1198 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1957 



for a by means of a nomograph to be described later. Next the quantities 



cosh a ^3(0") 



tpi^a) <p2(o') 



sm a, 



are plotted as functions of a- for various a. These plots can then be used 

 as follows to solve (98) for a given t. Solve (99) to obtain <to . From the 

 plot of <pz{(t)/(P2{(t) sin a compute 



r * If 1 I '^^(o-o) ; • \ 

 All = Al I 1 + — 7 — r t sm a 1 . 



Using the value ^i* for /i, compute o-i from (99) . With this value of o-i , 

 compute /i2* from 



fi-i = Ai I 1 + — — r t sm a. 



\ ^2(0-1) 



)■ 



etc. In this way a convergent sequence o-q , o-i , • • • , (7„ is generated. 

 Finally, from the plot of cosh (x/<p-i{<j) obtain T^ . 



The above iteration procedure sounds tedious. Actually, in most cases 

 the iteration is not necessary because a remains essentially independent 

 of time. Thus, the solution of (99) by means of the accompanying nomo- 

 graph will usually give the complete solution of the problem. 



E.2 Nomograph (Alignment Chart) for the Solution of Equation (99) 

 The relations 



Xi = 0, X2 = d, Xs 



yi = h, y2 = ztt:, 2/3 



10(^2 (ff) d sin a 

 1 + 10^2(0") sin a ' 



(fiia) sin a 



(100) 



10 ' "^ 1 + 10<P2((t) sin a ' 



where d is an arbitrary constant define parametrically three curves 



Vi = ViiXi), i = 1, 2, 3, 



which we imagine plotted on a cartesian (x, y) coordinate system. A set 

 of values h, e, and a determine three points (.r» , yi) {i = 1, 2, 3) which 

 lie on these curves. If these points lie on a straight line, it can be sho\Mi 

 that they satisfy (99). 



On the left-hand sides of Fig. 36 we have plotted the curves given by 

 (100) for various values of the critical angle a. The values of the parame- 

 ters h = wh/EA and e, which describe the curves yi = yi{xi) and , 

 ?/2 = y2{x2) respectively, are plotted on the indicated scales. Rather than 

 indicate the values of a along the curve 2/3 — y-iixz) we have for con- 



