PROCEEDINGS OF THE IRE

The Propagation of Radio Waves of

Frequency Less Than 1 kc*E. T. PIERCEt

Summary-The splified mode theory of propagation in a wave-guide formed by the earth and a concentric ionosphere of constantheight is applied to the experimental observations of Chapman andMacario for the frequency range between 100 cps and 1000 cps. It isdemonstrated that the discrepancies between the theory and thenighttime experimental results may be explained by modifying thetheory and postulating an effective increase in the ionosphericheight as the frequency decreases. This concept is also shown to benot necessarily incompatible with the results for day.

INTRODUCTION

INCREASING attention has been paid of late topropagationi at extra low frequencies (ELF). Atthese frequencies it appears that the propagation

cani be very simply explained in terms of a single modetravelling within the waveguide formed by the earthand a conicenitric homogenieous ionosphere at height h.

Accordinig to the mode theory of propagation in theformii developed by Wait,"2 only the zero mode is of im-portance at ELF. After certain approximations, the at-telnuation coefficient a in this mode can be expressed as:

7850 /f Ne2a h V' where wr = *_ (1)

h XT momy

In (1), a is measured in decibels per 1000 km, h is inkilometers, f is the frequency, w, is in mks units with Nand y the electroni denisity per ml and the electron col-lisional frequLency, respectively, for the homogeneousionosphere; the other symbols have the conventionalsignificance. It is evident fromn (1) that if the simpletheory applies, and h and wr are therefore not to be con-sidered as dependent upon the frequency of the wavebeing propagated, then V/f/a should be constant.

THE EXPERIMENTAL INFORMATION

Experimental values for the attenuation coefficientsat ELF are regrettably scarce. The only strong naturalsource of signals at these frequencies is the lightning dis-charge, and it is from studies of the propagation ofatmiospherics that Chapman and Macario3 have derived

* Original manuscript received by the IRE, May 4, 1959; re-vised manuscript received, December 1, 1959.

t AVCO Research and Advanced Development Division, Wil-mington, Mass.

1 J. R. Wait, "The mode theory of VLF ionospheric propagationfor finite ground conductivity," PROC. IRE, vol. 45, pp. 760-767;June, 1957.

2 J. R. Wait, "The attenuation vs frequency characteristics ofVLF radio waves," PROC. IRE, vol. 45, pp. 768-771; June, 1957.

3 F. W. Chapman and R. C. V. Macario, "Propagation of audio-frequency radio waves to great distances," Nature, vol. 177, pp. 930-933; May, 1956.

figures for the attenuation coefficients. Some valueshave also been obtained by Holzer, Deal, and Rutten-berg,4 but these have not yet been published in thegeneral literature; even the valuable information ofChapman and Macario is only available in summarizedform and not in the detail that could be desired. Itshould perhaps be pointed out that the single stationwork of Chapman and Macario is not an entirely satis-factory source of data. For instance, the observationsare limited to distances comparable with the wave-lengths at the lower range of ELF; this implies that nearfield analysis should be applied. Again in single stationwork, as compared with investigations using severalstations, the variable spectral content of the source can-not be eliminated, and it is therefore necessary toaverage many individual results. This procedure hasboth advantages and disadvantages. The conclusionsdrawn can never be as precise as those obtained frommulti-station recording on the same atmospheric. Onthe other hand it is notoriously misleading in geophysics,where many quantities show considerable variations, toconcentrate upon an individual example and ignore theover-all behavior.Chapman and Macario's experimental results do not

appear to have been examined on the basis of (1). Ac-cordingly, this has been done in Fig. 1 in which twocurves are plotted of v/f/a against f for day and nightconditions respectively, and in the frequency range of100 cps to 1000 cps. It is apparent that during day forfrequencies exceeding about 300 cps the constancy of\/f/la to be expected on the simple theory is realized; atnight, there is no indication of such a constancy; andboth by day and by night there are divergencies for thelower frequencies. The last effect is perhaps not entirelyunexpected, since (1) is really only valid for distancesexceeding about one or two wavelengths, and, as alreadyindicated, the experimental observations are limited towithin 3000 km (wavelength for 100 cps). Again atdistances "close" in terms of a wavelength, the electro-static component of the field due to a lightning dis-charge is significant. Indeed, a combination of these in-fluences would imply that the short range attenuationlaw should differ from that for greater distances. Thereis some suggestion of this in the results of Chapman andMacario, but the spread of the observations is too greatfor any precise conclusions to be drawn.

I R. E. Holzer, 0. E. Deal, and S. S. Ruttenberg, "ELF propa-gation," Proc. VLF Symp., Boulder, Colo., Paper No. 45; January,1957.

3291960

PROCEEDINGS OF THE IRE330

10

E000-o

z

o loo 200 300 400 500 600 700 800 900 1000

FREQUENCY f (c/s)

Fig. 1-Relation to frequency of Chapman and Macario's experi-mental values for the attenuation coefficient a.

Fig. 2-Variation of electron density N anid electroncollisional frequency, y with height.

95

90

8s5

THE REINTERPRETATION OF THE RESULTSNight

The quantity hV\Ir may be estimated directly fromionospheric data for the variation of N and y with height.The exact form of this variation is not known but thesituation has been summarized by Waynick5 who givesseveral references to the best available information. Acombination of this data is depicted in Fig. 2, while Fig.3 represents the variation of hV\/w with height, for bothday and night conditions, deduced from the curves ofFig. 2. It can be seen from Fig. 3 that hVIr increasesmonotonically with increasing height, but that the rateof increase is very slight by day for the ranige of height71 to 77 km.

It is interesting to accept Chapmani and Macario'sexperimental figures for Vf/a from Fig. 1, and then use

(1) to obtain the corresponding values of h \w, at vari-ous frequencies. In turn these values may be employedto define a series of heights since hV\W9r is monotonicwith height. These are indicated, for frequencies at in-tervals of 100 cps, by the horizontal lines on the nightcurve A of Fig. 3; for these night results the variationwithin the range 100 to 1000 cps is represented quitewell by the empirical relationi

H = (89 - 0.17V\/f), (2)

or with rather less accuracy by

H = 81.0 + -/) (3)

H is the new height parameter.The procedure outlined above is equivalent to postu-

lating that the simplified waveguide theory does notapply, but by introducing the slight modification that

5 A. H. Way nick, "The presenit state of knowledge concerning thelower ioniospherec " Pizoc. I RE! vol. 45, pp. 741-749; june, 195 7'.

w

80O

75

70

b6 -- - -

103

h %rFOj

Fig. 3-Variation of hV.\Cr with height.

h, a constant in (1), should be replaced by H whichvaries with frequency, a considerable measurement ofagreement is obtained between theory, the experimentalresults, and what is known of the constants of theionosphere. Physically the modification seems plausible.As indicated by Budden6 one would anticipate the pene-

tration of the fields of the waveguide modes into theionosphere to be greater the larger the wavelength, andthus (2) and (3) for the variation of H with f are nlotunreasonable. Again, the two layer model of Wait7shows that frequencies from 8 to 18 kc are effectively re-

flected from the lower layer, while frequencies less than3 kc penetrate to the higher level. Indeed, (2) and (3)may be regarded as an application to frequencies below1 kc of an extension of the Wait two layer model; themodel is now infinitely layered; that is, continuous. Eq.(3) is reminiscent of formulas associated with the skineffect, but too much significance should not be ascribedto this similarity.

6 K. G. Budden, "The propagation of very low frequency radiowaves to great distances," Phil. Mag., vol. 44, pp. 504-513; May,1953.

7 J. R. Wait, "An extension to the mode theory, of VILF ioniosphericpropagation," J. Geophys. Res., vol. 63, pp. 125-- 35; Malrch, 1958.

March

IGHT

~~/

100 ,/ X,//300 4zn<t 0

- ~~~500 0

9H~~~~~~_/~~~~~ //I~~~~~ /

/~~~~~~ /

000--- /0900 BO40

8000 --

-, .---~---B.DAY -

|X B Z

Pierce: The Propagation of Radio Waves of Frequency Less Than 1 kc

Day

At first sight it would appear that the experimentalresults for day are in good agreement with simple theoryusing a constant height, since the quantity \/f/a doesnot change appreciably in Fig. 1 between 300 and 1000cps. However, consideration indicates that becausehV\IWr is practicallv constant over an appreciable rangein height by day (Fig. 3), the experimental results areniot entirely inconsistent with the idea of a heightchalnginig with frequency as postulated in the precedingsection.

T'he argument is as follows. First of all, to clarifymlatters, the values of h\/Wor corresponding to the ob-served attenuation coefficients at 100, 300, and 1000cps, are plotted on curve B of Fig. 3. Now the graphsof Fig. 2 giving the variation of N and y with height areby lno meanis precise; for example, at 70 km, values of,y of between 1 X 107 and 3>X 107 collisions per secondhave been quoted in the past. Uncertainties of this orderimply that the curves of Fig. 3 could well be displacedat least within the limits indicated by the dotted lines.Stuch a displacement would have relatively little in-fluience upon the interpretation of the night results sincethe graph .4 on Fig. 3 changes little in slope. By day,however, the situation is entirely different. In particular,only a slight displacement is needed to bring the experi-menitally determined values of hV\Iwr for 300 and 1000cps oln to the steepest part of curve B. It is temptingto suggest that this is indeed the case since under thesecircumstances a change in Hof roughly 2.5 km between300 and 1000 cps as indicated by the night results,wouild correspond by day to a variation of onily about5 per cent in h/w/vr, and therefore a change also of thesame order in the experimentally deternmined v'fla. Thescatter and uncertainties in experimental results suchas those of Chapman and Macario are probably suffi-cienit to obscure variations of this magnitude. Thus it ispossible, without too great difficulty, to reconcile theconicept of a height varying with frequency with the ex-perimnental results both by day and by ilight. It mustbe admitted, however, that if the apparenit minimiumi ofabotut 600 cps for \/f/a by day (see Fig. 1) is a realeffect, then the derived H is not nmoniotoniic with fre-quency, and the reiniterpretationi may be saidl to fail forthe day results.

DISCUSSION

It has beeni pointed out how there are difficulties inthe interpretationi on the simple waveguide theory ofthe experimental results at night for the attenuation of

ELF radio waves during propagation. The difficultiesare removed by the physically plausible postulationthat the height of the guide increases as the frequencydecreases. This concept has also been shown to be notentirely inconsistent with the daytime results.The two layer model introduced by Wait has already

been mentioned. This model referred primarily to propa-gation in the VLF range, but Wait has also considereda second extension to the mode theory for ELF propaga-tion, being concerned with the indication from theexperimental work of Holzer, Deal, anid Ruttenbergthat there is a minimum of attenuationi at around 100cps. Incidentally, the first diagracm in the paper byChapman and Macario also suggests that the daytimeattenuation is, if anything, less at 125 cps thani at either100 or 160 cps, although Chapman and Macario in theirderived attenuation coefficients give a monotonic in-crease from 100 to 1000 cps. In his second model, Waitretains the concept of a waveguide with a sharplybounded edge at the lower ionosphere but adds an ex-ponential variation of N/ly decreasing upwards withinthe ionosphere. Strictly speaking, this pictuire is untrueat least up to the maximum of electroni dlensity in the Fregion; effectively, however, it may be legitimate overdistances comparable with a wavelength (3000 km for100 cps), although this is not immediately evident bear-ing in mind the comparatively slow decrease in N abovethe F2 maximum indicated by Sputnik observations andby whistler results for farther from the earth, togetherwith the uncertainties relating to the values for y. Thetapered exponential model of Wait can account for aminimum of attenuation at a frequency of the order of100 cps and in this respect has the advantage over theapproach advanced in the present paper. In general,however, the latter seemns to be more effective in ex-plaining the discrepancies, particularly at night, be-tween Chapmnani and Macario's results and those antici-pated on the simple theory.

It is, however, perhaps academic to pursue the de-velopment of propagationi theories at these low fre-quenicies while the experimenital observationis remain soscanty. Further and more detailed results are urgentlyneeded and these could be immedliately obtained in themanner of Chapman and Macario by work on atmos-pherics. Multistation observations would be even morevaluable.

ACKNOWLEDGMENT

The author is indebted to Professor R. E. Holzer forhelpful commeints.

1960 331