Long‐distance propagation in a stratified atmosphere with an even power n profile

Abstract

The propagation characteristics of the radio wave were analyzed in a nonuniform atmosphere in which there exist an inverted layer with an even power profile of a height under a standard atmosphere and the lowest layer with a linear profile. If the modified index of refraction obtained by introduction of the earth-flattening approximation has a 2n power profile of the height, the solution of the wave equation cannot be given by a closed form function. Hence, the rigorous solution was obtained by a series representation in accordance with the method by Frobenius. This series solution is found effective even if the wave equation has two turning points close to each other. The function value can be obtained within a time shorter than the one required for the numerical solution such as the one by the Runge-Kutta method.By continuation of the analytic solutions for each atmospheric model layer, the wave function was derived and the received electric field at a far distance was obtained by the mode theory. It was found that the received electric field depends strongly on the order n of the 2n-th power N profile as the atmospheric layer near the zero of the index of refraction slope is increased.

An estimate of the modified refractive index from the received field strength

“…As atmospheric models, the bilinear profile model and the evenpower N profile model are used in which there exists an inverted layer with the 2n-th power profile under a standard atmosphere with a lower linear layer [10]:…”

“…Then, the Hertz vector \ in Eq. (1) is given by [10] where the distance r is normalized as For the magnetic and electric vertical dipoles, respectively, ' m is given by where p m and p e are the magnetic and electric dipole moments, while P and H are the permeability and permittivity of air at the source. When K 1 and K in Eq.…”

“…To evaluate the method of estimating this refractive index, the bilinear model and 2n power N profile model [10] are adopted as the atmospheric models. Numerical examples of the refractive index distribution to be derived from the altitude distribution of the received electric field with each model are presented, and are considered simultaneously.…”