modal theory of long horizontal wire structures above the earth - part 2
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Olsen KuesterTRANSCRIPT
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Radio Science,Volume 13, Number 4, pages615•623, July-August 1978
Modal heory f longhorizontal irestructuresboveheearth,2, Propertiesf discrete odes
Robert G. Olsen
Department of Electrical Engineering, Washington State University, Pullman, Washington 99163
Edward F. Kuester and David C. Chang
Department of Electrical Engineering, University of Colorado, Boulder, Colorado 80309
(Received July 13, 1977.)
The characteristics of the discrete propagation modes on horizontal thin-wire structures located
above a dissipative earth are investigated. In addition to the well-known transmission ine mode,
a new root of the characteristic equation is found which is identified as a surface-attached
mode becauseof its close connectionwith the Sommerfeld pole (Zenneck wave) in some parameter
ranges. Under many conditions the surface-attached mode suffers substantially less attenuation
along the propagation direction than does the transmission ine mode. Numerical investigation of
the propagation constants of the two modes is made, and field plots for the modes for a variety
of wire parametersare presented.
1. INTRODUCTION
In the first part of this paper [Kuester et al.,
1978], the excitation of current on an infinite thin
wire over a finitely conductingearth by an arbitrary
source was considered, and it was demonstrated
that a unique decompositionof the currents nduced
on the wire into discrete and continuous modal
components is possible. In this part of the paper,
we examine the •properties of the discrete modes
more closely and show that, due to the singularities
of the function M(a) (where M(a) = 0 is the
characteristicequation for these modes), there are
generally two modes in the neighborhoodof ot =
1, and that for some parameter ranges the new
surface-attached mode can have substantially
lower attenuation that the well-known transmission
line mode. It is also found that, for some particular
values of the physical parameters, the two modes
have identical propagationconstants. Such a double
root of the characteristicequation s called a degen-
erate mode, and under these operating conditions,
conventional transmission ine theory breaks down
entirely.
Most of the existing literature on the wire-over-
earth problem has been in the context of ordinary
or modified transmission ine theory. However, at
the beginning of the present authors' investigations
Copyright 1978by the AmericanGeophysical nion.
(a preliminary announcement of whose results
appeared n 1974 [Olsen and Chang 1974]), it was
discovered that the behavior of the transmission
line mode and of a single spectrum of radiation
modes was inadequate to describe the input con-
ductance of an infinite antenna over a conducting
half space [Chang and Olsen, 1975]; it failed to
oscillate about the free-space value as the height
of the wire above the half space continued to
increase. Only by including the effect not only of
a second discrete mode, but also of an additional
branch of radiation modes was it possible to obtain
the correct behavior. Regardless, therefore, of
whether or not it proves easy to excite the new
mode individually or not, its effect must be ac-
counted for in order to achieve a proper under-
standing of wire-over-earth structures at high fre-
quencies.As in Part I of this paper, the frequencies
of interest are such that wires longer than 10 rn
or so are electrically long. Therefore, end effects
and discontinuities can be analyzed independently
(e.g., by Wiener-Hopf methods) and used to study
wires which are long but finite [Olsen and Chang
1976].
2. SINGULARITIES OF M(a)
In equation (16) of Kuester et al. [1978], it was
shown that the discrete propagationmodes satisfied
the characteristic equation M(a) = O, where
0048-6604 78 0708-0615501.00 615
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616 OLSEN, KUESTER, AND CHANG
M(•x)= Mo(•X- i• s o0M (o0/'qo• (1)
Mo o = •2 H•o) (•A) - Jo •)H• • (2•H)]
+ P(a;2n)- a:Q(a;2H) (2)
Mi (•) = •2n(ll••) -- J• •) {•2n•l' 2•n)
- e(a;2n) + a:Q(a;2n)} (3)
• = (1 -- a2)1/2; n= (n2 - a2) /2; Im •, Im •n• 0
and H = klh, A = kla are the heightand radius
of the wire normalized by the free-space wave-
number , w•e n is the complex efractivendex
of the earth. A propagationactor of exp(iklaZ --
i•t) is associatedwith the mode.
The surface mpedance (a) of the wire will
be analytic for all finite a since it is associated
with a structure bounded in the transverse plane.
Only the analyticityof P and Q, which were defined
• Part 1. as Fourier •tegrals with respect to a
variable •, remains to be determined. We recall
that P and Q are actually special cases of a set
of functions which determine the potentials n terms
of which all electric and magnetic ield components
above he earth canbe expressed,and are the values
at Y = 0 of
2 •ø xp(--UlX+XY)
(•;x, ¾)= •
lqT U 1 -[' U2
dX (4)
2 o xp(--UlXX )
(a;X, Y) = • •
lqT n u I -[- U2
where
dX (5)
Ul _- (•k2 _ 2)1/2} U --' •x -- •2n)1/2'Re Ul,U >' 0
(6)
Here X - klX and Y- klY are normalizedcoordi-
nates in the plane transverse o the wire.
It is required that branch cuts be taken in the
complex a plane to ensure hat fields and currents
given by Fourier integrals in the a plane decay
properly at infinity (since the sourceof excitation
considered by Kuester et al. [1978] is finite),
regardlessof how the a-plane integration path is
deformed. These cuts will define our proper Rie-
mann sheet. Any roots of the modal equation ound
in this sheet will correspond to discrete modes
whose fields decay as the distance from the wire
approachesnfinity in any transversedirection,and
will be denoted proper modes.
X-PLANE
+•
Fig. 1. Integration aths or Sommerfeldntegralsn X plane.
Let us consider irst P(a;X, Y). If [ variesaround
a semicirclein the X plane) = eiø,0 <_0 <_
,r, then the branch cuts of its integrandat _+[ in
the X plane move as indicated n Figure 1. Now
the integrand s the same along the contour of
integrationC for 0 = 0 and 0 = ,r except or the
segment etween [ and +[, where the signsof
u are opposite, nd thereforeP takes different
values for arg [ = 0 and arg [ = ,r. However,
the semicircle racedby [ correspondso a complete
loop n the X-plane roundoneof the branchpoints
a = + 1 (Figure 2). Since+ [ cannotactuallycross
over the real axis of the Xplanewhile the integration
contour C remains on the axis, the lines Im [ -
0 shown n Figure 2, whichare alreadybranchcuts
for the Hankel functions, are likewise cuts for
P(a;X, Y). By similarargumentst can be shown
that P possesses ranchcuts along the lines Im
(l - PLANE
/•- Im n=O
Fig. 2. Locations f a in complexplane;branchcuts m • =
0 and Im •n = 0.
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LONG HORIZONTAL WIRE STRUCTURES, 2 617
[, = 0. It is easily seen that the integrand for P
has no poles, and so we have completely charac-
terized its singularities.
Precisely the same argumentsas for P show that
Q also possesses he branch cuts Im [ = 0 and
Im [, -- 0. There is, however, also a pole in its
integrand, which may or may not appear in the
Riemann sheet in which the integration is taken.
The pole locations (there are two, symmetrically
located) are given by:
X= +hi,---- •B= +[n2/(n2+ l)-a 2] /•
-= [,• - ,•1 '/• (7)
where, or definiteness, e take X•, thusalso
a a, and the square roots to have nonnegative
imaginarypart. The notation a for the pole ocation
is chosen to emphasize he similarity to [ and [,,.
For a lossy earth, with 0 _< arg n _< 'rr/4, both
poles are indeed in the proper Riemann sheet
for all values of a. The intepretation of this pole
when 0/Oz = 0 (i.e., a = 0) is well-known [Batios,
1966]' it is the Sommerfeld-Zenneck urface wave
associatedwith the lossy nterface. When a differs
from zero, it is clear that the meaning of the pole
is that of the Zenneck wave travelling at some
(complex)angle with respect o the z axis in the
yz plane (the plane of the air-earth interface) so
that the z componentof the normalizedwave vector
is +_a,and they components +X•, recallinghe
expressions or the fields from Kuester et al.
[1978]). We emphasize hat in spite of the fact
that this portion of the total field may not dominate
in any given range, neither can it be neglected,
for it is often at least the same order of magnitude
as the entire field, even for dipole sources [Barios,
1966].
Let us then tracethe semicirclea = I[•l eiø•
0 _<0• _< rr.The polesmove as indicated n Figure
3. If n is not real, so that [ is not real simultaneously
with [a, then the values of Q corresponding o
0 = 0 and0 • = 'rrdiffer precisely y the difference
of the pole residuesat _+ [a. Since the semicircle
traced by [a corresponds o a complete oop in
the a plane aroundone of the points +-a, it follows
that in addition to _+ and +_n, he function Q(a;x, Y)
must also possessbranch points at _+a . For Y
_>0, Q may be evaluatedby deforming the h plane
contour of integration upward over the two branch
cuts; for Y _< 0 by deforming downward; in both
cases the residue contribution from the pole can
•-PLANE
+•j s (0= •.)
ß
+/JB 0 =
+/jB (O=O)
Fig. 3. Integration paths for Sommerfeld ntegrals n X plane
relative to pole location.
be computed explicitly to be
exp(--Ulp + iX•, rl) <8>
•B n4- 1
where l• = i/(rt q- 1) /2 -- i//•, the ootbeing
chosen o that Re(Ulp) ->0. It shouldbe noted
that if the integrationcontourwere indented o allow
the poles o come between t and the real axis (i.e.,
if we allow 0a > 'rr) the residue contribution 8)
would give wavesgrowingexponentially s I YI •
0%and would correspond o values of a no longer
on what we have designated the proper Riemann
sheet. inceor large n , (8)containsfactor 3,
it is tempting to drop it in approximatingQ, but
it should be noted that this term contains an inverse
square oot singularityas a --->+_ a• and cannot,
in general, be neglected.
Using these or analogousmethods,all the func-
tions of a defined as Sommerfeld integrals with
respect o X, whichwere encounteredn connection
with the excitation problem of Kuesteret al. [1978],
are found to have branch-cut singularities with
branch points at +_1, +_n,and +_et. In closing his
section, we should note that an inverse square-root
singularity imilar o that n (8) wasalsoencountered
recently n an analysisof open microstrip [van der
Pauw, 1976], and s due in that case o the influence
of a pole corresponding o the lowest-order TM
surface wave of a groundeddielectric slab.
3. NUMERICAL SOLUTION OF CHARACTERISTIC
EQUATION
In a numerical search for solutions of M(a) =
0 it is very time-consuming o repeatedly compute
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618 OLSEN, KUESTER, AND CHANG
the Sommerfeld integrals P and Q by numerical
integration, since this must be repeated for each
different value of ot tried. We have developed a
number of accurate analytical forms for P and Q
which employ series expansions aking much less
computer ime. Detailed derivationswill be present-
ed elsewhere; we give the most useful expressions
in appendixA. An essential eature is that the proper
singularitiesnear [ - 0 and [B - 0 (where the
most important modes are expected to be found)
are retained.
We consider first the bare, perfectly conducting
wire, for which s(a) = 0. Since or Inl- 0%
P and Q vanish, a solution of M o •) = 0 should
be expectednear • = 1 if Inl is large. This solution
should correspond to the well-known result of
transmission line theory first given by Carson
[1926]. Since, however, near • = aB, Q could
conceivably dominate the characteristicequation,
it also seems possible that a mode could exist in
this region, as well, which would not be found if
Q (or more specifically (8)) were neglected. Using
these rough estimatesas initial guesses, he charac-
teristic equation was solved usingNewton's method
to converge to the exact roots.
In Figure 4, results are shown for a relatively
highly conductingearth. The refractive index n used
here corresponds o a relative permittivity e/e o •-
10 and a conductivity of • •- 2.8 millimho/m at
0.02
O.OI
_ 0'2
0.3
0.2
• 0.3
1.00 I.O •
Re a
Fig. 4. Mode propagation constants or bare wire above lossy
earth as function of h/h: n = 5.52 + i4.53, a/h = 0.01.
a frequency of 1 MHz. In this case, as indeed was
found in all but one of the cases we studied, two
modeswere found. At large wire heights,one mode
is found in the vicinity of a a, while another is
found nearer • = 1. The latter can be shown by
perturbation arguments to agree quite well in this
parameter range with Carson's [ 1926] result, to
which a number of authors have made extensions
[ Wise, 1934, 1948; Kikuchi, 1957;King et al., 1974].
A mode closer to a a than to 1 will be designated
surface-attached because of the influence of the
pole in the Sommerfeld integral Q, while in the
oppositecase we use the general headingof struc-
ture-attached, indicating fields not spread out
along the surface. It is to be recognized that •
for the structure-attached mode approaches 1 as
h --> 0% in which limit it no longer functions as
a pole in the excitation problem, because of the
logarithmic singularity possessedby M(a) at [ =
0. This, of course, is known from the theory of
long antennas [Wu, 1969]. As the height of the
wire decreases, he structure-attachedmode gradu-
ally becomes surface-attached, while the surface-
attached mode becomes another kind of structure-
attachedmodewhosepropagationconstant for very
small h) is predicted by a result of Coleman [1950]
[see Wait, 1972;Chang and Wait, 1974] to approach
ot --- (n q- 1)/2] 1/2.The potential sefulnessf
the surface-attachedmode at low heights because
of its much smaller attenuation constant is evident.
Clearly, as a result of continuously varying the
height , a propagationonstant%,which s ocated
in the surface-attached region of the a plane can
move into the structure-attached region, and vice
versa. It is thus not possible to label a single mode
trajectory as having either of these properties.
Figure 5 examines a case where the earth is less
lossy. This refractive index corresponds more
closely o a higher requency, and somewhatdamper
earth, with e/eo -• 27, and • -• 56 millimho/rn
at 100 MHz. When the wire radius is 10 3 wave-
lengths (curves 2), the qualitative behavior of the
two modes s much the same as in Figure 4, although
the new location of a a has distorted the curves
somewhat. When the radius is decreased to 1.66
X 10 4 wavelengths,owever curves ), an inter-
esting phenomenon is seen to have occurred. The
structure-attachedmode at low heights passescon-
tinuously to a = 1 as height is increased without
ever attaining a definite surface-attached character.
The surface-attached mode, on the other hand,
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LONG HORIZONTAL WIRE STRUCTURES, 2 619
0.02 0.1
/ o.,//
/ /
O. Ol
I
o| i I
0.99 1.00
Fig. 5. Mode propagation constants for bare wire above 1ossy
earths unctionfh/h:n = 5.3+ iO.95'Oa/h 1.66
X lO-4,Qa/h.-'-.OX 103.
retains this property to some extent for all heights,
remaining n the vicinity of as. At some value of
a between these two, there must be a situation
wherein the two curves touch for some value of
h (probably around h /h = 0.15). This phenomenon,
where two roots of the modal equation become
one double root, is known as modal degeneration.
Examples of it have previously been found in the
earth-ionosphere waveguide [Krashnushkin and
Fedorov, 1972; Budden, 1961 and in the earth-crust
waveguide [ Wait and Spies,1972]. The full physical
basis for such degeneracy does not seem to be
understood at present, but some discussion of it
will be given in the concluding section. It should
be noted, however, that the exact degeneracydoes
not have to be achieved for the effect to become
noticeable; if the phase difference between two
propagating modes over a length I of wire has not
become significant compared to 2vr, the results will
be much the same as if they were degenerate.
Figure 6 shows results for a dielectric coated
wire (Goubau line) over the same relatively less
lossy earth of Figure 5. The dielectric has radius
a and refractive index nc, and the inner conductor
has adius . The surfacempedances (a) for this
case is given by Chang and Wait [1974]. Once
again, this time as a result of varying the refractive
index of the coating, we see that modal degeneration
has occurred, at nc somewhat less than 1.1, and
h/h about 0.25. The general behavior is much like
that of the bare wire, with the exception that the
structure-attached mode for large heights ap-
proaches a value of a larger than 1 on the real
axis, corresponding o the TM surface wave of the
G-line in free space.
It can thus be seen that Kikuchi's description
[Kikuchi, 1957; Chiba, 1977] of the continuous
transition between an earth return mode and a
surface wave mode can be modified by the existence
of a degeneracy. For sufficiently large coating
index, a variation in height does indeed produce
the transition between the structure-attached mode
and the surface wave. As the coating becomes less
dense, however, the continuous transition is inter-
rupted by the acquisition of surface-attached
properties.
4. FIELD DISTRIBUTIONS OF THE MODES
The fields of the discrete modes, the expressions
for which can be found by Kuester et al. [1978],
have been plotted for some special cases. For the
surface-attached modes it was found that much
more of the field shows up above the wire than
below it (and hence in the earth), as comparedwith
a structure-attached mode, so that the reason for
0.02
0.01
/5
0.2•
0.99 1.00
Fig. 6. Mode propagationconstants or Goubau ine above 1ossy
earth as function of h/h for various coating indices: n = 5.3
+ i0.95,/h = .006, /h= .005;h = 1.O, nc = 1.04,
Qnc= 1.1,Qn= 1.11,Ca= 1.25.
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620 OLSEN, KUESTER, AND CHANG
the smaller attenuation is simply that a smaller
percentage of the mode fields actually travel in
the earth.
In Figures 7 and 8, cross-sectional ield plots
are given for the two modes on a Goubau line above
the earth. The greater transverse spread of the
surface-attached mode as compared with the struc-
ture-attachedmode (in this case, predominantly he
G-line surface wave) can be seen. The fields of
the latter are all nearly linearly polarized, with the
exception of points very far removed from the wire.
On the other hand, the fields of the surface-attached
mode are elliptically polarized near the wire, but
more linearly polarized near the earth's surface,
where the E field is also more vertical, and less
attenuated with distance from the wire. Plots such
as these can be used in conjunction with the
excitation discussion of Kuester et al. [1978] to
design an efficient excitation scheme to select one
of the discrete modes in preference to the other.
This is a necessarydevelopment f practical utiliza-
tion of these modes is to be achieved.
Some further insight into the characteristics of
the modes can be obtained by examining their
asymptotic behavior in the transverse plane. Once
again, since the procedure for all field components
4.7
- I
y/X -0.2
9.0
8.5/13.0
I
-0.1
WIRE
0.2
0.1
00
Fig. 7. E-field lines for structure-attachedmode on Goubau ine.
Arrows give field strength along direction of major axis of
polarizationellipse. Numbers A /B are ratio of major to minor
axis, n = 5.3 + i0.45, h = 0.24h, a = .007h, b = .0lb, nc
= 1.25, ot = 1.0123 + i0.01134.
9.8
4.3
2.0
1.5
2.2
6.7 I0.0
I
WIRE
0.2
A/B=•o
. I
y/x -o.2 o
Fig. 8. E-field lines for surface-attached mode on Goubau line.
Arrows give field strength long directionof major axis of
polarizationllipse.Numbers /B are ratioof major o minor
axis,n = 5.3 + i0.45, h = 0.24h,a = .007h,b = .0lb, nc
= 1.25, ot = .9854 + i0.00577.
is similar, we shall consideronly the integral (5)
for Q(tx;X, Y). The asymptoticbehavior of Q is
foundfrom the propertiesof the functionappearing
in the exponent:
f(h) = -i(h cos • + iulsin •))
(9)
where 4) = arctan (X/Y), and the exponentof
(5) becomes(X 2 + y2)l/y(•k). Sincewe have
0 _• 4) -• 'rr for the integral (5), it is clear that
the saddlepoint h• for f(h) (where '(h•) = 0) is
located at • - • cos 4). As is well known [Felsen
and Marcuvitz, 1973], by deforming he integration
path into a steepest-descentath (f(
2
= s, -oo ( s (+oo), the major contribution o
(5) is made to come from the vicinity of the saddle
point, and s dominated y a factor exp[-(X 2 +
y2)l/•f(h•) = exp[i•(X2 + y2)1/2].Some om-
plication arises from the pole in the integrandof
(5), however, and different cases can arise.
Let us consider mode ocated t %1 such hat
I•al is smaller han I•[, and thus we identify this
as a surface-attachedmode. The singularitiesat
•, and • are shown n Figure 9, and the saddle
point lies on the dashed ine between +• and -•.
At 4) = 'rr/2 (observation point directly over the
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LONG HORIZONTAL WIRE STRUCTURES, 2 621
(•) xPLANE
ß
.
•' « ARG2 •'/4
Fig. 9. Steepest-descent aths in h plane for fields of discrete
mode%,.
wire) the steepest-descent ath (SDP) is located
at curve 1 in Figure 9, and no influence of the
pole s felt. At some ntermediate ngle b, he SDP
is locatedat curve2, andthe pole at h•,hasbeen
picked up (the detailed structure of the field in
this vicinity must be describedby Fresnel ntegrals
[Felsen and Marcuvitz, 1973]). Finally, near {b =
0, curve 3 is reached (observation point near the
surfaceof the earth), and, depending n the mutual
dispositionf • andh•,, he residueermcan be
as large as, or actually arger than, the saddlepoint
term, and thus the ground wave term due to this
residue forms the major part of the field of the
mode. Note that it is the difference in relative
locations of pole and branch cut that make this
possible, n contrast o the situationof spacewave
and ground wave for a Hertzian dipole. source
[BaHos, 1966].
A mode%2 such hat I<l is smaller han I<l
would be designated structure-attached, and the
situation in the h plane is shown in Figure 10. The
SDP's for the same angles as in Figure 9 are again
shown. In this case, however, not only is the pole
not picked up until a much smaller angle, but its
contribution is much smaller than that of the saddle
point, and it never forms the dominant part of the
field.
5. DISCUSSION AND CONCLUSION
A numerical solution of the characteristic equa-
tion for a thin-wire structure located parallel to
a lossy earth has shown the existence of a second,
heretofore unknown, propagating mode in the
proper Riemann sheet of the •x plane (i.e., whose
fields decay at infinity in all directions normal to
the wire). Since the only approximation involved
in the derivation of this equation was a thin-wire
assumption [Kuester et al. 1978], it must be verified
that the location of the second mode is not simply
due to this approximation. Pogorzelskiand Chang
[1977] have investigated the error involved by
including angularly distributed and azimuthally di-
rected currents around the wire surface, and show
that their effect is negligible for all •x except within
very small circles surrounding (x = •x and (x =
1. It can be verified using these criteria that none
of the data presented n this paper fall within these
circles, and so proximity effects contribute a neg-
ligible correction to our results.
The possibility of modal degenerationwhich has
been demonstrated by our results warrants some
additional discussion of the phenomenon. The de-
generacyof the kind found here is, in the language
of mathematical linear operator theory [Kato,
1966], algebraic but not geometric. This should be
distinguished rom the related phenomenonwhich
occurs, for instance, in closed circular waveguides
between TE and TM modes. The latter degeneracy,
which is both algebraic and geometric and could
be designated semi-simple following Kato
[1966], is accidental, a result of the high symmetry
of the waveguide. The fields of these modes,
however, due to their distinct polarizations, remain
mutually orthogonal even at degeneracy, and
sources can be constructed to excite one of the
modes at the expense of the other. There remain,
X-PLANE
G:) :
(•)/•/2-•
•/2-
'\\
Fig. 10. Steepest-descentpaths in h plane for fields of discrete
mode%,2.
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622 OLSEN, KUESTER, AND CHANG
in this case, two orthogonal ½igenfunctionsmodal
fields) correspondingo the•degenerate igenvalue.
From the construction of Kuester et al. [1978],
however, it is seen that as two modes of the present
structureapproachdegeneracy, heir fields tend to
become identical, and that at such a nongeometric
degeneracyonly onemodal field remains. However,
an adjoinedmodal ield appears ecauseheGreen's
function resolventkernel) possesses doublepole
at the degenerate igenvalue; his can alsobe viewed
as the interference produced by two distinct but
almost degeneratemodal fields.
Other researchers [dos Santos, 1972; Levitt and
Marx, 1974] have found a second mode on an
improper Riemann sheet (though their definition
seems to be different from ours). In Figure 11,
we compare our results (this time for fixed, un-
normalizedphysicaldimensions,dielectricconstant
of the earth, and conductivity)against hose of dos
Santos [1972] as a function of frequency. What
dos Santoscalled a proper mode agrees ather well
with our structure-attachedmode (the data point
at 12 MHz appears o have been misprinted n dos
Santos paper, and we have made what seems to
be the correct modification to the real part of a).
0.06 -
0.05
o.o4-
0.02 - AI2
148'-:'e'-'•x/3'
o.o, z
ol %,7o . ,
0.97 0.98 0.99 1.00 1.01 1.02 1.03
Rea
Fig. 11. Mode propagationconstants or bare wire above 1ossy
earth as functions of frequency. The dashed curve indicates
the motion of the branch point otB with frequency. a = 2.5
x 103 m, h = 1.0m, e/e = 15, = 10-2mh_o/m;
Carson-Colemanode,Qdosantos'roperode,{•)surface-
attached mode.
The discrepancy, which was largest at 6 and 12
MHz, can be attributed n part to readingdos Santos'
results from graphs, and from tables of numbers
having only three-decimal-place accuracy, as well
as to possibledifficulties in his numerical procedure
for integrating P and Q. Our surface-attachedmode,
however, has no analog in dos Santos' results; his
improper mode is even further removed from the
branch point at a• than is the structure-attached
mode, and does not show up on the portion of
the a plane exhibited in Figure 11. It is well worth
noting that at around 215 MHz, the surface-attached
mode disappears nto the branch cut Im [ - 0 and
passesover to an improper Riemann sheet. Although
by this time the wire is about 2/3 of a wavelength
over the ground, and the ground effect might be
expected to be small anyway, it is of interest that
the surface-attached mode does not always show
up as a proper mode, as in Figures 4-6. An exhaus-
tive search of all possible Riemann sheets for
improper modes has yet to be made, as does a
study to determine which, if any, of these can
contribute significantly to the field of a finite source.
APPENDIX A
The most useful formulas for P and Q valid for
the range of parameters examine in this paper are
these: For P,
P(ot;X) • (2•2n/N2)(l/[.X){- [(IX) 2/[.X]Ho(I)([X)
+ [i[XH(l ) (IX)- 2/-rr] 1 - i/[.] + al(--i[nX )
-- Yl(-i[nX)- 2i/'rr[nX } (A1)
for I /In I and IXI not too large, where H• is the
Struve function of first order [Abrarnowitz and
Stegun, 1965, p. 496]. For Q,
Q(a;X) = Qo(a;x) + Ql(a;X)
under similar conditions, where
Qo(o•'X)2rt2//•N2)[H•ol)(•X)'1-ig/h)e•x/a
2arcsin(X•,/•
He l)(1 h•,•X) + --
• (x,,/0
QI (oL;X)- (ih [/N 2h2n2)H(l) IX)
and
He(l•(a, ) = e 'H(o) t)dt
o
(A2)
(A3)
(A4)
7/17/2019 Modal theory of long horizontal wire structures above the earth - Part 2
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LONG HORIZONTAL WIRE STRUCTURES, 2 623
is known as an ncompleteLipschitz-Hankel ntegral
or Schwarz function [Agrest and Maksirnov, 1971;
Luke, 1962] for which seriesexpansionsare availa-
ble. In all the above, N 2 = n2- 1, h: = n: +
1, with Im (N) and Im (•) taken positive.
Results or solutionsof the modal equation using
these approximations have been checked against
those presented n this paper, which were obtained
using exact values of P and Q (computed by
numerical integration). For wire heights up to a
third of a wavelength, with n = 5.3 + i0.45, the
exact and approximate values of a differed at most
by 10 5 in either eal or imaginary art, andwere
frequently s close s 10 7 When he wire height
approached half wavelength, he error approached
10 4 (whichcouldgive substantialrrors n the
attenuation constant); however, alternative repre-
sentationswhich are accurate for larger heights can
also be derived. The solution of the characteristic
equationo withinan accuracy f 10 s tookabout
5 sec on a CDC 6400 when the Sommerfeld integrals
P and Q were integrated numerically, while only
0.1 sec was required when the approximate expres-
sions above were used.
Acknowledgments. This paper represents results of research
carried out since 1970 under partial support from the following
organizations: he US National Oceanic and Atmospheric Ad-
ministration (NOAA) under grants E22-58-70(G) and N22-126-
72(G), and Rome Air Development Center (RADC/ET) under
contract no. AFF19628-76-C-0099. The authors are indebted to
J. R. Wait and L. Lewin for many interesting and profitable
comments and discussion on the present work. Some of the
numerical results were obtained by S. Plate.
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