REGULAR COMBINATION OF ACOUSTIC ELEMENTS 263 



We have now that i] = pJZs where Zs is the impedance per unit area 

 of the sidebranch, or the ratio of the excess pressure to the linear veloc- 

 ity. Substituting this value in the above equation, we have 



We have also 



pi= pi, 



(12) 



where p-2 is the excess pressure in the conducting tube on the out- 

 going side. The equations are exactly equivalent to Kirchoff's laws, 

 and hence any equation for a combination of acoustic elements will also 

 apply to the combinations of equivalent electric elements. 



A slightly better approximation than the above has been obtained 

 by solving completely the case of three pistons placed in the sides of a 

 rectangular box. This corresponds closely to the condition considered 

 here, if we have rectangular tubes, since the waves can be considered 

 plane up to the junction point with little possibility of error. The solu- 

 tion obtained indicates that the main effect of the junction point is to 

 add an end correction to all the tubes entering the junction. For 

 example, we will measure the length of the main conducting tube, 

 between sidebranches, from the center of the sidebranches rather than 

 the edge, as the approximation given first would imply. Also the 

 length of the sidebranch should be measured from the center of the 

 conducting tube, rather than the edge. For other types of junctions, 

 different end corrections will apply to the sidebranch tubes. For 

 example if the width of the junction is large compared to the width of 

 the sidebranch, we should expect Rayleigh's theoretical value of .82 R 

 to apply where R is the radius of the sidebranch tube. Hence the equa- 

 tions for a junction are equivalent to Kirchoff's laws with the additional 

 proviso that end corrections shall be added to tubes entering a junction. 



The effect of a change of area of the conducting tube can be obtained 

 with the same assumptions as above. If we have one conducting 

 tube of area ^i, joined to a second of area S2, we can write 



^5, = ^,82 or Fi = F2, (13) 



where ^1 is the linear velocity in the first tube and ^2 in the second 

 tube. We have also that the pressures in the adjoining tubes are 

 equal. Hence 



p2 = pi and F2 = Fi. (14) 



This equation is of the same order of approximation as the second ap- 

 proximation given above for a junction, since we measure the length 

 from one change of area to the next change. 



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