REGULAR COMBINATION OF ACOUSTIC ELEMENTS 



279 



Equation (39) gives the relationship between pilp2 and pzlpi which 

 must be satisfied if the output impedance of one section equals the 

 input impedance of the next section. If we specify a value of p^lp-i, 

 then the value of p^lpi is determined. The impedance Z2 terminat- 

 ing the second section is also determined and hence the pressure ratio 

 of the third section, etc. Hence if we specify a value of p-ilp\, we also 

 determine the propagation characteristic of any other section in a 

 series of sections. The pressure ratios will not in general be constant 

 from section to section. 



We can write p^lpx — Ke~^ since this will represent any phase or 

 amplitude change. Similarly we can write p^jpi as K'e~^' . Sub- 

 stituting these values in (39), we have 





cosh r -f 



SiS:i — Si 



2SiS3 



^2 + ^3 \ e' 



2& 



K 



+ 



5i + 52 



2^1 



(40) 



K'e- 



Now if the value of 8 remains unchanged from section to section a 

 great simplification results, for in order to determine the overall pres- 

 sure ratio we have only to multiply the number of sections by 5. Hence 

 it is desirable to determine for what rate of taper this condition is met 

 and also how good an approximation it is for all rates of taper. 

 If we set 8 — 8' and multiply through by e~^, we obtain 



g—v _ 



(5. + 52)(52 + 53)\^^^j^^.^5.53-52^ 



251^3 



2S,S, 



Si + ^'2 r^, 



~2sr^ 



+ 



S^ + S-A 1 



2^3 J K 



Similarly the equation for the next two sections is 



,-... i( 



(52 + S,){S, + S,)\ , ,, , / 525-4 - 53- 

 cosh 1 + 



2525. 



25-254 



/ 52 + 53 \ 

 \ 252 / 



K' 



+ 



52 + 53 



252 



= 0. 



53 + 54 \ 1 



254 / K' 



= 0. 



K" 



