REGULAR COMBINATION OF ACOUSTIC ELEMENTS 267 



is always zero. Hence either the attenuation constant A is zero, or 

 the phase shift is zero, tt radians or some multiple of w radians. Now 

 since cosh A can never be less than 1 while cos B must lie between 

 + 1 and — 1 , then when the expression for cosh V is between — 1 and 

 + 1, the attenuation constant A is zero and cos B equals the expression 

 in (23). When the value of cosh V is outside the limits ± 1, the phase 

 shift is 0, X, or some multiple and the attenuation constant A is given 

 by the expression in (23). 



The specific characteristic impedance Zo, given in (23), can be shown 

 to be a real quantity within the transmitted band and an imaginary 

 quantity outside the transmitted band. 



The type of filter obtained with the structure shown in Fig. 2 de- 

 pends on the sidebranch impedance Z^. As long as Zg is of such a 

 value as to make the expression for cosh V greater in magnitude than 1, 

 an attenuation band occurs, while if cosh V is less than 1, a pass band 

 occurs. The cut-off frequencies of the band occur when cosh F = ± 1. 

 From equation (23) the cut-off frequencies occur when 



Z, = -^^— -cot(^-^j or Z,= --^^— tan(^-^j. (24) 



A . Low Pass Filter 



The model shown in Fig. 2 can be used to obtain the dififerent types 

 of recurrent filters possible by acoustic means. One of the simplest 

 types of filters in the electrical case is the low pass filter. No exact 

 analogue of this filter exists in the acoustic case, as every acoustic 

 filter has more than one band, but a filter which passes low frequencies 

 and attenuates high frequencies can be designed. 



Suppose that the sidebranch used is a straight tube closed at one 

 end. Then by equation (10), the impedance Zg, when the tube is 

 terminated in an infinite impedance, is 



Zs — Z^^ coth 0:2/, 



where Z^, and ao are respectively the specific characteristic impedance 

 and propagation constant of the sidebranch, and / its length measured 

 to the center of the conducting tube. Substituting this in the expres- 

 sion for cosh r and Zo, we have 



Zi.6'2 sinh 2a iL 



cosh r = ( cosh 2aiL + ~ 



2Zl,Si coth ao/ 



I 



/j Zl.S-j tanh a^L -^ /2^\ 



„ ^ , 2Zz....5i coth a4 



° ~ ^'- ' J Z^,.S2 coth aiL • 

 2Z/,,5'i coth atl 

 18 



