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This dissertation has been 63—1592 microfilmed exactly as received
POST, Robert Elgin, 1928THE FLAT-EARTH APPROXIMATION TO THE SOLUTION OF ELECTROMAGNETIC PROPAGA­ TION IN A STRATIFIED TERRESTRIAL ATMOS­ PHERE.
Iowa State University of Science and Technology Ph.D.,1962 Engineering, electrical
University Microfilms, Inc., Ann Arbor, Michigan

THE FLAT-EARTH APPROXIMATION TO THE SOLUTION
OF ELECTROMAGNETIC PROPAGATION Bï A STRATIFIED TERRESTRIAL ATMOSPHERE
Robert Elgin Post A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of
DOCTOR OF PHILOSOPHY Major Subject: Electrical Engineering Approved :
Signature was redacted for privacy. In Charge of Major Work
Signature was redacted for privacy. Head of Major Department
Signature was redacted for privacy.
Iowa State University Of Science and Technology
Ames, Iowa 1962

October 4. 1962
I hereby request that the changes listed below which I desire to make in the manuscript copy of the thesis submitted for the degree Doctor of Philosophy be approved.
Signed; Signature was redacted for privacy. The term exp (-jagp/2K2) be changed to exp (-jagkQp/2K^) in Equation 143. The term exp (-jagkQp/2K^) be added to Equation 168 inside the summation sign. The term exp [-(1.115 x 10 5)(l + a ^)^pj, be changed to
exp [-(9.65 x 10 ^)(l + a ^)^3pJ in Equation 194. The term exp [-(6.7 x 10 ^)(l + a ^)2^5pJ, be changed to
exp [-(5.8 x 10 5)(1 + a ^)2^pJ in Equation 266. These changes will affect the calculated curves plotted on Figures 2 and 4 by increasing the slope of these curves. Approved:
Signature was redacted for privacy. In Charge of MS})or Work

Signature was redacted for privacy. Head of Major Department

ERRATA

Signature was redacted for privacy.

ii TABLE OF CONTENTS

INTRODUCTION

PROPAGATION IN A STRATIFIED ATMOSPHERE

5

A. Definition of the Problem

5

B. Maxwell's Equations for the Inhomogeneous Atmosphere

8

C. The Magnetic Dipole

9

D. The Electric Dipole

13

E. The Flat-Earth Approximation

18

F. The Airy Integral

19

G. The Green's Function

23

THE COMPLETE SOLUTION FOR THE MAGNETIC DIPOLE

27

THE COMPLETE SOLUTION FOR THE ELECTRIC DIPOLE

43

COMPARISON WITH EXPERIMENTAL DATA

50

EXTENDING THE GREEN'S FUNCTION TO INCLUDE THE EFFECTS

OF ELEVATED STRATIFICATIONS

59

CONCLUSIONS

86

BIBLIOGRAPHY

88

ACKNOWLEDGEMENTS

90

1
I. INTRODUCTION
The problem of the propagation of electromagnetic energy around a spherical body where the diameter of the sphere is large with respect to the wavelength of the energy was first solved by Lord Rayleigh in terms of an infinite series of spherical harmonics. The solution was practical­ ly useless from an engineering point-of-view because of the large number
of terms required to approximate the final answer. Subsequently, G. If.
Watson developed a transformation which transformed the infinite series into a contour integral which is then evaluated by the method of residues. Die residues of the integral involve asymptotic expansions of the Hankel functions of order one-third or the Airy integrals. A second approach to the problem, and the approach which serves as the basis of this paper is that developed by Pryce (16) and Pekaris (15) in which the earth is as­ sumed to be flat and the atmosphere homogeneous. Hie solution now reduces to an infinite integral of zero-ordered Eessel functions. Schelleng and Burrows (17) proposed a model wherein the earth was assumed to have a modified radius of about 4/3 the actual radius and the earth's atmosphere was assumed to be homogeneous. This model was used to account for the gradient of the refractive index of the earth's atmosphere. All of these models refer to the so-called "normal mode" of propagation as distinguished from the "turbulent scatterer" theory of propagation of electromagnetic energy which will be discussed later. Very complete discussions of normal mode theory are included in: "Terrestrial Radio Waves" by Bremmer (3) and Volume 13 of the M.I.T. Radiation Laboratory Series: "Propagation of Short Radio Waves" by Kerr (12). In 1958, Carroll and Ring (4) published

2
the results of a very extensive investigation into the propagation of radio waves by the normal modes of the atmosphere. These results indicated that the classical "airless earth" modesi were not valid for calculating the effects of the internal reflections of a stratified atmosphere. In 1959 Tukizi (18) published a theory which, while using a different approach to the problem, achieved results which had excellent comparison with ex­ periment and, which are corroborated by this analysis. Tukizi's results indicate the utility of the normal mode theory in predicting the strength of the field in the diffraction region and especially in predicting the radial attenuation of the field in the diffraction region.
The term "diffraction region" as referred to by Tukizi (18) and a number of other authors stems from the theory that the presence of the field over the horizon from the source is due to a diffraction process caused by the curvature of the earth. Other authors call this same phe­ nomenon by different names; for example, Carroll and Ring (4) use the term "Twilight region". In this paper, the terminology which is favored by the proponents of the "turbulent scatterer" theories will be used; "transhorizon field". This will mean that field which is over the horizon from the source.
Hie other theory of propagation of short radio waves over the horizon was first introduced by Booker and Gordon (2) in 1950. Basically, the Booker-Gordon theory is that spherical or elliptsoldai anomalies in the refractive index act as scatterers of electromagnetic energy. These anomalies, often called "blobs", are supposedly located in the common volume of the transmitting and receiving antenna beams and serve to

3
scatter some of the transmitted energy down to the receiver. These 'mobs" are being formed continuously, reducing in size, and finally dissipating, and are a direct manifestation of the turbulence of the atmosphere. The turbulence can be described in a statistical fashion and there are sev­ eral theories concerning turbulence which serve as the basis for these arguments (9, 19, 20). The advantage of the turbulent scatterer theory is that the statistics of the turbulence lead directly to a statistical character for the field in the diffraction region. In nonoal mode theory, the atmosphere is assumed to be static, so that there is no statistical character to the transhorizon field. The turbulent scatterer theory is very attractive from a number of points-of-view, most of which involve the statistical character of the field in the transhorizon region. The most significant shortcoming in the turbulent scatterer theories is that the intensity of the turbulence, or the variations in the index of refraction in the "scattering blobs", is not sufficiently large to account for the fields observed since in most cases, the common volume of the antennas is very high in the troposphere.
At the present time the Booker-Gordon theory is most widely accepted with the majority of the work being done in correllating the meteorolog­ ical phenomena with the field in the transhorizon region. A number of good resumes of the Booker-Gordon theory and the various turbulence theories are available (2, 9, 19, 20).
The present analysis uses the flat-earth approximation applied to the normal mode theory of transhorizon propagation. Die results reported by Pryce (16) are acheived by a different technique and the analysis is extended

4
to include the effects of an inhomogeneous atmosphere with a constant gradient of refractive index. Finally, a technique for approximating an arbitrary profile of refractive index is developed and tvo model atmospheres are considered. Field strengths calculated on the basis of this analysis are compared with experimental data reported by Dinger, Garner, Hamilton and Teachman (5) with good results.

5
II. PROPAGATION H A STRATIFIED ATMOSPHERE
A. Definition of the Problem
The problem to be considered is that of the propagation of electro­ magnetic energy around a spherical earth from a source located at some point on or above the surface of the earth. The atmosphere of the earth will be assumed to be spherically stratified; that is, the index of re­ fraction is a function of radius. The coordinate system for the spherical earth is shown in Figure 1. The source is located at the point r = a + d, where a is the radius of the earth. Thus, d is the height of the source above the surface of the earth.
The analysis is concerned with the transhorizon field only, since the intent of the analysis is to develop an analytical technique •which will allow a piecewise-linear approximation to any profile of refractive index. The transhorizon field will be solved for both magnetic and electric di­ pole sources located in an atmosphere having a linearly varying profile of refractive index. This solution will be compared with the result ob­ tained by Pryce (16) for the case of a homogeneous atmosphere. This will be the case of a profile of refractive index with zero slope. The article by Pryce (16) is the basis of the present analysis since the "earth-flattening" transformation and the Airy Integral solution with its rapidly convergent asymptotic expansion is a very attractive technique. It should be pointed out that Pryce (16) solved for the cases of vertical and horizontal dipoles rather than the cases of electric and magnetic dipoles considered in this analysis. The essential difference lies in the fact that there are no

6 Figure 1. The spherical polar coordinates of the problem

7
azimuthal variations to consider when using the elementary dipoles as op­ posed to the more practical antennas of Pryce's analysis. The fact that the elementary antennas of this analysis are not practical antennas con­ figurations does not detract from the significance of the results because it is the ratio of the energy in the transhorizon region to energy which has traveled a corresponding distance in free-space which has direct ap­ plication in engineering problems. This ratio would be the same for any antenna configuration of a given polarization. Thus, it is a simple mat­ ter to estimate the strength of the field in the transhorizon region from the free-space field strength.
The analysis will begin with Maxwell's equations from which Hertzian vectors for both the magnetic and electric dipoles will be developed. The components of the electric and magnetic fields for both types of polari­ zation will be expressed in terms of the Hertz vectors. The inhomogeneous wave equation for the Hertz vector will be reduced to a pair of singledimensioned differential equations by the separation-of-variables tech­ nique. At this point the "flat-earth"approximation will be made. The "flat-earth" approximation is a transformation from spherical-polar co­ ordinates to cylindrical coordinates. After the transformation has been made, the two one-dimensional differential equations will be solved. The Hertz vector will be found by weighting each term of a complete orthonormal set of solutions in one variable by an appropriate Green's function of the other variable. The resulting expression will be the Hertz vector for a monochromatic elementary dipole of unit strength. The Green's func­ tion technique is well suited to making a piecewise or sectionally-linear

8

approximation to an arbitrary continuous curve. The resulting indefinite integrals are evaluated by means of the residue summation of the theory of complex variables. The asymptotic expansion of the Airy integrals allows the final expression, which is an infinite series, to be approximated by a few terms.

B. Maxwell's Equations for the Inhomogeneous Atmosphere

Maxwell's equations in spherical-polar form for application to an in-

homogeneous atmosphere are:

v x £(r, 9, f , t) = -n

^

V x 7~/(r, 9, </>, t) = a£(r, 9, /, t) + e(r)

^ ^

V • e(r)£(r, 9, /, t) = pQ V • >f(r, 9, /, t) = 0
Where £r(r, 9, pf, t) is the electric-field-intensity vector, 7/(r, 9, /, t)
is the magnetic-field-intensity vector, a is the conductivity of the med­ ium, pc is the conduction charge in the medium, e(r) is the dielectric constant of the medium; in this case, the permittivity of the medium is a function of the radial distance, and p is the permeability of the medium. The radiation will be assumed to be monochromatic of frequency œ; that is, the time-dependence can be written in the form; e^"*. Under this condition, Maxwell's equations can be written in time-independent spheri­ cal-polar form as;

9

V x E (r. oQ = -jauH (r, 9, /)

V x H (r, 9, <f>) = a E (r, 9, <f>) + jm (r) E (r, 9, <f>) 2
V • e(r) E (r, 9, /) =
V • if(r, 9, <f>) = 0

C. The Magnetic Dipole
A magnetic dipole of strength M is defined to be a loop of current of
magnitude I and radius r such that M = lim rl, and is characterized by r -*-o
J—>-00
the following field relationships:

Er = B,r He = 0 •
The formulation of the Hertz vector for a magnetic dipole in an inhomogeneous atmosphere shown here is due to Friedman (7). Maxwell's equa­ tions for field components with no azimuthal variations in a charge-free inhomogeneous non-conducting medium are:

V x E (r, 9) = -jcqj. H (r, 9) V x H (r, 9) = jof(r) E (r, 9) V ' £ (r) E (r, 9) = 0 V (r, 9) = 0

a
b 3
c
d

Ey Equation 3(c):

V . e(r)~E (r, 9) = 0 ,

10

s(r) E (r, 9) can "be written as the curl of a vector e(r)1T (r, 9). mat
ill
is;

e(r) E (r, 9) = -jo^i V x ^(rjTT^ (r, 9) »

4

The seemingly arbitrary form of Equation 4 is justified by the fact that it is desired to develop a Hertz vector which is applicable to a magnetic dipole. For this reason, the operation above and other somewhat arbitrary definitions in the equations to follow are justified. Using Equation 4 in Maxwell's equation, 3(b) leads to:

V x H (r, 9) - cu2n V x e(r)TT^(r, 9) = 0

5a

or V X [H (r, 9) - œ2H e(r)TTJr, 9)] » 0

The expression in parentheses can be written as the gradient of a scalar:

6

Solving Equation 6 for the magnetic field intensity H (r, 9), results in:

7

Equation 3(c) can be rewritten in terms of the Hertz vector as:

E(r, 9) = iffy Vxe(r)TTm (r, 6) .

8

Substituting Equations 7 and 8 into Equation 3(a) results in:

V x 77TT v x e(r)TT (r, 9) - co2^e(r)T[ (r, 9) - vY(r, 9) = 0. 9

t- ^ x /

in

in

11

Substituting Equation 7 into Equation 3(d) results in:

(Mi V • e(r) 1 \ m(r, 9) + V2 (r, 9) = 0 .

10

Friedman (7) points out that the divergence of Equation 9 will result in Equation 10, therefore it will be sufficient to consider only Equation 9. Assume that the Hertz vector is radially directed and given by the equation:

TTm (r, 6) ="ar r U^ (r, 9) .

Again, this somewhat arbitrary assumption is justified by the fact that it

leads to the proper field components. Actually, this Hertz vector must

reduce to the classical free space Hertz vector if the medium is assumed

to be homogeneous and it is known that the free space Hertz vector has the

same direction as the source dipole. In both cases, the source dipole is

oriented radially.

—ï

du (r, 9)

V x e(r)TTm (r, 9) =

e(r) —

,

11

and

,

==.

r,

v x^v X e(r)IT(r, 9)- -ar|?

.

ÔU

+ - Cot 9^

*4i %-$?} •

Ey Equation 9, Equation 12 must be equal to:

^ e(r)lT rUm(r,9) +% â^'9' +tfl

•

13

Equating the coefficients of the unit vectors if and results in:

12

, a^u

du__ n

^(D

— 9; -u ± ^ O ,—-U J-

>-<-> - TT U/ -

r dQ2 ' r

09 " ~ -

" "m " dr "

and

i aum + s\ iaC

?5~ë+5?Sê-?Sê'

15

Equation 15 can "be integrated once with respect to © to give

JflM) -

16

Substituting Equation 16 into Equation 14 results in:

i A

du

d2r U

17

Equation 17 is recognized as Helmholtz's equation in spherical polar co­ ordinates applied to a function with no azimuthal dependence. Thus, it is shown that for the magnetic dipole case, the equation to be solved is

V2 Um (r,©) + œ2n e(r) U (r,9) =0

18

regardless of the stratification of the medium. Once the scalar function U (r,9) has been determined the Hertz vector
TT"m(r,9) can be written and the components of the electromagnetic field
can be formulated. The field components are:

Er = 0
E9=°
dU

(a) Hr =

1 + H£(r)Um

d r

He = ?èe*

(d)

<e>

18

13

The equation "which must he solved to determine the components of an electromagnetic field propagating from a magnetic dipole located in an inhomogeneous atmosphere is:

.A i

du

0

d2rU

rr ~0~9f + rr COt 9 5dtiT + œ ^ €(r) r Umm + drT = 0 '

17

Assume a product-type solution of the form

Um(r,0) = R(r) 6(0) .

20

Substituting Equation 20 into Equation 17 results in:

TT~ + IT" + ^m2ne(r) +|- + cot 6|- = 0 .

21

2 The separation constant ^ is defined such that:

"

» .?

0 + cot e 9 + A 0 a o

22

m

and

,

/\2

R + — + (co2ne(r) - —|) R « 0

23

r

and

\2>0-

At this point it will be appropriate to develop a corresponding set

of equations applicable to the electric dipole.

D. The Electric Dipole

An electric dipole of strength M is defined to be two charges of mag­ nitude Q, having opposite polarity and located a distance d apart, such

14

that M - lin dQ .

d-"> o As in the case of the magnetic dipole, the time-dependence of the radiation will be assumed to be of the form, e^"*. Actually, the electric dipole is the radiating element, short compared with wavelength, which is used as an initial example in elementary texts on antenna theory (l). The formula­ tion of the Hertz vector for the electric dipole again follows the work of Friedman (7). Starting from Equation 3(d);

V -lf(r,9) = 0

H (r,©) can be written as the curl of a Hertz vector,

H (r,©) = V x 71^ (r,©) .

24

Substituting Equation 24 into Equation 3(a) results in:

V x E (r,©) = -jcqi V xTTe(r,0)

25a

or

V xlT(r,6) + jonTf^(r,e) = 0 .

25b

The expression in parentheses can be written as the gradient of a scalar <f>{ r,©).

E (r,9) + jonTTg(r,e) « V /(r,9)

26

Solving Equation 26 for the electric field intensity vector results in:

E (r,9) = -j<4iTT (r,©) + V <f> (r,9)

27

15

Substituting Equations 24 and 27 into Maxwell's equations 3(b) and 3(c) results in:

V x V x TTg(r,@) =» co2n e(r) 7Tg(r,9) + ja£(r) V / (r,9)

28

and

-jqi V • e(r)TT(r,9) + V • e(r)V pf(r,@) = 0.

29

Again, it is noted that the divergence of Equation 28 is Equation 29; there­ fore, it will be sufficient to consider only Equation 28. Assume that the Hertz vector to be radially directed and given by the equation:

Tfe(r,9) ="ar r Ug(r,0) .

30

Substituting into Equation 28 and equating the coefficients of the unit vectors"ar and aQ, results in the two equations:

du _ d^U

h h ^ oot 8 st + rJ?+ e(r)"e +

r

r d9

and

0

si

Integrating Equation 32 once with respect to 9 results in:

drU (r,9)

^ 'T9)- wun —i—

33

Substituting Equation 33 into Equation 31 results in:

h h 00t 8 s~t+

rj+ ro2|je(r)ue+

&jsirfy= 0

54

or

16
y!T co?Hb(r) ue 34

Equation 54 looks just like the corresponding equation which vas developed for the magnetic dipole except for the terms which are due to the gradient in the permittivity of the medium. This difference is to be expected since the Hertz vector is a measure of the electric field; and, since the elec­ tric flux density must be continuous, the electric field intensity must vary on account of the inhomogeneities in the permittivity of the medium. Just as in the case of the magnetic dipole, once the scalar Ug(r,6) has been determined, the Hertz vector is known and all the field components can be formulated. The field components for the electric dipole are:

a

^ = 0
H =0 r
dU
e " W

b
c 35
d e
f

In the electric dipole case, the equation which must be solved is:

17

.au . a^u _ e , -L

au , _ e , ,'d

à ti'+ ?1 §""îr + H r zài9? + r-2cot 9 r!+ mt*M u,

du

îM !ik. i lM u .o

si - ?

"e

Assuming a product-type solution, that is:

U (r,e) = R(r) 9(9)

36

Substituting into the differential equation and separating the variables

results in the tvo equations:

"

« .p

9 + cot 9 9 + A 9 = 0

37

e

2

R" + i(2 - r

)Rf + (a)2(i€(r) - — - y4)R - 0

38

e(r)

r

v '

2

.2

Where the separation constant ^ is defined such that > 0. The Equa­

tions 37 and 38 are quite similar to the corresponding equations for the

magnetic dipole, hovever, Equation 38 is more unwieldy than either of the

other equations. This analysis is confined to an atmosphere vith a lin­

early-varying profile of refractive index. This means that the gradient

terms of Equation 38 will be a number rather than some function of radius.

Thus, certain simplifying assumptions can be made during the analysis. In

particular, vhen considering the portion of the atmosphere in which the gradient of the refractive index is very small, Equation 38 reduces to the

equation developed for the magnetic dipole.

Having developed expressions suitable for determining the field com­

ponents of the electromagnetic wave propagating from either of the tvo

18
source types, the next step is to convert from the spherical geometry of the problem to the cylindrical geometry of the "flat-earth" approximation.
E. The Flat-Earth Approximation
The earth-flatting approximation is nothing more than a transforma­ tion from a spherical-polar geometry to a cylindrical geometry. The transformation equations are:
= a in r/a 39
p= a 9
Where p represents radial, distance or range and "^represents height. These are the same transformation equations used by Pryce (16) but reduced to the case of no azimuthal variations. The transformation is such that a cone with its apex at the earth's center becomes a cylinder and the sur­ face of the earth becomes a plane oriented such that the axis of the cylinder is normal to the surface of the plane. The introduction of the "earth-flatting" approximation is credited to Schelleng, Burrows and Ferrell (17) who, in an effort to simplify the analysis of the transhorizon field due to the curvature of the "rays" of energy from the source, decided to transform to a coordinate system where the rays became straight lines. This leads directly to the equivalent radius of the earth concept, or the so-called 4/3 earth radius. Subsequent work considered the earth to be flat which meant that the rays were bent upward. This is no problem if one is interested in a solution of the differential equation governing the propagation of energy as opposed to the ray-tracing technique. Pryce (16) credits the final form of the transformation equations to C. L. Pekaris

19
for the range transformation and to Prof. E. T. Copson for the height transformation. Pekaris has shown (15) that the error involved in making the small-angle approximation is less than 2$ for ranges up to one-half the radius of the earth. Copson pointed out that the height transforma­ tion » a £n r/a is preferable to the somewhat more intuitive 7^= r - a because the geodesies correspond to straight lines in the first case and only approximate straight lines in the second case.
The differential equation with >|, the height variable, as the inde­ pendent variable will be referred to as the height-gain equation. The differential equation having p, the radial variable, as the independent variable, will be called the range equation. Pekaris analyzed the error involved in approximating the height-gain equation by Stoke's equation and concluded that the error becomes quite large at moderate heights. Koo and Katzin (13) have shown that the height-gain equation can be made exact by making a change-of-variable so that the height-gain equation becomes Stoke's equation. It will be seen that the range equation transforms into
A Vessel's equation of order zero with a parameter /a. The solutions to
the transformed range equation will be zero-order Bessel functions and the solutions of the transformed height-gain equation will be Airy inte­ grals.
F. The Airy Integral The Airy integrals, signified by Ai(z) and Bi(z) are solutions of the differential equation known as Stoke's equation:

20

u y7 - z U(z) = 0

40

dz

The differential equation is of second order; therefore, there are two

linearly independent solutions, Ai(z) and Bi(z), defined by the integrals:

3

Ai(z) = -

x

it

cos (sz + 2— ) ds

41

and

00

Bi(z) =| ^ I exp (tz -|-)+ sin (tz +

dt

42

A complete discussion of the Airy integral is given in Jeffreys and Jeffreys (ll). The relationship between Airy integrals and the general solution of Stoke's equation, Bessel functions of order one-third, is given in Die Annals of the Computation Laboratory of Harvard University, Vol. II, entitled "Modified Hankel Functions of Order One-Third" (10). The Airy integral solutions are shown to be superpositions of the Hankel function solutions. The particular advantage of the Airy integral formu­ lation is rapid convergence property of the asymptotic series expansion of the Airy integral solutions. The asymptotic expansions and their associated intervals of convergence are listed below.

Ai(z)^_L t-i exB (_!,3/8)L . 1=5 z-3/2 + kl^ll z-3

8f?

3

I

1,48

2 2! 48'

for -it < arg z < +it.

1'7'13'5'=11'17 z_-9'/2 +. ••• 43 3Z 48

21

Bi(z)• ^ s"1'4 exp <§ Z3/2)[l + Ma z"5/2 +

Z":

44

. 1-7-13-5-11-17 „-9/2 ,

T

^

Z

T •e •

5! 48

for -l/3 jt < arg < + l/3 it.

When arg z = nj set z =5?exp (jit) , which results in:

Ai(z) =

sin (2/3%^ + it/4) " 9® =os(2/3^2

T

45

+ %/4)y

Bi(z) -

l/4[ï(?I cos (2/3^/2 + V4) + <Hf) sin(2/3f5^
+ *A)}

where

P(f)^l - ^^f3 + 1-7-13.19-5.11.13.19 + ___

47

21 482

4! 484

and

<*?>~P® ^"3/2 - W-15T'U f"9/2 +

48

3 J 48

It should be noted that the second term of eauh expansion is about onetenth of the first term for|| z = 1. The functions Ai(z) and Bi(z) are chosen as the fundamental pair in such a way that one of the solutions, Ai(z), would decrease exponentially along the positive real axis and that both of the solutions would be of equal amplitude and oscillatory, but differing in phase by it/2 radians, along the negative real axis. These

22

solutions are most suitable for application to propagation problems be­ cause the first condition provides a solution applicable to vave propaga­ tion in a lossy medium such as the earth while the second condition pro­ vides a solution which represents an outgoing wave at large heights which is one of the boundary conditions of this problem. In the particular problem at hand, the dissipative medium, the earth, is the region of neg­ ative and the medium suitable for lossless propagation is the region of positive 7^ . It will be seen that on making a change of variable in the height-gain equation to transform it into Stake's equation, an inversion of the axis will be made thus making the Airy integral solutions ap­ plicable.
Two identities involving Airy integrals which will be of considerable utility in the analysis to follow are;

Ai[z exp (j 2kit/3)] = exp(j krt/3)j^cos kit/3 Ai(z) Bi(z)J
and

sin krt/3 49

Bi[z exp (j 2krt/3)] = exp (j krt/3) fs sin krt/3 Ai(z) + cos krt/3

Bi(z)J

50

Complete tables of Airy integrals and their derivatives and a tabulation of the zeros of the Airy integrals and their derivatives are given in the Mathematical Tables Part-Vol. B (14). These tables allow ready evaluation of the expressions which will be developed for the field components.
The Airy integral solutions will be used to formulate a system of equations whose solution will be used to weigh each term of a complete

23

orthonormal set of solutions of the differential equation in the other variable. The resulting integral will "be the complete field solution,

G. The Green's Function
The solutions of the differential equations governing the propagation of electromagnetic energy around the earth must be modified to account for the singularity at the source point. The Green's function is a function of -which satisfies the boundary conditions at the earth's surface, repre­ sents an exponentially decaying wave inside the earth, represents outgoing radiation at great heights above the surface of the earth, and whose de­ rivative has the proper discontinuity at the source point.
Assume that a dipole of unit strength is located at a height d above the surface of the earth. It will be convenient to locate the 0=0 line through the dipole, such that the coordinates of the dipole are (a + d, 0). This point will transform into the point (7%.,0) in the cylindrical geometry of the flat-earth approximation. Hie point discontinuity is represented by Dirac-delta functions in the spherical-polar coordinate system. The differ­ ential equation which must be solved can be written as;

[D2(r,9) + k2]U « cf[r - (a + d)] (©)

51

where

D2(r,9)s" D2(r) + D2(e)

52

2 D (r,9) represents a second order differential operation which in the case of a homogeneous atmosphere or a magnetic dipole in a spherically atratified atmosphere is the Laplacian operator. In the case of an

24

electric dipole in an inhomogeneous spherically stratified atmosphere t.hp D (r,0) operator is not the Laplacian operator because of the grad e(r) terms. The point discontinuity will retain its essential characteristics through the transformation to the cylindrical coordinate system, thus Equation 51 transforms into:

ED,2(7t,p) + k'2 ]U =

S{p)

53

•Where the primed superscripts merely indicate that a transformation has been made. Since the dipole is monochromatic, the time-dependence can be removed from the source by assuming an exponential time-dependence of the form, e^^*. All of the field components have the same time-dependence so the e^^ can be factored out of each term, thus reducing the problem to one of static fields.
Hie problem can be solved by finding a complete orthonormal set of solutions of the equation:

[D'2(p) + ^'2] 0=0

54

Each term of the complete orthonormal set of solutions of Equation 54 will be weighted by a Green's function in the 7^ variable. To see that this procedure does indeed produce a solution of Equation 53, consider the separated differential equations after transformation. These transformed differential equations are:

[D'2(7[) + (k'2 --A'2)]R=5(7L-^)

55

and

25

lD:^(p)TÂ!^J 8 s ^(P)

56

A complete orthonormal set of solutions of the equation:

[D»2(p) + A2] 9=0, is a series expansion of a Dirac-delta function, that is

[D'2(p) +X'2] 2^ 9A.(p) 9^.(0) = £(p) ,

57

i»o

•where the superscript means complex conjugate. It is necessary to find

a solution to the equation

[D»2(x) + k»2 - v2]G =S(7i-nd)

58

Where G(7l,'>^) is the Green's function. The complete solution can toe -writ­ ten as:

U=

G(7LF\) 9a.(p) 9 A.(0) .

59

i=o

If Equation 59 is a solution of the differential equation, Equation 53,

then substitution of Equation 59 into Equation 53 should result in the

two-dimensional Dirac-delta function S S { p ) .

[D'2(^p) + k'2]

G(?t,^)9Ài(p)9Ai(0) =<^(M.-^)6(p) 60

1»0

Separa t e

D '2{\}p)

into

2 D1 (fy) +

2 D* (p) and

move

the

operation

inside

the

summation sign.

26

!>2ty,p) + k'2]

G(VO 6 (P) 6 (0) = 27^ I'D' (yi)

H=o

* ^i *i

3^0

61

+ D'^(p) + k2] G(%)@ (p)8 (0) A± "A £

2 Add and subtract ^' inside the parenthesis which contains the differential

operation and perform the indicated operation on the Green's function

[D'2C%) + k'2 -%'2 + D'2(p) +?l'2] n=6
This results in

0 (p) @*(0) 62 ^i Ai

[D'^(P)

% (P) % (0)

63

i=o

"

^i ^i

or

[D'2(p) +%'2]

9 (p) 9* (0)

i=o ^i

which, by Equation 57, is

<Si (%-%%)f(p). Thus proving the validity of the technique.
The next step in the analysis will be to solve the separated equa­ tions for the two source types. The first source type to be considered will be the magnetic dipole.

27

III. THE COMPLETE SOLUTION FOR THE MAGNETIC DIPOLE

The separated differential equations and the field equations govern­ ing the propagation of electromagnetic energy from a magnetic dipole in an inhomogeneous medium are:

9" + cot 6 0» + 7l2 9 = 0

64

m

R" + ——— + (u>2^e(r) - —% ) R = 0

65

r

r

E^ = 0

66

EQ = 0

67

= -jcoP- R8*

68

E = 8(R" +|RI + m2KG(r) R)

69

H = (Ri + R/r) 9'

70

ay = 0

71

The boundary conditions which must be satisfied at the earth-atmosphere interface or at any other spherically stratified discontinuity in the re­ fractive index require matching of the tangential electric and magnetic fields. Matching the boundary conditions at the earth's surface requires that:

<Vi =(Vn

72

and

28

fïT ) - /TJ 1

-

vD'i X"9'II

•where the subscripts I and II refer to the earth and the atmosphere re­

spectively. Substituting into the expressions for these tangential com­

ponents of electric and magnetic field results in:

Bi 6i

" Rn en

74

r=a

r=a

and

(Ri+ Vr' eî

(R|l + Rjj/r) 9-f-j-

75

r=a

r=a

When considering the field inside the earth, it can be shown that the same

differential equations which were developed to describe the fields in the

earth's atmosphere will be applicable provided e(r) is replaced by

e„ o
where a is the conductivity of the earth, m is the angular frequency of the source, and is the permittivity of vacuum. Transforming Equations 64 and 65 into the cylindrical coordinate system of the flat-earth approx­ imation results in:

A + icotp/a|i+^ e=0

76

dp

a

and

d^R t+ a

I* +

[cu^€(r) exp

(ZHJS.)

2, 2, +^7»^

R

=

0

77

d-n.

The range equation, Equation 76, will be simplified by using a series ex­

pansion for cot p/a, that is:

29

cot p/a ~-l/(p/a) - (p/a)/3 + (pv/a")/45 + ...

78

Equation 76 can be rewritten as:

As mentioned in Section II-E, Pekaris (15) has shown that the error in­ volved in ignoring the right-hand side of Equation 79 is less than twopercent at ranges up to one-half the earth's radius. The approximate range equation can be written as:

â-|+ i + fl2/a2 s s:0

80

ap2 pdp

This equation is seen to be Bessel's equation of zero order with a parameter
"A/a. The solutions are JQ(^p/a) and NQ(/lp/a), where J (flp/a) is the zero-

ordered Bessel function of the first kind and KQ(ftp/a) is the zero-ordered

Bessel function of the second kind. Because of the unbounded nature of
NQ^PA1) as p->0, it will not be an allowed solution; therefore, the solu­

tion of the range equation is:

9(p) = C JQ (Ap/a)

81

where C is an arbitrary constant. The second equation to be solved is the height-gain equation:

^§+ ™ + ("Aie(r) exp (2%/a) - ft2/a.2) R = 0

82

d>|_ a L

At this point, it would be appropriate to change from a formulation in

30

terms of the permittivity of the medium, tfr). to a formulation in terms

of the index of refraction of the medium, n(r). The relationship between

the permittivity and the index of refraction is:

2

HE = ZL

83

C

where C is the velocity of light. Equation 83 can be written

H€(r) = n2(r)/C2 which can be substituted into the height-gain equation to give:

4+ a E

n2(r) exp (2Va) - A2/a2) R = 0

84

a>C

^ cd

The atmospheric model which will be used in this analysis will be assumed

to have an index of refraction with a uniform slope, that is:

n(r) an o

+

|

£

(

r

-

a

)

r > a,

85

2 from which n (r) can be approximated by

n2(r)(%n2 + 2nQ gg (r - a)

r > a

86

provided ^ < < 1. Transforming Equation 86 into cylindrical coordinates results in:

n2(r) » n2 + 2nQ(|~) a (exp [Va ] - l)

87

The term (—) is not transformed because it is a number which would be known in any atmospheric model. Inserting Equation 87 into the height-gain equation results in:

31

A -1

r B^W&)- ij] (2V&)

= 0 Koo and Kàtzin (13) suggest that changing to a nev function

« 88

V a (r/a)1/28, R or R = V exp (-71/28.) . When this substitution is made, the height-gain equation becomes:

4+ [l + ^ [exp(Va) - 1]Jexp(2^a) - i^5t|Zàljv= o 89
2 22/2 where k = w n /C . The factor exp (Vl/a) - 1 vill be approximated by ?%/a and the term exp (27l/a) vill be approximated by 1 + 2W/a.. These approxi­ mations reduce Equation 89 to:

A [ka.flgsiMt^(i + 5.^]v.o

90

d%L

a2

*

*o

Making the substitution

-f = P"2//5 (a - pn)

91

•where

a„k=.^|Zi) a

and 2

P- 3 < 1 + M >

93

0

results in Stoke1 s equation:

32
It should he noted that in the process of making the change in variable, the axis undergoes translation and inversion; processes which lead to the proper Airy integral solutions as was discussed in Section II-F.
To get the function U, it is necessary to find a complete orthonormal set of solutions of the range equation. The solution must he finite at p = 0 and go to zero as p->°°. It has been established that the proper solutions are J^p/a), where the A's take on a number of values. A com­ plete orthonormal set of the solutions, applicable to a pipe of radius b and subject to the boundary condition that © be finite at p = 0 and 9 be zero at p = b, is given by the equation;
95
To expand this solution to the boundary conditions of the present problem, it is necessary to take the limit as b -><».
00 96
In the limit, the summation becomes an integral:
CO
97 o For this problem, p1 = 0, therefore:
CO
98 o

33

To see that this is a dirac-delta function, it vill "be necessary to evalu­

ate the integral, Equation 98. This can be done quite readily by means

of Laplace Transform theory.

CO

00

f \ J0 (a^) = lim ^2 T exp (-s%) JQ(^)AdA

99

J a 0

s —>"0 8, J 0

= lira

A J (^2. )]

100

s —^o a,

Talcing the Laplace Transform of the quantity in the brackets results in:

T^-Vr11 - h . 2 *8, 2.3/8

a

a (s + p /a )'

101

Taking the limit as s->o

S( a
p P % P %/p

p)

s-»o a2(s2+p2/a2)3/2

102

it is seen that the integral is zero everywhere except at p = 0, at vhich

point it is undefined. Thus, the integral is a Dirac-delta function. To see that the complete orthogonal set is normalized, it is necessary to

integrate the dirac-delta function over p.

CO

£(p) — - — 1 —

pdp " s'-ï'o Jo ?ÛVÂ¥75

103

= 1

104

llm

r 2. 2/ 2.1/2

s->o (s +p /a )'

o

thus proving that the set is normalized.

The next step is to find the Green's function. The differential

34

equation which must be satisfied is:

~(i + ~^)>l] V = 0

105

°

a

a

or

•which is transformed into Stoke1s equation upon making a change of variable

-g"= p"2/3 (a +

106

"where C£ and P> are functions of the medium. In the earth;

o - \

2

1

o

a

2k2
s P•- r ( 1 - 3 '

In the atmosphere:

0

7

108

a = k2 2k2
P =

+V4) a
+
o

109
110

In either case, the equation to be solved is:

a2

(—"ô G = (yi-Tl)

111

af

The allowed solutions from which the Green's function must be formulated

are the Airy integrals, Ai(-S^) and Bi(-Ç). The boundary conditions on the Green's function are that it must represent an outgoing wave for large positive (large negative £), an exponentially decreasing wave for large negative71 (large positive^), the solutions must be continuous at the earth's surface and at the source point, and have the proper discontinuity

35

in derivative at the source point. By considering the asymptotic from of

the Airy integrals, the appropriate solution for large positive is

SD[Bi(-^) - j Ai(-^)]. This is the solution which will be used above the

source point

Between the source point and the earth's surface,

the solution will just be a superposition of Airy integrals fg>Ai(-§) +
C.Bi(-^). Inside the earth, the appropriate solution would be Q_Ai(-g),

since this solution decays exponentially for large positive

The

boundary conditions are found by returning to Equations 74 and 75. Since

9 is not a function of the index of refraction, it is the same function

for all7%; therefore, Equations 74 and 75 can be written as:

Gi - Gn

112

and

dG

dG

d^+<va--«r V*

m

Hlus, the boundary conditions at the earth's surface require continuity

of the function and its first derivative. This leads to two independent

equations. The continuity of the Green's functions at the source point

results in another equation and the last equation which is required in order

to determine the coefficients of the Green's functions is obtained from

the equation:

,2

[—p + (a + PA)] G = <6(7I-VL) .

114

Integrating this equation once with respect to "7^over the interval from 7^ - £ to 7^ + e results in:

36

+e

= 1

115

V€ Now the system of equations can be written:

ffiM-g,) +CM(-§i)

0

+CBi(-|l)P^3 -S>[Bi(-fd) -,jAi(-fd)] P^3 = 1

_ Ai(-^) -03Ai(-?a) -CBi(-^)

116 = 0

aj^-fje1/3

-CBÎ(-^)^/3

= 0

To simplify the analysis, the following notation will be employed.

l i 7 -

*2d" Ai^"§:id^ B2d = Bl^"Çld^ A2a" Al^"^tla^ B2a = 4a" Ai (-#%)

Al ("%Id^ V 4a'Al ^"?Ia^
V Bl ^"§Ia^ AÎ (-%)
4a =

117

It will also be very useful to denote the function Bi( -5) -j Ai( by f(^ "which is in agreement with the notation used by Pryce (16). The system of equations can be written in matrix notation as:

37

r

-*<

0 0
V

a
'dO.

B
•<da

•v

-Baa

0

-ti" -vf 0

S i

Ijf
®
c
%

n
l
118
0
0

The Green's function appropriate to the region between the earth's surface and the source point is found by determining the coefficients^ and(2,.

G(%<\)=

2d

119

(A2dB2d~B2dA2d^

In the region above the source point, the Green's function is found by de­ termining the coefficient^.

f(-e
aQl>\)
2d (A2dB2d~B2dA2d^

;X->zX
120 '2d

It can be seen that a general Green's function can be formulated.

G(%<'%>) =

?(-&)

(A2dB2d"B2dA2d^2^ 121

Once the Green's function has been determined, U can be written as:

CO

U = j" exp (-n/2a) ^/a2 Jq (Ap/a) G(>(<,^>) d\

122

38

i

i

The expression A Bpfl-B^A,^ is the Wronskian of Stoke's equation and

is equal to l/jt for all arguments. To see that this is true, consider

the two solutions of Stoke's equation, Ai(-g) and Bi(-^). Being solutions

means that:

d

- ^Ai(-§) = 0

123

d2Bi(^

%Bi(-^ = 0

124

d@2

Multiply the first equation by Bi(-^ and the second equation by Ai( -Ç)

and subtract the two resulting equations to get:

Bi(-g d A1^ - Ai(^ d

a#

*9

= 0

125

This is exactly

a§

- M(^)

= 0

126

or

Bi(-g)

- Ai(-§| ) —= const.

127

Thus, the Wronskian is independent of . The field component which must be known in order to determine the
energy density in the atmosphere is the radial component of the magnetic field, Hr« Once H is known, it is a simple matter to compute the Poynting vector since the ratio of E to H must be the intrinsic impedance of the medium. The equation used to determine is:

H =

+ 7 I- + co2ge(r)] U

128

r dr

39

111 the cylindrical coordinates.

2

E = exp (-2ty/a)

+ - |r- + w2p.e(r) exp (2^/a)] U

129

r

dOt a ™

or
CO

H = exp ('-25(/a) exp (-^/2a) r

œ2[i€(r) exp (2%/a)]

% J"o (^p/a)G(^'%>) dA

130

a

The height-gain equation in cylindrical coordinates is:

+ -|- + œ2n€(r) exp(2^a) - /\2/a2] U = 0

131

from -which it can he seen that the radial component of the magnetic field

is:

00 a3
Er = exp (-5%/2a) ^
8,

(^p/a) G(X>,7^) d A

132

This integral must he evaluated in order to determine the field in the

diffraction region. To facilitate evaluation of the integral, it vill be

appropriate to change the variable of integration to

defined by f3 =>

l/3 Pg CKg. When this substitution is made the integral becomes:

K2 e%p (-5?y2a) F Ê___ (1 -^/K2)

[k^p(l -S^K2)^2]

-CO

J k a ,
Where K = (~g~) and kQ = on C.

df 133

The Green's function can be simplified somewhat by making an approximation

40

in the function defining propagation into the earth. That is:

Ai( -i,

= Ai(^+

^ _<^/2 Ai(^)

154

?L=o

fl-o

This is the large argument approximation to the Airy integral and, since

^ contains a large imaginary part due to the complex dielectric constant

of the earth, the asymptotic expansion is valid. The

) term is

approximated by ^. This is allowed because the wave is assumed to atten­

uate rapidly and only has appreciable amplitude in region of small.

With this approximation, the Green's function becomes

2 135
f(f+

Where g(4?) = Aiféf). This notation is used to show agreement with the re­

sults of Fryce since this is the notation that he used. If a further simpli­

fication is made by denoting

by T^, the Green's function can

be written as:

g (?K + g(^)

^2

f

+ f(^)

136

Substituting this into the integral defining results in:

-K2

H. = exp (-5)(/2| a)

(l

& Pg

g.T)vsf?

137

f (^T1+f(^)

41

Since the zero-ordered Bessel function decays very rapidly for imaginary argument, the range of integration can be extended to the entire axis without adding significantly to the integral. If the large argument form of the Bessel function is used, that is:

Jo [kQp(l -f/K2)1/2] — I^Çcos (kQp -fp/K2 - jr/4) 138

the integral reduces to: J"
Hr = exp (-5H/2a) J -»

cos(k^p-^p/K2-rt/4)G(f;^,%^)df

°

139

This simplification is possible because kQP is a very large number; about 10^ at a range of 50 miles and a frequency of 100 mes. Equation 139 cor­

responds exactly to Equation 12.7 of Eryce (16) except that a magnetic

field is considered here. The integral can be evaluated by the method of residues because the denominator of the Green's function has an infinite

number of zeros in the complex plane. Using the relationship

Ai [-%)exp(j%/3)] = GXP'

f (p)

140

the denominator of the Green's function and consequently of the integral, becomes:

f+

= 2 exp(-j«/6) Ai[f^expC0^/3)3^ + Ai[^exp(jit/3)] 141

Since Ai(z) is oscillatory along the line, arg z = jt/3, there are an in­

finite number of poles of the integrand, whenever

42

..'r

/. / - M

.. r

, t »n

cap \jA/vyj ^ t AxLa^ exp lj%/Ojj = U.

1é'id

Thus the integral can be evaluated by complex residue theory, and Eryce (16) has shown that the integral can be expressed as:

H = exp (-571/28.) k2py3(2it/k P)1^2 exp [j(3jt/4-k p)3

r

o c.

o

o •- v

s

(1 + TI as)f'2(as)

2 2

Where = 2k2/a, k2/a, k2 =

, and ^

•

43

r/. THE COMPLETE SOLUTION FOR THE ELECTRIC DIPOLE'

The case of propagation from an electric dipole located in an irihomogeneous medium is somewhat more difficult to analyze than the case of the magnetic dipole because the differential equations are more complicated. However, it will be possible to make simplifying approximations at appropri­ ate points to facilitate the analysis. The separated differential equa­ tions and the field equations governing the propagation of electromagnetic energy from an electric dipole in a spherically stratified inhomogeneous medium are:

9" + cot 9 9» +^9 = 0

R" +

2 Rl + (m2^c(r) - ~§ -

R = 0

144 145

Er - 5%[R" +

+ (<°2Nr)

e

146

E9 = ifrj (R/r + R,) 9' E^ = 0

147
148

H^ = 0

149

= 0

150

H, = -RQ»

151

e ffr) Equation 145 can be simplified somewhat by considering the term y .
2n'(r) This expression can be rewritten as —^trj^ means Actuation 85. lypically, n'(r) is a number on the order of 10 /meter near the surface of the earth; getting smaller at increasing heights, while n(r) is very close

44

to unity. Therefore.

€£|^| 2 x 10 ^/meter .

The function exp OVa) can be approximated by 1 + ^l/a, or exp (H/a)^ 1. The earth's radius a is a large number, about 6.5 x 10^ meters. The term

exp (Va) e (r) can

a

e(r;

neglected, but the term a '

e(r)

neglected. Thus, Equation 145 can be approximated by:

cannot be

^-§+ ^(1 d>i

^+ [o)V(r) exp (2Va) - A2/a2]R » 0 152

o

\

The change of variable suggested by Koo and Katzin (15) becomes

R = v exp( ->]/2a)b where b = (l - 2a^ ^ ), n o
and the height-gain equation becomes:

2 2

jk2 il + 2an'(r)[exp(Va) - 1]1 exp (2Va) -

dVj_ L 0 L

o

J

} V = 0 a j

155 Making the same approximations as in the magnetic dipole case and ignoring

2 2 the b /4a term as being too small with respect to the other terms, re­

sults in

+ [(ko -\} + ~r"(1 + |-

d-^

a

o

va 0

154

which is the same as Equation 90 -which was developed for the magnetic di­

pole case.

The boundary conditions at the earth-atmosphere interface requires

continuity of the tangential electric and magnetic fields, which means

45

that:

Riei

= Enen

155

r=a

r=a

and

R, 1_

i »

e„ <r+ Bi> ei

r=a

156 r=a

When considering the field inside the earth, it can be shown that the same differential equations "which were developed to describe the field in the earth's atmosphere will apply provided e(r) is replaced by e (l -
o The range equation for the electric dipole case is the same as the range equation for the magnetic dipole case, and is subject to the same boundary conditions. Therefore, the solution developed in the preceeding section will be applicable in this case also. Thus, the problem is re­ duced to finding the appropriate Green's function. The height-gain equa­ tion is somewhat different than the height-gain equation for the magnetic dipole case, so the electric dipole analysis will begin with consideration of the height-gain equation. In the cylindrical coordinate system, the height-gain equation is:

d^R
2+ 1(I - 8
dH,

§UpW(r)exp 0,/a) - *2/a 157

The Airy integral solutions will also apply in the electric dipole case, and the main difference will be due to the boundary conditions at the earth's surface. The boundary conditions require that

46

n -+ a "I "II

(r+4> = r^?+4i>

- ir>

o

These two boundary conditions lead to two independent equations. The other

two independent equations are the same as for the magnetic dipole case.

Thus the system of equations can be written as:

(a)

$>Ai(-£=Id) + CBi(-^Id) -^[Bi(-^Id)-jAi(-j^Id)] =0

®Ai(-?iia)en5 +CBi("lia)pn5
"

a AK-fj

lia' -CM-ÇliJa

a[^4> + A.(_g )]jg

e»

a

Cl)
1 ^= «•
(c)
= 0
(a)
pi

= 0 160

In matrix notation, and using the symbolism of the preceding section, the system of equation becomes

47

0

A^^

B,

-I-.-

1 ~G~

n

cu

0

= V2/5

-f pl/3

%

+1

2d 2

161

-A

-B

4.a

"2a

"2a

C

0

0

0

Evaluation of the matrix is simplified if one considers the relative sizes of the terms in the fourth row. A good pictorial representation of the Airy integrals of real arguments and their derivatives is shown on page B-16 of The British Association For the Advancement of Science Mathematical Tables (14). Here it is seen that Ai(x) is a very well behaved function, as is Ai(x) and both are of the same order of magnitude; Bi(x) and Bi(x) are well behaved for negative arguments but increase exponentially for positive arguments. In either case it is reasonable to ignore the term which is divided by a, the earth's radius. Under this assumption, the matrix equation becomes:

0

A2d

B2d

"f2d

~(f

0

0

V/3

+1

162

^a

-Â2a

0 "B2a

C

0

0

©

0

This matrix is xhe same as the matrix which was developed for the magnetic dipole case except for the term which is multiplying the derivative of

48

the solution in the earth. The solution of the preceding section can be

applied if

is substituted for

The general Green's function

for the electric dipole case can be written directly as:

= -TTs

4- ^ ^^(5»)

)] ^3

where

T2 " To determine the energy-density in the transhorizon region, the field component "which must be known to the radial component of the electric field, Er« The Equation which is used to determine E^ is:

E, = -

[B"+?(2 -

+

e=o

m

Consideration of the height-gain equation for the electric dipole case

2

R" + J (2 - £F^P- H' + [A>2|IE(R)

] E- 0

165

r^

indicates that the radial component of electric field is:

00

. r

^3

Er = œ \ exP("W2a) exp (->i/a) — jjTtp/a) Cfjf}

dA

166

or

E -ja exp(-a-b)?t/2a) A J (^p/a)G(f;)^,^ ) d%

167

R

M E(R)

4 "o a

The integrand of Equation 167 is exactly the same as the integrand of Equation 132 except that the Green's function in this case has T

49
substituted, for T„. The integral can be evaluated by means of complex
residue theory just as in the magnetic dipole case, with the result

_ -ja ezp[-(2tb)%/2a] A^at/k p)1/2 exp [j(3it/4-k p)]

œ e(r)

° d

0

0 s^l

f(yPa/^.>) f(a5+e2/^<) (l + T2 as) f (a2)

50
V= COMPARISON WITH EXPS.BŒ1AL DATA.
There has been a great deal of published and unpublished experimental •work concerning the strength of the electromagnetic signal beyond the radio horizon. In recent years, much of this experimental data has been taken between two fixed locations with emphasis on the statistics of the field in the transhorizon region. The statistics of the received signal would ideally be correlated with the statistics of the turbulence of the atmo­ sphere. In fact, herein lies the key to the preference of the BookerGordon (2) "scattering blob" theory over the traditional normal mode theory •which is the subject of this paper. The Green's function approach to the solution of propagation in a stratified atmosphere could permit the in­ clusion of a time-varying atmosphere by using a time-dependent Green's function. It would be assumed that the time-variations of the atmosphere occur much slower than the frequency of the source. This subject will be considered in somewhat greater detail as the Green's function solution is extended to include elevated stratifications in the next section. At this point it will be appropriate to compare a typical set of experimentally measured field strengths with the field strength calculated by using the results of Sections III and IV. The type of data which is most suitable for comparison with the theory of this paper is not that which is taken between two fixed points, but data •which is taken from a radial path of varying length. This leads to a plot of field strength versus distance. A good set of data for this is given by Dinger, Garner, Hamilton and Teachman (5).
To show the validity of the flat-earth approximation, a comparison

51
of the theoretical results of Section III with the data of Dinger, Garner. Hamilton and Teachman (5) will be made. This data was taken from a trans­ mission path extending from the south shore of Massachusetts, near the city of New Bedford, to a ship traveling along great circle courses to a maximum distance of 630 nautical miles from the transmitter. A 10-kw, 412.85 mc transmitter feeding a 28-foot paraboliod antenna and a 40-kw, 412.85 mc transmitter feeding a 60-foot paraboliod antenna were used, with the more powerful transmitter being used at the longer distances. The receiving antenna aboard the ship was a 17-foot paraboliod. Both the trans­ mitting and the receiving antennas were horizontally polarized. Both the transmitting antenna and the receiving antenna were located at a height of about 95 feet above sea level. A plot of the results of this investigation is shown in Figure 2. The theoretical model which is used for comparison will be magnetic dipole. Actually, the comparison will be made between the Hertz vector for free-space propagation and the Hertz vector for prop­ agation in a spherically-stratified atmosphere, which reduces to the free space Hertz vector in an isotropic medium. The free space Hertz vector for a unit dipole is:

The corresponding Hertz vector for a stratified medium is; by Equation 11,

it = r U

170

m

For fields at the earth's surface,

n = aU.

171

m

-120
-140
-160

FREE SPACE FIELD STRENGTH

2 -180
lu "200 t_oj LU CD -220
-240

EXPERIMENTAL FIELD STRENGTH
CALCULATED FIELD STRENGTHS
= 0

-260
50

100

150

200

250

300

350

400

450

500

5)0

NAUTICAL MILES

Figure 2. A comparison of experimental field strength vith calculated field strengths for linearly varying profiles of refractive index

53

The function U is given by equation

CO

U r % Jg (4p/a) G

a%

172

Or, upon making the change-of-variable

f= p-2/3(k2.*2/a2) ,

173

•where k = ^ and £ = 2k /a(l + — —) ,

o C

o' x n dr '•

o

H =

it n rf"

l/? ^ gg,•(g)i]L-i-e(?)

cos(k p+^p/SK2-ir/4) fg(?+Pp^<) - —

AOP UVOO

0

c.

f „,(%»)?,+f(%)

174

1 a?

This is the same integral "which vas evaluated in Section III so the result can be written directly as:

2^ exp (j 3it/4-kop)

o

/

f(as+Pî/^>)f(asR+P2p'/^<) 1
(1 + a,) f ' ^ )

exp (-j aop/2K2)

175

The experiment -was carried out over sea water at a frequency of 412.85 mc. Using this information to evaluate

54

2kc ,0xl/3( a ana/3

,1. PV|,-i/^.

+

2k2

176

- M - i^-i2/s ("rli
o
For sea water a = 4 mhos per meter and G1 = 81. Therefore,

1 ~ Hi " 1 " j 2,16

177

k = 8.65

178

o

K = 3.05 x 102

179

and

2k2

("a2"^3 = 2.87 x 10~2

180

Thus,

T a

' "a i /o •

(2.87 x 10 *)(1 - j 2.16)(a - j 20 x 10*)^^

181

The root of the Airy integral; a , is a small positive number. If consids
eration is limited to the first root, a % 2.34. Under these conditions
Equation 181 can he approximated as:

T, %

0^

ô

182

(2.87 x 10 )(1 - J2.16) (4.45 x 10'7-45 )

Thus, the integral expression for U can be simplified to:

u=|3'2' 1/51 fïârT ? 6 =X»5 W4 - v> Ç

fK+^<)
f,2(as)

exp (-ia p/SEC2) .

185

Pryce (16) has shown that with this simplification,

« -

' o

(j 311/4 - kop) 17 s

exp [-(^3+j)a P/2K2]

186

where

[lx) = exp (j Jt/3) Ai [-a + exp (j Jt/3) x]/Ai(-a )

and C<s = exp (-2j jt/3) ag

At great distances, the first term of the series is a good approximation, that is:

iui «

P y

s
i

2k p

o

In this case

I^P2^>) exp (-2.0249/2K2) . 187

56

2

T--,L//„O t 2kO x,-L//O„ /. a drixi/5

- --2 /. a_ dn.l/:ï

V-1- T n dr;

CL.OI JL 1U II t- — T—J • x n dr

o

0

188

= 28.9 meters.

Assume n = 1.000. o

Typically,

dn = Y= -3 to -4 x lO^/meter. dr

189

P^3 = (3.1 x 10"2)

190

^l\< = 1 1

= 0.93

where

= ko(1 + rs' o

Therefore :

(i + !- B)2/5
n dr o

^-(^)2/5d^^)2/5

191

V? 'V

no dr

9.1 x 10

The expression for magnitude U reduces to:

Uj = (3 x 10"2) -j^Ç7

(.93) exp

(1.0125)(1 + §)2//5p. o
9.1 x 10 192

Once the magnitude of U is known, the magnitude of the Hertz vector is easily obtained by multiplying by the earth's radius a

57

- a |U|= 6.35 X 10S(3 X 10"2)

^ (.G3)

-1.0125(1 + j|- g)2/3p

exp [

]

9.1 x 104

193

in]î>) |has been tabulated by Pryce (16) so that f^(.93) can be readily estimated, and is determined to be 1.04 or %1. Thus « can be evaluated as a function of p only.

= 7-8^ 10 exp [-(1.115 x 10"5)(1 + a ^)2/5 p] I?

194

The desired quantity to be plotted will be the ratio of the trans-

horizon field to the value of the field in free space where there is no

earth to influence the field. This ratio, when expressed in decibels, ends

up as the difference between the amplitude of the transhorizon field in

db and the free space field in db. This difference is plotted on Figure 2,,

for four values of the gradient. It is seen that the attenuation rate

would be well predicted gradient refractive index , of about -5x10

meter. This is not an unreasonable gradient above the ocean as is pointed

out by Tukizi (18) in his paper. The significant difference lies in the

fact that the experimental data is roughly 4 to 6 db stronger than pre­

dicted by the model.

There are a number of factors which could account for the 4 to 6 db

offset. It is doubtful that the approximations made by considering only

the first term of the asymptotic expansion of the Airy integrals would

contribute much more than a 10 to 15 percent error. One possible source

58
of the error lies in the fact that the plot of experimental data shown in Figure 2 is really nothing more than a smooth approximation to a great number of data points. Consideration of the plotted data points as shown in Figure 7 of the paper by Dinger, et al. (5) indicates that the data points cover a range of 10 to 20 decibels at any given distance. Thus, the 4 to 6 decibel discrepancy between the calculated curve and the "smoothed" experimental curve is not as serious as it looks since the calculated curve is still well within the range of the actual data points.

59
VI. EXTENDING xELE GREEN'S FUNCTION TO UNCLUI/E THE EFFECTS OF ELEVATED STRATIFICATIONS
The linearly-sloping profile of refractive index provides a very sim­ ple analytical result when considering the propagation of electromagnetic energy around a spherical earth vith a spherically stratified irihomogeneous atmosphere. This atmospheric model is not very acceptable because the profile of the refractive index becomes infinitely negative at large heights. This fact, plus the fact that the atmosphere is seldom such that the profile of the refractive index can be described by a simple mathe­ matical model, means that some technique must be developed whereby an arbi­ trary profile of the index of refraction can be approximated to a reason­ ably good degree of accuracy.
It will be shown that a Green's function can be formulated such that any profile of refractive index can be broken dovn into a series of straight-line segments each of vhich has a slope approximately equal to the slope of the continuous function vhich represents the profile of the re­ fractive index evaluated at the appropriate point. In this way any ana­ lytical function vhich describes the profile of the refractive index can be approximated by representing the earth's atmosphere as a series of con­ centric shells each of vhich has a linearly-varying profile of refractive index. The general solution can be formulated by matching the tangential components of the fields at the boundaries of the shells. Since the per­ mittivity of space is assumed to be continuous, the boundary conditions vill be satisfied if the Green's function and its derivative are continuous at the boundary. Each "shell" adds tvo more unknown coefficients which

60
must be determined if it is desired to know the complete Green's function applicable at any height. The fact that there are two additional coeffi­ cients in the complete Green's function means that a complete solution re­ quires two additional independent equations. These additional independent equations are given by matching the boundary conditions.
In general, it is not necessary to determine the complete Green's function applicable at any height, rather it is usually desired to know the field close to the surface of the earth due to a transmitter located over the horizon and close to the surface of the earth. This means that the Green's function applicable in the first strata above the earth's sur­ face is sufficient and it is only necessary to solve for the coefficients in this region. The rest of the atmosphere above the source and receiver can be represented by an arbitrarily large number of stratifications. In fact, it will be possible to express the result of the stratifications as a summation of effects and, since there can be an arbitrary number of stratifications, the summation can be extended to an integral. This tech­ nique will allow rapid evaluation of a profile which can be expressed as an explicit mathematical function. Thus, the effect of an exponentiallytapering profile of refractive index can be considered. It will be seen that the simple linearly-varying model is a good approximation to the ex­ ponentially-varying atmosphere which is a more satisfying model because the refractive index goes to unity at large heights. This technique is also very useful to determine the effect of layered pertubations in the refractive index.
If the atmosphere is divided into L concentric shells, each of which

61

has a different slope of refractive index, the Green's function will be

determined by solving a system of 2L equations for the unknown coefficients.

The boundary conditions are such that the solution in the earth represents

an exponentially decreasing wave, the solution above the

shell repre­

sents outgoing radiation, and the solutions and their derivatives are con­

tinuous at the boundaries in the atmosphere. The boundary conditions at

the earth-atmosphere interface are the same as those considered in the sim­

ple model having a linearly-tapered profile of refractive index. The com­

plete system of equations is written in Matrix notation as shown in Equa­

tion 196. To define the Green's function in the first stratum of the at­

mosphere it would be necessary to determine and (% since

G(7L<:\;0 -

).

195

It will be assumed that the receiver is not higher than the transmitter. This assumption is merely for convenience in analysis and other situations can be considered in the same fashion.
The coefficients C^ and will be determined by applying Cramer's Rule to the 2L x 2L matrix of Equation 196. In order to simplify the analysis the matrix will be written as shown in Equation 197.

62

0 0

1o o M 1

o o

/

|o° oH

OJ u

otO

o-d*

in o

CO u

o o o ooo o

o o o ooo o

o o o o o o o

o o o ooo o

-o oo

o o o o1 1

II

10 m

to CVJ H

1

C n-
C n-
C n

• o*

oG ofl oA
to

o
to CM
rH t-^1 ^cu I
o -^I

=H

o V

to H
rH

tO H

,H ^

H CD. i HH

-AI* •f-j ^

tO rH

to H
1—1
H Cûu

^

1

i.

Y f ~<t* ^

to CM

r—1 t-Q

I

eu

i - AJ>

to

to

rH CVJ

OJ -caCVJ

H CVJ en.

i i

to
rH rH

i—I rH

m wH'SH + +

to
ro rH i—I
H H
Y H i 1V O O O
tO
H O CO.
" c ° <° O

196
ooo oooo o o o o ooo oo o o ooo O O O O O O OI

63

o
o o

i—1 c
DoJ a

r—1 r4 H S3 Si S3 i—ii 10 0? i—1 s .si .S3 s;

o o

cvj OJ cvj

II I

S3

S3

S3 cvj

1

/ to"
1

c\T

r1-T

S3

/ dfl

dG

to to to to si ti S3 S3 irT "<* to" oT

o

o o

S3

O S3

in

to cvT

197

/ J* J> o o

m i

10 i

in

10

s ti c S3

r~- CIO to •d1

d*

S3 dà

o oo o

tmo OO d
O O dto
to to to
H OJ tO d d d
OJ OJ CVJ H CVJ tO
dd d
dd jd d o

ooooooooo
ooooooooo
ooooooooo ooooooooo ooooooooo

64

This n x n matrix, n = 2L. can be reduced to an n - 1 x n - 1 matrix by means of the relationship (6):

\ S
Bet y o

d2-n Bet (dAj,

198

Where d =

is the n - 1 x n - 1 matrix formed by deleting the nth

rov and the n^ column of the n x n matrix of Equation 197. S is the

n - 1 x 1 column matrix formed by deleting a__ from column n of the origi­

nal matrix and Tis the 1 x n - 1 rov matrix formed by deleting a from nn
rov n of the original matrix. Thus, to evaluate the determinant of the

denominator matrix, the n x n matrix can be reduced to an n - 1 x n - 1

matrix, and the result is a constant multiplying the determinant of an

n - 1 x n - 1 matrix.

Bet (Ben C^, C^) = a^n Bet(a^ -0)

199

where

all ai2 a21 a22

13 a
23

a nA

200

a n-3,n-3 an-3,n-2 an-3,n-l

n-2,n-S an-2,n-2 an-2,n-l

0

a a — _ a a ..

nil n-l,n-2 nn n-1,n-1

65

I0 0 0

0 0

0

0 0

Thus

HJ. ai2 al3 a21 a22 a23
a32 a33

0

0 0

0

0 0

0

0

a,a _ n-l,n n,n-2

an-ln ,nan,n-l_

an-3,n-2 an-3,n-l an-2,n-2 n-2,n-1
n-1,n-2 n-1,n-1

Where

a

= a

203

'n-1,n-2 nn n-1,n-2 n-l,n n,n-2

66

=a a

- a

a

'n-1,n-1 nn n-1,n-1 n-l,n n,n-l

204

Then

Det (Den C,, C.) = a2"n Det hi

1 2 nn

I

205

Where A^ is the n - 1 x n - 1 matrix defined by Equation 202. If the
i process is repeated on the n - 1 x n - 1 matrix A^, the result is:

Det (Den

C , C )= a'2-n 1' 2Z nn

3-n

a1

Det A'

n-1,n-1

2 '

206

,2 .
Where A^ is given by the matrix

H p r o

all

0 0 ai3

0 0 a21 a22 a23

0 a32

0

0

an-4,n-4 an-4,n-3 0

0

an-3,n-4

an-3,n-3

a2 n'

2

0

an-2,n-4

an-2,n-3

a n'

and

= a1

- a1

n-3,n-2 n-1,n-1 n-3,n-2 n-1,n-2 n-3,n-l

2

a

- a'

a

n-2,n-2 n-1,n-1 n-2,n-2 n-1,n-2 n-2,n-1 '

207
208 209

67

a' _ _ and a® _ . are as defined earlier. At. this st-Rov», it. is seen

U-i-jÛ-JL

ll-X}l±-e

~ '

that the matrix looks exactly like the original n x n matrix, that is non-

zero-entries in the last three positions of the "bottom two rows, except

that this is a n • 2 x n • 2 matrix. By continuing this process, it would

he possible to reduce the n x-n matrix to a 4 x 4 matrix with some multi­

plicative constant in front.

11 *12 a13 °

a81 22 23

Det (Den C^, C^) - ^ConstJ Det -

n-4

210

*32 a33 34

n

n-4

0

a42 43 44

n-4

n-4

The a^ and a.^ terms are modified from the original terms in the n x n

matrix because of the successive reductions in the size of the matrix.

The other terms of the 4x4 matrix are the same as the corresponding terms

in the n x n matrix and incidently are the same terms as in the 4 x 4

matrix for the linearly-sloping profile.

This matrix reduction technique can also be used in evaluating the

determinants of the numerator matrices to determine the coefficients

and Cg. The matrices, whose determinants are desired, are:

68

0

n I fN

"il

13

a,21 0 *23 °

0

0

1 a,33 34 a,35

0 a43 a44 a'45

0 0

a54 a55

l'îum = Det

211

and

*11 *12°

a21 a22 °

0

a32 1

0 *42°

0 00

0 00

îîum Cg = Det

° an-l,n-2 an-l,n-l an-l,n

0 an,n-0 2

a T
n,n-l

a nn

0 0 0 0

34 35 a
44 45 a54 a55 a64 *65

212

0

0 n-2,n-1

0 an-1t ,n-n2 an-1n ,n-.1 an-ln ,n

0& ^ „ a

a

n,n-2 n,n-l nn

69

It is noted that in either case, except lor the 4x4 sutmatrix in the upper left corner, the rest of the n x n matrix is the same as the cor­ responding part of the denominator matrix. If these matrices are re­ duced in the same vay that the denominator matrix vas reduced, the re­ sult is

0

0

all

a!3

0

0

Det (Num C^) =|Constj Det a21 S23

213

(1 n-4)

0

1

a53

a. 54

(n-4) 0 a
'43 44

and

0 0 all ai2

a21 a22 0 0

Det(Num C )=[ConstlDet 0 a32

(n-4) a, 34

214

0 a42

(n-4) 44

(n-4) (n-4)

Where a^ and ax

and the multiplica.tive constant are the same as

defined for denominator. Thus, the Green's function applicable to the

first strata above the earth is given by tie equation:

70

"1 1 HO n n

a21 ° a23 °
Ai(^5 +
° 1 a33 4r4)

° G =

° S43 aM-4)

0 0

(n-4) Bi(-#

32

34

0 a(n-4)

42

'44

215

all a12 al5

a21 a22 a23 a32 a33 a4-2 a43

(n-4) 34 (n-4) 44

At this point, it is appropriate to consider the terms a^ ^ and a^Y ^• Once these terms are known, the Green's function can be deter-
44 mined since all the other terms are known.
These terms will be evaluated by going back to the original n x n matrix and examining the modified entries as the matrix is reduced. In the first reduction', the modified terms are a'n-,1,n-1-, and a1n-1,n-d. These terms are given by the equations

a'

=a a

a

216

n-1,n-1 nn n-1,n-1 n-l,n n,n-l

=a a

- a ,, a

217

n-1,n-2 n,n n-1,n-2 n-l,n n,n-2

where

-l,n-l =

an,n-l ~ \ an,n-2 = \

218

an,n " "fL an-l,n = 'fLPL-l an-l,n-2 V L-1

It will be necessary to use the approximation

71

AA

1 /? 1 /? .

,

Pr' " = Pr' n (1 + 1/5 —^ )

219

PL-1

to simplify the ensuing analysis. Inserting the values in the expressions

for the modified terms results in:

i/3 aA_l,n_l = j%l

fr* B ^ -j \
ÔPL-1

and

•i-w - <45 <« +
L-l
-where W is the Wronskian of the differential equation from -which the solu­
tions were taken.
p
In the second matrix reduction, the modified terms are a _ _ and n-2,n-2
2 an-3 n-2' given ^ tile equations

an-2,n-2 Bn-l,n-l an-2,n-2 an-l,n-2 an-2,n-l

222

an-2,n-2 &n-l,n-l Bn-2,n-3 an-l,n-2 an-3,n-l

22^

a'n-x,n-x and a'n-o.,n-d are carried over from the first matrix reduction and

' ,1/3 a
n-2,n-2 " -AL-1 an-3,n-2 ~ ^L-l^L-l 224
an-2,n-l = "BL-1 &n-3,n-l = BL-1PL-1

Substituting into the equations for the modified terms results in:

72

'/?

» R-/ ~ an-2,n-2 = PL-1 WfL-l +

^ BL~1 " BL 4,-1'

225

3P L-l

a

= e2/3

n-3,n-2 L-l WfL-l +

^L_l ^ BL-1 " \4-1^

226

3 P.

L-l

If the thickness of a shell is reduced to infinistesiml dimensions as

"will "be done "when the final summation process is reduced to an integral,

the expressions for the modified terms of the n-2 x n-2 matrix can "be

further reduced because

t 0 and ^ ^

i ^—*• W.

Under these conditions:

2
-3,n-2~ ï-ï fi-l W P£-l t1 +

227

an-2,n-2' ^L-l W fL-l

228

In the third reduction, the result is an n-3 x n-3 matrix, "with the modi-

3

3

fied terms an-3»,n-,3 and an-„3,n-4.; given by the equations

3

2

a

2

an-3,n-3 &n-2,n-2 n-3,n-3 &n-3,n-2 %-2,n-3

229

3

_ 2

2

an-3,n-4 an-2,n-2 Bn-3,n-4 an-3,n-2 &n-2,n-4 '

230

vhere:

B' bV3 a

./ =1/3

n-3,n-3 L - 2 I . - 2 n - 3 , n - 4 L - 2 P L - 2

231

a

= B

n-2,n-3 L-2

a

= A

n-2,n-4 L-2

and

73

%" ^-2 ^+

)

232

3P.L-2

Substituting into the defining equations for the modified terms of the

n-3 x n-3 matrix and ignoring product terms in results in:

3

_ - *1/3

an-3,n-3 ~ "J PL-2

233

L-2

L-l

and

-w py? pys

n-3,n-4

L-2 ^L-l

234

L-l

1, L-2

The next matrix reduction will reduce the matrix to an n-4 x n-4 matrix

4

4

•with the

modified

terms

a . . and n-4,n-4

an-5c,n-4..

These modified terms re-

duce to:

an-4,n-4= "W

PL-2 fL-2

235

-W pVS pl/S pl/3

n-5,n-4

L-l L-2 L-2

L-l

L-2

236

It is evident that a pattern has been established and it -would be possible

(n-4)

(n-4)

to -write general expressions for a^_ and

. These general expres­

sions are:

a(;n,-4) = const f

44

2

237

= const pj/3 f. +

"2 3^ Pr2

p„3

+ fL ^L-1 'P'rL-l J

238

Since these two terms are the only non-zero entries in the fourth column

of the reduced matrices which define the Green's function, the const can

74

be factored out of each matrix and consequently cancel out of Green's function completely. The Green's function for the stratified atmosphere is similar to the Green's function developed for the linearly-varying at­ mosphere vith the difference being a result of the modified terra a3^"^* If the Green's function given by Equation 215 is evaluated in terms of the solutions of the last section, the result is :

G =

g(5f+

)

w sW

f(r)+yf?) -1

1 g(^f g'+rf/i
+ ^[g'^fe+P^Xa)s
w

4 ) 239

Where S is the stratification function for the atmosphere above the trans­

mitter

1/3 P'

f.ze f'43

S =

T= + %+ +

p.

pc

2

"3

'"5

PrLT -l J

240

Below the transmitter, the atmosphere is assumed to have a linearly-vary­

ing profile of refractive index. It should be noted that if S = 0, the

Green's function reduced to the Green's function developed in the linearly-

varying profile analysis. This result is reasonable since for this case

the stratification function is identically zero because the 43's are all

75

zero= This Green's function developed for this case is somewhat unwieldy, but in an actual example some simplifying assumptions can be made. At this point it is of interest to consider two examples. First, the effect of an exponentially varying profile of refractive index will be consid­ ered. The second example will consider the effect of a layered perturba­ tion in the profile of refractive index.
The first model which will be considered is the exponentially varying profile of refractive index. This profile will be defined by the equation:

n(r) = 1.000000 + 4 x 10-4£ 1 - exp E-(r-a)/l04]^ r > a

241'

It is desired to know the stratification function:

p1/3 r

S= —

3 + •"

242

P2

+

Po

f<L + 4/V 4^1

+

^

The various components of the stratification function are defined as fol­

lows.

+r I

)

243

o

r=rT

In the Airy Integral solutions, whose derivatives are part of the strati­

fication function, it will considerably simplify the analysis if the argu­

ment ^

is replaced by

This simplification intro­

duces an error of less than 20$. The increased ease of analysis is a

76

small price to Day for the error. The derivative nf t>»e Air".' integral solution f' + %)can be written as:

£'(Çf+ P^)= exp (jrt/6) Ai

P>Lj_) exp (jjt/5)] .

245

Equation 243 can be used to formulate:

2k

, a dn

L _ ~ u + r0 a? r=L

(3t "

n dr

o

r=r.

LU

246

2k a

2k

This equation is simplified by using

for all the P's. Equation 246

can be simplified to

= 4 x 10"8 — exp [-(r-aj/io*] [l - exp (-Ar/lO*)]. 247

pi

n

This can be written in approximate form as:

IT = 4a X 10

exp [-(r -a/104)] Ar

248

i

o

Or, setting r_^ - a = h_ and a = 6.35 x 10 meters, Equation 248 can be

written as:

_s

-p- = 2.54 x 10 exp (-10 ii^)

249

The stratification function can now be written as

g=P /

(jm/6)(g ^ ^

^ Ai[^exp (j^/3) + P^exp (j^/3)]

i=3

i

exp (-10

)

250

77

Tf t>>» =f™+ifi??.ticn:: arc tckczi to to very close together, llic ouiumation of Equation 250 can be extended into the integral

S = P1/5 2-54 x 10
5 w

00
\ Ai[-a + p1/3 exp(jit/3)l exp(-lO'li)dh ,
K

•where %exp(jrt/3) = -CC which is the root of the denominator of the Green's

function. The upper limit of the integral has been extended to infinity.

The number -d has not been determined as yet for this case, but it should

not be much different than the root ag of the linearly-varying atmosphere.

The derivative of the Airy Integral Ai(-C£ +

exp [jrt/3] ) can be

approximated by the first term of the asymptotic expansion

i

Ai(z) % — zz1'/4 exp (-2/3 zz3'/2)

252

2f?

Substituting this expression into the integral defining S results in:

25.4xl0"6 e;
B-P W

J I -QH-x exp( jîr/3)]1/4 exp [-2/3(-a

hQ

253

+ x eac3»(jit/3) )3/2 - 10_4h]^

l/3 where x = P zvj^ . To see the effect of the exponential atmosphere on the

transhorizon field it is necessary to estimate the magnitude of the strat­

ification function S. If S is not infinitely large, the linearly-tapering solution will be a reasonable approximation to the solution in an expo­

nentially tapering atmosphere. To get an idea of the size of the integral,

consider the behavior of the integrand at reasonably large values of x.

78 The integrand can he approximated as

ei: xp(j:r/l2) [P^afn(^)

exp [-j§[P^afn-^^ ]3/2]

254

exp f[pl/3a,^)]l/2 ^

The integrand oscillates rapidly with increasing h and it rises ex­ ponentially to some value then levels off and drops to zero as the exp
There will be some contribution to the integral for some values of x but even though the integrand rises quite rapid­ ly for intermediate values of x, the oscillations of the integrand occur with shorter and shorter periods so that the net contribution is probably quite small. This is a hueristic argument which is borne out by the com­ parison with experiment. An exact analytical solution would be desirable only in that it would determine a different root of the denominator of the Green's function, and once a different root of the denominator of the Green's functions were established, different gradients of the refractiveindex would be used to establish a correlation with experiment. This is all true as long as the integral is of a reasonable value. If the inte­ gral is infinitely large, then there would be only one root Ct and it would be zero. This would in turn mean that the field would not exhibit an exponential radial dependence which contradicts experimental evidence.
A second stratification example which will be considered is that of an isolated layer in the upper atmosphere. Consider the profile of the refractive index to be as shown in Figure 5. If the effect of the ex­ ponential atmosphere is ignored, it is necessary to consider only the

79

portion of the atmosphere from hQ to h^ + L. The stratification function can be written as:

h +L

S = exp (jit/6)\

Ai [-a + x exp(j%/3)] sin dh 255

i

Using an asymptotic expansion of the Airy integral for large x, the strati­

fication function reduces to

s = e*?(;)*/*)«# h

x1/4 sin 2/3

exp ( 3

sin

dh

/g) 256

Again, in order to evaluate this integral, a number of approximations will be made, namely:

xl/4^,Ol/5h)l/4

x

(P^Al)1/2

257

x"

Using the approximation; hQ > > L, the stratification integral can be written as:

IS T5L ^HO^1//4 exp^aOho)1/^ exp ^-J[G/3 (PHQ)3/2 - JT/4]^

L

exp [-j(P^3h )"Ly/'2h] sin

dh

O

Jj

80
~ (P1/5^)1/4 exp[ 3 a(Pho)1//2/2] exp£-j[2/3(p1/^h.o)3^2-it/4]J 258
The term (3 ' hQ L/2JT is, in general, much greater than one, therefore E-
quation 258 can be rewritten as:
259
zv 260
There are several significant features of this result which are worthy of note. First, the integral becomes larger as the height of the pertubation becomes larger. This is reasonable because the perturbation acts as a source of partially reflected wavelets. Thus, the higher the source is above the earth, the stronger is the resulting field. Secondly, the re­ flected wave is a function only of the end points of the perturbation, that is, a function of the width or thickness of the perturbation. This fact was pointed out by Friis, Crawford and Hogg (8). The reflected wave is directly proportional to the intensity of the perturbation in the

81
refractive index and becomes larger with decreasing T.. These character­ istics are to be expected since increased £P or decreased L mean that the gradients are greater.
Consider an example using the preceding theory. The Green's function A
defined by Equation 248 can be simplified by setting = 0, in

f(^+

g(^+

)

_

g(%9f(f+

J"'
Jg(^P^j}

261
The stratification function will be evaluated for the following parameters:
p1/3 -Sx lo"2
100 meters 262
h =10 meters o
^P = 1 0-3
Inserting these values, the stratification function S becomes:

|A 3 x 10"10 exp (15a) [exp (-jl732) - l]

3 x 10'10 exp (150!) (-1.94 + j 342)

263

6 x 10"10 exp (15a) exp (j 170°)

At this point one can draw some general conclusions which lead to an approximate form of the final answer. The complete integral is evaluated

82

"by determining the sum of the residues of the poles of the in+emrncL The poles of the integrand are determined by finding the zeros of the denominator of the Green's function. That is, the equation

- § [g(%9f(4f+ P^ ) - f(#g(#+ P^)] = o

264,

The functions f(^) and gVoO?(ff+

^) - f(^)g(^f+ P^^^t are of com-

g
parable magnitude, so the magnitude of ^ has to be about unity. This

would enable an estimate of the size of a.

6 x 10"10 exp (152)<% 1

a<vl.4

265

*

or

a = -1.4

s

Now, this value of ag can be used in the distance dependence of the Equation 194, developed earlier for the case of a dipole radiating into an atmosphere with a linearly-varying index-of-refraction. If the at­ mospheric perturbation of this example is combined with the model used to compare with the experimental results of Section IV, the resulting Hertz vector is:

a = 7,86 X 10 exp [-(6.7 x 10"6)(l + a ||)2/5 p]

266

A comparison of the results of Section IV with a plot of Equation 266 is shown in Figure 4 to indicate the effect of the stratified perturbation.
The model of the preceding example is considerably simplified in an effort to get an idea of the effect of the stratification. A more detailed

83

look into the effect of the stratification of the atmosphere would, reveal additional information which would be useful "but it is questionable whether this information is worth the additional analytical effort. For example, a root of the denominator of the Green's function has been determined which is a negative real number. The actual root of the denominator would not, in general, be a real number; rather, it would be complex because the stratification is, in general, complex. The effect of a complex root would be to put an additional sinusoidal p dependence into the solution. This is not unreasonable since it would be expected that the reflections from the stratified anomaly would set up an interference pattern in the radial direction. The complex root would not be expected to have a phase angle much different than % radians, because the term which was suppressed in the phase of the stratification integral was

exp

where a is a negative number. This means that the phase will change quite

rapidly with (%, since

is a reasonably large number. Thus, one

would expect that the phase of the root of the denominator of the Green's

function would be altered only slightly.

84 à
0 = /30+ ^(l-cos^p) a + h0 - r - a + h0+ L
Figure 3. The profile of a stratified anomaly in the refractive index

-120

-140

-160

-180
lu
-200
u_
w -220

EXPERIMENTAL RESULTS
CALCULATED FIELD STRENGTH r=-5.5X I0"6 UNITS

-240

FREE SPACE FIELD STRENGTH
CALCULATED FIELD STRENGTH y = - 5.5 X I0™8 UNITS AND AN ELEVATED STRATIFICATION
AT in KM A A = M~3

-260
50

lOO

150

200

250

300

350

400

450

500

5!:i0

NAUTICAL MILES

Figure 4. A comparison of the transhorizon field for two atmospheric models

86
VII. CONCLUSIONS
The earth-flattening approximation coupled with the Airy Integral solutions of the resulting differential equations provides a simple, readily understood technique for evaluating the strength of the fields over the horizon from a radio transmitter. The Green's function approach to the formulation of the complete solution allows considerable utility in extending the results obtained for the simple models of the earth's atmosphere. In this analysis, the Green's function is modified to in­ clude the effects of stratified perturbations in the refractive index of the atmosphere. Further, the Green's function can be formulated with an explicit time dependence which will permit the effect of slow temporal variations in the structure of the atmosphere to be considered. This modification would then bridge the gap between the "turbulent scatterer" theory and the normal mode theory.
The results of the analysis are compared with data taken from a transhorizon propagation experiment conducted over the ocean. There is good agreement between the modest amount of reliable experimental data and the results of this analysis; thus substantiating in large measure the valid­ ity of this technique. The possibility of using a time-dependent Green's function to develop a model for the time-variant atmosphere was not pur­ sued further because there is no experimental data with which to make a comparison.
It is impossible to go through analysis on a subject of this nature without slighting many ramifications of the problem. Such is the case in

87
this analysis. There are several areas of the investigation which should he carried through to a more detailed conclusion. These include the de­ tailed evaluation of the effects of perturbations in the structure of the atmosphere and the use of a time-dependent Green's function to include the statistics of a time-varying atmosphere into normal-mode theory. The rea­ son these topics were merely mentioned or given superficial treatment is that the subsequent analysis are considered worthy of individual reports. It is intended that this analysis will serve as the foundation on which to treat these topics in more detail.

88
VIII. BIBLIOGRAPHY
Aharoni, J. Antennae. Oxford, England. Clarendon Press. 1941.
Booker, H.G. and Gordon, W.E. A theory of radio scattering in the troposphere. Institute of Radio Engineers Proceedings. 38: 401-412. 1950.
Bremmer, H. Terrestrial Radio Waves. New York, N.Y. Elsevier Pub­ lishing Co., Inc. 1949.
Carroll, T.J. and Ring, R.M. Twilight region propagation of short radio waves by modes contained in the normal air. Massachusetts Institute of Technology. Lincoln Laboratory. Technical Report. TR-190. 1958.
Dinger, H.E., Gamer, W.E., Hamilton, D.H., Jr. and Teachman, A.E. Investigation of long-distance overwater tropospheric propagation at 400 mes. Institute of Radio Engineers Proceedings. 46: 14011410. 1958.
Finkbeiner, D.T. Introduction to matrices and linear transformations. San Francisco. W. H. Freeman and Co. 1960.
Friedman, B. Propagation in a non-homogeneous atmosphere. In Theory of Electromagnetic Waves: a Symposium, pp. 317-350. New York, N.Y. Interscience Publishers, Inc. 1951.
Friis, H.T., Crawford, A.B., and Hogg, B.C. A reflection theory for propagation beyond the horizon. Bell System Technical Journal. 36: 627-644. 1957.
Gossard, E.E. Power spectra of temperature, humidity, and refractive index from aircraft and tethered balloon measurements. Institute of Radio Engineers Transactions. Antennas and Propagation-7: 186-201. 1959.
Harvard University. Computation Laboratory Staff. Tables of modified hankel functions of order one-third and their derivatives. Cambridge, Mass. Harvard University Press. 1945.
Jeffreys, H. and Jeffreys, B.S. Methods of mathematical physics. 3rd ed. London, England. Cambrdige University Press. 1956.
Korr, D.E., ed. Propagation of short radio waves. New York, N.Y. McGraw-Hill Book Co., Inc. 1951.

«
89
13. Koo, B. Y.=C. and Katzin, M. An exact earth-flattening procedure in propagation around a sphere. National Bureau of Standards Jour­ nal of Research. 64D: 61-64. 1960.
14. Miller, J. C. P. The airy integral. In British Association for the Advancement of Science. Mathematical Tables, Part-Volume B. Cambridge, England. Cambridge University Press. 1946.
15. Pekaris, C. L. The accuracy of the earth-flattening approximation in the theory of micro-wave propagation» Physical Review. 70: 518522. 1946.
16. Pryce, M. H. L. The diffraction of radio waves by the curvature of the earth. Advances in Biysics, 2: 67-95. 1953,
17. Schelleng, E. B., Burrows, C. R. and Ferrell, E. B. Ultra-short-wave propagation. Institute of Radio Engineers Proceedings. 21: 427463. 1933.
18. Tukizi, 0. Diffraction theory of tropospheric propagation near and be­ yond the radio horizon. Institute of Radio Engineers Transac­ tions. Antennas and Propagation-7: 261-273. 1959.
19. Villars, F. and Weisskopf, V. F. On the scattering of radio waves by turbulent fluctuations of the atmosphere. Institute of Radio Engineers Proceedings. 43: 1232-1239. 1955.
20. Wheelon, A. D. Radio-wave scattering by tropospheric irregularities. National Bureau of Standards Journal of Research. 631): 205248. 1959.

90
IX. ACKNQWTiwi The author is indebted to Mr. I. H. Gerks of the Collins Radio Company •who suggested the topic and Dr. R. M. Stewart, his major professor, for valuable suggestions and discussion during the preparation of this manu­ script.