VTMOSlMIKKir. STK \TIKIC ATION AND RUFR \CTION 



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with the horizontal of 0.5° <>r less, arc strongly 

 affected by nonstandard refraction. This part of the 

 transmitter radiation is of paramount importance in 

 early warning radar and in communications. For 

 such applications of radar as gun-laying or search- 

 light control the effects of nonstandard propagation 

 arc usually negligible because the rays which reach 

 the target have emerged from the transmitter at a 

 fairly large angle with the horizontal. 



The progress of a ray through the stratified atmos- 

 phere is described by Snell's law, discussed in Section 

 17.1.4. When the angle a between the ray and the 

 horizontal is small 



cos a = 1 — — , 



provided a is expressed in radians. 



Introducing this into Snell's law for a curved 

 earth, equation (6), noting that n + h/a = 1 + M • 

 10~ 6 and neglecting second order quantities, it is 

 seen that 



(18) 



1 («! - a „2) = (M - Mo)10" 



Since a is the angle which the ray makes with the 

 horizontal it is equal to dh/dx, the slope of the ray. 

 Solving equation (18) for a, 



_ dh 

 dx 



= "\U 2 + 2{M - Af„)10- 



(19) 



These relations apply to any two levels provided a 

 and ao are the angles at the levels to which M and 

 Mo refer. 



variations of the modified index. Although this ray 

 tracing method is only an approximation of the true 

 solution of the wave equation, it can be used, subject 

 to certain limitations, for computing quantitatively 

 the strength of the field. The approximation breaks 

 down when neighboring rays cross each other and 

 form caustics. 



The method may be illustrated by the case of 

 standard refraction with k = %. As shown in Figure 

 21, draw the M curve with a slope ka = 4a/3. Let 

 the subscript 1 stand for the transmitter level (of 

 height hi). Pass a vertical line through the corre- 

 sponding point Mi of the M curve. Lay off the 

 distance ai 2 /2 to the left of Mi for a particular ray, 

 1, which emerges from the transmitter at angle ai 

 with the horizontal. In order to make a and M 

 comparable numerically, the factor 10 -6 should be 

 eliminated from equation (18) above. For this pur- 

 pose cr should be measured in the same unit as M, 

 that is, in 10~ 6 radian. The distance between M and 

 1 at any height h then is equal to (M — Mi) + ai 2 /2, 

 and by equation (19) the square root of twice this 

 quantity is equal to the slope of the ray at height h. 

 Hence, ray 1 starting downward from the transmitter 

 is bent more and more toward the horizontal as h 

 decreases. At point P this ray becomes horizontal 

 and from there on increases in slope with increasing 

 height. 



Ray 1' starting upward from the transmitter at the 

 same angle ai continues to curve upward more and 

 more rapidly as the height increases. Ray 2 is the 



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OR SEA LEVEL 



Figure 21. Rays in the standard atmosphere. 



Equation (19) provides a technique for tracing the 

 paths of rays emitted by a transmitter at various 

 angles with the horizontal, and it indicates how their 

 passage through the atmosphere is controlled by the 



horizon ray which represents the limit to which rays 

 can be directed by refraction. Beyond this lies the 

 diffraction region where ray tracing cannot be used. 

 To study the field in the diffraction region the original 



