G32 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1957 



7/ = and // = h, each with refloftion cooffiriont 



If the abrupt change in slope, the soUd Une in Fig. 3(b), is replaced 

 by a gradual change as indicated by the dotted lines, (2) still holds pro- 

 vided d < X/4A. For more gradual changes, d = 7iX/4A where n > 1 , 

 the reflection coeflScient is 



. irn 

 _ KX ^"^T 

 IGttA^ Trn 



~2 



and q varies with n between 5 = and 



^ ~ IGttA'^ ' Trn 



Smoothing of the boundaries reduces the value of q. 



It will be assumed in all the calculations to follow that reflection from 

 layers in the troposphere is described by (2). 



VARIATION OF STRENGTH OF LAYERS WITH HEIGHT 



The formula for the reflection coefficient includes the factor K, which 

 represents the change in the gradient of the dielectric constant at the 

 boundaries of the layer. A dielectric constant profile constructed of 

 many randomly positioned gradients is shown schematically in Fig. 4. 

 The variations are showTi as departures from the standard linear gradient. 

 Measurements indicate that the fluctuations of the dielectric constant 

 normalh^ decrease with height above ground. The changes in the dielec- 

 tric constant gradients associated with these fluctuations probablj^ vary 

 in a similar manner so that K is some inverse function of the height above 

 the earth. However, to simplify the computation of received power, to 

 be described later, we have adopted the cylindrical coordinate system 

 shown in Fig. 5, and it is convenient, then, to let K be a function of p, 

 the distance from the chord joining the transmitter and receiver to the 

 point in question. 



We assume, therefore, that 



P 



where Ki is the change in gradient at point .4 in Fig. 5 which, for a typi- 



