414 BELL SYSTEM TECHXICAL JOURNAL 



perhaps for special \'alues, the correct answer is obtained. W'e proceed on 

 this basis. 



Solntion 



To permit a more coherent presentation of the arguments, only their 

 general outline follows. More mathematical details are given later. 



We start with the formulas for (^ - (= i^) as given in Fig. 1. 



A 



The lirst operation is to show that, under comparable conditions, i.e., 

 X, r, n tixed, the TE Oniii modes give the highest values of P. That this is 

 j)lausib!e can be seen in a general manner from the equations as they stand. 

 For the TE modes, if ( — 0, the numerator of the fraction is largest. Also, 

 P simplities, and the denominator roughly reduces the e.xpression in square 

 brackets to the 1 2 power. Now compare this expression with those for 

 the TM modes. That for the TM modes (// > 0) is smaller because of the 

 factor (1 + R) in the denominator. Finally, that for the TAf modes (;/ = 

 0) is still smaller, because 1 < (1 + p-R-Y'-. 



This leaves only the TE Omii modes to be considered, and the next step 

 is to show that ;;/ = 1 is the most favorable value. Since the relation be- 

 tween P and ir is com{)licated, a j)arameter cp is introduced, with (p dehned 

 by 



tan (^ = pR. (3) 



The resulting parametric equations are: 



r 1 

 P = ^ ^^— (4) 



^TT .•? ,1.3 



COS v? + - sm (f 



p 



pr^ 1 



47r cos ip sm ip 



For each of the discrete values of r and n (;/ is related to p) then, plots 

 of P vs W can be prepared as shown in Fig. 2 for the TE 01 » modes. 



Inspection of Fig. 2 shows that the best value of Q does not correspond 

 to a minimum of W or a maximum of P for a given value of ;/, but rather to 

 a point on the "envelope" of the curves. To get the envelope, we assume 

 p to be continuous and proceed in the standard manner. It turns out that, 

 by solving (4) f(^r p in terms of 7^ /- and v?, substituting the resulting e.x- 



(9 IF 



pression in TF, and setting --- = an equation is obtained which, when 



^^p 



Sf)lvcd for <p, gi\'es the \'alucs of ^p which lie on the en\-clo]u\ 



