•-.j. jj aj cdu ox soanaaras

Y, N.W. Bldg

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OCT 5 1961

PB 161615

^ecknical vtote ^lo. 114

j5oulder laboratories

MODE CALCULATIONS

FOR

VLF PROPAGATION

IN THE

EARTH-IONOSPHERE WAVEGUIDE

BY

KENNETH P. SPIES AND JAMES R. WAIT

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
isnaaras 164

5Sy 0D7

THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

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NATIONAL BUREAU OF STANDARDS

technical ^iote

114

July 17, 1961

MODE CALCULATIONS FOR VLF PROPAGATION IN THE

EARTH -IONOSPHERE WAVEGUIDE

by

Kenneth P. Spies and James R. Wait

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.

DISTRIBUTED BY

UNITED STATES DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D. C.

Price $ 1.50

-1-

Table of Contents Pa-ge

I. Introduction 1

II. Calculations for a Flat Earth-Ionosphere Waveguide 3

1. Ionospheric Reflection Coefficients in the Quasilongitudinal (Q-L) Approximation 3

2. An Approximate Solution of the Mode Equation for

Perfectly Conducting Ground 6

3. Ionospheric Reflection Coefficient for Transverse

Magnetic Field 19

4. An Approximate Solution of the TM Mode Equation

for Perfectly Conducting Ground in the

Transverse Case 2l

5. Arbitrarily Dipping Magnetic Field 30

III. Calculations for a Spherical Earth-Ionosphere Waveguide

Using the Quasi -longitudinal Approximation 39

1. The Mode Equation 39

2. A Crude Solution for Perfectly Conducting Ground 40

3. Solution for Perfectly Reflecting Boundaries 48

4. Newton5 s Method for Solving the Mode Equation 49

5. Solution for Perfectly Reflecting Ionosphere and

Finitely Conducting Ground 53

6. Solution for Imperfectly Reflecting Ionosphere and

Perfectly Conducting Ground 55

7. Solution for Imperfectly Reflecting Ionosphere and

Finitely Conducting Ground 58

IV. Further Calculations for a Spherical Earth-Ionosphere

Waveguide 66

1. A More Accurate Form of the Mode Equation 66

2. Solution for Perfectly Reflecting Boundaries 69

3. Newton1 s Method for Solving the Mode Equation 71

Appendix A. Table of Inverse Tangents of Airy Functions 74

Appendix B. Some Formulae Involving Airy Functions 80

Appendix C. A Note on the Conductivity of the Lower 90

Ionosphere at VLF

_ , Q7 References

-11-

Ac knowledgmerit

The authors would like to thank Mrs. Eileen Brackett for her

extensive secretarial assistance in the preparation of this technical

note. Appreciation is also extended to John C. Harman and Victor J. Pfannenstiel for their help in the careful drafting of the many

illustrations.

Special thanks go to J. Ralph Johler and Mrs. L. C. Walters

for providing us with certain of their numerical data on reflection

coefficients. These were used in obtaining the results of section II- 5<

The research reported in this document has been sponsored in

part by the Advanced Research Projects Agency. The publication of

this report does not necessarily constitute approval by ARPA of the

findings or conclusions contained herein.

MODE CALCULATIONS FOR VLF PROPAGATION IN THE

EARTH-IONOSPHERE WAVEGUIDE

by

Kenneth P. Spies and James R. Wait

I. Introduction

The concept that radio waves are channeled between the earth

and the ionosphere as in a waveguide has been very useful at VLF

[Budden, 1953;Alpert, 1956; Wait, 1957]. Unfortunately, the computational aspects of the problem are quite complicated even when the

model is highly idealized. The difficulty stems from the grazing

nature of the modes of lowest attenuation. Some progress has been

made recently by utilizing higher order approximations for the various

spherical wave functions which enter into the problem. In this way the

influence of earth curvature has been fully accounted for. The detailed

theoretical aspects and essential derivations have been presented

elsewhere [Wait, I960, 1961] . Here the actual computational procedure

is outlined and some numerical results are presented. It is believed

that the methods used are of general interest and also have possible

application to propagation of acoustic and seismic waves in curved

layered media.

An essential feature of the techniques used is a simplified representation of the ionospheric reflection coefficient which is valid for

highly oblique incidence. This permits a perturbation procedure to

be applied to the equation for the mode characteristics. In the

examples employed here the ionosphere is represented by a sharply

bounded and homogeneous ionized medium with a superimposed

magnetic field. Initially the qua si -longitudinal approximation is
-2-

invoked since it permits a great simplification in the analysis. Its

validity has been discussed in detail elsewhere [Budden, 1961].

Generally speaking, it is good when the propagation is in the magnetic meridian or in polar regions for all directions of propagation.

These results are then introduced into the mode equation which involves the reflection coefficients &,, and . R. and the conversion

coefficients R and . R evaluated for a complex angle of incidence.

Here the earth is regarded as flat in order to simplify the calculation.

In the following section the earth 1 s magnetic field is taken to be

horizontal and transverse to the direction of propagation. This would

correspond to propagation along the magnetic equator. Then the case

of an arbitrarily dipping magnetic field is treated in a relatively crude

fashion. Most of the calculations mentioned above are then carried

out for a curved earth where the modal equation is a great deal more

complicated. Nevertheless, by making certain convergent expansions

of the Airy functions, the problem becomes tractable and can be treated

by perturbation methods.

For the convenience of those engaged in related investigations,

a table of inverse tangents of the ratio of two Airy functions is given

in Appendix A. A fairly extensive collection of formulas relating to

Airy functions is then given in Appendix B. Finally, for sake of

completeness, a short note on the concept of conductivity of an

ionized gas is contained in Appendix C.
-3-

II. Calculations for a Flat Earth -Ionosphere Waveguide

1. Ionospheric Reflection Coefficients in the Quasi -longitudinal

(Q-L) Approximation. - In the quasi -longitudinal (Q-L) approximation,

ionospheric reflection and conversion coefficients for a sharply bounded

homogeneous ionosphere have been given by Budden [1961] in the following form 2

(E + E er(C-C G

e)+(( t Pe -l)(C + C e)C

" R

" " K+ »J (cSc c ) + (JT ^e + 1) (c

o + ce

) C

R = <^^e»<c2 - C o C e'-^o'J e1"C o tCe» C

_ (2) 1 l '

(^ +^)(cz + c o c

e

) + ((x o ^ + i)(c

o + cjc

i2C <|JL C -ix C ) R =

o ° e e

f (3)

" i

(n +|x KC^ + C C ) + (|x (1 +1)(C +C ) C

o e o e o e o e

i2C(u C e -u C ) and R = 2__f 1_2 , (4)

(fJL +fx )(C +C C ) + ((J. M- +1)(C +C ) C o e o e o e o e

where C is the cosine of the (complex) angle of incidence

2 2 2

i - or = \i (i - c ) o

v o' and where -, _ (Snell's Law).

1- C = uf(l 1 e

x - CIe

The refractive indices u. and u. of the ionosphere are defined by the

o e relations

i - i H

2 =l-J_e ^ and u. 2 = 1 - -L e

J- o L

re L

CO

. L longitudinal gyro frequency where tan = = § =f—; - L- L v collisional frequency
-4-

where

L =

GO

CO

1_ co(v 2 + co

T2

)

cTs

and where to is the angular frequency and to is an effective conductivity

[Wait, I960] which is defined explicitly above. Note that R and

R both approach -1 as C approaches (grazing incidence), whereas

R and R both approach 0.

To facilitate solving the made equation for VLF propagation in a

flat earth-ionosphere waveguide, it is convenient to represent the

rather complicated reflection coefficients ..R.. and R by expressions

having the form

R = - exp [o1 C + c 2 C

2 + a3 C

3 + . . .] (5)

where the (complex) parameters Gj , c2 , c3

, . . . depend on ionospheric

properties, but not on C. The validity of such a procedure depends on

the fact that |c | is small for the most important modes at VLF, so

that R is near -1.

There is, of course , no unique way of choosing g^ , a 2 , c3 ,

• • • . The method finally adopted is the following. In

lo § i.R|

and R = e log R

ll R H ~

e

the logarithms are expanded in a Taylor series about C = to obtain

00 00

:XP I Ck C

L k =

and 1 R , = exp

Tc =

(6a)(6b)

where

1 r a K

zr 1 "^7^ (log i^n 1 acJ and Pk = ¥T

r9-

c = o ac

k ( lo§ i

R

i^

c=o

(7a) (7b)
-5-

The first few Taylor series coefficients for log ..R.. are

c = ITT o

(8a)

2 u. u.

' o ' e i ^o+^e L \p 2 n \T|J.2 - 1 J o

(8b)

cu =

2fJL [I o e 2 (JL fJL o e

1 o ' e

(8c)

do = 2Ti ai + Hnr L o e pnm 1 e

1 o e o "e "o

uT^l u z -

1

o e

, (8d)

and those for log R are

r o

(9a)

Pi =

|JL (J. o e (9b)

P* = G 2 (9c)

_ Pi 2 „ 2 6 Pi

p 3 = £U ((I'5

|JL ^

vr -1) (. o r 24 e ' V M- - Pi

2

) +

o e

[X fJL o e (9d)

The ionospheric reflection coefficients R and R can now be

approximated by using a few terms of the series (6a) and (6b),

respectively.
-6-

2. An Approximate Solution of the Mode Equation for PerfectlyConducting Ground - When the ground is perfectly conducting,

the mode equation for VLF propagation in a flat earth-ionosphere

waveguide is given by [Wait, I960]

(e

i2khC

- jjR,,) (e

i2khC

+ iRl) +

|| R

JL lR|| = 0. (10)

As a first approximation, coupling between the modes is neglected and

thus R = R =0. The mode equation (10) separates into the two

equations

e

1 C

- jjRjj = (for TM modes) (11)

i2khC

e + ,R, =0 (for TE modes) (12)

If the ionospheric reflection coefficient ..R is approximated by using

the first two terms of the Taylor series (6a) for log ,.R , one has

iR.. = - e ax C

where ax is given by (8b) . With this approximation, the solutions

of the TM mode equation (11) are

C n

= ik hl

li\ < n = 1

' Z > 3 ' •••), Sn=^TTC^ (13)

Curves of l/Re (S ) and -HIm(S)vs. H = %— have n n 2tt been computed when n = 1 for

B = H

{ ^-] = °- 02 ' °- 05 ' °- 10 ' °° Z0 and 0x =0°, 10°, 20°, 30°, 40°,

r i-'

50 ,60 . These results are shown in figures 1 through 11. (Only curves for (j)^ = 0° and L

= 60° have been plotted. )
7-

Likewise, if the ionospheric reflection coefficient R is

approximated by using the first two terms of the Taylor series (6b)

R - - e^ C

i

R

l

6

for log ,R.> one has

where (3 X is given by (9b). With this approximation, the solutions of

the TE mode equation (12) are

c m = ?onnk (m = 1, 2

' 3 "--' • (14)
FREQUENCY, kc/S

10 12 14 16 18 20 22 24

006

1.005

.004

CO

<d 1.003

1.002

1.00

1.000

4H

Fig. 1
FREQUENCY, kc/s

1.006

10 12 14 16 18 20 22 24

.005

1.004

S. 1.003

1.002

.00

1.000

4

H

Fig. 2
-10-

FREQUENCY, kc/s

1.006

10 12 14 16 18 20 22 24

1.005

004

(/>

CD 1.003

,002

1. 001

.000

4H

Fig. 3
-11-

FREQUENCY, kc/s

10 12 14 16 18 20 22 24

1.006

1.005

.004

CO

CD 1.003

1.002

,001

1.000

4"

H

Fig. 4
-12-

FREQUENCY, kc/s

10 12 14 16 20 22 24 .006

,005

004

COu 1.003

1.002

.001

000

4H

Fig. 5
-13-

FREQUENCY, kc/S

0.5

10 12 14 16 18 20 22 24

0.4

O

X

in£

i—

i

x

i

0.3

0.2

0.1

4H

3.5

3.0

2.5

2.0

S 1.5

:$ 1.0

0.5

£

O

O

o

-Q

<

Z>

z:

L±J I-5

Fig. 6
-14-

FREQUENCY, kc/S

0.5

0,4

2 0.3

E

i—

i

x

i 0.2

E

o

o

o

-Q

T3

O

5

Z>

-z. UJ

I-&

Fig. 7
15.

FREQUENCY, kc/S

0.5

12 14 16 18

0.4

2 0.3

inE

x 0.2

0.1

4H

• 3.5

3.0

t 2.5

2.0

1.5

E

_^

o

o

o

-Q

o

3

LJ

= = 1.0 <

E

j 0.5

Fig. 8
-16-

FREQUENCY, kc/S

6 18 20 22 24

E

o

o

o

-oo

UJ

!5

Fig. 9
17-

FREQUENCY, kc/S

CVJO

tn

E

o

o

o

\

_Q

O

h-<

Z>

UJ

Fig. 10
-18-

FREQUENCY, kc/S

12 14 16 18 20 22 24

2 5 °

o

15 <

_);z

LJH

1-

1.0 <

Fig. 11
-19-

3. Ionospheric Reflection Coefficient for Transverse Magnetic Field

The ionospheric reflection coefficient R given by Barber and

Crombie [19 59] for a transverse terrestrial magnetic field may be

written as

C - A

,R„ = C + A

c

2 + where A =

1+6 "1 + 52 - y

2

\ (6-1- 6* -a,

2

) - iy [l-c"]

l 2, 2

(15)

(16a)

(1 + 6)

2 - y

and 6 = i - , CO r y = «

+ — |tan I (East-to-West Propagation)

r

" ~ l tan

T l (West-to-East Propagation)

(16b)

Just as in the preceding section, expand log ..R in a Taylor

about C = to get

series

00

i Rll = exp ) ?i c

1^ =

where ck

~ ¥r[a^ < lo§ ji r

ii)]

(17)

(18)

c = o The ionospheric reflection coefficient ^R,, can now be approximated

by using a few terms of the series (17). The first few Taylor series

coefficients for log R are

aQ = lir (19a)
.20-

^[(l+5) 2 -72

I

i [(l+6)(6 + 6

2 -72 )]2-i 7

(19b)

a9 =

T2T "1 i (l+6)2-7 2

"(6 + 6* -?»)

7

L (1 + 6)2 + i-y + G

(19c)

(19d)

04 = (19e)
-21-

4. An Approximate Solution of the TM Mode Equation for PerfectlyConducting Ground in the Transverse Case - For a purely transverse and horizontal terrestrial magnetic field, the ionospheric reflection coefficients R and R vanish, so that TE and TM modes

are not coupled when the ground is perfectly conducting. The TM

mode equation for VLF propagation in a flat earth-ionosphere waveguide is then given by

i 2k h C „ n e - „R

n =0 . (20)

If the ionospheric reflection coefficient ..R.. is approximated by using

the first two terms of the Taylor series (17) for log R , one has

R = - e C

l C

)I K|| e

where ax is given by (19b).

With this approximation , the solutions of (20) are

C

n =i^T (" = 1.2.3,...) . (21)

kh - n imia j vs. ti = n

computed when n = 1 for

Curves of l/Re (S ) and - H Im(S ) vs. H = S have been n n' 2tt

B

=h-(--) = 0.02, 0.05, 0.10, 0.20 and = 0°, 30°, 60°.

r T

These results are shown in figures 12 - 19. Frequency and

attenuation scales corresponding to h = 70 km have been appended.
22-

i.ooe

.005

LOO 4

,003

.002

.00

,000

FREQUENCY, kc/s

10 12 14 16 18 20 22 24

ffl

4H

Fig. 12
23

1.006

FREQUENCY, kc/s

10 12 14 16 18 20 22 24

,005

1.004

<a i CD q:

003

.002

,001

1.000

4H

Fig. 13
-24-

FREQUENCY, kc/s

10 12 18 20 22 24

Fig. 14
.25.

CD

FREQUENCY, kc/s

14 16 18 20 22 24

1.000

4H

Fig.

6

15
-26-

0.5

FREQUENCY, kc/s

10 12 14 16 18 20 22 24

0.4

2 0.3

X

E

x

i

0.2

0.1

E

jz

o

o

o

\

-Q

<

z.LU

tttt

Fig. 16
.27

FREQUENCY, kc/s

10 12 14 16 18 20 22 24

0.5

0.4

O2 0.3

E

x

i

0.2

0.1

3,5

t: 3.0

# 2.5

2.0

.5

1.0

0.5

E

o

o

o

\

<

LU

4H

Fig. 17
.28.

FREQUENCY, kc/s

10 12 14 16 18 20 22 24

E

.^

o

o

o

\

-O

Z>

z.UJ

Fig. 18
0.5

-29-

FREQUENCY, kc/S

14 16 18 20 22 24

0.4

"o 0.3

§

E

x

0.2

0.1

ffi: 15

3.0

2.5

! 2.0

l\ 1.5

== 1.0

il 0.5

£

O

o

o

<

z:

UJ

fh-

<T

4H

Fig. 19
-30-

5. Arbitrarily Dipping Magnetic Field -

In general, the ionospheric reflection coefficients depend on the

direction of propagation with respect to the terrestrial magnetic field.

Johler [1961] has evaluated ..R... .R > ,,R . » .R., for a (real) angle of

incidence of 82 , various magnetic dip angles I, and directions of

propagation (j) (measured clockwise from north). These results have

been used to estimate the effect of variations in I and 6 on the ra

attenuation and phase of TM waveguide modes. This was done in the

following way. First, the ionospheric reflection coefficient ..R.. was

approximated by an expression of the form

,R„ = -e gC (22)

where the parameter a was chosen so that (22) agrees with the exact

value computed by Johler when C = cos 82 [note that when C =cos 90°,

(22) automatically reduces to the exact value] . Since ..R,. e ^ is

equal to Johler' s T , this requires that ee

l0g

e

l T eel ar§ (T

ee } " U

Re (a) = ~_ and Im(o) = — . (23a) (23b)

cos 82 cos 82

If coupling between TM and TE modes is neglected (equivalent to

setting

(| R = R |( = 0), the TM mode equation for VLF propagation in a flat earth-ionosphere waveguide becomes

i2khC

" j|R,| = • (24)

Using the expression (22) for R the solutions of (24) are

C =

< Zn

C Z ill i„ » 1 2 * \ n 2kh + i a (n- 1, 2, 3,...).
-31-

Now let

and

P = Attenuation with Magnetic Field

Attenuation without Magnetic Field

[im (S)] with Magnetic Field

[ Ira (S)] without Magnetic Field

(25a)

_ Phase Velocity Deviation with Magnetic Field

Phase Velocity Deviation without Magnetic Field

[_ ~ 1] with Magnetic Field

[_ -1] without Magnetic Field

[otct ~ 1] with Magnetic Field

= -5"M . (25b)

[_—._. - 1] without Magnetic Field

For the dominant mode (n = 1), curves of P and Q vs. direction of

propagation (0 ) have been computed for:

Magnetic dip angles . I = , 45 , 84.3 ,

3 3 3 Electron densities , N = 3(10) , (10) electrons/cm ,

Frequencies, f = 10, 22 kc/s

The results are shown in figures 2 through 24.
-32-

In the case of I = , P and Q were also computed for (j) =90

and = 270 -- that is, for a (horizontal) transverse terrestrial

cL magnetic field. In order to make the methods of computation uniform,

the exact formula (15) for the ionospheric reflection coefficient R

was first evaluated for C = cos 82 . The determination of a and subsequent calculations were then carried out according to the procedure

outlined above.

For I = 90 (vertical magnetic field), P and Q are independent of

the direction of propagation . In this case, P and Q have been

cL plotted as a function of frequency (figure 25).
-33-

<£a
.34-
35-
-36-

Q

0.99 J \- \ [^r:

- 1

60 120 240 300 360

4>c
37-

Q

.02

1.01

0.99

0.98

jftrottftftfflM

N = (IO) 3

1 = 84.3°

n

"T"

ffl+r^H K

60 120 240 300 360

*r

Q I

4>a

Q

gmmTMfclmlii&MM

60 240 300 360

4>a Fig. 24
38-

1.2

I.I

P 1.0

0.9 —

THTtffl E3

= 90° n

N = (IO)

;

! "±= i

I

I

I

I l -l-l N=3(IO) 3

0.8

10 12 14 16 18 FREQUENCY, kc/s

20 22

1.03

.02

Q 1.01

.00

0.99

10

[! i. 1=90° n=m

ffffitffHTFF

12 14 16 18 FREQUENCY, kc/s

20 22

Fi e . 25
-39-

III. Calculations for a Spherical Earth-Ionosphere Waveguide

Using the Quasi -longitudinal Approximation

1. The Mode Equation - A simplified form of the mode equation

for VLF propagation in a spherical earth-ionosphere waveguide has

been given by Wait and Spies [i960] . If the ionospheric reflection

coefficient R. is expressed as

i_ r 7 2h 21 R = - exp | Cl (C c + —)

J , (26)

L J

where the parameter czj is that computed previously (in section II) for

the quasi-longitudinal approximation, this mode equation may be

written as

2 ka ( C 2 4. ^)

3/\ i fll (C 2 + ^]

/Z

+ i log |^(t)-qw2 (t) Lwf (t) -q wj (t) _

- (4n-l) 1= (27)

ka2/3

2 where n is an integer and t = - (-_-) C ,

A restriction on this mode equation is that h/a « 1.
-40-

2. A Crude Solution for Perfectly Conducting Ground - When the

ground is perfectly conducting (q = 0), i log [ w^tj/wj (t)]may be

approximated by

i log = - arg[wM0)] +arg [wf (0)] = *

and if |C 2

| « — , if follows that

,„2 2h.3/2 „ t 2h.3/

2

r.

L 3 C

2

j - _2^2hY2 _ ,2h^

(C + 1T> * (—)

L1+ 2-7E-J and < C + —> w (—)

V2

r 1 C2.

a

The resulting mode equation can be solved at once for C to give

,2hV2 . ,h

l/2

2 (12n-5) ^ -^ ka (__) - i

, (_)

77^

ka <^) +i-t<£)

-r-t/2- (28)

Using a value h/a = 0. 01, curves of l/Re (S ) and - H Im(S ) vs. H = 4- '

x n' x n' 2ir have been computed when n = 1 for

B = -^ (—)= 0.02, 0.05, 0.10, 0.20 and = 0°, 60°.

These are shown in figures 26 through 30. Frequency and attenuation

scales corresponding to h = 70 km have been appended. Some curves hav

also been drawn to show comparisons between solutions for flat and

spherical earth-ionosphere waveguides (Fig. 31). The particular curve

labelled [1 -h/2a)] x Flat Earth is a semi-empirical result which has

been used previously for interpreting diurnal change of ionospheric

reflection heights [Wait, 1959]. For such purposes, it is a good

approximation (here h/a =0. 01).
-41-

12

FREQUENCY, kc/s

14 16 18 20 22 24

V)

<£ 0.999

Fig. 26
.42

1.002

(/) £ 0.999

0.998

FREQUENCY, kc/s

10 12 14 16 18 20 22 24

0.997

0.996

Fig. 27
43

1.002

FREQUENCY, kc/s

10 12 14 16 18 20 22 24

to £ 0.999

0.998

0.997

0.996

4 5 6 H

Fig 28
-44-

.002

.001

1.000

to £ 0.999

0.998

0.997

0.996

FREQUENCY, kc/s

14 16 18 20 22 24
-45-

10

FREQUENCY, kc/s

12 14 16 18 20 22 24

COO

X

IT)

I

I

o

o

o

\

-O

<

z>

h-<

Fig. 30
-46

in

CD

0.998

0.996

4H

Fig. 31
-47-

An exact solution of the mode equation (27) for perfectly reflecting boundaries (to be described in the following section) indicates

that the approximate method described above must be used with considerable caution. For one thing, the assumption that |C |« 2 h/a

is not always justified for n=l. In fact, for n = 2, the reverse is often

true, especially for the lower frequencies (around 10 kc/s). Also

when the condition |C I « 2 h/a fails, the approximation

i log[w£(t)/w{ (t)] ~ tt/3 is also very poor.

To obtain the corresponding "crude solution" for finitely conducting ground, one may approximate'

i log wfe(t) - q w2

(t)

Lwf (t) - q Wj (t) j

by expanding the logarithm in a Taylor series about t = 0, then

neglecting terms involving t2 and higher powers to get

i loa

! ~WM*) - q w2(t)

g

|wf(t) -q Wl (t) y + i loS

1 -n

W2(°? -, q wnuri

1 q ww>

2/3 Q

2

/ wl(°) - w2(°) \ r*

(I q w7iT5y H1 q wtrbT)

Using this approximation, the method of solution described above

gives

-, 7i,3/2 ,,1/2 (12n-5)--| k a(^ . i a, <£j - i ^

" ka , 2hYV«. ^f72

ka(—) +i-f<—) -H-2-) V/\8,

(30)
-48-

where 6 = log rHSm and 5l -liSRff 5Ell . (31)

u q

wf{0) i q w;(0) M 4 ^n°r

)

However, further effort along this line does not seem warranted,

since there still exists the difficulty that |C 2

| « 2 h/a is not always

true (nor is the approximation (29) always realistic).

3. Solution for Perfectly Reflecting Boundaries - When R = + 1

(a- = oo or q = 0) and R. = -1 (ax = 0), the mode equation (27) may be

written as

^

Z + ^ = 2-^— [{4n -

1) ^ + 2 tarT

1 (^f*|.)]

. (32)

For perfectly reflecting boundaries, the modes propagate without

attenuation, so that real solutions of (32) are desired. For real t,

"-l^--**--1 ^' ' < 33'

where the inverse tangent is a continuous function of t such that

tan

1 (l!l^)=- ff

First approximations to the solutions of (32) were obtained for n = 1 and n = 2 by graphical means -- plotting the right- and left-hand

members of (32) vs. c , then reading off the abscissae of the points of

intersection. Two further approximations were then obtained by

using the "method of false position. "

1 In appendix A, this function is tabulated to six decimal places at

intervals of 0. 1 from t = - 10. to t = + 5. 0.
•49-

Curves of (— -1) vs. frequency have been computed for h = 60, 70,

80, 90, 100 km, The frequency range covered extends from 8 kc/s to

30 kc/s» [Using graphical means, curves have been interpolated for

h = 65, 75, 85, 95 km.] These results are shown in figures 32 and 33.

For h = 60, 80, 100 km, curves of ( 1) vs. frequency were also

computed using the crude method of section 2 [set Gj = in (2$)] when

n = 1. A comparison of the crude solution with the exact solution of

this section is shown in figure 34.

4. Newton's Method for Solving the Mode Equation - Let z = C

and write the mode equation (27) as

F(z) =0 (34)

2 2h.

3/2 2h2/

2

where F(z)= T

ka(z + —) + i Cl (z + —

)

+ ilog r^t|- qW2(

/

t

t 6 \'-(4n-l)^ (35) |w1'(t) -q Wl (t), v 'Z v

' and

t = - (~) z .

According to Newton's method, if z is an approximate root of (34),

a next approximation zx is given by

Zi

1 = z + Az

o where

Az = - Zi£°) .
50-

0.010

0.008

0.006

0.004

0.002

u 0.000 >

-0.004

-0.006

-0.008

-0.010

8 12 16 20 24

FREQUENCY, kc/s

28

Fig. 32
51-

0.016

0.014

0.012

0.010

0.008

<-> 0.006

>

0.004

0.002

0.000

0.002

0.004

8 12 16 20 24 28

FREQUENCY, kc/s

Fig. 33
52

i

to

cr

0.010

0.008

0.006

0.004

0.002

0.000

-0.002

-0.004

-0.006

-0.008

-0.010

16 20 24

FREQUENCY, kc/s
-53-

Differentiating (35) with respect to z and using the relati on

wf(t) w2

(t) - wx (t) w|(t) = 2i

1/2

2b. y >:>,, , . Zh" 1/2

;ives F»(z) = ka(z + -£i) +i^L(z + ±ti)

ka 2/3 2{t-q2)

[wf(t) -q Wl (t)][wj(t)-qw2 (t)]

so that Az =

3/2

2 2h "'

" 2h — ka (z + —) + ic, z + —) + i log

3 o a

x o a

&

w»(t ) - q w.(t ) 3 _, o

2 o _w,(t ) - q w, (t )_ o o

(4n-l) £

2h?. ,ka

2/3

ka(z +—) + i -2M z +—)+(-^) o a £ o a £

2(t - q* ) o

[ wj (tj - q wx (t J] [ wj, (t J - q wa

(t J]

(36)

where _ ,ka/

t = - (^5-) z . o

x 2 ' o

Further approximations can be obtained by successive applications

of Newton's method.

5. Solution for Perfectly Reflecting Ionosphere and Finitely Conducting Ground - For a perfectly reflecting ionosphere, ax =0,

so that the mode equation (27) becomes

3/2 2,, , 2h/ . . [wi(t) - q w, (t) ,. ,, tt n ^ ka (z + ) +1 log I —2-i_£ 2.—l±J - (4n-l) -j =

Lwf(t) - q Wl (t)-

(37)
54-

where z = C . If z is an approximate root of (3 7), Newton's method

then gives a next approximation, z x = z + Az

where 3/2

Az =

rw£(t

o

) - q w2 (t

Q

)

-5- ka (Z + ) + i log | r-rr-, jrr-r 3

v o T

a ' e |w{(t) - q wx (t

Q

) 2 < t o

" qZ)

IT - (4n-l, £

i / . 2h.2 ka ka(z

o +_) +(^-)

27T

[wi(tQ

) - q Wl (t

Q

)] [w»(t

Q

) - q w2 (t

Q

)]

(38)

If, for the first approximation, one chooses z to be the solution of

o the mode equation (32) for perfectly reflecting boundaries, so that

2 2h 3/2r w2<yi TT

then in the expression for the second approximation z 1 = z + Az,

Az is given by

i log

Az =

W

?JK i - i log

wjf(to

) j * rw2 (t

Q

) - q w2 (t

Q

)

-

| wf(t ) - q Wl (t

Q

)

_i

t / . 2h. 2 ,ka ka(«

o +—) +(T

2/3 2(t

Q -q2

)

(39)

[wf(t

Q

) - q wj (t

Q

)] [w2 (t

Q

) - q w2 (t

Q

)]

For any further approximations, however, one must use (38) to

evaluate Az.
-55-

Choosing z to be the solution of the mode equation for perfectlyreflecting boundaries, a single application of Newton's method has

been used to compute curves of ( 1) vs. frequency and attenuation

c -3

(db/1000 km) vs. frequency for n = 1 when q- = 5 x 10 mho/m and

h = 60, 70, 80, 90, 100 km. The frequency range covered extends

from 8 kc/s to 30 kc/s. Using graphical means, curves have been

interpolated for h = 65, 75, 85, 95 km. These results are shown in

figures 3 5 and 36.

6 - Solution for Imperfectly Reflecting Ionosphere and Perfectly

Conducting Ground - For a perfectly conducting ground

q = (cr = oo), so that the mode equation (27) becomes

_ka(z + _) +1Cl (z + _) + 1 log -^ - (4n-l) j- = (40)

where z = C . If z is an approximate root of (40), Newton's method

then gives a next approximation zx = z + Az

where Az = -

2 2h3/2 2h* r W2(to

) 3

ka(VT» +ic^Zo +

-a-)-,- ilo

g[_^l^"J

- (4n-l)

IT

_1 2h 2 r/3 2 t ka (zv o +—)a ' + i ^L

2

(zo + ^)% a ' fa) x 2 ' wf(t ) w|(t )

(41)

If, for the first approximation, one chooses z to be the solution of

the mode equation (32) for perfectly reflecting boundaries, so that

2 , . , 2hV2

. . 3

ka ( z

Q

+—) + i log

1V o -I

TT - (4n-l) £ =
56-

0.010

0.008

0.006

0.004

0.002

o 0.000 >

-0.002

-0.004

-0.006

-0.008

-0.010

12 16 20 24

FREQUENCY, kc/s

28

Fig. 35
-57-

16 20 24 FREQUENCY, kc/s

Fig. 36
-58-

then in the expression for the second approximation Zj = Z

q + Az,

A z is given by

1 ci ( Z

o +

"a-)

Az = r r-T 273 Z~t ' * 42)

, 2h 2 a, . L 2h. 2 ka. ' ~ o ka(z + T)+iJ-(z+r) + (-=-) a ' 2

x o a 2

[u'(to

)] 3+[V(t

o

)] 2

For any further approximations, however, one must use (41) to

evaluate Az.

Choosing z to be the solution of the mode equation for perfectly

reflecting boundaries, a single application of Newton's method has

been used to compute curves of (— - 1) vs. frequency and attenuation

(db/1000 km) vs. frequency for n = 1 and n = 2 in the absence of a

5

terrestrial magnetic field when co = 2 x 10 and h = 60, 70, 80, 90, to r 100 km. Again, the frequency range covered extends from 8 kc/s to

30 kc/s. Using graphical means, (— - 1) vs. frequency curves have

been interpolated for h = 65, 75, 85, 95 km for n = 2. Attenuation

vs. frequency curves have been interpolated for these same values of

h. These results are shown in figures 37 through 40.

7. Solution for Imperfectly Reflecting Ionosphere and Finitely Conducting Ground - Using Newton' s method, solutions of mode

5 equation (27) have been obtained for n =1 when h = 70 km, co = 2 x 10 , r and cr =1, 2, 5, 10, 20 millimhos/meter. Again, the frequency range

extended from 8 to 30 kc/s. Initially, z was chosen to be the solution

5 °

of (27) when co = 2 x 10 and cr = oo. However, for the smaller conr g

ductivities at the lower frequencies, it was found that such a choice

of starting values did not result in convergence, so the computing

technique was modified in the following manner. The solutions for

<r - 20 millimhos/meter were obtained first by choosing z to be the

solution of mode equation (27) for perfectly conducting ground and
-59-

5

co = Z x 10 . The solutions for cr = Z0 were then used as starting

r g

s values to obtain solutions for <r =10 and these solutions were in turn

g

used as starting values for cr =5, and so on. In every case, one

g

application of Newton' s method was sufficient to produce adequate

convergence, though a second application was carried out as a check

on the computations. The results of these calculations, in the form

of curves of (— -1) vs. frequency and attenuation (db/1000 km) vs. frequency, are shown in figures 41 and 4Z. Calculations for

cr = 4 mhos/meter (corresponding to sea water) were also carried

out, but the results were practically identical to those for c = oo.
-60-

0.010

-0.010

12 16 20 24 FREQUENCY, kc/s

28

Fig. 37
-61-

0.016

0.014

0.012

0.010

0.008

o 0.006

>

0.004

0.002

0.000

-0.002

0.004

12 16 20 24

FREQUENCY, kc/s

28

Fig. 31
-62

16 20 24

FREQUENCY, kc/s

Fig. 39
-63-

16 20 24

FREQUENCY, kc/s

28

Fig. 40
64.

0.010

0.008

0.006

0.004

0.002

cj 0.000 >

-0.002

-0.004

-0.006

0.008

-0.010

16 20 24 FREQUENCY, kc/S

28

Fig. 41
• 65.

16 20 24

FREQUENCY, kc/S

28

Fig. 42
• 66-

IV. Further Calculations for a Spherical Earth-Ionosphere Waveguide

1. A More Accurate Form of the Mode Equation - A more accurate

form of the mode equation for VLF propagation in a spherical earthionosphere waveguide is given by [Wait, I960]

w£(t_) -qw2 (tj rwny + qwi (t.H _.

LW^(t

a

) -qwj (tjJ Lwj(t

c)+qw2 (tc

)J

$ - i 2 n it e " = e (43)

where $ = 2(v - y ) - 2 (p - p ) , (Re (S) < 1) C d (_ a.

= 2yc

- 2 p c (KRe(S)<l+ —

)

kaS j (ka) 2_

' (ka S) 2

3/2

^a

=

ka

kaS

(kaS) : 1/2

dx

ka S (kc) 2 - 1 (kaS) 2 J

3/2

kc

r

kaS

x (ka S) 2 x2 J

1/2

dx

and

!<-y3/2

K - - <1 Pa>

2/3

c 3

2 (-t) 3/2

t = - (| P ) 2/3

c

v 2

K c'

c = h + a
-67-

Equation (43) is not restricted by the condition h/a « 1. By carrying

out the above integrations and a certain amount of algebra, this mode

2

equation may be written (if z = C )

i log * 1 ' + i loe 2 I £_

-wi(t ) - q wx (t )J Lw^(t

c

) + q. w2 (tc

) + § - 2nTT =

(44)

where t = v 2 /3

(1-z)z/3

9/ , ^ 2h A h2

t - ,kaf/

3 Z + - + F

(1-z)'

and

§ = 2ka{ /z+ ~+ — - •£"+ ^2 !cos

_1

(n/1-TT)- cos'Y-^

\' "I

2 ka —3— 1- z £.*£)3/2

- z 3/2

. (ReCz)>0)

= 2 ka , 2h h2 . -1 /^/^^^ \

+ -=- + - ^-Z COS [ - r— ) 2 ka

M +r /_

z + —+ _ 3/2

1-z

(Re(z)<0).

The (complex) inverse cosines above are made unique by requiring them to vanish as their arguments approach unity. Since the

arguments are in fact close to one, an infinite series representation

convenient for this range is obtained by considering cos

_1

(l-u) where

ul is small. Now
68

cos (1-u) = T

- sin (1-u) = T

1-u

-3

dt

o ^^

so

cos (1-u) = -j - dt f

dt 1 _. P dt

o 1-u 1-u

Introducing a new variable of integration by means of the relation

w = 1 - t, one gets

cos (1-u) = d w

"* \j 2 w - w2 n/T~ o

u w

-_1 W ^ (1-4) dw

Expanding the right-hand integrand into an infinite series (using the

binomial theorem) and integrating term-by-term gives the desired

result:

-1 — r 1 (1)(3) , (1)(3)(5) o os (1-u) = n/Zu 1 1 + u +

v

'

K

' u

2 +

v M M ' u

3 +

!_ 1!(3)(2) 2 2i(5)(2) 4 3!(7)(2) 6 (45)

This result is now used to obtain infinite series for

cos ( n/1 - z j and cos
-69-

Expanding n/I - z into an infinite series (again using the binomial

theorem), one has

cos (n/1 - z) = cos (1-U! ) (46a)

1 1 ? (1) (3) 3 ,(1) (3) (5) 4 , where ux = z + z2 +

l '— • z3 +v M 7 V ' z

4 + . . . . 1!(2) 2!(2) 2 31 (2)

3 41 (2)

4 (46b)

Now = i - a N

(l-ui ) = 1 1

! h h L /i^ h\"l It — - -"

Thus cos

1 /\^Tz~ >

1 +- / a

= cos" (l-u2 ) (47a)

where u2 = (!+£) (J+u,). (47b)

2. Solution for Perfectly Reflecting Boundaries - When the ground

is perfectly reflecting, q = 0; when the ionosphere is perfectly reflecting, q. = oo. The mode equation (44) then becomes

i log

rw£(t

a

)

-wf(ta

)J + i log ! wi(tc H

Lw2

(t )- + $ - 2n-rr = 0, (48)
70-

Again, real solutions of (48) are desired since the modes propagate

without attenuation when the boundaries are perfectly reflecting. For

real arguments,

i log

-w£(t

a

)

LwJ(ta

)

= - 2 tan

i[-

v'<Vi

Lu'(t )J

(49)

and

i log

wi(t

c h

lwz(U J = 2 tan

v c' u(t ) x c'

(50)

where the inverse tangents are continuous functions of t such that

tan

-l rv(0)

_| _ tt Lu'(0)J

and tan ! —L_L

--u(O) - = +

TT

The mode equation for perfectly reflecting boundaries can now be

written as

-irv(Vl -ir v,(

tan

- r M cH V] J

I - tan

Lu(t )J

x c _u»(t ) J

v a'

+ j-i -niT = (51)

where t , t and $ are defined in the previous section [following (44)] .

In appendix A, these functions are tabulated to six decimal places at

intervals of 0. 1 from t = -10. to t = + 5. 0.
-71-

The solutions of the mode equation (32) should be a fair approximation to the solutions of the mode equation (48) above. With these

solutions as a guide, a single application of the "method of false

position" has been used to obtain a further approximation for n = 1 when h = 60, 100 km. These results are shown in figure 43.

As expected, they differ but little from those obtained by solving the

mode equation (32).

3. Newton's Method for Solving the Mode Equation - Write the

mode equation (44) as

F(z) = (52)

r wi(ta) - q. w2< ta

)l r Wl' (tc

) + qi where F (z) = i log I + i log | Wl ( V

LwJ (ta

) - q wx (t

a

)J LwJ (tc

) + q; w2 (tc

) J + §(z)-2nu

(53)

and t , t , $(z) are defined following (44). Then according to Newton's

3. C

method, if z is an approximate root of (53), a next approximation zx is given by

= z + Az

o

F(z

Q

) where Az = - F' (z

Q

)

Differentiating (53) with respect to z and using the relation

wj(t) w2

(t) - Wl (t) w'(t) = 2i
72-

gives

F»(z) =

_ .ka

2 (-2-

)

2/3

^

(ta -q2 )(l- r

z)

(1-z) (l-z)"/J

l[wf(t )-qWi(t ).][wj(t )-qw2

(t )]

'V>(*- ^) 4 (* +" C *1 3

v 4ra + §)

[wMV+qiWx^)] [wj(t

c )+q.w2 (tc

)] cTz"

where

d§ = ka

n/t

cos ! — _ ) - cos (n/1 - z) a

1 3/2

, 2h , h2 "| 1 2 . 1 r/ , 2h , h2 \ 3/2

(1-z): a6 /

(Re(z)> 0)

kaj _—_ cos "

/

I \fi~z v 1 +

1 / 4lTi\ l / 2h . h^

-

i / 2 + +

1 \! a a 2

3 (1-Z)* V

a

1 / 2h ^ h*\

fz + + ) a2 ./

3/2

(Re(z) < 0)
73

0.010

0.008

0.006

0.004

0.002

o 0.000

>

-0.002

-0.004

-0.006

•0.008

-0.010

12 16 20 24 FREQUENCY, kc/S28

Fig. 43
-74-

Appendix A +

t tan

"v(t)"j

_u(t)J

6

2m

-1

tan

rvur

V(t) m

0. 0. 523 599 +4 603 -0. 523 599 + 11 929

0..1 0.462 765 4 59 5 -0. 517 681 + 11 671

0.2 0. 406 521 4 560 -0. 500 215 + 10 745

0.3 0. 354 832 4 500 -0.472 147 + 9 044

0.4 0. 307 637 4 409 -0.435 170 + 6 612

0. 5 0. 264 845 4 288 -0. 391 666 + 3 720

0.6 0. 226 335 4 134 -0. 344 452 + 771

0. 7 0. 191 953 3 950 -0. 296 397 - 1 79 7

0. 8 0. 161 515 3 734 -0.250 015 - 3 690

0.9 0. 134 806 3 492 -0.207 183 - 4 825

1. 0. Ill 585 3 228 -0. 169 050 - 5 276

1. 1 0. 091 589 2 946 -0. 136 099 - 5 215

1.2 0. 074 538 2 6 59 -0. 108 305 - 4 839

1.3 0. 060 145 2 364 -0. 085 318 - 4 289

1.4 0. 048 117 2 077 -0. 066 609 - 3 679

1. 5 0. 038 168 1 799 -0. 051 582 - 3 086

1.6 0. 030 021 1 541 -0. 039 649 - 2 535

1. 7 0. 023 418 1 29 8 -0. 030 264 - 2 056

1. 8 0. 018 118 1 082 -0. 022 948 - 1 646

1.9 0. 013 905 893 -0. 017 290 - 1 302

2. 0. 010 589 724 -0.012 946 - 1 023

2. 1 0. 008 002 583 -0. 009 635 - 796

2. 2 0. 006 002 463 -0. 007 128 - 614

2. 3 0. 004 469 365 -0.005 242 - 472

2.4 0. 003 304 284 -0. 003 833 - 357

2. 5 0. 002 426 220

--- -0. 002 786 - 272

t These functions, expressed in degrees, have also been extensivelytabulated by Miller [1946] . Note that

tan

-1 v(t)

L u(t) J

x(t) and tan

-1 v'(t)

Lu'(t)

= +(t),

where x(t) and ^(t) are the functions given in Miller 1 s table. (The

authors are grateful to Nelson A. Logan for pointing out this

relationship. ) (Also, see footnote on page 76. )
• 75.

t -irv(t)]

tan . , 6

2 -1

tan

~v'(t)]

[y (t) j

6

2

2. 5 0. 002 426 + 222 - 0.002 786 - 275

2.6 0. 001 770 168 - 0. 002 014 - 206

2. 7 0. 001 282 129 - 0. 001 448 - 153

2.8 0. 000 923 096 - 0.001 035 - 113

2.9 0. 000 660 073 - 0. 000 735 - 085

3.0 0. 000 470 052 - 0. 000 520 - 060

3. 1 0.000 332 039 - 0. 000 365 - 045

3.2 0. 000 233 029 - 0. 000 255 - 033

3.3 0.000 163 020 - 0. 000 178 - 022

3.4 0. 000 113 015 - 0. 000 123 - 017

3. 5 0.000 078 Oil - 0.000 085 - Oil

3.6 0. 000 054 007 - 0. 000 058 - 008

3. 7 0. 000 037 00 5 - 0.000 039 - 007

3.8 0. 000 025 004 - 0. 000 027 - 003

3.9 0.000 017 002 - 0. 000 018 - 003

4.0 0. 000 Oil 003 - 0. 000 012 - 002

4.1 0. 000 008 000 - 0. 000 008 - 001

4.2 0. 000 005 001 - 0.000 005 - 002

4.3 0. 000 003 001 - 0. 000 004 + 001

4.4 0.000 002 000 - 0.000 002 - 002

4. 5 0. 000 001 001 - 0. 000 002 + 001

4.6 0. 000 001 000 - 0.000 001 - 001

4. 7 0. 000 001 - 001 - 0. 000 001 000

4.8 0. 000 000 + 001 0. 000 000 - 001

4.9 0.000 000 000 0. 000 000 000

5.0 0. 000 000 000 0. 000 000 000
76-

t

. -lp(t)"l tan — —

Lu(t)J

6

2m

-lrv'(t).-

tan ; —,-J~T Lu'(t)J

6

2m

0. 0. 523 599 + 4 603 -0. 523 599 + 11 929

-0. 1 0. 589 033 4 595 -0. 517 678 11 698

-0. 2 0. 659 059 4 568 -0. 500 116 11 159

-0.3 0. 733 650 4 525 -0.471 424 10 463

-0.4 0. 812 764 4 474 -0.432 278 9 716

-0. 5 0. 896 350 4 410 -0.383 414 8 977

-0.6 0.984 345 4 343 -0.325 564 8 286

-0. 7 1. 076 682 4 270 -0.259 417 7 655

-0. 8 1. 173 288 4 192 -0. 185 602 7 096

-0.9 1.274 086 4 115 -0. 104 680 6 594

-1. 1. 378 999 4 036 -0. 017 151 6 163

-L. 1 1.487 948 3 958 +0.076 550 5 778

-1.2 1. 660 855 3 879 +0. 176 038 5 442

-1.3 1. 717 641 3 801 +0. 280 976 5 147

-1.4 1. 838 229 3 728 +0.391 068 4 888

-1. 5 1.962 545 3 655 0. 506 054 4 659

-1.6 2. 090 516 3 582 0.625 704 4 455

-1. 7 2. 222 070 3 515 0. 749 813 4 274

-1. 8 2.357 139 3 447 0. 878 199 4 110

-1.9 2.49 5 6 56 3 3 84 1. 010 698 3 964

-2. 2. 637 557 3 320 1. 147 164 3 833

-2. 1 2. 782 779 3 265 1.287 465 3 711

-2. 2 2.931 265 3 203 1.431 479 3 602

-2. 3 3. 082 955 3 151 1. 579 097 3 503

-2. 4 3.237 796 3 097 1. 730 219 3 409

-2. 5 3. 395 734 3 044

J

1. 884 752 3 326

Note: Where fourth and higher differences are not negligible, their

effect on interpolated values may be taken into account by using

"modified second differences, " 6^ , instead of the usual second

difference, b

2

. (A good discussion of this technique, known as "throwback, " can be found in Kopal, Numerical Analysis (John Wiley,

New York, 1955). The appropriate second differences, when used in

Everett1 s interpolation formula, should give results correct to within

at least one or two units in the last figure tabulated.
77

t -1tan

"v(t)l

_u(t)J

6

2 -1

tan "V(t)l

_u'(t)J

6

2

-2. 5

' 3. 395 734 + 3 045 1. 884 752 + 3 327

-2.6 3. 556 717 2 998 2. 042 612 3 247

-2.7 3. 720 698 2 950 2. 203 719 3 175

-2. 8 3. 887 629 2 904 2. 368 001 3 105

-2.9 4. 057 464 2 860 2. 535 388 3 044

-3. 4.230 159 2 820 2. 705 819 2 982

-3. 1 4.405 674 2 778 2. 879 232 2 926

-3.2 4. 583 967 2 739 3.055 571 2 875

-3.3 | 4. 764 999 2 701 3.234 785 2 825

-3.4 4.948 732 2 666 3.416 824 2 776

-3. 5 5. 135 131 2 631 3. 601 639 2 732

-3.6 5.324 161 2 596 3. 789 186 2 690

-3. 7 5. 515 787 2 564 3.979 423 2 650

-3. 8 5. 709 977 2 533 4. 172 310 2 611

-3.9 5.906 700 2 501 4.367 808 2 573

-4. 6. 105 924 2 473 4. 565 879 2 540

-4. 1 6. 307 621 2 444 4. 766 490 2 505

-4.2 6. 511 762 2 416 4.969 606 2 473

-4.3 6. 718 319 2 390 5. 175 195 2 441

-4.4 6.927 266 2 363 5.383 225 2 413

-4. 5 7. 138 576 2 338 5. 593 668 2 384

-4.6 7. 352 224 2 315 5. 806 495 2 355

-4. 7 7. 568 187 2 289 6. 021 677 2 329

-4. 8 7. 786 439 2 267 6.239 188 2 304

-4.9 8.006 958 2 246 6.459 003 2 279

-5.0 8.229 723 2 222 6.681 097 2 2 54
78-

t -lf-v(t)-| tan ——— 1 l_u(t) J

6

2 -1

tan

rv'(t)-i

U»(t)J

6

2

-5.0 8. 229 723 + 2 222 6. 681 097 + 2 254

-5. 1 8.454 710 2 202 6. 905 445 2 232

-5. 2 8.681 899 2 180 7. 132 025 2 209

-5. 3 8.911 268 2 162 7. 360 814 2 187

-5.4 9. 142 799 2 142 7. 591 790 2 166

-5. 5 9. 376 472 2 121 7. 824 932 2 146

-5. 6 9. 612 266 2 105 8. 060 220 2 125

-5. 7 9. 850 165 2 086 8.297 633 2 106

-5. 8 10,090 150 2 067 8. 537 152 2 088

-5.9 10. 332 202 2 052 8. 778 759 2 069

-6.0 10. 576 306 2 034 9. 022 435 2 051

-6. 1 10.822 444 2 017 9. 268 162 2 034

-6.2 11.070 599 2 003 9. 515 923 2 017

-6. 3 11. 320 757 1 985 9. 765 701 2 001

-6.4 11. 572 900 1 971 10. 017 480 1 984

-6. 5 11. 827 014 1 955 10. 271 243 1 969

-6. 6 12. 083 083 1 942 10. 526 975 1 9 54

-6. 7 12.341 094 1 927 10. 784 661 1 938

-6.8 12. 601 032 1 912 11.044 285 1 924

-6.9 12. 862 882 1 900 11.305 833 1 910

-7.0 13. 126 632 1 885 11. 569 291 1 896

-7. 1 13.392 267 1 872 11. 834 645 1 881

-7. 2 13. 659 774 1 860 12. 101 880 1 870

-7.3 13.929 141 1 848 12.370 985 1 855

-7.4 14. 200 356 1 833 12. 641 945 1 843

-7. 5 14.473 404 1 823 12.914 748 1 830
-79

t tan_1

E("t)]

6

2 , -irv'(t)-| tan

Lu'TtTJ

6

2

-7. 5 14.473 404 + 1 823 12.914 748 + 1 830

-7.6 14.748 275 1 811 13. 189 381 1 819

-7. 7 15. 024 957 1 799 13.465 833 1 806

-7. 8 15.303 438 1 786 13. 744 091 1 794

-7.9 15. 583 705 1 777 14.024 143 1 783

-8.0 15. 865 749 1 766 14.305 978 1 771

-8. 1 16. 149 559 1 753 14. 589 584 1 761

-8. 2 16.435 122 1 744 14.874 951 1 750

-8.3 16. 722 429 1 734 15. 162 068 1 738

-8.4 17.011 470 1 722 15.450 923 1 729

-8. 5 17.302 233 1 713 15. 741 507 1 717

-8.6 17. 594 709 1 703 16. 033 808 1 709

-8. 7 17. 888 888 1 693 16.327 818 1 697

-8. 8 18. 184 760 1 684 16. 623 525 1 689

-8.9 18.482 316 1 674 16.920 921 1 678

-9.0 18. 781 546 1 666 17.219 995 1 669

-9. 1 19. 082 442 1 654 17. 520 738 1 661

-9.2 19.384 992 1 647 17.823 142 1 650

-9.3 19.689 189 1 639 18. 127 196 1 642

-9.4 19.995 025 1 629 18.432 892 1 632

-9.5 20. 302 490 1 620 18. 740 220 1 625

-9.6 20. 611 575 1 613 19.049 173 1 616

-9.7 20.922 273 1 604 19.359 742 1 607

-9.8 21.234 575 1 596 19.671 918 1 600

-9.9 21. 548 473 1 588 19.985 694 1 590

-10. 21. 863 959 20.301 060
-80-

Appendix B

Some Formulae Involving Airy Functions

The Airy functions u(t) and v(t) are linearly independent solutions

of the differential equation [Miller, 1946; Fock, 1946]

d

2 f

T-r - tf =0 dr

i.e., u

M

(t) = t u(t) and v

rr(t) = t v(t).

Infinite Series Representation:

u(t) = -^

yi (t) + —2 y2 (t)

Wr(2/3) WT{\/3)

= 1.089 929 069 yi (t) + 0. 794 570 425y2 (t)

n/tT Jtt~ v(t) = yi (t) - —

y2 (t)

^/Tr(2/3) N/Tr(i/3)

= 0.629 270 841 yi (t) - 0.458 745 449 y2 (t)

where

v It) 1 + -L t» +

(1) (4)

t e +

(1) (4) (7) 9 (1) (4) (7) (10) , a Yi(t) -1 + 37 t + -j|— t + - t + — t +

v, (t , - t + i- t

4 +

(2)(5)

t

7 +

(2)(5)(8)

t " +

(z)(5)(8)(11)

t 13 + y2 w - t

4 , 7j

c 10jt 13 , t

These are valid for all t.
-81-

Of course, y^t) and yz(t) are also linearly independent solutions of

the differential equation satisfied by u(t) and v(t). In fact, yi(t) and

y2 (t) are obtained when the differential equation is solved by the method

of Frobenius.

Wronskian Identity:

u'(t) v(t) - u(t) v'(t) = 1

Definite Integral Representation:

u(t) = -r=- \ iexp(- - x

3 + tx) + sin (— x

3 + tx) [dx

\pn

it) = -j=- \ cos (-^ x

3 + tx) d:

Representation in Terms of Bessel Functions:

If principal values are taken

UTT [ T. 2 3/2 x , T .2 3/2,

V'T j 1 .^ 1 !* , + V3 ( 3

t ]

v(t) = I -l/3 ( 3

t

> ^l/S^ 1 } K, ,. (4~ t 3/2

) 3tt ^1/3 V 3

/ .\ !*~t [ t / 2 3/2, T . 2 3/2 v 1 u( "t) = Jt J-i/3< l* ) "Wi* >1

V ° L l/3 V 3
-82-

( t} ~

3 L

" 1/3 ( 3 *

) + J

l/3 ( T t }

u'(t) = K-tTT V 3 V3<32t3/2 > +I -Z/3<F

/2

>

VW-f-t I /i t 3 /z.

t / 2 . 3 /2. /z/S^ ) - I -2/3 ( 3

t

>

IT u'(-t)=./^-t

[

J Z/3 <1 ' V2

> *

J

-Z/3 <l

^

v'(-t) = ^t J

2/3<i'

3/2 >-

2/3<F/2

>

Asymptotic Expansions:

Large positive numbers: f|t| — oo, |arg 1 1 < — ~|

U(t) 174" exp ( 3

t

' t

'

v(t) %-77T-P(-T t3/2 )

u

» m ~ 1/4 .2 3/Z

\t) ~ t exp (— t

( j t"' ")

v»(t)

t l/4

, —— 2 3/Z, exp ( - - t )
-83

Large Negative Numbers: |t| -*- oo, arg (-t) < —r

u(t) *

(-t)

174

cos f(-t)

3/2

+ i

v(t) " -774

c-tr

sm |(-t» 3/2 ^

/ %l/4 . [2 . .3/2 TT u'(t) * (-t) ' sm j (-t) + 4

v'(t) » - (-t) / '"" cos }(-t) 3/2 + f

] ,

In general,

Wl (t) = u(t) - i -v(t)

w2 (t) = u(t) + i v(t)

Contour Integral Representation:

ooei2n/3

Wi(t)

n/1F A

exp(tz - — z

3

) d

wi (t) = *r=- \ z exp (t z -— z

3

) dz

z plane

2n/3

> 00

C, contour

t

It should be noted here that V. A. Fock [1946] defines wx (t) =u(t) + iv(t)

and w (t) =u(t) - i v(t).
.84.

w2 (t) = t=- ^ exp(tz--z3 )d

i C 1 w2

' (t) = -7=— \ z exp (t z - — z

3

) dz

\/Tr J 3

ooe 2tt/3

Wronskian Identity:

wf (t) w2 (t) - wi (t) w2

8 (t) = i 2

Representation in Terms of Hankel Functions:

, . - i 2tt/3 / ir t wx (t) =e '

/ - — H.

(2)

3 l/3 f(-t)

3/2

+ i Zir/3 !~-ST TT

' 1' w2 (t)=e -- H

l/3 f(-t)

3/2

. /, , (2) r w1'(t)=e I-- H

2/3

|(-t) 3/2

(1) >!. Ff-(-t) + i tt/3 / it t 3/2

3 ^2/3 [3"
• 85-

Asymptotic Forms:

arg t J < —: w^ (t)

1 ,2 3/2,

exp (jt ) 74

t| 00, |arg(-t) | < y~: TMt (t) ~ -

• / x l/4 /^ 3/2

i'(t) » t

' exp (- t

'

T iu/4

wf(t) » tf' * exp (~ f' ")

2

-^

(-t>

i74 exp Ti |,-t) 3/2

,. . . , .1/4 T i ir/4 w^t) » ±i(-t) e

' exp

2 T.if(-t)

3/2

±i(-t)

1/2

Wl (t)

Values on Special Rays:

. +iTr/3. -iir/3 . +iir/3 +iu/6 . wx (r e ) = e w2 (-r) w2 (re ) =2e v(-r)

. + i 2u/3 . „ - i tt/6 . . Wl (re '

) = 2e ' v(r* , + i2u/3 + iir/3

w2 (r e ) =e wi(r)

, -iTr/3. _ -itr/6 ,

v

. -iTr/3. +iir/3 . . wi(re ) =2e v(-r) w2 (re ) =e Wi(-r)

. -i2ir/3 -iTr/3 wi(re ) =e w2 (r)

-i2ir/3 + i tt/6

w2 (r e ) =2 e v(r)

+ i tt/3. + i-rr/3 , ,

J ) =e wz1 w\(r e ) (-r + i,r /^ -7.- iTT /6

w „•

2'(-r) w2'(re ) =2e v'(-r)
-86-

+ i2u/3 +iTr/6 +i2ir/3 -iii/3

(re ) =2e v'(r) w^re ) =e w{(r)

/ " ilT /3 x 9 + i Tt/6 ,, v ,, -iw/3. -iTr/3 (re ) =2e v'(-r) w2 (re ) =e wi(-r)

-12tt/3 +iTr/3 -12tt/3 -iTr/6 ,. wi (re ) =e w2 (r) w2 (re ) =2e v'(r)

Wi

w/

Derivatives

d w

If w(t) is any solution of the differential equation ——

2— - tw =0,

then

w

(2)

(t) = tw(t) (Note thatw(n)

(t) = -^ w(tf)

dt

w

(3)

(t) = w(t) + tw' (t)

w (t) = t

2 w(t) + 2w' (t)

w

(5)

(t) = 4t w(t) + t

2 w'(t)

(4)

(t) = t

2

(t) = <

w

(6)

(t) = (t

3 + 4) w(t) + 6tw'(t)

w

(?)

(t) = 9 t

2 w(t) + (t

3 + 10) w'(t)

( 8)

w

V '(t) = (t

4 + 28 t) w(t) + 12 t

2 w *(t)

w

(9)

(t) = (16 t

3 + 28) w(t) + (t

4 + 52 t) w'(t)

w

(10)

(t) = (t

5 + 100 t

2

) w(t) + (20 t

3 + 80) w«(t)
87-

w

(11)

(t) = (2 5 t4 + 280 t) w(t) +(t 5 + 160 t

2

) w«(t)

w

(2) (0)=0 w

(5)

(0) =0 w

(8)

(0) =0

w

(3) (0)=w(0) w

(6)

(0) = 4 w(0) w

(9)

(0) =28 w(0)

w

(4)

(0) =2 w'(0) w

(?)

(0) =10 w« (0) w

10)

(0) =80 w« (0)

A Taylor Series Expansion:

If w(t) is any solution of ~T"2" - tw =0, then

d

2 w

w(t + h) = a (t, h) w(t) + (3(t, h) w' (t)

w' (t + h) = c« (t, h) w(t) + pf (t, h) w1 (t)

where

a(t, h) =1 + t a2 (h) + a 3 (h) + t

2 a 4(h) + 4 t a5 (h) + (t

3 + 4) a 6 (h)

+ 9t2 a7 (h) + (t

4 + 28 t) a8 (h) + (16 t3 + 28) a9 (h)

+ (t

5 + 100 t

2

) a.10 (h) + (2 5 t

4 + 280 t) a.n(h) + . . .

(3(t,h) = sl

x (h) + t a 3 (h) +2 a.

4(h) + t

2 a5 (h) + 6t a6 (h) + (t

3 + 10) a7 (h)

+ 12 t

2 a8 (h) + (t

4 +52t) a 9 (h)+(20 t

3 +80) a 10 (h)+ (t 5 + l60t2

) a^ (h) +
-88-

a« (t,h) =t ai (h)+a2 (h)+ t

2 a 3 (h)+4t a4(h) + (t

3 +4) ag(h)+9t2 a^h)

+ (t 4 + 28 t) a7 (h) + (16 t

3 +28) a8 (h) + (t 5 + 100 t

2

) ag (h)

+ (25 t4 + 280 t) a l0 (h) + . . .

6'(t,h) = 1 + t a2 (h) + 2 a 3 (h) + t

2 a4 (h) + 6 t a5 (h) +(t 3+ 10) a 6 (h)

+ 12 t

2 a7 (h) + (t

4 + 52 t) a8 (h) + (20 t

3 + 80) a9 (h)+ (t 5 + 160 + t

2

) a10 (h) +

h

k

and a.(h) = — (k =0, 1 , 2, . . . ) . k k!

Note that an (h) = ^ a, . (k =1, 2, 3, . . . ) . k k k-1

Formulas Involving Complex Arguments:

Let z = x + i y where x, y are real. If w(z) is any solution of the

differential equation

-2~ - Z W = 0,

d

2 w

dz<

the following expressions are fairly convenient for computation when

y is small and x is not too large:

w(z) = [9 (x, y) w(x) + 0(x, y) w1 (x)] + i [£(x, y) w(x) + n(x, y) w1 (x)]
89-

where

Of 1-1 X A x y

4

(x

3 + 4)y6 (x4 + 28x)y8 (x5 + 100 x

2

) y

10 lX,y; "

2 24 720 40,320 3,628,800 '"

^X' y; 12 120 3360 181,440

*'

tt i Z

3 +£if_ x

2 y

7

( 4x3 +7)y9 (5x4 f 56 x) y

11 ^x, yj -

6 3Q 56Q 90,720 7,983,360

xy* x

2 y

5 (x3 +10)y7 (x4 + 52 x) y

9 (x5 + 16 x

2

) y

11

, n(x, y)-y- 6 12()

- 504Q 362,880 " 39,916,800

and

w'(z) = [9'(x,y) w(x) + 0»(x,y) w'(x)] H [g» (x, y) w(x) + V (x, y) w' (x)]

where

9

. (x Vl _ j£ + ELX! *

2 y

6 + < 4 x

3 + 7) y

8 (5x4 +56x)ylQ

1

' y' " " 2 6 80 10,080 725,760 '"

(kit ^LL. x

2 y

4 (x 3 + 10)y6 (x4 + 52 x) y

8 (x 5 + 160 x

2 )-/

9

(x, y)-l2 24 ?20 + 40,320 " 3,628,800 "

£

. (x vi xv i-il +

(* 3 + 4 >y

5 U4+Z8x)y7 +

(*5+ ioo*2 >y

9 6 lX' y' " Xy " 6 120

" 5040 362,880 " ••

ti ix, y; -

3 2Q 42Q

t

ig ^ i44

Note: An interesting discussion of Airy functions and relations among the

many differing notations is found in an unpublished report (dated Dec. 1959)

by Nelson A. Logan of the Lockheed Aircraft Co. , Sunnyvale, Calif.
-90-

Appendix- C

A Note on the Conductivity of the Lower Ionosphere at VLF

In the theory of ionospheric propagation of radio waves, it is

nearly always assumed that the collision frequency of electrons with

neutral particles is independent of electron energy. In fact, the

Appleton-Hartree equations were developed on this basis. It has

been suggested recently [Alpert et al. 1953; Sen and Wyller, I960;

Phelps, I960] that the calculation of the propagation constant for a

weakly ionized medium should take into account this energy dependence.

While the theory has been so generalized by Sen and Wyller [I960] and

Johler and Harper [1962], it seems worthwhile to present a somewhat

simplified account of the consequences of a linear dependence of the

collision frequency on electron energy. Furthermore, this sheds

some light on the validity of describing the electrical characteristics

of the lower ionosphere in terms of a conductivity.

W. P. Allis [19 56] and others have related the components

of the dielectric tensor to integrals involving the angular frequency

co, the gyro frequency of electrons 00 , the electron plasma frequency,

the electron density N, the normalized electron energy distribution

function f Q

, the frequency of mementum transfer collisions of electrons

with gas molecules v(u). When a constant and uniform magnetic field

is applied in the z direction the dielectric constant of the ionized

medium is a tensor of the form
-91-

where

q = i (e L

- e R

)

e" = e p

.

In this, the dielectric constants e , e and e may be written in the

J_i XV. Jr form

€ L

= e( w + wH

)

e p

= e(u>)

where, according to Molmud [1959],

4rr f u

3/2 9 fo

i e(0) a, = «r(0) = " T eo w

o

2 J v (u)+ifi ~^T dU ' o

In the above, o" (fi) is a generalized conductivity which is a function of

a generalized frequency £2.

In the case where the electrons are in thermal equilibrium with

the gas, the energy distribution of the electrons is Maxwellian and

given by

f = (e/nk T) e o
-92-

In the case of weakly ionized dry air, the collision frequency v(u) is

now believed to be approximately proportional to the electron energy

u. In fact, Phelps [I960] has shown that, for low energy electrons in

nitrogen,

v(u) = 1. 2 x 10" ? N(N

2 )u sec"

1

where u is in electron-volts/c. c. For present purposes, we will

just set v (u) =a u where a is a constant. Furthermore, a normalized

collision frequency v % is chosen such that

vj = ak T/e

where k is the usual Boltzmann constant, T is the absolute temperature in degrees Kelvin and e is the electronic charge.

Using the simplifications described in the preceding paragraph,

it is seen that

3/2 " 3/2 , ,(0) .12. « u . C--') \ -^75 e

" "

a/vi *(**) • 3 o o \jt vxJ J au + ifi V vxs

o

This is essentially the formulas given by Phelps [I960], Sen and

Wyller [I960] and others. As they have indicated, the integral can

be expressed in terms of the "semi-conductor integral" E (x)

P

defined by

00 E (x) = 7-j-r- V aP

(g s + x^)"

1 e" C

da

P (pi) J

o
-93-

These have been tabulated by Dingle, Arndt and Roy [19 56] for integral

and half-integral values of p in the range -1/2 to + 5. It easily follows

that

e "

2 - 5 W = -V[! E

5/2

(x) - ixE

3/2

(x)

J

where x = 0>/v l

.

If, on the other hand, v (u) had been replaced by a constant v , as is conventionally done ,we would have arrived at the standard

result _ e co^ o-(O) = °

V +ifi ' o

This can be written in the form

<r(«) = e

where (3 = JL

vo

CO 2 and go =

r

00 ° r 1 + i (3

The parameter co occurs often in the theory of VLF propagation

which is usually formulated on the assumption of an energy independent collision frequency. Furthermore, at VLF co < < co so that

rl

co H

p « ± — . o
-94.

The parameter co HoTT /v also occurs consistently in the presentation

of theoretical results. Thus, it appears that a convenient way to

illustrate the implications of the energy dependent collision frequency

is to define effective values (co ) and (3 as follows

e r e

Thus

(co ) = co a (x)

where

and a (x) =

e

00 "i v i

! E

3/2

(x)

r 5

2

E

5/2 U) + [x E

3/2

(x)

-, s

Also, it is seen that

x E

3 /2

(x)

Pe (x)=

5

2

E

5/2

(x)

where, as above, x = —V,
95.

Using the numerical values of E , (x) and E / (x) given byDingle, Arndt and Roy [1956], the values of a (x) and p (x) are plotted as a function of x and given in Table C-l. It is seen that for small

values of x, g(x) asymptotically approaches the constant value of

2/3, whereas (3 (x) is asymptotically approaching the value 2x. On

the other hand, for large values of x, the respective asymptotes are

0. 4 and x/2. 5.

If the collision frequency was chosen to be independent of energy,

the corresponding values a (x) and 6 (x) would be simply 1. and x, e q

respectively. It is thus concluded that a linear energy dependence

for the collision frequency is not going to lead to any essential

modifications to the theory of VLF propagation. In fact, most of the

numerical results on the characteristics of the VLF modes can be

adapted directly to the energy dependent case if u> and the ratio

co /v are given their more general meaning.
96

Table C-l

X a

e

(x) (3(x) 1 e

0.01 0. 66546 0.01755

0.02 0.66363 0.03322

0.03 0. 66164 0.04776

0.05 0.65738 0. 07451

0. 1 0. 64665 0. 13256

0.2 0. 62691 0. 22772

0.4 0. 59561 0.37771

0.6 0. 57184 0. 50127

0. 8 0. 55300 0. 61080

1.0 0. 53771 0. 71148

1.2 0. 52487 0. 80578

1.6 0. 50441 0.98308

2.0 0.48921 1. 15113

2.5 0.47428 1.35171

3.0 0.46293 1. 54763

3. 5 0.45416 1. 74084

4.0 0.44701 1.93203

5.0 0.43617 2.31099

6.0 0. 42862 2. 69037

7.0 0.42315 3. 07048

8.0 0.41911 3.45146

9.0 0.41579 3. 83384

10.0 0.41360 4.22m

11.0 0.41177 4.60788

12. 0.41011 4.99500

13.0 0.40893 5.38480

14. 0.40783 5. 77448

15. 0.40679 6. 16362

16. 0.40626 6. 55735

17. 0. 40545 6.94835

20.0 0.40395 8. 12884
-97-

References

Al 1 pert, Ya. L. , Computation of the field of LF and VLF radio waves

over the earth 1 s surface under natural conditions, Radiotechnika

i Elecktronika, 1, 281 (19 56).

Barber, N. F. , and D. D, Crombie, VLF reflections from the ionosphere

in the presence of a transverse magnetic field, J. Atmos. Terrest.

Phys. , 16, 37 (1959).

Budden, K. G. , The propagation of very low frequency radio waves to

great distances, Phil. Mag. , 44, 504 (1953).

Budden, K. G. , Radio Waves in the Ionosphere, (Cambridge University

Press, London, 1961).

Fock, V. A. , Diffraction of radio waves around the earth 1 s surface,

Acad, of Science, U. S. S. R. (1946).

Johler, J. Ralph, Magneto -ionic propagation phenomena in low- and

very-low-radiofrequency waves reflected by the ionosphere,

J. Research NBS 65D (Radio Prop. ), No. 1, 53 -66 (Jan. -Feb. 1961).

Miller, J. C. P. , The Airy Integral, giving tables of solutions of the

differential equation y

1

' =xy (Cambridge University Press, 1946).

Wait, J. R. , On the mode theory of VLF ionospheric propagation,

Rev. Geofis. Pura e Appl. , 37, 103 (1957). (Paper presented at

International URSI Conference of Radio Wave Propagation, Paris,

Sept. 1956.)

Wait, J. R. , Diurnal change of ionospheric heights deduced from phase

velocity measurements at VLF, Proc. IRE, 47, No. 5, 998 (1959K

Wait, J. R. , Terrestrial propagation of very-low-frequency radio

waves, a theoretical investigation, J. Research NBS 64D (Radio

Prop.), No. 2, 153-204 (Mar. -Apr. I960).
-98-

Wait, J. R. , A new approach to the mode theory of VLF propagation,

J. Research NBS 65D(Radio Prop.), No. 1, 37-46 (Jan. -Feb. 1961),

Wait, J. R. , and K. Spies, Influence of earth curvature and the

terrestrial magnetic field on VLF propagation, J. Geophys.

Research, _65, No. 8, 2325-2331 (I960).

References for Appendix C

Allis, W. P. , Handbuch der Physik, 21, 413 (Springer-Verlag, Berlin,

1956).

Al' pert, la. L. , V. L. Ginzburg and E. L. Feinberg, The propagation

of radio waves (Moscow, 1953).

Dingle, R. B. , D. Arndt and S. K. Roy, Semiconductor integrals,

Appl. Sci. Research, B6, 155-164 (1956).

Johler, J. Ralph and John D. Harper, Reflection and transmission of

radio waves at a continuously stratified plasma with arbitrary

magnetic induction, J. Research NBS 66D (Radio Prop. ) No. 1,

(Jan. -Feb. 1962).

Molmud, P. , Langevin equation and the ac conductivity of nonMaxwellian plasmas, Phys. Rev., 114, 29-32 (April 1959).

Phelps, A. V. , Propagation constants for electromagnetic waves in

weakly ionized dry air, J. App. Phys., 31, 1723-1729 (Oct. I960).

Sen, H. K. , and A. A. Wyller, On the generalization of the AppletonHartree magneto-ionic formulas, J. Geophys. Research, 65,

3931-3950 (Dec. I960).
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