Asymptotic Solutions

8/3/2019 Asymptotic Solutions

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AN ASYMPTOTIC SOLUTION FOR BOUNDARY-LAYER FIELDS NEAR A

CONVEX IMPEDANCE SURFACE

Paul E. Hussar and Edward M. Smith-Rowland

IIT Research Institute185 Admiral Cochrane Dr.

Annapolis, MD 21401

Short title: ASYMPTOTIC SOLUTION FOR BOUNDARY-LAYER FIELDS

Correspondence and proofs to: Dr. Paul HussarIIT Research Institute

185 Admiral Cochrane Dr.

Annapolis, MD 21401


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ABSTRACT

An analytic representation for fields (E,H) that, for wavenumber k, satisfies the

Maxwell equations to order k-2/3 within a suitably-defined boundary-layer neighborhood is

provided for the case of a general doubly-curved convex impedance surface. This

solution is an ansatzconstruct obtained via heuristic modification of a residue-series

solution to a corresponding circular-cylinder canonical problem with an infinitesimal

axial magnetic dipole excitation. The field components are in the form of creeping-ray

modal series written as functions of geodesic-polar and normal coordinates (s, ,n)

appropriate to the vicinity of a general convex surface. Adaptation of the canonical

solution to the general case begins with a transformation from the native cylindrical

( , ,z) coordinates of the canonical solution to a system ( , c,s) defined by cylinder-

surface geodesics. The transformed canonical solution is further modified by

replacement of corresponding factors deriving from the metric and curl operators in the

( , c,s) and (s, ,n) systems, and by pervasive application of a substitution previously

employed in a more limited way by Pathak and Wang. The physical content of the

substitution process is that the creeping-ray attenuation along the geodesics occurs

independently of the surface normal curvature transverse to the geodesic direction.

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1. INTRODUCTION

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The Uniform Geometrical Theory of Diffraction (UTD) [1] provides accurate

asymptotic representations of electromagnetic fields both close to and far from smooth,

convex perfectly conducting surfaces that are characterized by electrically large principal

radii of curvature but are otherwise arbitrary. The UTD representations are obtained by

heuristically adapting asymptotic solutions for cylinder and sphere canonical problems to

a general convex surface geometry. We say, therefore, that the UTD representations are

obtained via the canonical-problem method. In the near vicinity of the surface, an

alternative method is also available to determine the asymptotic field behavior. This is

the boundary-layer method [2], which involves construction of the solution in terms of

stretched coordinates, followed by matching with the solution in the region exterior to

the layer. It has been demonstrated [3] that the boundary-layer method can be employed

to reproduce, in a more rigorous fashion, the essential features of UTD creeping-wave

propagation over a smooth convex conductor.

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In contrast, the solution for doubly curved impedance surfaces obtained by the

boundary-layer method [2] realizes the expected functional behavior in the context of a

single mode. This solution results from consistency conditions obtained through fourth

power in k-1/3by applying the Maxwell Equations in general coordinates to E and H

components written in terms of Zauderers asymptotic expansion. The treatment of the

boundary conditions is, however, only through order k-1/3 and relies on the assumption

that the normalized surface impedance or its inverse is of this order. The resulting

solution exhibits distinct electric and magnetic creeping rays, a feature observed in the

three-dimensional impedance-cylinder canonical solution only in limiting cases. Since

both of the recently obtained formulations thereby face important limitations, further

efforts are required to adequately describe the field behavior in the vicinity of convex

surfaces that are not perfect conductors.

In this paper, we will describe a new creeping-ray modal solution for fields (E,H)

in the close vicinity of a doubly-curved convex impedance surface. Our solution is

roughly complimentary to those provided in [4], though for the surface-impedance case,

because the modal-series representation we employ is primarily useful in cases where a

surface ray extends some distance into the shadow region. This solution has been

obtained by the canonical-problem method as an adaptation of a solution for fields

radiated near a circular cylinder by an axial surface magnetic dipole. It embodies the

same GTD prescription realized by the deep-shadow representation in [5] while avoiding

the kind of surface-property restriction that appears in [2]. The solution is expressed in

terms of geodesic polar coordinates s and and surface normal distance n, such that the

origin of coordinates may be identified as the location of a source. Though the

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origination of the solution is heuristic, the formal rigor of the boundary-layer method is

recovered because the boundary conditions and the Maxwell equations in the boundary-

layer region are both satisfied through order k-2/3.

Construction of the new solution occurs via heuristic modification of a dipole-

excited-cylinder solution transformed from its native cylindrical coordinates to a new

system defined in terms of cylinder-surface geodesics and a radial coordinate. Certain of

the modifications are introduced to reflect a further transformation between these

cylinder-specific coordinates and coordinates appropriate to the neighborhood of a

general surface. Such modifications are guided by a comparison of the metrics and curl

operators in different systems. Elsewhere, we introduce a heuristic construction that

employs the UTD generalized torsion factor [6] and is consistent with the GTD principle

of locality. For the case of an infinitesimal, axially oriented dipole on an impedance

circular cylinder, the mode-dependent constant roots noted previously are defined by pole

locations that are the zeros of a denominator function that depends on the angle between

the cylinder axis and the surface geodesic. We find that if this angle is re-expressed in

terms of the UTD generalized torsion factor, evaluation of the resulting generalized

denominator function restricted to the case of an impedance sphere produces the correct

pole locations for the creeping-ray modes on the sphere. We are thereby provided with a

substitution method for interpolating between cylindrical and spherical geometry that can

be used to obtain local-geometry-dependent mode-specific root values at points along a

geodesic over an arbitrary surface. The crucial point that we will demonstrate is that

introduction of this substitution preserves both the Maxwell equations and the boundary

conditions through next-to-leading order.

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Since our aim here is simply to obtain analytic fields (E, H) that satisfy the

Maxwell equations in geodesic-polar and normal coordinates on a general surface, only

the total field components as they occur in the cylinder problem need to be considered.

These components contain an overall factor equal to the sine of the polar angle. This

factor is understood in terms of the dipole pattern and ignored. Otherwise, we begin

our solution construction by applying the substitution method described above to certain

terms in the canonical-problem total-field solutions in such a way that the leading-order

behavior on a general surface is specified. In particular, GTD locality is satisfied by

replacing exponentiated terms involving the Fock parameter, , multiplying cylinder

roots by exponentiated integrals involving roots that vary with the local geometry along a

geodesic. The Maxwell equations and the boundary conditions are satisfied at leading

order in this process and are thereupon employed to provide constraints on the next-to-

leading-order terms.

In Section 2, we review the circular-cylinder canonical problem. Construction of

an asymptotic representation for dipole-excited fields in a suitably defined boundary

layer of a circular impedance cylinder relies on standard approximation techniques

including steepest-descent evaluation of axial-wavenumber integrals. We find that two

possible canonical solutions that differ at next-to-leading order can be obtained

depending on whether differentiation of potentials to obtain fields occurs before or after

steepest-descent integration.

In Section 3, the construction of a solution for general convex surfaces begins

with the adaptation to general geometry of certain lead terms that appear in the

cylinder-problem asymptotic solution independently of the order of operations. While

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neither of the canonical cylinder solutions satisfies both the Maxwell equations and the

boundary conditions through next-to-leading order, we are able to specify the remaining

terms in our solution in such a way that both of these conditions are observed. The terms

at next-to-leading order are further restricted by the requirement that the solution be a

well-behaved function of the surface impedance.

In Section 4, we assess the quality of our solution via a comparison with the

perfect-conductivity UTD solution extended into the boundary layer.

Section 5 contains concluding remarks.

2. IMPEDANCE CYLINDER WITH AXIAL SURFACE DIPOLE EXCITATION

For the problem of computing fields in the vicinity of an impedance circular

cylinder aligned with the z axis and subject to an axial magnetic dipole excitation, it is

sufficient to consider vector potentials of the form

zA A0=

(1a)

and

zF F0=

(1b)

where A0 and F0 are scalar potentials conveniently written in terms of normal modes via

the spectral-integral representation

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( )

( )

( )

( )

=

=

ddee,,

,,

4

1

z,,F

z,,A n2j)zz(j

n2

0

0

F

A

(2)

The source coordinates are given by ( , ,z ), the field point is located at ( , ,z),

and, for < , the transformed potentialsA andFare expressed in terms of normal-

mode coefficients a0-, a0

+, f0-, and f0

+ as

( ) ( ) ( ) + = +

)2(

0

)1(

0 HaHa,,A

(3a)

and

( ) ( ) ( )+= +

)2(

0

)1(

0 HfHf,,F

(3b)

with

22k =

(4)

and k representing the wavenumber. For the case of a magnetic source of strength M0, a0-

is 0, while f0- is given by

( ) 0)2(

0 MH4

jf

=

(5)

When > (3b) is modified via the replacement

( ) ( ) ( )

)2()1(0)1(0 HH

4

MjHf

(6)

with the result that the scalar potentials A0 and F0 satisfy

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[ ]( ) ( ) ( )

=

+/zzM

0

F

Ak

00

02

0

2

(7)

The remaining coefficients, a0-, a0+, and f0+ are obtained from the boundary condition

HnEnn = s

(8)

which is applied at the cylinder surface and is expressed here in a general way in terms of

the surface normal n (= in the cylinder specific case), with s defining the surface

impedance. In general, for an isotropic medium characterized by permittivity and

permeability , the fields associated with the potentials A and F are given by

FAH

+=j

1

(9a)

and

AFE

+=j1

(9b)

while in the present case (9a) and (9b) apply with = 0 and = 0. For

convenience, the fields (9a) and (9b) are listed in component form in the Appendix A1.

High-frequency asymptotic representations applicable in the close vicinity of the

cylinder are obtainable for the potentials in (2) and for the fields in (9a) and (9b) by

following a sequence of familiar steps. These steps include use of the Poisson

summation formula to transform the angular-wavenumber integrals into residue series

involving poles


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functions, and evaluation of the axial-wavenumber integrals via the method of steepest

descent. The resulting potential/field representations can be associated with a ray picture

in which the rays follow cylinder-surface geodesics. For simplicity, we will restrict

consideration to rays that are travelling in the + direction and have not completed any

encirclements of the cylinder. A near-vicinity or boundary-layer region within which the

Airy-function approximation for the Hankel functions is appropriate is defined by the

requirement that y 1 where

gm

kny =

(10)

with n representing the normal distance from the field point to the surface and mg =

(k g/2)1/3, g being the normal curvature of a cylinder-surface geodesic associated with

the given field point. We choose to write the solution for the cylinder problem in terms

of coordinates ( , c,s), which are related to the usual ( , ,z) coordinates via

( ) ( )[ ] 2/1222 bzzs +=

(11a)

and

( )

= zz

btan 1c

(11b)

where b is the cylinder radius.

For = b, the cylinder radius, application of the above sequence of steps to the

potentials in (2) results in the asymptotic representations

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( ) ( )

( ) ( )

( )p

p

Dsin

eywqb

msinkcosm

sGKA

c

j

p2p2

p

cc

p

020

+

=

(12a)

and

( ) ( )( ) ( )

( )p

p

D

eywqsGjKF

j

p2p1

p

010

=

(12b)

for A0 and F0, while corresponding field components in the coordinate system ( , c,s)

can be obtained by applying (9a) and (9b) to (12a) and (12b) to give

( ) ( )( )

p

p

D

eksGjKH

j2

p

02

=

( )( ) ( )ywqcoscot

ks3

jmy2

m

qcossin

kb3

mcossin p2p2c

2

c

p1

cc

2p

cc

+

( ) ( )

+

+

ywmqcos3

1cos2

kb

mcossin

bp2p2c

3

c

p

cc

(13a)

( ) ( )( )

p

p

D

eksGKH

j

1

2

p

02c

=

( ) ( )ywqcoskb3

m4sin

bp2p1c

2p

c

2

2

+

( ) ( ) ( )

+

+

ywqcoscotks3

jy2qcos

kb

mcossin

bp2p1ccp2c

2p

c

2

c

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(13b)

( ) ( )( )

p

p

D

eksGKH

j

2

2

p

02s

=

( ) ( )ywqcossinkb3

m2cossin

bp2p1cc

p

cc

2

2

( ) ( ) ( )

+

+ ywqcos

ks3

jy2qcossin

kb

mcossin

bp2p1cp2cc

pcc

2

(13c)

( ) ( ) ( )p

p

p01 D

jke

sGKE

=

( ) ( )ywqcos3

11

kb

msin

bp2p1c

2p

c

+

+

( ) ( ) ( )

+

+ ywqcoscot

ks3

jy2qcossin

3

11

kb

mcossin p2p1ccp2c

2

c

2p

c

2

c

(14a)

( ) ( )( )

p

p

c D

eksGjKE

j

1

p

01

=

( )( ) ( )ywqcoscot

ks3

jmy2

m

qcossin

bx p2p2c

2

c

p1

cc

+

( ) ( )

+

+

ywmqcos

3

4cos

kb

mcossin

bp2p2c

3

c

p

cc

2

2

(14b)

and

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( ) ( )( )

p

p

D

eksGjKE

j

2

p

01s

=

( )( ) ( )ywqcosks3jmy2m

qsinbx p2p2c2p1c2

( ) ( )

+ ywmqcossinkb3

m2cossin

bp2p2c

2

c

p

c

2

c

2

2

(14c)

where m = mgsin , w2 is a Fock-type Airy function and is the Fock parameter given

by mgs/ g. In (12a)-(12b), (13a)-(13c), and (14a)-(14c) we employ the notation

( ) 4/j01 eM2

1K

=

(15a)

( ) ( ) k/KK 012 =

(15b)

and

( )s

esG

jks

0

=

(16)

along with

( ) ( ) ( )p2cp2p1 wCsinjmwq +=

(17a)

and

( ) ( )p2p2 wq =

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(17b)

where 0 is the free-space impedance and C = s/ 0. In addition,

2/1

c

2

c

2

2

1 sincosb

+

=

(18a)

2/1

c2

c2

2

2 cossinb

+

=

(18b)

and

= 1b

b

2

(18c)

are geometrical factors that arise due to the choice of the ( , c,s) system. Finally, the

roots, p, are defined as the zeros of D( ) where

( ) ( ) ( ) ( ) ( ) ( ) ( )= + 2222 wqwwqwD

(19)

with

( )( )

+

+=

22

c

ccc

C

1C/

sinkb

cotmsinkb41

C

1C

C

1C

2

sinjmq

(20)

The field components in (13a)-(13c) and (14a)-(14c) include all terms of size

within k-2/3 of the leading-order term in each of the functions w2( p-y) and w2 ( p-y).

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Such terms are obtained exclusively from the action of differential operators upon the

exponential functions and on the functions w2( p-y) and w2 ( p-y) appearing in the

potentials A0 and F0 in (12a) and (12b). Because the c derivative of a root, p, can be

expressed as

( )

( )[ ] ( )p

qq

q

2

pp

p

p

=

(21)

we will assume that p/ c is of order k-1/3, though we note that, since p/q

is

of order k-1/3, this assumption can be violated when p is sufficiently close to q( p). As

a consequence of taking p/ c to be of order k-1/3, derivatives acting on the

denominators in (12a) and (12b) will in general not contribute terms to the fields at any

order considered. For completeness sake we point out that these denominators can

exhibit problematic behavior when the square-root term in (20) is close to zero.

It is easily verified that the field components (13a)-(13c) and (14a)-(14c) obtained

by differentiating the potentials (12a) and (12b) through order k-2/3 represent an order k-2/3

approximate solution of the Maxwell equations in the ( , c,s) system governed by the

metric

=2

2

212ij

Ts0

Tss0

001

g

(22)

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where T is the surface-ray torsion equal to sin(2 c)/2b. The curl operator corresponding

to the metric (22) is provided in component for in Appendix A2. On the other hand, the

boundary condition (8) is satisfied only through leading order when the fields are

represented by (13a)-(13c) and (14a)-(14c). Alternatively, the differential operators in

(9a) and (9b) could have been brought under the integration signs in (2) and forced to act

upon the integrands prior to the steepest-descent evaluation of the integral over axial

wavenumber. Changing the order of operations in this way results in field components

which comprise a higher-order solution of (8) but which satisfy the Maxwell equations

only to leading order. For the purposes of our solution construction, it is important that

changing the order of operations leaves unaffected the first or lead term proportional to

w2 ( p-y) and the lead term proportional to w2( p-y) throughout (13a)-(13c) and (14a)-

(14c).

3. APPROXIMATE MAXWELL SOLUTION FOR A CONVEX SURFACE

We seek to modify the cylinder solution (13a)-(13c) and (14a)-(14c) in such a

way as to obtain a solution valid within the boundary layer of an arbitrary smooth convex

surface. In the neighborhood of a general smooth surface, it is convenient to employ a

system (s, ,n) consisting of geodesic polar coordinates and a normal coordinate. For the

specific case of a cylinder, these coordinates are related to the earlier ( , c,s)

coordinates via n = - b and = - c, with s representing, in either case, the distance

traveled along a given geodesic from the origin or source locus. According to (10), the

condition y 1 restricts the cylinder solution to a boundary region where n/ g is of

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order k-2/3. We define the boundary-layer region for an arbitrary smooth surface more

stringently by requiring that n/ g, n/ t (where t is the surface radius of curvature in

the direction normal to the geodesic), and nT are all of order k-2/3.

A solution through order k-2/3 for both the Maxwell equations and the boundary

condition (8) is obtainable neither by direct modification of (13a)-(13c) and (14a)-(14c)

nor by direct modification of the alternative component representation derived by

reversing the order of differentiation and steepest-descent integration applied to the

potentials (2). Since the lead term proportional to w2 ( p-y) and the lead term

proportional to w2( p-y) are unaffected by changing the order of operations, we will

assume that these terms are suitable for adoption into a generalized form. While these

lead terms are not strictly leading order terms, all the leading-order behavior in (13a)-

(13c) and (14a)-(14c) is nevertheless included. Following generalization of these lead

terms, the solution-construction process will be completed by introducing and solving for

unknowns that are of size k-2/3 relative to leading order in terms proportional to w2 ( p-

y) and to w2( p-y)

The field components we seek are solutions of the Maxwell equations

HE 0j=

(23a)

and

EH 0j=

(23b)

in the (s, ,n) system for which we adopt the linearized metric

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( )

( ) ( )

+

+

=

100

0n2

1s

s

s

nsT2

0s

nsT2n21

gt

4

2

2

2g

ij

(24)

where ; (s) is the divergence factor. For an arbitrary vector-valued function v (s, ,n),

the components of the curl operator corresponding to the linearized metric (24) are given

by

( ) ( )( ) s

sn

t

2

sn

v

n

vnT2

v

/n1s

s+

+

+

= v

(25a)

( ) +

+

=

s

v

/n1

1

n

vnT2

n

v n

g

sv

(25b)

and

( ) ( ) ( )( ) n

s

t

s

gn

v

/n1s

s

s

vv

s

snT2

s

v

/n1

1 +

+

+

+=

v

(25c)

where s, , and n consist of terms that do not involve differentiation of the field

components. We merely indicate the existence of these terms here because, inasmuch as

there will be field-component derivatives in (23a) and (23b) of order k larger than the

components themselves, the terms will yield no contribution at any order of interest.

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Clearly, the presence of the geometrical factors 1, 2 and in (13a)-(13c) and

(14a)-(14c) can be associated with the form of the metric (22) and curl operator

(Appendix A2) in the ( , c,s) system. We expect that the role corresponding to that

played by these factors within the canonical solution should be played, within a solution

adapted for an arbitrary convex surface, by corresponding factors appearing in the metric

(24) and curl operator (25a) (25c). In other words, we begin construction of a solution

for a general surface by taking the lead terms proportional to w2 ( p-y) and to w2( p-y)

in (13a)-(13c) and (14a)-(14c) and making the substitutions

t

1n1

+

(26a)

g

2

n1

+

(26b)

and

n2

(27)

In addition, we may observe that factors of b/ in (13a)-(13c) and (14a)-(14c)

should be associated with the metric (22) and curl operator (A2.1)-(A2.3) due to the role

of the determinant derived from the metric tensor in defining the curl components. In

view of the fact that the determinant derived from the cylinder metric is given by gc =

s2 2/b2, while the determinant derived from the linearized metric (24) is given by

+

+

=

ts4

2

ln2n2

1s

g

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Making use of the additional definitions

( )( )

( ) ( )[ ]2g

g

,s,sT1

,s,sb

~

+

=

(31)

and

( )( ) ( )

3/1

2

,ssin,sb~k

,sm~

=

(32)

we are allowed to rewrite q

in (20) in the form

( )( )

+

+

=

22

p

pC

1C

sinb~k

cotm~sinb~k

41C

1C

C

1C

2

sinm~jq

(33)

so that cylinder-specific quantities no longer appear. The generalized form (33) for the

functions q

can be employed in the adaptation of (13a)-(13c) and (14a)-(14c) to general

convex-surface geometry by replacing factors of the form exp(-j p) with factors of the

form exp[-j p(s, )] where

( )( ) ( )

( )sd

,s

,s,sm,s

s

0 g

pg

p

=

(34)

with p(s , ) satisfying

( ) ( ) ( ) 0w,s,qw p2pp2 =

(35)

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(36a)

( ) ( )( )

+

=

t

j2

p02

n1

/D

eksGKH

p

p

( )( )

( )( ) ( )

+

++

+

p

H

p2

2

ts

p2

p2

t

p1ywcos

/n/n1

qyw

/n21

q

(36b)

( ) ( )( )

p/D

eksGKjH

j2

p02n

=

( ) ( ) ( ) ( ) ( )

+++ p

Hnp2

ts

p2p2

p1 ywcosm~/n/n1

qywcosm~

q

(36c)

( ) ( )( )

+

=

s

j

p01s

n1

/D

kesGKjE

p

p

( )( )

( ) ( ) ( ) ( )

++

++

p

E

sp2p2p2

s

p1ywcosm~nTq2ywsin

/n/n1m~

q

(37a)

( ) ( )( )

+

=

t

j

p01

n1

/D

kesGKjE

p

p

( )( )

( )( )

( ) ( )

+

+

++

p

E

p2

t

p2

p2

s

p1ywcosm~

/n21

qywcos

/n/n1m~

q

(37b)

and

( ) ( )( )

p

p

/D

kesGKE

j

p01n

=

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( )( ) ( ) ( ) ( )

+

++

p

E

np2

2

p22

s

p1ywcosqyw

/n/n1

q

(37c)

where H, sH, nH,

E, sE, and nE are unknowns involving terms in w2 ( p-y)

and w2( p-y) that are assumed to occur at order k-2/3 higher than the corresponding

leading order terms. The overall factors of are in agreement with the cylinder and

sphere canonical solutions, and have the same significance here as within the UTD.

Conditions on the unknowns may be obtained by applying the Maxwell equations (23a)-

(23b) in the (s, ,n) system. Table 1 summarizes the order behavior resulting from the

action of the differential operators in (23a)-(23b) upon the various objects that appear in

(36a)-(37c). Only terms included in Table 1 are taken into account, all other terms being

considered to be of higher order. The fact that no s-derivative acting on the Airy function

appears in Table 1 carries the implication that g/s is being taken to be at least as

small as order k-1/3. This point will be discussed when we compare our solution with the

ordinary UTD solution for the case of perfect conductivity. Applying the prescription

just outlined leads to the requirements

( ) ( ) ( ) ( )ywq3

jycosq

ksp2p1

s

s

2p2

p2

pHs

=

( ) ( ) ( )pEp2p1ppnk

1ywq

kscos

b~

k

m~2

+

+

(38a)

( ) ( ) ( ) ( )ywsinqksm~

cosb~

kp2p1

ppp

Ep

Hn

+

=

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( ) ( )ywcossinqb~k

m~

p2p2

p

2

(38b)

( ) ( ) ( )ywcosqm~3

jy

m~

q

ksp2p2

g

g

p1p2

pEs

=

( ) ( ) ( )pHp2p2p2

p

nk

1ywcosqm~

kscos

b~

k

m~2

+

(38c)

and

( ) ( ) ( ) ( )ywcossinqks

cosb~k

m~

p2p2

ppH

p

E

n

+

=

( ) ( )ywsinqb~k

m~

p2p1

p

+

(38d)

being imposed upon the unknowns. These requirements constitute a consistent set,

meaning that a Maxwell solution in the form of (36a)-(36c) and (37a)-(37c) is available.

Table 1. Order of Contributions Resulting from Various Derivatives

Derivative Acting on Action (k 1/3)

Initial

Order

Final

Order

/s e-jks k 0, k -2/3 0, k-2/3

/n w2( p-y)k 0, k -2/3 0, k-2/3

/n w2 ( p-y) k1/3 0 k-2/3

/s ( ) ,sj pek1/3 0 k-2/3

/2 ( ) ,sj pe k1/3 0 k-2/3

/2 w2( p-y) k1/3 0 k-2/3

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where the coefficients 1, 2, 3, and 4 are required to be of order k-1 if the

contributions from the unknowns are to be of order k2/3 smaller than the corresponding

leading-order terms. Given (17a) with substitutions, (17b), and (35), it is apparent that

(39a) and (39b) together with (40) result in identities between second order polynomials

in q( p). Terms in these identities may be equated according to the power of q

( p) to

give relations between the coefficients defining the unknowns via (40). While this

process does not uniquely specify the unknown coefficients, we require the solution to

exhibit well-behaved dependency on the normalized surface impedance, C, in particular

as C 0 or C . This restriction, along with the relations between the unknown

coefficients resulting from (39a) and (39b), and the Maxwell conditions (38a), (38b),

(38c), and (38d) are all satisfied if the unknowns are chosen as

( ) ( ) ( ) ( )ywqks3

yjq

kscos

b~

k

m~2p2p1

g

g

2

p2p

2p

pHs

+

=

( ) ( )ywqks

cosb~

k

m~2p2p1

p2

p

+

(41a)

( ) ( ) ( ) ( ) ( )[ ]ywqywqcotks

p2p1p2p2p

2

pH

=

(41b)

( ) ( ) ( )ywsinqksm~

cosb~

kp2p1

p2pp

Hn

+=

( ) ( ) ( )ywsinqks

m~cot2cossin

b~

k

m~

p2p1p

2p

++

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(41c)

( ) ( ) ( )ywqm~ks3

yjp2p2

g

g

2

pEs

=

( ) ( )ywcosqm~ks

cosb~

k

m~2p2p2

p2

p

+

(42a)

( ) ( ) ( )yw.qm~sinks

cotb~

k

m~2p2p2

p2

pp

E

+

=

(42b)

and

( ) ( ) ( )ywcosqsinb~

k

m~cot

ksp2

2p2

pp2

pEn

=

( ) ( )ywqcotks

sinb~

k

m~

p2p1p

2p

+

(42c)

The solution for the unknowns in this form appears to be almost unique. Addition of

an identical next-to-leading order term proportional to yw2 ( p-y) either to both E

and nH or to both nE and H would leave satisfaction of the Maxwell conditions and

the boundary conditions unaffected.

Finally, we note that all of the terms in our solution, including the lead terms in

(36a)-(36c) and (37a)-(37c) as well as the additional terms (41a)-(42c), contain either at

least one of the quantities cos , T, or p/ , all of which are zero in the case of

spherical geometry, or the factor q1( p) which is zero for spherical geometry when p

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again employing geodesic polar coordinates centered at the source locus, we observe that

the leading order UTD magnetic field is -directed and given by

( )

( ) ( ) ( )( )

( )p2p2

p

j

p

1

t

2/1

gg0

4/ksj

w

ywen

1sm0m2

kss2

ejk

H

p

+

=

+

dp

(43)

and that violation of the Maxwell equations is apparent from

( )( )

( )( ) ( )

2/1

gg0

4/ksj

2 sm0m2

kss2

ejk

k

1

=+

dpH

( )( ) ( )

+

+

ywsk3

jy2yw

n1

w

ep2

g

g

p2

1

tp2p

j

p

p

(44)

In (43), dp represents an infinitesimal magnetic dipole source, is a -directed unit

vector at the source, is the usual Fock parameter, and the roots, p, satisfy w2 ( p) =

0. We have employed the residue-series representation of Fh and have included the

geometrical factor (1+n/ t)-1 to extend the UTD surface field component H in a way

that best agrees with the Maxwell equations. The violation of the Maxwell equations is

represented by the term involving g/s, under the assumption that g/s is of order

unity. Similar terms result from the action of the curl operator on the field components

obtained in Section 3 but were excluded by imposing the restriction that g/s be of

order k-1/3. Under this restriction, the solution in Section 3 satisfies the Maxwell

equations through terms in both w2 ( p-y) and w2( p-y), which are smaller by k-2/3 than

the leading term in each function. Under the less restrictive assumption that g/s is of

order unity, the solution is Section 3 observes the Maxwell equations through the order

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k-2/3 terms in w2( p-y), which are the order k-2/3 terms in the solution as a whole, provided

that w2 ( p-y) and w2( p-y) are taken to be of comparable size. In either case, we

observe that violation of the Maxwell equations occurs at the same order in our solution

as it occurs in the UTD solution for conducting surfaces extended into the boundary

layer.

5. CONCLUSION

An approximate asymptotic solution for the fields (E,H) in the boundary layer of

a smooth convex impedance surface has been obtained by applying a substitution process

to a canonical solution for the fields in the vicinity of an impedance circular cylinder

excited by an axial magnetic dipole. The solution appears in a creeping-ray modal format

that includes exponentiated integrals involving mode-dependent root values that vary

along surface geodesics. For each mode and for each point along a given geodesic, a

corresponding root value is determined by the surface impedance and by geometrical

quantities defined for a doubly-curved convex surface with distinct principal curvatures.

The substitutions that transform the impedance-cylinder roots into roots that describe

propagation over an arbitrary impedance surface involve the UTD generalized torsion

factor and are based on the physical assumption that to low order such propagation occurs

independently of the surface radius of curvature transverse to the direction of

propagation. This assumption receives strong support from the fact that if in place of our

substitution involving the UTD generalized torsion factor (cot cT g), alternatives

involving the geodesic-transverse radius of curvature (cot c1/T t or cot c

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g1/2/ t1/2) had been employed, neither the correct root spectrum for spherical geometry,

nor satisfaction of the Maxwell equations to order k-2/3 would have been obtained.

In a sense determined by the Maxwell equations, our solution appears to represent

a level of accuracy roughly comparable to the ordinary UTD solution for perfect

conductors. Extension of the present result into a dyadic Greens function may, in the

future, provide a fully satisfactory UTD field representation for general convex

impedance surfaces. We note that our Maxwell solution exhibits asymmetry between

source and field-point coordinates. While an approximate solution not linked to a

specific source excitation is not subject to reciprocity constraints, construction of a

Greens function in UTD format will nonetheless require enforcement of reciprocity,

presumably by some symmetrization procedure, such as in [4].

Finally, results comparable to those presented here but applicable in cases of

alternative boundary conditions (i.e., single layer coatings) have already received some

discussion [7].

APPENDIX A1

FIELD COMPONENTS IN CYLINDRICAL COORDINATES

The component form of (9a) and (9b) for a cylindrical system and for z-directed

potentials appears in numerous places such as [8], but it is reproduced here for

convenience. The , , and z components ofE and H are given in terms of A0 and F0

by

=

00

2 F1

z

A

j

1E

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(A1.1)

+

=00

2 F

z

A

jw

1E

(A1.2)

0

2

2

2

z Akzj

1E

+

=

(A1.3)

z

F

j

1A1H 0

2

0

+

=

(A1.4)

z

F

j

1AH 0

2

0

+

=

(A1.5)

and

0

2

2

2

z Fkzj

1H

+

=

(A1.6)

APPENDIX A2

CURL OPERATOR IN THE ( , c,s) SYSTEM

For an arbitrary vectorv, the components of the curl ofv in the cylindrical

( , c,s) system are given by

( )s

v

bs

v1v

s

1T

s

vv

s

b3

s

2c11

c

s2

1

ccc

+

= v

(A2.1)

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( ) s2g

121t

s211 vv

TT2

vvT

s

vbc

c

=

v

(A2.2)

and

( ) s2g

c1t

2

c

2s21s v

TT2v

v

s

vT

vb

2

c

+

+

+

=

v

(A2.3)

References

[1] Pathak, P. H., Techniques for High Frequency Problems,Antenna Handbook,

Y. T. Lo and S. W. Lee eds., Chapter 4, Van Nostrand-Reinhold Co., New York,

NY, 1988.

[2] Bouche, D., F. Molinet and R. Mittra,Asymptotic Methods in Electromagnetics,

Springer-Verlag, Berlin Heidelberg, 1997.

[3] Andronov, I. V. and D. Bouche, Asymptotic Expansion of the Electromagnetic

Field Induced by a Dipole on a Perfectly Conducting Convex Body,Journal of

Electromagnetic Waves and Applications, Vol. 9, No. 7/8, 1995, pp. 905-924.

[4] Munk, P., A Uniform Geometrical Theory of Diffraction for the Radiation and

Mutual Coupling Associated with Antennas on a Material Coated Convex

Conducting Surface, Ph.D. Dissertation, The Ohio State University, 1996.

[5] Syed, H. H. and J. L. Volakis, High-frequency Scattering by a Smooth Coated

Cylinder Simulated with Generalized Impedance Boundary Conditions,Radio

Science, Vol. 26, Sept-Oct 1991, pp. 1305-1314.

[6] Pathak, P. H. and N. Wang, Ray Analysis of Mutual Coupling Between

Antennas on a Convex Surface,IEEE Trans. Antennas Propagation, Vol. Ap-29,No. 6, Nov 1981, pp. 911-922.

[7] Hussar, P. E. and E. Smith-Rowland, Formal and Computational Aspects of theCreeping-Ray Problems on a Singly Coated Double-Curved Convex Surface,

2001 North American Radio Science Meeting, July 2001 (to be published).

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[8] Harrington, R. F., Time-Harmonic Electromagnetic Fields, McGraw-Hill, New

York, NY, 1981.

Asymptotic Solutions

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