REGULAR COMBINATION OF ACOUSTIC ELEMENTS 287 



Substituting these values in equation (53), we obtain the equation 



r 



p2 = '-^\ pi 

 X-1 



Sin 



cos J, (X2 — Xi) + 



( ^ (X2 - Xi) j 



CO 



Fi 



— i-^ VPoTP sin ^ {x2 — -Xi) I , 



¥2 = —-1 Vi\ cos -^ (.T2 - Xi) - 



Xl \ 1 Ly 



X 



sin ( -^ (x2 — Xl) j 



^ (55) 



C 



Xo 





If we introduce two lengths ti and eo defined by tan (co/C)€i = (w/C)xi 

 and tan (co/C)e2 = (co/C)x2 and take account of the fact that the imped- 

 ance as defined here must be multiplied by io: to correspond to the 

 impedance defined by Webster, then it is evident that the above 

 equation corresponds to the relation given by Webster.^- ^° 



It is interesting to compare the relations obtained above involving 

 the assumptions introduced in Section II with the solution involving 

 no assumptions. This can be done for the conical horn, since its 

 solution can be obtained using spherical waves. In Section lll-D, 

 the impedance looking into a conical horn was obtained when an 

 infinite impedance terminated the horn. If we set F2 = in the last 

 of equations (55) and solve for the ratio of pi/Vi, it is evident that the 

 impedance agrees with that given in Section III-D. Hence it is evident 

 that both methods give the same solution. 



Many other types of tapered filters can be solved in a similar man- 

 ner, but no more will be considered here. 



V. General Network Equations and Network Parameters 



We can combine a number of symmetrical structures to form a 

 general network. For any symmetrical structure we can write the 



i"The solution for the conical horn has been discussed in more detail by I. B. 

 Crandall, "Theory of Vibrating Systems and Sound," D. Van Nostrand, 1926, p. 152. 



