1184 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



SO that 



7 = — a)/Zo€o — {io:h/gQ)Vt, (526) 



then (524) becomes, approximately, 



- 1 1 + ^» - ^ 



^0 Co. 



,2 , ,. _ , / A/I , Ae 



.Mo Co 



L \J"o eo/_ 



(527) 



In all cases of interest we shall find that (A/x/mo + Ae/eo) is smaller than 

 or at most of the same order of magnitude as Tf/io)jJogo . Hence the 

 differential equation (523) takes the approximate form 



dif 



2 I . _ - / Am , Ae 



Mo Co 



H, = 0, (528) 



where v] is determined by the two-point boundary conditions (521). 

 The variations of the stack parameters appear in (528) only in the 

 term (A/z//Zo + Ae/eo), which is some as yet unspecified function of ij. 

 For convenience we shall write this term in the form 



^ + ^' = -^*>(i/), (529) 



Mo Co oinogoa- 



where C is a dimensionless parameter and <p(y) is a function whose aver- 

 age value over the stack is zero, and whose maximum absolute value 

 will usually be of the order of unity. It is worth noting that if the con- 

 ducting and insulating layers all have equal permeabilities, then (529) 

 becomes 



Co Za^do 



where 5i is the skin depth in the average conducting layer and ^o is the 

 average fraction of space filled with conducting material. If we solve 

 the differential equation for different values of the scale factor C but 

 the same (p(y), we can calculate the effect of stack nonuniformities of the 

 same type but different amplitudes, or the effect of nonuniformity in the 

 same stack at different frequencies. In the latter case C is directly pro- 

 portional to the frequency. 



The final step in the transformation of the differential equation (528) 

 will be to reduce it to dimensionless form by the substitutions 



