96 ACOUSTICAL ELEMENTS 



where p = density, in grams per cubic centimeter, 



c = velocity of sound, in centimeters per second, and 

 Si = cross-sectional area of the infinite tube, in square centimeters. 

 Equation S.SS is the mouth impedance of the exponential horn. Equa- 

 tion 5.53 then becomes 



2^1 = TT m „. . ■ . 777. 5.56 



_ p<:rcos {bl - 6) 4-isin {bl)~\ 

 Si\_cos {bl + d) +jsin {bl)\ 



For b = 0, equation S.S6 is indeterminate. To evaluate take the deriva- 

 tive of the numerator and denominator with respect to b and set b — 0. 

 Then the expression for the throat impedance becomes 



pc 



. bn Im 

 Im . Im 



5.57 



Below the frequency corresponding to ^ = 0, /^ is imaginary. This por- 

 tion of the range may be evaluated by employing the standard formulas 

 involving complex quantities. 



The impedance characteristics of two exponential connectors with a 

 flare cutoff of 100 cycles (that is ^ = at 100 cycles) is shown in Fig. 5.7. 

 Below 100 cycles the throat impedance is the same as that of the infinite 

 pipe. However, at the high frequencies the throat impedance is the same 

 as the surge impedance of a pipe of the diameter of the throat. In order 

 to effect a constant transfer of impedance with respect to frequency over 

 a certain frequency range the cutoff of the connector must be placed below 

 the low frequency limit of the frequency range. 



5.23. A Horn Consisting of Manifold Exponential Sections -". — The 

 efficiency of a horn loud speaker is governed, among many other factors, by 

 the throat resistance. To obtain the maximum efficiency at any fre- 

 quency the effective reactance of the entire vibrating system should be 

 equal to the effective resistance. This, in general, means that to obtain 

 maximum efficiency the throat resistance of the horn should be propor- 

 tional to the frequency, since the reactance is primarily mass reactance 

 and, therefore, proportional to the frequency. Practically any throat 

 impedance frequency characteristic may be obtained by employing a horn 

 consisting of manifold exponential sections. 



26 Olson, H. R, Jour. Soc. Mot. Pic. Eng., Vol. 30, No. 5, p. 511, 1938. 



