n Bureau ui ovcuiutiiua

library, N.W. Bldg

SEP 2 9 1965

^ecltnical ^lote ^o. 319

NUMERICAL VALUES OF THE PATH INTEGRALS

FOR LOW AND VERY LOW FREQUENCIES

By

LESLIE A. BERRY AND MARY E. CHRISMAN

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical <v2cte 3/9

ISSUED September 2, 1965

NUMERICAL VALUES OF THE PATH INTEGRALS

FOR LOW AND VERY LOW FREQUENCIES

Leslie A. Berry and Mary E. Chrisman

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

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Table of Contents

Page

Abstract iv

1. Introduction 1

Z. Description of figures Z

Z. 1 The path integrals 3

Z. Z The ground wave 6

Z. 3 Normalization 6

3. Reflection coefficients 6

3. 1 Isotropic ionosphere 7

3. Z Anisotropic ionosphere 8

3. 3 Extrapolation of reflection coefficients .... 9

4. Sample calculation 10

5. Formulas used 11

5. 1 The path integrals 1Z

5. Z The ground wave 17

5.3 The Airy functions, W (x) 18

6. References Z0

Figures Z3-105

in
NUMERICAL VALUES OF THE PATH INTEGRALS FOR

LOW AND VERY LOW FREQUENCIES

Leslie A. Berry and Mary E. Chrisman

ABSTRACT

Numerical values of the path integrals for the first five hops

are given as a function of distance from 1000 km to 8000 km, for

frequencies of 10, Z0, 30, 60, 100, 150, and ZOO kHz,, for sea paths,

and for land paths with conductivities of 0. 01 and 0. 001 mhos/m,

assuming reflection heights of 60, 70, 80, 90, and 100 km. For

convenience, ground wave curves for the same parameters are given.

Using these values, and any ionospheric reflection coefficients, the

individual time modes, and the total field can be calculated simply.

The calculation is valid near the caustic and in the shadow region.

The path integrals can also be used to extract ionospheric reflection

coefficients from experimental data.

Key words: path integrals, wave hops, low frequencies, propagation,

diffraction.

IV
Numerical Values of the Path Integrals for

Low and Very Low Frequencies

Leslie A. Berry and Mary E. Chrisman

1. Introduction

This paper presents numerical results, in graphical form,

which make possible simple full-wave calculations of LF and VLF field

strengths. The theory used is analytically and numerically equivalent

to VLF mode theory, but the concepts are asymptotically related to HF

ray-hop theory, or geometric optics. The ray theory itself is not

adequate at low frequencies because diffraction around the curvature

of the earth is important for these wave lengths. The cutback factor

[Wait and Conda, 1958] can be used to correct the first hop for

diffraction, and there have also been attempts to provide diffractive

corrections for the higher order hops [Norton, 1959] but these

corrections are as difficult to calculate as the full wave hops [Berry,

1964; Berry and Chrisman, 1965], and more difficult to use in LF

field strength calculations.

The theory used here was suggested by Rydbeck [1944] and

Bremmer [1949], and developed in detail by Wait [l96l]. Wait discussed

the propagation characteristics of the hops and techniques for evaluating

them, and Wait and Conda [l96l] evaluated the path integral for the first

hop. Berry [1964] showed the propagation characteristics of the first

three hops graphically, and Berry and Chrisman [1965] developed

detailed methods for calculating the path integrals. This paper is

essentially a numerical supplement to the last mentioned paper.
The vertical electric field is written

E = E + Y Y. I. , (1) r o L j j

J = 1

where E is the ground wave, y. i s the ionospheric reflection coeffio J cient, and I. is the path integral [Wait, 196l], Each term of the series

in (1) is called a wave hop since it reduces, in the saddle point approximation, to the corresponding ray-hop of the geometric optics series.

Thus the familiar geometry of ray theory is used, as shown in figure 1. The vertical electric dipole source is located on the surface of the

spherical earth of radius a, with conductivity o, and permittivity e. The field is found on the surface at a great circle distance d = a6. The

ray paths of the first two hops are shown for a reflection height h. The

jth hop is incident on the earth at an angle T., and on the ionosphere at

^

TT an angle cp.. Its phase path length is D.. The lit region is T. <— ; n

J J J 2

T. = — is the caustic; and beyond the caustic is the shadow region. In

j 2

the shadow region the "ray path" is assumed to run along the surface

of the earth.

Section 2 describes the graphs of the path integrals and of the

other parameters necessary to use them. Sources of ionospheric

reflection coefficients to be used in (1) are discussed in section 3. A

sample calculation, illustrating the use of the graphs, is given in

section 4, and the formulas and methods of calculation are given in

section 5.

2. Description of figures

The main results are the amplitudes and phase lags of the path

integrals, I., j = 1 , . . . , 5, for frequencies of 10, 20, 30, 60, 100, 1 50,
and 200 kHz. Three kinds of paths are assumed; sea water (a = 5,

e = 80) where a is the ground conductivity, in mhos/m, and e is the

dielectric constant relative to free space, and two land paths (o = 0. 001,

e = 10, and o = 0. 01, e = 15). The curves are given for reflection

heights of 60, 70, 80, 90, and 100 km. It is hoped that this set of

parameters covers most cases of interest, and that the points are

sufficiently close together as a function of frequency and height to

allow interpolation. The space between the earth and the ionosphere is

assumed to be free space, and the actual earth' s radius, a — 6. 367 (10 )m,

is used.

Also included are graphs of the ground wave, E , for the same

set of parameters (figures 6, 7, and 8); a graph of coscp as a function

of distance parametric in height (figure 3); a graph to determine D

(figure 4); and a graph to convert measured field strength to the

equivalent dipole current moment (figure 5).

2. 1 The path integrals

Figures 9 through 2 3 show the calculated path integrals. The

figures are grouped first by conductivity; then each group is arranged

by ascending reflection height. The wave hop number, j, is indicated

by the digit after the decimal point; e. g. , figure 14. 3 shows the third

hop. Each figure shows both the amplitude (divided by the frequency

in kHz) and the phase lag, 8., of the wave hop for the set of frequencies

mentioned above. The phase lag is defined by

8 = - [ph E + kd + tt/2} for the ground wave, (2)

and

8. = - (ph I. + kD. + n/2} for the wave hops. (3)

J

l J J There is, of course, an ambiguity of an integral multiple of 2n. This

definition was chosen so that B. -» for perfectly conducting earth in

the saddle point (ray-hop) limit [Berry and Chrisman, 1965].
Figure 14. 1 shows the amplitude and phase of I x for a path

conductivity of 0. 01 mhos/m. For this reflection height the caustic is

at d = 1740 km -- beyond this is the shadow region. However, the

first wave hop propagates into the shadow region by diffraction. Deep

in the shadow region the propagation characteristics are much like a

ground wave as demonstrated by Wait [l96l]. Thus, the diffraction loss

increases with frequency and is less for higher conductivities (compare

figures 9. 1 and 19. 1).

The amplitude of I2 for the same parameters is shown in

figure 14. 2. There is a relative minimum in the amplitude near

3000 km. This is caused by the pseudo-Brewster angle in the ground

reflection coefficient [Bremmer, 1949]. The minimum is very sharp

for a frequency of 150 kHz, showing that there is nearly perlect

matching of characteristic and surface impedances for this set of

parameters [Wait, 1962], Corresponding to the sharp minimum in

the amplitude, there is a sudden phase shift of 180°. The location of

the psuedo-Brewster angle varies with frequency, conductivity, and,

of course, reflection height. The pseudo-Brewster angle minimum

also occurs on the third and higher hops at appropriately greater

distances.

From geometrical considerations it is clear that the width of

the neighborhood of the pseudo-Brewster angle and of the caustic region

is proportional to the hop number. Figures 9. 4 and 9. 5 show that the

very low frequency high-order wave hops have an oscillatory behavior

in the vicinity of the caustic and the pseudo-Brewster angle. These

are perhaps related to the interference fringes of optics, although, if

so, it is not clear to the authors why they only occur for the high-order

hops.

4
Figure 19. 1 illustrates an interesting phenomenon that occurs

on sea paths -- the amplitude curves cross over each other just beyond

the caustic. The same crossover occurs in the geometric optics cutback factor [Hyovalti, 1965]. In figure 19.3, there appears to be a double crossover in the third hop. The first is due to the frequency

variation of the location of the pseudo-Brewster angle, and the second

is the true crossover beyond the caustic.

Bremmer [1949] shows that the ground wave at a distant point

does not increase monotonically with conductivity. Rather, for a fixed

frequency, the field first increases with conductivity, goes through a broad maximum, and then decreases slowly with increasing conductivity.

The path integrals behave the same way. For example, for h = 60 km,

d = 7000 km, and f = 20 kHz, |l x (o = . 001) | = 2. 85 (10

-11

), [la (o = . 01) | = 3.79(10 -11

) , and |

I

a

( a = 5. )| = 3. 5(10~ n

). The phase lag, 8., is a very sensitive indicator of computational

accuracy since the first few significant digits are lost in (3). In some

cases, especially hops 4 and 5, there is a slight discontinuity in the

phase lag curves where the computer switches from the saddle point

approximation (23) to numerical integration of (17). The worst example

of this is the top curve in figure 18. 5, where there is a discontinuity of

about 40°. In most cases, the discontinuity is less than 5°, and it is not

difficult to draw a smooth curve to eliminate the discontinuity.

There are undoubtedly features of the wave hop calculations of

more interest to some readers than those described above. It is not

the intent to exhaust the possibilities here -- the calculations are being

presented in hopes they will be used and studied by others interested

in LF and VLF propagation.
2.2 The ground "wave

For completeness, the amplitude and phase lag of the ground

wave are shown in figures 6, 7, and 8, for the same set of parameters

used to calculate the path integrals. The only difference is the distance

range covered -- the ground wave is important only at shorter distances,

so the curves are shown from 100 km to 3500 km.

2. 3 Normalization

The calculations have been made for a unit dipole current

moment. (Refer to (19) of section 5. ) Figure 5 shows the field of such

a dipole at one mile as a function of frequency. The field strength

increases below about 30 kHz because the induction and electrostatic

fields extend out to one mile at these frequencies. The equivalent

dipole current moment to be used for an existing transmitter is found

by dividing the value given by figure 5 by the measured field strength

at one mile.

3. Reflection coefficients

To use (1) to predict field strengths, ionospheric reflection

coefficients must be available. Calculation of such coefficients is

outside the scope of this paper; however, several sources of reflection

coefficients will be listed, and instructions given for their use. In general, the reflection coefficients are functions of angle of

incidence, cp, which varies with distance and reflection height according

to [Bremmer, 1949]:

coscp = [ a (l - cos-|:)+ h] £2a(a + h)(l - cos-^)+ h

2 ]~* (4)

J '-] 2j 22ji
The subscript j indicates the hop. Figure 3 shows cos cp as a function

of d/j for various heights.

For propagation into the shadow region, the reflection coefficient

for maximum cp (minimum cos cp) is used. A useful approximation for

the great circle distance to the caustic is

d = 225/h~ j , (6)

where h and d are both in kilometers, and the angle of incidence for

this distance is given approximately by

cp = 89. 9 - Jh~ degrees, (7) c

where again h is in kilometers.

3. 1 Isotropic ionosphere

If the effect of the earth1 s magnetic field is ignored, the

ionosphere is isotropic, and

Y

J = T

ee

j = „V (8»

where T is the ratio of the incident field in the plane of incidence to ee

the reflected field in the same plane. (The T notation is used by

Bremmer [1949], Johler and Harper [1962], etc. ; the R notation is

used by Budden [l96l], Wait [1962], etc.) For a sharply -bounded,

homogeneous, locally plane ionosphere [Bremmer, 1949],

Tl I 3—

i cos cp - / 1 - (sin cp/ \)

T

ee

= " ' <?>

ti. cos cp + 7l - (sin cp/ri.)'2I 1

Wo

2

where r\. = 1 - i U) /uu, uo = ——— , uu is the plasma frequency, and

l r r V+iiu ' v is the collision frequency.
Wait and Walters [l963a,b,c] show many results for continuously varying models of the ionosphere.

3. 2 Anisotropic ionosphere

If the effect of the earth' s magnetic field is included, y. is

J given by

Y, 0C12 | T_ T_ | | jjR,, „Ri

! =

ttsi a2 2 :

ee em

-R T -R T I -R ,R„ -R ,R , m me m mm j mJ- ml _L

(10)

where R is the Fresnel ground reflection coefficient for horizontal

m

polarization [Bremmer, 1949]:

cos T - a/h

2

- sin2 T R = ; g

, (11)

cos T + Vn - sin T g

where -n is the index of refraction of the ground, given by (21). In ( T is the ratio of the incident field perpendicular to the plane of mm

incidence to the reflected field perpendicular to the plane of incidence;

T is the ratio of the incident field in the plane of incidence to the em r reflected field perpendicular to it; and T is the ratio of the incident

me

field perpendicular to the plane of incidence to the reflected field in the

plane of incidence.

Explicitly,

Yi = T

ee

v2 =T-RT T , and

ee m em me

Y-,=T 3 -2RT T +R 2 T T T . (12)

,J ee m em me m em me mm
Clearly, (12) reduces to (8) if the terms containing the coupling

coefficients T and T are neglected. This is usually a good em me

approximation at grazing incidence for LF and VLF radio waves.

Johler, Walters, and Harper [i960] show reflection coefficients

for a sharply bounded, homogeneous anisotropic ionosphere with

arbitrary magnetic field, and Johler and Harper [1962] calculate them

for a continuously stratified ionosphere with arbitrary magnetic field, as do Walters and Wait L 1 963 H for a horizontal, transverse magnetic

field.

Belrose [1964] shov/s average magnitude and the seasonal

variation of measured reflection coefficients for several LF and VLF

frequencies over various paths.

3. 3 Extrapolation of reflection coefficients

It is sometimes convenient to extrapolate a reflection coefficient

measured at a given frequency and distance to other frequencies and/or

distances. The following empirical formulas have been suggested.

For an exponential model ionosphere and very low frequencies,

Wait and Walters [ 1963a] have shown that

m _ -a-, cos cp icco cos cp ., . T = - e

x ^ e ^ , (13)

ee

where cca and Og are positive constants (for a fixed frequency). This is

a very good approximation especially over moderate variations of cp. Allcock [1955] suggested that

IT I = ota f cos cp , (14) ee

which can be used to extrapolate in the frequency domain. Belrose

[1964] has compiled data that tends to confirm (14) in the winter, but

not in the summer.
4. Sample calculation

Using (2), (3), (17), and (42), (1) can be written

00 > E -i te-1(kd+,T/2 >{iEo|-iPo + y iv.i ii.i e

- i(e

j + k(D

r

d) - ph V} J J

j = l (15)

All quantities in (15) except | Y.I and phY. are shown in figures in this

paper, as discussed in section 2. Sources for | Y. I and ph y. were

discussed in section 3.

A sample field strength calculation is described below for the

parameters:

f = 20 kc/s, h = 70 km, a = 0. 001, d = 1000 (200) 4000 km.

For simplicity, an exponential, isotropic ionosphere is used so that

Y. = T

J (see (8)),and [Wait and Walters, 1963a]

J ee T - - exp((-3 + 3. 5i) cos cp), i. e. , ee

|Y | - e

" 3j COSCP

, andphY. - j (tt + 3. 5 cos cp). (16)

J J

(1) For each distance and each hop, cos cp is found from

figure 3. Since the first hop caustic is at d = 1880 km, cos cp

is held constant for the first hop for all greater distances.

Similarly, cos cp is constant for the second hop beyond

3760 km. Only three hops are necessary for this distance

range.

(2) |y.| andphY. are calculated using (16).

(3) |e I and 6 are read from figure 6. |I. | and (3.,

j = 1 , 2, 3, are read from figures 10. 1 , 10. 2, and 10. 3

10
(4) For f = 20 kHz, k= .419. (D. - d) = j Q

DJ " d

~) is

found from figure 4. Note that p. is given in degrees while

k(D. - d) and phy. are in radians.

J J (5) The individual terms of (15) are calculated and the sum

of the series is found.

(6) If desired, the series can be multiplied by Io^e

as shown in (15). This was not done in the sample calculation.

The calculated amplitudes of the wave hops, the ground wave,

and the total field for this example are shown in figure 2. As would be

expected, it is not necessary to consider the ground wave at great

distances. Indeed, for this daytime model, reasonably good results

are obtained by including only the two largest wave hops at each point.

At night, it would probably be necessary to include three, four, or more,

hops.

The path integrals can also be used to extract ionospheric

reflection coefficients from measured data [Berry and Chrisman, 1965].

5. Formulas used

A previous paper [Berry and Chrisman, 1965] discussed the

calculation of the path integrals in detail and compared the methods of

calculating them. The formulas used are repeated here for completeness,

11
5. 1 The path integrals

The path integral I. is given by

I. - (-DW1" 74 2^ & ( 1+.t)'» e^ ^"'^ ^

dt , (17)

J a/ sin fl J _ j+1 ,

.

.

VsinB 7

c Exj+1

(t)

uo where k = 2tt/X = — , uo = 2nf, and c is the speed of light. c

1 ka x

/ x= v6, z =-^3" , and v= ( —) . (18)

K = 11.96 lol^jK v

2

, (19)

where I £ is the dipole current moment.

En (t) = W,'(t) - qW, (t), where (20) k k k

q = -vjl- n

2

/ n

2

, and (21)

g g

2 2 / T| = e- i|ioC a/^, where eis the relative permittivity of the earth

and o is its conductivity.

The Airy function, W (x), is discussed by Spies and Wait [l96l] and

JK formulas for its calculation are in section 5. 3.

where y = kh/v.

The saddle point approximation to (17) is

12
• 2 2 L_Z_

4xj

where a = 4\

y, " X

, (24)

2 4, 2

3

/o 4 3

Q = - xa

2 + - j (y + a

2

) /2

- - j a

3

, (25)

R - aM(i s ) - iqL ( is ) . L(-is) , ?M

aM(-is) + iqL (-is) L(is) '

K '

s =|a3

, (27)

H(a) = (l-a2 z ) 5/2 L

2 (is) {^j"/ \ (28)

and L(x) and M(x) are defined in (55).

The saddle point approximation is valid if a » 1. For most cases of interest H (a) — 1 , in which case Wait Ll96l]

has shown that (23) is the Jth geometrical optics ray-hop.

The integral in (17) can also be calculated by summing the

residues at the poles, t , such that

Ej. (t g

) = 0. (29)

The calculation is intricate since the poles are of order j+1, but it can

be done on a computer with the following formulas:

Rewrite (17) as

t =r nJ- 1^ iTT/4 e

"

r D(t) e2 (t) F

J (t) dt

I. — (-1) 4Ke i , g —

j Vsin 6 J J +1 /^x

(30)

t j_i in/4e"ikd

r = -1 J AW

4Ke J -J— dt

Vsin 6 B.(t)

J

5 -3Xt

where D(t) = (1+ f- zt) e

' , since |zt|«l. (31)

13
Let Res(j,t ) be the residue of the integrand of (17) at t . Then s s

I. = (-l)J_1 8ni Ke

._ / . -ikd ro in/ 4 e v

) Res(j,t ) V sin 6

s = o

(32)

b.

J°

J

JO Jl J2

b. b. b. . JO Jl J2

b. b. . Jo Ji

JJ

JJ

b.

Jo

jj-l

'jJ-2

b. b

Jo Ji

, (33)

where a. (t ) = A.

(n)

(t )/n!, and b. (t ) = B.

(n+j+1)

(t )/(n+j+l)!. (34) jn s jn s

The superscript (n) denotes the nth derivative.

Using Liebnitz 1 s rule for the nth derivative of a product

[Kaplan, 1952],

n

ww

-I

n! (m) (n-m)

m!(n-m)! g (35)

m - 1

the necessary derivatives of [E2 (t)F(t)] and

A.(t) = A. (t)[E(t)F(t)]

J J -1 2 (36)

are found recursively. The necessary derivatives of D, E2 , and F for

the first five I. are

J

14
n( n)m - / • \ n ~ 1 _ixt r 5 D (t) - (-ix) e Lng-z ix(l+ f- zt)]

E2'(t) = tWa (t) - qW2

/ (t),

E/(t) = tW2 (t) + (l-qt)Ws (t),

E2 (s)

(t) = (t2 -q)Wa (t)+ (2-qt)Ws(t),

E2 U)(t) = (4t-qt2 )W2 (t)+ (t2 -2q)W2

'(t), and

F "(t)

F<

3

> F

(4)

(t)

F

(5)

(t)

(37)

(38)

E2 (5)

(t) = (4-4qt+t3 )W2 (t) + (6t-qt2 )W2 (t);

F-'(t) = 2i[W2 (t-y)]"

2

, = -2GF'(t),

= {-2(t-y) + 6G2

} F'(t) (39) = {-2+l6(t-y)G-24G3

} F '(t), and

= {l6(t-yf+20G-120 (t-y)G2 + 120G4 }F'(t).

In (39), G = W2 (t-y)/ W2 (t-y); a single prime indicates a first derivative

and a double prime indicates a second derivative.

15
The b. can be found by repeated use of (36) and (29), but are

given explicitly here for the first five hops.

b = (E x'(t )) J+1

, j= 1.....5 , JO fa

b.

/

,

S J4

"+ 1 " = J—-(Ei(t ))

J EX (t ), j = 1.....5,

Ji 2

(3)

i+1 ( 4) + TT El <V V j =

3

' 4 ' 5 .

i - 2 »« . i (3

)

Ei(t )b. =V Ej (t ) b. t-frEi' '(t ) b. (40) 8 "» »V u

j3

T

12

^

*V u

j2

>55 =-^E1

, (t s )(E 1

// (t s

)) 5 +|(E1

# (t s

))2 (E 1

,, (t s

))3 E1 (3)

(t s

)

+ |(E1'(t jfE^t )(Ei

(3)

(t ))2 OS fa S

+ |(Ei<t ))3

(Ei'(t )fE1 {4 \t)

O S fa fa

+ ^(E1'(t s

))* E1 (3)

(t s

) E1 (4)

(t s

)

+ l<E1'<t s

))* E1"(t

s

>E1 (5)

Jt

s

>

and b. = 0, if j < n.

16
The necessary derivatives of Ex (t ) are

s

El'(t ) = (t - q

S

) Wi(t ), s s s ^x(t

B

) = Wx

(t

g

),

El

(3)

(t ) = (t 2

- q

2

t + q) Wi(t ), s s s s Ex

(4)

(t ) = (4t - 2q2

) Wx (t ), (41)

s s s Ei '(t ) = (t

3

- q

2

1 3 + 2t q + 4) Wn (t ), and

s s s s s Ei

(6)

(t ) = (9t 2

- 6q2

t + 6q) Wx (t ). s s s s

In general, the saddle point approximation is valid in the lit

region except near the caustic; numerical integration of (17) is

necessary in the vicinity of the caustic on the lit side; and the residue

series is used in the shadow region.

5. 2 The ground wave

In the notation used in this paper, the ground wave is ["Wait,

1962]

-in/ 4 e E =2iKe ' —

-ikd

Vsin6 E x (t)

. ,. A -ikd °° D(t , -in/ 4 e V s)

D(t)Wl(t)

. dt (42)

= - 4TTKe

^•' , * - Y =—; , (43)

V

rIinl U t - q

s

s = o

where, again, t is a zero of E^t).

s

17
5.3 The Airy functions, W (x) k

Let [Spies and Wait, 1 96 1 ]

CO M-14.S (l)(4)(7)...(3n-2) 3n

Yi(t) - 1 +^ 7J^T[ t » an<* (44)

so that

Then

where

(3n)l

n= 1

M-+4.X- (Z)(5)(8)...(3n-1) 3n+l

y2(t) - t + l —onTiyi

t (45> n= 1

oo 'ftl-Y (l)(4)(7)...(3n-2) 3n-l

.(t) -£ I3n-ni * ' yi^ and (46) =

Z, (3n-l)!

n= 1

W(t)=1+ £

(2)(5)(8)...(3n-l)

t3n ^

n= 1

Wk(t) = u(t)+ (-l)

k

iv(t), (48)

u(t) = (aj yi (t) + a2 y2 (t))/3

,

( 49)

v(t) = alYl (t) - a2 y2 (t), (50)

aa = —-

, and a2 = (51)

V9r(|-) 3vTr(^-)

These convergent expansions are adequate for small |t|. For large |t|,

the plane is divided into four regions:
Region I: |ph t| < 20°

Region II: |ph tl > 100°

Region Ilia: -100° ^ ph t ^ -20°, and ( 52 ) Region nib: 20° ^ ph t ^ 1 00°.

3/

In region I, let z = §- 1 , then [Olver, 1954]

W

k

(t)= t~^[e Z (L(z)+ (-l)

k

|

e"

Z

L(-z)], and (53)

W^(t)=t^ [e

Z M(z)+(-l)k+1 i e'

Z

M(-z)], (54)

where

00 00 L(z)=£ U. z"j

, M(z) = Y V.z"j

, (55)

J=o j=o

U = V = 1 , and for j > 1

Ub =

(Zj+ l)(Zj+ 3)

: ..(6,i-l)

^ = _ u

6j±l

^ J j!(2l6)J J J 6J" 1

3/

In region II, let z = f- (-t) 2

, then

Wk

(t) = (-t)"4 exp[i(-l)

k

(z+^)] L[(-l)k

iz], and (57)

W^(t) = (-t)T exp Li(-l)

k

(z -^ )] M L(-l)

k

iz]. (58)

19
3/7

In region III, let z = f- (t) , then, in Ilia

Wk(t) = t"

4 [e

Z L(z) + i(k-2)e"

Z L(-z)], and (59)

i z„ ,, . . „ „. -z. W^(t)=t 4 [e M(z) -i(k-2)e M(-z)]. (60)

In region Illb

i z_ , . , ... . . -z. W (t)=t" 4 [e L(z) + i(k-l)e" L(-z)], and (6l)

W '(t) = t 4 le\A(z ) - i(k-l)e

_ZM(-z )]. (62)

These formulas are asymptotically valid in regions larger than those

given here [Olver, 1954], but greater computational accuracy is

achieved in the restricted regions.

6. References

Allcock, G. McK. , (1955), Ionospheric absorption at low and medium

frequencies in The Physics of the Ionosphere, Phil. Soc.

London . Belrose, J. S. (1964), The oblique reflection of CW low-frequency

radio waves from the ionosphere, Propagation of radio waves

at frequencies below 300 kc/s, AGARDograph 74 (Pergamon

Press, New York, N. Y. )

.

Berry, L. A. (1964), Wave hop theory of long distance propagation of

low frequency radio waves, Radio Sci. J. Res. NBS , 68D> No. 12, 1275-1284.

Berry, L. A. and M. E. Chrisman (1 965), The path integrals of

LF/VLF wave hop theory, Radio Sci. J. Res. NBS, 69D,

No. 11.

20
Bremmer, H. (1949)> Terrestrial radio waves. (Elsevier Publishing

Co. , New York, N. Y. ). Budden, K. G. (1961), Radio waves in the ionosphere (Cambridge at

the University Press, London).

Hyovalti, D. C. (1965), Private communication.

Johler, J. R. and J. D. Harper, Jr. (1962), Reflection and transmission of radio waves at a continuously stratified plasma

with arbitrary magnetic induction, J. Res. NBS 66D,

(Radio Prop. ), No. 1, 81.-99.

Johler, J. R. , L. C. Walters, and J. D. Harper, Jr. (I960), Low

and very- low-frequency model ionosphere reflection

coefficients, NBS Tech. Note. No. 69 . Kaplan, W. (1952), Advanced Calculus (Addison-Wesley Publishing Co. , Inc., Reading, Mass.) . Norton, K. A. (1959), Transmission loss in Radio Pr opagation - II,

NBS technical Note No. 12.

Olver, F.W.J. (1954), The asymptotic expansion of Be ssel functions

,

Roy. Soc. of London, Phil. Trans. A, 247 , 364-367.

Rydbeck, O.E.H. (1944), On the propagation of radio waves, Trans.

of Chalmers Univ. of Tech. , 34. Spies, K. P. and J. R. Wait (1961), Mode calculations for VLF

propagation in the earth-ionosphere waveguide, NBS Tech.

Note No. 114.

Wait, J. R. (I960), Diffractive corrections to the geometrical optics

of low frequency propagation, Electromagnetic Wave

Propagation (Academic Press, London).

Wait, J. R. (1961), A diffraction theory of LF sky-wave propagation,

J. Geophys. Res. 66, No. 6, 1713-1724.

21
Wait, J. R. (1962), Electromagnetic waves in stratified media

(Pergamon Press, New York, N. Y.).

Wait, J. R. and A. M. Conda (1958), Pattern of an antenna on a curved

lossy surface, IRE Trans. Ant. Prop. AP-6 , No. 4, 348-359.

Wait, J. R. , and A. M. Conda ( 1 96 1 ), A diffraction theory of LF skywave propagation - an additional note, J. Geophys. Res. 66,

No. 6, 1725-1729.

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

waves from an inhomogeneous ionosphere, Part I, Exponentially

varying isotropic models , J. Res. NBS 67D (Radio Prop. ), No. 3, 361-367.

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

from an inhomogeneous ionosphere. Part II, Perturbed

exponential model, J. Res. NBS 67D (Radio Prop. ), No. 5,

519-523.

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

from an inhomogeneous ionosphere. Part III, Exponential

model with hyperbolic transition, J. Res. NBS 67D (Radio

Prop.), No. 6, 747-752.

Walters, L. C. and J. R. Wait (1963), Numerical calculations for

reflection of electromagnetic waves from a lossy magneto

plasma, NBS Tech. Note No. 205.

22
u

oO

U

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23
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-II _

2000 3000 4000

Figure 2,

DISTANCE, KILOMETERS

Amplitudes of the total field, ground wave, and first

three wave hops showing the sample calculation of

Section 4.

24
I

0.9

0.8

0.7

0.6

0.5

0.4

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0.1

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DISTANCE, d/j , KILOMETERS

2500 3000 3500

Figure 3, Cosine of the angle of incidence of a ray versus

distance for various reflection heights.

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