This paper contains the results of theoretical investigations of the electrical field radial component |Er| and phase ψ as well as the mean phase velocity \̄gn and the differential velocity vd of VLF and LF waves in the spherical Earth-ionosphere waveguide. It is taken that the ionosphere is homogeneous and isotropic and the conductivity of the lower boundary—the Earth is infinity. The calculations are made for frequencies ƒ = 5-50 kHz and for distances D = 100–10000 km. Two theoretical methods are used: the hop-method for D = 100–5000 km and the mode-method for D = 1000–10,000 km. It has been shown that for distances D = 100–4000 km the hop-method is more convenient and for distances D ≈ 2000–10,000 km the modemethod is more useful; for the overlapping distances the both methods give practically identical results. The influence of different number of hops on the field amplitude and the phase has been considered. The detailed analysis of the dependence of phase velocity \̄gn on distance has shown that the maximum deviation of \̄gn from C is of the order of 3 per cent at distances D ≈ 100–1000 km and 0.2 per cent when D ≈ 10,000 km. A detailed considering of the dependence of \̄gn/c on frequency for different fixed distances has led to new phenomena unknown till now. At different fixed distances D for some ‘character frequencies’, namely, for the selected ionospheric parameters at frequencies f = 12.9–13 kHz, D ≈ 515 km; f = 22–22.05 kHz, D ≈ 2700 km; f = 26.7–26.9 kHz, D ≈ 525 km; f = 42.5–43 kHz, D ≈ 2900 km, the phase velocity \̄gn/c changes by leaps.

The paper gives a brief description of the results of experimental investigations and some theoretical calculations of propagation of e.l.f. and v.l.f. waves in the Earth-ionosphere waveguide in the frequency range from 60 c/s up to 30 kc/s.