NEW YORK unjv:p.s:ty

INSTITUTE OF MATHEMATIC/.L SCitNCES NEW YORK UNIVERSITV

LiDRARY 'n-JsmTUTEOF?-' \T.c,'d.mmeE£

^ ^averly Place, New York 3, N. Y.

NEW YORK UNIVERSITY ^ ^.^ IW ^ Y^ *, N. *.

WASHINGTON SQUARE COLLEGE OF ARTS AND SCIENCE

MATHEMATICS RESEARCH GROUP

RESEARCH REPORT No. EM-57

ON THE ELECTROMAGNETIC FIELD EQUATIONS

IN THE IONOSPHERE

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by

HERBERT B. KELLER

CONTRACT No, AF.19(122)-*2

SEPTEMBER 1953

Manufactured in the United States for New York University Press

by the University's Office of Publications and Printing

NEW YORK UNIVERSITY

Washington Square College of Arts and Science

Mathematics Research Group

Research Report No. EM-57

ON THE ELECTROMAGNETIC FIELD EQUATIONS IN THE IONOSPHERE

by

Herbert B. Keller

Herbert B. Keller

Morris Kline

Project Director

The research reported in this document has been made possible through

support and sponsorship extended by the Geophysics Research Directorate

of the Air Force Cambridge Research Center, under Contract No. AF-19

(122)-42. It is published for technical information only, and does not

necessarily represent recommendations or conclusions of the sponsoring

agency.

September 1953

New York, 1953

Abstract

This paper consists of two notes on the equations governing the

propagation of electromagnetic wares in the ionosphere. In Part I the ten-

sor properties of the ionosphere are derived by a method due to van der Wyck

(tlj » PM ). While we introduce only minor improvements on his method, it v;es

felt worthwhile to make these results generally more available*

Pert II shows the equivalence of various forms of the propagation

equations. A general scheme is indicated for finding simplifying transforma-

tions of the equations in their first-order from. In particular, we bttain

that transformation which yields Rydbeck's coupled equations, and give an

interpretation.

TABLE OF CONTENTS

Part I

iS£S.

1

1

2

7

1, Introduction 8

2, General Formulation 8

3» Normal Incidence 10

h» Oblique Incidence ih

Eeferences 17

1.

Introduction

2.

Formulation

3.

Standard Derivation

k.

Rigorous Derivation

References

Part II

-1-

PAEI I

nSE lENSOR FROPERIIES OF THE lONOSPHEEE.

1. Introduction

It is well knovm that the linearized macroscopic field eq^iations

in the ionosphere involve tensor properties of the medium. The most accurate

derivation of these equations shoiild start with the microscopic field equations,

and apply the methods of statistical mechanics to yield the tensor field equa-

tions for the macroscopic fields (see[l]). Some such attempts have "been made

by Hartree [2] and Darwin [3] for a case in which the ionosphere may he describ-

ed by scalar properties. However, the general derivation in which tensor proper-

ties arise seems quite difficult and remains to be carried out. In place of this

we are, at present, forced to accept derivations in which the averaging is less

detailed. Many such derivations have been given but they are not as rigorous as

might be expected, and contain numerous assucrptions which are not explicitly stat-

ed. In the present note we present a modification of a derivation due to van der

Wyck [4J in which the averaging is explicit. The main assumptions are stated and

it is shown that the resultant tensor properties of the ionosphere (when some of

van der Wyck's simplifying assumptions are eliminated) are just those obtained by

the less rigorous methods, dis may serve as a justification for the use of the

usual derivations. Por comparison, one of the standard derivations, essentially

that of Baner jea [5] , is included.

2, Tormulation

The general macroscopic field equations are

^7K E - - ^^

(1)

o'C

together with the relations

(2) ? = c^ + ?

T = 0-?+ t .

-2-

Here jul ,c and cr have their usual meanings. P, the polarization, is a measure

of the electrical " stress" of the medium, which is proportional to the average

displ£icement of the electrons from their equilibrium positions, Z^ the total

current density, consists of the usual conduction current density, cTE, and a con-

vection ctirrent density, I, which is the average flow of free charge density.

The ionosphere is represented "by free space, i.e., e = /t- = 1 and

o^ = 0, in which there is a given density distribution, N(x,y,z), of free

electrons. Heavier particles (ions) are assumed to be present, and we assvune

their effects may be taken into account by means of the considerations given

in Section h, A static magnetic field, H_,, due to the earth, is also present.

The problem is to derive expressions for the macroscopic quanti-

ties P and I in terms of the field conrponents and ionospheric properties. The

equations of motion for charged particles in the presence of electromagnetic

fields mast play a major part in this derivation; these equations are introduc-

ed below.

?, Standard Derivafcion

In the standard derivation it is assumed that 1=0 and that

(3) ? = N e ?,

where p is the average displaxiement of an electron from its equilibrium position

and e is the charge on an electron. To coOpute p one considers the equation of

motion for an '• average" electron:

C) ?*"? - S[?-%] = l^ .

where m is the mass of an electron, and where we have used the linearizing ap-

proximation Jh]

(2.18)

cl\(z)

■'T, ■ ,

2 o

c

te)1v^'=-&)s^-(a+^)

Note that again +^. (z) and i\Az) are the eigenvalues of the A(z) now given by (2.19).

There are now three distinct co-upling factors:

7

1 -/AV'

>U7

1 -z^fv"

1 -/x7'

for the system (2.9). They reduce to one if ^/X = 1 and then we have the case of

Section 3. If in eq, (2.9) we use the values given in (2.25) and the first of

(2.13) and then eliminate ^ . and ^ , we ohtain

dV(z)

-[x|*a2-;-^h2*^]n,

= - ac + c -/Uab - ^^ I n^ - I c +/^ "b ri

cL TT,(z) /U cLtt (z) ,

Here we have introduced

(2.28) a =

/^ r

2(1 -yUV^)*

h =

(1 -/x-T^)' " (1 -/Xt2)

The equations (2.27), while equivalent to (2,2l), are by no means simpler and seem

to have only the virtue that they reduce to the I^dbeck equations (2.18) when

A* = 1.

-17-

KEPEHENCBS

[i] 0,E.H. Eyd"beck; On the propagation of waves in an inhomogeneous medium;

Trans, of Chalmers Univ. of Tech., Ko. 7^ (19^).

[2] E.B. Keller; Ionospheric propagation of plane waves. Math. Res. Group,

N.Y.U., Wash. 3q. Coll. of A. and S., Res. Rep. No. EM-56(l953).

[3] H.B. Keller and J.B. Keller; On systems of linear ordinary differential

equations. Math. Res. Group, N.I.U., Wash, Sq[. Coll. of A. and S., Res.

Rep. No. E^33 (1951).

[k\ V.L. Ginsberg; On the influence of the terrestrial magnetic field on the

reflection of radio waves from the ionosphere. Journal of Physics (U.S.S.R, )

Vol. VII, No. 6 (19^3).

[5] M.V. Wilkes; O'blique reflection of very long wireless waves from the

ionosphere; Proc. Roy. Soc. 187, p. 130-1^7 (19^7).

Date Due

BEC 22-61

■ ■ 1

H^2t Bfs:

MAfi - P K :

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PRINTED

IN U. S. A.

4

EM-

V

I'.TYU

' Em-

:j3\\ev

Lei iRr

c. 1

c. 1

On the electroin ap- ne t ic

field equations in

Jry,r, ^r>^ 1

KYU

EI/i-

57

.1 /

Keller

On the electromagnetic field

equations in the ionosphere.

i

II

1