276 



BELL SYSTEM TECHNICAL JOURNAL 



characteristic impedances, but whose conducting tube areas increase 

 in some regular manner, a tapered filter is obtained whose character- 

 istics dififer from those of a recurrent filter. The distinguishing prop- 

 erty introduced by a tapered filter, in addition to its filtering property, 

 is a transformer action which increases the pressure by a given ratio 

 and decreases the volume velocity in the same ratio, or vice versa, 

 thus giving a transforming action and a complete transmission of 

 power over the pass band. This is a useful property, if acoustic sys- 

 tems of different impedances are to be connected together. Horns 

 are the limiting cases of tapered acoustic filters and hence their study 

 has considerable practical importance. 



The typical section of a tapered filter considered here is one built 

 up from two symmetrical structures with the same propagation con- 

 stants and characteristic impedances per square centimeter, but with 

 different cross-sectional areas. If we use any of the recurrent filters 

 discussed in Section III, then, for example, since 



cosh r — 



cos 



2coL 



S2 sin 



2coL 



2Si cot y; 



for the low pass filter, to keep the same value of r when we vary the 

 conducting tube area it will be necessary to keep the ratio of the areas 

 constant and to leave all values of L and / the same. Similarly for the 

 other types of filters. 



If r/2 is the propagation constant of each of the symmetrical 

 structures, Z^ the characteristic impedance per square centimeter for 

 each structure, Si the cross-sectional area of the first structure and 52 

 that of the second, we can write three sets of equations for the two 

 structures and the junction point. These are 



pi" = pi 



p2 = Pl 



pi cosh 2 — 



Vi Y sinh ^ , 



Vi cosh -^ — ^-^ sinh -^ , 



Vi" = Vi', 



cosh -7:— Fi"-^sinh-;r 

 z 02 2 



I 



Vo 



Fi"cosh^-^ 



Z Zo 



■ sinh -z 



