906 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



letting ti and ti approach zero at a finite frequency will also hold for finite 

 thicknesses if the frequency is sufficiently low. 



We shall let 6 denote the fraction of the stack which is occupied by 

 conducting material, so that 



d = h/{k + ^2), (89) 



w^here at present t\ and h are both infinitesimal. Then the stack may be 

 regarded as a homogeneous, anisotropic medium, characterized by an 

 average dielectric constant e perpendicular to the layers, an average 

 permeability /i parallel to the layers, and an average conductivity g 

 parallel to the layers. Sakurai has treated such an artificial anisotropic 

 medium, and from his formulas we find that when the layers are al- 

 ternately conductors and insulators, the average electrical constants are, 

 to a very good approximation, 



6 = 62/(1 - 6), 



M = 0M1 + (1 - ^)M2 , (90) 



g = Ggi- 



Sakurai has also shown that the average values of the electrical con- 

 stants may be used in Maxwell's equations for the average (macroscopic) 

 fields, due regard being paid to the orientations of the field vectors with 

 respect to the laminae. 

 For the plane stack, these equations read 



dHJdz = iwlEy , 



dHJdy = -gE, , (91) 



dEy/dz - dEz/dy = tco/i^x , 



where the bars denote average values. By analysis exactly similar to that 

 carried out at the beginning of this section for a homogeneous, isotropic 

 medium, we may find the relation between the tangential field compo- 

 nents E: , Hx at the two surfaces of a stack of infinitesimally thin layers. 

 (The bars representing average values may be omitted, since the tan- 

 gential components of E and H are continuous across the boundaries of 

 the layers.) We obtain a matrix relation analogous to (55), namely 



EiO) \ /ch T(S K sh T(s\ /E{s) ' 



1 M ■' ^^^^ 



H(0) I \-- sh Tts ch T(S / \h(s) 



" T. Sakurai, J. Phys. Soc. Japan, 5, 394 (1950), especially Section 3. 



