REGULAR COMBINATION OF ACOUSTIC ELEMENTS 271 



factor, {Zi — Zt)I{Zj -\- Zr), while the current transmitted to the 

 terminating side of the junction will be the unmodified current in the 

 line multiplied by the factor IZf/iZj + Zt) where Zj and Zj, are 

 respectively the impedances looking towards and away from the source 

 at the junction. We see then that the second and third factors are 

 transmission factors, determining respectively the transmission from 

 the input impedance Za to the inserted structure, and from the inserted 

 structure to the output impedance Zb- The first factor is the inverse 

 of the transmission factor determining the transmission from the im 

 pedance Za to the impedance Zb- The fourth factor is the transfer 

 factor and gives the reduction in volume velocity due to attenuation. 

 1 he fifth factor has been called the interaction factor, and it gives the 

 change in volume velocity in the termination due to repeated reflec- 

 tions of the volume velocity within the structure. All of these factors 

 reduce to 1 except the transfer factor when Za = Zb = Zo. It will be 

 noted that all factors except the transfer factor cancel out if Za = Zo, 

 or Zb = Zo. 



The effect on the pressure due to inserting a filter can be shown to 

 be given also by equation (30). 



If the terminating impedances are resistances about equal to an 

 average of the resistance value of Zo, the effect of these is generally to 

 introduce some loss in the pass band, when the characteristic impedance 

 differs materially from the terminating impedances due to a reflection 

 of the sound wave at the junction points. Since the characteristic 

 impedance of a non-dissipative filter goes either to zero or infinity 

 at the cut-off frequency, the effect of the reflection loss is generally to 

 narrow the pass bands of the filter. 



The effect of dissipation, when we take account of the viscosity 

 effects by equations (20) or (21), is two-fold. It changes slightly the 

 position of the band in the frequency range, due to a small change 

 in the velocity of propagation. This is generally negligible. The 

 other effect is to introduce attenuation in the pass band, due to ab- 

 sorption and dissipation of the sound wave. 



B. High Pass Filter 

 An analogous type of high pass filter, which will attenuate the low 

 frequencies and pass the high frequencies, can be made from the 

 structure shown in Fig. 2 by using side tubes which are open on the 

 outer end. The termination at the end of an open tube has been 

 shown by Rayleigh ^ to be a mass with some resistance due to radia- 

 tion. We could substitute this relation in equation (10) to determine 



' Rayleigh, "Theory of Sound," Vol. II, p. 106. 



