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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
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.