176 



DIFFRACTION BY TERRAIN 



If the ground near the transmitter or receiver is 

 smooth, however, it becomes necessary to take 

 ground reflection into account. This may be done 

 by introducing an image transmitter and receiver. 

 The field is then the sum oi four components whose 

 relative phase must be calculated (see Figure 11). 



Earth cur\'ature will he neglected throughout the 

 present section. 



8.3.2 



Criterion for Roughness 



It is difficult to establish a ciuantitati\^e criterion 

 for the roughness of a surface. From the viewpoint 

 of radiation theory, the effect of a rough surface is 

 to scatter incident radiation diffusely in all directions 

 with no preference for the direction of regular reflec- 

 tion, whereas a smooth surface will reflect the inci- 

 dent radiation according to Snefi's law. In radio 

 work, the effect of diffuse reflection is to weaken the 

 radiation scattered in the direction of the receiver 

 so much that its intensity may be neglected com- 

 pared to the direct ray. A moderately rough surface 

 will give a coefficient of reflection intermediate 

 between zero and unity. A surface will be optically 

 smoother as the incident radiation approaches 

 grazing, and even surfaces that are comparatively 

 rough geometrically may then give partial reflection. 



A rule taken from optics and known there as 

 Rayleigh's criterion has been used successfully in 

 radio practice. Assume that the roughness is pro- 

 duced by numerous small elevations above a le\'el 

 surface and let H be the typical height of such an 

 elevation. The difference in path between a ray 



Figure 7. Geometry for Raj'leigh's criterion for 

 rough ground. 



reflected from the ground and a ray from the top 

 of the elevation is 2AB in Figure 7, which is eciual 

 to 2H sin i/- or 2H^l/ approximately for small angles i/-. 

 The difference in phase between the two rays is 

 2H\p{2ir/\). The criterion now requires that the 

 surface be considered as rough when this phase 

 difference exceeds 45 degrees, or ir/4 radians. 



Hence the critical value of H is given by 



= - ,oiH = , 



X 4 16i/' 



with \p in radians and 



H=^6X 



(8) 



(9) 



with \p in degrees. The surface is considered smooth 

 or rough according to whether H is smaller or larger 

 than this value. 



Sometimes it is convenient to refer to the field 

 pattern that would be present over a reflecting 

 surface. This is done by introducing a new^ variable, 

 the lohe number 



n = l^i!^ (10) 



X 



{hi transmitter height above the ground), where 

 n = 1, 3, 5, etc., correspond to the angle of the first, 

 second, etc., maxima in the lobe pattern and n = 0, 

 2, 4, etc., to the nulls of the lobe pattern. Introduc- 

 ing n into enuation (8), the criterion assumes the 

 form 



H = 



h 



4:71 



(11) 



Although the criterion is approximate and gives no 

 more than an order of magnitude estimate, it is rather 

 surprisingly well fulfilled in radio practice. Experi- 

 ence has shown that when the differences in level 

 which constitute roughness are of the order indicated 

 by these ec^uations, the reflection coefficient is 

 reduced to a small fraction (about one-fifth) of the 

 value calculated for an ideal surface. 



*^ * Diffraction by a Straight Ridge 



Assume that the ground intervening between the 

 transmitter and receiver is everywhere rough, so 



Figure S. Diffraction by a straight ridge. 



that all ground reflection may be neglected. For 

 the sake of computation, the ridge is replaced by a 

 vertical screen of height ho above the line TR. The 



