. The Bell System technical journal . clature is indicated in Fig. 17. The length of the element of path, hh\ equals rdOsin if (1) and the time required for the wave to traverse it is dt = rdd V sin ip nr ddc sin ^ (2: 2 It is inferred from a statement made by Jouaust, Froc. I. R. E., Vol. 19, p. 487,Mar. 1931, that for his experiments between France and Corsica p/ro would haveto be 5 or less in order for the direct ray to be unobstructed. Our figure of 4 thereforewould indicate that his is an optical path. We do not believe, however, thatan optical path is a necessary or sufficient condition
. The Bell System technical journal . clature is indicated in Fig. 17. The length of the element of path, hh\ equals rdOsin if (1) and the time required for the wave to traverse it is dt = rdd V sin ip nr ddc sin ^ (2: 2 It is inferred from a statement made by Jouaust, Froc. I. R. E., Vol. 19, p. 487,Mar. 1931, that for his experiments between France and Corsica p/ro would haveto be 5 or less in order for the direct ray to be unobstructed. Our figure of 4 thereforewould indicate that his is an optical path. We do not believe, however, thatan optical path is a necessary or sufficient condition for strong signals, although itcertainly does help to make them probable. ULTRA-SHORT WAVE PROPAGATION 157 (f is velocity of light and « the refractive index, which is assumedapproximately equal to one at point a). But by Pedersens equation 9 7ir sin <p = To cos \{/. f3) Hence dt = n^r tqc cos \{/ dd. (4) Now, Eccles ^^ has shown that when the dielectric constant varieswith r (distance to center of earth) as follows: n = «+i (5). Fig. 17the solution for the path of the ray is ro cos (sd — \p) ^ r cos \p. (6) Here 5 is a constant and ro is the distance to the center of the earthfrom a, the arbitrarily fixed point of reference in the path, ro is there-fore not very different in our case from the radius of the combining (5) with (6) we find that nr = /o cos xf/ cos {sd — xp) which when substituted in (4) gives fo cos xp dd dt = c ^0 Electrician, 71, 969-970 (1913). cos^ {sd — yp) (7) (8) 158 BELL SYSTEM TECHNICAL JOURNAL integrating which from 6 = Q to 6 we obtain t - to ^ ro_cosj/ ^^^^ ^^^ - V) + tan ,A]. (9) If 5 in (5) is made somewhat less than one in absolute value and isnegative, we obtain a fairly good approximation of the distributionactually encountered. The exponent 5+1 has then a small positivevalue and from (5) 1 r (10) dnjdr n{s + 1) Since n is very close to unity and since we may assume p/r = (Appendix II), we find that 5 = — This value
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