92G THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



which is directly proportional to frequency (and independent of con- 

 ductivity). For/ ^ /a the stack acts much like an infinitely thick solid 

 plate, for which 



Zo(7o) ^ (1 + i)/gA = (1 + i) VWT^ , (183) 



an impedance again proportional to -\/f. 



The real parts of the approximate expressions for surface impedance 

 may be plotted on log-log paper, where power-law relationships are 

 represented by straight lines, to give quite a good idea of the way in 

 which the stack resistance varies over the entire frequency range. To 

 show how the exact resistance departs from the approximate formulas in 

 the transition regions, we have calculated the resistance of a particular 

 stack over the full frequency range from equation (172), and also the 

 resistance of the corresponding solid plate from the formula 



Zoiyo) = (1 + OVmT^ coth [(1 -f i)V^mgJTr], (184) 



and plotted the results, together with those for an infinite plate and an 

 infinite stack, in Fig. 7. The actual numerical values were chosen solely 

 for ease in plotting, and are of no particular significance. It should be 

 noted that the exact curves oscillate slightly around the asymptotic lines 

 in the transition regions. For example, the resistance of the laminated 

 stack is actually higher than the resistance of the solid plate at certain 

 frequencies slightly above /s . These oscillations appear clearly in the 

 numerical results, but are scarcely visible on the plots because of the 

 logarithmic compression of the upper ends of the frequency and re- 

 sistance scales. 



We shall next obtain an expression for the rate at which the surface 

 impedance of a laminated stack begins to depart from its dc value as 

 the frequency is increased. For this purpose we must expand the various 

 factors appearing in equation (170) for Zo(7o) in powers of ©. Using the 

 expansions (163) and (164) which have already been derived for r, A^i , 

 and K2 , it is a matter of straightforward if tedious algebra to show 

 that: 



e =1--^U- ^ U-f---, (I80) 



6 360 



in 

 sh nF = — ; 



■2V3 



■ . e^ _ (175n' - 48) e 

 ^ 30 12600 



, (187) 



