508 BELL SYSTEM TECHNICAL JOURNAL 



Using these and the general relations (95), we can obtain 



I y sin (joyh'(y)dy 

 Jo 



t = "-^' . ('07) 



f 



Jo 



sin o:yh'(y)dy 



which is independent of A. 



Since (if B is not everywhere zero) dB/dco is positive when A = o 

 according to Theorem I, and since by (107) it is independent of A 

 (a constant), it will be positive whatever the value of A. Hence, B 

 increases with frequency in such transducers. 



Appendix III 



Propagation Constant and Iterative Impedance Formula for 

 General Ladder, Lattice and Bridged-T Types 



These formulae apply to the general types of structures shown in 

 Fig. 2 and should be used whenever it is desired to take into account 

 accurately the actual physical impedances. Network designs which 

 follow the methods given in this paper are made under the assumption 

 of invariable lumped elements. In constructing physical networks 

 according to such designs, however, certain departures from this as- 

 sumption unavoidably make their appearance and must be taken into 

 consideration whenever extreme accuracy is required. The departures 

 include dissipation in coils and condensers, distributed capacity in 

 coils, as well as inaccuracies due to manufacture. 



Some of these formulae have been given in previous papers but all 

 can be derived readily either by the method given in B. S. T. J., 

 January, 1923, p. 34, or by that in B. S. T. J., October, 1924, p. 617. 



Ladder Type: 



coshr = 1 + i-^ (108) 



The iterative impedances at different terminations are: 



At full-series = Ki -\- |si, 

 At full-shunt = Ki - |zi, 



Lattice Type: 

 and 



At mid-series = Ki = VzizT+izi^, 

 At mid-shunt = K2 = Z1Z2/K1. 



(109) 



coshr = 1 +-r^^^ (110) 



422 — 2i 



iC = Vi^2. (Ill) 



