nomo(;k\i>iii<: soi.i tions koii the standard cask 



97 



ami slide-rule calculation is extremely convenient. 

 For each of the selected values of b, we have pre- 

 pared a nomogram connecting Y, B, and a. Although 

 there are six adopted values of b, the expressions for 

 b = %, %, ';,, % coincide, so that four charts 

 suffice. A representative sample of these charts, for 

 b = 0, is given in Figures 2 and 3. Connect each 

 value of Y, for which a contour is desired, with the 

 value of B on the appropriate chart, according to 

 the value of b (or k). Read off the corresponding 

 value of a. 



Having determined a for a given point on the 

 coverage chart, we now calculate S from the nomo- 

 gram in Figure 4, with a, db, and S as variables. 

 For S measured in units of 1,000 yd, we have 



2Q-tfS/20 _ 



10 3 



1 



[' - (rhs)'] 



y/~ \914 S 



+ 



4 - 3c 



r 



A typical example for the selected values of b is 

 shown, as before. 



Finally, we must calculate hi. For heights we have 



H = h, + (1 



(3.281) (914)' ffl 



2r 



+ 



(3.281) 2 (914) (150) (n - b) (- 2 + 3a) 

 hi / mc a 2 



S. (2) 



This equation, unfortunately, has too many variables 

 for nomographic solution in a single step. We first 

 define a quantity C, such that 



C 



(3.281) 2 (914) (150) (n - b) (- 2 + 3a) 



hi /mc OL' 



Obtain the simple product hif mc , which is a charac- 

 teristic of the set. Then use the nomogram of Figure 

 5 to obtain the values of C for the selected ranges of 

 k and a. Then we can determine H from the nomo- 

 gram of Figure 6, for each value of S and C. Finally, 

 from a nomogram (not shown here), representing 

 the equation 



hi = H - (1 - a)^ 



we determine h 2 . Actually, for much of the range, 

 a ~ 1 and hi ~ H. 



For the upper lobes considerable simplification is 

 possible. We may omit all the steps involving cal- 



culation of a. We determine the various B's as before. 

 Then, as long as B » 1 we employ the equation 



10-*/ 10 = 191 /_L_Yy 

 U w \9USJ X 



[^ + ( 1 -F 2 ) sin2 ' r6 ]- 



This equation gives S directly for each decibel value 

 and assumed value of b. The nomogram for this 

 problem appears in Figure 7. We then obtain H from 

 equation (2), with a set equal to unity. 



// 



(3.281) (914) 2 

 2r 



>S 2 



, (3.281) 2 (914) (150) c 

 + - -£> 6 



(3) 



In equation (3) we have written C instead of C. 

 For much of the range, wherever B is very large, 

 we may take C ~ C. If greater accuracy is desired, 

 we may compute C" directly by the equation 



, = (3.281) 2 (914) (150) k ( J_ 



6/m.fcl \ B> 



It is interesting to note that equation (3), apart 

 from the correction factor (1 — 1/B 2 ), which merely 

 serves to improve the accuracy of the result, is 

 familiar to many in the construction of so-called 

 "fade charts." These diagrams depict merely the 

 lobe minima (and sometimes also the maxima). If 

 we set b = we get the former, and if we take 

 b = Yi we determine the latter. 



The total number of lobes N is approximately 



JV = 



2hi 



hi f„ 



(150) (3.281) 



2.03 X lO-'hiU 



for hi in feet. These will be distributed over an angle 

 of 90 degrees. Hence A, the average angle per lobe, is 



A = 



9CT 



N 



4°43 X 10 4 



/mc hi 



Near the horizon, however, the angle per lobe A is 

 somewhat smaller, to wit: 



A = 



360° 



ttN 



5°64 X 10 4 



/mc hi 



