JOURNAL OF RESEARCH of the Nationa l Bureau of Standards-D. Radio Propaga tion

Vol. 64D, No.4, July- August 1960

Mode Theory and the Propagation of ELF Radio Waves I

James R. Wait

(F ebruar y 18, 1960)

The mode Lheory of propagation of electrom agnetic w aves at extremely loll' frequen­cies (ELF) ( 1.0 t o 3,000 cycles per second) i s treated ill t his paper . StarLill g wiLh t he r epresentation of t he fi eld as a sum of modes, approxim at e formulas arc p resenLed for t he attenuati on and ph ase constants. Ce rtain al ter nate represen ta tions of t he ind i v idual m odes arc ment ioned. These are used as a bas is for desc ribing t he ph ys ical behav ior of th e fi eld at lar ge distances from the source, par t icular l y ncar t he a nt ipode of t he source . A L t he ~ ho r te r d istances, wh ere t h e range is comparab le to t he \yavel ength, t he sph eri eal-ear t h mode seri es is besL t rall sformed t o a seri es in volv ing cy li ndri cal w ave fun ct ions. This l atter f orm i s Ll sed Lo evaluate t he near fi eld behavi or of Lhe var ious fi eld co mponent.

The eff ect of the earLh's m agnetic field i s also eV ~L l u a ted using a quasi- longiLudinal a pproximation. III general i t i s indicat ed t hat if t he gy l"Ofrequency is less t han t he effec­t i ve value of t he co lli sion frequency, t he p resence of t he ear t h 's magneti c fi eld may be neg lec ted for Ii;LF. When t his condi t ion is not met th e attenuation may be increased somew hat. The infl uence of an inhomogencous ionosphere i s also bri efly co nsidered and, fin all y, t he propagation of E LF pulses is t reat ed . It is suggested t hat ce rt ain observed characteri stics of E LF w a veforms m ay be attribu ted to t he inclination of t he current cha llnc l in t he li gh t ning discharge.

1. Introduction

Th e prop<lgfLtioll of electrOlnfLgnetic WfLves at extremely low frequ encies (ELF) has received consid erable fLtLention recently [1 to 5]2 Unfor tuna tely, the experim ental da La in this frequen cy range (1.0 cps to 3,000 cps) is s t ill rath er limited [8 to 15]. Th er e is liLtle doubt, however , that th e bulk of the energy is transferred via a waveguid e mod e . Th e bounding surfn.ces of this waveg uide are tbe grou nd and th e lower edge of Lhe ionospheric It region . J n th eoreticfLl fLpproaches to tb is problem , it is usually assum ed til a l theiono­sp here is homogeneous in its electrical properties . On comparing t heore timl aLlenuatioll rn,tes based on this ass umption , it is apparent that the model mus t be refin ed to Lake ftccount or the varifLtion of the electron densit~T with height. Such an extension to th e mode th eory has been ma~e r ecently [16] . The inJiuence of the earth's magnetic field is fLlso expected to be important at ELF sin ce the waves are reJiected at a level where the collision frequen cy is of the same order as the gyrofrequency for electrons. The ionosphere is thus expected to be anisotropic. Unfortunately, the experimental data is not sufficient to confirm whether the anisotropy is significant.

Another feature of ELF propagation is that the distance between source and observer may be comparable to the wavelength. For example, at 300 cps the wavelength is 1,000 km. Many of the experiments are carried out at distances of this order and the computed a ttenuation rates must be considered with caution.

It is the purpose of this pfLper to survey th e mode theory as it applies to ELF propaga­tion and to develop several extensions. Many of the results are presented in graphical form as an aiel to application.

1 Contribu t ion from Cen t ro l Radio Propagation Laboratory, Xal ional Bureau of Stand ards, l3 ould er, Col o. rape r prese nted at Confe rence 0 11 t he Propagation of ELF Ra(l io Waves, Jan. 25, 1960.

2 F igures in brackets indicate the literatu re references at the f' ll d of this paper.

387

2" Basic Theoretical Model

The assumed theoretical model is taken to be a homogeneous conducting spherical earth of r

radius a surrounded by a concentric conducting ionospheric shell of inner radius a+ h. It is '1 convenient to introduce the usual spherical coordinate system (r, 8, rf». Using this model and assuming a vertical electric dipole source located at 8= 0, r= a, the expression for the vertical electric field, E" is given by an expression of the following form [4 , 16]

where

IdST} '" v(v+ l) E r= 2kha2 L: On - . - P.( -cos 8)

n=O Sin VTr

I - average cm-rent in the source dipole, ds- Iength of source dipole,

1J- intrinsic impedance of air space ~ 12071" ohms, P.( - cos 8)- Legendre function of argument - cos 8 and (complex) order 11,

(1 )

1I +~=ka S n where S n, for n = O, 1, 2, .. "' is determined from the boundary condition and is described below, and

1 on'" sin 2kh On "'1 /2 for n = O, ~1 for n = l , 2, 3, ....

1+ 2khOn '

The individual terms in the above series correspond to the waveguide modes. In the general case, the factor S n is obtained from the solution of a complicated transcendental eq uation which involves spherical Bessel functions of large argument and complex order. Certain as­pccts of this problem have been discussed by Schumann [1] and also the author [4]. For a homogeneous earth and a homogeneous ionosphere, both assumed isotropic, it has been shown that S n may be approximated b~r [17]

{ (7r11)2 1 [ I . /:},kh ] 2} t S ,,= 1- kh 4 l +-y 1+ 4t (?rn)2 (2)

where

in terms of the refractive indices, N g and N i , of the earth and the ionosphere respectively. This equation is valid subject to the condition I/:}, Ikh< < 1. The sign of the radical is chosen in the above equation which makes the real part positive. The values of S n then are located in the fourth quadrant of the complex plane. ' iVben n = O, the above simplifies to

(3)

Now, since I/:}, Ikh< < 1, the radical in eq (1) can be expanded for n > O to yield

S [ ( 7I"n)2 . 2/:},Jt e n= 1- kh -t kh TOr n = l , 2, 3, (4)

Furthermore, if in addition,

[/:}, I < <1_(7I"n)2 kh kh

388

I ~

it is permissible to write, for all 11,

[ (7r'n)2Jl E [ (7r'n)2J_; Sn~ 1- kh 2 -i 2kh Ll 1- kh 2 (5)

where Eo= l, En= 2 (for n = l , 2, 3, .. . ). This latter eq uation is valid when the "waveguide modes" are not near cutoff and is in the form given by Schumann [2]. It i quite analogous to the standard results for the propagation constant in a rectangular wavegu ide with finitely conducting walls [18, 19] .

3. Behavior of the Modes for the Spherical Earth

For purposes of computation of the mode series, several implifications can be made. The asymptotic expansion for the Legendre function , given by

p (-cos O)~(-~-)~cos [(v+~) (7r- O)-"!..J (6) • 7rV SIn 0 2 4

IS valid if Ivl> > 1 all d 0 llot neal' 0 or 7r [20] . Thus, t be ~modes are simply proportional to

--:-.1_ ; cos [lwSn (7r - O) _"!..] (Sill 0) i 4

which apart from a co ns tan t factor can be identified as the lineal' combination of two peripheral waves. These have the form

1 -ikaS 0 --- e n (sin O) t

and

__ 1_ e - ikaSn(21r -O) ei1r /2

(sin O)! where 0< 7r.

These waves are travebng in opposing directions alo ng the two respective great circle path aO and a (27r- 0) , from the source to the observer. It is noticed that there is a 7r/2 phase advance which the wave traveling on the long great circle path picks up as it goes through the pole 0= 7r. The lineal' combi nation of Lhese two traveling waves is to form a standing wave pattern whose distance 8m between minimums is approximately given by

subject to

1m Sn« Re Sn-

The preceding asyulptotic expansion for P v (-cos 0) is not usable at and in the vicinity of the pole 0= 7r . In this region a suitable representation is given by [20]

where 1/ =(2v + 1) sin (( 7r - 0) /2). J m (1/), for m = O, 1, 2, and 3, is the Bessel function of first type of argument 1/ and order m. ' Vhen 7r - 0 is small, the first term is usually suffieient and further­more

1/ ~(v+D(7r- O) ~kaSn(7r- O).

389

Thus for mode n, the field in the neighborhood of the pole is proportional to the Bessel function

It is then not surprising to see that the first t.erm of the asymptotic expansion of Jo is the same as that of P v (- cos 0) .

At the lower end of the ELF band, where the wavelength is large compared to the height of the ionospheric reflecting layer, the electric field is essentially radial and only one wave­guide-type mode is significant. The fieJd is thus expressed by the fiTSt term of the mode series which reads

where v + t~kaSo and

Ids 7J v(v+ 1) E r = 4kh 2 - .--Pv(-cos e) a sI n V7r

(8)

in terms of the relative refractive indices Ni and Nu of the homogeneous ionosphere and the homogeneous ground, respectively. Now as mentioned above, the factor PvC - cos 0) may be replaced by an asymptotic expansion if kaO or ka(7r - O) is somewhat greater than unity. The field in this case may be regarded as two azimuthal-type traveling waves. Furthermore, at t he pole (0 near 7r) where the second of these restrictions is violated, it is possible to use an equivalent representation which correctly accounts for the axial focusing. An alternate viewpoint which is suitable at ELF is to consider the field as a superposit ion of cavity-reso nator type modes. I t is expected t hat such a representation would be very good when the circumference of the earth is becoming comparable to the wavelength. A suggestion of this kind was apparently first put forth by Schumann.

The starting point is the expansion formula

SIll V7r

1 00 , 2n+ 1 -; ~ Pn(X) n(n+ l )-v(v+ l) (9)

where the summation is over integral values of n. This result follows directly from a formula given by ::\Iagnus and Oberhettinger [20] (p . 57) which is valid for v7"'O , ± 1, ± 2, ... , and o ~ 0< 7r. The electric field, for h/a< < 1, is thus written

where X = cos O. The early terms of the series are then proportional to

and so OIl.

P o (x) = 1, PI (x) = cos 0, P z(x) = !(3 cos20-1I,

Retaining just the first term it is seen that

Ids 1 47ra2Ech iw

(10)

(11 )

which is independent of o. Clearly this corresponds to a concentric spherical capacitor ener­gized by a current Ids /h resulting in a constant voltage hE; between the plates. On rewriting eq (11) in the form

390

i t' is see n that

C = 47ra2€o e 11,

which can be identified as the capacity between the sp herical s urfaces whose areas are both 47ra2 within the approximation h/a< < 1.

4. Earth-Flattening Approximation

The poss ibility that t he CUl'vature of the earth may be neglected is now investigated . To simplify field calculations at short distances, it would seem desirable to transform the mode series to a form where the fU'st term c01'l'esponds to the model of a flat earth and t he s ucceeding terms are corrections for curvature. This approach has been used by Pekeris [21] and more recently by Koo and Katzin [22] in their investigations of microwave duct propagation. From their work, it may be shown that

where v+ l /2= k8nP a lld p= aO. HJ2) and H J2> are Hankel functions of the seco ndkincl of order zero a nd two , respectively . When the great circle distance p is reaso na bly small compared to the earth 's radius a, only the first term in the expansion need be retained.

Fo r co nve lli ell ce In what follows , it is co nvenient to express the field co mpon e llLs as a ra t io Lo t he quantit)T

( 1) ~ 1207r) ;

Eo is t he radiation field of the so urce at a distance p on a perfectl.\~ co ndu cting ground . Thus, for both the source a nd the observer neal' t he gro und, it is not difficult to s how that

Ez= liVEo where

(13)

where

8 '" 7r P ikp ~ • 8 }'T(2) (k8 ) = }lg h e ~ U n n :Il " p (14)

a nd

wbere

(J 5)

where terms containing (p/a)2, (p /a) \ etc., have been neglected. When kp> > 1, corresponding to the " far-zone ," the above expressions Illay be sinl plified

since the Hankel fun ct ions may be r eplaced by the first term of their as.\Tmptotic expansion . This leads to the compact result

rWJ r 8'1 l 1 . [ 27t'P 7rJ n 8 ~(p/f-.. )2 , );'"'- :;- -.0 . l - 8~/N e - ihSnP/)..

(h/f-.. ) f ~ U n : oJ T 8~

(! 6)

which is valid for p> > f-.. . As expected , the ratio of W to T for a given mode is 8 n which for low order or grazing modes is of the order of unity. The ratio of 8 to T, quite generally, is - l iN n which is very small compared to unity; in fact, it vanishes for a perfectly co nducting ground as it must.

391

_J

L

The excitation by a horizontal electric dipole may be treated in a similar manner. A com­plete discussion of the theory is given elsewhere [17]. The expression for the vertical field Ez of the horizontal dipole Ids which is located at the origin and oriented in direction 4> = 0 may be written in the form

(17)

where S and Eo are defined above. By using thc reciproci ty theorem, this result can also be deduced directly from eq (15). Comparing the above formula for Ez (for a horizontal dipole) with corresponding formula for E" for a vertical dipole, it may be easily shown that

Ez (for hori~onta~ dipole) '" cos </> (1 __ I_ )t ~ cos </>

Ez (for ver-tIde dIpole) No N: - N g (18)

being valid in the asymptotic or far zone. In terms of t he ground conductivity IT g and dielectric constant~g,

This is a small quantity. The above formulas for TV, S, and Tare valid only if p< <a. If the first curvature correction

term is included, it is a simple matter to show, in the " far zone" , that this amounts to multi­plying the right-hand side by the factor

1 (p)2. 02 1+- - or 1+- · 12 a 12

Now, returning to the mode series for a spherical earth and using the far field approximation for the Legendre function, it turllS out that lV, S, T have precisely the same form as in eq (16) except the factor (O/sin O)t occurs in the right-hand side. Noting that

_ 0_ 2 '" !!.._ . . . 4 6 ( )1 2

sin 0 = 1+12+tmms 111 0 , 0, etc.

it is apparent to a second order that the derived curvature correction is the same as from the Hankel function series.

5. Distance and Frequency Dependence

Certain features of the mode series are best illustrated by a graphical plot of field strength versus distance under vario us values of the parameters. Since N u> > N ; it may be safely assumed for ELF that 6. = l /N ;. Then on the assumption that the ionosphere is behaving like a conductor, it is possible to write

(19)

where W5 (plasma frequency)2 W=-= .

T v collision freq ucncy

The above relation for N; is an approximation which is valid when the collision frequency v between electrons and neutral ions is much less than the angular frequency w. Since v is of the order of 107 in the lower E region , this is certainly valid at ELF. The (angular) plasma frequency Wo is given by

where N is the electron density, e and m are charge and mass of the electron, and ~o is the dielectric constant of free space. w" which is proportional to the ratio N/v, is believed to be of the order

392

!h w 0 ::> '::: z

'" <t :;;

of 10 5 in the lower region of the Elayer. At least this appears to be the effective value deduced from vlf observations for highly oblique incidence [23]. At ELF it appears that the waves are reflected at higher levels in the ionosphere and the effective value of W T is somewhat larger, being possibly of the order of 106•

It should be noted in passing that W T may be replaced by <5i/ ~O where <5; is the effective con­ductivity of the ionosphere and ~o is the dielectric constant of free space.

To indicate the val'ifLtion of the field as a function of distance, the magnitude a nd phase of Wand T are shown in figures 1 to 4 for the typical values: wr= 5 X I05 and h= 90 km. Actually IvV! and ITI, both divided by the square root of t he distance p (in kilometers), fLre plotted since the curves become linear at larger distances. The slope of these linear parL of the curves are proportional to the attenuation rate (in decibels per I ,000 km) as usually defined. It is seen immediately tbat at the shorter distances, the curves are no longer traight. This im­mediately indicates that co nsiderable caution should b e exercised in computing attenuation rates from spectral analyses of "sferics ." For example, it is only wh en the d istance p exceeds about one-sixth of a wavelength, is it permissible to assume that log (I WI/v;) 01' log (IEzl X v p) vary in a lineal' manner with distance p. Similarly, the phase of W (or E z) varies in a linear manner only when the distance exceeds about one-half wavelength. Similar remarks apply to the magnetic field variations with dis tance.

'Orrr-----~ ----- ---

W r '" 5 x IO~

1.0

01

0.5 t z 50c/s

'00

01 200

400

0.02

0.01 O'----:-1OO-"'C.OOc:---£OO=-~aooc:---:'IXll:::-"-:':O'1OO::---:'400:::----='£OO::----::'aoo::---O:lOOO.

DISTANCE. p (km l

J.°II 1.01 \

'.0

---I

0.1\~~ ~

'~'::\~ '::: z ~ 0.1

::> 0.01

0.05

0.03

0.01

DISTANCE , P (kml

FIG URE 1. 'The normalized magnitude of the elecll'ic fi eld as a f unction of distance f rom the source for h= 90 km.

F I G UR E 3. The nor malized magnitude of the magnetic field as afunction of distance Jrom the sou Tee fo r h = 90 km.

( p is expressed in kilometers in both ordinate and abscissa) .

1.a

1,4 I <C>

~ 2.0 , w (/) 1.6 <! I "'-

'.2

na

OL

f = 50c/s '00 200 400 800 '600

0100400 £00 a B ~ ~ ~ 1000

DISTANCE, P (kml

FWCRE 2. . The normalized phase (lag), i n radians, oj the electric field as a function of distance fo r h = 90 kilt .

393

La r-~-rr---'--------------'

1.6

I..

w' (/) 1.0 sOO <! I "'-

o.a

0.6

I O·\L-----::c1OOr---:.OO::-------::c600r---:800::---':::IXllr---:'1OO::---c':::400r--::'600::---':::aoor---::1OOO

DISTANCE, P (kml

FIG U RE 4. The normalized phase (lag), in mdians, of the magnetic fi eld as a function of distance for h = 90 km.

~I~ w-0 ~

~ Z l"> <1 ::;;

At the extremely short distances where p is somewhat less than the ionospheric reflecting height (i.e., 90 km), the magnitude of W varies essentially as the inverse square of the distance and the phase approaches 180 0 • In this same region the quantity T varies as the inverse of the distance and the phase approaches 90 0 • The general characteristics of the field at these extremely short distances are in accord with the near field behavior of a dipole on a flat ground plane when the presence of the ionosphere is neglected. In this rather trivial case

( i 1) W= 1- kp-k2p2 (20)

and

T=(l-~} (21)

To shed further light on the nature of the ELF fields in the waveguide, the radial impedance of the wave is now considered. Noting that

(22)

which is by definition the (normalized ) impedance of the wave looking III the r adial or p

direction . First it should be noted that for kp > > 1,

W T~l

and for kp < < 1 and p < < h,

For the in termediate and interes ting range, kp is comparable to unity and p is somewhat greater than h. Using the numerical values of Wand T mentioned above, the ratio IWITI and the phase (lag) defined by arg T- arg Ware plotted in figures 5 and 6 and as a function of distance for various frequencies. As before, wr = 5 X I05 and h= 90 km.

Vi Z <1 is <1 !!'. :;: l"> 0:: <1

I-

l"> 0:: <1

W <n <1 I Cl.

OJ 1.6

1.000 1000

DISTANCE , p(km ) DISTANCE, p(km)

F I GURE 5. The magnitude of the radia(wave impedance. FIG U RE 6. The phase (lag) of the radia( wave impedance.

394

The impedance ratio vF/T is independen t of t be frequency spectrum of the so urce provided, of co urse, it may be represented by an equivalent vertical electric dipole. In fact, t his complex ratio could be easily calculated from the frequency spectra of the waveforms of the vertical electric field and the horizontal magnetic field of an atmospheric. The observed vtLriation of magnitude or phase of WIT as <L fun ction of frequ ency should then provide a basis for distance measuring. Such a scheme, while admittedly crude , requires only one rece iv ing station equipped with a vertical whip and a loop antenna.

6 . Effect of the Earth's Magnetic Field

In the preceding it has been tacitly assumed that the ionosphere i behaving as an isotropic homogeneous conductor. In this case, t he refractive index Ni of the ionosphere may be written [24]

where (plasma freq u en cy)2 (collision frequency)

(23)

'Vb en the earth's magnetic field is steeply dipping, it is appropriate to invoke the quasi-longi­tudinal approximation of Booker [24]. In this case the refractive index is double valued, one corresponds to the ordinary and the other the extraordin ary, thus

(24)

wh ere

tan 7 Wr, longitudinal compon ent of gyrofrequency

/I collision frequency and

Using this model for the ionosphere, the reflection coefficient for a harp boundary ha been derived by Budden [25]. Adapting this r es ult to ELF it has been shown [J 7] that t he formula for S n has the same form as the isotropic medium if Ll is defined by

1 (iw) t Ll = N g + [IT cos 7/ 2. (25 )

Since [lr= wT cos 7 it is possible to write the results in the same form as the isotropic ionosphere if we set

where

(N.) _ [ (wT)e ll] t . 'l eU- .

~w

(Ni)ell and (Wr)eff are the effective values of Ni and w" respectively.

Specifically,

cos 7

(WT)err~WT (cos 7/2)2'

395 ·

(26)

In other words, the effective conductivity of the ionosphere is modified by the factor cos T/ (COS T/2)2 which varies from unity to zero as T varies from 0 to 7r/2. Thus, the attenuation is increased as a result of a steeply dipping (or vertical ) magnetic fi eld. In fact

attenuation with magnetic field cos (T /2) attenuation without magnetic field'" (cos T)t •

(27)

If T«l (i.e., WL«V ), this ratio becomes unity and the influence of the earth's magnetic field vanishes. At the level in the ionosphere where ELF waves are reflected , it is expected that wand v are comparable, both being of the order of 106• Assuming that they were actually equal, T becomes 45 ° and the ratio of the attenuation rates is l.05 . Thus, it is only when WL is greater than jJ does the earth's magnetic field appreciably influence the attenuation. The ratio is plotted in figure 7 as a function of tan T or WL / V . This illustrates the situation clearly.

o 0 --' --,w ~~ LL 6

u uf::: f:::w wZ Z <.0 5 <.0<1 <12 2

f-

i= 6 4 -I 5':::: Z5 Qz ~Q ::Jf­z<1 W ~ 2 f-w ~f­~

v.5 I.e 5 10 20

WL/ll OR ton T

50 100

FIGU RE 7. The infl uence of eaTth magnetic fi eld on attenuation.

The abscissa is the ratio of the vertical or longitudinal component of the gyrofrequency, WL , and the collision frequency v.

The preceding r esults are subject to the validity of Booker's quasi-longitudinal approxima­tion of the Appleton-Hartree equation [24]. The validity of this approximation requires that

(28)

where WL and WT are the longitudinal and transverse components of the (angular ) gyrofrequency. Clearly this condition is violated when the transverse component of the earth 's magnetic field is large such as for propagation along the magnetic equator. This case, however, has been considered by Barber and Crombie [26] who derived explicit results for the reflection coefficient at a sharply bounded ionosphere with a purely transverse magnetic field. Adapting their results to ELF, it is not difficult to show that D. should now be replaced by [17]

(29)

where

396

N ow at ELF, W«W r and thus X is very close to unity sin ce W T and jJ are of the stUne order of m:Lgnitude. 3 Consequently, a transverse magnetic fi eld appears to have a negligible effec L on t he attenuation and the phase for ELF4

A reasona ble conclusion from Lh e above is tha t only Lhe vertical compoll ent of the earth 's magnetic fi eld is effecLi ve in ELF propagation. FurLh ermore, th e ionosph ere is effectively a n isotropic conductor even in the presence of the earth 's mn.gneLie fielel .

7. Effect of an Inhomogeneous Ionosphere

The elecLron densiLy in the actual ionosphere usually in crea es lI'iLl 11 eighL i 1\ Lh e I t.' H'giol l. Furthermore, the collision fr equency decr eases with he ight. Thus tb e efrecLiY(' valu e o f Lh refractive index cannot be ass umed constant. All lLpprolLch is to le L UJC rcfr:1C'l ive index increase or decrease from some initial value N at height h ill a mono toni c fashion ." C hoosin g an exponenti al variaLion the refractiv e ind ex as a fUllc['ion of heigh L if; ex pli citl y

N( z)= 1.0 for O<z<h

= N exp [(z - h)/l l for z>h (:-;0 )

where 1 is a scale facLor. If l> O, Lhe rcfnLcLive index is in creas ing with heig hl llml , if l <"'O, th e refractive ind ex is decr easin g with h eigh t. It has been s!JOInl elsewhere [1 GJ thl1 I I he fac l-or Q has the form s

Q K o(iNkl )

if l> O Kj (iNlcl )

(8 \ )

and

Q I o( - i Nkl)

if l< O. I l( - iNlcZ)

whcre 10,11, K o. andI(] Me modifi.ed B essel fun cLions with LiJeir convenLio nal m elwing . The preceding resul ts flre valid for IN I> >1 whi ch is well satisfied a t. ELF. 1'0 th e same

approxima tion

-v [1 . wrJ} (Wr)! -; ,, / 4 1 '" - ~ - ~ - e W W

wbere wr =w5/li in terms of Lhe plasm fL freque ncy w~ and collision frequ e ncy v. LllfLt the imagin al'Y and real parts of the propagation constfLnt lcSo arc giv en b)'

a nd

where

1m lcSo=-.l (~)! P(x) 2h 2w r

Re lcSo=k+~ (~)! P' (x) 2h 2wr

P(x) =IQI sin G-arg Q) ~2

P' (x) =!QI cos (: - arg Q}v'2

(38)

I L th en follows

(34)

(:35)

3 :~ote the positive values of WT corrC'spond LO propagatio n from cast·tD-\\'C~ i while nC!!fl.tivc yal ucs corres pond to propagation froll1 wc~t -to- e!1st. " rr WT js somewhat greater than v, it js sC'C' n tha.t to a first ordl'l'

X~J-(wTM (i w/w,)112 whieh will modify the Re So to a sligh t extent bui not 1m So.

'In ihis scction the inUuenee of the carih's ma-;nl'li c nrld is nrglrcicd.

544328-60--7 397

and \\IJlcrc

X = lklNI = WI CWT/W)~ .

Thc lUHctiolls F(x) and PI(X) approach unity if :1: is sufficiently large . This limiting case cor­responds to thc homogeneous ionosphere. Thus P(x) is the modification of the attenuation and pI (:c) is the modification of the phase resulting from a nonhomogeneous ionosphere. It may be observed that the attenuation is generally lower if the refractive index is an increasing function with height. On the other hand, the attenuation is increased where the refractive index is dccreasi ll g with height.

To illustrate the behavior of tIle attenuation rates as a function of frequency, they are plotted in figures 8a and 8b in terms of decibels per 1,000 km of path length for a frequency scale frOtH 50 cps Lo 1.5 kc. Valucs of the scale distance l are indicated in kilometers. The height h of the lower edge of the ionosphere is taken to be 90 krn. The values chosen for wr are 105 and 10" "'hi ell arc not inconsistent with previous work.

- --r 'I (0)

~ 0 I 0 0.8 Q 50 "- 100 D 06 ro "Q

05 -100 50 h 0 90 km

(j 04 30 wr: 10 6

20 OJ

; =10

01 005 008 0.1 0.1 OJ 04 0.6 0.8 I 15

FREOUENCY, kc

FIG um; Sa . Altenllalion at ELF fo1' an exponential profile, 11 = 90 kin and <;; ,= 106•

10 (bl

..2 = - 20

10 -30 8

-50

~ 0 0

J - -100 Q "-D "Q 1 -

<Xl

" 100

50

0.8 30

0.6 .i = 20 ~p 0.5 -0.4

0.050.06 0.08 0.1 0.1 OJ 04 06 08 I 1.5

FREOUENCY, kc

-- CALCULATED

- --- MEASURED (CHAPMAN AND MACARIO)

FIG U RE 8b. A ttenuation at ELF f 01' an exponential p"oftle, h = 90 bn, -;;', = 105•

The dashed curve corresponds to nighttime experim cntal data of Chapman and Macario.

It is seen that lhe curve for l= 00 corresponding to a homogeneous ionosphere has a slope of 1/2 as it should. Depending on the sign of l and its magnitude, the slope may be modified considerably for Lhe inhomogeneous ionosphere. The dotted curve in figure 8b corresponds to nighttime experilllental data from Chapman and Macario [27] who obtained it from the spectral analyses of large numbers of atmospherics recorded in London. Clearly this experimental curve appears to fit fairly well on the l = 30-km curve, at least in the range 100 cps to 1.5 kc . 'rhere is 110 reason Lo expect any better fit than this because of the idealized profile assumed. Fmthermore, the atmospherics analyzed by Chapman and "NIaeario were often CJuite near the receiving location in terms of a wavelength.

8 . Propagation of ELF Pulses

In Lile foregoillg discussion, it has been assumed that the source is time harmonic. In most cases of practical interest, the current in the source dipole is of a transient nature such as a surge. The radiated field s are also transient in nature. While the resulting waveforms may be transformed to the frequency plane via spectral analyses, it is often more convenient to study the wlLveforms themselves [28 to 31].

398

The source is assumed again to be equivalent to a vertical electric dipole on the earth's sUl'face, but now its current-moment is a function of time and is denoted pet) which is zero for t< O. At a distance p along the earth 's surface, t he resulting vertical electric field is ez(t) and the horizontal magnetic fi eld is h.p (t). The Laplace Transform of the CUl'rent moment is denoted Po(s) or more explicitly

where S is the Lransform variable and may be formally identified with iw. Also

E z(s) = L ez(t )

and

(36)

where L is the Laplace-Transform operaLor defined above. Now iL is assum ed that Lhe im­portant frequencies in Lhe spectra are sufftcienLly low that only the zero-order mod e need be retained. Thu

(37)

wh ere

SO= [l +.9 (~)~J ~ h SfT i

and wilere Ko and KI are modified Bessel functions of argumenL sSop/c. to ti cally represen ted by

These may be a ymp-

K ( 2)! - x [ AD Bo ] . o(x)~ 7rX e 1+-X+ x2+ . ..

and

T7 () (2 )~. -x [ Al BJ ] Ll. I X ~ 7rX e 1+-X+ ;}+ .. .

for large x, where

A o= - 1/8, B o= 9/128, AJ = 3/8, B I=- 15/128.

Thus

where x has been replaeed by sSop/c in the exponent but is replaced by sp/c elsewhere. justified since So is near unity. In fact, the binomial expansion

So= l+~ ~ 2 _ _ ~ ~+_ ~ ~ + ( ) J. 1 ( )2 1 ( )3 ( )3/2 2h SfT i , 8 h SUi 16 h SfTt • ••

(38)

(39)

(40)

This is

(41)

may be truncated beyond the second term in most cases of interest, although the third term is retained since it doesn't complicate matters.

The current moment is now taken to be an impulse, that is

544328- 60-

pet) = Poo(t)

399

where oCt) is the unit impulse function . Thus

The transforms may now be readily inverted

e~ (t) J.L (C)! Po . [pc (eo)] [ (tl) Ao(ctl) (tl) Bo(Ctl)2 (tl) ] - TJh~(t)~27r2h 2p /33/2exp 811,2 (J i P (j + AI P Pa 7i + BJ P Pb 7i + ... u(t - p/c)

where

a nd

P(T)=(2~- I) (~yI2exp ( -~t)

Pa(T)=2(~ y l2exp ( - 4~)

(1)312 ( 1) 1 J (1) P b(T) = 4 - exp -- -2 - 7r i erfc -T 4T T 2 2T!

I 1 P EO "2 ( )

1

t = t - p/c and /32=- - . 211, (J i

(42)

(43)

(44)

(45)

It is immediately seen that if (ct' / p) < < 1, the field responses are both proportional to peT). To illustrate the nature of this function, it is plotted in figure 9 as a function of T. To facili­tate the application a multiple scale is included which relates actual time t ' in microseconds with the parameter T for distances p of 1,000 to 4,000 km. The height of the ionospherir, reflecting layer is taken as 70 and 90 km. The quantity B indicated on the curves is related to the ionospheric conductivity by

B=_I _='2'7rc. 60(Jih wTh

The values of B shown, namely 0.05 and 0.1, are typicftl. For 11, = 90 km, these correspond to W T values of 4 X I05 and 2X I05, respectively.

The responses ez(t) and h",(t) to a general source p et) can be expressed in terms of the im­pulse responses e; (t) and 11,; (t), respectively, by using the convolution theorem. Thus

1 It ez(t) =- p(t-T) e~(T) dT'U(t - p/c) Po pIc

( 46)

and similarly for h",(t) . For example, if a special analytical form for pet) is chosen such as

(47)

the convolution integral may be evaluated to give

(48)

where

The function peT) was defined above and is the same as the one plotted in figure 9. Now, however, T is to be identified with t' h rather than t' //3 . The special form for pet) given above is a unidirectional pulse which reaches its maximum value at t= tm and decays to 7 percent of

400

1

I -

!::; 0..

0 r

9 1-\ 8

1 / 6

\ 5

4

3

1

I

0

I I II --

e·O ·Os

Q ,

10 QOS

·1 V> -0 c 0 u

'" V>

e 10' u

~ '" E

f= ..:

10'

10' ~_L----::-!::--.l...C-'--:~ ___ -;:-;--~ 0.01 0.04 0.1 0.4 1.0 4.0

T

FlGUlm 9. T mnsient l'es ponse of the ze" o-or-del' mode lOT

an 1:mplllsive veTticai dipole sow ·ce. ~rhc 10wcI' chari facilitates conversion fro m the parameter 'T' to actual

t ime l',

its maximum value at t = 3tm . If the duration , 3tm, of the source pulse is much less than (3

the response e,(t) approaches eO (t). This is not surprising since p et) approaches Poo (t) if t", approaches zero.

The general effect of the finite duration of the ource pulse is seen to "stretch" the waveform of the electric and magnetic field. This effect has been discussed at some length in a recent paper by the author entitled "On the theory of the slow-tail portion of atmospheric waveforms ." This is to appear in the July issue of the JourllfLl of Geophysical Research.

While most cloud-to-ground lightning strokes may be represented by a vertical electric dipole, it is believed that cloud-to-cloud discharges are better represented by a horizontal electric dipole [32) . In the ELF portion of the spectrum the height of the discharge is very small compared with the wavelength, thus eq (17) is valid. If the source moment is again p(t) then the transform for the vertical electric field, in the far zone, is given by

E,(s)~~ (~).~ (£)! sPG(s)e -Ssop/c (to)! cos cf> 27rh 7r P (T i

where cf> is the angle ubtended by the horizontal dipole and the direction to the observer. Letting the source be an impulse [i. e., p (t) = Poo(t)), t he transform may be readily inverted to

e,(t)~2:2p (2CpY ;~2 (::Y Q(T) (49)

401

where

Q(T) = T~/ 2 (2~-3 ) exp ( - ;T} (50)

and where, as before,

T= t' /(3 and (32=- -- . J. p (EO)t 2h U i

An immediate generalization is a source which may be imagined as an inclined electric dipole. Lightning discharges from cloud-to-ground and cloud-to-cloud would be seldom purely vertical or horizontal. In view of the small dimensions of the discharge paths in terms of a wave­length, it is permissible to replace the inclined channel by superimposed vertical and hori­zontal electric dipoles. The resulting responses of the radiated field are thus obtained by super­position. For example, if the current moment of the vertical electric dipole is Poo(t) then the current moment of the horizontal electric dipole is gPoo(t) where 9 is a positive or negative dimensionless number. For lightning strokes with long horizontal sections, 9 could be large compared to unity [32]. As before, the direction cp = O corresponds to the direction of the horizontal dipole component.

Invoking the above mentioned principle of superposition, it readily follows that the far zone expression for the inclined dipole is given by

ez (t)~;7r~h (2cpY [P(T) + GQ(T)](3-3/2

where

(U ')~ h G= g -2. - cos cp. Ug P

To indicate the bahavior of the waveform, the quantity

SeT) P(T) + GQ(T)

l +IGI is plotted in figure lOa and lOb for both positive and negative values of G. The quantity SeT) characterizes the waveform of the inclined dipole energized by an impulsive current. The special case G= O, such that S(T) = P(T) , corresponds to a vertical dipole source, whereas the special case G= <Xl corresponds to a purely horizontal dipole source. Such a range of G values can be expected since cloud-to-ground and intra-cloud discharges are primarily vertical , whereas the cloud-to-cloud discharges and air flashes are mainly horizontal and may have either polarity.

100 20

(0 ) (b)

80 0

60 -20 >-(f)

w -40 (f) 40 z

0 Q (f) w -60 0: 20

0 -80

- 20 L--L---'--L.Ll.J...l..ll_---,-L---,-L...L.l:::'--'-~__=' -100 L----'----'-----'---L.LJL.Lll_---::-'::---'-...LJ:::'--'-~__=' 0.01 0.02 0.05 0.1 0.2 0.5 1.0 2.0 0 .01 0.02 0.05 0.1 0.2 0.5 1.0 2.0

T

FIGURE 10. Transient j'esponse of the zero-order mode for an inclined dipole source. Wben 0=0 tbe dipole is ver tical and wben G=± <» the dipole is borizontal.

402

The modification of the pulse shape of the ELF waveforms, as a result of the inclination of the current channel, would appear to be an important factor in interpreting observed data. For example, Pierce [9], Lieberman [29], Tepley [3 1], and J ean [33] have all observed ELF waveforms which are strikingly similar to those shown in figures lOa and lOb. For example, the observed reception of a pulse which has a second half-cycle of the same magnitude as the first half-cycle can only be reconciled with an inclined so urce with G equal to about ± 0.2 . For certain values of G (e.g., small negative val ues), a third half-cycle of relatively small amplitude is also produced .

9. Concluding Remarks In the present theoretical study it has been assumed that the ELF signal propagates entircly

in the space between the earth and a concentric ionosphere. It is quite possible that whistler type propagation may also be importan t. Furthermore, so urces of ELF energy may be prese nt in the exosphere due to the possible existence of ionized hydrogen whose gyrofrequency is of the order of 900 cps [34]. Other types of ions such as sodium may also be significant as pointed ouL by Aarons [14] . D espite these extraneo us factors, the evidence that the "slow tails" arc propa­gated in the earth-ionosphere waveguide is overwhelming.

It is the opinion of the author that the only really ques tionable assumption is the neglect of heavy ions on the co nstitutive proper ties of the lower edge of the E region or the top of the D region. Normally, it would be expected that the co nstiLutive properties of the ionosphere would only be influenced by the electro ns because of the extremely large ratio between the ma ses (i.e., of the order of 107). However, if the ratio of the number of heavy ions to the number of electrons is of the order of 103 or 10\ as has been suggested by Scott [35] for the Arctic E layer , the influence of heavy ions may be significant at frequencies less than 100 cps. In parLicular , it should be noted that the gyrofrequency of the sodium ion is approximately 30 cps. However , if the operating frequency is somewhat greater than the gyrofrequency of heavy ions, their influence is expected to be minor.

The author is indebted to Mrs. Nancy Carter and Mrs. Alyce Conda who carried out many of the calculations, to Mrs. Patricia Murdock who typed the manuscript, and to Mrs. Barbara Bolton who prepared the illustrations.

10. References [1) iV. O. Schumann, iJbcr die Ausbrcitung schr langer elckLrischer Wellen um die Erde und die Signale der

Blitze, N uovo Cirnento I X (1952). [2) W. O. Schumann, iJber d ie Oberfelder bei del' Alisb reitling la nger, elektriseher Wellen im System Erde­

Luft-Ionosphare und2 Anwend ungen (horizon taler und se nkrechter Dipol), Z. angew. Phys. 6, 34 (195'1). [3) VV. O. Schumann, Dber d ie Strahlung langer Wellcn des horizontalcn D ipols in dem Lufthohlraum zwischen

Erde und Ionospha re Z. angew. Phys. 6, 225 (1954). [4) J . R . Wait, On t he mode theory of VLF ionosphe ri c propagation, R eview Geofis ica pura e applicata 37,

103 (1957) (paper prese nted at Intern . URSI Conf. R ad io Wave Prop. Paris, France, Sept. 1956). )- [5) L. Liebermann, Anomalous propagation below 500 cis, Symp . Prop. VLF R adio Waves, 3, paper no. 25,

Boulder, Colo. (1957). [6) R . E. Holzer and O. E. Deal, Low a udiofrequency electromagnetic s ignals of natural origin, Nature 177,

536 (1956). [7) O. E. Deal, The observation of very low frequ ency electromagnetic s ignals of nat ural origin (unpu blished

Ph. D. Thesis, Univ . Calif. , Los Angeles, Calif. , 1956) [8) F. W. Chapman and 'V. D . Matthews, Alidiofrequency spectrum of atmospherics, Nature 172, 495 (1953). [9) F. H epburn and E. T . Pierce, Atmo pheri cs with very low frequency componen ts, Nature 172, 837 (1953).

[10) H . F. Willis, Audiofreq uency magnetic flu ctuations, Nature 161,887 (1948). [11) J. Aarons and M . H enissar t, Low frequency n oise in t he range 0.5 to 20 cis, Nature 172, 682 (1953) . [1 2) R . Beniot, Low frequency radio wave noise of t he eart h 's magnetic fi eld , C. R. Acad . Sci. 242, 2534 (1956). [13) P . A. Goldberg, Electromagnetic phenomena of natura l origin in t he 1.0- 150 cis band, Nature 177, 1219

(1956) . [14) J . Aarons, Low frequ ency electromag netic radiat ion 10 to 900 cis, J . Geophys. Research 61 , 647 (1956). [1 5) J . M. Watts, An obser vation of audiofrequcncy electromagnetic noise during a period of solar disturbance,

J . Geophys. R esearch 62, 199 (1957).

403

[16] J. R. Wait, An extension to the mode theory of VLF ionospheric region, J . Geophys. R esearch 63, 125 (1958) .

[17] J . R. Wait, Terrestrial propagation of VLF radio waves, J. R esearch NBS 640 , 153 (1960). [18] H. Motz, Electromagnetic problems of microwave theory (Methu en & Co., London, 1(51). [19] H. R. L . Lamont, Waveguides (Metheun & Co., London, 1942). [20] W . Magnus and F. Oberhettinger, Special function s of mathematical phys ics (Chelsea Publishing Co.,

New York, N.Y., 1(49). [21] C. L. Pekeris, Accuracy of the earth-flattening approximation in the theory of microwave propagation,

Phys. Rev. 70, 518 (1946). [22] B. Y.-C. Koo and M. Katzin , An exact earth-flattening procedure in propagation around a sphere, J.

Research NBS 640 , 61 (1960). [23] J . R . Wait, The mode theory of VLF ionospheric propagation for finite ground conductivity, P roc. IRE 45,

760, (1957). [24] J . A. Ratcliffe, Magneto-ionic theory (Cambridge Un iv. Press, 1(59). [25] K. G. Budden, The reflection of very low frequency radio waves at the surface of a sharpl y bounded

ionosphere with superimposed magnetic field, Ph il. Mag. 42, 504 (1951). [26] N. F. Barber and D . D . Crombie, V.L .F. reflections from the ionosphere in the presence of a transverse

magnetic fi eld, J. Atmospheric and T errest. Phys. 16, 37 (1959) [27] F. W. Chapman and R . C. V. Macario, Propagation of audiofrequency radio waves to great distances,

Nature 177, 930 (1956). [28] W. O. Schumann, Uber die zeitliche Form und das Spektrum ausgesendeter Dipolsignale in einer d ielectri­

schen Hohlkugel mit leitenden Wanden, Verlag Bayerisc hen Akad . Wiss ., Munich (1956). [29] L . Liebermann, Extremely low freq uency electromagnetic waves, II,propagation properties, J. App!. Phys.

27,1477 (J956). [30] J. R. Wait, Propagation of very-low-frequency pulses to great distances, J. R esearch NBS 61, 187 (1958)

RP2898. [31] L. R. T epley, A comparison of sferics as observed in the VLF and ELF bands, J . Geophys. R esearch 64,

2315 (1959). [32J E. T . Pierce, The development of ligh t ning discharges, Quart. J . Roy. Meteoro!. Soc. 81, 229 (1955). See

also, H. Norinder and E. Knudsen, Analysis of daylight photo~raphs of lightning discharges, R ecent Advances in Atmospheric Electricity, p. 503 (1958).

[33] A. G. J ean, private communication. [34] J. W . Dungey, The physics of the ionosphere, p. 22!J (The Physical Society, 1(55). [35] J . C. W. Scott, The gyro-frequency in the arctic E layer, J. Geophys. R esearch 56, 1 (1951).

10.1. Additional References H. Bremmer, Terrestrial radio waves (Elsevier Publishing Co., Amsterdam, Netherlands, 1949). W. H. Campbell, A study of micropulsations in the ear t h 's magnetic fi eld, Sci. Rep. No.1, Nom 233 (47), Univ.

of Calif. Inst. of Geophys., Los Angeles, Cal if. (April 1(59). D. D. Crombie, D ifferences between east-west and west-east propagation of VLF signals over 10llg distances, J. Atmospheric and T en ·est. Phys. 12, 110 (l!J58) . H . J. Duffus, P . W. Nasmyth , J. A. Shand, and C. S. "'hight, Sub-audible geomagnetic fluctuat ions, Nature

181, 1258 (1958). R. M. Gallet, A very low frequency emission generated in the earth 's exos phere, Proc . IRE 47, 211 (1959). A. L. Hales, A possible mode of propagation of the "slow" or tail component in atmospherics, Proc . Roy . Soc.

A, 193, 60 (1948). C. O. Hines, Heavy-ion effects in aud io-frequency radio propagation, J . Atmospheri c and Terrest. Phys.

11, 36 (1957). (Shows that whistler ray direct ions call differ marked ly from the di rection of the earth's magnetic fi eld when heavy ions are considered).

H . Kaden, Die Reflex ions und Sch irmwirkung metallischer Hi.Ulen in einer ebellen elektromagnetischen "Veli e, Sonderdruck Arch E lek Ubertragungen, p. 403 (1957).

'V. O. Schumann and H. Konig, Uber die Beobachtung von "atmospherics" bei geringsten Frequenzen, Natur­wissenschaften 41, 183 (1954).

' V. O. Schumann, Uber elektrische Eigenschwingungen des HohJraumes Erde-Luft-Ionosphare angeregt durch Bli tzentladungen, Z. angew. Phys 9, 373 (1 !J57).

L. R. O. Storey, A method to detect the p!'esence of ioni zed hydrogen in the outer atmosphere, Can. J . Phys. 34, 1153 (1956) . (Discusses the effect of protons on the propagation of whistlers in the exosphere, collisions are neglected.)

A. D. Watt and E. L . Maxwell, Characteristics of atmospheric noise from 1 to 100 kc, Proc. IRE 45,787 (1957).

Additional numerical data are available in, Field strength calculations for ELF radio waves, by J . R. Wait and N. F. Carter, NBS Tech . Note 52 (PB161553) 1960; available from t he Government Printing Office, Washington 25, D .C., for 50 cents.

(Paper 64D4- 72)

404