given mode canbe determined by a single integral, allowing one to obtain eitherEz or Hz, based on known currents. Mathematically this can be ex-pressed as follows: Omitting the source singular term, one can write(using Tai's notation)

56

-Figure 3-1. Basic geometry for engine scattering model.

AP

n

A

Figure 3-2. Coordinates system and dimensions forscattering from an axial conducting cone.

57

(e nt"0-- ff' -;e ( ,1 N +k-)Ne n(+kx)+n =0 m=1 n " XX o

+ -Re (+k )Mne' (;k )

for z <>z (3-3)

Gm2(ZiR') = x gel

I - I C n 0, 2^ 0,, ^(k~feX ;

n=0 m=l X kXIx 0 0

+ XNe (+k )Ne'. (;kU)2ki o - 1 np

for z z' (3-4)

where

j= J'/a X = Jnm/a

nm nj =- 1 a 2Xk - 2 = r k X = k2 anm

Ip: a2l-2/nm)n(Jn )2 X) :aJn(Jnm)2

k= k

C = n=O2 n=1,2,3...

IJ(h) represents TM electric fields or TE magnetic fields; 1R containsan axial component

RW(h) represents TE electric fields or TM magnetic fields so thatR is purely transverse.

58

For example, for a single TE mode, we can obtain

Hz,n,m = -"ff 9m2(RR)J() dv'

- 42i .n(tk ) fff Ren (;k ).(') dv' (3-5)4wi 2kI1 0 n~ 1 0 nj

and similarly calculate E using Gel* The elements of Gel and Gm2are written out in Appendix C.

Because of the difficulty of and restrictions on previoussolutions for current, a new technique of solving for currents wasexplored. A few cases have been previously solved with great effort.For example, [Tesche (1972)] solved the problem of a skewed wire betweenparallel plates by using images. [Wang (1978)] used the dyadic elec-tri2 Green's function to solve for the currents in an arbitrarilyshaped dielectric body inside a rectangular waveguide. [Harrington(1961), pp. 402-406] gives a variational technique to find stationaryformulas for scattering using approximate current distributions.He applies this to a post in a parallel-plate guide. Unfortunately,this method requires computing the self-reaction of the assumedcurrent distribution, which in most cases is equally as difficultas solving the problem exactly, since it requires computing theelectric field generated by the assumed current in the source region.Hence, no computational advantage is obtained over solving directlyfor the actual current by the method of moments, or some similartechnique.

In principle, the problem can be solved by the method of momentsby assuming the current distribution to be a collection of a series-of pulses with unknown weighting coefficients, calculating the elec-tric fields generated by these pulses, and adusting the coefficientssuch that the tangential electric field vanishes as the surface ofthe obstacle. The problem with this approach, as pointed out by[Wang (1978)], is that for coplanar field and source, the dyadicelectric Green's function converges extremely slowly, if at all,in addition to the problems involved with the source singular term.Convergence of the residue series is enforced by the axial propagation

m1mz~z'I wihbcmlz-z'factor e , which becomes e , for large enough m.For coplanar source and field, this factor becomes unity. In Wang'scase the problem was solved by summing the resultant series (withoutthe convergence factor) in closed form. In the case of the circularwaveguide, we have a Fourier-Bessel series whose summation in closedform is not readily apparent. For example, the z component of Ge1is given by

59

ia mlz-z'I. 2- e 3 nmelz co 0n: elnnm Jnm2aJn(np/a)Jn(nP'/a)cos n¢ cos n '.

elz,za en*~ f 2 , 2 43 (n p/) ni nn=Q m=l ax k 3nO ( ) a n ln ~o ~csn'

(3-6)

Taking asymptotic forms for J and Jn(), assuming n=l, for coplanarsource and field point, we obain

Ji I ' (M+;4)W (3-7

Jl(x) m V± cos(x-3f/4) (3-7

-. cosk c s ' I(l+ os((m+, )wp/a-3I/4) cos((m+ )i p'/a-3/4).

elz,z k~2 m ~~

(3-8)

It seems that even if p and p' are different, this series does notconverge. The series does converge for real objects after inte-grating over volume currents of finite extent, since integrationover p' results in dividing the asymptotic terms by (m+ )), andintegration over z multiplies the element by the differential dz,forcing the coplanar elements to make an infinitesimal contribution.The reminder of the integral is well behaved, due to the convergencefactor for noncoplanar points.

This extremely cumbersome process seemingly cannot be avoided.However, poor convergence in the source region suggests a similarproblem in free-space scattering, and the superior convergence ofthe magnetic field integral equation (MFIE) over the electric fieldintegral equation (EFIE) for most obstacle scattering problems.to.-to fore- Sr -ffuto,We therefore proceed, analogous to the MFIE o orce s = n x H

just outside the surface of the conducting body and n x Htotal= 0just inside the surface. As with the MFIE, taking the mean at thesurface results in dividing the current by two. Assume the currentconsists of a collection of pulses of arbitrary weighting

is akP(r rk)Uk (3-9)

where ^ is a unit vector tangent at the surface at rk, and p(T-rk)is loca~ized near rk. Then

60

'RS(R) = fff Gm2(RR).Js(,) dv'

= Z ak ff gm2(RRk')'uRP(k-) ds' (3-10)k

Enforcing the boundary condition

-ftotal = x -i + ^ x

n xi = -Js - x s

n x fi(R) = u n x i m u (fR'.k -r' ds'

(3-11)

We can then enforce this condition as many times as necessary toobtain the required number of equations to solve for the a k's. Theadvantage of this approach is found only by careful examination ofthe convergence properties of Ge1 and Gm2. The least convergent

term in Gel goes asymptotically as m sin mx, but the least convergent

term in Gm2 goes as I sin mx. After integrating over the surface,

we obtain sin mx or cos mx, which are convergent. Hence, we canm mwork with surface, rather than volume currents, and construct arelatively simple code. Adjitionally, there appears to be no disputeconcerning the behavior of G in the source region.

m2With the geometry shown in Figure 3-2, we define

LP +bz

FL 2-2

1 L 0 b -bP+L2 (3-12)2 2 7 2L+ b 0100 1 0

Assume the current consists of the sum of pulses

61

N-s L akt P(P-Pk) +k2. bk P(PPk) (3-13)

k=l k=1

where p(p) is a triangular pulse function. This current distributionis shown in Figure 3-3. Then using Equation (3-11), we obtain

n x Hi ak[t P(P-Pk) - n x f m2( ,').t p('-fk )ds']

+ + k k) x JJ 'm2 (R,R')'^ P(R'-rk) ds.

(3-14)

Next, to employ Galerkin's method, we generate 2N equations by multi-plying through by each basis function and integrating over the surfaceS, to obtain

JAui.n x If Wi( )p(-F ) ds =

N

akIbk[ fP (7-7 k)P(R-rq dS (ui'l)-ui'-nxfffn(T-_rq)gm2 ( ' ' '

N k1

• p('- rk)dSds] (3-15)"

q =1,2,...N

i = 1,2

where Ul u2 = t "

Thus 2N linear equations for the 2N unknown coefficients a and bkcan be obtained, which can be straightforwardly solved by Tinearalgebra.

In order to further speed computation, elimination of unknowncurrents in the backplane can be obtained by using the method ofimages. Instead of the original problem, we introduce an image such

62

04

a 3 P4

a.-P

Figure 3-3. Current pulses for moment method solutionof scattering by cone.

63

that the image of the cone has image currents on it, and the imageincident field is assumed to be present, as shown in Figure 3-4.

This automatically forces nxEtotal=O and n.Htotal=O in the backplane,hence reducing the amount of coplanar integration necessary. Withthis technique, we obtain g solution for the currents on the cone,then integrate these with Gm2 to obtain the H-field in the backplane,hence the currents in the backplane. The current distribution thusobtained is shown in Figure 3-5. It would be extremely difficultto measure this experimentally, and constraints of time and moneyprohibit this verification.

The fields in the region of the cone-base termination can beapproximated in the spirit of the WKB approximation, if the coneis slender enough, and the cone diameter varies slowly enough withaxial position. At each axial position on the cone, it is assumedthat appropriate TE fields exist for an infinite coaxial line ofthe same inner diameter. Since Ez=O, this guarantees x E = 0 onthe cone and outer waveguide walls. The radial wavenumber is thencomputed by solving the characteristic equation

J (pb) Yn(pa) - Y(jib) Jn(ua) = 0 (3-16)

for the smallest positive p, with b the inner diameter and a theouter diameter. The values of 11 are plotted in Figure 3-6. Next,the scalar wave equation for H is approximated by assuming thatv(z) varies slowly enough so t~at we can neglect the coupling ofV(z) through the radial function. Suppressing the azimuthal depen-dence, let Hz = R(p,z) Z(z). The scalar wave equation

1 a (aHz _ n2 32H z 2 (3-17)

Pap a 2 Hz +--z 2 + k 2Hz 0

is approximated by setting

aR(p,z) = 0 (3-18)

az

so that the wave equation reduces to

'2R + I R (,2_ n Z(z)*(k2-02)R(p)Z(z)0R(p) 2 0.

(3-19)

64

11Jc~-4bJ0 -

4a)0,'UE

a MJ- 0~

t!~ILfl

0,

4*1 -, 1.-a)

.4-'

w .4-'LIZ U04

----------------------------------------------- -A -~ 0UL0

'4-.

4- S.--~ .4-)

-~ a)

a)

S.-

0,

U-

LI

65

20

00

r Hi

010

20 1020

00 0 0 0?

W 90

W 0

1.0M 0

0

0 I I I I0 1.0 2.0

z/o

-80-

-ISOL

i d 0 0o0e 0 0 0 0 0 0 o

ta

W O t 0 I I I I I I, 1.0 2.0

Z

-66

Figure 3-5. Currents induced on cone in circular waveguide withincident TEll mode, L/a=2.0, b/a=O.5, ka=2.0 solution by

dyadic magnetic Green's function.O : Jtan

66

1.8-

1.4 1.7-

1.6

1.5

1.3-

N INNER/(

Figure 3-6. Eigenvalue of TF mode for coaxial waveguide as afunction of innP conductor radius.

67

Since it was assumed that R(p) was an appropriate combination ofBessel functions, this reduces to

a2 Z 2_ 2+ (k )Z = 0 (3-20)

Since p(z) is a known function, based on the local radius, thisequation can be integrated numerically to obtain Z(z). The resultsof this numerical integration are given in Figure 3-7. Finally,

to normalize the radial function properly, fIR(p,z)1 2 da must beconstant locally at all cross-sectional planes. Subsequently, HHe, and H can be determined at the inner conductor, and hence tgesurface c6rrents J tan and J, which are shown in Figure 3-8.

There are several limitations to this approach. Since H, = 0on the inner conductor but H 0, n • 0. Maxwell's equationsare only satisfied approximately, not exactly, since the couplingof v(z) through the radial functions was neglected. The behaviorof the fields near the tip of the cone was approximated very looselysince it was assumed that the field consisted only of incident andscattered TEll modes immediately beyond the tip.

68

il

68

2.0

.40

00

0 1.0 2.0

Figure 3-7. Solution for axial variation of fields for TE1 mode

in circular waveguide with cone. L/a=2.O, b/a=0.5, ka=E .

=(ka)2 _ (pa)2

OZ(Z)

69

z1.0 2.0

J/Hi-.5-

-2.0 J J ton

Figure 3-8. Currents induced on cone in circular waveguide withincident TEl ode, L/a=2.0, b/a=O.5, ka=2.0, solution by

N ~coaxial approximation with slowly varyingcenter conductor.

70

CHAPTER IVCONCLUSIONS AND DISCUSSION

The problem of predicting the radar cross section of jet intakecavities in a spectral region where the cavity aperture is of resonantdimension has been approached via the exact Wiener-Hopf solutionfor a semi-infinite waveguide. Exact (numerical integration) RCScomputations have been presented for the semi-infinite circular wave-guide for electrical diameters from zero to three wavelengths. Thesecomputations smoothly join high frequency asymptotic results andalso serve to define the region of validity for the simpler asymp-totic models. Therefore one of the matrices in a generalized scat-tering matrix development of the scattering by a finite loaded circularwaveguide has been completed. In the process, the nontrivial relation-ships between two earlier studies of the semi-infinite cylinder havebeen developed and certain errors in a publication from one of thesestudies have been corrected.

The coupling coefficients relating an incident plane wave tointernal waveguide modes have been extracted from earlier studies,recast in a more convenient form, and evaluated in the resonanceregion. The predominance of the TE modes over the TM modes in powerabsorption for axial incidence has been demonstrated. The relativeimportance of the five lowest order modes for non-axial incidencehas been demonstrated graphically. The relationship between thecoupling coefficients and the radiation patterns of the waveguidemodes has been established via the Lorentz reciprocity theorem.The relationship thus demonstrated is that the radiation patternand the coupling coefficient for a given mode at a given frequencyand polarization are proportional. This is of particular importanceto RCS calculation since a given waveguide mode will couple to andradiate efficiently in the same direction and with the same polari-zation.

Modelling of the leading surface of a jet engine as a coneon a flat plate inside a circular waveguide, we have computed thecurrents on this obstacle, by using the method of moments and thedyadic magnetic Green's function appropriate to the interior of acircular waveguide. The reflection coefficient for an incident TEllmode has been calculated based on these currents at selected fixedfrequencies.

It therefore remains to apply these results to Equation(1-8) to solve the radar scattering problem in a self-consistent

71

fashion. This is done for some illustrative cases. First, fixinga perfectly conducting flat plate at ten radii down the waveguidefrom the mouth as shown in Figure 4-1, the RCS has been computedwhile varying the diameter from zero to 1.5 wavelengths. The normal-ized RCS is shown in Figure 4-2. In Figure 4-3, the normalized RCSis shown for the case when the waveguide diameter is one wavelengthand the flat plate varies from two to ten radii down the waveguide.

Discussion

The calculation of scattering from the inlet mouth and couplingto internal modes has been performed modelling the inlet as a circularduct with infinitely thin walls and a knife-edge rim. In practiceinlet rims have a finite wedge angle, and are very rarely circular.The major inadequacy encountered in calculating rim scattering forthe knife edge by ray optics proved to be the underestimate of thepeak just above cutoff of the lowest order propagating mode (comparingFigures 2-4 and 2-5). Unfortunately, for arbitrary geometries, thereseems to be no simple way to patch up ray-optics to solve this problem.Since the frequency range this occurs in is low, it may be possibleto evaluate this region by moment method techniques. Although thiswas not done in this study, we can see that the ray optics approxi-mation does work well provided we are sufficiently above cutoff ofthe lowest order mode. For circular geometries, this condition isthat the diameter is greater than one wavelength.

Because of the reciprocity relations demonstrated in SectionII-C, it is clear that discussion of radiation patterns and couplingcoefficients are redundant. The calculation of radiation patternshas been discussed by [Weinstein (1969), pp. 139-150]. Weinstein'scomments on the use of Huygen's principle for calculation of radiationpatterns indicates that in the transition region (near cutoff)Huygen's principle performs poorly, and worse for TM modes than TEmodes. As Figures 2-13 and 2-14 indicate, the relative importanceof TM modes to backscattering from cavities is very far below thatof TE modes, and limited to angle far from the forward direction.For all practical purposes, forward scattering in the transitionregion can be calculated using the TE1 1 mode, adding the TE2 1 andTE01 modes as requirements may dictate. The reason for this canbe grasped physically by considering Figure 1-3. For modes higherthan m=l and for all TM modes, the oscillation in sign across thewaveguide cross-section forces the average interaction to a verysmall value, regardless of incidence angle. For m=l modes, the vari-ation due to cos n dependence can be seen to lead to an optimumangle, from which the aperture fields appear to be nearly of thesame sign.

For non-circular geometries, the primary conclusion we mightdraw is that the appropriate low-order TE modes will dominate scat-tering in the forward direction. Since Wiener-Hopf solutions are

72

................................-

E'Q

Figure 4-1. Geometry for self-consistent scattering problemusing flat plate.

73

) >4- f

C

41

a) -

.- 4-'

--

CMu

0~~ 0 Oo"

742

)

4 j

CCALf

0 L)

LL.

C--,

C~j 0 O

b-M

75~

not available for these arbitrary geometries, it will probably be nec-essary to construct moment method solutions for near cutoff frequen-cies. Higher frequencies can probably be adequately approximated byray-optic solutions as done by [Pathak and Huang (1980)], for example.

The impulse response waveform given by Figure 2-6 indicatesthat rim backscatter is really only significant for times of t<5a/c.Furthermore, this short time response can be reasonably approximatedray-optically, confirming our earlier statement that high-frequencyscattering can be adequately modelled ray-optically. The low-frequency scattering, which is not ray-optic, corresponds to thelong-time response, which is typically of little interest.

Having enumerated the difficulties in calculating scatteringcoefficients for object inside waveguides, it becomes clear thatmuch work remains to be done. The use of the magnetic field dyadicGreen's function and appropriate boundary conditions provide someimprovement over use of the dyadic electric Green's function, butit is still very expensive computationally. The possibility ofazimuthal asymmetry (for example, a blade structure) has not yetbeen addressed. One might model the blade structure axisymmetricallyby using boundary conditions in the backplane E^ = 0 and J = H9 = 0and at the outer wall J (P=a) = - H (p=a) = 0. This allows radialcurrents and azimuthal E-fields to exist, but not radial E-fieldsor azimuthal currents. It further forces the axial current to vanishat the tip of the blades. For a structure with 28 blades, for example,it would require an extremely large diameter waveguide before signifi-cant azimuthal currents could flow on the blades. One problem inimplementing this study would be that the image procedure used inChapter III to eliminate coplanar integrations could not be used,and it would be necessary to integrate coplanar source and fieldpoints. Another approach would be the use of the free-space Green'sfunction, to analyze the structure. In this case it would benecessary to also set up currents and enforce boundary conditionsat the waveguide walls, thus vastly increasing the number of currentelements required. Another possible approach is to segment theobstacle structure and use waveguide modes appropriate to coaxialwaveguides near the obstacle. However, the convergence propertiesof this approach are unknown.

76

REFERENCES

Abramowitz, Milton and Irene A. Stegun, (1964) Handbook of MathematicalFunctions, U.S. Government Printing Office, Washington, D.C.

Bowman, John J., (1970) "Comparison of Ray Theory with Exact Theoryfor Scattering by Open Waveguides," SIAM J. Appl. Math., 18(4),pp. 818-829.

Chu, L.J., (1940) "Calculations of the Radiation Properties of HollowPipes and Horns," J. Appl. Phys., 11, pp. 603-610.

Chuang, C.A., C.S. Liang, and Shung-Wu Lee, (1975) "High-FrequencyScattering from an Open-Ended Semi-Infinite Cylinder," IEEE Trans.Ant. Prop., AP-23(6), pp. 770-776.

Collin, Robert E., (1960) Field Theory of Guided Waves, McGraw-HillBook Co., New York.

Einarsson, 0., R.E. Kleinman, P. Laurin, and P.L.E. Uslenghi, (1966)"Studies in Radar Cross Sections L - Diffraction and Scattering byRegular Bodies IV: The Circular Cylinder," Radiation Laboratory,University of Michigan, Ann Arbor, DDC #AD-635186.

Harrington, Roger F., (1961) Time-Harmonic Electromagnetic Fields,McGraw-Hill Book Co., New York.

James, G.L. and C.J. Greene, (1978) "Effect of Wall Thickness onRadiation from Circular Waveguides," Electronics Letters, 14(4),pp. 90-91.

Jones, D.S., (1955) "The Scattering of a Scalar Wave by a Semi-InfiniteRod of Circular Cross-Section," Phil. Trans. Roy. Soc. London, serA., 247, 499-528.

Jones, D.S., (1964) The Theory of Electromagnetism, Pergamon Press,MacMillan Company, New York, pp. 594-602.

Kao, Cheng C., (1970a) "Electromagnetic Scattering from a FiniteTubular Cylinder: Numerical Solutions," Radio Science, 5(3), pp.617-624.

Kao, Cheng C., (1970b) "Currents on a Semi-Infinite Tube Illuminatedby Electromagnetic Waves," Radio Science, 5(5), pp. 853-859.

77

Kouyoumjian, R.G., and P.H. Pathak, (1974) "A Uniform GeometricalTheory of Diffraction of an Edge in a Perfectly Conducting Surface,"Proc. IEEE, 62(11), pp. 1448-1461.

Lee, Shung-Wu, Vahraz Jamnejad, and Raj Mittra, (1973) "Near Fieldof Scattering by a Hollow Semi-Infinite Cylinder and Its Applicationto Sensor Booms," IEEE Trans. Ant. Prop., AP-21(2), pp. 182-188.

Levine, H. and J. Schwinger, (1948) "On the Radiation of Sound Froman Unflanged Circular Pipe," Phys, Rev. 2nd ser., 73(4), pp. 383-406.

Mittra, Raj and S.W. Lee, (1971) Analytic Techniques in the Theory

of Guided Waves, MacMillan Company, New York, pp. 107-108.

MIttra, R., S.W. Lee, and C.A. Chuang, (1974) "Analytic Modelling

of the Radar Scattering Characteristics of Aircraft," Univ. of IllinoisElectromagnetics Laboratory, Urbana, DDC #AD-773685.

Moll, John W. and Rolf G. Seecamp, (1970) "Calculation of Radar Reflect-ing Properties of Jet Engine Intakes Using a Waveguide Model," IEEETrans. Aero. Elec. Sys., AES-6(5), pp. 675-683.

Morse, P.M. and Herman Feshbach, (1953) Methods of Theoretical PhysicsPart I, McGraw-Hill Book Co., New York, pp. 978-992.

Narasimhan, M.S., (1979) "A GTD Analysis of the Radiation Patternsof Open-Ended Circular Cylindrical Waveguide Horns," IEEE Trans.Ant. Prop., AP-27(3), pp. 438-441.

Noble, B., (1958) Methods Based on the Wiener-Hopf Technique, PergamonPress, Los Angeles.

Pathak, P.H. and C.C. Huang, (1980) "An Analysis of the Electromag-:netic Fields Backscattered From a Jet Intake Configuration," Proceed-ings of the 1980 Radar Camouflage Symposium, 18-20 November 1980,sponsored by USAF Avionics Laboratory and Martin Marietta Corp.

Pearson, J.D., (1953) "The Diffraction of Electromagnetic Waves bya Semi-Infinite Circular Waveguide," Proc. Cambridge Phil. Soc.,49(4), pp. 659-667.

Tai, Chen-To, (1971) Dyadic Green's Functions in Electromagnetic TheoryIntext Educational Publishers, Scranton, Pa.

Tai, Chen-To, (1973) "Eigen-Function Expansions of Dyadic Green'sFunctions," Math. Note 28, Univ. of Michigan Radiation Laboratory,Ann Arbor.

Tesche, F.M., (1972) "On the Behavior of Thin-Wire Antennas and Scat-terers Arbitrarily Located Within a Parallel-Plate Region," IEEETrans. Ant. Prop., AP-20(4), pp. 482-486.

78

Wanq, Johnson J.H., (1978) "Analysis of a Three-Dimensional Arbi-trarily Shaped Dielectric or Biological Body Inside a RectangularWaveguide," IEEE Trans. Micro. Th. Tech., MTT-26(7), pp. 457-462.

Weinstein, L.A., (1969) The Theory of Diffraction and the FactorizationMethod, Golem Press, Boulder, Colo.

Witt, H.R. and E.L. Price, (1968) "Scattering From Hollow ConductingCylinders," Proc. IEE (London), 115(1), pp. 94-99.

Yaghjian, A.D., (1980) "Electric Dyadic Green's Functions in theSource Region," Proc. IEEE, 68(2), pp. 248-263.

<4

1

79

APPENDIX AEVALUATION OF WIENER-HOPF FACTORIZATION FUNCTIONS

The factori2-,ion functions L+(a), and M+(a), are defined asbeing functions which are analytic in the upper half of the complexa-plane, which also satisfy the equations

L+(c)L+(-a) = iJn (ya)H~l)(ya) (A-1)

M+(a)M (-a) = 1iJ(ya)Hl)'(ya) (A-2)

where y = k2 - a2 (Einarsson et al, Equations (5-33), (5-34)).The exact function can be evaluated by numerical integration [Mittraand Lee, (1971), p. 107, Eqs. (9.8)]o The approximate evaluationof these functions proceeds in two different regions, for which theBessel functions are approximated differently. Section A-1 dealswith exact numerical integration. Section A-2 deals with low-frequency

:1 approximations. Section A-3 deals with high-frequency approximations.

A-1. Exact Numerical Integration

The exact expression for L+(c), is given by

= e l00+ic tn[-,iJn(a i Hl) (a k-)]dZ}L+(a) = exp T,- _f z-a- ic

Z-ci (A-3)

where -Im(k) < c < Im(a) < Im(k).

Before proceeding further, however, we note that it would be verybeneficial to not have to integrate out to -. We note that, forRe(z)>Re(k), the arguments of the Bessel functions become essentiallyimaginary. From Abramowitz, Equations (9.6.3) and (9.6.4)

- i ni

Hl)(ia )k = - e Kn(af ) (A-4)

80

AD-AIl 496 AIR FORCE INST OF TECH WRIGHT-PATTERSON AFS O$4 F/9' 21/5BXELECTROMAGNETIC SCATTERING BY OPEN CIRCULAR WAVEGUIDES.(U)1900 T W JOHNSON

UNCLASSIFIED AFIT-CI-80-69D NL

m iillEEEE/IEE2N2

illlinllllll

I rJ iav' 7 ) e= lna7iT(A5

n

where Kn( ) and In( are modified Bessel Functions.

Taking asymptotic forms (Abramowitz Equations (9.7.1) and (9.7.2))

inn inne- eT - ii2 T na zV_2 ) e7 In(a) z2_ 2 )

II

j e - K a z 2 k e -k2

eaVk

s'2 e

1 (A-6)a z2k 2

We are integrating n(az -2-k2 ) from k to -, but would prefer to inte-grate ln(1) from k to =, since it is trivial. Hence, we form theeq uat io n

-ia V L+()L+(-a) = iryaJn(ya)Hl)(ya) (A-7a)

V-ia k+a)L+,(a) /-ia(k-a)L+(-a) = nYaJn(ya)H 1 (ya) (A-7b)

We can evaluate this by the same integral, except that the argumentof the logarithm becomes very close to 1 for Re(z)>Re(k). Takingc+0

Ina7kaF4L+(cl)kz2exl{(Uk2-f2)' = Z-a dz

(A-8)

81

Similarly

-i a ) ( z-

(A-9)

These integrals have the advantage that we can essentially integrateon the range where -(ka+C) < z < (ka+C) and the total error thusintroduced can be made negligible by proper choice of C.

The complex value of k requires some thought. The argumentintroducing this imaginary part is given by [Einarsson (1966),p. 1473. The correct answer is defined as the one obtained in thelimit where Im(k) - 0. For the purpose of numerical integration,two routes present themselves. We can either allow k and a to havesmall but finite imaginary parts, or we can attempt to integratein a symmetric fashion about the singularities at z=a and z=k, whenthey lie on the path of integration, and add in the appropriateresidues. In fact, the first alternative was chosen. It was evidentthat the factorization functions must be smooth, hence slowly varyingas functions of complex k and a. Also, experimentation with valuesof loss tangent indicated the results were relatively insensitiveto the loss tangent. Hence, for purposes of numerical integrationcomplex values of k and a (being k0 and ao) are defined by

ko = (1 + i.005)k (A-i0)

00 = a + i.0025 k (A-11)

The results of numerical integration for values of n (the order ofthe Bessel functions listed as n in Table A-l) ranging from 0 to3 are presented in Table A-l. It was found, by way of confirmation,that there was excellent agreement with both DC and asymptotic forms.The $M+ function is defined by

$M+( a) = a(k + a)M+( a) . (A-12)

It remains finite as k * O.

82

Table A-iVALUES OF FACTORIZATION FUNCTIONS L+ AND M+ COMPUTED BY

NUMERICAL INTEGRATION

n D/1X / Re L+ Im Re M+ Im

0 ,.. i.. I.7539 0.'444 1. 355 0.'355. 1.S,.3 9.595 1 .D03 ..v7IB

0 U.1 3. 1.4793 O.9b1 l.%9b 0.10380 ,i.i 1 I.3!?i 0.9575 0.9941 1]329

3. . 1.27; 94. 9 5 n.9772 0.15991 i.,. I.1929 t' .93.,3 q.595 0.1 ?5

u I 11. 31'- v ^ .727 0.13n35°, . d !.1435 0 1ql 1 .1579- .,, ,,1.-l J 9 "; 9

'. o .733 iu 3.2 1 . 9 .. 44 C. :451 0.2704

0 ." ,.,0. 2? qG19 0 . 0 -. 27 1 .7?991v, 6 - isC 7" 5 4. is.759 .r21 . 9 5? 0 .3214

~0 0. 5., 1 .o5 i° 1.7 7 5 0 .25052

% 1 '.. .1?5 917 C. 9.Y42 0.3169So. C 1.?O) '3.155 C' .c,25 i 0.3577

Jo 3 .o .,2,j 703i 'X9574 f). 3z395 n.702 0.7953 0.3995

C, ,1, 1.2 0v 5., 5.74 ?0 0.4077o. 0--3 .974 1.39;i 3.3923

0 .. , . .; S 09420 992 55 4.(359

. C.. . 'l .'7377 C.9294 1.4571u .,' , .' 72 ?.5553 P.74,75 1.4,572

v ,7. ,- .72 f). 5-j 7 '-',67 q3 0.4,542u,. i.u, t. 5 7 '-. 5,t?, (..5225 n . 4 4 9

j 2 ,. 0.-,552 0.95q2 . 95 95 0.51971: 3-) D I.719J 0.g9. 0.53260,3Z95 3,5125 n.7142 Z35

U 1o o oi12 C.5379 0 5 3 ( o5042i j.2, L.o '.3 117 4:,t, C ' .5 5 0 oC. 431u .5 i., 0.,7 . 0.5151 .4 572

I 6.-131 3e ~4' 1.62 35a, C,. ,-o 4L? S 15 .5537 .6 95 3 0.5949u 35. ,. . 51 4. :)39 .5977 C.554f>

.o C .4152 9.5191 0.5140u .. o . ).595 9.37?5 (7,.4544 .4763L 0 l.C 0.2 o2 0.3541 0.4243 0.44320 jo -.1 Oo- :3t) 0.4265 C,597 0.69!0 3.7 Oo Os I-t3 5 . 3 4C n.5ti9 M.5200

li 7 c . .190, .3003 C.4734 0.5533'51 - 0s b C.?2J2 T,2o,15 0,4173 3, 49703 .7 0. '.?4:#7 0.2713 f.1799 0.4515

0 3.? i .??l 9?.>45 n.3512 0 .414 j

83 HIS PAGE IS BES T QUALITY PRAC

y F-b I k .. . . l - - DI

n D/X /k Re L+ Im Re M+ Im

1 3.1 0.3 1.355 0.036o f.1419 3.07353.1 0.2 1.)23j 0.' 71* 0.?C?2 2.5930

-. 1l 3.', I '9 1 39 0.2420 2o23313 ?oJ ?.4 1.94 o~ 9,: 0.V ?.QU 0.9772 0.1534 0.2$90 1.7 b70

I -... i:.i:. 0.1525 0.3033 1.5039I .2 C.CU 1.r7?7 C.1303 0.2021 1.5047SJ.2 .0 1. 1575 0. 2352 1.2795

i-e .t 0. ' 4 0.231- 5.565 1.1 16:i-.... oT 9.45 1 0.?74 n.2713 0.993t0. D.;c?7 0.29QI 0.2521 0.997b

4 ,.I 1.0 %.9 ?0 ".321, 0.2902 0.9205U. .3 1. T77:) 0.2552 0.199t 1.0155

... P992 C.. 3165 0.2145 0.9614o 3. .. q52 0.3577 0.2342 0.7506

I ,.3 .. 5 5.: 7 4 0.39?9 ?495 0 5b'31. . 0.79S 3)qt rOao15 0.50521 7 4.2 0.7 0 *.4 -77 0.2712 0.5551

. .. 1 3? 9 0.3923 0.124C 0.73901 c.. O.2 0. 95o p.4359 .Ib650 0.,6173I ,,.3.-, *.929i 0.4571 0.1991 0.5351I ,.' 0.7,t75 0. 4 _uc n.2242 0.4793

u , 0. t C. 793 0.4532 o.?427 4.43715 . '2.522 0.44,90 0.C?553 0.4 04

.5 C.. s 2.5197 0.1496 .4751G .2 .- 0.53?t 0.1250 0.3925

I .s C. t 0.7142 0.5235 0.1797 0.345b1. 3Ib C. t 3'7o C. 5042 Ca.76C 0.3195

I j.7 . :. C. b50 0.4-.30 .2395 0.3020o 2.5 1.3 ".5151 .:572 0.2542 0.2913. . C.%2' 4 0.5235 0.1 b14 0.0234

1no 0 ..2 05.s95j 0.594b 0.2742 0.05?6i t.,, . 't '.5-)?7 C.554b 0.3234 0. r5901 . t.o 0.5130 0.514C C. 3455 0. 119b1 3.d '. 4 ,:,44 ).4762 n.3537 0.1454i . 0.4243 0.4432 0.3523 0.1545SC3. 0.0 .i 73 3.591 f .f912 0.1715

1 .. 7 0.2 0.5019 0.520 0.4952 0.13441 3.a -f 0.72733 T. 5533 C .04 355 0.179:i J.1 0.0 0.-173 0.4970 O. 4 C21 0.20QBi ,.. . 0.373v - .4515 .4357 0.22931 .7 1.3 .3!2 .149 0.09 0.2394

Z I

n D/X a/k Re L+ Im Re M+ Im

., 0. 0.7131 0.' C04 0.1359 4.4633- 3.1 C.k 0.712t 0.0124 0.075 3.72rn,2 . . t :.71! > 02f,3 0,1253 3.1912z j 0'. 3.0 C.7097 0.0359 0.1535 ?.7953

I .1 .: 0. 7 0 7 3 0.0474 0.175C 2.4993i 1. 0 C.73 4 ,t 0.0595 n.1936 2.2434

0 L,.u ., .73?7 0.%p51 0.9327 2.1703, C,.z 0.7244 0.0313 ". q 3 l. 13b

3.2 L.,t C.7 "3- 0.555 C.1299 1.50102 . , 0.7155 .9 5 0.1573 1.3731

/ 0.2 0,=.,-,/ n , 93D 0.1735 1 .2279C. i. .t, : c. 11 q 0. 19 45 1.1125

2 .- g. Zd7 :)e D 0 -.')2 ?Z T.0577 1.3339

2 . 3.2 0.7,.3-t 0.1o3 0.1107 1.1b47-U.4 0.7372 ?.09l9 n.14 59 1.0099

to 0.7177 0.1310 0.17'12 C. 89 4

2 U TI .t9951 0.1536 0.1 7b 0.9050

2 0.1 1.T .7 3 4 .j 10. 20,'> 0.7353Z ,. o f. 0?3 ).r537 0.0O72 0.9923

S 0, .2. 7755 .1137 0.1394 0.94432 &.-, . + .A 0.1591 0.1b 4 0.7395

j., L 0.7 Z5 0.1930 n.1Itb 0.55959.5733 0.21o7 0.1974 0.5995

1.- I. 0 3.17 .2 3?o 0.2067 C. 54 b2 L.i 0. 0.9 34 ? 0.1171 0.1311 0.77652 j. ..2 3.7 52 0.1594 n.lb03 0.5559- h,.131.'+2.3,. 0.?257 0.1794 0.5556

.u C.. 5 32 C.?557 0.1929 0.52530 ., . 0.5357 T.2769 0.2C26 0.4796

2 30 .. 0.'75 0.?90 0.2097 0.4418S .3. 0.537 0.1945 0.1391 0.5;97

5 3.b s. 0.7783 0.2571 n.loO5 0.55412 0.0 Co .7J,.3 C.7947 0.1779 0.O4622 .. 0.5412 0.3157 0.1915 0.43622 6s :.o) 3 0 .324b 0.2013 0.39852 1 1.) ).5 7 0.3252 0.C 91 0.3700k .7 .1 0-3513d P.2955 n. 1147 0.55032 7. 0 1.; C.7,9o 0.335,. 0.''425 0.4634e L., 0. a 559: 0.359L C. I b 0.4047

i oC 1 0.55 o .3 3o 0.Ie41 0.3541. 0.7 u.- 0.52b7 0.359b 0.1961 0.-351

7 1. u 0.47?2 0.3 4. 0.?C41 0.3137

THIS PAGE IS BESI QUALIfl PRACTICABLE

85 ~ u nU±M lu 1DC85

n D/X a/k Re L+ Im Re M+ Im

.. 0.014i 0.3b4b 1.0738 0.44230.2 0. '%14 :1 0.407 0.1142 0.3671

2 we ,. 0.547j . 09.1 0.1535 0.321b

e J.3. Os5424 1.3451 0.1771 0.29352 Z.: .- n. 4t)% 0.3791 0.1922 0.2753

1.0 .2 2 o .352 n. ?U12 n.2625Z . .C 0.7o2 % 4 63b 0.0291 0.2900

2 ,.i ,. '.5Z5Cj '.4655 2 .37 0.24

2 4. 4. 1521 0.22102 T.. 4,. . 57 ? 1.4111 C.! z14 0.2131

' ~ ~ ~ C t.- . } .:7 e 9.362 n.1975] n.2Oqb

e . , . 9 0.3555 0.2070 0.?092.b721 .55C 0.1793 0.0219

2 5 3., 5a 0.5''o 0. 5C4 0.2514 n.n569

e. C.. 3t% 0. 115 0. ?934 n.1232. ).. . C.34ce, 0.375, .1l 0.1435

iA.L 0 321i 3.43v C.4/3b I,. 1570

; s. L 5 0. 593 0.37 7 0514C' .~ 4, 4. 1 :S .5157 0. 3 77 0.11'49

i.i 1., "1. 37o 0 .3452 0.3752 0.12

1. .i C.2? '1b~ 3.49 -y7, 355 0o173

2 ..13 . 5 7.353 0.33?9 0.1970E ~ ~ ~ ~ ~ ~ f 3. . ., - .39o3126 C .2032

e .2 O.. 0.433t1 C. 5U73 .375 0.1305./ ,,.c " 14-' 2 .502Pi 0.4-n .1 -)"A .. , .~C. ,.717 ?.4255 C.34L7 0.?25

8675- 0 . "A 71-. 3

t o .,C. -:)4 f .33 4 33?C r. -342.L., 7. C. * ' 3 05 2 .3C63 I. ? 313

-. I.l , . i2 C.. 5 7C 1 9 117 1. 2 5r . ] - 0 . 2,. ? 3 . 4t>4 1. 4B 9 1. ?57,,

'.. '.-, . 43 5 .3 55 C.3915 0 .27 n2z., 3. n.1334 Po. 3354 .3, 79 0. ?577

0.L.o .7 2 .3uAC 0.3142 0. ?5B9C. ., . 2"') 0.2?7 9 C.?594 0.?414

86

n D/, c/k Re L+ Im Re M+ Im

. 0. -. 0.5791 0.000 0.,,14 5.4965.. 2 0.5790 0.0052 0.0775 4.55(5

2,. ,0.9737 0.0 i?4 n.1 '49 3.9269

3 ,.1 0.s 0.5191 0.015 0.125Z 3.4371. .3 9.57'- 0.0247 0.1'429 3.056b

3 .i . 0.3755 ". 0317 0.1556 2.7525.' 5.: .5 , ').1011 % t 211 2.7215

5.0 . 5 3.d .13: (.0b44 2 .26853 I 5.19 C, .025t 0.0959 1.94513 . .5 0a .391 .llo 1.7053

,,.'577o 0.0502 0.1390 1.5195, i.j 1.0 0.513 0. "o 1y 0.1525 1.37,13

u- I-. .505 1.00"9 P. o1j52 1.78172 -. -.? 0.)3 '.(213 lo 16 ? 7 1 .4653

7 3 . .;,Y / 0.1)41 I .fl-5j 1 .27723 7 r 7 ". 1)5?9 .12()2 1.1215

..I.b 0.5170 0.0775 ".1397 1.00203 J. . .5::1 0 C.0940 C.1529 0.9069S ,.,, .. , 17,1?J 0.0040 0.190 1.29q5

S,1.. 1 0.a 0.531 0.0333 n.)572 1.0 9590. 3 D.0 CC'lb 0.9353

3 5; 4, _1 . "i1 1.124b 0. 525 7D. 5.1 9.5752 r.1030 0.14 ?2 0.7 407

S..o, i.' .1274t 0.155, 0.573b,,-. ,,., 0.5353 .) ?4 n. 0304 0.99973 .2 0.5265 n.0523 n.C791 0.4393

0 It) 0. .DI!; 17 .Oo7l .110, ,n. 7 279

5 6.! 0 5SP2 0.1170 0.1321 0.54573 d .- :. ,717 4.!'19 n .1490 0.55? 73 i . 0 .595,5 0.1 o2 C.1595 .5329

Lo C.5s l 0.29 b 0.0510 O. 79 41).o . e.5,55 0. MbOO C.94b 0.5735

. .. ' 211 1.1211 0.1225 0.5S95.3. 6. 0. r. 91 7 ,.15 4, 0.14,24 0.5261

.. 0 C. 5b15 0.1759 1557 0.47753 o.s 1.0 0.5311 7 .1-) (N.1551 0.4395

0.? ..0 0.9b: C. 1o 0.0779 0.5527

I .? .. s4 0.1193 0.1137 0.5592to 2. 23 0.1c35 0.1355 .4930

3 Z.7 V.0 0.5331 0.195o 0.1529 0.44343 J.1 ,3. s 2. 4 3:5 r).2157 0.1533 0.4050

7 j.1 1.3 0.5j7t 9.22)- 0.1707 0.3747

I'HIS TAGE IS 11, 7, .ULITY FRAO'ICAB"

I u DDC -

87

n D/X c/k Re L+ Im Re M+ Im

.1 0,: 0.. .7,,5 1 .10 3} .1331 .5 59u

,,.= .. 313 O.lQ7 0.1293 0.4Bl9

. W04 0.515 0.2119 0.146S 0.4261

. 0.5b54 0.237t 0.1591 0.35453 J. : u. 0.5145 0.251 Y.1571 0.3527

,. L O C) 0.473o 0.2594 0.173b 0.3273 ,., .O . 7453 C. I o7 0.1,-J 9 0.4964.

i.-, ,.2 . 721 L.2 37 n.1311 0.1,255".4, 0.o23 C.?540 0.1474 0.376 4

3 ,.i C. ~ r.5,'1 0.2757 (.1595 0.3411

3. j1 O.2d11 V. I b2 1.3 130-.-,3 :s n. 7- b)1 0.1747 0.2925

0 7 .9. 2 733b 0.1019 0.44521.., G.2 ".~ '.7G 0.12 5 ? C.3755

1.., J.. 0.571- 0.3C94 0.1439 0.33ni

1.J . C. 519 0.312 C.15 92 r. 30C,.44,37 0.3C5R C.It7 0.27950, C., 5. O." 7 .2 9 5 0.1737 0.2519

o C.u 0.7j32 ').3053 0.)751 0.3593±.i U.2 0.61 0.34 1 0'.1103 0.324C

. 3. 5e7 D. 3 5 3 0.1355 7 .2945is ... , , 0 " . 3 5b n. 1546 0. 20'14,

.wc7 0.0325 ".1561 0.24413 t -L.C J. 0.3043 0.1725 0.325

i.-c GOj 0.6-Y7 -. 3619 O .') 43 5 n. 30713 . 0.5015 .3913 0.0969 0.2536

- , .".475t> 0.3753 0.1332 0.?273i.4 J.o 0.,11v, 0.3517 C. 1554 0.2139

3 .2 i.: .3o55 0.3277 0.1691 0.2062I.e i.C 0. '3A 0.3050 0.1743 0.2009

3 i., V. C.542 1. 4517 0.0203 .1 6451.:, -. 2 % 5CPI 0.4302 0.099b 0.1430

1 1.3, 0.42 2 0.3939 0.1455 0.14333 i.1 u.0 .3* 0.3573 0.1703 0.1463 i.i .3252 7.3269 0.1516 0.15533 1.3 1.0 0.2,Y72 0.3017 0.159 0.15o2

o 3. L..5o 1 D .5051 (.2114 0."2 61.o z. 0. 433 ".4519 0.?5?1 0.0733 .- t,^ .3 t.4 3 .3914 .2755 ".1072

164 0 0 . 17 5 .33 n 7. 730 0.1325S i., o. 0.2*73 1.315 0.2b _7 0.1475

L .- .(. , 0. 2.? 5 - .9 2c92. n 7.?507 0.15S2

88

n D/X aIk Re L+ Im Re M+ Im

.3 C. 471 1 ". 5393 C.3339 0.1655

1.5 .o3> .4. 54T 0.3, 4 0.1233

13 .z Dt.?9i 0.317 3 n.2b 0.1513

ro0.274 9. 3 0.3:19 .-171

3 . C.5 37 0.304,5 .no' n.1539

.13 j ,53 1, 294 ? 6 039 $5 ?4

- 0 . 3 , .7 '.' 9 . 7 C . 9 4 6 2 M. 4 0 3 0 0 .1 2 4 9

Z.'t . ,3i.-,l".2351b .3 5

-:?51 .o~n .3370 0.2039

53 ., -. 37 5 0.312 f .7,7 5 0.2095,.0o 1. . 5 . o 3 0. 2 5 1 '. ?0 49

C. .. 54, 2 95t)7 0.4b334 C.939.?. O. 2L "7 , 0 0. 37 5 0. 2341

C .c C. C 75 " . 1)O 2. 7 9 1 1 P- 3655

-7 2. .2?a b.!5 7 ? 5 3 4 C.2 27 5

.,7 .1 73 c.241 0.2425 1.'175

89

A-2. Low-Frequency Approximations

Derivation of IDC' forms for the factorization functions isnot difficult. They can be obtained directly from the small argumentapproximations for Bessel and Hankel functions. For n 0 0, we get(Abramowitz, Equations (9.1.7) and (9.1.9))

L+(a)L+(-a) = RiJn (ya)H l)(ya) (A-13a)

n n

n

Zii -i "l) (A-13b)~n (T) n

1 (A-13c)

1L+. ), . .- (A-14)

M+(a)M+(-a) niJn(ya)H(1)'(ya) (A-15a)

1ya n-l7T_-_ i n! n

t it (n-l)! "7 n+ I b2 (A-15b)

-n

-n (A-15c)a2(k 2_ 2 )

M+ a) Za-Tr (A-16)

For n = 0, we note that

Jo(Z) = - Jl(z) (A-17a)

H(l)'(z) - H)(z) (A-17b)

90

Hence,

M+(n=O,k,a) =L+(n=l,k,a) (A-18)

From Abramowitz Equations (9.1.7) and (9.1.8), for n = 0

L+(i)L+(-i) = ( iJo(ya)Hl)(ya) (A-19a)

ffi.l. 2i n(ya) - 22n ya (A-19b)

-ln(a 2(k 2_a2)) (A-19c)

The factorization of this DC form is non-trivial. Based onrWeinstein

(1969), P. 335-36] we may approximate L+() z i, In a/2k k+) for<< k << 1.in(2ka)

Having thus obtained the first low-frequency approximationwith ease, it is extraordinarily difficult to obtain a better approxi-mation analytically. Lee, Jamnejad, and Mittra present a simplederivation. Unfortunately it does not agree with the results ofnumerical integration. This can be seen by comparison of theirformulas with the plot in Figure A-1, which shows the trajectoriesof L+(k) and M (k) for n=l computed by numerical integration. Theinclusion of additional terms from the small argument approximationdoes not improve matters. The approach taken, therefore, is thatfor small values of ka, the values of L+ and M are computed bynumerical integration, and interpolation is used.

It turns out that, for L (n=l,x==k), the values over a surprisinglylarge range (0 < ka < 2) can be approximated by

(1-i'0584 ka)I+i.68 ka)

However, this does not admit any generalization, nor does it possess

any theoretical justification.

A-3. High Frequency Asymptotic Approximations

Using the large argument approximations for Bessel and Hankelfunctions, we obtain (from Abramowitz Equations (9.2.17) through(9.2.20))

91

Im

1.

05

0 0.5 1.0 Re

Figure A-1. Factorization functions L+, M+, $M+ for n=1, a=k.Arrows indicate increasing k.

9?

* L+(c)L+(-t) = wrij (ya)H~')(ya) (A-21a)

=Trim (-ya)cosO (Ya)Mn (ya)e e n(ya) (A-21b)n n n

= ii 2 i2en(ya)=-(M (ya)) (1+e )(A-21c)

M+(cz)M+(-x) = lriJl(Ya )H~1).(Ya) (A-22a)

= iriN (ya)cos (Ya)N (ya)e i~~a (A-22b)n n n

(N (Ya) (1ae (A-22c)

Where, from Abramowitz, Equations (9.2.28) through (9.2.31)

M (z)2 L 1[ + ___ I] (A-23)

O(z) z _ (n + 1 + n 1 (A-24)

Nn(z) 2r 2__ (A-25)~ ~ L 8z2

For L+(cz) we observe that we can separately factor the two expres-sions, making L+(cx) the product of two factors, both of which areanalytic in the upper half-plane.

L()=LI(a)LII (a) (A-27)

93

2|

L+(o)L+(-(): _ Mn,(Ya)

i2 In2-_ 4 '+ 8 a2(k2

2t2_1

k2 4n

2

8a

'P8

Hencey by anpcto

2/ 22 4n2- 2_1____ ( 2k a +

k + 1 a

Hence, by inspection &i (kaK n1 ca 2 A-)

'a(kt a(k+) (A-29)

+ vr/4(k-k2+! a JkQa)

Similarly

I Mri N(_a))M+(a) = N( 2

a, 2 k2 _a2 4 2_3l~~i 2 ~ ( 2 a2

(k a 2 i2 T- aa)(A-30)ak2 a (ka2 ) A-0

94

2a2 4 n2"3 +

M (c) ei r/4 8ka +aa

,a(k+a) a(k+a)

In evaluating LI (a) and M+I(a), we follow the derivation of Mittra

et al, (1975) .

Lii~,.II i2B (ya)

L 1()L (-a) = 1 + e n (A-32a)

+ (a)M+ (-a) = 1 + e 2 n (ya) (A-32b)

The exact expression for L+I(a) is given by

J L+(a) exp - z- a dz (A-33)

Let

.n(l + ei2n(Ya dz (A-34)

Then the identity

co m+l mln(l + x) 1 (A-35)

m=l

can be used to obtain

I OD1m~ i2m n (ya)

m=l Za f e dz (A-36)

provided Im(en) a 0.

95

Let

e 2mG (ya) dIM f eo dz (A-37)

where

6n(-ya) -.rya - (n IT + 842_ (A-38)

ya a Fk~' (A-39)

Let

z =k sin-u (A-40a)

dz =k COST dr (A-40b)

ya = Z2 k2sin2r ka COST (A-40c)

The contours of integration in the z and Tr plane are shown in FiguresA-2 and A-3, respectively. The contour in the T plane is selectedso that ya has the right sign, and the integrand vanishes at bothends of the contour. The contour in the T plane is then deformedto the steepest descent contour. Since no poles are passed overin this deformation (for Re(ci)>O) the integral is unchanged. Hence

i2mO (kacosT)=m f jkin kcosi dT

f COST i2mBn (kacosT)f cst e ndT (A-41)CSPsinT -f

with B (kacostr) =kacosT - n + 14 7 +4n2-lsi(A42

Let XL 4n2_1 (A-43)

96

IImz

C aX

Rez

Figure A-2. Contour of integration in the comfplex z-plane.

97

c SDP

ak

I-s iTi Ret

Figure A-3. Contour of integration in the complex T-plane.

98

Im is an integral in the form

2 F(r) e dtCSDP

where F(T) =COST (A-44a)sinT - k

K = ka (A-44b)

i2mn (kacost)jn f(T) = ka

i n XL

= i2m(cosT - (n +) 7a + LL (A-44c)

This can be readily evaluated by asymptotic techniques. We makea first approximation by ignoring the pole near the saddle point.

XLsn

f'(Ts)=0 i2m (-sinT + X ) (A-45)COS T

Ts = 0 (A-46)

2Then f"(T s) = i2m(-cosT + 2-cos )

cos3

= -i2m(l-XL) (A-47)

The asymptotic value of the integral is then

I',re ) e(S F(TS) (A-48)

where 7 is the angle of the CSDP through the saddle point.

99

i2me (ka)f(s) = kTa- (A-49a)

F(r k (A-49b)

f"t(T' s -i2m(l-X L) (A-49c)

Thence

i 2m n (ka ) e-in/4 V 2mka1_(A)50)I m -P e e ((-X L ) aA50

m i2men (ka) -i/4

I=m -(-)m e e (k) fM= 1 m a (m1F- XTL

k e ir/4 O - l1m ei men(ka)k- e~ e a _L

aaXL m=1 m3/2 (A-51)

The indicated sum is shown by [Chuang, Liang, and Lee (1975), p. 773]to be a modified Lerch function. They present a development of itsprzperties and a transformation to simplify its evaluation.

i2me (ka) i2 mpm -l em3/2 :m:nl e 2 : L(p,3/2) (A-52)

M--l m M=l I mT

where +imy + i2men(ka) = i2lmmp

e n(ka) 1or p = - + T (A-53)

This function has two useful and simple properties. Since theparameter p appears only in the exponent, and

100

e i2I1I1p+i2n = ei P

*L(p+l,3/2) =L(p,3/2) .(A-54)

Furthermore e i2-m(l-p) = e- 2 -ump

L(l-p,3/2) = L*(p,3/2), where *denotes the complex conjugate.

*1 (A-55)

By using these relations, it is possible to transform any value ofp to the range 0 <p <S 0.5. In this range, the series solution isextremely accurate

i (1-v7r/2 mL ~rpl-v) i m (\-,)(2,rp)m' 6

L~p~v)(2,ffp \T- +m=0 A-6

For \r=3/2, this gives

L(p,3/2) = -21 vrp.(l-i) + I (R n + ilnp)pn (A-57)n=0

where values of.Rn and I n are given by Table I of Chuang, Liang,and Lee (Note sign error in Equation (35) of Chuang et al (1975)corrected here).

We can similarly find an asymptotic form for M+(cz). The analysis

is identical with 'n (ka) replaced by on(ka) and XL + XM = 4n 2+3 24,(ka) 8(ka)2

This results in p =+ 1/2. Otherwise the expression isunchanged. I

+.1@ I exp ( IiLM- (A-58)

Where

I e-ail/4 L) L(p,3/2) p =+1/2 (A-59a)

101

k / iT 4n(ka)IM= e- i /4 L(p,3/2) p - +f 1/2 (A-59b)

M -M

A comparison of these results with those obtained by numerical inte-gration is instructive. For a=k, the agreement is very good downto the first zero of Jn(ka) or J;(ka) (for L+ and M+ respectively)

if the proper form of B n(ka) or n(ka) is chosen. This is illustrated

in Figure A-4 for M+(n=l, a=k); the exact value of M is comparedto that obtained by using the two term and three term approximationsto * (ka). It can be seen that a rather sharp minimum occurs forthe Ixact form of M+ for that value of ka for which J'(ka)=0. Forthe two-term and three-term forms of 4l(ka), the minium occurs atthe value for which cos~l(ka)=O. Thus, the more accurate On or @nshould, in general, lead to a more accurate approximation for L+or M+.

L and M are also functions of c/k; we must explore the ac-curacy of this approximation when this ratio varies. This is impor-tant since a/K = cosB for scattering, radiation, and coupling problems,

3nmand o/k =1 - ( ) for coupling and radiation of waveguide modes.

In Figure A-5, the value of L.(n=l) is plotted in the complex plane,where the main curve traces out the real and imaginary parts of L+with opk and D/k as a parameter. The dashed curves then indicatethe trajectories which occur when varying a/k while holding ka (orD/X) fixed. It is immediately evident that the preceding approxi-mation does not, in general, capture the nature of this behavior.The reason for this can be sqen from Figure A-3. As c/k varies,the pole located at T = sin (a/k) moves closer to the saddle pointat Ts=0. We totally ?gnored this pole originally. Without belaboringthe point, it will only be stated that rederiving the approximationincorporating the effect of the pole produced only a marginal im-provement. It was therefore necessary to include two terms in theasymptotic expansion. This derivation follows.

With the same definitions of C,f(T) and F(T), we have theasymptotic approximation

102

0.9 \0.8 \ EXACT

0.7- 2 TERM0.6-\ 3 T ERM

0.5

\j-0.4I+

II0.3

1.5

c1.0--

0.5-

0 I I I I I0 0.5 1.0 1.5 2.0 2.5

D/k

Figure A-4. Factorization function M+ for n=l, a=k,calculated by numerical integration and asymptotic

approximation based on two and three termexpressions for phase of

Bessel function.

103

> 4-

£0

) W.

0. L. X

*- (V E.4*-.- aw+ 4-~

0.,- Eutf

-0 (U uL

co >~0, c; 4- 0.-)~ 4-3

-~ Eu

4-)

.0 x o.>

K%

0 Q) 0

UL-

104

Kf(Ts) iSIm - e SV eF n(,s)

[13f (T)(T)5"T+ iA (1 - (KA)) 14 F(Ts) 3f"() 3

f f,,,C S F,,(T S)'(T (A-60)

Sf'(T S) 1!where f(X) is the transition function defined and discussed by[Kouyoumjian and Pathak (1974), p. 1453]. A is a distance parameter,relating to the separation of Ts and Tp, defined by

A = i(f(Ts) - f(Tp)) (A-61)S p' I

J The saddle point is unchanged, so Ts=0. The pole is located at

T= sin-a (A-62)

We thus derive Table A-2.

105

TABLE A-2

SUMMARY OF ASYMPTOTIC FUNCTIONS AT SADDLE POINT IN T-PLANE

Function form Value at T 5=0

Xqf(T) i2m(cos -(7 + l4)-r + Li2m(1_(n + 1)r + XL

COS ST4'Fa L

f (T) i2m(-csT+X L 0-o~

COS T

f () i2m(sin +X L r- ~6-cos T)) 0COS T

f (iV)(T) i~m(cos +XL CO)-0sT2 i~m(1+5XL

F(T) COST - k/asiflT-c/k

P() a/k siriT-1 _k 2

(sin-r-a/k)2 C)

Fie (T) -a/k silTCOST-(a/k) 2 cosT+2cOST k~ 2()

(s inT-a%/k) 3 a1 a

106

ifTS fTp

=i.i2m(COST n( + 1)71 + L~r -(COST (n + +)7 L~i)

(A-63)

COSTS1

2m1X 1 -(a/k)2 + X )

2m(2 V, I-(a/k)2

2 1- ,a/+1 (A-64)

Jei2meO (ka) 2

+ iA1-ikaA)[~- k 3i2m [1+5L .i2m[ +L

3(i2m[-1+XLI)

+ 2 k2 .!(1-2(.- 2

co i2m(-I+X L)j A6a

107

k ~ m 1/2e mO(ka)

1+5X+XL

1 T-0 kaA)141(-I+X L) 2+ _IX(16b

Def ine

A = m6

6 = 2 I1-(a/7k- 1 + X L( 1(tk 2-(A-66)

k 2A0=-

3+9XL + 8(-) (1-X (-7Ao 8(1-X (A67

A ek i 1/4 (A-68)I

Im .pfemn){ + A06(1-q1(mka6)) (A-69)

m+

I m= 1

m~ (-A,) ei 2ma(ka) (+ (-~k6)

Ae n 112mO(+(ka)

m= m Ai

.Al (L (p,3/2)+A 06G(p,q)} (A-70)

108

where

G(p~q) m- n 31 'e ~~P1-..(mq)) (A-7l)m= 1

1 p= O k)+ 1/2 (A-72a)

q = kais (A-72b)

We could follow exactly the same analysis to obtain M+, except that

en (ka) n(ka)

4n +3 (-3XL -I Xm = (-3

L ~ 8(ka) 2

These formulas produce a very good agreement with the results ofnumerical integration, down to cL/k=0.2.

109

APPENDIX BSUMMARY OF WIENER-HOPF COUPLING AND SCATTERING COEFFICIENTS

The coordinate system chosen in these equations is that des-cribed in Chapters I and II. Hence, although the results are derivedfrom Einarsson et al (1966), they have been transformed into ourcoordinate system.

B-1. Direct Scattering From the Rim

These coordinates are cast in the form

Is

E~ S S Ei~~i 00 0(B-1)

Where both Es and E1 are expressed by

etk . + ^E) (B-2)r (E

+ikr swhere e is for E

-ikr ie is for E

The expressions presented in Appendix B are all asymptotic in thesense that they are far-field and valid only far from the waveguidemouth (kr >> 1)

S2i cos Jn (kasine i) an (kasines)S60 - n s sineiL+(kcos-i) sinOsL+(kcosOs)

(l'cosei) (l'c°Ses) f2

2(cosOi+cOSs) ln 2 (B-3)

110

i1

2i J' (kasineO. I~ains

~ COSOS n=O M4+(kcoseiT) M+(kcoso5)T

(l+cose1)(l+cose5 ) f[2(cose.+cose6sT + n(B4

4i O J~ (kasinO) J'(kasinei) ~-o k1+cose) s~~5f sine L+(kciis-Os) M+(kcosei) 2

(B-5)

0_____ JI(kasine5 J~ (kasine.) f

(bS4 e =-k(l+cos6 S) n=1 lf 5 s +T- Ts-s sine iL + kco-sO 7 1_f 2

n

(B-6)

Where

oi 7 r is assumed

f nL+(k)(B7

en = 2 n 1,2,3... (B-8)

It was found that Chuang et al (1975) contained several misprints.A correction letter is reproduced on the following page.

AP If-44 Gal. I

Correction to "'High Frequency Scattering from an Open-Ended Semi-Infinite Cylinder"

C. A.CHUANG, MEMBER, IEEE,CH ARLES S. LIANG,MEMBER, IEEE, AND SIIUNG-\U LEE,

SENIOR MEMBER, IEEE

In the above paper I , there were six sign misprints.

1) Time factor should have read exp( -iwt).

2) In (I), x should have read (-x).3) In (4), (1 + cos 0) in numerator should have read (I -

cos 8).4) In (7), (cos 8 + cos 80) should have read (-I)(cos 0 +cos 00).,5) In (11), (a - &') should have read (a' - a).6) In (35), (1 + i) should have read (I - i).

the above sign misprints do not affect any otherequations or numerical results.

The authors wish to thank T. W. Johnson and D. L. Moffattfor bringing some of these errors to their attention.

C. A. Chuang is with Aeronutronic-FordWDL, Palo Alto, CA. Heis now at 552 Solitaire Dr., Indialantic, FL 32903.

C. S. Liang is %%ith the Fort Worth Division, General Dynamics, FortWorth, TX 76101.

S. W. Lee is with the Department of Electrical Engineering, Univer-sity of Illinois, Urbana, IL 61801.

I C. A. Chuang, C. S. Liang, and S. W. Lee, IEEE Trans. AntennaPropagat., vol. AP-23, pp. 770-776, Nov. 1975.

112

A,

B-2. Coupling of Incident Field to Waveguide Modes

With the same definition of the incident field (as in Equation(B-2)) the axial fields within the waveguide can be represented asfollows:

AnmC°S n Jn(JnmP /a) -icanmZ

E Z n=O m=l Anmsin n - r(J e (B-9)

nnm

B e sin no jn(jip/a) _icmZz o Om n m ) e nmlz

o n=O n nmn (B-10)

Jn(Jnm )=0

J (j m)=O

k 2

2, 2 Jnm2

nm : ma2

The top line of Equation (B-9) indicates that E. producesEz with cos no dependence

The bottom line of Equation (B-9) indicates that Ei producesE with sin no dependence

-e i ma n L+( rn i(kasiInGj) f 2 (anm+k) (l-cosO:)A = Enm a +(kcose i) kasin i + 2(ani )

nmi. 1-f nm-kcosOin

(B-1l)

113

A4in in L+ (a fm

nm kcnm a (l+cosoB) M4 kos 1 1'ki 1'

4 n m M+(,'im) Jn(kasinei)~

B +4i kctnm (1 2 L+(kcose1)T kasinei 1-f2813

a nm nmn

jnm

f2 (k-a' )(l+cose.)_______________________ m 1(B-14)

2- + ?(CEos6i-ctnM)

A B-3. Radiation Patterns From an Open Waveguide

ji The incident field is assumed to be a single waveguide mode,either TE or TM, with axial fields given by

E n -P/a) cos 0~ e m (8-15)

H 1 = ~ ( H n / cosn0 e nm(B-16)

The radiated field is given by

114

CE c e- ii En 1 kr(-)

OCE OHL Inm ij

CE (i)nl nk 2 ka 3cos 0 s L+(ctnm) Jn(kasinO S)c E 2inm L+(kcosU ) kasinO

f2 (am+k)(1-coses)1I [ -***n + 2(mko~) j(B-18)Y 2( %M (kcos )k-f z

nn

(B-19)

c (i)nl a 2 ksin nof 5 L+(%xm) J'kasine fm n1cs5 +kcs~ (B-20)

C, -(-) 1 kao fl5 M(c) J'(kasineH2(1+cose )(aci-k) WM-(kcos'e5) n s

E 7*~ + 2(kcosO -Ar) (B-21)

115

B-4. Internal Reflection and Mode Conversion

With incident field as described in Section B-3, the scatteredfield is given by

Jin(intp/a) -ian csz FE nH nm

e~ m mt L

Es = sn¢ T n jFL,- H r

(8-22)

0n(jn'P/a -ianZ nE HE ]oHn e sin n H Rcos no

SHnm

(B-23)

Rn= J~k 2' ' Ff (k+ia )(k+i.aR _E n L+,(anm) L+(ani ) fn +( nm) (an9)

janmn lf 2 2k(nm+n)

(B-24)

nH OJntk+(c m)L+(ang) fnTm9 (B-25)'n9ak-oLm) 1-f2

n

ka(k+n) fnTm'tn£ n2 L+(atnm)M +((anP) l'f2 (-6

2 1~~n

~~2)nH k(k+atn.)M+(inm)M+ (n) nf (k-amt(k' ( -27)

Rmi = fl J-121 " kc~t (B-27)m7 (k-i2)k -kanan)

116

APPENDIX CELEMENTS OF DYADIC GREEN'S FUNCTION

FOR A CIRCULAR WAVEGUIDE

The dyadic electric Green's function, omitting the source-singular term, is given by Tai (1971). It is assumed, in this appen-dix, that the azimuthal dependence for Ep, E, , 3J, andH iscos nh, and for E , Hp, and H is sin n. This zimuthal depen-dence is understo d ad suppresse. The summation over n is alsosuppressed.

0 imi (RRn n %mn JnJnm) 2 [ ETE]

i nml ZZslC+ e.2,. ,2- 2 [a ETM] (C-1)

inm )n 2 nm k

2 Jn(Jnm/a)Jn(jnmPs/aL) -P$ njn Jn(Jn'p/a)JETE=00" PPS a P

nj ~ ~ ~ sa ' J'j'/))j' a 2nm n(3nm /~n(3nms/a nm2n J '' + ̂ ^ J(jp/a)J'(j'p/a)a a nJnm nJ nms

(C-2)

117

2.2n~ ~ ~ in/a)- nm~n nm n_______~EM ~ ~ J;(jnmP/a)Jn-(jnm s /a)-$ nm -a J Vnmp/a) P

+pz ia sgn(z-z )2! Jinm m/)J~ mnm Sa J.im/~nimsa

2~n jn J(ji~a) p/a) 2A J2 J(ip/ a)Jn(jnmPs/a)

am p n nm s+ap 5

ja 2'mps/a)+#n

a p n P

ian 2~ Cin p/a) j p a

nm -7 gn(z-z) .2 U p i~np/a)

ia sgnnmJ(J) p/)n3_^A nm snm l~n~s

a

+jj -.!2 zj ( ) p/ )j 2 (j p /a) ( 3a^ n nm n nms

mzz - nm i~mpaJnmsa (C-nm3n

(C-4)

118

n jnm Jn(irim/a)IaUHTMP i~nm z z5 a p nnns

lctwisgnz-z) 2 Jn(inmp/a)Jn(jnmps/a)

PZ-nm n nm Jipa

a -P Jn nmps/

a

sgnz-n) 2J'jpaa(j p Ia)nm m n nis

ia~nms gn(z-z5) a J.(j nmp/a) p m

inm Iz '(nm/ani nm~s/a) (C-5)

GHTE pp icin,'sgn(z-z) a Jn ,np/a) PS(~ ~5 a

-P^ nm snm

a2 Jn(npaj'j~sa

',P1nm s PPS

_A^ 'n Ji (j r~ /a) ' r p Ia)44 -a s ~nm pg~- )jIJ

.nj *2 j (jlp /a)+ -fra Jn(ir~/a) n rnp s

a '

119