3.^8 BELL SYSTEM TECHNICAL JOURNAL 



Is that of an anti-resonant circuit, as shown on Fig. 8. There are 

 two possible impedance characteristics for the series arm, in relation 

 to the lattice arm, which will give a single band filter. One of these is 

 obtained by letting the series arm have an infinite impedance when the 

 lattice arm has a zero impedance, which results in a low pass filter. 

 The second relation — which is that shown on Fig. 8 — is obtained by 

 letting the series arm have an infinite impedance when the lattice arm 

 has an infinite impedance. The pass band is between zero frequency, 

 and the frequency at which the lattice arm resonates. 



In a similar manner, the other types of acoustic filters can be ana- 

 lyzed. 



C. Side Branch Impedances 



The possible types of side branches can be divided into two classes, 

 those which are entirely enclosed, and those which are open to the air. 

 The first kind are characterized by a series capacity, while the second 

 kind always have a shunt inductance. 



One of the simplest side branch impedances is a short tube open on 

 the end. The first approximation to this side branch is an inductance, 

 as shown on Table I, No. 1, equal to the total distributed inductance of 

 the tube. This approximation holds well if the product of the tube 

 length by the frequency, is not too large. A longer tube, open on the 

 end, can be represented by an inductance and capacity in parallel as 

 discussed in Section A and shown on Table I, No. 2. A tube closed 

 on the end can be represented by an inductance and capacity in series 

 as shown on Table I, No. 4. 



When these tubes are used as side branches, an additional factor 

 comes in — an end correction. That is, the side branch must be con- 

 sidered as extending into the main branch for a distance proportional 

 to the radius, because a motion of air in the direction of the side 

 branch, occurs in the main branch. The value of this effect has been 

 investigated by Rayleigh, who found that this effect can be calculated 

 by increasing the length of the tube by a length equal to .785 times the 

 radius. Another correction applies to an open ended tube, which has 

 been determined experimentally as .57 times the radius. Hence the 

 length of an open ended tube must be considered as 



/' = / + (.785 + .57)r. 



A straight tube can give all the combinations of side branch imped- 

 ances, but one of its dimensions is necessarily limited, namely the 

 area. For the area cannot become larger than the area of the main 

 tube, since otherwise it could not be connected to the main tube. By 



