Quadrature formula for an attenuation function of a field over the path with square inhomogeneous distribution

A ground wave field that takes into consideration a square distribution of geometrical and electrical inhomogeneities when source and receiver are on Earth-s surface is considered. As a rule, a change of inhomogeneities is considered for modeling of the radio wave propagation only in direction to the receiver. But in some cases, a necessity of taking into consideration of inhomogeneities situated aside of the propagation line is appeared. To take into account a square distribution of inhomogeneities within some first Fresnel zones is offered for calculating of an attenuation function. One-dimensional attenuation functions are calculated by means of one of known methods on each line that are similar to Fresnel zones. The numerical method will be presented in the form of a totality of one-dimen¬sional attenuation functions.

The problem is reduced to two-dimensional Kirchhoff integral equation. Elliptic coordinates are used for a convenience. By using approximated expressions for distances in elliptic coordinates that are appropriate in the essential area for propagation, the integral is reduced to the form of a totality of one-dimensional attenuation functions. Choosing an essential integration interval that guarantees a convergence of the integral and using asymptotic expansions of functions we come to quadrature formula for calculation of an attenuation function taking into account two-dimensional inhomogeneities.

As an example, calculations of an attenuation function of the ground wave field propagated along an inhomogeneity in the form of forest massif that has other electrical properties than properties of the path of propagation are shown. Usage of the offered quadrature formula showed a necessity to take into consideration of the square distribution of inhomogeneities for more precision predicted of the EM field.