340 BELL SYSTEM TECHNICAL JOURNAL 



using other types of side branches, this difficulty can at least be 

 partially eliminated. For example, a concentric tube closed on the 

 end is, to a first approximation, equivalent to an inductance and 

 capacity in series, and it can be made to have a larger area relative to 

 the main branch tube, than can the straight tube. 



The choice of the forms of the structures to give the simplest 

 impedance elements, is large. For example, Stewart represents a 

 shunt inductance and capacity in parallel, by a concentric tube closed 

 on the end, and a straight tube open on the end, joined together to the 

 main conducting tube at a common point.^ Other methods for 

 representing two elements are shown on Table I. In these structures, 

 the equivalent length and equivalent areas have been calculated cor- 

 responding to these values for a straight tube. These elements have 

 been calculated by calculating the impedances looking into the 

 structures and taking the second approximations. 



D. Design Formulx for Acoustic Filters 



Using the side branch impedances shown in Table I, in the lattice 

 network shown by Fig. 7, the resulting characteristics can readily be 

 obtained. A large number of multiband characteristics can be secured 

 by using various combinations of side branches, but only five single 

 band filters (to the degree of approximation considered here) have been 

 found. The attenuation characteristics of these filters and the design 

 formulae for them are shown on Table II. In designing a filter, it is 

 usual to obtain the dimensions in terms of the singular frequencies 

 which determine the action of the filter. One other parameter appears, 

 Zo, which represents the characteristic impedance of the filter at the 

 mean frequency of the band i.e. fm = V/1/2. It is usual to match, 

 approximately, the impedance terminations of the filter to the value Zq. 



All of these filters have been calculated for side branch tubes, of con- 

 stant cross section but any of the other side branches shown on Table 

 I can be used by employing the equivalent values of /' and 5 shown 

 there. 



The frequency /a appearing in the filter No. 1 has no significance for 

 the attenuation constant. It determines the frequency at which the 

 characteristic impedance equals infinity. Considering the loss caused 

 by inserting the filter between two impedances equal approximately 

 to Zo, an additional loss occurs at the frequency /„, due to a mismatch 

 of the impedance of the filter and the terminating impedances. Filter 

 No. 4 of Table II is similar to No. 3 except that it has twice the attenu- 

 ation constant. It is then equivalent to two sections of the No. 3 

 filter. 



* See for example Journal of the Optical Society, July 1929, page 18. 



