286 



BELL SYSTEM TECHNICAL JOURNAL 



To express p<i and Vi in terms of p\ and Vi and these three param- 

 eters, we can obtain the equations. 



p2 = 



F.- 



n + 1 



n + 1 



- piS, 



^°'''' + {ii^)'-''] 





sinh b 



V, 



cosh 5 + 



\ZnP + Zn.p 



1 sinh b 



■7 I \ ^ 1 



X 



sinh b 



Zu. 



Zr^ + i^TSi 



7 o 



>)1 



(53) 



X 





7 '^ — 7 



+ cosh 5 



^z^,'' + 2/ 



as can readily be seen by comparing these expressions with the equa- 

 tions, Pi = PrA - (Fi/5i)Zo5; Fo = FiC - {pi/Zo)S,D. In the above 

 expression the letter / indicates that the impedances Z^j, and Z^, are 

 to be taken at the input junction, while the letter indicates that they 

 are to be taken at the output junction. 



The effect of this type of tapering is to change the propagation con- 

 stant scarcely at all, but to lower the characteristic impedances in the 

 neighborhood of the cut-off frequencies. This tends to produce large 



reflection losses and hence effectively the band is narrowed_^ A 



transforming action equivalent to a transformer of turns ratio ^ISl|S2 

 occurs as before. 



To obtain the equation for a straight horn, we let Si, a typical area 

 of the horn, equal 



S, = n'K = K'{niy- = K'ixif, 



where nl = Xi, the distance from the apex of the horn, and / the 

 length of an individual section. T becomes iul/C, and Zj^^ and Zj^., are 



^ip^ypin J, I ^p^iypi -^ Xi 



'Ri 



1 -{- in-^l 



1 -{- t-^xi 



and 



(54) 



Zr,, 



•V/>o7P^7.^i 



. CO 

 1 + * ^ Xi 



