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BELL SYSTEM TECHNICAL JOURNAL 



Now since tt = -rr and ^p^ = -^ , we have 



Z2 



^2 



(36) 



Z2/S2 and Zi/Si then are respectively the terminating impedance and 

 input impedance necessary to give a structure specified by the factors 

 A, B, C, D the pressure ratio pi/pi- To solve for the input impedance 

 we take the first of equations (34) and obtain 



Z,= 



ZaB 



(37) 



A - 



Hence by virtue of (36), the terminating impedance Z-z/Si becomes 



Z2 



S2 



Zo „ 



pi 



(38) 



- C 



Equations (37) and (38) state that there is a relation between the 

 input impedance and the pressure ratio, and the output impedance 

 and the pressure ratio. When one is specified and the constants of 

 the section Zq, A, B, C, D are given, the others are known. 



Suppose now that we wish to join a second structure of this type 

 to the first, assuming that the cross-sectional area at the junction is 

 the same for both. We must have now Z2, the specific output im- 

 pedance of the first section, equal to Zi', the specific input impedance of 

 the second section. Hence we can write. 



S2 -Tf 



or -^B 



A' 



B' 



h 



P2 



c 



where the primes refer to the constants of the second section and where 

 P3IP2 is the pressure ratio of the second section. Substituting in the 

 values of A', B, B' , C, we have 



{Sy -\- S2)(S2 + S,) 



cosh r + 



•Ji-JS — 'J2 



2S,Ss 

 S2 + S, 



2S, 



)^ 



^j_+_^\ p^ 



2S, J P2 



(39) 



