National Bureau of Standards

Library, S.W. Bldg

NOV 2 7 1963

\^&cknical <ylote 92o. 205

NUMERICAL CALCULATIONS FOR REFLECTION

OF ELECTROMAGNETIC WAVES FROM

A LOSSY MAGNETOPLASMA

LILLIE C. WALTERS AND JAMES R. WAIT

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Luther II. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

technical ^©te .205

Issued November 21, 1963

NUMERICAL CALCULATIONS FOR REFLECTION

OF ELECTROMAGNETIC WAVES FROM

A LOSSY MAGNETOPLASMA

Lillie C. Walters and James R. Wait

Central Radio Propagation Laboratory

National Bureau of Standards

Boulder, Colorado

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.

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington, D.C. 20402

ERICE : 5 CENTS

Contents Page

Abstract 1

1. Introduction 2

2. The Model 3

3. Presentation of Results 5

4. Discussion and Concluding Remarks 9

5. Appendix 10

6. References 14

Acknowledgement 15

Figures 16

in

NUMERICAL CALCULATIONS FOR REFLECTION OF

ELECTROMAGNETIC WAVES FROM A LOSSY MAGNETOPLASMA

Lillie C. Walters and James R. Wait

Extensive numerical results are presented for the reflection

coefficient of a horizontally stratified ionized medium. The profiles

of electron density and the collision frequencies are both taken to be

exponential functions. The d-c magnetic field is taken to be horizontal and transverse to the direction of propagation. The specific

results described are applicable to the oblique reflection of VLF

radio waves in the D layer of the ionosphere for propagation along

the magnetic equator. It is confirmed that the reflection coefficient

is non-reciprocal in both amplitude and phase. For a wide range of

the parameters, the magnitude of the reflection coefficient is greater

for west-to-east propagation than for east-to-west propagation.

The extensive graphical data in the present paper are to be

regarded as supplementary to the paper "Reflection of Electromagnetic

Waves from a Lossy Magnetoplasma" which contained only a small

sample of such calculations.
1. Introduction

The lower ionosphere is primarily responsible for the propagation of VLF radio waves to great distances. In theoretical treatments of this problem, it is often assumed that the lower edge of the

ionosphere may be represented by a sharply bounded and homogeneous

ionized medium. When the earth's magnetic field is included, the

medium is rendered anisotropic. If the vertical inhomogeneity (or

horizontal stratification) of the ionosphere is also considered

simultaneously, the situation becomes very complicated indeed.

In this technical note a special case of a horizontal stratified

and anisotropic ionosphere is considered. Specifically, the earth's

magnetic field is assumed to be purely transverse to the direction of

propagation. Strictly speaking, this is applicable only to the situation

when the path of propagation is along the magnetic equator. However,

the characteristics in this special case prevail at other latitudes if

the transverse component of the field is appreciable. At least this

is borne out by a numerical study of the sharply bounded ionosphere

for an arbitrary magnetic dip angle [Johler, 1961]. In any case, the

resulting simplicity of the differential equations for the limiting case

of a purely transverse magnetic field encourages one to consider this

situation in more detail. In particular, it is desirable to investigate

the influence of gradient of both the electron density and collision frequency. In much of the previous work on this subject the collision

frequency has been assumed constant.

-2-
In a study of the recent literature [e. g. , Belrose, 1963], it is

found that both the electron density N(z) and the collision frequency

v(z) vary approximately in an exponential manner with height z. For example, in the undisturbed daytime ionosphere we may assume

that

N(z) = N exp (bz) , (1) o and

v(z) = v exp(-az) , (2)

where a and b are positive constants and z is some specified

level in the ionosphere. From a study of the experimental data

[Belrose, 1963], it appears that, if the reference level is 70 km

2 , above the earth's surface, N ~ 10 electrons/c. c. and

7 -1 v ~ 10 sec . The gradient parameters are then expected to be

given approximately by b ~ 0. 15 km (± 0. 1) and a ~ 0. 15 km (± 0. 02).

The quoted values of these constants must be considered tentative and

certainly subject to change. Furthermore, it must be understood that

significant departures from the exponential shape are to be expected

under disturbed conditions.

2. The Model

The situation is shown explicitly in figure la. A vertically

polarized plane wave is incident at angle 9 on to a horizontally

stratified ionosphere. The z axis is taken to be positive in the

upward direction. At the reference level z = 0, the electron density

and the collision frequency have values designated by N and v , o o respectively. The "scale height" which is equal to 1/b or l/a, as

indicated in figure lb, is of the order of 6 km for both of these profiles.

-3-
As mentioned above, these are typical of the daytime D layer for

both the N and v profiles. For this model, the reflected wave is

also vertically polarized and thus the reflection coefficient is

described by a single (complex) quantity R.

The lower ionosphere, which is idealized here as a stratified

ionized medium, may be regarded as an electron plasma. The

(angular) electron plasma frequency go is thus given by

co a = 3.18xl0

9

X N , (3) o

where N is the electron density in electrons per c. c. and go has

o dimensions of radians per second.

The continuous profiles of N(z) and v(z) are replaced by a very large, but finite, number of steps. In other words, the inhomogeneous medium is replaced by a stack of thin homogeneous layers.

For purposes of discussion, there shall be P such layers while a

typical layer is the p'th layer. Thus, p ranges from 1 to P

through integral values. Somewhere at a sufficiently negative value

of z, the medium may be regarded as free space. This level is

denoted z = - z . o The problem may now be solved by an application of non-uniform

transmission line theory [Schelkunoff, 1943; Wait, 1962], Thus, the

reflection coefficient for vertically polarized waves, referred to the

lower edge of the bottom slab, is given by

o C + A

-4-
where A = Z /r\ and where Z is the input impedance at the

1 o 1 bottom of layer number 1. Now, Z may be expressed in terms of

Z which, in turn, may be expressed in terms of Z . The process

is continued until the topmost layer is reached where Z is assumed

P

known. The details of this derivation are given elsewhere [Wait, 1962;

Wait and Walters, 1964].

The required number of layers is best determined by studying

the stability of the solution as the number is increased. Because of

the relatively long wavelength involved, and because of the finite

losses in the medium, the solution converges nicely as the number

of layers is increased.

A typical profile with its layer approximation is shown in figure

N(z) No

lc. The conductivity parameter —;—- = exp [(b + a) zl is

v(z) v

is plotted against the vertical distance z above or below the reference

level at z = 0. Between z = -z and z = T, the upper edge of the

top layer, the medium is divided into P homogeneous layers of

N(z) width hj , hP h h , h . The quantity ——— is replaced

in each layer by a constant value. The values of T and the h 's P

must be arbitrarily chosen for the computations, and the selection of

these constants for various given parameters X , the free space wavelength, C, the cosine of the angle of incidence, a, and b is discussed

in the appendix.

3. Presentation of Results

The final results of the numerical calculations are presented

in such a fashion that the phase of the reflection coefficient R is

referred to the level z = 0. Thus, by definition

R = fR exp (i 2k C z )] . (5) o o

Z — 00 O

-5-
Physically, this means that the observer at z = - z is sufficiently

far below the ionosphere that the medium may be regarded as free

space. In practice z is chosen to be large enough that the phase of

R does not vary with further changes in z . For the case here, z o o was of the order of 40 km. This particular normalization of the

phase has been used on previous occasions [Wait and Walters, 1963].

Following the usage in previous papers [Wait and Walters, 1963],

the quantity co = u

z {0)/v {0) is specified. In particular,

4 1 oo/co = l/Z at 15 kc/s, or co = 6 n x 10 sec . The "effective"

r r conductivity tr of the medium at this level z = 0, is then given by

6 -6 /

cr = e co ~ 1. 7 x 10 mhos/meter. With exponential-type profiles,

e o r the fixing of the parameter of co is not an essential restriction. It r

is a simple matter to shift the reference level from z = to any

other value if desired.

The parameters of the problem are thus X, C, b, a, and

Q, = co /v where co is the (angular) gyrofrequency. In order to

display the relative influence of these quantities, it is desirable to

plot the amplitude and phase of R as a function of co /v from -3

to + 3 for a range of values of X, C, b, and a. It should be noted

that X is in km, C is dimensionless, while b and a have dimensions

of km . Consequently, the scale length in the present problem is

the kilometer. By changing this scale, the results may also have

significance at higher frequencies.

In figures 2a and 2b the magnitude of the reflection coefficient

|R| and the phase of R are plotted as a function of co /v . Negative

values of the abscissa correspond to propagation from west to east

along the magnetic equator. The cosine of the angle of incidence is fixed

at 0. 1. Thus, the angle is highly oblique, being only 5. 7 from grazing.

-6-
For long-distance propagation of VLF radio waves, such highlyoblique conditions prevail. For the curves in figures 2a and 2b, the

collision profile is chosen so that the collision parameter a = 0. 15 km

and the wavelength X = 15 km correspond to a frequency of 20 kc/s.

For these curves, the electron density parameter b takes the values

in the range from 0. 1 to 0.5 km . It is evident that for oo_/v = 0, To

the steep gradient of electron density is associated with maximum

amplitude of reflection. However, when co /v is finite, this may no

T o longer be the case. In fact, the asymmetry of the curves about

co_/v = is a measure of non-reciprocity in the reflection process. To

As indicated, the reflection coefficient for propagation from west to

east is greater than for propagation from east to west. There is

also some non-reciprocity in the phase curves, but it is not great.

A very similar set of curves is given in figures 3a and 3b where the

conditions are the same except that C =0.2.

In figures 4a, 4b, 5a, and 5b, a set of curves shows the influence

of varying the collision frequency parameter while keeping the electron

density parameter fixed at b = 0. 15 km . For these curves, as before, C = 0. 1 and 0. 2, and X = 15 km. It is evident that the steeper

gradient of the collision frequency corresponds to larger reflection

coefficients.

In figures 6a to 8b, |R| and the phase of R are shown as a function of to /v for various frequencies in the range from 6 to

T o 100 kc/s. For the curves in figures 6a and 6b, C = 0. 1, b = 0. 15,

and a = 0. 15 while, in figures 7a and 7b, and figures 8a and 8b,

t In fact, the attenuation of the dominant mode in the earth-ionosphere

waveguide at VLF is approximately proportional to 1 - |r| for highly

oblique incidence [Wait, 1962].

-7-
C =0.2 and 0.3, respectively. The tendency is for the reflection

coefficient to be diminished at the higher frequencies or shorter

wavelengths. In this case, the medium is acting like a good absorber

rather than a reflector. It is to be noted that at the steeper angles

of incidence (i. e. , C = 0. 3), there are some complicated phenomena

which are probably related to internal reflections within the medium.

In figures 9a and 9b, |R| and the phase of R are plotted for

different values of the angle of incidence. For these curves, \ = 15 km,

b = 0. 15 km , and a = 0. 15 km . In general, it may be seen that

the reflection coefficient is diminished for the steeper angle of incidence. It is rather interesting to note that the asymmetry ( or nonreciprocity) in the phase curves is more pronounced at the steeper

angles.

Finally, in figures 10a and 10b, |r| and the phase of R are

plotted for various values of the collision frequency parameter

a for X = 15 km and C =0.1. These curves differ from figures 4a

and 4b in that here the parameter 8 = b + a is fixed, rather than

just a. In other words, the profile of N/v as a function of z is

fixed while the gradient of v is changed. In the isotropic case,

where go = 0, it is interesting to note that R is determined only

by the gradient of N/v. However, for a finite gyrofrequency, the

situation is changed significantly. Similar curves are shown in

figures 11a and lib where C = 0. 2. In general, the non-reciprocity

is accentuated when a is diminished. For example, if v were

assumed to be a constant, the dependence of the gyrofrequency is

much greater than for a collision frequency which varies with height.

In much of the earlier work [e. g, Budden, 1955] on full wave solutions

in ionospheric radio waves, it is often assumed that v can be regarded

as a constant. Clearly, such an assumption may lead to very misleading results.

-8-
4. Discussion and Concluding Remarks

The numerical results given here should provide some insight into

the nature of reflection from an inhomogeneous ionized medium. The

nature of the dependence on N, v, and co is quite complicated. Nevertheless, it appears that the sharper gradients of electron density are

usually associated with higher reflection coefficients. The dependence

on the collision frequency profile is not so clear-cut.

In nearly every case it may be seen that the presence of the

transverse magnetic field causes the reflection coefficient |R|

to be non-reciprocal. Furthermore, for a wide range of the parameters,

|R| is greater for west-to-east propagation than for east-to-west

propagation. This is in accord with experimental data of Round, et al.

[1925] who observed that, for propagation over distances of the order

of 6000 km, signals from VLF transmitters to the west are received

more strongly than from those to the east. This observation has also

been confirmed by Crombie [1958] in a series of field strength measurements in New Zealand and by Taylor [I960] who analyzed the waveforms of atmospherics.

It is interesting to note that Budden [1955] deduces, from a full

wave solution, that the directional dependence is just opposite to this.

Although his model is not the same as the one considered here, it is

difficult for the authors to accept the validity of his results in this

regard. However, it is possible that, because of the complexity of

the various phenomena, a reversed trend may emerge for certain

special conditions , particularly for the nighttime ionosphere [Rhoads,

et al. , 1963], It is also worth mentioning that Budden [19 55] has

some qualms concerning the accuracy of his numerical data at small

values of C.

-9-
5. Appendix

The principal object of the calculations is to find Z the

normal wave impedance at the lower edge of the bottom layer.

The reflection coefficient R , for vertical polarization, is then v

obtained from (4). The numerical method used is briefly outlined

here.

The impedance Z is dependent upon Z , the impedance

of the layer p above it. Therefore, Z is dependent upon all

of the Z's of the layers above the first. Because of this property,

an iterative process, using a digital computer is appropriate for

the computations. In such a process a good starting value of Z

is desirable. Then the impedance Z for the layer next to the top

one may be expressed in terms of the wave impedances for the

upgoing and downgoing waves. These wave impedances can be

calculated at once as they are functions of given parameters. Thus,

with a starting value of Z available, the impedance for each

successive layer down through the first can be found [Wait, 1962].

In these calculations, the quantities T and h must be

P

chosen correctly (see figure lc). The distance T should be chosen

such that,' if it is increased, the value of the reflection coefficient

is unchanged. From economic considerations, however, this

distance should be as small as possible consistent with accurate

results. Furthermore, the individual layers of thickness h must

P

be small enough to well approximate the given conductivity profile.

Thus, small intervals are chosen near the top of the medium where

-10-
the conductivity profile is rapidly varying, while h is gradually

P

increased at the bottom of the profile. Again, for the sake of

economy, the h 's should be chosen as large as possible consistent with the required accuracy.

It is also necessary to determine the distance from the reference level to the lower edge of the bottom layer, z = - z . The

o lowest layer must be far enough down in the medium to effect

convergence. An answer "converges" to the value R at a distance

z = -z if decreasing z leaves R unchanged to the required

number of digits. The value of R at this distance is considered

the "free space" value of R to the accuracy desired.

When the earth's magnetic field is neglected corresponding

to Q = 0, the problem reduces to the one considered previously

by the authors [Wait and Walters, 1963], using the same exponential

profile. Because the formulation differed in the two papers, a check was possible when Q - 0. The results from both types of

computation agreed to within five digits wherever they could be

compared.

To develop a satisfactory procedure to find R using the

layer method, the problem under the isotropic assumption

(i. e. , Q = 0), was first programmed for horizontal polarization.

Because R , the reflection coefficient for horizontal polarization,

ii / 2it2 C\

may be expressed in closed form |R | = exp ( - ) , it was

possible to check the values of R obtained by the layer method.

h

Then the values of T and the h ' s were adiusted until the answers

P

calculated by the two methods agreed to five digits.

-11-
The program was then written using the step-by-step process for

vertical polarization. An important check here utilized the fact that

for C = 1, the values of R are identical for vertical and horizontal

polarization. Extreme values were used for the various given

parameters and thepaths printed out step by step for vertical and

horizontal polarization. By studying these print-outs, it was found

that the variation of C, with other given parameters held constant

did not. greatly alter the choice of the arbitrary constants necessary

to effect convergence to the correct value. Thus, the comparison

of the case for C = 1 with the closed form was used as a criterion

to determine T and the h 's for all the C's. If the value of R , p v using T = 4 km, agreed for C = 1 to four digits with the closed

form answer, this was considered satisfactory. If it did not agree,

T was set equal to 10 km and finer intervals of h were used.

P

For T = 10 km, the values agreed in every instance with the

closed form for C = 1 to five digits. As a further check, especially for C ^= 1, for a few cases, T was set equal to 20 km,

and these answers agreed in every instance to five digits with

those calculated using T = 10 km.

When Q == 0, it was found that the convergence of R was at

v

least as rapid as when Q, = 0. Thus, as a starting point, the same

choice of T and the h ' s was used for the anisotropic cases.

P

Following is a small table showing the distance z required for

convergence of R to five digits for a typical set of parameters . for X =15 km with fi = :

-12-
C = 0.1 C = 0. 2 C = 1

z Q (km) z Q (km) z Q (km)

0.3 50-55 40-45 40-45

0. 5 30-35 25-30 20-25

L 10-15 10-15 -

2 5-6 4-5 - From these results, it was found that if 8 and I are constant,

the variation of C does not modify the required value of z greatly.

Because the medium is slowly varying near - z , large intervals of

h , requiring few steps, may be used in this region. Thus, if

only C of the given parameters varies, it is feasible to use the

estimate of the distance z suitable for the smallest C for all the

o C's. Similarly, if only the wavelength is allowed to vary, there is

little effect on the required value of z . For 8=0.3 and C = 0. 1, o between \ =10 km and \ =25 km, the value of z varies from

o about 50 to 55 km, a distance of only five kilometers for the whole

range.

As might be expected of the given parameters, the value of 8,

the gradient of the conductivity change, has the most critical effect

on the choice of the arbitrary constants T and the h 's. For

P

8 > 0. 3 and T = 4, the initial h 's were chosen 0. 02 km and

P

gradually increased to 0. 5 km, corresponding to a total of about 200

layers. The manner of transition of the size of the layers through

the medium is not particularly critical. Several different combinations

of numbers of various step sizes were tried with the same final results. For 8 5 0. 3 and T =10 km, the initial h 's were 0.005 km

P

with steps increasing to 0. 5 km, corresponding to about 1300 layers.

The choice of the arbitrarily chosen parameters T and the h 's P

-13-
was probably on the conservative side in some cases, but as the

computation was very fast, and accuracy was guaranteed, this

seemed a worthwhile procedure.

The general technique described above was also used for perturbed conductivity profiles superimposed on the exponential conductivity profile. There was a provision in these cases for taking

smaller intervals of h when the effect of the perturbation was

P

encountered and continuing with these until the exponential profile

became dominant again [Wait and Walters, 1963]. This technique

can be applied to various types of profiles in addition to those referred to with satisfactory results.

6. References

Belrose, J. S. (1963). Present knowledge of the lowest ionosphere,

a chapter in Radio Wave Propagation (ed.by W. T. Blackband),

Pergamon Press, Oxford.

Budden, K. G. (May, 1955). The solution of the differential equations

governing the reflexion of long radio waves from the ionosphere

II, Phil. Trans. Roy. Soc. (London), Series A 248 , 45-72.

Crombie, D. D. (1958). Differences between east-west and west-east

propagation of VLF signals over long distances, J. Atmos. and

Terrest. Phys. 12, 110-117.

Johler, J. R. (Jan. 1961). Magneto-ionic propagation phenomena in

low- and very-low-radiofrequency waves reflected by the ionosphere,

J. Res. NBS 65D, 53-61.

Rhoads, F. J. , W. E. Garner, and J. E. Rogerson (1963). Some experimental evidence of direction effects on VLF propagation (private

communication)

.

-14-
Round, H. J. T. , T. L. Eckersley, K. Tremellen, and F. C. Lunnon

(1925). Report on measurements made on signal strength at

great distances during 1922 and 1923 by an expedition sent to

Australia, J. IEE 63_, 933-1011.

Schelkunoff, S.A. (1943). Electromagnetic Waves (Van Nostrand,

New York).

Taylor, W. L. (July, I960). VLF attenuation for east-west and

west-east daytime propagation using atmospherics, J. Geophys.

Res. _65, No. 7, 1933-1938.

Wait, J. R. (1962). Electromagnetic Waves in Stratified Media

(Pergamon Press, Oxford, and Macmillan, New York).

Wait, J.R. , and L. C. Walters (1963). Reflection of VLF radio

waves from an inhomogeneous ionosphere, Part I. Exponentially

varying isotropic model, J. Res. NBS67D (Radio Prop.), No. 3,

361-367, May-June 1963; Part II. Perturbed exponential model,

No. 5, 519-523, Sept. -Oct. 1963; Part III. Exponential model

with hyperbolic transition, No. 6, Nov. -Dec. 1963.

Wait, J.R. , and L. C. Walters (1964). Reflection of electromagnetic

waves from a lossy magnetoplasma, J. Res. NBS 68D (Radio

Prop. ), No. 1 (Jan. -Feb. ) (VLF Symposium issue).

ACKNOWLEDGEMENT:

The work in this paper was supported by the Advanced Research

Projects Agency, Washington 25, D.C. , under ARPA Order No. 183-62.

-15-
N(z) OR (Jo (z

<i

'

i rnV 2-0

REFL. WAVE

FIGURE la - ILLUSTRATING OBLIQUE REFLECTION OF VLF

RADIO WAVES FROM AN INHOMOGENEOUS (HORIZONTALLY

STRATIFIED) IONOSPHERE WITH A TRANSVERSE D-C MAGNETIC

FIELD.

N(z) = N EXPIbzl

1/(2) \l I

l\

1 \1 \

1

1

vti) = M

1

1

1

1

1 4

-6km 6 km

FIGURE lb - SKETCH OF THE PROFILES OF ELECTRON DENSITY

N(z) AND THE COLLISION FREQUENCY v(z).

FIGURE lc - THE STEP APPROXIMATION TO AN EXPONENTIAL

PROFILE.

16
1.0

1

1 | 1 1

x = 15 km C = QI

0.9 i^Si. _ a = 0.15 km-1

cr 0.8 r—

^03 N.

h-UJo

u.

u.UJo

o

z

0.7

0.6 b

-02 ^N/V

0.15 ^^""^Ox'v

= 0.10 km" 1 ' 0\.

-

o

1-oUJ

_lu_

Ijj cr

0.5 - ^\V^->^

v V

0.4

i 1

-

-I I 2

<

i

QMAGNETIC FIELD PARAMETER, ft FIGURE 2a - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON ELECTRON DENSITY PROFILE FOR C = 0.

-100°

-120° -

-140'

1

' 1

! 1 X = 15 km

C= ai

a = 0.15 km"'

h = oao km"'

______ rvtB _ OZ^ —

- 03___- - b35Kni__—

I - I i |80<-3 -I I MAGNETIC FIELD PARAMETER, ft FIGURE 2b - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON ELECTRON DENSITY PROFILE FOR C = 0. 1

<r O.i

1 r

6 .^5;

7

ujo "~-^2l

ii 0.6 u_

iii ' 02 o<_> 0.15 zo 0.4 b = 0.1 km" 1 1-

l)

III

1 llUJ

cr

0.2-3-2-10 I 2 MAGNETIC FIELD PARAMETER, XI FIGURE 3a - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON ELECTRON DENSITY PROFILE FOR C = 0. 2

-iff

-80°

COUJ

UJ

cruj -100° o

-120° —

-140'

1 1

1

'•

X = 15 km

C = 0.2

- ^o£&-~~--^

a = 0.15 km"'

- o£-~~"~~"^^"""^ ^ry-

- J^^^^'^ yj -

- ^-""

>/

- J*^&^~

i

-160°

- |80 -3 -I I MAGNETIC FIELD PARAMETER, ft FIGURE 3b - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON ELECTRON DENSITY PROFILE FOR C = 0. 2

17
06

-2-10123 MAGNETIC FIELD PARAMETER, ft

FIGURE 4a - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON COLLISION FREQUENCY PROFILE FOR C = 0. 1

i r

S^Sikm

i -3 -2-10122 MAGNETIC FIELD PARAMETER, ft F^URE 5a - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON COLLISION FREQUENCY PROFILE FOR C = 0. 2

(f) a

o

IdQ

q:

<

1 1

1

1

X = 15 km

C = 0.i b = o.l5km"i

o!_^—

—"

1 ——

n = 0.2 KrtL

1 1 , I I 3 -2 -1 1 2 3

-160'

MAGNETIC FIELD PARAMETER, ft FIGURE 4b - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON COLLISION FREQUENCY PROFILE FOR C = 0.

1

o

COX

a.

-40c

-60<

-140°

1 1 1

x = 15 km

b = 0.15 km"1 C=0.2

- of

6/

-

J^^^

- O^-—-^^-

-

1 1

i

- -2-10 I 2 3 MAGNETIC FIELD PARAMETER, ft FIGURE 5b - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON COLLISION FREQUENCY PROFILE FOR C = 0. 2

18
cr 0.

8

-2-1012 MAGNETIC FIELD PARAMETER, ft

FIGURE 6a - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON WAVELENGTH FOR C = 0. 1

120'

40° -

-40°-

-I20

1

-160°

-200°

1

1 C = O.l $ = 0.3 km"' o = 0.15 km"' X=3km

- -

- -

6

-

8

;

10 —

-

15 — on ^25

X = 50 km \40

... 1 1 1 1 1 -2

2 MAGNETIC FIELD PARAMETER, ft FIGURE 6b - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON WAVELENGTH FOR C = 0. 1

0.3

0.6 h

h

III Oo

04

<-" o

H(>

lil _l

li 0.2 hi

tr

UJUJ

UJQ

1

_____40—-

1

i C = 0.2

j8 = 0.3 km"' a = 0.15 km"1 -

~—^ir^^^

15 ' - _

10

___;

1

8

6

s

X = 3km 1 1 -3 -2-1012

MAGNETIC FIELD PARAMETER, ft FIGURE 7a - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON WAVELENGTH FOR C = 0. 2

a*r

'

! 1

C = 0.2 £ = 0.3 km"1 240" a = 0.15 km"'

200" ±^£—

- r

160° — ~

120" -

80* -

j2———

_

40°

0°

^_6

__8

-40°

10

-80° ^~~~"^

15 ^

-120°

. leno ~~

go —

"x=50km,

" ^40

3-2-10 I 2 3 MAGNETIC FIELD PARAMETER, ft FIGURE 7b - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON WAVELENGTH FOR C = 0. 2

19
0.8-

(

>

ll OB

ll. X

<_> o 0.4 1-oIII

1 llUJ 0.2

1 1 | 1 1

C - 0.3

/3 = Q3 km

a = 0.15 km" _

~—~^5 ^^\.

-

- -

15 ^""--.^^O^V

-

10 ^^-C^^s:

^

' 6 " —

1 1 ^t= =f 1 ! — , -2-10 I 2 3 MAGNETIC FIELD PARAMETER, Si

FIGURE 8a - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON WAVELENGTH FOR C = 0. 3

3-2-10 I 2 3 MAGNETIC FIELD PARAMETER, SI

FIGURE 8b - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON WAVELENGTH FOR C = 0. 3

or

0.6-

0.5-

0.4-

a 0.3 _i

u_UJ

'0 = 0.1 ' 1

1

X = 15 km

*"— /3 = a3knrr' ^^^ a = 0.15 km" - 1

____QI6

- ____Q2

- -

- —-_C^Q3

-

1 1 1

I 1

- 0.2

-3 -2-1012 MAGNETIC FIELD PARAMETER, Si

FIGURE 9a - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON ANGLE OF INCIDENCE.

UJ

ce(5

UJQ

CE

00<

X

-40'

-120'

-160'

1

x = 15 km

1

1

1

/3 = 0.3 km"1 a =0.15 km-1 "

C --^

~^Z~~— O.I6 _

C=O.I

1

-

1 1 1 3 -2 -1 2 3 MAGNETIC FIELD PARAMETER, SI

FIGURE 9b - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON ANGLE OF INCIDENCE.

20
u.a \9.

1

1

1 1

x= 15 km

c=o.i

0.8 /3=0.3kmH

cc ^^ 07 =——S&i _

UJ OI5 o

ll a = 0.2 km-1

llh 06 _ ?5^^^ \.\ ^ ^^-^

U

2 \\ \ \ \\ \ ^

\— \ \ \ *-•

III 0.5 - \\ \ \ 1 llIII ±

0.4

m 1 1

1 1 \ \

-3 -2-10123 MAGNETIC FIELD PARAMETER, ft

FIGURE 10a - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON COLLISION FREQUENCY PROFILE WHEN THE

N/v PROFILE IS FIXED FOR C = 0. 1

o.e

0.7-

0.6

0.5-

0.4-

0.3

0.2

0.1-

\ '

' '

1 l

x = 15 km

N?- /3 = Q3 km"1 \p C = 0.2

^vv \

-

^^—-^£/ N. \ -

O.lT ^~\N\

a = 0.2 km-l ^aS^

- \\\ \ ^^^ \\ \ ^ ^^^ \\ \ ^ ^^ \\ \ ^ \ \ \ "N. \ \ \ "^ \\ \ ^

- \ \ ^s. _ ^

-120°

-130° -

cc -140'

-150'

-160°

1 1

1 ' ' x = 15 km

j9 = 0.3 km* 1

-

C=0.i ^

~~\" figs -4// -

^JS / -2-10123 1 1 1

MAGNETIC FIELD PARAMETER, ft

FIGURE 10b - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON COLLISION FREQUENCY PROFILE WHEN THE

N/v PROFILE IS FIXED FOR C = 0. 1

MAGNETIC FIELD PARAMETER, ft FIGURE 11a - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON COLLISION FREQUENCY PROFILE WHEN THE N/v PROFILE IS FIXED FOR C = Z

MAGNETIC FIELD PARAMETER, ft FIGURE lib - REFLECTION COEFFICIENT CURVES ILLUSTRATING THE DEPENDENCE ON COLLISION FREQUENCY PROFILE WHEN THE

N/v PROFILE IS FIXED FOR C = 0. 2

21

GPO 837-445



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.

In general, each section carries out specialized research, development, and engineering in the field indicated by

its title. A brief description of the activities, and of the resultant publications, appears on the inside of the

front cover. WASHINGTON D.C.

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Dielectrics. High Voltage. Absolute Electrical Measurements.

Metrology. Photometry and Colorimetry. Refractometry. Photographic Research. Length. Engineering Metrology.

Mass and Volume.

Heat. Temperature Physics. Heat Measurements. Cryogenic Physics. Equation of State. Statistical Physics.

Radiation Physics. X-ray. Radioactivity. Radiation Theory. High Energy Radiation. Radiological Equipment.

Nucleonic Instrumentation. Neutron Physics.

Analytical and Inorganic Chemistry. Pure Substances. Spectrochemistry. Solution Chemistry. Standard Refer- ence Materials. Applied Analytical Research. Crystal Chemistry.

Mechanics. Sound. Pressure and Vacuum. Eluid Mechanics.. Engineering Mechanics. Rheology. Combustion

Controls. Polymers. Macromolecules: Synthesis and Structure. Polymer Chemistry. Polymer Physics. Polymer Characterization. Polymer Evaluation and Testing. Applied Polymer Standards and Research. Dental Research.

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Inorganic Solids. Engineering Ceramics. Glass. Solid State Chemistry. Crystal Growth. Physical Properties.

Crystallography.

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Instrumentation. Engineering Electronics. Electron Devices. Electronic Instrumentation. Mechanical Instru- ments. Basic Instrumentation. Physical Chemistry. Thermochemistry. Surface Chemistry. Organic Chi lar Spectroscopy. Ele- mentary Processes. Mass Spectrometry. Photochemistry and Radiation Chemistry.

Office of Weights and Measures. BOULULK, COLO.

CRYOGENIC ENGINEERING LABORATORY

Cryogenic Processes. Cryogenic Properties of Solids. Cryogenic Technical Services. Properties of Cryogenic

Fluids. CENTRAL RADIO PROPAGATION LABORATORY

Ionosphere Research and Propagation. Low Frequency and Very Low Frequency Research. Ionosphere Research. Prediction Services. Sun- Earth Relationships. Field Engineering. Radio Warning Services. Vertical Soundings Research.

Troposphere and Space Telecommunications. Data Reduction Instrumentation. Radio Noise. Tropospheric Measurements. Tropospheric Analysis. Spectrum Utilization Research. Radio-Meteorology. Lower Atmosphere

Physics.

Radio Systems. Applied Electromagnetic Theory. High Frequency and Very High Frequency Research. Fre- quency Utilization. Modulation Research. Antenna Research. Radiodetermination.

Upper Atmosphere and Space Physics. Upper Atmosphere and Plasma Physics. High Latitude Ionosphere

Physics. Ionosphere and Exosphere Scatter. Airglow and Aurora. Ionospheric Radio Astronomy.

RADIO STANDARDS LABORATORY

Radio Standards Physics. Frequency and Time Disseminations. Radio and Microwave Materials. Atomic Fre- quency and Time-Interval Standards. Radio Plasma. Microwave Physics.

Radio Standards Engineering. High Frequency Electrical Standards. High Frequency Calibration Services. High

Frequency Impedance Standards. Microwave Calibration Services. Microwave Circuit Standards. Low Frequency

Calibration Si Joint Institute for Laboratory Astrophysics -NBS Group (Univ. of Colo.).