Extraneous Frequencies Generated in Air Carrying Intense 



Sound Waves * 



By A. L. THURAS, R. T. JENKINS and H. T. O'NEIL 



The exact equation of the propagation of plane sound waves in air is not 

 linear and consequently harmonics and combination tones are generated. 

 The pressure of these extraneous frequencies in terms of the fundamental 

 pressure, frequency, and distance from the source has been mathematically 

 determined by Rayleigh, Lamb and others. These equations have been 

 applied to an exponential horn. 



Measurements of the second harmonic and combination tones have been 

 made at various points within a long tube, and in front of an exponential 

 horn. Measurements, in general, agree with theory, but the absolute values 

 are lower than the calculated values. 



RECENT developments in horn type loud speakers ^ for high 

 quality reproduction of intense sounds necessitate a consideration 

 of the more exact equations of wave motion if distortion due to the 

 generation of extraneous frequencies in the air of the horn itself is to be 

 avoided. Similar considerations may be of some importance in con- 

 nection with the pick-up of intense sounds. 



The propagation of waves of finite displacement has interested 

 physicists for more than a century. In 1808 Poisson derived an 

 equation which shows that, in general, a sound wave cannot be 

 propagated without a change in form and consequent generation of 

 additional frequencies. This distortion is caused by the non-linearity 

 of air; that is, if equal positive and negative increments of pressure are 

 impressed on a mass of air the changes in volume of the mass will not 

 be equal ; the volume change for the positive pressure will be less 

 than the volume change for the equal negative pressure. An idea of 

 the nature of the distortion can be obtained from the adiabatic curve 

 AB for air as given in the familiar volume pressure indicator diagram 

 (Fig. la). The undisturbed pressure and specific volume of air are 

 indicated by point PqVo. Any deviation from the tangent through 

 this point causes distortion and consequent generation of extraneous 

 frequencies. The theoretical magnitudes of the waves of extraneous 

 frequencies are obtained from a solution of the exact differential 

 equation of wave propagation in air. The solution shows that the 

 pressure of the second harmonic frequency, which is generated in the 

 air, increases with the frequency and the magnitude of the fundamental 



* Published in the January 1935 issue of the Jour. Acous. Soc. Am. 

 ' E. C. Wente and A. L. Thuras, "Loud Speakers and Microphones," Bell Sys. 

 Tech. Jour., 13, 259 (1934). 



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