UJ. hTI/CUlfUCUUS

J. Bldg

JUL 1 1 1966

^ecknic&l ^iote 92c. 335

ZONAL HARMONICS IN LOW FREQUENCY

TERRESTRIAL RADIO WAVE PROPAGATION

J. RALPH JOHLER

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards is a principal focal point in the Federal Government for assuring

maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. Its responsibilities include development and maintenance of the national stand- ards of measurement, and the provisions of means for making measurements consistent with those standards; determination of physical constants and properties of materials; development of methods

for testing materials, mechanisms, and structures, and making such tests as may be necessary, particularly for government agencies; cooperation in the establishment of standard practices for incorporation in codes and specifications; advisory service to government agencies on scientific and technical problems; invention and development of devices to serve special needs of the Government; assistance

to industry, business, and consumers in the development and acceptance of commercial standards and

simplified trade practice recommendations; administration of programs in cooperation with United

States business groups and standards organizations for the development of international standards of

practice; and maintenance of a clearinghouse for the collection and dissemination of scientific, technical, and engineering information. The scope of the Bureau's activities is suggested in the following

listing of its four Institutes and their organizational units. Institute for Basic Standards. Electricity. Metrology. Heat. Radiation Physics. Mechanics. Applied Mathematics. Atomic Physics. Physical Chemistry. Laboratory Astrophysics.* Radio Standards Laboratory: Radio Standards Physics; Radio Standards Engineering.** Office of Standard Ref- erence Data. Institute for Materials Research. Analytical Chemistry. Polymers. Metallurgy. Inorganic Materials. Reactor Radiations. Cryogenics.** Office of Standard Reference Materials. Central Radio Propagation Laboratory.** Ionosphere Research and Propagation. Troposphere

and Space Telecommunications. Radio Systems. Upper Atmosphere and Space Physics.

Institute for Applied Technology. Textiles and Apparel Technology Center. Building Research.

Industrial Equipment. Information Technology. Performance Test Development. Instrumentation.

Transport Systems. Office of Technical Services. Office of Weights and Measures. Office of Engineering Standards. Office of Industrial Services.

* NBS Group, Joint Institute for Laboratory Astrophysics at the University of Colorado. ** Located at Boulder, Colorado.
NATIONAL BUREAU OF STANDARDS

technical ^ote 335

ISSUED April 13, 1966

ZONAL HARMONICS IN LOW FREQUENCY TERRESTRIAL

RADIO WAVE PROPAGATION

J. Ralph Johler

Institute for Telecommunication Sciences and Aeronomy*

Environmental Science Services Administration

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.

Formerly the Central Radio Propagation Laboratory of the National Bureau of Standards;

CRPL was transferred to ESSA in October J 965, but will temporarily use various NBS

publication series pending inauguration of their ESSA counterparts.

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

Price: 25<t

Contents

Abstract 1

1. Introduction 2

2. Theory of Zonal Harmonic and Geometric Series 6

3. Plane Reflection Coefficients 11

4. Inhomogeneous, Anisotropic Ionosphere 12

5. Discussion of a Sample Calculation 17

6. Conclusions 18

7. References 19

Figures 1 through 3 21

in

Zonal Harmonics in Low Frequency Terrestrial

Radio Wave Propagation

J. Ralph Johler

If 7?

n and Tn are ground and ionosphere reflection coefficient

matrices and 1 is the unit matrix, then for the terrestrial waveguide

boundaries, a geometric series \l + 2 7?^ T^ | for each zonal harmonic

i =o mode, n, results in a rapidly converging zonal harmonic series. Thus,

by an interchange of the order of summation of the two series, the zonal

harmonic series can be employed to calculate each term of the geometric

serie s.

This method of analysis provides a practical solution to the propagated LF, VLF, ELF terrestrial waveguide field. Thus, geometric

series terms (often called wavehops) can be calculated quickly and

accurately with the zonal harmonic series. The summation of the

geometric series then provides an efficient method for calculating the

total field (or waveguide mode sum). The simplicity of the analysis is

an attractive feature when compared with the complex integral method.

Computation speed is obtained on the large scale computer by use of

recursion formulas for the Legendre and spherical wave functions. It is noted that the terms of the geometric series in the high frequency

limit give the geometric -optical rays.

An improved definition of the reflection coefficient matrix is

presented which accounts for the reflection process in a manner which

can be justified both mathematically and physically. In fact, the entire

reflection coefficient matrix for the anisotropic ionosphere can be

retained as a variable of the zonal harmonic summation process with

resultant improvement in computation accuracy. Demonstrative computations indicate considerable advantage in the method as an alternate

approach to the propagation problem.

Key Words: Extra low frequencies, geometric -optics, geometricseries, LF, VLF, ELF mode theory, low frequencies,

terrestrial radio wave propagation, very low frequencies,

zonal harmonics.
1 . Introduction

In a previous paper [johler, 1964] the notion of a zonal harmonic

series for each term of the geometric series representation of the

terrestrial radio wave field was introduced. The upper boundary of the

terrestrial radio waveguide for low frequency radio waves (Johler

and Berry, 1964) can be characterized by the reflection coefficient

matrix T,

T = T

( E

) T

( s

)

-1- e e * e m T

( S

) T

( s

)

in 6 in ni

(i.i)

1 (s ) where Te e is the reflection coefficient for vertical polarization

(TM-mode) of the incident and reflected waves on the boundary g - a + h,

figure 1, where a is the radius of the terrestrial sphere, and h is the

(s) height of the guide. Similarly, Tm refers to the vertical magnetic

( S ) (s) incident and reflected wave s (TE-mode). Also, Tem and Tm e are abnormal

components resulting from the anisotropic nature of the ionosphere. For

simplicity, it is instructive to assume the ionosphere to be isotropic,

( £) and the excitation TM. Then, T = T^ e\ Also, for the ground,

7? =

(s) r: ' o

R

()

(1.2)

(s) / s\ where Re refers to vertical polarization of electric vector and R^

refers to vertical polarization of the magnetic vector. For the TM exci

tation of the terrestrial waveguide with isotropic reflection at the ionosphere,

7? = R

( s

) (1.3)

or dropping the subscript, e,

i? = R

()

(1.4)

Superscript (s) is here employed to remind the reader that the spherT

cal convergence -divergence or focusing factor has not as yet been

removed from the reflection coefficient. See (1. 10).
The classical zonal harmonic solution [johler, 1964], can be

written for terminals S and O in figure 1, on the ground, r = a, for the

particular case of the vertical electric polarization, Er

,

E. = -^ 'r ~ i^2 _ 4 ^

£

n(n+l)(2n+l)Pn kfa 4 4n

(cos 0) cfa K

n =0

x{l + R(

S >^}{1 + Tr>^}{l-Ri! »T<S) r- I 1 ' 5 ' bl a

(n = 0, 1, 2, 3. . . )

Here I t is the intensity of the electrical point source dipole, S, figure 1,

in ampere -meters which can be for convenience taken as unit, 10^= 1. The permeability of space is |J.

, |J. = 4tt(10~ ) henry /meter and c is

the speed of light (c ~ 2. 997925 (10 8

) m/s). kx is the wave number,

UL) . kx = — n-L in air where r^ ~ 1. 0001 to 1. 0003 and the frequency f = uj/2tt.

uu In vacuum,

^ =1, k = — . The subscript n on T n and Rn reminds the

reader that these quantities are calculated for each n. Pn (z) is the

solution of Legendre's differential equation,

< L -''l^-^^^HUP.I^O. (1.6)

(l , 2 ) The abbreviations Q 1& ' and ^ a are, letting kx a - z,

ka = C n ( z )=^/— Hn + i ( z ) I 1 - 7 )

'ia = K(Z) = /^ J n + i(z) (1-8)

(l 2 ) where J n+ i(z) and H^ + 'i (z) are Bessel and Hankel functions of order

9 2 (12)

nf-g- and argument z and H^ may be of the first or second kind.

-3-
The reflection coefficients R; ' and T^ contain focusing or

convergence -divergence factors. Thus, the spherical reflection coeffic

ients will be designated by Rn and Tn , dropping the superscript (s), or,

after substitution in (1.5) of the exact relationship,

*n(«) =i[ci

l)

(z) + d

a W] (1.9)

R

( S

) _ -Cia Rn R - (a) > bl a

(1.10)

T

( 2) (> _ ^cii

n T "

r

(l)

n (1.11)

Here, the spherical reflection coefficients are defined,

and,

Rn =

T.

c

(1)/ k c (1)/

'si a Ki ^3»

Ci

(1) k2 r

(i)

c

(3)

' k c

(l) '

'

'si a %_ 'oSa

r

( 2

) k2

(i)

Wia "=>2a

ill.^ ^3e

( 2 ) k3 r

(a)

£i I ksLa -c

(1)/ k c

(2)

r(i) k3 r( 2

) 36

(1.12)

(1.13)

where C~ ' ' (z) = f- C

1 * ' ~ '(z). dz

The wave number, k3

, is,

k3 =

(!) UU. uo(v+iO))

(1.14)

-4-
2 Ne 2 for a plasma frequency squared IU N where N is the number density

e m

of the plasma (N = Ne el/m3 for an electron plasma and m = me the elec

tronic mass) and £ = — . The electron-neutral collision frequency

c Ho

is v, sec

-1

. The wave number of the ground, k2 , is given by,

W / • °"M- o c

2

where the permittivity, € = € £ 2 . The conductivity is given by O,

mhos /m.

The geometric series expansion was obtained from the denominator

in (1. 5),

i-kW _1 = 1 + £

[r1

S) t1

S) ] , (1.16)

3=1

l ( s) ( s )i which converges absolutely, if | R* T^ < 1.

This basic idea has been employed in integral form in the complex

n-plane by Bremmer [1949] and Wait [ 1961 ]. Indeed, Wait [l96l]

suggested a rigorous type of geometric -optics applicable to low frequency propagation. Berry [ 1964a, b] and Berry and Chrisman [1965]

developed elegant computation methods based on this idea using

asymptotic formulas for the spherical wave functions (1. 6), (1. 7), and

(1. 8) in the complex n-plane. In effect, the method presented in this

paper extends the classical geometric-optics [Johler, 1961; 1962]

to great distance (> 3000 km). Such an approach is an alternate to the

waveguide mode solution described in detail by Wait [1962], The

asymptotic formulas for the spherical wave functions [Berry, 1964a, bj

are intricate to program for a large scale computer. On the other hand,

along the real axis of the complex n-plane, the spherical wave functions

(1. 6), (1. 7), and (1. 8), for integer values of n, can be calculated with

great speed and simplicity on a computer with standard recursion formulas. Furthermore, the recursion formulas give an exact rather than an

asymptotic solution. This paper, therefore, presents an alternate

approach to the calculation of the low frequency fields propagated in the

terrestrial waveguide. Finally the solution obtained leads to an improved

method for introducing the anisotropic ionosphere reflection coefficient

matrix.

5-
2. Theory of Zonal Harmonic and Geometric Series

In the previous paper [johler, 1964] the propagated field (vertical

polarization of the source) was written as the series,

E r = Er ,o+I 'r, 1 (2.1)

J=i

where the ground wave, E r

, can be regarded as the zero order (j = 0)

term of the series,

[i c i l f Tr(„\ r

(2) r

(l)

n + r )

n =0

and a particular ionosphere wave, j,

CO lipc i i y r

(2) r

(l)

Er n + r r

> J ~ 8n kp A

F(n) Cia Cla(1 +RJ

n =0

(2.2)

rr (i) r

(a). kliL . Zl-6 L (3) (1). _Rn ]

j -i

T. (2.3)

where Fn = n(n+l)(2n+l) Pn (cos 9). Here the order of summation of

the zonal harmonics and geometric series has been interchanged. This

does not impose any serious mathematical restrictions, since the j -series

and the n-series are absolutely convergent.

The Legendre function, Pn (z), can be calculated exactly for integral

n from the recursion formula,

p„ + 1 (z) = <^i^p„(z)-^ipn . a(z ). (2.4)
where,

Po(z) = 1 (2. 5)

and,

Px (z) = z. (2.6)

Both integer and non-integer values of n can be used in a formula

[Bremmer, 1949] based on the saddle point approximation of the

integral of Sommerfeld (letter z = cos 6).

2 cos

Pn (cos 0)

(n+!)9 - J_j

V2TT(n+l) sin

(2.7)

This formula becomes inaccurate for sin ~ and n small. This

formula (2. 7) was checked with formulas (2. 4), (2. 5) and (2. 6) in a computer subroutine. The agreement was excellent. The recursion formulas

were used, however, for the computations illustrated in this paper.

The spherical wave functions can also be calculated from the

recursion formulas,

(1,2) 2n+j_ (1,3) r

( 1 »a). x

, ? o, ^n+l ( z ) = : C n ( z ) - C n -1 ( z )> ( 2 - 8 )

starting with the values,

Ci

1 '' 3

\z) = =Fi exp [±iz] (2.9)

C

(_

X

i ,8)

(z) = exp [±iz]. (2. 10)

Using a computation technique of Goldsteing and Thayler [1959], a computer program was written to calculate the spherical wave functions

(2. 8), (2. 9), and (2. 10). An extensive comparison was made with a program developed by Berry [ 1964a] using the asymptotic formulas

(designed for the complex n-plane) along the real axis of the n-plane.
Excellent agreement was found. The recursion formulas (2. 8), (2. 9),

and (2. 10) are of course calculated with much greater speed and accuracy

on the computer.

The ratio of the derivative of a spherical wave function to the

function can also be determined by recursion [Johler and Berry, 1962],

#'"><.) 1 n

e ••'(.)

" n

z

(i , 2 )' d.'i

; (z)

Cn-l ( z )

z (2.11)

where,

= ± i.

For complex argument, z = k2a, the Debye approximation can be

employed,

(i)

C; ' (ks a) _, / n(n+l) _

x tfW, V

^2 (2.12)

This very simple approximation becomes quite excellent since v n(n+l)

does not grow as large as |

k

2 a | , since the series converges at n ~ kx a.

is

7 (k a) Alternately, the function yn „

2—- can be readily used in (2. 11) by Mksa)

/ 2\

( 2 \

7 replacing the C n and Q function with \|j n and \|^ functions. Thus,

CJ

1} (k2 a) K'(k2 a)

(!) Ci

; (k2 a)

(2.13)

'n(k2 a)

Equations (2. 2) and (2. 3) can be readily generalized for elevated

terminals, with the source S situated at b = a + h s above the ground and

the observer O situated at r = a + h , instead of the surface r = b = a illustrated in figure 1. Thus, the ground wave can be written, r > b,
(1)

.(i)

, A

2

) CI a' 1-(a)

'r,0- 8rT k

2 r

2 b

2 A

F(n) l>lb + Gib (a)

Rv

n =0 CI

(2. 14)

and the ionosphere waves can be written,

E.

, 1 8tt Iq1 Y F<ni [r

(l) + c!2 r

( 2

) R

n -0 Gi

a

k^r

( 2

) x

! c Hr. +^ch) ]p j Ri _i Ti

Ci a

(2.15)

Here the abbreviations, cfr

' 3)

= cf ' 2 \^j. r); dV * * = d

1 ' * ^b), . . . , have been employed.

The series (2. 2) is the zonal harmonics representation of the

ground wave and converges quite slowly. Thus, approximately 15 kx a terms are required to get a reasonably accurate answer. At 10 kc / s this would be 20, 000 terms. Hence, the zonal harmonic series (2. 2)

should be replaced by the classical ground wave theory. An efficient

computer program has been developed for this purpose Johler [1962].

Once the ground wave (2. 2) has been removed as a problem, the second

term of (2. 1) and the higher order terms can be readily evaluated with

zonal harmonics. This can be made evident by introducing the Wronskian

(Johler and Berry, 1962 ; page 768).

c«

l) ' c<

2 >'

c

(,

l) c<2)

2i, (2.16)

into the factor 1 + Rn in equation (2. 3). Thus,

1 + R,

2i

c

(1) c

(2) c

(2) k c

(1) *sia f^i_ <s2a A2

)

' k2 (!) 2a

(2. 17)
Thus, the £^ a

' Q{ a in (2. 17) cancels the G la £ la in series (2. 2), and as n

grows large, advantage cannot be taken of the fact that Q1& £ la grows

large. This is the reason for the notoriously slow convergence of the

harmonic series [Johler and Berry, 1962], On the other hand, the

ionospheric waves converge rapidly as n-4cx a since the factor (1 + Rn ) replaces (1 + Rn ) in (2. 3). It is noted that Rn

-» -1 as n grows large.

This would lead to computation problems if (2. 17) were not employed.

In any case approximately 1400 terms at 1 kc/s or approximately kx a terms required. In fact, 1500 terms gives very high accuracy (> 6

significant figures).

Although this is still a large number of terms, a simple recursion

method for calculating each term allows the electronic computer to sum

the series as quickly as it could calculate the smaller number of complex

terms used in the evaluation of an integral in the complex n-plane. The

complete field can then be calculated by a summation of the geometric

series (2. 1) which requires only a few terms if the upper boundary is

lossy. However, 100 terms or more may be required if the boundaries

are not very lossy, but this adds little to the computation time. In (2. 3)

the factor,

M) Jfl)

Fc (n) = -iii- ^pexp(icp) (2.18)

Gla Gig

(1) (2)* is of unity magnitude since £

* a

' = £ la for real arguments kx a and kx g. Also, as n becomes large, cp is a phase angle which approaches zero,

Lim Fc (n) = 1. (2. 19)

The analysis procedure described above which in essence removes the

ground wave from the problem, permits the practical use of zonal

harmonics techniques to frequencies as great as 300 kc/s. Of course,

the linear dependence on kx a makes the technique more efficient at

10 kc/s and lower. Below 2 kc/s the zonal harmonics can also be used

for the ground wave without any significant increase in computation time

on a computer. In the case of elevated terminals (2. 13) and (2. 14) the

convergence is improved by the factor f — J

-10-
Thus, the factor (see (2. 14) and (2. 15)),

"sib + bi b / 2 \ ^n

&1 a

= [1 +RB ] Cl

(x)

(2. 20)

if b = a (receiver only elevated). For n > k-j^a, Johler and Berry [1962]

used a more quickly convergent series based on an approximation of

Watson [1919], or using (2. 17),

(2)

(1 +Rn )(c1

2

r

) Rn + ^

q[

where,

t 1

)

exp (is

4ik£a

2n+l

exp(icp) + R

^1 r "sia

Cl

( 2

) c

(1) bl a

(2.21)

Hence, the terminals S and O on the ground require the greatest number

of terms. It is also noted that a large number of cases, say 100 distances,

can be calculated simultaneously since the distance factor or Legendre

function is merely multiplied by the frequency dependent factor, term-byterm to introduce the distance dependence.

3. Plane Reflection Coefficients

The ionosphere spherical reflection coefficient (1. 13) can be

replaced by the Fresnel reflection coefficient, which, for vertical polarization can be written,

cos ^-(tr^O

2

cos CP

t + 1 -(^sincpj

(3.1)

where [Johler and Berry, 1964]

cos qjj = l

: Ci

(")'

c

(*)

1 n ( n+ 1 ) 1 ~ —r. rrr (kig)'

(3.2)

11-
or,

Vn(n+1) _, sin CD. ~ -~r* (3. 3) 1 ki g

with the stipulation that sin cp

t = 1 if —— > 1. This latter stipulation kig

causes only a small loss in computation precision since near n = kx a the

series converges abruptly (see (4. 10) below), and the spherical reflection

coefficient reaches a value of -1. The plane reflection coefficient, Te e , exhibits the same asymptotic behavior of n->°°. The introduction of plane

Fresnel reflection coefficients permits use of available reflection

coefficients for an anisotropic plane plasma developed by Johler and

Harper [1962], Johler [1963], and Wait and Walters [1963]. Fresnel

reflection coefficients are not necessary to the isotropic case (3. 1)

except to demonstrate numerically that the procedure causes only small

loss in computation precision. The capability of using Fresnel reflection

coefficients is an important consideration in the anisotropic ionosphere,

since available reflection coefficients are restricted to the Fresnel type.

4. Inhomogeneous, Anisotropic Ionosphere

The generalization of equation (2. 3) to the inhomogeneous,

anisotropic ionosphere is similar to methods introduced in previously

published work [Johler and Berry, 1964], A consequence of the presence

o f the anisotropic ionosphere is the existence of TE (transverse electric)

propagation in addition to the TM (transverse magnetic) propagation,

notwithstanding the fact that the source is TM. Equation (2. 3) can be

re -written,

&i a &1

Let p = i ^

^LL

A2

) M)

tla tig

* = Re

, „ Pn

-12
R.

L o -l J' n L

T e e Tx e m Tme Tmm

Pn = P

o n 7 r i oi

J'

J=

R Lo iJ

Then,

(p^*.^)'"

1 ?^ = P

J [

^ C J X J y^ z

j J

(4.2)

Then it can be readily shown that,

Cx = Te e C2 = Re T^ + R T T

o e iq ee em me e © e m mm ein me (4.3)

These results (4. 3) were given by Johler [l96l]. These results (4.2)

and (4. 3) can be deduced from equations given by Johler and Berry

[l964]--see equations on page 11; (43) and (52) on pages 8 and 9.

The solution for the vertical electric field, Er

, can be written,

[Berry, 1964b],

E.

|i o c l l

8TT k^a 4

CO £

F(n) C^CxVd+Re)

M+Pn^.

n =0 \1 ~ Me.Jn

(4.4)

The ratio of the determinants in (4. 4) can be expanded in a geometric

type series by Berry [1964b]; Johler and Berry [l964]--(43) and (52),

pages 8 and 9,

-13-
I-Z + Pn^l x = \l- pA.n^l U+ Pn^nl (4-5)

i-P„^e. n ^nl

or,

1+ Pn^n

1 ~ P n *. n^n

= U + (*+*.,,) £ (Pn^n^Jpi"1 ^!. (4-6)

5=1

This, of course, leads to (4. 2). Berry [1964] pre -multiplied (p n Tn )

j

by Bl ~ which would cause the second term of C2 to read - Re i?m T e Tm e. Similarly, the higher order Cj's would be altered. Whereas this permits

removal of a factor y . from the integral (or in this paper the summation),

such that Yj = Ri _1 Cj, the procedure, although a good approximation,

does not seem to be physically real, since the second term is a TE wave

and does not require a TM reflection coefficient, Re , (see figure 2),

The expansion (4. 6) converges absolutely for all n. Thus the

expansion of the denominator (1. 16) of the left side of (4. 6) is the geometric series which converges if,

PlA <n rn | < l,

which is certainly true for all physical values of H e , p n

, and Tn, if n

is not too large. However,

Lim p n Tn = - 1,

Lim 7?

e f n = - 1

,

n —> oo

hence (4. 5),

Lim |i+ Pn Tn | = 0,

14
which multiplies into the geometric series (4. 6). Hence (4. 6) converges

absolutely for all values of interest, i. e.

Pn^e T < 1

The geometric series representation permits the introduction of

local reflection coefficients. This is illustrated by analogy to geometric

optics in figure 1. Thus, the first ionosphere wave, j = 1, has a reflection point [l, 1 ], located in the middle of the propagation path. In

the region of this point, it can be conjectured, most of the reflection

occurs. This becomes evident from an examination of the angle of

incidence, cp

t , on the ionosphere of the spherical wave [Johler and Berry,

1964], determined by

cos cpj r Rei

or

r(

2 )'

Re

n(n+ 1)

(klg) 2 (4.7)

sm

Vn(n+1)

For cp

t ~ 0, or n small, the contribution of the terms of the harmonic

series is small. As sin cp

t approaches unity, the field is determined.

Thereafter, the series converges rapidly as a consequence of the behavior

of (2. 17) previously discussed. Therefore, within a small range of

values near kx g the field is primarily determined. Similarly, the local

reflection points [2, 1], [2,2], [3,1], [3,2], [3,3], [4,1], [4,2],

[4, 3], and [4, 4] can be defined. The anisotropy and non-homogeneity,

of course, causes the calculation of Cj to be quite complicated. Thus,

instead of (4. 2), use,

11 (Pn.K^.K^.n.KKPn,^ )= fl P K [^ *<

=2 l. y i ^

i K =2 K =1

(4.8)

where for j = 1

,

(Pn.l^n,!) = Pi

Cl Xll

y, Z X J

-15
Explicit expression has been given by Johler [l96l] which can be written

Ca = T ee (l,l)

C2 = Tee (2,l)Tee (2,2)Re (2,l) + Rn (2, 1 ) Te , (2, 1 ) Tm e (2, 2)

C3 = Re (3, 2) Rffl (3, 1) Tee (3, 3) T em (3, 1) T, . (3, 2)

+ Re (3,l)Rm (3,2)T ee (3 ) l)TeB (3,2)TIBe (3,3)

+ Re (3,l)Re (3,2)T ee (3,l)T ee (3,2)T ee (3,3)

+ Rffl (3, 1) RB (3, 2) Tmffl (3, 2) T em (3, 1)

T

ffle (3, 3) (4. 9)

Since higher order C.'s are quite complicated, it is perhaps simpler

to calculate C, from the general matrix formula (4. 8). The detailed

reflection, figure 1, can also be altered by the introduction of a local

reflection height at each point [1,1], [2, 1 ], and [2, 2] • • •

. This permits

an extension of the theory to take account of irregularities of the ionosphere. The limitations of this procedure are not treated in this paper,

but the procedure is intuitively evident from the rays depicted in figure 1. It should be emphasized here that the analysis presented here does not

depend upon the geometric -optics depicted for tutorial purposes in figure 1. The solution is exact since the waves fill all of the space shown. The

significance of the geometric -optical ray limit has been discussed elsewhere, for example, Johler [1964] and Wait [1961].

The error introduced in (4. 9) by using available plane (Fresnel)

reflection coefficients for real angles of incidence, cp

t , is small compared

with errors introduced by lack of knowledge of the ionosphere model, for

example, but the error is not mathematically negligible. Some improvement can be obtained by multiplication of (4. 8) by a spherical correction

factor, CF, based on the spherical reflection coefficient for infinitely

conducting ionosphere,

These functions are, of course, available from the computer code written

for (2. 2), and (2. 3). However, the best procedure would be the introduction of the complex angle of incidence, cp

t f (see (3. 2)) into the calculation

of the Fresnel ionosphere reflection coefficients directly. Thus, cos cpt in the analysis procedure of Johler and Harper [1962] is replaced by the

complex number after equality sign in (3. 2).

-16-
5. Discussion of a Sample Calculation

Figures 2 and 3 illustrate a sample calculation using the zonal

harmonic- -geometric series double summation at 1 kc/s and 20 kc/s.

An entire calculation of this type requires approximately two minutes at

10 kc/s, (or perhaps less with optimization of the computer code) on the

electronic computer. The ground wave, j = 0, was calculated with the

aid of a standard computer code using the classical ground wave theory.

At frequencies less Lhan 2 kc/s, (2. 2) becomes efficient, and should be

employed. The final answer, E, employed 20 terms. However, graphii cal accuracy can be obtained with fewer terms. Since kx g ~ 1340, 1400

terms of the harmonic series were employed in figure 2 and 2800 in

1400

figure 3. In practice, it is usually only necessary to form E at 1 28(30 n=10QQ

kc/s and E at 20 kc/s. . . , (2. 3) since the small n terms ordinarily

n=20Q0

contribute a negligible amount to the total field. This results in approximately 1/5 reduction in computer time. It is, however, necessary to recur se from n = in (2. 8) and (2. 11) but (2. 7) can in this case be used

to replace (2. 4). The E curve and curves j = 0, 1, 2, 3 agree with

previous results of Berry [ 1964b] using the integral form. The number

of distances calculated by any summation can be increased to 100 or more

without appreciably increasing the computation time, since Pn (cos 0) is a distance multiplier for each frequency dependent term which can be

calculated with great speed using (2. 4) or (2. 7). The latter is quite

accurate, except at extremely low frequencies where small n is important or at distances, sin 6 ~ 0.

It is of interest to note the characteristic behavior of each term of

the geometric series at great distance. Thus, the attenuation with distance approaches a ground wave slope. However, since full wave type

solution is employed, this attenuation with distance does differ slightly

from the ground wave curve.

The complicated behavior of the particular case, f = 10 kc/s,

N - 56 el/cc, j = 4, 5, 6, in the region immediately preceding the

diffraction region is of particular interest. The apparent standing wave

appears to be a consequence of the fact that a full wave solution has been

employed. Thus, the geometric optics solution would yield a simple

pseudo-Brewster angle type cusp in the curve. Here, however, a multiplicity of waves exist and would be expected under some circumstance to

yield a more complicated behavior near the pseudo-Brewster angle.

17-
6. Conclusions

A new approach to the calculation of LF, VLF, ELF fields has

been demonstrated. The solution permits introduction of existing plane,

anisotropic reflection coefficients locally as a function of a real or complex angle of incidence. This permits retention of the more exact method

of calculation in which the anisotropic ionosphere reflection coefficient

matrix is left inside the summation or integration process without an

excessive increase in computer time. The method appears to be tractable

up to frequencies of approximately 300 kc/s although the efficiency is

reduced at LF and MF. The method is very efficient at VLF and ELF.

The method also can be developed to introduce local irregularities and

local electrical constant changes at both the ground and ionosphere.

The author is indebted to his colleague, R. L. Lewis, for the

design of the computer subroutines based on the recursion formulas.

The author is also indebted to L. A. Berry for the design of the

asymptotic subroutines for the Hankel functions against which the

recursion formulas were checked. Finally, the author wishes to note

the able assistance of Carlene M. Lilley with many of the details of the

zonal harmonic program design.

-1
7. References

Berry, L. A. (June, 1964a), Computation of Hankel functions, NBS

Technical Note 21 6.

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

of LF radio waves, Radio Sci. NBS/USNC-URSI 68D, No. 12,

1275-1282.

Berry, L. A. , and M. E. Chrisman (Nov. 1965), The path integrals of

LF/VLF wave hop theory, Radio Sci. J. Res. NBS 64D , 1469-1480.

Bremmer, H. (1949), Terrestrial radio waves- -theory of propagation

(Elsevier Publ. Co. , Inc. , New York, N. Y. ).

Goldstein, M. , and R. M. Thayler (1959), Recurrence techniques for

the calculation of Be ssel functions, MTAC XIII (13) , 102-108.

Johler, J. R. (Oct. 1961), On the analysis of LF ionospheric radio

propagation phenomena, J. Res. NBS 65D (Radio Prop. ), No. 5,

507-529.

Johler, J. R. (Apr. 1962), Propagation of the low -frequency radio

signal, Proc. IRE 5_0, No. 4, 404-427.

Johler, J. R. (1963), Radio wave reflections at a continuously stratified

plasma with collisions proportional to energy and" arbitrary

magnetic induction, Proc. International Conference on the

Ionosphere, 1962 (The Institute of Physics and the Physical Society,

London), 436-443.

Johler, J. R. (Jan. 1964), Concerning limitations and further corrections

to geometric -optical theory for LF, VLF propagation between the

ionosphere and the ground, Radio Sci. J. Res. NBS/USNC-URSI

68D, No. 1, 67-78.

Johler, J. R. , and L. A. Berry (Sept. -Oct. 1962), Propagation of

terrestrial radio waves of long wavelength- -theory of zonal

harmonics with improved summation techniques, J. Res. NBS

66D (Radio Prop. ), No. 6, 737-773.

Johler, J. R. , and J. D. Harper, Jr. (Jan. -Feb. 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.

-19-
Johler, J. R. , and L. A. Berry (Oct. 1964), A complete mode sum for

LF, VLF, ELF terrestrial radio wave fields, NBS Monograph 78.

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

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

Wait, J. R. (1962), Electromagnetic wave in stratified media (Pergamon

Press, Oxford).

Wait, J. R. , and L. C. Walters (Sept. -Oct. 1963), Reflection of VLF

radio waves from an inhomogeneous ionosphere, J. Res. NBS

62D (Radio Prop. ), No. 5, 519-523.

Watson. G. N. (1919), The transmission of electric waves- -round the

earth, Proc. Roy. Soc. London ACXV (95), p. 546.

20
L CM —

llliljlt

^'-?y'-}yy^y^.-'\ v'.-W

^ ^ ^

rci

0)

O, rcj CO QJ o

tio

^1O CO

£ .2

^ O

>^

cti

,—

i

u

COO

•H

CJD cu

CJ

+3 GO ° £

o

Oh

co

CDU

o

o .2 M n7

GO O

GO rC! O 0) n

*

CU ^

X ° O a;

II TJ "I O

u

GO

21-
2000 4000 6000 8000

DISTANCE, km

10,000 12,000 14,000

Figure 2. Illustrating components of the rigorous geometric series

j = 0, 1, 2, 3. . . together with complete field, 2, for terres

trial waveguide propagation at 10 kc/s. i

-22-
9

3

n-13

—

\

1 1 1 1

— p —

— —

= \ \\ 2 \ \ \ - / ' —

zz— \ ^si~ 3 ^^

-=

— I i-l ^.,= 4 —

——

—

— 1=1 — 1

—

— fj = 3 - —

i=2 —

=—

j = 4 f = 20 kc/s

—=

—

0- = 0.005 , e 2

= 15

h = 675km, N=56 ELECTRONS /cm 3 — — v = 1.6 (I0 7

) 20

X = I —

=

J j = l

I J =1 AMPERE- METER

j=0 — — —

1 1 1

i 2000 4000 6000 10,000

DISTANCE, km

Figure 3. Illustrating components of the rigorous geometric series

j = 0, 1, 2, 3. together with complete field, Z, for terres

trial waveguide propagation at 20 kc/s.

GPO B57-896

23


U.S. DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20230

OFFICIAL BUSINESS

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCI