282 BELL SYSTEM TECHNICAL JOURNAL . 



occurs when cosh F = ± 1/cosh / while in the recurrent structure, the 

 band occurs when cosh F = ± 1. One result of this is that no low 

 pass filter exists in exponentially tapered structures. 



Considering now the pressure and volume velocity ratios when 

 1 = cosh 5 = — 1, the absolute value of e'^ is 1. Hence over the 

 band the ratios of pressure and of volume velocity from section to 

 section are respectively e~' and e' or -ylSi/S-z and -sSi/Si. Hence one 

 section multiplies the pressure by a ratio ^lSl|S2, and the volume 

 velocity by the factor ^lS2|Sl. Therefore a tapered structure of this 

 kind is equivalent to a transformer of turns ratio ■yfSJS'i, and a filter 

 of somewhat narrower bands than for the filter repeated recurrently. 



To specify completely a filter of this type requires three parameters. 

 Two such parameters have been developed above and are 8, the pro- 

 pagation constant of a tapered filter, and Zj^^, the specific recurrent 

 impedance in one direction. These are given by 



cosh 8 = cosh F cosh /, 1 



Vtanh^ / + sinh^ F - tanh / ] ^ (45) 



^Ri — -^0 



sinh F 



We take as the third parameter Z^^, the specific recurrent impedance 

 in the opposite direction. We can readily determine that Z^.^, the im- 

 pedance looking in the opposite direction from that used to specify 

 Zn^, but obtained at the same junction point, is 



_ Z()B 





It is desirable to have the same propagation constant serve for the 

 two directions, hence we let pi'lpi = K^e-^. Since K represents a 

 transformer change of the pressure in one direction, we find, when 

 going in the opposite direction, that the pressure should change by 

 the inverse of K, so Ki = 1/K. Substituting these values for pijpi, 



Hence for an exponentially tapered filter 



^ _ V r Vtanh^ t + sinh^ F + tanh / 1 f.^. 



Zr, - U — — . (47L 



L smh F J I 



In terms of the parameters, 5, Z^^ and Z^^ we can express pi, pi, V2, 



