PB 151383

jDouUer laboratories

^0.24

RADIO WAVE PROPAGATION

IN AN INHOMOGENEOUS ATMOSPHERE

BY JAMES R. WAIT

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical <7iete

24

September 10, 1959

RADIO WAVE PROPAGATION IN AN

INHOMOGENEOUS ATMOSPHERE

By

James R. Wait

This material was previously issued as a

project report in October, 1958.

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature. They are for sale by the Office of Technical Services, U. S. Department of Commerce, Washington 25, D. C.

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ABSTRACT

A self-contained treatment of the theory of radio waves in an

inhomogeneous atmosphere is given. The refractive index is assumed

to vary with height above the earth in a monotonic fashion. Variation

according to an exponential law is used for illustration of general principles. For this case, rigorous series formulas are developed for the

distance to the horizon for an elevated point in the atmosphere.

ON RADIO WAVE PROPAGATION IN AN

INHOMOGENEOUS ATMOSPHERE

1. Introduction and Formulation

It is the purpose of this paper to review, and extend, work of

1 2 3 4

Watson , Eckersley and Millington , Bremmer , Vvedensky and

5 Fock on propagation of radio waves in an inhomogeneous atmosphere.

In particular an attempt will be made to present a unified treatment which

includes the analyses of the above authors and shows that 'Ithe distance

to the horizon" is a valid concept in diffraction theory even when the

atmosphere is not homogeneous. The earth is to be represented by a homogeneous sphere of radius a and the source is to be a vertical

electric dipole which for the moment, is located near the earth's

surface. The usual spherical coordinates (r, 9, 0) are chosen with

the source oriented along the polar axis and the surface of the earth

is defined by r = a. The dielectric constant relative to free space is

denoted by K and is understood to be a function of r and decreases

monotonically with increasing r. The resultant magnetic field has only a single component, H ,, because of the assumed polar symmetry in the problem. As a further

consequence of the symmetry, the electric field component E . is zero.

It is convenient to express the remaining field components in terms

of single scalar function U which is actually the radial component of a Hertz vector. For a time factor exp (iojt), this leads to

H = ik

° K -Vi ' <*>

_ 120rr 9 / . a 9U\ , E = =—7,— -K-Tp ( sin 9 -5-75- ) , and

r r sin 9 9 9 V d 9 ' (2)

E

„ 120tt 8 /__ 8U \

9 = XT Tr. (

K r 1T0 < 3> where k = Zit/wavelength in vacuo.
- 2 -

These field components satisfy Maxwell's equations if the function U

satisfies

uV[4-^ (Kru» r sinfl ~d!T

sin0 -|y- 1 -J k! E i U = ©

Following the work of Foe k , it now proves to be convenient to

introduce an auxiliary quantity U 1 which is defined by

U» = e"

lka0 K r N/sin U

and consequently the equation for U satisfies

(4)

(5)

9

2 U'

9 r

z 2ik 9 U' a dO + *(2\ o

- J£\ ur /

K 1 9 U«

K 9 r

1 "9 2 U' (l +1 N

) U«

1 4 r sin2

y (6)

where K' = dK/dh, K is the value of K at r = a, and k = k. n/k" o o

is the effective wave number at the surface of the earth.

Since E - ZH, at r = a it follows that

au 1 a r ik (K ) 2 ZU' at r = a

o

(7)

6-8 where Z is the effective value of the surface impedance

The right-hand side of equation (6) can be replaced by zero. The

justification for doing this is based on the fact that

9U'/9 r = Order of (kU'/M)

and

9U'/90 = Order of (kaU'/M ) 1/3 where M = (ka/2) ' is a large quantity. It is also possible to

simplify equation ( 6 ) by noting that

(1

! /r 2

) =* 2h/a .
- 3 -

Consequently U' satisfies

where s = ad is the great-circle distance measured along the earth's

surface. The boundary condition at the surface of the earth then

becomes

l 8U'/9k = +ik(K) 2 ZU! (at h = 0) . (9)

The function U', which must satisfy equations (8) and (9)»

should also have the proper singularity at the source. That is it should

behave as

-ikR

U = ^~

(1 +R

v

) (10)

near the source, where R is a Fresnel reflection coefficient of the

v

ground for vertically polarized waves. For h < < s, and assuming

that our approximate boundary condition is valid, R s s + (h2 /2s)

and thus Rg ffi;g . (11)

Consequently, the required behavior of U' near the origin is

kh2

i -1 -i -t—

U'^K (a/s) 2 (2h/s) (h/s + Z) e

S

(12)

2. Equivalent Earth Radius Concept

The equivalent earth radius a will now be defined in the following

manne r

1a

e -1 +

a

K'

o 2K

o where K'

o = TM\= o

(13>

This equation is usually justified on the basis of geometrical optics.

It can also be demonstrated in a more rigorous fashion by rewriting

equation (8) in the form

l^-Zik-W+^lhd+tfU. =0 (14)
- 4 -

Wh6re

a

e / K " K o

g = -2K~ ( ET~^ " Ko

Now if the atmosphere is linear then by definition K' = (K - K )/h

and y = and consequently

8

2 U' -,„_ 8U1

. 2k2 W~ a

" 2ik TTi + 4^ hU' = (15)

But, on the other hand, if K =K for all h then from equation (8) or

(14),

B

2 U» -,„_ 9 U1

. 2k2 STpr- " akTT- a + ^~ h U' = ° (16»

This strongly suggests that propagation in a linear atmosphere of

radius a is equivalent to propagation over an earth of modified

radius a with a homogeneous atmosphere. It is therefore concluded

that the effective earth radius is indeed a valid concept. This conclusion has also been reached by many other authors, including

3 5 9 Bremmer, Fock and Miller .

3. Extension to Non-Linear Atmosphere

In many practical cases of interest, however, K is not a linear

function of height, particularly at great heights. Then the function g

is non-zero and is itself, in general, a function of height. To further

analyze this case, the revelant equations will be put in dimensionless

form. The following parameters are first introduced

i

i

1 / ka \ 3 1

-i (_e \ (K j 3 (z/120tt), and T o 1/6

w = IT" (-TTIo \ <
- 5 -

The equation for W is

_|^L . i|it»|H g|W = (17)

and the boundary condition at the earth's surface now becomes

-JPL + qW = 0aty = 0. (18)

Furthermore, as x tends to zero, W should behave as (2/\Tx) • exp (-iy2 /4x).

It is now desirable to first consider the solution of equation (17)

when y can be neglected compared to unity. This corresponds to an

atmosphere which is linear or near linear. The revelant equation is

now

a

2 w aw

sw - i-xir + y w = ° • (19) ix

This can be solved by the method of separation of variables. That is,

assuming

W = X (x) Y(y)

one finds

12

1 d^Y(y) _ . _1 dX(x) _

(20)

where t the separation parameter. Thus

JET + itx = (21)

and

4^- + (y - t) Y = . (22)

Solutions of the first equation are

X (x) = e ±itx

(23)

and the second equation has solutions of the form

Y(y) = W(t - y) (24)

where W(t) satisfies Airy' s equation

d

2 W(t)

"oT^

t W(t) = . (25)
- 6 -

The appropriate solution for the x dependence is chosen such that

outgoing waves are only retained. Therefore

X(x) = e"

ltx

. (26)

Furthermore, the radial part of the solution (i. e. y dependence) is

chosen so that the phase of Y(y) is a decreasing function with height

(i.e. phase lag increases with height). At this stage, it should be

noted that Airy's equation is closely related to Bessel's equation of

order 1/3. In fact

I ?A

W(t) = const, x t 2 Z

l

/ 3 [(2/3) (-t) '

] (27)

where Z . /_ is any cylindrical Bessel function or linear combinations

thereof. In order to satisfy our requirement that the phase lag be an

(2) increasing function with height, the Hankel function H. /~ of the second

must be used. A particular solution which is suitable from the above

standpoint is

W(t) = Wx(t) = e"

i2ir/3

(-ut/3) 2 H

( ^3

[(2/3)(-t) 3/2

] (28)

where the constant is chosen to be consistent with the work of Fock and

Leontovich. In fact, the latter authors represent W,(t) as an integral

of the Airy type

l C° t 3/3

»1« = ^

Je12,/3 * ' dZ « 29»

where the contour is a straight line from infinity along the straight

arg. t = 2tr/3 to the origin and then out to oo along the real axis,

(i.e. arg. t = 0).

To satisfy the boundary condition at the earth's surface, one must

choose

[W'jft-y) - qWjIt-y)] = . (30)

A numerical study of this equation indicates that the roots t = t

(s = 0, 1, 2. . .) are in the fourth quadrant of the t plane. The

solution is then made up a linear combination of terms of the type

— lXt c ,.,» , Wjft 8 -y).
_ 7 -

It can be seen immediately that an equivalent representation is

r> e _ixt Wj(t-y)

Wl = V J W' (t) - q W.(t)

dt < 31 > C

where C is some contour which encloses the poles of the integrand

(i.e. at t = t ). The factor y, assumed constant, is to be chosen

such that W, has the proper singularity as x approaches zero. It is

to be expected that behavior of W, for small x is to be determined by

the behavior of the integrand for large values of t because of the

-ixt

factor e » But, when t is very large, it follows from the known

asymptotic behavior of the Hankel functions, that

W, (t-y) 1 i W[(t) - q Wjt) " ± (t)" 2 exp[-y(t) 2

] (32)

where the upper (+) sign is to be employed for 2ir/3 = arg t > -tt/3

and the lower (-) sign for -ir/3 > arg t ^ 4tt/3. The contour C is now

drawn such that one branch is along the negative imaginary axis

(- i oo to 0) and the other along the real axis (0 to oo). For small

values of x, then

C -ixt e^ W ~ v \ e

e n_ dt (33)

-ysTT

+ v \ e"""' -1 dt

f -ixt e nTT

These integrals can be transformed to the form

00 j

e

" p2 dp = *nr/z (34)

and thus l^ . y

2

w ~

2

y (-s—)" e **

(35)

1

and therefore, if y = (i/tr) 2
- 8 - W as 2 x" 2 e (36)

as x approaches zero.

The required solution can thus be written

e~

ixt W.(t-y)

W = (77 Jc

Wj(t) - q w(t)

dt • < 37>

In this particular case the solution for U becomes

-i* s -ika0 -ika0

U = — i V s -£-, V (38)

a(0 sin 0) 2 ay

where V = x

2 W

/ixv r

e

" xt w

i<*-y) V = ("T ) Jc WF> - q WlW

dt • < 39>

Attention is now returned to the case of an inhomogeneous atmosphere

wherein g(y) is no longer negligible compared to unity. The more

general equation for W must then be retained

4jr~ ' * -%T + y [ l + g(y)] w =

°

' (40>

Proceeding in an analogous fashion as in the linear case, it is readily

found that solutions are of the form

W = e"

lxt F (y,t) (41)

where F(y, t) must satisfy

d2

^

Y,t) + [y-t + yg(y)] F(y,t) = . (42)

The solution is then expressed as an integral

•c r-^- + qrn

L J v=o
- 9

which is constructed so the poles of the integrand are the roots,

t = t , of the equation

9F(y,t)

9y + qF(y,t) = (44)

y=0

and the constant term outside the integral is chosen so that it reduces

to the corresponding expression for W, in the linear atmosphere

(i.e. g = 0). The corresponding (formal) solution for the function U

is thus written ., n U = i V

where

V = H-li

a (0 sin 0)

2

-ixt _. ,. e F(y,t)

L. -Jv=o

dt

(45)

(46)

4. Asymptotic Form of the Solution

Various asymptotic approximations are now investigated. For

large values of the real part of (y-t + yg), the function F (y,t) of

equation (40) is given approximately by the W.K.B. solution of

equation (40). This is

.-iir/4

F(y,t) a Ce

[y-t + yg (y)]4

exp

_ -i r [ uL Jt

t + ug(u)] 2 du

(47)

where C is a constant and the lower limit of the integral is arbitrary.

It should be noticed that if g =

F(y,t) = C Wx (t-y)

which for large t-y behaves asymptotically like

F(y,t) C e -iir/4

exp [-{^ du (48)

(y-t) 4 in accordance with equation (28).

Equation (46) for V in the case of an inhomogeneous atmosphere

can now be simplified by noting F(y,t), where it occurs in the denominator,

can be replaced by W, (t- y) and therefore 3F + qF>

y = W'

x (t) + qWj(t).
- 10 -

In the numerator F can be replaced by the asymptotic form given by

equation (46) where y is large. Thus

V = -

where

1 / x T (0i II?

1 y - t + yg(y)

-ifl

WJ(t) - qWjtt)

dt (49)

Q = xt + \ n/u - t + ug(u) du . J

o The saddle point, determined by d^/dt = 0, is thus given by t = -p

2 where y

f

du

- 2x = . (5

Jo ^/u + p2 + ug(u)

Thus, following the prescription of the saddle point method

-too 2x 8p

where

and

F( " p) = 3T I

ipt

F (-p)

dt

W»(t) - qWj(t)

ft c = -xp' it- + p

2 + ug(u)] 2 du

(51)

(52)

(53)

The integral F(-p) has been discussed on previous occasions and

10 numerical results are available . For large negative values of the

argument (i.e. p > 0) it was noted that

F(-p) =

p l

P

{q exp.(-ip 3

/3) (54)

and therefore

e

-iJi. f2x _^P.^ 2 Zp

; exp . ( _ ip

3

/3) By / p + iq ^"^"» "r ' "' * 55*

must be the geometrical optical approximation and consequently, the

factor 2p/(p + iq) can be identified as 1 + R(a) where R(a) is the Fresnel

reflection coefficient for a plane wave incident at an angle a on the

boundary. The parameter p is then related to a by
- 11 -

1

p= (ka /2) 3 cos a .

In the case of the linear atmosphere (i. e. g(u) = 0)

fl. = -xp

2 + \ (u + p2

) 2 du (56) J o m p

3

/3 + 2y

3/2

/3

for p small compared to x and y. In this same case

2 P = -^^ <57)

and consequently i_2

£* 1 (-£")

When g(u) = 0, the formulas thus reduce to those for a homogeneous

atmosphere with a modified earth radius.

In the case of the inhomogeneous atmosphere equation (50) for p

can be solved approximately to yield

p = 4- \

d"

- x (58) C ^ N/ U + Ug(u)

under the condition that x and y are large but the difference x - NTy"

is finite. Thus

x \

(** ^r) ' = ~ r = 1 (59)

V

dy

) [y + yg(y)l*

and y

Q & p

3

/3 + V [u + ug(u)] 2 du . (60) Jo For application to an actual atmosphere, it is necessary to

evaluate the integral y

p + x = 4 r

du

(6i) c J

o V u + ug(u)

where

a

g = TEC

o (^ -«oWhen the dielectric constant varies in an arbitrary, but montonic, way
- 12 -

with height, the integral can be evaluated readily by numerical

methods. Some simplicity is obtained if it can be assumed that the

variation is exponential in form. For example if

K - K

q = (Kj/c) (1 - e"

ch

) (62)

it is seen that K decays exponentially from the value K at the surface

(h=0). The constant c is of the order 0.13 for an average atmosphere

and c measures of the rate of decay of the dielectric constant with

13 increasing height. Before proceeding further in this direction, it is

probably preferable to show the connection of the above results with

those of geometrical optics.

5. Distance to the Horizon

The distance s to the horizon from an elevated point in the

atmosphere is often used conveniently as a parameter in comparing

one type of vertical refractive index profile with another. Using a method

based on a stationary phase principle.it readily follows from Bremmer^

(and others) that

h

b = n c -»l a (63) Jo r ^[rn(r)]» - n.a*

where r = a + h, n(r) is the refractive which is a function of r and n

is the surface value [ i. e. ntt = n(a)] . In the case of exponential atmosphere

so that

n = 1 + be" ch

= 4K (64)

n. =n] h=Q = 1 , = sfK

Q

The term under the radical cai. oe written

a a

^a2

^a*

( • "-") ( * ¥)' - (>

* ') ]

]

1 + 2be" ch

+ 2-5- - 1 - 2b

a

i*. + 2b fe'

ch

- 1

under the assumption that b < < 1 and h/a < < 1. To the same
- 13 -

n 1 approximation —- in the integrand can be replaced by . Consequently,

the integral becomes

11 o - J

- g (65) ° \/(2h/a) + 2b (e"

ch

- 1)

which is equivalent to equation (61) derived from the full theory (with p = 0).

Now, it is convenient to break the integration into three parts, such that

h h» h" h =h'"

(66)

o ^h» h

and thus

s = s' + s" + s IM

In the first range (0 = h = s')

h»

£

dh

(67)

^J

(2h/a) + 2b Ach) + Sr-^

3 which is valid if h 1 is chosen such that the (ch) /6 term in the expansion

-ch

of e is negligible compared to ch. This is satisfied if

(ch1 ) 2 /6 < < 1

(For example if, ch 1 = -j- then (ch 1 ) 2 /6 = l/24) . Writing h' ... f dh

N/(2h/a)- 21 '

> n/KTI + ah)

(68)

we find

where

G =

\T2/a - 2bc n/g

be 2 (2/a) - 2bc

1 / ch'

-F- \l 1 + Gh«

tanil s/ 1 x kl (69)
- 14 -

If the effective earth's radius is defined by

1 _ 1 . K«

where

e

K = (1 + b) 2 & 1 + 2b

and , y i v 2

K- = ,* 6+b."

h =

and thus

s -2bc

J_ = J_ . be

a a

e so that

1 \[Za/a

«' = tt4^^ ^"J-rnip < 70>

where

c /J_ J_

N^7a"

e ^a " a

e

if ah' < < 1

s« =*

*

-s/ ah' = n/ 2a h' (71)

V 2 a/a e

' e which is the usual approximation for the distance to the horizon from

a height h' in a linear atmosphere. The hyperbolic tangent formula

above is actually the distance to the horizon from a height h1 in an

atmosphere defined by

n = 1 + b (1 - ch + (ch) 2 /2) (72)

for = h = h'. This is an approximation to

. , , -ch

n = 1 + be

as mentioned above.

We shall turn our attention to the third portion (i.e. h"5h=h'"=h );

'h»« \/(2h/a) + 2b (e"

ch ^1)
- 15 -

Now the term in the radical is written in the following form

-ch

(2h/a - 2b) + 2be- ch

= (2h/a - 2b) ["l + ^/a - 2b 1

' (74)

Letting u = (2h/a - 2b) it follows that

1

= J

A

n

(2b» n

kl ,n A

(2bT

e -chn

-ch N«-i I j— A n n

2bc \ 2 ^u n i > * u

uf 1 + ) n=0, 1, 2, 3. . .

-cnba

e x

n

cna

u

un+1/2 (75)

where A =1, A, =

o 1

1 . _ 1.3

T' A2 ~ ~2Ti"

a 1-3-5 A 1-3-5-7 , A- = - -*—a—7- * A. = t—^

r—a— ana so on

3 2« 4» 6 4 2 . 4 • 6 • 8

provided 2b e -ch

u < 1 . This is satisfied if 2b < h/a ^ h"/a or if

-4 h" > 2ba s 2x3x10 x 6360 ^ 3. 9 km for any value of c. -ch

Actually, this is more stringent than necessary since e < 1 . „, cna Thus u - —£— u

s'" = Y AJ2b)ne- cnba

C

e _T/ , du(a/2).

n

n=0, 1,2...

u" uE+T7T

Now consider the integral which occurs in the above expression,

00 Vm

(pu)= j"

u u

: -pu

m+1/2 du

(76)

(77)

and integrate it by parts successively to yield the recurrence relations

Vm(P«) = (m - 1/2)

Vm-l<Pu» " (5 - 3/2)

-pu

m

u

e r

n/r - p vm

mZJ/T - P V„ , (P

L U

and so on, until

lu)l

(78)
- 16 -

where

where

V

2

( PU

» = TV2)[^ PV1 (PU)

00 _ _ 1

Vl ((3u) = I ^-TfT du = (-J-)*

erfc <^> (79)

oo erfc (n/^u) =

u u

2

I

^

^/pu nTtF

dx (80)

is the complement of the error integral which is extensively tabulated."

Finally

(81)

14

i»i = T ^A

n

(2b)

n

e- nbca ["V^pu") - V

n (puJ

n

where

(3 = SJ^l , U" = 2 (— - b)

and u = 2(^2 -b)

It would appear that in most cases h* can be taken equal to h" and

chosen so that h1 = h" = 4 km which is a convenient upper limit for the

quadratic profile and a lower limit for the expansion formula. Thus

s = s' + s ,n . —ch An alternative is to return to the integral s" and expand e to

terms of third order. This leads to

h"

1 dh :ii =

where

N/(2/a) - 2bc

be 2 Y %7I

N/h (1 + Yh)(l + 6h)

be

(82)

(F^y

and 6 =

if

-r " bc

This integral is expressible in terms of elliptic integrals (see Pierce

No. 53 5 for example).

14
- 17 -

6. Concluding Remarks

The present analysis provides a strong justification for the validity

of an equivalent earth radius concept when the refractive index of the

atmosphere varies in a linear manner with height in agreement with

the conclusions of Bremmer , Norton , Miller and Northover

The effect of the non-linearity becomes important at great heights as

pointed out by Millington and others ' who have suggested that

the theory for a linear atmosphere could be modified by simply relocating the optical horizon point. In the present paper a complete

mathematical justification is given for this step provided that the

variation of the refractive index with height is monotonic.

The idea that the structure of the diffraction fields in a monotonically

varying or tapered atmosphere is similar to that for a homogeneous or

18 airless atmosphere is in apparent contradiction to Carroll's theory

of "radio twilight". However, the model he uses is not smoothly

tapered since there are discontinuities in the gradients of the refractive

index. In the bilinear model, for example, the atmosphere is represented

by a layer of dielectric with a constant gradient up to about 30,000 feet

after which the gradient is zero. Carroll obtains his solution by matching

19 the appropriate wave functions at the atmosphere-free space interface

In the free space region (above 30,000 feet for the bilinear model) only

outgoing waves are allowed but at the top of the dielectric graded layer

the two types of wave functions which are used can be identified as upgoing and downcoming waves. In the present treatment which uses a modified W.K.B. method, the solution in this region does not have this

double character.

Other models used by Carroll and Ring ' such as the inversesquare and trilinear profiles can be objected to on similar grounds.

It is of interest to note that the inverse-square profile is somewhat

better in that the discontinuity is in the second derivative at the bounding

surface and the computed attenuation, as expected, is greater. The

trilinear model has two piece-wise linear sections with interfaces at

15,000 and 90,000 feet and the mode computations show less attenuation
- 18 -

which is probably a consequence of the strong reflections from the

lower interface.

It is the opinion of this writer that the Carroll and Ring computations

are correct for the models specified. However, they cannot be regarded

as valid solutions for propagation in a monotonically tapered layer. It

is probable if the model were chosen so that the higher-order derivatives

of the refractive index are continuous at the interfaces, the mode solutions

and the modified W.K.B. solutions should approach one another. Unfortunately, the poor convergence of the resulting mode solutions will

make this ultimate comparison very difficult.
- 19 -

References

1. G. N. Watson, "The Diffraction of Radio Waves by the Earth",

Proc.Royal Society London, V. A95, pp. 83-99, 1918, V. A95,

p. 546-563, 1919.

2. G. Millington, "The Diffraction of Wireless Waves Round the Earth",

(A summary of the diffraction analysis, with a comparison between

the various methods), Phil. Mag., S. 7, V. 27, No. 184, pp. 517-542,

May, 1939, and T. L. Eckersley and G. Millington, Phil. Trans.

Roy. Soc. No. 778, Vol. 237, pp. 273-309, 1938.

3. B. Van der Pol, H. Bremmer, "The Diffraction of Electromagnetic

Waves from an Electrical Point Source round a Finitely Conducting

Sphere, with Applications to Radiotelegraphy and the Theory of the

Rainbow; The Propagation of Radio Waves over a Finitely Conducting

Spherical Earth", Phil. Mag. S. 7, Vol. 24, No. 159, pp. 141-176,

July, 1937, S. 7, V. 24, No. 164, pp. 825-864, Suppl. , November

1947, S. 7, V. 25, No. 171, pp. 817-834, Suppl., June, 1938, S. 7,

V. 27, No. 182, pp. 261-275, March, 1939, and H. Bremmer,

"Terrestrial Radio Waves", Elsevier, New York, 1949.

4. B. Vvedensky, "The Diffractive Propagation of Radio Waves",

Techn. Phys. U. S. S. R. , V. 2, pp. 624-639, 1935, V. 3, pp. 915-925,

1936, V. 4, pp. 579-591, 1937.

5. V. A. Fock, "Diffraction of Radio Waves Around the Earth's Surface",

Jour. Phys. U.S.S.R. , Vol. 9, pp. 256-266, 1945, Jour. Theo. Exp.

Phys. , Vol. 15, pp. 480-490, 1945 and M. A. Leontovich and V. A.

Fock, "Propagation of Electromagnetic Waves Along the Earth's

Surface", Jour. Phys., U.S.S.R., Vol. 10, pp. 13-24, 1946.

6. M. A. Leontovich, "Approximate Boundary Conditions for the

Electromagnetic Field on the Surface of a Good Conductor", Bull.

Academy Sciences, U.S.S.R. serie physique, Vol. 9, p. 16,

1944. (in Russian).

7. G. D. Monteath, "Application of the Compensation Theorem to

Certain Radiation and Propagation Problems", Proc. Inst. Elect.

Engrs. Vol. 98, Pt. IV, p. 23-30, 1951.

8. J. R. Wait and W. J. Surtees. "Impedance of a Top-Loaded Antenna

of Arbitrary Length over a Circular Grounded Screen", Jour.App.

Physics, Vol. 25, No. 5, pp. 553-555, May, 1954.

9. W. E. Miller, "Effective Earth's Radius for Radio Wave Propagation

Beyond the Horizon", Jour. App. Phys. Vol. 22, pp. 55-62, Jan. , 1952.

10. J. R. Wait and A. M. Conda, "Radiation Pattern of an Antenna on

a Curved Lossy Surface", Trans. I. R. E. , Vol. AP-6, pp. 348-359,

October, 1958.
- 20 -

11. K. A. Norton, "The Calculation of Ground-Wave Field Intensity

Over a Finitely Conducting Spherical Earth", P.I.R. E. , V. 29

pp. 623-639, December, 1941, and "Transmission Loss in

Radiowave Propagation", Pt. II to be published.

12. L. J. Anderson, "Tropospheric Bending of Radio Waves", Trans.

AGU Vol. 39, No. 2, pp. 208-212, April, 1958.

13. B. R. Bean and G. D. Thayer, "On Models of the Atmospheric

Radio Refractive Index", to be published in Proc. I.R.E. (Federal

Govt. Issue, 1959).

14. B. O. Pierce, "A Short Table of Integrals", Ginn and Co. , 3rd

Rev. Ed. 1929.

15. G. Millington, "Propagation at Great Heights in the Atmosphere",

The Marconi-Review, Vol. 21, pp. 143-159, 1958.

16. F. H. Northover, "Long Distance V. H. F. Fields", Can. Jour.

Phys. Vol. 33, pp. 241-256, May, 1955.

17. Personal communications from K. A. Norton, J. W. Herbstreit

and H. Bremmer.

18. T. J. Carroll, "Internal Reflection in the Troposphere and

Propagation well beyond the Horizon", Trans. P.G.A.P. Vol. 4,

p. 19, Dec. , 1952.

19. T. J. Carroll and R. M. Ring, "Propagation of Short Radio Waves

in a Normally Stratified Troposphere", Proc. I.R.E. , Vol. 43,

pp. 1384-1390, Oct. ,1955.

20. T. J. Carroll and R. M. Ring, "Twilight Region Propagation of

Microwaves by a Monotonically Tapered Layer of Air Dielectric",

L'onde Electrique Vol. 37, pp. 471-479, May, 1957 (in French).
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