MINIMUM PERCEPTIBLE DIFFERENCES 331 



the case of the higher harmonics introduced by Class B or pentode output 

 systems, a fraction of a per cent is noticeable on direct comparison tests 

 with systems in which these higher components are very low as, for 

 example, Class A output systems. 



13.15. Combination Tones and Nonlinear Transducers. — In most of 

 the discussions in this book the elastic restoring force of the elements of a 

 vibrating system have been considered to be proportional to the first 

 power of the displacement. If a second power term is included the element 

 is asymmetrical, the restoring force being different in magnitude for positive 

 and negative displacements. According to most investigators the struc- 

 ture of the ear is of such an asymmetrical character. 



Assume that the displacement of the nonlinear element may be expressed 

 as follows 



y = ap + bp" 13.1 



where p = the actuating force, 



a = a. constant, and 



b = a. constant. 

 Suppose two harmonic forces pi = po cos wi/ and p2 = po cos C02/ are 

 impressed upon the system. The total force on the nonlinear element is 



P = pi -\- p2 = po cos COi/ + po cos CO2/ 13.2 



Substituting equation 13.2 in 13.1 the resulting equation may be put in 

 the form 



y = apo cos coi/ + apo cos co2/^ + ^bpo"^ cos 2a)i/ + ^bpo^ cos 2co2/ 



+ bpo^ cos (cOi + CO2)/ + bpo"^ cos (wi — CO2)/ + bpo"^ 13.3 



If the element under consideration is the ear it will be seen that six dif- 

 ferent frequencies will be heard as follows: the first primary frequency 

 coi/27r, the second primary frequency w2/27r, the second harmonic of the 

 first primary frequency coi/tt, the second harmonic of the second primary 

 frequency 002/ t, the summation frequency (coi + w2)/2t, the difference 

 frequency (coi — 0)2) /27r. The last term bp^ represents a steady force and 

 produces no sound. 



Combination or sum and difference tones may be produced in any non- 

 linear system as, for example, an intense sound in the air, in the throat of 

 a horn, by overloaded vacuum tube amplifiers, by diaphragms and by 

 suspension systems. 



13.16. Minimum Perceptible Differences. — The minimum perceptible 

 difference in frequency ^^ is of interest in any type of sound reproduction 



2^ Shower and Biddulph, Jour. Acous. Soc. Amer.^ Vol. 3, No. 2, Part 1, p. 275, 

 1931. 



