936 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



curvature effects into account would require a considerable amount of 

 numerical calculation. Equation (98) of Section III provides an explicit 

 expression for the surface impedance of a cylindrical stack of infini- 

 tesimally thin layers in the presence of dielectric mismatch, in terms of 

 Bessel functions of complex argument; but if the layers are of finite 

 thickness we can at present do nothing better than multiply out the 

 matrices of the individual layers step by step. 



The variation of the surface impedance of a laminated stack with 

 frequency over the full frequency range is not quite so simple in the 

 presence of dielectric mismatch as when Clogston's condition is exactly 

 satisfied, but a somewhat analogous discussion may be given. As in the 

 preceding section, we consider a plane stack of n conducting layers 

 each of thickness h , where n(i = Ti , and backed by an infinite-imped- 

 ance surface. When the mismatch parameter is k, the three critical 

 frequencies are: 



/2 = V3/(7rMi6fi/iTi\/rT3^^0 



, (235) 



= A/3n/i/VrT3^- (2^1 = Ta), 



/s = 3/(7rMig/i) = 3n'/i (^1 = V35i). 



In the range ^ / ^ /2 , the surface impedance of the stack is ap- 

 proximately constant, being given by 



Zo(7o) ^ l/giTi . (236) 



In the range f-> ^ f ^ fz , we have 



Zo(7o) ^ K, , (237) 



where Ki is given by (217) provided that k is small compared to unity. 

 For infinitesimally thin layers the upper critical frequency fs is infinite, 

 and we have for / ^ /2 , 



Zo(yo) ^ I k p(l - i sgn k)/gA 



. (238) 



= (1 - t sgn fc)\/7rMi I k I f/gi , 



which is proportional to s/f. If the layers are of finite thickness but 

 k = 0, we have the result obtained in the preceding section, 



Zo(7o) ^ (l/\/3 + i)TiXitJ, (239) 



which is proportional to / up to the critical frequency fo . If neither the 

 mismatch parameter k nor the layer thickness h is zero, then the surface 



