national Bureau of Standards

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OCT 1 3 1964

NBS MONOGRAPH 78

A Complete Mode Sum for LF, VLF, ELF

Terrestrial Radio Wave Fields

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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UNITED STATES DEPARTMENT OF COMMERCE • Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

A Complete Mode Sum for LF, VLF, ELF

Terrestrial Radio Wave Fields

J. Ralph Johler and Leslie A. Berry

Central Radio Propagation Laboratory- National Bureau of Standards

Boulder, Colorado

National Bureau of Standards Monograph 78

Issued October 1, 1964

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402 - Price 20 cents
Contents

Page 1. Introduction 1

2. Theoretical solutions 2

2.1. Isotropic model ionosphere 4

2.2. Anisotropic solution with constant radial magnetic field 5

2.3. Reflection coefficients 8

3. Computational methods 12

4. Discussion 13

5. Conclusions 18

6. References 19

7. Appendix 20

Library of Congress Catalog Card Number: 64-60055
A Complete Mode Sum for

LF, VLF, ELF Terrestrial Radio Wave Fields'

J. Ralph Johler and Leslie A. Berry

Contribution From the Central Radio Propagation Laboratory,

National Bureau of Standards, Boulder, Colo.

Previous papers about VLF, ELF propagation in the earth-ionosphere waveguide have emphasized

the characteristics of the individual modes. In this paper, emphasis is on the total field found by sum- ming all modes vt^hich contribute appreciably. Solutions are given and calculations are made for a homogeneous isotropic ionosphere, a homogeneous ionosphere with a constant radial magnetic field imposed, and an ionosphere with a magnetic field of arbitrary dip angle. In the isotropic case, one mode is sufficient to describe the field at ELF, but at the upper end of the VLF band, two or more modes may propagate 7500 km. When a magnetic field is imposed on the ionosphere, three abnormal modes or fields are generated at the anisotropic boundary. These abnor- mal modes have little effect on the total field in the VLF band, but they change the ELF field appreciably. Indeed, in the absorption region between VLF and ELF, the abnormal modes may become

dominant beyond 1000 km.

1. Introduction

The object of this analysis is a more accurate description of the field of the LF, VLF, ELF radio wave in the space between the ionosphere and the terrestrial sphere. A number of investigators

have devised theoretical methods to this end; indeed, considerable emphasis has been placed on

the development of field-strength curves, both as a function of distance and frequency. Wait and

Murphy [1957] pubhshed field-strength curves computed with the aid of the geometric-optical

theory, applicable to short distances, and Wait [1957, 1962] and Wait and Carter [1960] developed

curves with the aid of the residue methods, or "mode theory," which is a full-wave solution for an assumed model, but employed the Debye [1910] approximation for the spherical-wave function.

This approximation was used previously by March [1912] and von Rybczynski [1913] for the

groundwave problem and is, as pointed out by March [1912], not valid in areas corresponding to the

most important terms in the residue series at frequencies greater than 8 or 10 kc/s. It is interest- ing to note that von Rybczynski [1913] analyzed and corrected March's [1912] paper in the use of

this approximation. Budden [1961a] considered the fuU-wave solutions or "mode characteristics"

assuming a flat model for the terrestrial sphere and ionosphere, with the imphcation that this procedure simphfied the analysis. Alpert [1961] finds the mode sum for a flat waveguide and then

multiplies by a geometric factor to correct for sphericity. Wait [1961] computed the attenuation

and phase velocity of the least attenuated residue for certain specified models, including in the

solution some of the effects of anisotropy, and Wait and Spies [1963] considered the amplitude of

the "excitation factor" of the first two residues with a perfectly conducting earth and ionosphere

and of the first residue for a finitely conducting ionosphere.

The possible multimode character of the guide has been noted by Wait [1962a, b]; however,

since large scale computer techniques have not been available, httle has been done to sum aU

modes. Indeed, at distances less than about 3000 km, it is not sufficient to consider only a single

such mode, if the frequency is greater than about 15 kc/s, for a daytime-noon ionosphere model

employed by Johler and Berry [1962, 1963] or at frequencies greater than about 4 or 5 kc/s for a corresponding nighttime model ionosphere. Johler and Berry [1962, 1963] computed LF, VLF,

ELF fields by summing the classical series of zonal harmonics and by summing the residues for

' This paper was presented at the VLF Symposiutn, Boulder, Colo., on August 12, 1963.

1
the contour integral of the Watson transformation (called "modes" by Wait [1962a, b] and by

others). The present paper extends the previous work of Johler and Berry [1962] to take account

of anisotropy. Curves are presented for the field which include all residues or modes of signifi- cance to the computations.

2. Theoretical Solutions

The electromagnetic propagation of radio waves in an anistropic medium is described by

Maxwell's classical equations. For the continuous time-harmonic solution, with a frequency,

—

d ^ . d . E — i(oE, — H = i(oH,

at dt V xE + fjLo i(oH = 0

V X H - [e] • ^=

0, (1) —> —> where E and H are the electric and magnetic intensities (rationalized mks units) and /jlq, to are the

permeabiUty and permittivity (€o= /-to where c is the speed of light) of space and [e] is the di- electric tensor of the anisotropic medium. In the isotropic, nonconducting case.

[e] = eico = eoito in free space. (2)

The dielectric tensor in an anistropic medium, employing the Cartesian coordinate systems,

X, y, z, figure 1, can be written [Johler and Harper, 1962; Johler, 1963a, 1963b]

v + i (iu + iu„) v+ i (uu - iDh)

a- sin I ^

-g- sm I V+i(U)+UUh) v+i(uj U)h)

5- COS I COS I v+i(ii)+uu„) v+i(uu-u)„)

[e] = eo sini

v + i (iu + (Uh)

sin I sin'^ I v+ i (01 - UJ„)

cos^ I v+ id) ' v+i((u + ai„) ' v+i(ii)-uj„ i sin^ I sin I cos I h sin I cos I h sin I cos I + ^ + i v+iu) v + i(iju+uj„) v+i(uu-uuH)

V + i ( U) + uUh ) — cos I v + i (u; - uj„)

sin I cos I h sin I cos I h sin I cos I + £ + i v + itu v+i(ou+tu„) v+i(uu-uj„) i cos^ I v + iuu ^ V + i{iij + uUh) ^ v+i(iD-ujH) sin^ I g cos^ I

The motion of the electron is described by the Langevin equation,

mio^+ mgV+ eiVx 1b) + eE = 0 (4)

where the complex, frequency-dependent parameter, g, is assumed to be a constant, g = v [Johler,

1963a] where^ Wh = fJ-o ej^lm, the gyrofrequency for electron charge, e, mass, m, and the magnetic

field, J^— or the magnetic induction B = (Xo^. 0)'^= Ne'/eom, the angular plasma frequency

squared, / is the magneti^ dip measured from the horizontal, A^is the electron density of the plasma

and the magnetic field, jCis assumed to be oriented in the y-z plane.

If a plane wave,

(5)

where

E = A exp iu)t .(O

T7D = x sin (/), sin + y sin </), cos (/)„ + 2^

2
Figure 1. Cartesian coordinate system and spherical geometry.

is assumed as a solution of Maxwell's equations in such a medium, where (f)i is the angle of incidence

relative to the vertical z-direction, and is the magnetic azimuth measured clockwise from the

y-z plane containing the magnetic vector, with a dip angle / from the horizontal, a quartic in the

complex number, ^, is deduced [Johler and Walters, 1960; Booker, 1939]

a4i^^ + a3^^ + a2i^' + arC'+ao = 0. (6)

The coefficients of this quartic are given in appendix 1, and a complex index of refraction, iq, of the

medium is defined,

Equation (6) is analogous to Booker's [1939] quartic in classical magneto-ionic theory, because

Booker first attached physical significance to the four roots as: ordinary (o) and extraordinary (e), upgoing (i) and downgoing (r) propagation components, 17 = 17^,' p. The downgoing components

can arise from boundaries (rnore than one) or electron density gradients in the z-direction. Thus,

in the homogeneous medium, only upgoing ordinary and extraordinary propagation components

exist.Attempts to solve Maxwell's equations in an anisotropic medium in coordinate systems other

than Cartesian, such as spherical, r, 6, (f), figure 1, coordinate systems lead to intractable equations by conventional methods. However, for propagation theory in the space between the plasma

and the terrestrial sphere, approximate solutions can be worked out and tested, which task is an

objective of this analysis.

The spherical geometry of the model propagation media is shown in figure 1, together with

the transmitter, receiver, and boundaries. The origin of a spherical coordinate system (r, 6, (f)) is the center of a homogeneous terrestrial sphere with a radius, a, and a wave number,

, at /

. o-A^qC^ ^. = -^e.-.— (8)

where €2 is the dielectric constant relative to a vacuum, or the permittivity, 6 = 6260 and cr is the

conductivity (ohms "Vmeter) of the terrestrial sphere. The space between the ionosphere and

the terrestrial sphere is represented electrically by the wave number,

^i=-T7. (9) c

where tji is the index of refraction (tji 1 for vacuum, rji ~ 1.000338 for air at the surface of the

earth).
A vertical electric Hertzian (electrical point source) dipole is located at 9 = 0, r = 6(a < b < d), and the field is to be calculated at any point (r, 6, cj)), although the specializations in this analysis

confine the computations to the space {a < r < d). Azimuthal symmetry will be assumed

2. 1. Isotropic Model Ionosphere

When the earth's magnetic field is neglected, the ionosphere can be characterized by the

isotropic wave number

0}{v + ioj) (10)

The solution for this case is well known [Watson, 1918]

The vertical electric field, in particular, can be written, using the Watson transformation,^

477 hi

where P„{cos B) is the Legendre function.

t;5 — -)(Vs)Pt-s--i/2(— cos d) Avs-ii2

cos VsTT A- 1/2]

J V

-

(11)

^Az) = \l-Y Ji + \l-i (z)-

(12)

(13)

H[!\^l2\s the Hankel function of ith kind, order f + |, and argument, z, and A+ 1/2 is Bessel's function. i}l\z) represents an outgoing, radial elementary spherical wave and l,[}\z) represents an

ingoing, radial, elementary spherical wave, and ^n{z) = \ [^*„^'(2) + ^'f(2)]

1 i^fihb) C,!'(kia) I}l\k,r) Cl\ha)-C\!ik^r) ^*f)(^,a) 0

An= (14) 0 i'^i^k.d) C'Jikrb) ClHkrr) ^<,V(A:,cO-^(;)(^,r) C^l\k,d} 1

0 ^!,'^,ik,d)Cl\k^b) i^^^{k,r) ^,iL\(^i^{';,'(A-ir) ^f_\(A,rf) |p«+^

Rn = «//,; ik2a)/ilj„{k2a). (15)

where

and

2 For purposes of graphical presentation in this paper, /o/ = 47r/jUflC = 3.34 (10"-) ampere meters.

4
Finally

D„ =

1 h J.

, n

-r-Kn-r-.

0

0

iha) 0

0

1

ki_

-\- — ks ^" kid

(17)

The zeros, Vs, of D1-112 = 0, are ordered or numbered so that | Im (vm) | < | Im {v„) | if m < n, so the residue series is convergent, since Im (vs) < 0 and

Pv,-ll2{—C0S 6) cos VsTT

Vs sin Q,

1/2exp [—i(vs)0]. (18)

provided Ivsl > > 1 and d is not near 0 or tt. Therefore, Re {vs) is related to the phase of a single

residue, 5, and the attenuation rate is [Attenuation] « = 1.364 \ Im {vs) \ dB/1000 km.

" ^ \li)(Vs)Av-ll2. ,

. . r r 1• i t-i i

• The quantity IS the excitation lactor 01 the residue 5. 1 hus this quantity

dv

D,.

weights the significance of the residue.

The expression (11) is exact except for a line integral

2k

i f'^ v{v'-i}P,-il2{-cos 6>) COS VTT

1/2

1/2

(2) (1) dv{r^b), (19)

which is zero if only /?[ 1/2 is an even function of v [Berry, 1964a]

This requirement (R1—112 even) is much less stringent than the conditions given by Wait [1963b,

1962b], i.e., that the impedance boundary conditions are exact at the ionosphere and the earth or

that \k2\ »ki and |A;3| >> ^1. Since (11) without the integral becomes exact as the earth's conductivity approaches infinity, the integral represents the field in the medium a < r < (/(the space

between the ionosphere and the terrestrial sphere) resulting from waves which have escaped after traveling through the medium r < a (terrestrial sphere). The neglect of this integral (19) in ter- restrial radio wave propagation is therefore a very accurate approximation, since waves traveling

in the ground are very highly attenuated before again emerging into the space a<r<d.

2. 2. Anisotropic Solution with Constant Radial Magnetic Field

A constant, radial magnetic field (/ = =h90°, (3)) is now imposed on the ionosphere so that it is no longer isotropic. The permittivity tensor, (3), then becomes,

[€\ = €olOJ

1-

0

1- s

where

5 = -col

(20)

(1) (21)

5
and where the upper or lower sign (±, T) is taken for a dip angle / — — 90 or +90°, respectively

(the dip or inclination angle is measured from the horizontal by convention). Wait [1963a] de- rived a mode equation for this model (vertical magnetic field, / = + 90°, i.e., directed in the — f

direction). Following Wait's procedure but writing the solution first as a series of zonal harmonics,

then making the Watson transformation, the Hertz-Debye potential is found in the space a< r< d. The vertical electric field (11) now becomes

77 {vl — DvsPvs -i/2(— cos 6) Bvs -1/2 cos VsTT (22)

where Bv and Ev are 6X6 determinants.

4^^ 0 0 0 0

Ci'Kkrd) -1 -1 0 0

C^Uk^d) C[}l,{k^d} ^0 + ^20:0] Ci[Ue + C20C ^^-id' 0

0 0

'«o V_~

ki kid_

Ue V

ki kid_

micid) C'^Ukid)

0 0 — tto — ae C[!Kkrd} C['\kid}

0 0 0 0 b[}>

^1 ^ kia

k^ kia i^f(ha)-i['^_\{ha).

Ci =-—, {ei = T]\) and G =ga>.

Zg is the ground impedance,

Zg = Z(t

and Yg, the ground admittance,

^0

1/2

cr + i(x>eQ€i

cr + iweoCo 1/2

"

(T + ia)eoe2

1-

cr + ia;eoe2

1/2

1/2

(23)

(24)

(25)

(26)

(27)

in which Zo ~ 12077 ohms is the impedance of space, uq and Ue correspond to the ordinary and

extraordinary waves, either magnetic or electrical solutions of an assumed planar form in the

ionosphere: A exp [— Uo, p^], where A is an appropriate constant. By analogy with the Booker

[1939] quartic, (6), uq and Ue are solutions of

+ {s\ + 5i)u2 + s\si - si = 0, (28)

such that Re («o, e) > 0 (upgoing propagation components), where

6
si = k'i--X\ (29)

to

5| = J(fJ-X2^, (30)

~d^- ^^^^

ao,. = — (32)

OigUo,e

(35)

Equation (28) is comparable to (6), the Booker quartic, when 7 = 90°, ipa = 0. The first two columns of Bv are,

- ClKk,d)C!Mk^b) i^lKk^r)C^l\k^d) - C['Kk^r)Cf{k^d)

-(,^f-,ikid)l'lKk^b) C^«(^ir)^<iV'^i^-^';,'(^iOC'!\(A:i(i)

0 0

0 0

0 0 (36)

and columns 3 through 6 are the same as the corresponding columns of Ec, (23). Equations (24) and (25) result from the use of the impedance and admittance boundary conditions [Wait, 1963a] ,^ He = YgE>p

Ee = ~ZgH^ r=a. (37)

If the usual boundary conditions are used, and Bv are 8X8 determinants.

Hr, He, and E^ are no longer zero, since a weak TE field is excited by the corresponding TMfield (originally excited by the vertical electric source current) at the boundary of the anisotropic

ionosphere. If the magnetic field is zero, (22) reduces to (11) if only

and

p,.-x/.--.Vl-(^) (38)

k^ ia)€oZg = ij^^. (39)

Equations (38) and (39) are good approximations for practical cases at VLF.

'' For more details see Wait [1963a]. Note that the function A'^'U) defined by Wait is identical to $'^'(z). The latter notation is used here since it has over 50

years of historical precedence [March. 1912] and has been used by other more recent authors, e.g., Fok [ 1946] and Northover [1962].

7
2. 3. Reflection Coefficients

Reflection coefficients for a plane ionosphere with plane wave incidence have been treated

by Johler and Walters [1960], Johler and Harper [1962, 1963], and Johler [1963a]. A method

is now discussed in which these Fresnel type reflection coefficients are introduced into the spherical wave solutions of Maxwell's equations. The x-y plane becomes the surface of the ionosphere;

y corresponds to 6, figure 1, z to r and x to — 0. The dielectric tensor (3) is employed to deduce a

reflection process for waves in the guide (a < r < d). Dimensionless field ratios are defined, Johler and Berry [1962],

no = ^s— 0' =^ ^ = —

no ^ _ £j: ^ £^ ne , /r(e) r(.e)

(40)

(41)

where o and e refer to ordinary and extraordinary, respectively. These ratios are functions of the

angle of incidence and the connecting relationship is

&m<pi = (n + y2)lkid. (42)

The fields which satisfy Maxwell's equations (1) in the guide {a < r < d), can now be written,

(a<b<d) V ki l^,i{kib)Qn'{kir)

Hr = S [en &'ikir) +f„ il,„ik,r)]

I, kxjJLQlO)

'-

E6 = 2.- w.'v^ +6„dr'('^^i'-) + c«0„(/:ir)

» ki lilfuikib) C>P (^ir)

He = I.^[e„ i^ik^r) +/„ ,^,;(A,r)]

r<b

r>b

r< b r> b

n "^1

B

n fJiolCO

X^Vikrb) iPnikir) iPnik.b) ^\fHk,r)

+ bnCn'if^ir) + Cnlpn{kir)

r<b

r>b,

(44)

{a<b<d) (45)

(46)

(47)

(a<b<d) (48)

where

A =

B =

n(n + l)(2n + l)P„(cos 0)

f'b-'

sin 6')(2n+l)P'(cos 0)

hl/J-oC

rb'

477-

hlfJ^oC

For the waves in the ground, k\ is replaced by k> (8), source terms containing 6„ are zero, and

in'KkiO) waves also vanish. The constant c« is replaced by a,, and h,„ the boundary constants

for waves in the ground for the electrical and magnetic solution, respectively. Waves in the

ionosphere are written, as the product of a field component F and a boundary constant.

for ordinary and extraordinary components. Similarly,

8
H^,n = dnF^^} (49)

In the ionosphere the known solutions of the field ratios (40) and (41) are introduced to yield boundary conditions which satisfy all tangential field types which can exist at the boundary,

£"61. 1 = Ee, 3

H<p, I — H^, 3

He. : = Qe.nH,. 3 = ^<W.'s + Qi'rlH!^:, • (50)

These boundary conditions can be written in matrix form using the abbreviations = [}^\kia).

^2 ^1

^1.

4>ia

k,

(//id

k,

k,

^1

^id

k,

A. k-z

a,i

bn

Cn (— l/'lft) l\d

dn,o

ki d n,e

en

1 fn

1 hn

In the guide (a<r<d) the fields which uniquely satisfy Maxwell's equations (1) for the specification of the sources, can now be determined with the aid of the boundary constants, which can

be written

^U, + bn = (^W - R%U)(l - RUlrmWn

Cn = (1//16 - RlWi- n,a - R%T%,J - RmTl^TUDn

en^{4'ib-RlCib)R',„TlJDn

fn = (llflO + Rlill>)njDn

where,

Dn (1 /?|7'|p)(l R%,T%„^) R^Rf,^T^„^Tf,^p

or in matrix notation, D„ is determinant of the matrix

[1]

7?^, 0

0 /?? •Ts Ts

' me ^ mmJ

(52)

(53)

9
The spherical type reflection coefficients are,

C^Kk^a) k^ iSmha) + Rn — 1 ^ ^ 1 ijj'nikia) ^ ifjnikia) ki "

k-z ^n(kia)

^1 " -^2

Ts

^ e

^ Hi/

ipnikid)

xjjnikid)

ifjnikid)

^nikid)

where, for the case ipa = 0 in plane wave, plane ionosphere solution.

(54)

(55)

1

C(Cn)

0

0

-1

mnmVpm

_0(o)

-1

me

0

0

1

(56)

1

Ciiljjn)

0

0

0(0)

I • /Vme

' Vem

-1

I • /Vme

' Vem

-1

-Q(e) Vme

1

Ci(C«)

0

0

(57)

Tmm ^

1

C,(i//„)

0

0

I • /Vme

' Vem

-1

)(0) 'me

)(e) Ve

irjeV^I^'

)(e) me

0

0

C,(^«)

1

(58)

and,

rme = AA =

0

0

CiiCn)

1

1

Ciiljjn)

0

0

I • /V me

' Vem

-1

Vme

Q(0)

I • /Vme

' Vem

-1

)(0) 'me

+ iT?e

(e) em

Vme

+ tT7e

)(e) 'me n(e)

Ve»n

)(e) 'me

0

0

Ciiljjn) 1

0

0

C#n)

1

(59)

10
Ciil^n) = l

, ~ COS 4>i

cos 4>i {ktd}^

(60)

(61)

and in the Watson transformation, n = fs-1/2. A vertical magnetic dipole or an elementary line magnetic current, Iml, volts-meters, is equivalent to an elementary current loop loO, ampere-(meters)^, w^here O is the area of the loop in square

meters. Iml^ /J-oifoIoO.

The magnetic dipole source solution can be obtained from (43) to (48) by moving the source functions ^ib^tr, ^ib^i'r , or ^ir'/'ift, ^I'r i/'i6 from (43), (45), and (48) to (44), (46), and (47) respectively, and

replacing lol by ikJoO. The fields propagated by such a vertical magnetic dipole are also equivalent to the fields propagated in the equatorial plane of a horizontal electric dipole.

The Watson transformation can be introduced to facilitate computation, which for the particular case of the vertical electric, Er, field can be written.

4-77 lol -Itt

r[ cos {vsTt)

\'s-l/2(— cos 0) X

Ks-idl^s-U-iikir) + c[,^-ii2^vs-il2(kir) (62)

dv

D1.-1I2

where 6'. = Dibv, c'v = CiDv, and the poles v — Vg are the roots of the mode equation (53),

Dvs-ll2 = 0. Computations can be made more convenient by introducing the exact relationship

Wz) = He'W + C'(2)]

(63)

(64)

into (51). The result is that ijjnix) functions are replaced by ^*^'(^) functions in D„ and in the defi- nitions of and (54), (55), and the quantity in square brackets in (43) is (assuming r = a = b)

y(2) — ffs rwri^) in —Ts j?s vi Ts \ j's Ts jfs n-1

>-(l) e / felaSla I V-^ ^ mm^^mJ\^ y{2) ee> >.(2) -* em-* me"-m / '^n

which agrees with the result obtained by Wait [1963b] using impedance boundary conditions

instead of (50). Thus, the impedance boundary conditions [Wait, 1963b] are.

'Ee Zu -Zri He

.-£'<*. _ Z21 Z22.

Z\2

E,_ Z2^ Z22 _ where the impedance Zy must be expressed with reflection coefficients and the cosine of the angle

of incidence Ct,

Zn = — 2 TmeZolD

Zi2= — CiZo[(l — Tmm)ii ~ Tee)~ TemTme]ID

Z21 =Zo[(l + 7'mm)(l + Tee) — TemTmeVDCi

Z22— 2 TemZo/D

where D — {1 — Tmm){i + Tee) + TemTme- It is, therefore, concluded that the impedance boundary

conditions and the conditions stated (50) are equivalent.

The analysis leading to (43) to (48) and (62) is clear for propagation in the guide [a < r < d)

subject to some restrictions, such as ^= 0, 4>a = 0, 180°, / — ±90°. For cases 7^ 0, (ha^O,

o(p d(p

11
180°, I ¥=± 90°, the approximation accuracy can be tested by a summation of the spherical har- monics [Wait, 1963b] in the variable </>.

3. Computational Methods

The solutions in section 2 were programmed for a large-scale digital computer. This allows accurate computation of the functions involved, makes summation of all significant modes feasible,

and provides for study of the field variation with any parameter.

The functions are computed using (12) and [Fok, 1946]

where ^^^(tanh w — w), and cosh w = vlz. The approximation for H[!'Hz) involving the Airy integral function, Wk{z), which was introduced by Fok [1946], is the first term of a series in ^1—

representing the right side of (65), so it is equivalent to (65) when v = z. However, (65) is uniform

over a wide range of v and z. Berry [1964b] gives detailed instructions for computing (65) and

finds the magnitude of the relative error is about 10"^/|z|.

Taking the derivative with respect to order, one finds that A . ^ exp [(- 1)^-+ Vi/6]

+ iw exp [(- l)''+'27Til3]m% (- iO (66)

The determinates are computed numerically by reducing them to equivalent triangular de- terminates using the Crout forward solution and then taking the product of the diagonal elements

[Hildebrand, 1946]. Full advantage is taken of the zeroes in the corners. Let Dniv) be an n X n determinate whose elements are functions of v, and let D^'\v) be the n X n

determinate which is identical to Dniv) except that the elements of the tth row (or column) are replaced by their derivatives with respect to v. Then ^Dniv) = f^D^i^ ivy (67) ^ i=l

Equation (67) is used to evaluate the denominator of the excitation factor, and in (68) below.

Given a A:th approximation, 1;^', to a zero, Vs of Dv-nt, a (A:+ 1) approximation is obtained with

Newton's iteration [Hildebrand, 1956]

,

+ 1) =: „(/c) !^J/2_

dv "s-"^ The iteration converges quadraticaUy to Vs if vi'^'Hs sufficiently close to Vs. Initial approximations to the Vs are obtained several ways. Contours of \Dv-ii2\ can be drawn

in the complex f-plane and the zeros approximated graphically. This is the most foolproof way of

insuring that all significant zeros are found, but is slow, tedious, and expensive.

For frequencies less than about 15 kc/s, kia is an adequate first guess for fi, and first guesses

for higher-order modes for the isotropic case can be obtained from analytic expressions given by

Wait [1957].

For frequencies above 15 kc/s, a first guess of kia frequently converges to v-z- To find the zeros in the anisotropic case the zeros for the corresponding isotropic case are found for both vertically and horizontally polarized source dipoles (TM and TE modes). These are

12
usually adequate first approximations for the anisotropic case, as shown in table 1, which gives the zeros for /= 10 kc/s, h = 70km, N = 630, and v = W.

Table 1

• Isotropic TM modes

Anisotropic modesIsotropic TE modes

1 331 K.Ci a (\A .' lOol.oy — o.U'+ I 1313.68-2.14 / 1313.17-1.79 i'

1260.09-22.49 / 1262.34-21.4 i 1222.85-8.64 i 1220.62-7.07 /

1079.42-48.77 / 1077.35-49.5 i' 10.56.52-23.9 i 1048. .54- 18.43 i

760.67-70.60 / 739.34-83.7 i

If the Vs are known for some set of parameters, they can be used as first approximations for a case in which one (or more) of the parameters has been changed by a small increment. The parameter(s) can be incremented again and again until the desired set is reached. When this is done

with only one parameter, the result is the variation of the field with that parameter, which is often

of interest. (See, for example, figs. 2, 3, and 11.) The Legendre function, Pv(— cos d} = Pv[cos (tt — 0)] is computed from [Fok, 1946]

Pv(cos 6) =

r(t;+l)

Y{v + 3/2) V27r sin d

[ e'-^G, + :/2( 1 - x) + e

-

+ i/2(x)]

(69)

irhere

and

4) = e{,v + h)-TTi^,x =

le" 2 sin 0

'

x +

p.32

221!(i;+l) 242!(2;+l)(i; + 2) +

p.32 (2n-l)2

2^"nliv+l)(v + 2) . . . {v + n

-x" +

(70) The series dix) converges if v is not a negative integer and |:x:|<l, i.e., 3O°<0<15O°. If 6 is outside

this interval, (69) is an asymptotic representation. If only the first term of (70) is used, (69) reduces

to (18), and the relative error is one percent or less if \v sin d\ > 10. Equation (69) is adequate at VLF for all points more than 50 km from the source and antipode and can be used down to 15 km

at 30 kc/s.

When 6 is near tt [Magnus and Oberhettinger, 1949],

P.(-cos e) = Jo(cl>) + sm' +

Acf>) O sin'' 2

(71)

where 0 = (2v + 1) sin source.

TT-e

Equation (71) is useful for studying fields near the antipode of the

4. Discussion

The computer program described in section 3 was used to compute the field for a variety of

parameters. Three models of the lower ionosphere (sharply bounded type) were used extensively:

Model I: h = 67.5 km, N = 56 electrons/cm^, v=1.6{ W);

Model II: /i = 75km,/V=160electrons/cm3, i/ = 4.5(10«); and

Model III: A = 70 km, A^ = 630 electrons/cm^, 10^.

Model III is a model used by Wait [1957] (corresponding to (o, = 2(10'') in Wait paper). In his early

work, Wait [1957] used the Debye approximation [Debye, 1910; March, 1912]

13
for all the values of v and z. Equation (72) is inadequate at frequencies above approximately 10 kc/s

for the first or most important mode (using Wait's terminology). It is interesting to note that the

inadequacy of this approximation was discussed by March [1912] for sin c^, =—^— = 1, yet he used

the approximation. Also, von Rybczynski [1913] noted the more accurate Hankel approximation (65) and used this as a basis for more accurate corrections to March's [1912] wrork. However,

for z large and v small or v large and z small the approximation is quite excellent. The use of

the approximation (72) is imphed by the use of plane reflection coefficients at the ionosphere

m)ii- ^ ^' ^^^^ However, in this case it is a comparatively simple matter to

replace cos 0; with the" functions, Cn'i^id)ICn{kid), instead of the approximation real cos or complex cos 0; = , / 1 _ /

' V \ kid J More recently, Wait [1961] and Spies and Wait [1961] used an approximation containing

Airy integrals which is nearly equivalent to (65). The characteristics of the least attenuated mode

were calculated. This is quite accurate [Berry, 1964a]. However, one mode is not adequate to de- scribe the complete field which satisfies Maxwell's equations at either intermediate or great dis- tances and higher frequencies, figures 2, 4, 5, and 6, for daytime models of the lower ionosphere

(highly absorbing models such as model I). In the case of a nighttime model even more modes

are needed at all distances, figure 2. Since the prime objective of the analysis of this paper is to find the field indicated by Maxwell's equations, and develop accurate techniques to calculate such

a field rather than attempt to study the physical significance of modes, all modes of significance

to the calculation were summed in an ordered series. The attenuation rate of the more important modes as a function of the electron density, A'^, in the homogeneous, isotropic model of the ionosphere is shown, figure 2, for 25, 10, 4, and 1 kc/s.

Notice that, for a fixed frequency, a given attenuation rate can be found for as many as three dif- ferent values of N, whose magnitudes are very roughly in the proportion 1:10:10^. This explains

why early investigators [Watson, 1918, 1919; Bremmer, 1949] deduced a high conductivity for the ionosphere, and why values of A'^ as different as those in models I and III have been considered more recently.

At VLF, an increase in electron density can cause an increase in attenuation rate for model I (daytime-noon model) while at night the attenuation rate is decreased, figure 2. The broad, nearly

flat minimum in the attenuation curves, together with the insensitivity to a magnetic field pre- viously mentioned, helps explain the observed stable characteristics of VLF propagation in the

vicinity of 10 kc/s and indicates that accurate determination of ionospheric characteristics using

VLF radio waves may require sophisticated techniques. The attenuation rate seems to be most

sensitive to the assumed height of the ionosphere, figure 3. This insensitivity to the detailed characteristics of the lower ionosphere suggests use of higher frequencies, such as LF (100 kc/s) which

are more sensitive to ionosphere changes rather than VLF (10 kc/s), to study experimentally and

theoretically the detailed characteristics of the lower ionosphere.

Figures 4, 5, and 6 illustrate the amplitude of the field and the phase correction or phase lag,

(pc=| ^TgEr \ —kiad, of the vertical electric field as a function of distance from the source, for

30, 20, 6, 1, and 0.1 kc/s, for three models of the ionosphere. At 30 kc/s the field shows large

oscillations, even at 7000 km. The attenuation of the second mode is nearly as small as that of

the first mode at this frequency, while the excitation factor of the first mode is Vs to Vio that of

the second mode. Thus, the two modes interfere with each other, even at great distances.

The decrease in the excitation factor of the first mode (s— 1) with increasing frequency was noted

by Wait [1962b].

At the point at which the curve for the amplitude of the field as a function of distance shows

14
100

100

1 r

f = 10 kc/s

CT = 0.005, «2 = I5

h, km

Figure 3. Attenuation rate of first three modes as a

function of altitude, h, of model ionosphere boundary

at 10 kcjs.

15
"0 1000 2000 3000 4000 5000 6000 7000 DISTANCE, km

Figure 4. Amplitude and phase of the vertical electric

field, Er, as a function of distance for model I.

0 1000 2000 3000 4000 5000 6000 7000

DISTANCE, km

Figure 6. Amplitude and phase of the vertical electric

field, Er, as a function of distance for model 111.

10"1-9.

10-12'

0 1000 2000 3000 4O00 5000 6000 7000 DISTANCE, km

Figure 5. Amplitude and phase of the vertical electric

field, Ef, as a function ofdistance for model II.

10-15

10

8

6

4

2

0

-2

-4

-6

-8

-10,

1 1 1 1 1 1 1

1

1 1

\—1—1 1 [ 1

o-= 0.005, £2 = 15 h = 67.5 km

N = 56 , !/ = 1.6 (107) ISOTROPIC

« —.- « VERTICAL MAGNETIC FIELD /

M 1

\ /

\\ \ /

\\

\\

\\ 1

1 1 1 II 1 1 1 1 1 1 ,1

I 1 1 1 1 1 1 1

1

1

1 ' 1 I

r_2000yi}--—

1 1 1

i

16

0.1 0.2 0.3 0.5 0.7 I 2 3 5 7 10 20 30 FREQUENCY , kc/s Figure 7. Amplitude and phase correction of the vertical

electric field as afunction offrequencyfor model I.
Figure 8. Amplitude and phase correction of the vertical FIGURE 9. Amplitude and phase correction of the vertical

electric field as a function offrequencyfor model II. electric field as a function offrequencyfor model III.

a sharp cusp, there is an abrupt change of approximately tt radians in the phase correction, (f)c. This occurs in the 20 kc/s case, figure 4, at approximately 800 km, and in the 30 kc/s case, figure 6, at approximately 5300 km. The cusps occur when the first two modes are exactly out of phase

and have almost the same amplitude.

Figures 7, 8, and 9 show the amplitude of the field and the phase correction as a function of

frequency at 2000 and 4000 km for the three models of the lower ionosphere both with and without

a vertical magnetic field. All three models show an absorption band, the position of which can

be changed by altering the assumed electron density, for example. The presence of the vertical

magnetic field has a small effect in model I, figure 7, and very little effect at VLF in all models.

However, figures 8 and 9 show a decrease in the field at ELF when the vertical magnetic field is present. Wait and Carter [1960] noted this effect and stated that it would occur when the col- Hsion frequency, v, is comparable with the gyrofrequency of the electron gas. Model II, figure 8, most nearly satisfies such a requirement and indeed shows the most decrease in the amplitude of

the field. The presence of the static magnetic field tends to increase the amplitude of the field

at VLF, figures 7, 8, and 9. Also, in figures 8 and 9, the magnetic field causes a "step" in the

amphtude curve as the amplitude of the field decreases into the absorption region from higher

frequencies, and the minimum field strength occurs at a lower frequency. The "step" in the curve is caused by interference between the first TE-mode (transverse electric) and the first TMmode (transverse magnetic). Ordinarily, the excitation of the first TE-mode is approximately

0.01 that of the first TM-mode. Physically, this is understandable, because the TM- or normal

modes are excited by the vertical dipole sources, while the TE- or abnormal modes are coupled

in at the anisotropic ionosphere boundary. However, at about 3 or 4 kc/s in models II and III, figures 8 and 9, the excitation of the first TE-mode was comparable with that of the first TM-mode,

17
and its attenuation rate was greater, so that they interfered with each other. Figure 10 illustrates the amphtude of the field as a function of distance for two such cases. The change in the slope

of the curve clearly indicates the change in normal or abnormal mode dominance.

The effect of arbitrary dip or incUnation, /, of the magnetic held vector, on amplitude of the

field is shown, figure 11, as a furlction of dip angle at 10, 1, and 0.1 kc/s. The change in the amplitude of the field with dip angle is more obvious at 1 kc/s. Comparison of figures 6 and 11 shows

dip angles between 30 and 60 deg exhibit virtually no difference between the isotropic and anisotropic curves at 10 kc/s (VLF).

As a consequence of the numerical and analytical insight gained by the above described analysis, the procedure to foUow in introducing reflection coefficients is evident. Considerable caution

is necessary since the Debye type approximation in the guide {a < r < d) (64) is again not accurate

for reasons described above. But in this case, there is httle difficulty calculating the spherical wave functions Cn^{z) and Cn^'iz). The analysis presented in this paper has been a theoretical-mathematical study based on physical premises and a number of phenomena have been predicted. Verification of these phenomena

have not as yet been made under carefully controlled experiment.

5. Conclusions

The effect of anisotropy in the ionosphere on the field of LF, VLF, ELF radio waves can be

described theoretically for the region between the ionosphere and the earth by following conventional isotropic analysis procedures and introducing the magneto-ionic theory with the aid of

reflection coefficients. In addition to a considerable number of normal modes, abnormal modes

must also be summed to completely describe the field. The LF portion of the long wavelength

spectrum is more sensitive to ionosphere modes and changes in electron density, magnetic field, etc., than the VLF. The most important parameter at VLF is apparently the height of the model

ionosphere boundary for the cases studied in this paper. The results of this paper suggests the

work be extended by introducing more accurate reflection coefficients which are based on an

ionosphere in which the electron density varies with altitude and distance.

The authors are indebted to their colleague, James R. Wait, for his critical review of this work.

18
6. References

Alpert, Ya. L. (1961), Chapter on long waves from book. Ionospheric Propagation of Radio Waves (Translation by the

late Dr. Alex Nennsberg) NBS Translation T5-60, 61. Berry, L. A. (1964a), Some remarks on the Watson transformation and mode theory. Radio Sci. J. Res. NBS/USNC-URSI

68D, No. 1, 59-66.

Berry, L. A. (1964b), Computation of Hankel functions, NBS Tech. Note 216 (1964). Booker, H. G. (1939). The propagation of wave packets incident obhquely on a stratified doubly refracting ionosphere,

Phil. Trans. A237, 411. Bremmer, H. (1949), Terrestrial radio waves; theory of propagation (Elsevier Publ. Co., New York, N.Y.). Budden, K. G. (1961a), The wave-guide mode theory of wave propagation (Prentice-Hall, Inc., New Jersey). Budden, K. G. (1961b), Radio waves m the ionosphere (Cambridge at the University Press, London).

Debye, P. (1909), Der Lichtdruck auf Kugeln von beliebigem Material, Ann. Physik (Leipzig XXX, 57-136. Debye, P. (1910), Math. Ann. 17, p. 535 (Minutes of meeting by Academy of Science, Munich, 1910 according to March

(1912). Fok, V. A. (1946), Diffraction of radiowaves around the earth's surface. Academy of Science, USSR, Translation by Morris

D. Freidman.

Hildebrand, F. B. (1946), Introduction to numerical analysis (McGraw-HiU Book Co., New York, N.Y.). Jahnke, E., and F. Emde (1945), Tables of functions, 4th ed. (Dover PubUcations, New York).

Johler, J. R. (1963a), On radio wave reflections at a continuously stratified plasma with collisions proportional to energy and

arbitrary magnetic induction, Proceedings of the International Conference on the Ionosphere, Imperid College (2-6 July 1962) London, England, PubHshed by Physical Society and Institute of Physics, Chapman & Hall Ltd, London.

Johler, J. R. (1963b), Radio pulse propagation by a reflection process at the lower ionosphere, J. Res. NBS 64D (Radio

Prop.), No. 5, 481^99.

Johler, J. R., and L. A. Berry (Nov.-Dec. 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 L. A. Berry (1963), On the propagation of long wavelength terrestrial radio waves — two theoretical tech- niques, Agardograph 74, Propagation of Radio Waves at Frequencies Below 300 kc/s, 101-129 (Pergamon Press, London).

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.

Johler, J. R., and J. D. Harper, Jr. (Mar. 1963), On plasma collision frequencies proportional to energy in the radio wave

reflection and transmission process, NBS Tech. Note No. 164. Johler, J. R., and L. C. Walters (May—June 1960), On the theory of reflection of low- and very-low-radiofrequency waves

from the ionosphere, J. Res. NBS 64D (Radio Prop.), No. 3, 169-285. Magnus, F., and F. Oberhettinger (1949), Special functions of mathematical physics (Chelsea Pubhshing Co., New York,

N.Y.). March, H. W. (1912), Uber die Ausbreitung der WeUen der drahtlosen Telegraphic auf der Erdkugel, Ann. Physik, Vierte Folge XXXVII, 29-50. Northover, F. H. (1962), The diffraction of electromagnetic waves around a finite, perfectly conduc ing cone. Quart. Journal Mech. and App. Math. XV, Pt. 1, 1-9.

Poincare, H. (1904), Sur la diffraction des ondes electriques: a propos d' un article de M. Macdonald, Proc. Roy. Soc. London A72, 42-52.

Poincare , H. (1910), Sur la diffraction des ondes hertziennes. Rend. Circ. Mat., Palermo, t. 29, p. 169. Spies, K. P., and J. R. Wait (1961), Mode calculations for VLF propagation in the earth-ionosphere waveguide, NBS Tech. Note No. 114. von Rybczyriski, W., Uber die ausbreitung der WeUen der drahtlosen Telegraphic auf der Erdkugel, Ann. Physik, Vierte Folge XLI, 191-208 (Leipzig, 1913). Wait, J. R. (June 1957), The mode theory of VLF ionospheric propagation for finite ground conductivity, Proc. IRE 45,

No. 6, 760-767.

Wait, J. R. (Jan.-Feb. 1961), A new approach to the mode theory of VLF propagation, J. Res. NBS 65D (Radio Prop.), No. 1, 37^6.

Wait, J. R. (July 1962a), Introduction to the theory of VLF propagation, Proc. IRE 50, No. 7, 1624-1648.

Wait, J. R. (1962b), Electromagnetic waves in stratified media (Pergamon Press, Ltd., Oxford). Wait, J. R. (May-June 1963a), Concerning solutions of the VLF mode problem for an anisotropic curved ionosphere, J. Res. NBS 67D, (Radio Prop.), No. 3, 297-302.

Wait, J. R. (1963b), The mode theory of VLF radio propagation for a spherical earth and a concentric anisotropic ionosphere. Can. J. Phys. 41, 299-31.5. Wait, J. R.. and N. F. Carter (Mar. 1960), Field strength calculations for ELF radio waves, NBS Tech. Note No. 52. Wait, J. R., and A. Murphy (June 1957), The geometric optics of VLF sky wave propagation, Proc. IRE 45, No. 6, 754—760.

Wait, J. R., and K. P. Spies (Mar.-Apr. 1963), Height-gain for VLF radio waves, J. Res. NBS 67D, (Radio Prop.), No. 2, 183-187. Watson, G. N. (1918), The diffraction of electric waves by the earth, Proc. Roy. Soc. London AXCV, 83. Watson, G. N. (1919), The transmission of electric waves round the earth, Proc. Roy. Soc. London ACXV, 546.

19
The coefficient of the quartic (6)

Appendix

are

Inhere,

ao = (a2-l)^

5-- + (a2-l)

1 5-2

+-

.5 52-/12 5(5<h\ 1 5-1

«i = 2 -7-^— (a^ — 1), '5(52-/12)

a2= 2 1--

03 = 2

52 -/l2

+

hi

04 = 1 —

5(52-/12)

5(52 - /l2)

Ci Sec2 (/);,

^ (a2-l) +

A|af 5-2 1 5(52-/12) 52-/12 5

5(52-/12)'

5 = -0}

h =

N L

0)H(O 0J%

'

0)

hL=—h sin /,

hT = h cos /,

aL = sin (/), cos 0a, ar = sin sin c^a,

a = sin .

The solution of Maxwell's equation (1) for plane waves in a Cartesian coordinate system con- sidered locally at the model ionosphere [Johler and Berry, 1962] and [Johler and Harper, 1962]

,

for the dimensionless field ratios.

i me r-i —

Pern

Ez E r Ey Ee _ Hz Hr

Hx Hip

Qn

Qe.

Ex

Ey

Eifi To —_ [Li ~ III Hx Hf

can be written

bu Pme 610

b-n b22_ _Qme _ + = 0

20
where,

bn

b-n

622

610 620

an Oil (221 ai3 023

a-zi — —

031

023 033 033

fll2 022 «22 a32 O32

in which it has been assigned that the solution of equations (1) and (4) in Cartesian coordinates

yields, upon eliminating V and H,

On O12 Ol3 E,'

021 O22 O23 = 0, Oai O32 033. E,

where.

On = l — o| —

O12 = aiaT — i

oi3 = art, + i —

031 = orC — i

O33 = 1 ~ o^

021 = aLaT + i

022 = 1-4'

023 = Oz,C +

032 = 01,4 +

a

hihr

s{s^-h^y

hi

Also,

where.

Cn C12

C21 C22

Pern +Cio Qem C20 = 0,

Cn du dn dzi

C12 di3 di3 C?33 C21 d2i dsi dsi

C22 dts daa d33

Cio di2 d\2 C?22

C20^ di2 _dz2 dso

respectively, where the

simultaneous solution of equations (1) and (4) in Cartesian coordinates

yields upon ehminating V and E,

where,

'dn d\2 di3

d2i d22 d23 = 0, d32 dss.

2~ C3— 1] d21 — UtIDaC^DbAo]

c?i2 = Oii)c[4^^i- 1] + OLOrf £>^4

di3 = -[alDci + C,DA^']

d22 = DaB3C - arDs [i,^ A,-l] - DaDb

d23 DAC[a^B.^l]+ai.aTDBi

21
where

and

c?3i = arDB [ag — C3 — 1] + aLarDcC

dai = — aila'^B— DcBs]

fl?33 = arZ^fiCi— aiDclar- B,— 1] —DbDc

— s

A.2 — ihi A3

ihr

iht

B2 B,

5(52 - K") — ihr _ hihr _ {s'^-h^)

5(52 - K")

Da =A3C + arAx -\-a1A2 + Ut

DB = aTC + B:iC + aLB2 + ai

He = arCi + atCo + Cs^ + ^ .

In particular, explicit expressions for Qme and Pme are,

Qme " l-a2-

5--

5(52 -/;•-) 52 -/i2

+

52-A2

hihr

5(52 - /l2

hrhL l-a2-^2.

52-A2

5

ari — i

CLlCIt ~ I ari — i

. hi

l-a2-- ^

5(52 -/l2)

l-a2-^2_

52-A2 52 -/l2

22
All of the equations and text of this publication were composed on the keyboard of a photocomposition system, eliminating all hand insertion of equation characters.