The transmission of electric waves round the Earth

On the hypothesis that the Earth consists of an imperfectly conducting sphere surrounded by infinite homogeneous dielectric, I have recently obtained a complete solution (in a form adapted for numerical computation) of the problem of determining the effect at a distant point of the Earth’s surface due to a Hertzian oscillator emitting waves of a definite frequency. Previous investigators had obtained approximations (some of which were incorrect) to the dominant terms of the series which represents the effect due to the Earth, but the earlier approximations cease to be valid in the neighbourhood of the antipodes of the transmitter. On this hypothesis the absolute value of the Hertzian function (with the time-factor suppressed) is roughly proportional to (sinθ )-½ exp (- 23⋅94λ -⅓ θ ), whereλ is the wavelength measured in kilometres,θ and is the angular distance from the transmitter. Whenθ is nearly equal to π, the factor (sinθ )-½ has to be suppressed. This formula does not agree with results obtained experimentally. The numerical factor 23⋅94 is much too large, so that, asθ increases, the magnetic force decays much less rapidly than the theory indicates; and it has also been suggested on experimental grounds that the actual state of affairs is represented much more closely when the factorλ -⅓ is replaced by the factorλ -½ .