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 THE PEJfNsnVANlA STAl» 
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 92o. /S2 
 
 A NOTE ON ANTIPODAL FOCUSSING 
 
 JAMES R. WAIT 
 
 U. S. DEPARTMENT OF COMMERCE 
 NATIONAL BUREAU OF STANDARDS 
 
THE NATIONAL BUREAU OF STANDARDS 
 
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NATIONAL BUREAU OF STANDARDS 
 
 technical ^ote fS2 
 
 Issued August 20, 1963 
 
 A NOTE ON ANTIPODAL FOCOSSING 
 
 James R. Wait 
 Central Radio Propagation Laboratory 
 NBS Boulder Laboratories 
 Boulder, Colorado 
 
 NBS Technical Notes are designed to supplement the Bu- 
 reau'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 o£ Documents, U. S. Government Printing Office 
 
 Washington, D.C. 20402 
 
Digitized by the Internet Archive 
 
 in 2012 with funding from 
 
 LYRASIS IVIembers and Sloan Foundation 
 
 http://www.archive.org/details/noteonantipodalfOOwait 
 
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 geo- 
 graphic 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 aLjl963] 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 imjjair 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 = 0^ while the effective lower boundary of the 
 ionosphere is h. For some concentric region p S p , it is assumed that the ionosphere height h 
 and its electric properties are constant. Furthermore, the surface impedance of the earth' 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 
 
 E^'^^p.^.z) = y A J (kpS)e^"^*^f (z) , (1) 
 
 /^ nn m n m, n 
 
 m= - ■» 
 
 where J is the Bessel function of the first kind of order m and areument 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 
 
 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, 3/3 </> = 
 and then 
 
 E^"\p,d),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, 
 
 f = — ikpS-in/4, -ikpSin/4 
 
 ^„<^'=i ~ ^^ '- 1^ '— ■ "I 
 
 which can be regarded as the superposition of two travelling waves. 
 
 At the radial distance p = p , the field has the form 
 
 o 
 
 +0O .. . ; . 
 
 E(p ,(/),z) = y A J (kp S) e'"''^ f (z) . (4) 
 
 o /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 ,(/), z). Then, the resulting form of K(p,<j) , z), 
 
 when p < p , will be discussed, 
 o 
 
 - i m' . 
 
 By multiplying both sides of equation (4) by e and integrating with respect to (p , over the 
 
 range to 2 n , one is lead readily to the formula 
 
 2n 
 
 ^ = ?u t\vo 9^ C E(p ,</.,z)e-^"^'^ d.?. , ■ (5) 
 
 m 2nj(kpb) J o 
 
 
 
 -2- 
 
if one notes that 
 
 2n 
 
 r i(m-m')</. ^^ ^ r 
 
 ^ I if 
 
 2n if m = m' 
 
 m ?^ m' (6) 
 
 Thus, formally, the coefficient A may be found from the specific form assumed for the electric 
 
 m 
 
 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. Ignoriag any diffraction effects, it will then be supposed that the great 
 circle paths in the angular region (j)^ < (j) < ^^ are paths of complete absorption. Thus, over this 
 interval on the ring p = p , 
 
 2 
 
 -'V*-' • (^s)=-'"'"'"-"'"-o. <" 
 
 \which is the contribution from the long great circle paths which are propagating from the direction 
 n +(/ij>0>n + 02 away from the antipode. 
 
 Using equation (5), it now follows that 
 
 2n 
 ^ ^^ J (kp S) e-^"^*^ d.^ 
 
 m 2n J (kp S) 1 J o' ^o 
 
 m o >- 
 o 
 
 IW 
 
 ) - J (kp S)|e"'"''^ d0 
 o o J 
 
 
 where 
 
 (p ) . J (kp S) . - (^-^-^^ e^^^^Po^ - "/^> (9) 
 
 o oo v2TTkpSy ^ ' 
 
 The integrations can be carried out readily to yield, for m ?i 0, 
 
 -3- 
 
m 2nJ (kp S) i 
 
 m o (<: rr 1^^ p s)"^ 
 
 o 
 
 i(k p S - n/4) - im<i_ 
 
 , - i (k p„ S - n/4) i m n - i ] 
 + e o e e 
 
 '^o 
 
 A A . (10) 
 
 m 
 
 where 
 
 0^ = (<^: + 0^)72 . 
 
 and 
 
 ^ ^ sin (mA(^/2) 
 m (nn A(/)/2) 
 
 In view of the already imposed condition that | kp S| >> 1, the Bessel function J in the denominator 
 
 o m 
 
 nnay be replaced by its asymptotic expansion. Thus, for m ?^ 0, 
 
 imTf/2 . . 
 
 ^^ ^ -^T^ e-^""*^o A./.A . (11) 
 
 Within the same approximation, it readily follows that 
 
 A0 
 o ~ 
 
 A - 1 - 2^ • (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 
 
 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,,^) . JJkpS) - qe^kpScos(0-0„) _ ^^3j 
 
 where q = A(/)/2tt, 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 (j) = (j) . 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 
 
E(p.(^) s J(kpS) -\ q e^kpScos{(/.„-(/.3) ^ ^^^^ 
 
 O / / s 
 
 s 
 
 where the summation over planes incident at various angles (^ = (/) . 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 (x) - q e , (15) 
 
 o 
 
 is considered, ■where x = kpS = 2 rr p/x in terms of the effective >vavelength 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 (/) . Because of certain symmetry properties, the curves are identical at (j) and 
 180 - 0. 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 <j) = (and 180 ) are almost identical to the ideal antipodal field 
 pattern characterized by the Bessel function J (2TTp/x ) 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 beconne 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. C, M. 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 19 59). 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 (19 58); 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- 
 
^yo 
 
 Fig. 1 Coordinate system for antipodal region. 
 
 -7- 
 
^ 
 
 ^ 
 
 •i-i 
 
 

 o 
 
 Li_ 
 
 UJ 
 U- 
 
 < 
 
 Q 
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 CL 
 
 3 
 LJ 
 
 q: 
 
 tuO 
 
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 GPO 835-528 
 
 -9 
 
U. S. DEPARTMENT OF COMMERCE 
 
 Luther 11. Hodges, Secretary 
 
 NATIONAL BUREAU OF STANDARDS 
 A. V. Astin, Director 
 
 THE NATIONAL BUREAU OF STANDARDS 
 
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