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SEP 9 1963

^ecUnlcai v^ote 92o. /«

A NOTE ON ANTIPODAL FOCUSSING

JAMES R. WAIT

-

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NATIONAL BUREAU OF STANDARDS

'Cechnical ^ote 182

Issued August 20, 1963

A NOTE ON ANTIPODAL FOCUSSING

James R. Wait

Central Radio Propagation Laboratory

NBS Boulder Laboratories

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U. S. Government Printing Office Washington, D.C. 20402

Price 150

A NOTE ON ANTIPODAL FOCUSSING

James R. Wait

There has been some interest shown recently in the practical significance of the antipodal

focussing in VLF propagation. If the earth and the ionosphere were perfectly concentric spherical

surfaces, theory indicates that the amplitude of the field should build up to a maximum at the geographic antipodal point of the transmitter. In the language of optics, this point may be called an axial

caustic.

Experimental evidence of some manner of focussing at the antipode was obtained by Round,

Tremellen, Eckersley, and Lunnon [1925] in an early series of measurements. Their field strength

data were obtained at frequencies in the range from 20 kc/s to 30 kc/s. More recently, Crombie [1958]

and Bickel, et aL,[1963] have found clear evidence of antipodal focussing. As predicted by theory [e. g. , Wait, 1962], the field strength E was found to vary approximately as the inverse square root of the distance from the antipode. There was also some indication that a standing wave pattern existed in the antipodal region. Unfortunately, this pattern did not always exhibit a consistent form and the spacing

between the nulls departed considerably from the half -wavelength expected on the basis of simple theory.

It has been indicated by Crombie [1963] and by Bickel [1963] that asymmetry in the earth-ionosphere

cavity may impair the focussing to some extent. Generally,- they used arguments based on geometrical

optics.

It is the purpose of the present paper to discuss this matter further. To shed some light on the

subject, a very simple perturbation of the ideal model is considered. Essentially, it is assumed that the illumination of the antipodal region is non-uniform.

The region around the geographic antipode is depicted in Fig. 1. For present purposes, the antipodal region may be regarded as a radial transmission line and thus local cylindrical coordinates

(p, <j) , z) are convenient. The earth' s surface is z = ; while the effective lower boundary of the ionosphere is h. For some concentric region p = p , it is assumed that the ionosphere height h

and its electric properties are constant. Furthermore, the surface impedance of the earth 1 s surface

at z = is regarded as a constant. This assumption of local uniformity around the antipode greatly

simplifies the subsequent discussion.
An appropriate form of solution for the vertical component of the electric field, for a waveguide

mode of order n, is given by

+ 00 E

(n) (p,0,z) = Y A J (k P S)elm *f (z) , (1)

/_, m m n m, n

' m= - °°

where J is the Bessel function of the first kind of order m and argument kp S in which k S m n n

is the propagation constant for a waveguide mode of order n. The function f (z) is a height-gain m, n

function for mode n and its specific form is of no concern at the moment; however, in general, it depends on both m and n. If the field in the antipodal region were perfectly symmetrical, d/d = and then

E

<n) (p,0,z) = A J (kpS)f (z). (2) o o n o, n

This particular situation has been discussed extensively in the literature [e. g. , Wait, 1958, 1962; Norton, 1959]. In most applications to VLF propagation to great ranges, it is permissible to consider

only one mode and thus the affix n may be dropped in what follows.

Equation (2) has a clear physical significance when J is replaced by the first term of its asymptotic

expansion. Thus for |kpS| » 1,

o

-=— ikpS -in/4 -ikpS in/4

-In-kjhr

~ ^ — • < 3 >

which can be regarded as the superposition of two travelling waves.

At the radial distance p = p , the field has the form

o+» E(p,0,z) = Sa J (kp S) e lm0

f (z) , (4) o l_j m m o m

m=-°°

where the notation has been simplified in accordance with the discussion above. On physical grounds we will now assume something about the variation of E(p , 0, z). Then, the resulting form of E(p, , z), o when p < p , will be discussed,

o By multiplying both sides of equation (4) by e and integrating with respect to 0, over the range to 2 rr , one is lead readily to the formula

2tt

1 m 2tt J (kp S) m o

j" E(p

o ,0,z) e" im0 d0 , (5)

-2-
if one notes that 2n

2n if P »<m-m')« d0 „ f „ I if m * m' (6)

Thus, formally, the coefficient A may be found from the specific form assumed for the electric

field around the ring p = p .

As a simple example, which leads to an embarrassingly simple formula, it will be assumed that

the illumination on the ring (at p = p ) is constant except for a finite sector. Such a situation could

be envisaged by imagining that the energy, propagated from the transmitter, passed, in part, over a very lossy region of the earth. Ignoring any diffraction effects, it will then be supposed that the great

circle paths in the angular region 0j < < 2 are paths of complete absorption. Thus, over this

interval on the ring p = p ,

- i(kp S - tt/4)

which is the contribution from the long great circle paths which are propagating from the direction

rr +01 ></>>rr + 2 away from the antipode.

Using equation (5), it now follows that

21T Am m'frp. I_(kp_S)e—V

m o

o

TT J (kp S) \ J

J (k|

m o *- o

+ f (Vtp) - J (kp S)le- im0 d0

\. L o o o J

01 + 2 y [E "<p

> - j

o (kpo

s)]e

" im</,d 4

01 TT+0s +

3

TT+0!

where

e^p ) - j (k P s) . - G-nhnr)

2 * ±i(kp

° s " n/4)

(9) o o o v<Jnkp by

The integrations can be carried out readily to yield, for m ^ 0,

-3-
A

1 1 j'^i(k p^S - n/4) ^-im0o m

= " 2nJ (kp S) (2n k p s)It'

-i(kp„S-n/4) imn _i„.

+ e vpo e e

r ° A (|iA • (10)

where

O

= (*! + 0s)/2 >

and

sin (m A 0/2)

m " (mA0/2) *

In view of the already imposed condition that | kp S| >> 1, the Bessel function J in the denominator

o m

may be replaced by its asymptotic expansion. Thus, for m ^ 0,

i mn/2 , Am " --^^ e

" lm0° A ^ Am • (U) m £ TT m

Within the same approximation, it readily follows that

Ad) A

q . 1 - jl . {12 )

Employing the above explicit formulas for the coefficient A , the resulting behavior of the field

in the vicinity of the antipode may be deduced from equation (4). To illustrate the nature of the problem,

it is assumed that A can be replaced by unity. This would be well justified if A is small compared m

with unity and if kpS is not too large. At the same time, the height-gain function f (z) is replaced by m

unity. With these simplifications,

E(p,0) » Jo (kpS) - qe

ik P Scos(0-0o

)

t (13)

where q = A0/2JT. it is quite clear that the total field is the superposition of the ideal symmetric

antipodal field and a plane wave, of relative amplitude q, incident from the direction </> = . The

negative sign preceding q is to indicate that the plane wave field is to be subtracted from the ideal or undisturbed antipodal field.

In a certain sense, the plane wave term can be regarded as the most basic type of perturbation. In

fact, a generally perturbed field could be written in the form

-4-
E(p.0) B J(kpS) - Y q e ikpScos(«o -0s

) (H)

o / , s

s

where the summation over planes incident at various angles = 0. Furthermore, the coefficient

q may be complex to account for relative phase shifts between the perturbing plane waves and the

ideal antipodal field.

In order to characterize the amplitude behavior of the resultant field for a single disturbing plane wave, the function

E(x) = J

q

(x) - q e r

, (15)

is considered, where x = kpS = 2 tt p/\ in terms of the effective wavelength X . While strictly

speaking, S has a small imaginary part [Wait, 1962], it may be regarded as real for the present

discussion.

The magnitude of E(x), expressed in db, is plotted in Fig. 2 for q = 0. 1 for a range of azimuth angles 0. Because of certain symmetry properties, the curves are identical at and

o

180 - 9. Also, although not shown explicitly on the curves, symmetry exists about the direction of 0=0. Thus, the results in Fig. 2 are also applicable for negative angles.

The curves, in Fig. 2, for 0=0 (and 180 ) are almost identical to the ideal antipodal field pattern characterized by the Bessel function J (2np/\ ) alone. Here, the variation along the radial

is in the direction of propagation of the perturbing wave. At other angles, it is apparent that the pattern becomes distorted and the spacing between nulls is modified to some extent.

The effect of varying q, the amplitude of the perturbing wave is shown in Fig. 3. For these curves, is fixed at 45 . It is evident that for the larger q values, the deep nulls disappear. In

this situation the symmetric J antipodal field can be regarded as a perturbation to the plane wave

field.

The graphical results shown in Figs. 2 and 3 are intended to indicate merely that the field pattern near the antipodal may become extremely complicated. Many other intriguing patterns can be

generated by using other combinations of the parameters. Also, equally interesting, are the resulting

phase variations near the antipode which may be calculated from equation (15).

I would like to thank Mrs. CM. Jackson for carrying out the calculations shown graphically in Figs. 2 and 3.

-5-
References

Bickel, J. E. et al.(1963). Personal communication.

Crombie, D. D. (1958). Differences between east-west and west-east propagation on VLF signals over

long distances, J. Atmos. and Terrest. Phys. _12_, 110-117. Crombie, D. D. (1963). Personal communications.

Norton, K. A. (June 1959). Transmission loss in radio propagation, Part II, NBS Technical Note

No. 12. Round, H. J. T. , T. L. Eckersley, K. Tremellen, and F. C. Lunnon (1925). Report on measurements

made on signal strength at great distances during 1922 and 1923 by an expedition sent to Australia,

J. Instn. Elect. Engrs. 63_, 933-1011. Wait, J. R. (1958, I960). Diffractive corrections to the geometrical optics of low frequency propagation, Proc. Liege Conference on Wave Propagation (1958); also published in Electromagnetic Wave

Propagation, edited by M. Desirant and J. L. Michiels (Academic Press, I960).

Wait, J. R. (1962). Electromagnetic waves in stratified media, Chapters VI and VIII (Pergamon Press,

London, New York).

-6-
*~p

Fig. 1 Coordinate system for antipodal region.

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