PB 161553

9io. 52

FIELD STRENGTH CALCULATIONS

FOR E.L.F. RADIO WAVES

BY JAMES R. WAIT AND NANCY F. CARTER

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical ^VLote

52

March I960

FIELD STRENGTH CALCULATIONS FOR E. L. F. RADIO WAVES

by

James R. Wait and Nancy F. Carter

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

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-1-

TABLE OF CONTENTS Page

Abstract 1

I. Introduction 1

II. Basic Theoretical Model 2

III. Earth-Flattening Approximation 4

IV. Distance and Frequency Dependence 6

V. Effect of the Earth's Magnetic Field 10

References 14

Figures 17
FIELD STRENGTH CALCULATIONS FOR E. L. F. RADIO WAVES

Abstract

The mode theory of propagation of electromagnetic waves at

extremely low frequencies (1.0 to 3000 c/s)is briefly reviewed in this

paper. Starting with the representation of the field as a sum of

modes, approximate formulas are presented for the attenuation and

phase constants. At the shorter distances, where the range is

comparable to the wavelength, the spherical-earth mode series is

best transformed to a series involving cylindrical wave functions.

This latter form is used to evaluate the near field behavior of the

various field components. The effect of the earth's magnetic field is

also evaluated using a quasi-longitudinal approximation which is

particularly appropriate for propagation in the magnetic meridian.

In general it is indicated that if the gyro frequency is comparable or

greater than the effective value of the collision frequency, the

presence of the earth's magnetic field may be inaportant for E.L.F.

In this case the attenuation may be increased somewhat. The influence of a purely transverse magnetic field is also considered.

This technical note is actually a numerical supplement to the

paper "Mode Theory and the Propagation of ELF Radio Waves, " by

J.R. Wait, J. Research N. B . S. , 64D, .... July/Aug. I960; however,

for sake of completeness the relevant theory is briefly presented.

1. Introduction

The propagation of electromagnetic waves at extremely low

frequencies (E.L.F.) has received considerable attention recently

[1-5] . Unfortunately, the experimental data in this frequency range

(1.0 c/s to 3000 c/s) is still rather limited [ 8-15] . There is little

doubt, however, that the bulk of the energy is transferred via a waveguide mode. The bounding surfaces of this waveguide are the

ground and the lower edge of the ionospheric E region.
-2-

An important feature of E. L. F. propagation is that the distance

between source and observer may be comparable to the wavelength.

For example, at 300 c/s the wavelength is 1000 km. In fact, many

of the experiments are carried out at distances of this order and

account must be taken of the near field effects.

It is the purpose of this note to present computations based on

the mode theory as it applies to E.L.F. propagation. It is also

hoped that these results will also be of assistance in interpretation

of future experimental data.

II. Basic Theoretical Model

The as sunned theoretical model is taken to be a homogeneous

conducting spherical earth of radius a surrounded by a concentric

conducting ionospheric shell of inner radius a + h. It is convenient

to introduce the usual spherical coordinate system (r, 9 , (f>)

. Using

this model and assuming a vertical electric dipole source located at

Q = Q r = a, the expression for the vertical electric field, E , is

r given by an expression of the following form [4, 16]

E = 14^ y 6 Mll±il P (- COS 9) (1) r Zkha'^ /_, n sm v tr v

' ^

' n = where I - - average current in the source dipole;

ds - - length of source dipole;

T| - - intrinsic impedance of air space = IZOir ohms;

P (- cos 9) - - Legendre function of argument - cos 9 and

(complex) order v; V + -T-- = kaK S_t where S_, _ . for n = 0, 1, 2, . . , , n n

deternmined from the boundary condition and is

described below; and ^ 1/2 for n = = 1 for n = 1, 2, 3 . 'n The individual terms in the above series correspond to the

waveguide modes. In the general case, the factor S is obtained from

5

n ^ 1 sin 2 kh C

1 +

^

^ Z kh C
-3-

the solution of a complicated transcendental equation which involves

spherical Bessel functions of large argument and connplex order.

Certain aspects of this problem have been discussed by Schumann [ 1] and also the author [ 4] . For a homogeneous earth and a homogeneous

ionosphere, both assumed isotropic, it has been shown that S^ may be

approximated by [17] n s.-[-c^y^^^,K7l^^]y (2)

where A = ^+

1 ^

g

in terms of the refractive indices, N and N. , of the earth and the

g

1 ionosphere, respectively. This equation is valid subject to the condition j a| kh « 1. The sign of the radical is chosen in the above

equation which makes the real part positive. The values of S then

are located in the fourth quadrant of the complex plane. When n = 0,

the above simplifies to

S =

o

1

. A A

"ZTOi (3)

Now, since JA| kh « 1, the radical in equation (1) can be expanded

for n > to yield

S = n

1 - irn

TEE

2 . 2A - 1 2 er

for n = 1, 2, 3 (4)

Furthermore, if in addition,

A « 1 Trn \

kE '

it is permissible to write, for all n,

i € 2 Sn" 1 - Trn^'

lEE

n

ZTcH

/irn \'

(erj (5)

where €„ = 1. e =2 (for n = 1, 2, 3 . . . ). This latter equation is

valid when the "waveguide naodes" are not near cut-off and is in the

form given by Schumann [ 2] . It is quite analogous to the standard

results for the propagation constant in a rectangular waveguide with

finitely conducting walls [18, 19] .
-4-

III. Earth-Flattening Approximation

The possibility that the curvature of the earth may be neglected

is now investigated. To simplify field calculations at relatively short

distances, it would seenn desirable to transform the mode series to a

form where the first term corresponds to the model of a flat earth and

the succeeding ternas are corrections for curvature. This approach

has been used by Pekeris [20] and more recently by Koo and Katzin

[21] in their investigations of microwave duct propagation. From,

their work, it may be shown that

P (- cos 0) 1 / o \2 / \ -^ = H<2)^kS p) - ^(-^) H^2'(kSp) (6) sm V IT o ^ n'^' 12 I a / 2 n^' ^ ' 4 ft + terms in (p/a) , (p/a) , etc. , where v + l/l' = kS p and p = a0. H^' and FT^' are Hankel functions n^ ^ o 2 of the second kind of order zero and two, respectively. When the

great circle distance p is reasonably small compared to the earth's

radius a, only the first term in the expansion need be retained.

For convenience in what follows, it is desirable to express the

field components as a ratio to the quantity

Eo = i (nA) I ds (e"'^P)/p, (ti = 12077);

E is the radiation field of the source at a distance p on a perfectly

conducting ground. Thus, for both the source and the observer near

the ground, it is not difficult to show that

E = W E z o 00 where W = - itr^ e'^P 7 6 S^ H^2)^kS p) h i_j n n o ^ n^ (7)

n=0

E = - S E

p o

OO where S ^ ^ ^

e^^P ^

6^ S^ k}^^ (kS^p) (8) ^ n=0

and H . = - T E /ti 9

o' '
-5-

where T =' - TT f e ikp

00 (2)

> 5 S H^2' (kS p) = - N S (9)

n=o

where terms containing (p/a)"^ , (p/a)"* have been neglected.

When kp » 1, corresponding to the "far-zone, " the above expressions may be simplified since the Hankel functions may be replaced by the first term of their asymptotic expansion. This leads to

the compact result

W

S

T

^^'"^-^1 «n

3/2

n=o

n

n ' g

n ___

- iZirS pA (10)

which is valid for p

» \. As expected, the ratio of W to T for a

given mode is S which for low order or grazing modes is of the order

of unity. The ratio of S to T, quite generally, is - l/JST which is very

small connpared to unity; in fact, it vanishes for a perfectly conducting ground as it must.

The excitation by a horizontal electric dipole may be treated in

a similar manner. A complete discussion of the theory is given elsewhere [17] . The expression for the vertical field E of the horizontal

dipole Ids which is located at the origin and oriented in direction = may be written in the form

E = S E cos (11) z o where S and E are defined above. By using the reciprocity theorem,

this result can also be deduced directly from equation (9). Comparing the above formula for E (for a horizontal dipole) with correspondz ing formula for E , for a vertical dipole, it nnay be easily shown that

E (for horizontal dipole)

z E (for vertical dipole)cos

N

/ 1 \2 cos ~ ( ^ "1^/ ^ "n:S

(12) z g ^ '\ ' g

being valid in the asymptotic or far zone. In terms of the ground
-6-

c onductivity <r and dielectric constant c , o o

g g g - g

This is a small quantity.

The above formulas for W, S, and T are valid only if p « a.

If the first curvature correction term is included, it is a simple

matter to show, in the "far zone", that this amounts to naultiplying

the right-hand side by the factor

^ / \ Starting from the mode series for a spherical earth [ 17] and using

the far field approximation for the Legendre function, it turns out

that W, S, T have precisely the sanne fornn as in equation (10) except the factor {9/sin9)^ occurs in the right-hand side. Noting that

1 2 ey .!+«!+•terms in• ^, 0" , etc, -^ -^ "sHT^ ~ ^ TT

it is apparent to a second order that the derived curvature correction

is the same as from the Hankel function series.

IV. Distance and Frequency Dependence

Certain features of the mode series are best illustrated by a

graphical plot of field strength vs. distance under various values of

the parameters. Since N » N. it may be safely assumed for E. L. F.

that A = 1/N. • Then on the assumption that the ionosphere is behaving like a conductor, it is possible to write

-i -^^

where

_ cOq^ _ (plasma frequency)^

r V collision frequency
-7-

The above relation for N. is an approximation which is valid when the

collision frequency v between electrons and neutral ions is much less

7

than the angular frequency o). Since v is of the order of 10 in the

lower E region, this is certainly valid at E.L.F. The (angular)

plasma frequency co,, is given by

2 e^N

00, €o m

where N is the electron density, e and m are charge and m.ass of the

electron, and€„ is the dielectric constant of free space, oj , which

is proportional to the ratio N/v, is believed to be of the order of 10

in the D layer or the lower region of the E layer. At least this

appears to be the effective value deduced from V. L. F. observations

for highly oblique incidence [ Z3] . At E. L.F. it appears that the

waves are reflected at higher levels in the ionosphere and the effective value of CO is sonaewhat larger, being possibly of the order of

10^

It should be noted in passing that co may be replaced by (r-Ao

where q-. is the effective conductivity of the ionosphere and ^ is the

dielectric constant of free space.

To indicate the variation of the field as a function of distance,

the magnitude and phase of T and W are shown in Figs, la to 4c for a range of values of co and h = 90 km. The curves denoted co = oo correspond to a perfectly conducting ionosphere. These are

presented for purposes of comparison with the set for the more

6 5 reasonable values of co . These are 2 x 10 and 5 x 10 . Actually,

|t| and |wj are both divided by the square root of the distance p

(in kilometers) and plotted since the curves become linear at

larger distances. The slope of these linear parts of the curves is

proportional to the attenuation rate (in db per 1000 km) as usually

defined. It is seen immediately that at the shorter distances, the

curves are no longer straight. This immediately indicates that

considerable caution should be exercised in computing attenuation

rates from spectral analyses of "sferics. " For example, it is only
-8-

when the distance p exceeds about one -sixth of a wavelength, is it permissible to as surae that log (|w|/'"Tp~) or log (Ie | x 's/p ) vary

z in a linear manner with distance p. Similarly, the phase of W(orE ) varies in a linear manner only when the distance exceeds about onehalf wavelength. Similar remarks apply to the magnetic field

variations with distance.

To shed further light on the nature of the E.L.F. fields in the

waveguide, the radial impedance of the wave is now considered.

It is noted that E

= -Z (14) z =

^ which is by definition the (normalized) impedance of the wave looking

in the radial or p direction. First it should be noted that for kp » 1,

^^ 1 T

and for kp « 1 and p « h,

W ^ 1 T ikp

For the intermediate and interesting range, kp is comparable to unity

and p is somewhat greater than h. Using the numerical values of T

and W nnentioned above, the ratio | w/ T | and the phase (lag) defined

by arg T - arg W are plotted in Figs. 5a to 6c and as a function of

distance for various frequencies. As before, h = 90 km and the same

range of co values are chosen.

The impedance ratio W/T is independent of the frequency

spectrum of the source provided, of course, it may be represented

by an equivalent vertical electric dipole. In fact, this complex ratio

could be easily calculated from the frequency spectra of the wavefornas

of the vertical electric field and the horizontal magnetic field of an

atmospheric. The observed variation of magnitude or phase of W/T

as a function of frequency should then provide a basis for distance

measuring. Such a scheme, while admittedly crude, requires only

one receiving station equipped with a vertical whip and a loop antenna.
-9-

At the very short distances where p is of the order of the

ionospheric reflecting height (i.e. , 90 km), the mode series

representation for the fields is very poorly convergent. However,

alternate representations are available which converge very rapidly

in this case [ 17] . For co = oo, these are

T = m

m=o

_t^Vl-^^e-i'^<'m-P'

m

(15)

m

and

00 W = m=o

L r ) [( - .:

1 \ i

; " RF-m J

/2mhV

\ m

[6- 3i

" krm

'p^+( 2mh)2

1 = 1, e = m

= 2(m

(16)

ik(r -p) m

where r = m

Individual terms in these series correspond to the contribution from

images of the source dipole. These particular expressions are exact

on the assumption of perfectly conducting planes, (i.e. , oj = oo and

cr = oo). It is known, however, from the mode calculations that at

g

short distances (i.e. , for p < JOO km), the numerical values of the

functions T and W are only very slightly dependent on the finite value

of u) and o" . Thus (15) and (16) are generally applicable.

To illustrate the nature of the fields at short ranges, the magnitude and phase of T and W are shown plotted in Figs. 7a to 8b. The

solid curves correspond to equations (15) and (16) for h = 90 km. The

dashed curves correspond to equations (15) and (16) when h = oo; this

is the sanae as taking only the first term in the series. At the very

short ranges, it is seen that the solid and dashed curves merge together. Apparently in this region, the reflecting layer at 90 km can

be ignored. At the greater ranges, the T and W functions both

approach unity for the case of no reflecting layer (i.e., h = oo).
-10-

Corrections for earth curvature resulting from diffraction would,

of course, modify these dashed curves at ranges exceeding about

500 km. It may be concluded from the information plotted in Figs.

7 and 8 that only for extremely short ranges (i.e. , less than about

30 km) is it permissible to neglect the presence of the ionospheric

reflecting layer in field strength calculations at ELF.

V. Effect of the Earth's Magnetic Field

In the preceding it has been tacitly assumed that the ionosphere

is behaving as an isotropic homogeneous conductor. In this case,

the refractive index N. of the ionosphere may be written [23]

1

N. s

1 i.

i _£. I s / .JL ' (17)

0) I 10)

where

_ cof _ (Plasma Frequency)^

r V (Collision Frequency)

When the earth's magnetic field is steeply dipping, it is appropriate

to invoke the quasi-longitudinal approximation of Booker [23] . In

this case the refractive index is double valued, one corresponds to

the ordinary and the other the extra-ordinary, thus

Nj^ r

1 - i exp (± It)

O) (18)

where

tan T =

and

L, _ longitudinal connponent of gyro -frequency

V ~ collision frequency

CM? O = j_ . ^

(v2+cu|^)2

Using this model for the ionosphere, the reflection coefficient for a

sharp boundary has been derived by Budden [24] . Adapting this result to E.Li.F. it has been shown [17] that the formula for S has the ^ ^ n
-11-

same form as the isotropic medium if A is defined by

_i

g V "r

Since O = u) cos t it is possible to write the results in the same

r r ^ form as the isotropic ionosphere if we set

where

(Ni)e£f. =

^ r' efi. ^ ico

(21)

(N.) rr and (u) ) ,^ are the effective values of N. and co , respectively.

* reff. r'eff. i r ^ •'

Specifically,

^ r'eff- r (cos t/2)2 '

In other words, the effective conductivity of the ionosphere is modiCOS T

fied by^ the factor -, /-, ^„ which varies fronn. unity to zero as t

(cos t/2 )2 -' varies from to ir/Z. Thus, the attenuation is increased as a result

of a steeply dipping (or vertical) magnetic field. In fact

Attenuation with magnetic field ^ cos (t/Z)

Attenuation without nnagnetic field ~ , ,1 ^ (cos T ) 2

If T « 1 (i. e. , (ii « v), this ratio becomes unity and the influence

of the earth's magnetic field vanishes. At the level in the ionosphere

where E. L. F. waves are reflected, it is expected that co and v are

comparable, both being of the order of 10 . Assuming that they were

actually equal, t becomes 4 5 and the ratio of the attenuation rates

is 1.05. Thus, it is only when o) is greater than v does the earth's

J—

I

magnetic field appreciably influence the attenuation. The ratio is

plotted in Fig. 9 as a function of tan t or oj /v. This illustrates the

situation clearly.

The preceding results are subject to the validity of Booker's

quasi -longitudinal approximation of the Appleton-Hartree equation[23]
-12-

The validity of this approximation requires that

4

where cj^ and co are the longitudinal and transverse components of

the (angular) gyro-frequency. Clearly this condition is violated when

the transverse component of the earth's nnagnetic field is large such

as for propagation along the magnetic equator. This case, however,

has been considerrd by Barber and Crombie[25] who derived explicit

results for the reflection coefficient at a sharply bounded ionosphere

with a purely transverse magnetic field. Adapting their results to

E. L. F. it is not difficult to show [17] that A is now given by

A=g)* X (23)

CU/l CO v2co/ VVW

X ^J \ r^\ r r^ \r

2

\ 2 '^rp 2

, , . CO \ T oj 1 + 1 — _ r ' r The real and imaginary parts of A are plotted in Figs. 10a and 10b as

a function of the frequency parameter 5x 10 x f/co where f is expressed in kc/s and co is in (sees) . The various values of a)_/v are

r X

shown in the curves; positive values of the ratio correspond to

propagation from east-to-west while negative values correspond to

propagation from west-to-east. Now the attenuation of the dominant

mode in nepers per unit distances is given by - kS where S is given

by equation (3).

Using the above results for A the attenuation coefficient q, expressed in db per 1000 km of path length, is plotted in Fig. Ha for

h = 90 km. It is immediately evident that the attenuation is only

slightly raodified by the presence of a transverse magnetic field

unless, of course, co_p is considerably greater than v. The phase T

velocity of the dominant zero-order mode, relative to free space, is
-13-

simply l/Re S . Denoting this dinriensionless quantity by p it is

plotted in Fig. lib for h= 90 km. It is apparent that the presence of a transverse magnetic field does have an influence on the phase

velocity.

Acknowledgment

We would like to thank John Harman and Barbara Bolton for

preparing the illustrations and Eileen Brackett for typing the

manuscript.
-14-

References

[I] W.O. Schumann, TJber der ausbreitung sehr langer

elektrischer wellen unn die erde und die signale des blitze,

II Nuovo Cimento, IX^ (1952).

[2] W.O. Schumann, Uber die oberfelder bei der ausbreitung

langer, elektrischer wellen im system erde-luft-ionosphare

und 2 anwendungen (horizontaler und senkrechter dipol),

Zeits fur ange, Phys. , 6, 34, (1954).

[ 3] W.O. Schumann, Uber die strahlung langer wellen des

horizontalen dipols in dem lufthohlraum zwischen erde und

ionosphare, Seits fur ange Phys. , 6, 225, (1954).

[4] J.R. Wait, On the mode theory of VLF ionospheric propagation. Review Geofisica Pura e Applicata, 37, 103, (1957),

(paper presented at International URSI Conference of Radio

Wave Propagation, Paris, Sept., 1956).

[5] Li. Liebermann, Anomalous propagation below 500 c/s, Proc.

Synn. on Propagation of VLF Radio Waves, 3, paper no. 25,

Boulder, Colorado, (1957).

[6] R.E. Holzer and O. E. Deal, Low audio-frequency electronnagnetic signals of natural origin. Nature, 177, 536, (1956).

[ 7] O E. Deal, The observation of very low frequency electromagnetic signals of natural origin, unpublished Ph. D.

Thesis, The University of California, Los Angeles, 1956.

[8] F. W. Chapman and W. D. Matthews, Audio -frequency

spectrum of atmospherics, Nature, 172, 495 (1953).

[ 9] F. Hepburn and E. T. Pierce, Atmospherics with very low

frequency components. Nature, 172, 837, (1953).

[10] H. F. Willis, Audio frequency nnagnetic fluctuations. Nature,

161, 887, (1948).

[II] J. Aarons and M. Henissart, Low frequency noise in the range

0.5 to 20 c/s. Nature, 172, 682, (1953).
-15-

[12] R. Beniot, Low frequency radio wave noise of the earth's

magnetic field, C. R.Acad. Sci. , 242 , 2534, (1956).

[13] P. A. Goldberg, Electromagnetic phenomena of natural origin

in the 1.0 - 150 c/s band, Nature, 177 . 1219, (1956).

14] J. Aarons, Low frequency electromagnetic radiation 10 to

900 c/s, J. Geophs. Res., 61, 647, (1956).

15] J.M. Watts, An observation of audio-frequency electromagnetic noise during a period of solar disturbance, J. Geophs. Res., 62^ 199. (1957).

16] J. R. Wait, An extension to the mode theory of VLF

ionospheric region. Jour. Geophys. Res, , 63, 125, (1958).

17] J. R. Wait, Terrestrial propagation of VLF radio waves,

J. Research NBS, 64D, 153, (I960).

18] H. Motz, Electronnagnetic problems of microwave theory,

Methuen and Company, 1951.

19] H.R.L. Lamont, Waveguides, Methuen and Company, 1942.

20] C.L. Pekeris, Accuracy of the earth -flattening approximation

in the theory of microwave propagation, Phys. Rev. , 70, 518,

(1946).

21] B. Y. -C. Koo and M. Katzin, An exact earth -flattening

procedure in propagation around a sphere, J. Research NBS,

64D, 61, (I960).

22] J. R. Wait, The mode theory of VLF ionospheric propagation

for finite ground conductivity, Proc. IRE, 45, 760, (1957).

23] J. A. Ratcliffe, Magneto-ionic theory, Cambridge Univ. Press,

1959.

24] K.G. Budden, The reflection of very low frequency radio

waves at the surface of a sharply bounded ionosphere with

superimposed magnetic field, Phil. Mag., 42, 504, (1951).

25] N. F. Barber and D. D. Crombie, V.L.F. reflections from

the ionosphere in the presence of a transverse nnagnetic field,

Jour. Atmos. Terr. Phys. , 16, 37,(1959).
-16-

[ 26] J. R. Wait and H.H. Howe, Amplitude and phase for ground

wave propagation in the band ZOO c/s to 500 kc, NBS Circular

574, 1956.

Figure Captions

Figs, la, b, c. The normalized magnitude of the magnetic field as

a function of distance from the source (p is expressed in kilometers in both ordinate and abscissa)

Figs. 2a, b, c, The normalized phase (lag), in radians, of the

magnetic field.

Figs. 3a, b, c, The normalized magnitude of the electric field as a

function of distance from the source.

Figs. 4a, b, c, The normalized phase (lag) of the electric field.
Q

3.0

2.0

1.0

0.7

0.5

0.3

0.2

0.1

0.07

0.05

0.03

0.02

0.01

t "

" 1 1 11 1 1 1 1 1t 1 1 1

Gl), =00

1

il\ \^ h = 90km =

P^^^^^

f = 50nA -

100

-

200 iV 400 — V 800 " A^ 1600 ~ y

-

! i 1 1

1

Fig. lo

i 20O 400 600 800 1000 1200 1400 DISTANCE, p(km)

1600 1800 2000

200 400

1—\—\—I

\ \ n

1000 1200 1400 1600 1800 2000 DISTANCE, pikm]

1

\—\—\—\—\

\ r~r-r

0.01

Fig. Ic

,11 I I I \ \ L J L

2X 400 600 1000 1200 DISTANCE, p (km)

1400 1600 1800 2000
cr<

I uJ<

X

Q.

1 \ I r

Fig. 2a

I I I

I 200 400 600 800 1000 1200 1400 1600 1800 2000 DISTANCE, ^(km)

1.4 -,

1.2 -

LxJ

C/) 1.0 <

X

0.8 -

0.6 -

0.4

—

1 : 1 1 1 11

1 I 1 1 1 1

1 1

1 -

OJr =2x 10^ -

\ h = 90 k m -

_ \ A c^ - A

;;;;;^-—

-^_^____^^—

z

_^.—

j:= -

__

Fig. 2b

—

—

1

I !

,

1 1 1 1 ! 1

'

__

200 400 600 800 1000 1200 1400 1600 DISTANCE, p (km)

2000

1 \\ T 1 \ \ \ r

J I \ L J \ \ \ L

Fig. 2c _^ \ L

200 400 600 800 1000 1200 DISTANCE, p (km)

1400 1600 1800 2000
^

UJQ3

<

0.03

0.02

0.01

h =90 km

f =50c/s

100

200

400

800

1600

Fig. 3a

200 400 DISTANCE, p (km)
gl^

LlI O

CD<

0.03

0.02

0.01

OJr = 2 X 10°

h =90 km

f =50c/s

100

200

400

800

1600

Fig^ 3b

200 400 600

1400 1800 2000 DISTANCE, p (km)

ih

UJQ

1-<

800 1000 1200 1400 DISTANCE, p (km)
1 \ \ \

\ \\ \ \ \


 cr<

I

LlT <X

Q.

h = 90 km

1200 DISTANCE, p (km)

2000

DC<

I ul<

X

Q.

1200 DISTANCE, /3 (km]

1400

200 400 600 800 1000 1200 1400 DISTANCE, ;o(km)

2000
Figs. 5a, b, c The magnitude of the radial wave impedance.

Figs. 6a, b, c The phase (lag) of the radial wave impedance.

Figs. 7a, b The magnitude and phase of the normialized magnetic

field at short distances. The broken curves correspond to no reflecting layer.

Figs. 8a, b The magnitude and phase of the normalized electric

field at short distances.
ILUQ

<

7

5

-\\

1^J 1 H II I — \ \

cjr = 00 :

_\

\\\ h =90km —

3

2 \ \^

\ V

1.0

0.7 ^s \ —

V. \ ^^^^ Z

0.5 — /^ —

F ig. 5a /

0.3 — —

0.? 1 L 1 1 1 1 1 1 1 1 1 1 1 1 1 1

20 30 50 70 100 200 300 500 700 DISTANCE, /)(km)
LJQ

ID

C5<

0.3

0.2

Fig. 5b

0.1 50 70 100 200 300 500

DISTANCE, ^(km)

2000

03

0,2

Fig. 5c

1

50 70 100 200 300 500 700 1000 DISTANCE, /5(km)

2000
8.0

7.8

7.6

<J)

-z. 74

<

Q

< \1

(T

7n ^

C3

cr 6.8 <

1 6,6 1-

C3

6.4 <

^ 6.? LU

CO

< 60

_L

Q5.8

5.6

5.4 20 100 200 300 500 700 1000 DISTANCE, /5(km)

2000
8.0

7.6 „ 7.4 in

-z. <

7?

LJ<Q-—

70 $

CO

q: fia <

1- 6.6

(.5

<X. < 6.4 LU

CO< 6.i! X

Q.

1 1 1 1 1 1 [ 1 1 11 11 1

\^\ \ \ aJr=2xiO^ \\\\ h = 90knn :^^^

50 70 200 300 500

DISTANCE, /)(km)

2000

<

<

CC<

cr<

UJ

CO<X

Q.

50 70 100 200 300 500 700 1000

DISTANCE, /3(km)

2000
5 10 20 30 50 100 200 300 500 1000

DISTANCE,

p (km)
if)

<

ct: <

I

CO

<

CL

10 20 30 50

DISTANCE,/) (km)
IjJ Q

Z)

1.0

0.7

0.5

0.4 5 7 20 30 50 70 100 200 300 500 700 1000

DISTANCE,/) (km)
3.4

iz50c/s

CO

<

CD

q:

LU

CO

<

CL SOLID CURVES

FOR CUr= 00

DASHED CURVES

\.

10 20 30 50

DISTANCE,/) (km)

100
Fig. "^ The earth's -magnetic -field-correction to the attenuation

factor for propagation in the magnetic meridian.

Figs. 10a. b The real and imaginery parts of the factor A as a function of normalized frequency (for a purely transverse

magnetic field).

Figs. 11a, b Attenuation and relative phase velocity for a purely

transverse field.
C\J

05o

o

o

o

5 10 20 (jJ^/V OR tan T

0.0

0.08 —

0.07

<

Li_ o

hq:

g

>-

a:<

CD

<

0.06 —

0.05 —

0.04

0.03 —

0.02

0.01 50 200 fX^I^INc/S

500 1000
E

j^

o

o

o

"O

a

<

LlI

00.

H

O

_J

LlI >

UJ<

X

CL

LJ>

LJ

3 -

2 -

< —

0.5 50

1.0

0.96

0,94

0.92

50

100 200 500

FREQUENCY, c/s

1000

70 200 300 FREQUENCY, c/S

700
I

U.S. DFJPARTMENT OF COMMERCE

Frederick H. Mueller, Secretary NATIONAL BUREAU OF STANDARDS

A. V. Astiii, Director THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards at its major laboratories in Washington, D.C., and

Boulder, Colorado, is suggested in the following listing of the divisions and sections engaged in technical work.

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