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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

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).



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57



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Keller

On the electromagnetic field
equations in the ionosphere.




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II





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