EXTRANEOUS FREQUENCIES 169 



Exponential Horn Theory 



The second harmonic generated in any short section of a horn is 

 approximately the same as that generated in a tube of area equal to 

 the mean area of the section of the horn. Therefore, from the tube 

 equation and the expression for the change in pressure due to the 

 divergence of the horn the magnitude of the generated second harmonic 

 pressure at any point along the horn can be obtained. 



The r.m.s. value of a small excess pressure in an exponential horn 

 of section 5 = ^oe*"^ is attenuated according to the law 



P = Pte-'"^i'~ or dPIdx = - mP/2, 



where Pt is the r.m.s. pressure in the throat of the horn (at x = 0). 

 The index of taper m is equal to 47r/c/c where /c is the cut-off frequency 

 of the horn. 



From equation (12), the rate at which the r.m.s. value of the second 

 harmonic increases along a tube is 



dP2 T + 1 Pi 



CO 



dx 2(2)* ypo c 



= KPl 



If it is assumed that the same expression represents the rate of genera- 

 tion of second harmonic along a horn, and that both the fundamental 

 and second harmonic diverge in the same manner, the complete 

 differential equation for P^ becomes 



where Pi and Pn are, respectively, the r.m.s. fundamental pressures 

 at the point in question {x) and in the throat of the horn. The solution 

 consistent with the condition Po = at :x; = is 



P2 = [i^Pf,/(m/2)][e-'"-'2 - e-"-]. (18) 



Since Pi = Pne""'^^^, the ratio of the second harmonic pressure to the 

 fundamental pressure at any point x in the horn is thus 



Pi mjl 2yl 2 ypo c 



* According to this equation the magnitude of the second harmonic pressure 

 generated in the air of a horn is 6 db lower than that given by Rocard's equation 

 previously published in the paper "Loud Speakers and Microphones" (Reference 1), 

 page 264, equation 1. 



