630 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1957 



Case 3. Layers of Intermediate Size 



If the layer dimensions, are such that c is large but 6A is small, com- 

 pared with the Fresnel zone dimension, the recei^•ed power is given by 



^^ = ^^ 2XW *^' " 



In the atmosphere, c and b are likely to be about equal, on the average, 

 and we have for this case, \/2aX < h < ■\/2aX/A. 



All three of these cases may be present at various times, since the 

 structure of the atmosphere changes from day to day. However, for the 

 purpose of the present study. Case 3 is considered most prevalent and is 

 assumed in all the calculations to follow\ 



Many of the numerous layers that are assumed to contribute to the 

 received power are not necessarily horizontally disposed, they may be 

 oriented in any direction. Therefore, reflection in the direction of the 

 receiver can take place from layers located both on and off the great 

 circle path. If there are N contributing layers per unit ^'olume in the 

 region V common to the radiation patterns of the transmitting and re- 

 ceiving antennas, then for Case 3, 



« 



_ ArAnNb 



2\'a' 



f A'q'dV (1) 



Jv 



In this relation it has been assumed that the layer size and the number 

 of layers per unit volume remain sensibly constant throughout the com- 

 mon volume. 



The integration process reciuires expressions for the reflection coefii- 

 cient q and the grazing angle A of the layers in the common volume. These 

 quantities are derived in the following sections. 



REFLECTION COEFFICIENT OF A LAY'ER 



The reflection coefficient of a plane boundary (Fig. 2) separating two 

 media whose dielectric constants, relative to free space, differ by an 

 increment de is given by Fresnel's laws of reflection. For both polariza- 

 tions, the plane wave reflection coefficient of the boundary is 



q = de/^A- 



1 



Fig. 2 — Reflection at a l)()undary between two homogeneous media. 



