92 ACOUSTICAL ELEMENTS 



stituted in equations 5.51 and 5.52 above. The constants may be elimi- 

 nated and the impedance of the throat in terms of the mouth impedance 

 may be obtained. The final result is 



_ pcr S2ZA2 [cos (^/ - e)] +jpc [sin (l?I)] l ^^^ 



^^' S, \_jS2ZA2 [sin {bl)] + PC [cos {bl + d)U 



where .S*! = area of the throat, in square centimeters, 

 S2 = area of the mouth, in square centimeters, 



/ = length of the horn, in centimeters, 

 Za2 = acoustic impedance of the mouth, in acoustic ohms, 

 6 = tan^^ a/b, 

 a = m/2, and 

 b = 1^4^ - m^. 



The impedance characteristic of a finite exponential horn is shown in 

 Fig. 5.4. From this figure a direct comparison may be made between a 

 conical and exponential horn of the same dimensions. These characteris- 

 tics show that the exponential horn has a definite low-frequency cutoflF 

 above which the throat resistance increases rapidly and becomes a con- 

 stant. On the other hand, the throat resistance of the conical horn in- 

 creases slowly with frequency and shows no definite low-frequency cutoff^. 

 Furthermore, the impedance frequency characteristics of the exponential 

 horn show a larger ratio of resistance to reactance. For these reasons the 

 exponential horn is more desirable and accounts for its almost universal 

 use in horn loud speakers. In view of its widespread use it is interesting 

 to examine some of the other characteristics of exponential horns. 



5.21. Throat Impedance Characteristics of Finite Exponential Horns ^5. 

 — The throat acoustic impedance characteristic as a function of the 

 mouth, with the flare and throat kept constant, is of interest in determining 

 the- optimum dimensions for a particular application. The impedance 

 characteristics of four finite horns having a cutoff of 100 cycles, throat 

 diameter of 1 inch and mouth diameters of 10, 20, 30 and 40 inches and the 

 corresponding infinite horn are shown in Fig. 5.5. These results may be 

 apphed to horns of a different flare by multiplying all the dimensions by 

 the ratio of 100 to the new cutoff frequency. The cutoff frequency of an 

 exponential horn is given by 



2co = mc 1 5.54 



25 Olson, H. P., RCJ Review, Vol. 1, No. 4, p. 68, 1937. 



