CONCLUSIONS 



259 



the layer height /) necessary to give infinite conver- 

 gence as a function of the range (plotted on right- 

 hand scale). 



■J7.2 



ROUGHNESS EFFECT 



The most apparent difficulty with the picture 

 presented so far is that the layers actually are not 

 perfectly smooth. In order to take that fact into 

 consideration, it was assumed that the layer was 

 composed of a large number of plates set at various 

 small angles about the horizontal according to a 

 Gaussian distribution. As in other parts of this 

 problem, variations are considered only in the plane 

 of transmission, since the effect of sideways deviation 

 would cancel out. This reduces the problem to one 

 of two dimensions only. Each plate is further assumed 

 to retain its original curvature. 



A beam falling on a patch of these plates would 

 be reflected in such a way as to spread the energy 

 at the receiver in a vertical pattern similar to the 

 Gaussian distribution of the plates. It is only neces- 

 sary to integrate this curve over the width of the 

 antenna to find the fraction, L, of the total energy 

 that will be useful. L will be a function of the 

 probable value of the deviation of the plates, the 

 range, and the antenna width. 



With the rough layer assumption, there will be 

 some plates correctly oriented at each part of the 

 layer to reflect energy into the receiver. Therefore, 

 a third factor, M , must be included that is the ratio 

 of y/j3, where y is the total angle subtended by the 

 layer that can reflect rays to the receiver, y would 

 be limited by the optical horizons. /3 is the angle 

 subtended by the receiving antenna when reflection 

 is from a plane surface; i.e., essentially, free space 

 conditions. 



The net convergence factor C must be the product 

 of these three quantities K, L, M. In this case, K 



must be the mean value of K averaged for various 

 points of reflection. In order to integrate the expres- 

 sion for the mean value of K, it is necessary to substi- 

 tute for sin </> in equation (2) . 



/ b , x , b , y \ 

 ^^{x + Ya + y + i)' 



which gives 

 K 



1 - 



(xvY- 



R 1 (2ab + xy) 



(5) 



(6) 



This expression is easily integrated if the product 

 xy is used for the variable and Xiy\ = Xi(R — Xi). 



Example. The preceding developments have been 

 applied to the one-way link of the U. S. Navy Radio 

 and Sound Laboratory at San Diego, which has been 

 extensively studied. High subsidence layers are 

 common for this region. The probable value of the 

 deviation of a reflecting plate from horizontal was 

 taken as 0.1° as an engineering approximation. In 

 this case, C equals 43, assuming a reflection coeffici- 

 ent of 1. If the reflection coefficient is not UDity, its 

 value as a function of angle of incidence must be 

 multiplied into the equation. 



Since K, L, and M can each vary through con- 

 siderable limits, C can vary through a very wide 

 range of values. 



27.3 



CONCLUSIONS 



The statistical treatment of the roughness is not 

 always applicable, since a finite number of plates 

 would actually be engaged in reflecting energy. 

 Hence, the received signal would vary almost ran- 

 domly with time as the orientation of the plates 

 changed slightly. This could produce marked fading 

 and peaks of large amplitude. Primarily, however, it 

 would explain signals of the magnitude of free space 

 signals or higher. 



