AD-A112 371 OO STATE UNIV COJ.UMBUS ELECTROSCIENCE LAB F/6 20/3HIGH FREQUENCY SCATTERING BY CURVED SURFACES.(U)PU 74 P H PATHAK, R J MARHEFKA. W D BURNSIDE N6226--72-C-0354

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HIGH FREQUENCY SCATTERING BY CURVED SURFACES

P. H. PathakR. J. MarhefkaW. D. Burnside

TECHNICAL REPORT 3390-5

June 1974

Contract N62269-72-C-0354

1 Naval Air Development CenterWarminster, Pa. 18974

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HIGH FREQUENCY SCATTERING BY CURVED SURFACES Technical Report

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-ESL 3390-57. AUTHOR(e) S. CONTRACT OR GRANT NUMBER(&)

P. H. Pathak Contract N62269-72-C-0354R. J. Marhefka

* W. D. Burnside9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

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Department of Electrical EngineeringColumbus, Ohio 43212

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19. KEY WORDS (Continue on reverse side it necessary and identfy by block number)

Curved surface diffraction Wing mounted antennaHigh frequency analysis Fuselage scatteringTransition regionNear zone scattering

~ 20. ABSTRACT (Continue on reverese ide It necessary and Identify by block number)

This report deals with the geometrical theory of diffraction (GTD)solution to the problem of plane wave high frequency (h.f.) scattering bya smooth, perfectly-conducting cylindrical surface. According to the GTD,

1 the surface diffracted rays accurately describe the field behavior deep inthe shadow region behind the obstacle; whereas, the incident and reflectedrays adequately describe the field in the lit region sufficiently far fromthe optical shadow boundary. These ray solutions fail within the trans-ition region adjacent to the shadow boundary; hence, an approximate h.f.

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I20. solution is developed for describing the field within the transition region.The usefulness and the accuracy of this transition region solution isillustrated by analyzing the h.f. plane wave scattering by a 2-0 cylindri-cally capped thick half plane, and a 2-D circular cylinder (for which anexact solution is available for comparison), respectively. The presenttransition region solution is in a form which facilitates its generalizationto treat 2-D, and 3-D convex surfaces of non-constant curvature. A simpleexample of this generalization is illustrated by analyzing the h.f. planewave scattering by a 3-0, hemispherically capped semi-infinite circular

tpcylinder. The work described in this report is important to the analysis ofthe problem of h.f. scattering by a finite elliptic cylinder excited by afinite source; this problem is directly relevant to a study of the effectof an aircraft fuselage on the performance of wing mounted antennas.

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I7

ABSTRACT

This report deals with the geometrical theory of diffraction (GTD)solution to the problem of plane wave high frequency (h.f.) scattering bya smooth, perfectly-conducting cylindrical surface. According to the GTD,the surface diffracted rays accurately describe the field behavior deep inthe shadow region behind the obstacle; whereas, the incident and reflectedrays adequately describe the field in the lit region sufficiently far fromthe optical shadow boundary. These ray solutions fail within the trans-ition region adjacent to the shadow boundary; hence, an approximate h.f.solution is developed for describing the field within the transition region.The usefulness and the accuracy of this transition region solution isillustrated by analyzing the h.f. plane wave scattering by a 2-D cylindricallycapped thick half plane, and a 2-D circular cylinder (for which an exactsolution is available for comparison), respectively. The present transitionregion solution is in a form which facilitates its generalization to treat2-0, and 3-D convex surfaces of non-constant curvature. A simple exampleof this generalization is illustrated by analyzing the h.f. plane wavescattering by a 3-D, hemispherically capped semi-infinite circular cylinder.The work described in this report is important to the analysis of the problemof h.f. scattering by a finite elliptic cylinder excited by a finite source;this problem is directly relevant to a study of the effect of an aircraftfuselage on the performance of wing mounted antennas.

I

1| ii

I

1I

TABLE OF CONTENTS

Page

I. INTRODUCTION 1

If. HIGH FREQUENCY SCATTERING BY SOME SPECIAL2-D, SMOOTH, CYLINDRICAL SURFACES 2

A. Transition Region Solution for the H.F. PlaneWave Scattering by a Circular Cylinder 6

B. H.F. Scattering by a Perfectly-Conducting2-D, Rounded-End Thick-Screen 23

III. HIGH FREQUENCY SCATTERING BY A 3-D,

SMOOTH CONVEX SURFACE 39

IV. CONCLUSIONS 47

Appendix

I EVALUATION OF THE INTEGRALS FOR THEFIELD IN THE TRANSITION REGION 49

II KIRCHHOFF DIFFRACTION BY A HALF-PLANE 55

REFERENCES 60

iii

h A .. . .....

II

j I. INTRODUCTION

On-aircraft antenna pattern performance is very much dependent on

I ithe scattered field from the fuselage. This point was emphasized in the

analysis of fuselage mounted antennas[l]. For certain sectors of the

Ivolumetric pattern, the fuselage may even play a more dominant role forantennas mounted off the fuselage. This is especially true in the

' Jazimuth plane for wing mounted antennas. For many years scale model

measurements have been used to study these effects; however, such

measurements are both time consuming and expensive. On the other hand,41 numerical solutions have been successfully applied for the past few

years to study designs and locations for fuselage-mounted antennas.

i Fundamentally, the analyses developed for fuselage-mounted antennas can

* ibe extended to handle off-fuselage antennas.

Based on the numerical techniques previously developed[l,2,3], it

is proposed that similar solutions be investigated for wing-mounted

antennas. Specifically, the near-zone scattered field solutions for a

I finite flat plate and finite circular cylinder have already been

developed[l]. However, these models are not as versatile as desired.

Further the finite cylinder is limited to electrically small radii due

to use of a modal solution. Consequently, it is proposed that a high

frequency solution for sources in the near zone of a finite elliptic

I cylinder be employed. This model allows for variations in the cross-

section of the theoretical model and is not limited to lower frequencies,

i.e., to electrically small bodies.

1 iThe theoretical solutions necessary to analyze this problem are

developed in this report in terms of the solution to the scattering by a

two-dimensional circular cylinder. The geometric optics solution

satisfactorily handles the lit zone; whereas, the geometrical theory of

diffraction can be used in the deep shadow, However, both solutions fail

on and near the shadow boundary; hence, an approximate asymptotic high

II

frequency result is developed for analyzing the field within the transitionregion adjacent to the shadow boundary. Further, the solutions developed

herein can be extended to handle arbitrary curved surfaces including the

three-dimensional case. These solutions will provide the necessary

theoretical background to complete the study of a finite elliptic cylinderwhich will be treated in a future report.

The canonical problem that is treated involves the calculation of

the total high frequency field which is observed at a finite distance

from a convex, cylindrically shaped diffracting obstacle when it is

illuminated by an incident plane wave. This problem is directlyequivalent to the relevant problem of calculating the radiation pattern

of a source located at a finite distance from the same smooth, convexdiffracting obstacle via the reciprocity theorem for electromagnetic

fields.

II. HIGH FREQUENCY SCATTERING BY SOME SPECIAL2-D, SMOOTH, CYLINDRICAL SURFACES

This section deals with the high frequency scattering of electro-

magnetic waves by a smooth, two-dimensional perfectly-conducting cylindrical

surface. In this study, special emphasis is given to the region near theshadow boundary. A solution is desired which can be generalized (based on

high frequency approximations) to treat the scattering by three-dimensional

surfaces without having to solve additional canonical problems. The general

problem of the diffraction of waves by smooth, convex surfaces for grazingangles, (i.e., in the transition region adjacent to a shadow boundary) is

difficult to solve and it has been the subject of investigation by variousauthors[4] who considered some general cases; however, owing to the mathe-

matical complexity of their solutions, their results do not appear to be in

a form tractable for application to engineering problems of practical

interest. For the special case of high frequency plane wave diffraction by

a perfectly-conducting circular cylinder, Goriainov[5] has obtained results ii'

ii

I!

for grazing angles which are in a form suitable for numerical calculations.

Wait and Conda[6] extended Goriainov's high frequency results to treat the

plane wave diffraction by a circular cylinder with an impedance type

boundary condition; they also included the case of spherical wave incidence.

Most of the previous works on curved surface diffraction for grazing angles

Iincluding the work in References [5] and [6] make use of Fock integrals

introduced by Fock in his fundamental work on diffraction by smooth surfaces

I(many of Fock's papers are listed in Reference [4]). Wait and Conda[6]

indicate that their high frequency (h.f.) solution for the plane wave

I diffraction (for grazing angles) by a circular cylinder can be interpreted

as being the superposition of the result based on Kirchhoff diffraction

theory, and a second term which is viewed as a correction to the Kirchhoff

theory. This observation by Wait and Conda[6] constitutes the physical

basis for the development pursued in this report to arrive at a result

*, which could be simply extended to treat the h.f. scattering (in the

transition zone) by arbitrary 2-D and 3-D smooth, convex, conducting surfaces.

* IThe decomposition of the field scattered by the 2-0 cylinder into a sum of

a Kirchhoff diffraction term and a surface diffracted term (viewed as a

correction to Kirchhoff theory) as observed by Wait and Conda[6] was also

indicated by Fock (see pp. 194-196 of Reference [6]). In his work on the

3-D scattering problem of the Fresnel diffraction by a sphere for the special

case of source and observer raised to small heights above the sphere, Fock

expressed the scattered field in terms of a superposition of two integrals;

one of these integrals happened to be independent of the electrical properties

of the surface, and the other explicitly contained the surface properties

, Iand its associated electrical properties. Fock then showed that the integral

for the scattered field which is independent of the electrical properties

* I reduced to a Fresnel edge diffraction term; this term corresponds toI Kirchhoff diffraction. On the other hand, the integral which depended on

" the electrical properties of the surface was shown by Fock to yield the

_| surface diffracted term (which is viewed as a correction to Kirchhoff

diffraction). Consequently, it is believed that the basis of our analytical

j development given in this report is justified in view of the above remarks.

!3

Since the circular cylinder has two penumbral regions on its surface

as shown in Fig. 1, one needs to isolate the contribution to the field

y /"P (p,4)TRANSITION FIELDSREGION LIT / POINT

. (REGION

SHADOW __Pu

BOUNDARY

SHADOW REGION 0i

L . PENUMBRA

SSHADOW

BOUNDARY p

TRANSITIONREGION

Fig. 1. Geometry for the plane wave scatteringby a circular cylinder.

from each of these penumbral regions in order to allow an extension of

the circular cylinder analysis to treat the h.f. waves diffracted from

a single penumbral region, such as that which occurs on a smooth convex

surface of the nose of an aircraft when it is illuminated by a wing

mounted antenna. One notes that the angular extent of the penumbral

regions and the transition regions (which lie adjacent to the shadow

boundaries) is approximately 2(ka/2) 1 / 3 in radians (ka = circumference

of the cylinder in wavelengths). In Section II-A below, it is shown that

the h.f. field contributions arising from each of the two penumbral regions

on a circular cylinder can be identified, and separated in a manner whichenables one to calculate the h.f. field within a transition region which is

4

j . .... .

I

'~I

I associated with a single penumbral region on the cylinder. Such a decompo-

sition of the circular cylinder solution facilitates the analysis of the

problem of h.f. plane wave diffraction by the convex portion of the 2-0,

perfectly-conducting, rounded-end, thick-screen configuration of Fig. 2.

-: j TRANSITIONREGION

LIT REGION

SHADOW Iu

BOUNDARYP\ PENUMBRA

J2 0 d1

S HADO WI REGION

G O 00

Fig. 2. Geometry associated with the plane wave scatteringby a cylindrically tipped thick half-plane.

D One notes that the configuration of Fig. 2 consists of a half cylinder of

radius (a) which is smoothly joined to a thick half plane whose thickness

is 2a. The curve defining the boundary of the configuration in Fig. 2 is

not closed, and therefore there exists only one penumbral region on the

. j convex portion of this scattering geometry as illustrated in Fig. 2. The

results for the h.f. scattering by the canonical configuration of Fig. 2

j are given in Section II-B; it is conjectured that these results can be

extended in a systematic fashion to treat the h.f. scattering by arbitrary

j convex conducting surfaces if one employs the hypothesis valid at high

I5

I

II

frequencies, namely that h.f. diffraction and reflection are localphenomena (i.e., h.f. diffraction and reflection mechanisms are localized

to the immediate vicinity of certain points on the surface of the scatterer,

which correspond to points of diffraction and reflection, respectively). Anextension of the results in Section II-B for the 2-D canonical configuration

2 of Fig. 2 to analyze the problem of h.f. scattering by a 3-D, perfectly-

conducting hemi spherically capped, semi-infinite circular cylinder geometry

is indicated in Section III. The more general extension of the results of

Section II-B to handle arbitrary curved surfaces will be the topic of a

future investigation.

A. Transition Region Solution for the H.F. Plane WaveScattering by a Circular Cylinder

Let the total field, u in a region external to the 2-D perfectly-

conducting circular cylinder of Fig. 1 be the superposition of the incident

field (ui ) and the scattered field (us ). In the case of TMz excitation,the electric field associated with the incident plane wave (ul) is entirely

parallel to the axis of the cylinder (i.e., along z); furthermore, thetotal electric field denoted by u is also entirely z-directed. Similarly,

ifor the TEz excitation of the cylinder, u and u represent the incident

and total magnetic field intensities, respectively, which are entirely

z-directed. Let us denote the value of u for the TMz case, and let uh

denote the value of u for the TE case. Then, us = ui+s satisfies the 2-Dh h

homogeneous, reduced wave equation in the region exterior to the cylinder,

i.e.,

(1) (V2 + k2) us = O,h

subject to the boundary conditions;

(2a) us =0 on the cylinder,

or

6

I I73

I (2b) h - 0 on the cylinder.

In addition,

(3) us must satisfy the Sommerfeld radiation condition as p-p for anh

I eiWt time convention which is assumed and suppressed.

I The incident plane wave field is given by:

i =ikx(4) us e

h

The quantity k present in the above equations refers to the free space

S I wave number. Since Eq. (4) satisfies the reduced wave equation, it

follows from Eq. (1) that us must also satisfy the reduced wave equation.

IAn application of the 2-D Green's theorem to the region surrounding the

cylinder of Fig. 1 yields: ui 127T G 0o(PIP-) GoP p ) uP )

(5) u(P) = u (P) + joiG (PI'IP-) p =a ado'

where G (PIP') is the 2-D free-space Green's function:

1 (6) Go(P)P') - LH(2 ) (k- o').

0 ~ 4- o

' The parameter (-) denotes the field point position vector of P = P(po);

and P' denotes the cylindrical surface position vector of P' = P' ')

with p' = a on the cylinder. Clearly, the integral term on the RHS of-'Eq. (5) is the scattered field us . Then, from Eqs. (2), (3) and (5)

(7a) uS(p) = _ (P IP ') au ' 5 =aS 0 p

1 and

I (7b) uS(p) = 27r (P') .p,-ado',h o [hp=a

1 7

I

with the understanding that

i S(8a) us= u + u for the TMz case,) s u

and

(8b) uh = ui + us , for the TEz case.

It is well known[4,5] that a valid asymptotic h.f. approximation

for the surface field which is induced within the penumbral regions on

the surface of a circular cylinder, when it is illuminated by a plane wave,

is given by the following set of equations:

aua 2 -ikaio ,~ -ikaS](9a) ka - ( k e + J(4) e 2!L 'ii p'=a )=0 k L 2

and

SFA -ikal -ika2

(9b) FUhiP1 9 g(Cl) e I + g(92 ) e LL bhp'=a LOL

The functions which appear above are the universal Fock functions defined

by ;A(10a),(10b) e- d= gKz er1 i2 t 4-'r'J1 ~ d-r

in which the Fock-type Airy function w2(T), and its derivative w2(T) are

defined by:

_iaIb 1-E e" dz ; w; (.) -z e 3dz.(lla),(llb) w2 (i) - J r2 -~T r2 ~e z

The Fock parameters 41 and t2 are given by:

(12a),(12b) *1 = aZ/ 3 ; ; 2 = _ 2

8

II

with = - Tr/2 + 2i7r, and 2 = 37T/2 - + 2iri. The contours of

integration rI and r 2 for the integrals in Eqs. (lOa,lOb) and (lla,llb)

I are shown in Fig. 3.

V I- T OR ImZ

rI12 Rer OR ReZ

I

I Fig. 3. Contours of integration for the Fock and Airy functions.

The two terms on the RHS of Eq. (9a) and Eq. (9b) are commonly viewed

as two oppositely traveling creeping waves (associated with the surface

current) on the cylinder which have encircled the cylinder k times. If

I (ka)I/31r >> 1, the effect of the multiply encircling terms is negligible

and the inclusion of only the I=0 term is sufficient. In this report, one

is concerned with eventually applying the appropriate circular cylinder

results to the semi-infinite geometrical configuration of Fig. 2; hence,

it is meaningful to retain only the Z=0 term in the present analysis.

I Employing the far-zone approximation to G (PIP') leads to the3aG 0(PIP,) 0following results for G (PIP') and 5p,

[G (PP) i e- (kp -T/41 e ikacos( -O')(13) [Go(P P')]p':a 2 -

I9II

(14) [DG O(PIP'fl cosk 2 ' e- i(kp-ir/4) e ikacos(o-o')

The far-zone approximation implies that kip - pl' >> 1 and p >> (P'=a) in

Eqs. (7a,7b).

It is now possible to obtain asymptotic approximations to theintegrals in Eq. (7a) and Eq. (7b) for the scattered fields subject tothe restrictions indicated in Eqs. (13) and (14). Employing Eqs. (9a)and (13) in Eq. (7a), and expanding the exponential part of the integrandabout a stationary point allows one to write the integral in terms offunctions which are tabulated; thus, one is led to the following result for

< < (ka)-1/3.

(15) u5(P) e -icf2 ei~r/4 sik~-l +

2w/ ~ ka~l3 ~ -ika p -ika ijj

A similar asymptotic solution for the integral in Eq. (7b) is obtained byincorporating Eq. (9b) and Eq. (14) in it's integrand; thus,

(16) us(P) -eikp LF e' i/ 4 sinka(o-r) 1 +

+ e iwf/4 2 ( k a)1/3 *( ) e -ika * I + q* ( 92 e - i 1

The results indicated in Eqs. (15) and (16) were obtained earlier byGoriainov[5], and Wait and Conda[6). The universal functions p*(C) andq*(4) are the compl.ex conjugates of the p(4) and q(4) functions defined

by Logan[4,5,7) (p*(t) = G(t,q+-) e in/4 and q*(t) = G(4,q=O) eifl/ 4,

where G(4,q) has been introduced by Wait[6]). The functions e- -i n/4

* 10

and e- i / 4 q*(4) are plotted in Figs. 4a and 4b, respectively. The

I arguments 41 of the p*(41) and q*(41) functions are given by:2 2 2

* 0.8

0.6

1 0.4-

1 0.2-

1 §0

CL

1 -0.2-

* 1 -0.4

1 -0.6

-0.8 I-3 -2 1t 0 1 2

Fig. 4a. Plot of e ~/ *4 s4bsdoLogan's tabulated data[7] for p(t).

0.6

0.4

0.2

0 -

-1.01-0.4

-0.6 .1

-0.8 1

-3 -2 -I 0 I 2

-ilT/4Fig. 4b. Plot of ei q*(C) vs 4 based onLogan's tabulated data[7] for q(&).

(17a,17b) tl : ( a )1/3 1 ; = (_a)l/3 '2

with Tr1 - , and '2= -

The details of the asymptotic approximation to the integrals in Eqs. (7a)

and (7b) do not appear to be readily accessible in the literature, and are

therefore given in Appendix I for the sake of completeness.

It has been pointed out by Wait and Conda[6] that the first term on j

the RHS of Eqs. (15) and (16) corresponds to Kirchhoff diffraction;

whereas, the remaining terms may be viewed as a correction to the Kirchhoff

12

I

Itheory. The Kirchhoff result is independent of the electrical properties

of the surface; it is obtained by assuming that the field scattered

by the cylinder of Fig. I is equal to that scattered by a strip of the

same geometrical cross-section (i.e., strip width = diameter of the

cylinder) which is located at x=O as in Fig. 5a. It is possible to provide

j Ia ray interpretation for the scattered fields given in Eqs. (15) and (16);

the Kirchhoff term is composed of the field associated with the Kirchhoff

3'' Idiffracted rays which emanate from the edges P1 and P2 (of the strip) as

illustrated in Fig. 5a; whereas, the terms involving p*( I) and q*( l )

are the fields associated with the ray which is launched by the incident

I field at P1 and shed along the forwiard tangent at P3 after creeping adistance PIP on the cylinder (PIP 3 = a lPi ) as illustrated in Fig. 5b. A

I pseudo-ray interpretation[8] is possible for the terms p*(4 2 ) and q*( 2)

as illustrated in Fig. 5b, wherein the ray associated with these terms is

I launched at P2 ' and is shed along the backward tangent at P4 after creeping

a distance P2 P4 on the cylinder (P2P4 = aj 4 2 1 = aj-,); this ray system

unlike the ones described above, does not satisfy the generalized Fermat's* U principle which is employed in Keller's geometrical theory of diffraction[9],

*but it is nevertheless useful in visualizing the geometrical coordinatesI ikaV 2 -ikaV 2which are associated with tne terms p*(g 2 ) e and q*(2 ) e . In

Fig. 5a, u k(P) denotes the Kirchhoff edge diffracted field emanating from1I Pl; similarly, u4P) denotes the field arising from the edge P2 so that

SI (18) u (P) + u (P) -sinka(o-w) F2 iff/4 e - i k p((_8) U I (P + '(p for o Tr

I corresponds to the Kirchhoff result which is contained in Eq. (15) and Eq.

(16), respectively,

One is now able to identify the scattered fields which arise from

* 3 the two isolated penumbral regions shown in Fig. 1. Specifically, the

scattered field at P (in the close vicinity of *=r) arising from the

I penumbral region associated with P1 is:

1 13

_7 7

,-,-STRIP OF WIDTH 20

2 P2 (0,-0

(a)

Y

q*(Ca)

(b)

Fig. 5. Physical interpretation for the results in Eqs. (15) and (16).

14 71

di

IP' I

(Iga) u (P,P 1 ) u,(P) - e 1 4 2 (-a)1/ P_(,) e Ia e

I for the TMz case and o ;,

s( ki T -ikIa, ikp(19b) Uh(Pl) Ul(P) e k (-)1/3 q*(4i) e I e

Ji for the TEz case and o I

Similarly, the scattered field at P emanating from the penumbral region

I associated with P2 is:

-ika ikp! (20a) Us(P' ,(P 2) - * -- p 2 ) e ,

for the TMz case and 0

k i~r/4 J-2-ik ikP(20b) us(P,P 2 ) -k e-i/ 4 F(a)l1/3 q 2)e 2 e

for the TEz case and •

Re-writing the RHS of Eq. (18) for 0-", one obtains the following:

(21) sinka(4-l) e iTr/ 4 e-ikp e - ika(0--n)

e ika(0-iT) 1 e -i7/4 e- i kp

(0 (-7t) J Y_ VlPk

One is now able to associate the first term on the RHS of Eq. (21) with

u ; likewise, the second term on the RHS of Eq. (21) corresponds to U2 .1 e ika( -) ot

3 However, the individual terms ± (0_7), on the RHS of Eq. (21) are

unbounded at O=w (even though the LHS of Eq. (21) remains finite);

i consequently, they must be replaced by the bounded Kirchhoff result for

15I

.. . .--- II . ., = l_ ". , -_..:L-- .. . . ._ I" ,

.14

ku which is deduced from the result for the Kirchhoff diffraction by a

conducting half-plane. The u term is now viewed as being the h.f. field

diffracted from the edge P1 of a half-plane located at x=O for --<y<a.

Similarly, the u2 term is viewed as being the diffracted field from the

edge P2 of a half plane located at x=O for -ay<a. In doing so, one assumes

that the h.f. fields diffracted by the strip in Fig. 5a is the superposition

of the fields associated with rays diffracted from the edges P and P2 of

two overlapping half planes (with their common domain of overlap = width

of the strip); this assumption is reasonable since ka is assumed large. The

Kirchhoff result for the field diffracted by the edge P1 of a half-planewhen a plane wave is normally incident on it is given in terms of a KirchhotffJ

edge diffraction coefficient, Dk as

e-ik (p-asi n¢)k i e..kPs ~(22) u, (P) r ui(P 1) DR()

where ui(P) is unity, and

(23) DR(q) - e f 7 (- 2-) F(X)

with

(24a) F(X) = 2iJ ]vX-,e e-XT dT

and

(24b) X = 2kpcos .

ul(P) of Eq. (22) with Dk(¢) as in Eq. (23) approaches the following

result as O-f:

(25) Uk (P) e'ika(1") \F e-iir/4 e ikp F(X)

16

I

<Iwhich is identical to the first term on the RHS of Eq. (21) except for

3 the additional factor F(X).

I F(X) - (VX - 2Xe i T/ 4) ei(Tr/4 + X

as p-*1, thereby ensuring a bounded result for ul(P) of Eq. (22) at the shadow

I boundary corresponding to O=iT. The derivation of the result in Eq. (22)

is summarized in Appendix II. The function F(X) was introduced by

SHutchins and Kouyoumjian[lO] to describe the h.f. diffracted fields in

the transition regions of the shadow and reflection boundaries associated

with a perfectly conducting wedge illuminated by a plane wave. A plotkindicating the behavior of F(X) is given in Fig. 6. The field uk(P) is

similarly expressed in terms of D k as

(26) uk () D-ik(p+asin4)2(P)_ u'(2 Dk(2k )--

w " h " ±ikasin >which is also finite at *=%. The exponential phase factors e in

Eqs. (22) and (26) arise from choosing the origin as the phase reference.

The results in Eqs. (22) and (26) facilitate the calculation of the

Kirchhoff diffracted fields uk and uk over a greater angular region about

the shadow boundary than just within the immediate neighborhood of the

shadow boundary at 0=w. It can be shown by a careful limiting operation

that a superposition of Eqs. (22) and (26) also leads to the result on the

LHS of Eq. (21) for € ir. In this limiting operation, the fields in Eq. (22)

| and Eq. (26) are initially assumed to emanate from P1 and P2 ' respectively

to the far-zone field point P (in the vicinity of o=w) along nearly parallel

ray paths which converge at P. Next, the field corresponding to this nearly

parallel convergent ray system is allowed to approach the parallel ray

system (of Fig. 5a) in the mathematical limit. One may thus employ the

field representations of Eqs. (22) and (26) for the uk and uk terms present

in Eqs. (19) and (20), respectively, to calculate the Kirchhoff diffraction

contribution to the scattered field us over a larger ingular domain aboutShs

0=7r. However, before the results of Eqs. (19) and (20) for us can beh

1 17

-t (S3380930) 3SVHdiAO a 0 in 0 o

Ln * W ) V) N N -n -0 0

100

t

01

.18

~I

extended to calculate the scattered field over this larger angular domain

about =ir, one must modify the terms involving p*(4) and q*(4) (which are

viewed as corrections to Kirchhoff diffraction) in a manner corresponding

I to that employed for ul in Eqs. (22) and (26). From Eqs. (A-10) and (A-24)

of Appendix I: 2

(27a) e() = L e-r V(t) d + 1

I v' r ~ QW2 ) . 2V

where

(27b) p*( if the operator Q = I

2q*(t) if the operator Q =

and 4 = 1 or 42" The function QV(T) is directly related to the Miller

type Airy function as described in Appendix I. Clearly, the Krchhoff

correction terms in Eqs. (19) and (20) may be expressed in terms of

I '( ) in Eqs. (27a) and (27b) as:

(2a)-ill/4~ , (2IkadI/3 i ^-ika p e -

e -in4 e- (ka)/3 e-ika e -ikp e i w dr -

(28al- L T4b),

- (unbounded Kirchhoff result of Eq. (21) for

with t =l corresponding to = l as before. The unbounded Kirchhoff

I result for 2 Oi is seen from Eq. (21) to be:

I )eika(o-w) e- i/ 4 e-ikp(28 b ) t - ') p

I The integral over the contour r1 in Eq. (27a) is unbounded at 4=0 (which

corresponds to the shadow boundary at *=f), but (4 ) in Eqs. (27a) and

I (27b) (and therefore the result in Eq. (28a)) is bounded[4,7] because the

I 19

I

singularity in the 1-term of Eq. (27a) exactly cancels the singularity

in the integral (over r,) when 4-0. It is then obvious that the Kirchhoffcorrection terms in Eqs. (19) and (20) can be appropriately modified to be

useful over a larger angular domain about the shadow boundary (=r), if the

unbounded Kirchhoff result for * in the close vicinity of 7 in Eq. (28a) is

replaced by the unbounded Kirchhoff result valid over a larger angular

domain about 0=7. This is accomplished by noting from Eq. (22) that the

-!factor F(X) keeps D(4) bounded at *=ff, and:

Cbounded Kirchhoff unbounded Kirchhoff xdiffraction field) (diffraction field x F(X).

Thus, Eq. (28a) which is valid for *% is now modified by replacing

-ikal 2 e-i kp

e- iw/4 2 (ka)l/3 () e ek2

VP-

with

-i kaW1

(280 e- i1/4 F ka)l/3 2 e-ikp

rp2 2rff t2

+ e-/' 4 s±l e -ik(masinf)sine ~p

Combining the Kirchhoff diffraction results of Eq. (22) and Eq. (26)with the Kirchhoff correction terms of Eq. (28c) yield an expression forus(P) which is useful over a larger angular domain about *=i (in contrasthto Eq. (19) and Eq. (20) which are good only in the close vicinity of O=w):

20

I (29a) us (P ,P1 E .i ±r F(X1)-l] eik(asin9))2 2ikV

I ~-ika*i

*~ 2 e e1 I___L 2 - 2 -J v~

Iwith X1 2kp cos 4/2, and X = 2kp co (21-ff4)/2.

fu 5(PP1) corresponds to the scattered field at P arising from the* 2

penumbral region associated with Pior P2. Also

us(PP 1) when P*( ,), for the TM~ case

I(29b) u5(PP 1) = 222 u h(PP), when (1 q*( ,), for the TE z case.

* 12 2 2The total hM. field at a point P arising from the vicinity of P1 on the

I cylinder may be written as:

U) + us(PP 1) on the lit side (O<ff)(30a) u5 ( P) u5(PP ) on the shadow side ( >n)

*for the TM zcase,

and

(3b) Uh(PP ) rWul(P) + uS(PP) on the lit side (O<w)(30) h(,P) ~'~IuS(P'P) on the shadow side (0>7)

for the TE z case.

The total M. field at P arising from the vicinity of P2 on the cylinderis similarly expressed as.

21

U + uS(P,P2) on the lit side (0<7)" :i (30c) Us(PP2) U

2) us (P 'p on the shadow side (>i)

for the TMz case,

and

su(P) + u(PP 2) on the lit side (0<7)(30d) u (P ) + h Uh(P

h ,2 u (PP2) on the shadow side (¢>r).

for the TE case.

The total h.f. field at P in the transition region is the sum of the

fields arising from the vicinity of P1 and P2 on the cylinder, thus:

(31a) u U(P)=us(P,Pl) + Us(PP 2) for the TM case

and

(31b) Uh(P) = Uh(P,Pl) + Uh(P,P2) for the TEz case.

The results of Eqs. (31a) and (31b) will be employed in part B which

follows, to calculate the fields in the transition region associated

with the canonical problem of Fig. 2. One notes that the results in

Eqs. (29a) and (29b) are based on a direct generalization of the results

in Eqs. (19) and (20). The accuracy of these results will be indicated in

the next section wherein they are applied to specific problems. Possibly,

a more accurate asymptotic h.f. result for uS(P,Pl) (when P is not2

necessarily in the close vicinity of 0=ir shadow boundary) obtained from

first principles might involve terms that are not a simple superposition

of the effective edge diffracted field term, and a surface dependent field

term as in Eq. (29a); such an asymptotic solution based on mathematical

rigor rather than on simplifying physical assumptions to bypass mathematical

complexities appears to be a difficult task. However, judging from the

22

numerical results obtained from the use of Eq. (29a) in the following

sections, it appears that the present solution yields sufficiently accurate

values for engineering applications.

B. H.F. Scattering b a Perfectly-Conducting2-D, Rounded-End hick-Screen

The canonical configuration of interest is illustrated in Fig. 2.

I It is desired to calculate the total h.f. field surrounding this 2-D

rounded-end, thick-screen configuration when it is excited by a normally

I incident electromagnetic plane wave of the TM or the TE type. Thesolution to this canonical problem is of relevance to a study of the

I effects of the aircraft fuselage on the radiation from wing mounted antennas,

U as indicated previously. Figure 2 illustrates the junctions J and J2

resulting from the smooth join of the half-cylinder onto the thick half

plane. In the present analysis, the diffraction effects arising from the

junctions J and J2 are not included for the following reasons:11) The actual fuselage of an aircraft will be approximated by

I this model for which these junctions will only be artificially

added to our model in that they are not present in the actual

* I aircraft structure.

2) The junction effects are important only if the penumbral

I region lies in the neighborhood of the junction, or if the field

point lies in the neighborhood of the reflection boundary associated

I with the junction. It is clear therefore, that in the present

problem, which deals with normal incidence and with aspects within

I and near the transition region, the junction effects are negligible.

" IThe total h.f. field in the lit region for the problem in Fig. 2 is*adequately described by the fields associated with the incident and

reflected rays of geometrical optics, whereas, the h.f. field in the shadow

123

I

region is accurately described by the surface diffracted rays of Keller's

geometrical theory of diffraction[9] (hence by the GTD). However, the

geometrical optics field description fails near and on the reflection

boundary (which is also the shadow boundary in this case). On the other

hand the GTD field description requires an increasing number of surface ray

modes to accurately describe the field near the shadow boundary thereby

making the GTD result cumbersome to employ. Thus, within and near the

transition region adjacent to the shadow boundary, one may employ the

result developed in part A to calculate the total h.f. fields there. The

specific results for the lit, transition, and shadow regions associated

with the canonical configuration of Fig. 2 are indicated below.

Field Description in the Lit Region - The total field in the lit

region is a superposition of the incident field u and the reflected field

ur predicted by geometrical optics, in which

(32a) ui = us eikX, for TMz case,h TEz

and,

(32b) u r = Ur = u (PR) R Pr e-iks

h h hr

where the subscript s is used for the TMz case, and the subscript h is

used for the TE case as before. The result in Eq. (32b) is in a ray-

optical form[ll] which is valid in the near zone. PR denotes the pointiTof reflection in Fig. 7a, and u (PR) is the value of the incident (TZ)

hzfield at PR* The reflection coefficient, Rs is defined by:

h(32c) Rs = - 1, for the TMz case,

(32d) Rh = + 1, for the TEz case.

24

_________________________________________ I

-I

II

0 U'

I

iS I Fig. 7a. Reflected ray geometry.

- s denotes the distance from R to the field point P in the lit zone. The*I quantity Pr within the radical in Eq. (32b) refers to the caustic distance(in the plane of reflection) associated with the reflected wavefront; r

• I is found to be:

a cos 0 0I (32e) Pr - 2The total field at a point P in the lit zone is thus given by:

u3(p) = a cos° -iks TMz

(32f) hsP u~)s us(PR) ( al) .acos o + 2s ; for TEz case.h h h 0I

25

I

If far zone results are desired, one must enforce the condition s >> P

in Eq. (32b). The result in Eq. (32f) is valid in the lit zone for aspects

sufficiently far from the reflection boundary (which occurs when PR coincides

with P) "

:* Field Description in the Transition Region - The total field in the

transition region may be immediately written down from Eqs. (30a) and (30b)

given in Part A. It is conjectured here that the fields in the transition

region of Fig. 2 are identical to those in Eqs. (30a) and (30b) which

arise from the penumbral region associated with P1 on the circular cylinder

of Fig. 1. This conjecture is based on the h.f. approximation which allows

one to consider h.f. diffraction as being a local phenomenon. The

results in Eqs. (30a) and (30b) for the TMz and the TEz cases, respectively,

are valid in the far zone; however, since h.f. diffraction is a local

phenomenon, one may also directly modify the far-zone results of Eqs.

(30a) and (30b) to obtain the following near zone transition fields at P:

(33) Us (P)--us (P-( )e sina 2:_ F •

h h "d

e-ika' e ie/4 ] Liks)l

with the understanding that '( ) P*(R) for u s(P) and (t) = q*( ) for

Uh(P); also H(ir-a) is the Heaviside step function whose value is unity if

c<ir and zero if a>w. The near zone result of Eq. (33) is obtained by

simply replacing the far zone ray distances by their corresponding near

zone ray distances. The quantities s, , and in Eq. (33) are given in

terms of the near field coordinates by:

)so in the lit portion of the transition region(34a) s in the shadowed portion of the transition region

26 I

. . . . ... . . . . . . . . . . . { , , -' -i I ...... .. . ..... . . ...L " ( ll l I]

iI

I (ka I/3 l-pp , in the lit portion of

(34b),(34c) - 2 ' the transition region.2 i s , in the shadowed part of

V the transition region.

where the angles @, and s and the distances s. and ss are indicated in

l jFig. 7b below. The F(X) function appearing in Eq. (33) is defined earlier

J P(LIT)

tI

a j

Fig. 7b. Equivalent ray geometries for the fields in the lit andl shadowed portions of the transition region associated :

IU

with the shadow boundary at P1.

l in Eq. (24a). The new argument (for near zone calculations) of F(X)

g is clearly:

(35a) X = 2kd cos2 ~

g 27

I

where a is the angle (measured in the cw sense) subtended by i- Kirchhoff

diffracted ray (directed along PP) and the negative of the incident ray

direction (i.e., along + x).

ig in the lit part(35b) = in the shadowed part.

The distance d is measured from P1 to P and

d in the lit partL4(35c) d

)d = in the shadowed part.

The quantities a., as, d and ds are illustrated in Fig. 7b.

The transition region solution discussed above must serve a two-

fold objective, namely that it should provide a sufficiently accurate h.f.

field description within the transition region; secondly, it should blend

smoothly with the geometrical optics solution in the lit region, and also

blend smoothly with the GTD solution in the shadow region. Some numerical

calculations which indicate the accuracy of the present results are provided

later.

Field Description in the Shadow Region - For aspects in the deep

shadow region, the scattered field (the incident field is zero within the

shadow region) is given a ray optical representation valid in the near

zone. The result based on GTD is given as[ll]:

Sh

h s .pa.i ka s s iks(36) u(P) (P) u(P ( e p Dh(P e

h h = h

~sS h a -iks TMi. S -pks'iasDh P

)e ,fr z

= h (Pl ) e P Ss) Dh(P , for TE case,

P=D z

28

I

since us(Pl) = 1. The geometrical details are given in Fig. 7c for the

3 h

, I ui

P, u

1 'P

I

Fig. 7c. Ray geometry for the GTD fields.

I surface ray modes which traverse a path PIPDP to the field point. Even

though the inclusion of an infinite number of GTD surface ray modes is

indicated in Eq. (36), it is generally found that sufficiently accurate

results for the field are obtained by summing only the first five modes

(corresponding to p=l to p=5), provided the field point P is far-removed

from the transition region. D and h are the cylinder diffraction coef-

ficient and attenuation constant for the p-th surface ray mode, respectively.

I D.R. Voltmer (in his Ph.D dissertation, Ohio State University, 1970) has

obtained diffraction coefficients and attenuation constants to orderI 2 ) 2/3 (here Pg = local radius of curvature of the surface along the ray

path) which are valid for smooth convex surfaces of non-constant curvature

j (this includes the circular cylinder, and the sphere as special cases); a

1 29

I

table of his diffraction coefficients and attenuation constants is, also,

available in Reference [8]. The superscripts s and h on the diffraction

coefficient and attenuation constant denote the TM and the TE cases,z Z

respectively.

Numerical Results - The numerical results for the near field patterns

of a circular cylindrically tipped thick half plane (as shown in Fig. 2)

illuminated by a TMz (acoustic soft) or TE (acoustic hard) plane wave have

been calculated via the formulas developed above. These results are

illustrated in Figs. 8 through 10. The geometrical optics field shown

as a dot-dashed line, consists of the incident and reflected fields up to

the shadow boundary. The transition field, shown as a solid line, starts

at 0=900 and is shown into the deep shadow region. The surface ray mode

field (obtained by summing over only the first 5 modes) is shown as a

dotted line and is valid in the deep shadow region. Note that the

geometrical optics field and the transition field blend well in the angular

range 900<o<1200 . The transition field and the surface ray mode field for

the hard cases blend in the neighborhood of 0=180°. The soft cases do not

blend as quickly. The phase for the above near field patterns also blended

well; however, it is difficult to illustrate and will not be shown here.

The validity of the above high frequency solutions is demonstrated

by applying them to the problem of plane wave scattering by a circular

cylinder (as shown in Fig. 1). The results are then compared with the

exact solution for the problem which is found from the eigenfunction

I If expansion in Figs. lla and llb. The high frequency solution is composed

of the superposition of fields from the two penumbral regions associated

with P1 and P2 using the appropriate results (of part B) in their

respective regions of validity. The good agreement lends confidence to

the present high frequency solutions.

30

GEOM. OPTICSK I-TRANSITION

000'SURFACE RAY MODES

>110

13

--- GEOM. OPTICS

-TRANSIT IONo.e.eSURFACE RAY MODES

.......

fa.

* Fig. 8b. Near field pattern for cylindrically tippedthick half-plane, soft case, ka=lO, kp=1OO.

32

7I7

--- GEOM. OPTICSI - TRANSITION

o o 9 SURFACE RAY MODES

V 10

Iw* 0

0- I2Iu

'I>_j-3

I Fig. 9a. Near field pattern for cylindrically tippedthick half-plane, hard case, ka=2O, kp=5Q.

*33

--- EOM. OPTICS-TRANSITION

s .. SURFACE RAY MODES

-1

Iw40

-t3

Fig. 9b. Near field pattern for cylindrically tippedthick half-plane, soft case, ka=20, kp=50.

34

--- GEOM. OPTICSI~- TRANSITION0000*SURFACE RAY MODES

V IV

.1101

-2

-3iI3

thick half-plane, hard case, ka=40, kp=100.

1 35

--- GEOM. OPTICS-TRANSITION

e SURFACE RAY MODES

Fig. l0b. Near field pattern for cylindrically tippedthick half-plane, soft case, ka=40, kp=1OO.

36

*900 EXACT SOLUTION

- - GTO SOLUTION

-1

I18000I 5 X FIELD POINT

IY0 TE INCIDENT

2 tx PLANE WAVE

Fig. Ila. Comparison of high frequency solution with* exact solution for near field pattern ofI circular cylinder, hard case, ka=20, kp=5O.

I3

90 EXACT SOLUTION-- -GTD SOLUTION

CID

50X

TM INCIDENTL'JAY4/71 X PLANE WAVE

Fig. llb. Comparison of high frequency solution withexact solution for near field pattern ofcircular cylinder, soft case, ka=20, kp=50.

38

III. HIGH FREQUENCY SCATTERING BY A 3-D,I SMOOTH CONVEX SURFACE

As developed in the previous section the transition solution can be

divided into two terms. These are the Kirchhoff term, which is independent

of the polarization and curved surface properties, and a second term which

is dependent on the surface curvature and associated electrical properties.

The usefullness of this solution is that each of these terms can be system-

I atically modified to three-dimensional surfaces (based on the high frequency

assumption that diffraction is a local phenomenon). One notes that the

geometrical optics and geometrical theory of diffraction solutions are

well-known for the three-dimensional case as presented in Reference [11].

S I In this section, the near zone scattering by a spherically capped

semi-infinite circular cylinder when excited by a normally incident plane

wave is examined as shown in Fig. 12. The object of this study is to

illustrate the three-dimensional nature of our high frequency solution and

to verify that it blends nicely with the geometrical optics field. The

field surrounding the configuration in Fig. 12 is given below in the lit,

I transition, and shadow regions in a manner similar to that indicated for

the canonical configuration of Fig. 2, in part B of the previous section.

3 Once again, the diffraction effects arising from the junction of the hemi-

1spherical cap and the cylinder are ignored for reasons indicated previously.

-I Field Description in the Lit Region - The geometrical optics solution

composed of the incident field (ul) and the reflected field (ur) observed

at a near zone field point P in the lit zone is explicitly given by:

(37) (P) i +u(PRs a2cSo e- iks TMz

(37) Us( R) (acoso +2s)(a+2scOSo forTE case.h h h h 0 0 z

The point of reflection PR and other geometrical parameters (such as a, s,

c'0 etc.) are the same as in Fig. 7a. The reflection coefficient Rs wash

39 1.II

! P( LIT)

o S*

_W U'

P (SHADOW)

Fig. 12. Geometrical configuration for the hemisphericallycapped semi-infinite circular cylinder.

previously defined in Eqs. (32c) and (32d). us(PR) denotes the value of

u at the point of reflection PR on the hemispherical cap.h

Field Description in the Transition Region - The total field in the

transition region for the 3-D geometry of Fig. 12 can be obtained by

generalizing the 2-D results for the geometry in Fig. 2 which are indi-

cated in Eq. (33). This generalization from 2-0 to 3-D is based on the

ray-optic description for the fields; analogous generalizations have been

previously indicated elsewhere[8,9,11,12] for certain antenna and scattering

problems. We obtain

40 .]

II

I (38a) u5 P%~PHi ~ + eiir/4 [F(X)-i] a e- ikd)

(F ('42(ka) 1/3 eika e- -ff4FW - I -iks f(s))

I where H(-c), '( ), F(X), a, s, d and i have the same meaning as in Eq.

(33). The quantity f(s) is given by:

b - in the illuminated region (a<)I (38b) f(s): . -

FdLn C 0 in the shadow region (ci'r).

The geometrical details are the same as those indicated in Fig. 7b. The

quantity "o is the spatial spread of the field on the surface of the

hemispherical cap along the ray path P1Ps of Fig. 7b. In particular, dnois the width of two adjacent rays at Pl; whereas, dn i the width of the

same two adjacent rays evaluated at P. The quantity is given by:

(38c) sinIe in the lit penumbra region (=i/2 - 14,,J) i.e., for 6<1

where 0 l{1tit h shdwpnmrwr e h in the shadow penumbra region (=-/2 + J*s1) i.e., for 6>1

Also, the caustic distance p appearing in Eq. (38a) is the radius of

curvature of the diffracted wavefront in the plane_L x-y plane, but contain-

ing the ray shed from the surface; it is given by

(38d) p = atane.I_Incorporating Eqs. (38c) and (38d) in of Eq. (38b) leads

to '~dn s P+s| to

1 41

I

1

(38e) s (as - ne+scosOe

Thus, p>O when e=elit, whereas p<O for e=esh. If

-F+s) 0

then

sP+s) ei/

because a caustic has been crossed. One notes that FLdn has been replacedby unity in the lit part of the transition region (see Eq. (38b)) because

it is conjectured that the fields scattered into the lit portion should

be independent of the spatial spread factor of the rays along PIPL on the

surface (this fact is directly evident in the reflected ray description for

the lit region).

In order to keep the Kirchhoff diffracted term bounded one notes

that the X in F(X) of Eq. (38a) should be

=2k [~P+d)](38f) (e2d) cos 2L/2.

L ~e J

The quantity pe is the caustic distance for the Kirchhoff diffracted ray

in the plane containing the diffracted ray and the edge of the curved

screen at Pl; it is computed easily via the formula for caustic distances

for curved edges given by Kouyoumjian[12].

Field Description in the Shadow Region - The iffracted ray field with-

in the shadow region at a field poin + P in the near zone is obtained by

extending the 2-D result of Eq. (36) to the 3-D result in the usual manner[ll].

42

6L

Iss haS k S d

h S- d 0A5 (39) us(P),,tj(P) 0 ~P)hP)P j ijn is .oPeis3 h h ~~h p n

H TMzIfor TE case

Sz

s h a' s-h - s ha -iks

where u I(P I,

h

and dn r- a

dln -ST-P- ') s (asi ne+scose)

as noted previously in Eq. (38e). The diffraction coefficient Dpand theh

ateuto constant ah are indicated in Reference [8] for the sphericalp

surface. The geometrical details pertaining to the GTD ray description ofI Eq. (39) for us(P) are the same as those in Fig. 7c. If

1 h

sP~s < 0,STTS

I then 0 p

1must be interpreted as

due to the crossing of a caustic.

43

It is evident from the results in Eqs. (38a), (38b), and (39) that

a caustic of the diffracted rays is present for =Tr (and a=-sin - ain

Eq. (38a)). One notes that on crossing the caustic which occurs at O=w

(i.e., along the negative x-axis),two additional diffracted rays which

creep around either side of the cylindrical surface also contribute to the

field at P; the field of these rays is easily calculated by GTD. The

caustic field may be assumed to result from the radiation by equivalent

half-ring electric and magnetic currents in which the half-ring is formed

by the locus of the points of diffraction of the rays shed from the

hemispherical cap to the field point P for which O=w. The equivalent ring

current concept[8,14,15,16] proves to be a useful tool for evaluatingthe proper field behavior at and in the vicinity of a caustic.* Such an

alternative field representation for the caustic region is necessary

because the simple field description of Eqs. (38ab) and (39) fails in

this region (the results of Eqs. (38a) and (39) become infinite at the

caustic). The transition from the single diffracted ray arriving at P

in the shadow region (whose field is given by Eq. (39)) before crossing the

caustic to the three diffracted rays which arrive at P in the shadow region

after crossing the caustic (the three rays result from the ray corresponding

to Eq. (39) shed from the hemispherical cap to P plus the two additional

rays which creep around either side of the cylindrical surface before

being shed towards P) will not be treated in the present report. This

transition is described by the solution obtained from the equivalent

current concept for the caustic region. The caustic field analysis will

be completed in a future report.

Numerical Results - The near field patterns for the hemispherically

capped cylinder of Fig. 12 for a radius of ka-lO and observation distance

of kp=lO0 are presented in Figs. 13a and 13b, when this structure is

illuminated by a TEz or TM type plane wave, respectively. The variousz zfields are represented as done previously for the 2-D problem. As can be

seen the transition solution blends well with the geometrical optics fieldfor both the hard and soft cases. The presence of the caustic at *=w,

*A study of the caustic due to the curved screen Kirchhoff diffraction term(i.e., for a=sin-1 a/d in Eq. (38a) when O=w) will also be completed in afuture report.

44

GEOM. OPTICS

-TRANSITION

e e *.. SURFACE RAY MODES

510

I Fig. 13a. Near field pattern of hemispherically cappedcylinder, hard case, ka=lO, kp=1OO.

1 45

__GEOM. OPTICSTRANJSIT ION

**@SURFACE RAY MODES

t-t

IC

iw

Fig.13b Nea fild ptten ofhemspheicaly cppecyidesotcae k~Ok0lQ

46

20U

however, interferes with the blending of the transition solution with the

I field of the surface ray modes. As stated above, the singularity due to

the caustic can be easily corrected; however, this has not been done here

I but will be completed in a future report. When this caustic correction is

incorporated, it is expected that the blending of the transition solution

with the surface ray GTD solution will be as good as that indicated earlier

for the 2-D problem.

Ii IV. CONCLUSIONS

The object of this report has been to develop the complete solutions

I for the fields scattered by curved surfaces illuminated by near zone

antennas. Specifically the high frequency solutions for curved surfaces

*have been completed for several years except for a transition solution

which is needed around the shadow boundaries. An approximate transition

g region asymptotic solution is presented and developed in such a form that

| it can be extended to arbitrary curved surfaces. Combining this solution

with the geometrical optics term in the lit region, and the geometrical

1theory of diffraction term in the shadow region, should provide a completehigh frequency solution for the scattered fields of antennas illuminating

curved surfaces in the near zone.

j The solution is presented in terms of the thick half plane with

a cylindrical cap showing how the various terms blend together in each

region. The transition region solution developed here (Eqs. (33) and

(38a)) can be shown to analytically reduce to the GTD surface ray mode

solution as F1X)iI in th6 shadow region sufficiently far from the shadow

I boundary. de have not yet shown that this transition solution can

analytically reduce to the geometrical optics reflected field as F(X)4I

j in the illuminated region sufficiently far from the shadow boundary.

However, the numerical results presented indicate that the transition

j region solution does indeed blend very nicely with the GTD and the

I47

geometrical optics solutions. Although the present transition region

solution appears to be fairly accurate, it would be desirable to obtain

a transition region solution from a more rigorous asymptotic analysis of

the canonical circular cylinder scattering problem.

Even though the solution presented describes the one term as being

a Kirchhoff solution, one can, also, show that this term can result

using the GTD edge diffraction coefficient instead of the Kirchhoff edge

diffraction coefficient. Using this approach the values of the transition

solution do not change much near the shadow region, but they might provide

improvements away from the shadow boundary. This approach will be investi-

gated for the spherically capped cylinder in the caustic region in that

in this sector the Kirchhoff and GTD solutions are observably different.

While the transition solution is being completed, the solution for

a finite elliptic cylinder will be developed. This solution will provide

a versatile model for the fuselage when the antenna is not mounted directly

on it, i.e., such as wing or pod mounted antennas. It can, also, be used

to examine the scattering effects of stores, etc. With the completion of

this solution, one should have all the necessary theoretical tools to

analyze any high frequency on-aircraft antenna pattern using a reasonable

model representation of the actual aircraft.

48

-- i

S I

IAPPENDIX I

EVALUATION OF THE INTEGRALS FOR THEFIELD IN THE TRANSITION REGIONl

* A summary of the details concerning the asymptotic evaluation of

l the integrals in Eqs. (7a) and (7b) of Section II is provided in this

appendix. The TM case (correspondinq to Eq. (7a)) is considered first;* z

it is followed by the TE case (of Eq. (7b)).

TM case: Employing Eqs. (9a) and (13) in Eq. (7a) leads to:

1~~2ti F2 -i(kp- kao(-'

l S~) 44 4)ikacsL-

I (A-1) us(P) e 0 L- ka 2 =

2 -a/3i4 -ika- k1 e + )e ado'.

Let

) Go r ) -ika~l+ikacos(o-o')- (A-2) II - J og{led'

=0

(A-3) 2 e ika 2+ikacos(o-o '). (a-3) 12 -- " (92)e d'

then Eq. (A-l) may be written as:

1i 2 -i(kp - ) (2 -/ 2

- (A-4) u5(P) - e- NPe 2) [1+ I 2

with the understanding that only the t=O term is of interest for the reason

{ I mentioned in Section II. The integral I,, of Eq. (A-2) is evaluated first.

-One begins by incorporating Eq. (lOa) in Eq. (A-2); thus:

I_ 49

~I

I

-. I

27r -ilT -ika(o'- 2-)+ikacos(o-o')(A-5) 1 ido' di(T ) e- 1 d =0 0 r 2

with C (r )a . The sum on X can be incorporated in the 0'

integration if one changes the limits of integration to --xo'<-, since the

stationary points occur at *'=€-(4L-l)i/2-z'. Expanding the exponentialeika[(4'-w/2)-cos( ¢- € ')] in Eq. (A-5), about the stationary point 4'ecorresponding to Z=O only (since multiple encirclements are neglected), and

retaining terms only to the third power in (0' ) one obtains:

_ka/3( -T/2).r -ika 3

11 L d Jr di wF eika(o-'r)e T6 -Os2 Ior

ka )1/3

(A-6) Il dT W

2

after re-arranging terms, and interchanging the orders of integration.

Introducing a change of variables:

(A-7) (T )I13 ,- s ) y,

allows one to simplify Eq. (A-6) to:

ka 133

(A-8) 11 eika(') Jrd () e ka-1/3 3- +3

Sw2 dy e

50

i

One notes that the integral on y yields the Miller type Airy function[l3]

I denoted by AiCTr); specifically:

3

(A--- )5 dy e (r+ 2VITAi (T 2 V (-)

I Defining the function p*(&) introduced in Sectirn II, namely[4]:

( A-10) P* (4) = 1 f~ ~ T 1r2

one ay ow ewrie E. (-8)in terms of Eq. (A-9), and Eq. (A-la) as:

I(A-11) I~ 2e 2/ir

Iwhere P2

I (A-12) 2l-(al3~

A similar analysis for 1 2 in Eq. (A-3) leads to:

(A-13) 1 2 ' 2eik(P~~ Tr )/ [P*('2)-I 2V r 2

where

I (A-14) C a

IIncorporating Eqs. (A-11) and (A-13) in Eq. (A-4) leads finally to:

* I (A-15) u s(P) k ~ e~/ ikC +

+ elW ?Z ~f(La)l/3p*( l + p*(C)

I 51

where I and *2 are defined in Eqs. (17a) and (17b). The result inEq. (A-15) corresponds to that indicated in Eq. (15).

TEz case: Employing Eq. (9b) and Eq. (14) in Eq. (7b) leads to:

27r k -i (kp- 14- ekaos4,-o,(A-16) uS(P) % a d*' \ cos(o-+') e e

0

(4 1) e + g(i2) ek=0 J

Let:

r r2; -ika~ip+ikacos( -')(A-17) 2 - = o g(4, ) e cos(O-O')dO';

(A-8)2 e -ky kcsO )cos (O-O')dO',

then Eq. (A-16) may be written as:

(A-19) uo(P) 7 (e) e ) + 2.

Following the procedure for the TMz case, one may write 1l in terms ofthe stationary point at *s=*-w/2 (corresponding to the L=O term) as:

(A-20) ?l g(Cl )W€"s)eik ( do-i - € € )

1 R i

where cos(€-€') in Eq. (A-17) has been approximated by ( '-,s) in theneighborhood of the stationary point, and the limits of integration have been

changed to #' I<" as done for the TMz case. Incorporating Eq. (lOb) intoEq. (A-20) gives:

52

• Jli

t II

(ka)1/3 iiT )

-i -) 3_)i ka (Or)_i a 0 )3(A-21) J, l_ " do'f dr e ,, • (o' 's)e-s

I e-ika(o-'r 1) r e

f' dT ' W

k, .3

" 3 Introducing Eq. (A-7) into Eq. (A-21) yields:

_ ka 1/3 ]3

(A-22) d, w (e)-2/3_ dyye

I One recognizes the integral on y to be related to the derivative of theAiry function Ai(r); namely[13]:

(A-23 1 dy~ - i ( - Y+ Y )

(A-23) -dyye 3 a 2iVT Ai '(t) = 2i V'(r).

j One also defines the function q*(C) introduced in Section II; namely[4]:

(A-24) q*( ) =-L ei4- dlw T)_ + 1

Il / 2- 24

j Incorporating Eqs. (A-23) and (A-24) in Eq. (A-22) finally gives:

i (A-25) ?l eika(- ) 2 i (.a- 2/3 E*(1) - 1

I with tl defined in Eq. (A-12). A similar analysis for 12 leads to:

I (A-26) 1'2 eik( .~v2i (ka)-2/33[*(42)- __

1 53I

with t defined in Eq. (A-14). Employing Eqs. (A-25) and (A-26) in Eq.

(A-19) gives

e' C i{r/4 sinka(O-ir)(A-27) us(P) --ep L{ e( i+

+ h ) -ikaip -ikai 2 }l+ e-i,/4 F k ()"'3 *(, e 1+ q*(C2) e

2 2'

with *1 defined in Eq. (17a) and Eq. (17b). The result in Eq. (A-27)2

corresponds to the desired result indicated in Eq. (16).

54 li

APPENDIX II

I KIRCHHOFF DIFFRACTION BY A HALF-PLANE

A plane wave field u is incident on the half-plane at y=O for

* 1x>O as indicated in Fig. A-I. In this appendix, the expression for the

I';'i( p, 4

.I o y

I HALF -PLANEU K,

IFig. A-i. Half-plane geometry.

: " Ifield diffracted by the edge (at 0) of the half-plane is derived by using

a Kirchhoff approximation. Only the significant details of the analysis

are indicated. It may be verified that an application of the 2-D Green's

theorem to the region surrounding the half-plane leads to the followingj integral equation for the electric surface current density, Js induced on

the half plane when a plane wave consisting of a z-directed electric field

I (TMz case) is incident on it[17l:U

(A-28) u(P)-- - kj G(x,y0x',O) Js (x',o) dx'

" 55

l I

where G is the 2-D free space Green's function, and

0+

i wjs(Xt ,O) au (x',O) au (x',0)

(Note: surface current density Js is z-directed.) u(xy) represents thescattered z-directed electric field surrounding the half-plane. The geo-metrical optics approximation allows one to let

-1, sine eikx'cos¢'1 W1

and

-1 au (x',0) .

Employing this approximation in Eq. (A-28) gives:

(A-29) u(P) % - iw Jo G°(x'yjx"O0) oe i x c s dx'

G 2iksin¢ J G(x,yx',O) eikx 'c os¢ ' dx'.00

(Note: k = wv free space wave number, and Z° = free space wave

impedance = iVp-/.) The representation for G chosen here is differentfrom that used in Reference [17] for solving the same problem (however,

the various alternative representations for G0 may be obtained via the

use of the techniques for construction of Green's functions which are also

given in Reference [17]); hence, the method for solving Eq. (A-29) differs

from that in Reference [17]. In the present analysis, the followingexpression is employed for Go:

ikxXX' 1-ik k2_k2 ly-y' I(A-30) Go e dk ewa 0

oU W1-K 2 2x

56 i

I

Incorporating Eq. (A-30) in Eq. (A-29) gives:

e x ik x'+i kx'cos$'

(A-31) u(P) 27r dkx dx' e-k2_k 2 o

The integral

Sdx' ik x'+ikx'cos4'Sdx' ex

O

in Eq. (A-31) gives the result

| ik +kcosW'

xV (Here k = I - ik2 where the loss k2 << kI and k, > 0. The loss is added2

for ease of analysis; k2 may be set equal to zero in the final solution.)

# 1 Thus, Eq. (A-31) becomes:

n -ik2k 2ylj-ik x

(A-32) u(P) -iksin dk e x

2r k~ Fk k2 (k x+kcoso')dI

3The integral in Eq. (A-32) may be evaluated by the steepest descent method

which includes the Pauli-Clemmow modification[10] (for a pole near a saddle

point) after employing the polar transformations kx = kcosa, and ky = +ksina.

The integral in Eq. (A-32) is thus transformed into a contour integral in

the complex c-plane as:

isino' I e - i k pc os (o,; o ) O^

(A-33) u(P) C coscs da

jC cos+coso'<2

The contour of integration C for Eq. (A-33) is shown in Fig. A-2. Since

' : nv12 in the present problem, Eq. (A-33) becomes for 0<0<n (the results

for n<<27c are symmetric about ;=n):

1 i57

ir e -ikpcos(-)

(A-34) U(P) dICosaThe integration contour C may be deformed onto the steepest descent

path (SOP) through the saddle point at a=;--Q as indicated in Fig. A-2.

IMO 01 CONTOUR C

SADDLE POINT

AT G, s

REFLECTED WAVE POLESDP AT C1- IF

Fig. A-2. Contour of integration in the complex a-plane.

The reflected wave pole is crossed if as <7r/2 and its contribution (residue)

must be included. The saddle point contribution to SOP gives the desired

diffracted field; without giving details of the SOP evaluation (which are

available for the wedge diffraction problem in Reference [10)), the result

for the diffracted field ud is:

* (A-35) ud, T2. i(k- F(X)

coso > '

where F(X) is defined in Eq. (23) of Section II; here X is given by:

2i(A-36) X = 2kpcos -

58

Converting the result of Eq. (A-35) (which is valid for the coordinate

I system of Fig. A-l) to a result that is valid in the coordinate system of

Fig. A-3 in which *=*- jand *'=€'- j= 0 (for 0' = ), one obtains:

I : ui

I

I HALF- PLANE

Fig. A-3. Transformed coordinate system for the half-plane problem.

-A-37) udp 2 F 2 I "/4 2 eikP(A3) P in.' _0 T ef F(2kpcos e -) P

i( eikpD" Opk() e r

in which, the Kirchhoff diffraction coefficient, Dk() is

(A38a) 00€) = -i-n (X

l where

| (A-38b) X 2kpcos2 0.

The result in Eq. (A-38a) checks with that given in Eq. (22) of Section II.

I The ud(p) for the TEz case is assumed to be the same as in (A-37); this

assumption is reasonable within the transition region near 4=r, where the

I Kirchhoff result is relatively insensitive to polarization effects.

I59

REFERENCES

1. Burnside, W.D., "Analysis of On-Aircraft Antenna Patterns," Report3390-1, August 1972, ElectroScience Laboratory, Department ofElectrical Engineering, The Ohio State University; prepared underContract N62269-72-C-0354 for Naval Air Development Center.

2. Marhefka, R.J., "Roll Plane Analysis of On-Aircraft Antennas," Report3188-1, December 1971, ElectroScience Laboratory, Department ofElectrical Engineering, The Ohio State University; prepared underContract N62269-71-C-0296 for Naval Air Development Center.

3. Burnside, W.D., "Analysis of On-Aircraft Antenna Patterns," Report3390-2, May 1973, ElectroScience Laboratory, Department of ElectricalEngineering, The Ohio State University; prepared under ContractN62269-72-C-0354 for Naval Air Development Center.

4. Logan, Nelson A., "General Research in Diffraction Theory," Vol. 1,Report LMSD-288087, Missiles and Space Division, Lockheed AircraftCorp., December 1959. (This reference contains an extensivebibliography dealing with previous work on the diffraction of wavesby smooth convex surfaces).

5. Bowman, J.J., Senior, T.B.A. and Uslenghi, P.L.E. (Eds.), Electro-magnetic and Acoustic Scattering by Simple Shapes, North HolandPublishing Co., 1969, (p. 103 and p. 112 list Goriainov's results).

6. Wait, J.R. and Conda, A.M., "Diffraction of Electromagnetic Wavesby Smooth Obstacles for Grazing Angles," Jour. of Res., N.B.S.,Vol. 63D, No. 2, Sept.-Oct., 1959, pp. 181-197.

7. Logan, Nelson A., "General Research in Diffraction Theory," Vol. 2,Report LMSD-288088, Missiles and Space Division, Lockheed AircraftCorp., December 1959. (This reference contains tabulated valuesof the p(t) and q(4) functions).

8. Pathak, P.H. and Kouyoumjian, R.G., "The Radiation from Aperturesin Curved Surfaces," Report 3001-2, December 1972, ElectroScienceLaboratory, Department of Electrical Engineering, The Ohio StateUniversity; prepared under Grant NGR 36-008-144 For NASA LangleyResearch Center, Hampton, Virginia. (pp. 22-23 contain a descriptionof the pseudo-ray system introduced by Kouyoumjian for the Fock currentin the lit part of the penumbra region on a smooth convex surfaceexcited by a plane wave.)

9. Keller, J.B., "Geometrical Theory of Diffraction," J. Opt. Soc. Am.,Vol. 52, No. 2, February 1962, pp. 116-130.

60 7

II

10. Hutchins, D.L. and Kouyoumjian, R.G., "Asymptotic Series Describingthe Diffraction of a Plane Wave by a Wedge," Report 2183-3, December1959, ElectroScience Laboratory, Department of Electrical Engineering,The Ohio State University; prepared under Contract AF19(628)-5929 for

Air Force Cambridge Research Laboratories. (AD 699 228).

1f. Kouyoumjian, R.G., "Asymptotic High Frequency Methods," Proc. IEEE,

53, 1965, pp. 864-876.

12. Kouyoumjian, R.G. and Pathak, P.H., "The Dyadic Diffraction Coefficientfor a Curved Edge," Report 3001-3, August 1973, ElectroScience Laboratory,Department of Electrical Engineering, The Ohio State University;prepared under Grant NGR 36-008-144 for NASA Langley Research Center.(A paper on this subject is currently in preparation for publication.)

13. Abramowitz and Stegum (Eds.), Handbook of Mathematical Functions,National Bureau of Standards Publication, June 1964, p. 478.

14. Ryan, C.E. and Peters, L., Jr., "Evaluation of Edge-DiffractedFields Including Equivalent Currents for the Caustic Regions," IEEETrans. Ant. and Prop., Vol. AP-17, No. 3, May 1969. (See correctionsto this paper in March 1970 issue of IEEE Trans. AP.)

15. Ratnasiri, P.A.J., Kouyoumjian, R.G. and Pathak, P.H., "The WideAngle Side Lobes of Reflector Antennas," Report 2183-1, March 1970,

ElectroScience Laboratory, Department of Electrical Engineering,The Ohio State University; prepared under Contract AF 19(628)-5929for Air Force Systems Command. (AFCRL-69-0413) (AD 707 105)

16. Burnside, W.D. and Peters, L., Jr., "Axial Radar Cross Sectionof Finite Cones by Equivalent Current Concept with Higher OrderDiffraction, Radio Science, Vol. 7, No. 10, October 1972, pp. 943-948.

17. Felsen, L.B. and Marcuvitz, Radiation and Scattering of Waves,Prentice-Hall, Inc., 1973, pp. 721-724.

1.

I.61

* *1