180 DYNAMICAL THEORY OF SOUND 



(26) is proportional to x, and to the square of the ratio of the 

 velocity of the piston to the velocity of sound. This latter ratio 

 may in practice be exceedingly small, but as we travel to the 

 right the correction continually increases in importance, until at 

 length the neglect of terms of the third and higher orders 

 would no longer be justified. This is what we should expect 

 from the results of Earnsnaw's investigation. 



When the motion of the piston is simple-harmonic, say 



/(0 = acosnt, (29) 



the formula (28) gives 



g = a cos n (t- -} + (v + V n9a * % \ _ C os2rc (t- -}l. (30) 



\ C/ oC \ C/j 



The displacement of any particle is no longer simple-harmonic, 

 but consists of a part independent of t together with two 

 simple-harmonic terms, one having the frequency of the 

 imposed vibration (29), and the other a frequency twice as 

 great. This illustrates the implied limitation to infinitely 

 small motions in the usual theory of forced oscillations ( 17). 



Again, if the given vibration of the piston be made up of 

 two simple-harmonic components, say 



f(t) = ! cos nj + a 2 cos n 2 t, (31) 



we find 



f = ttj cos nj (t - - j + a 2 cos n z (t -J 



f 1 ( / #\ 



- x \ n?a? + n 2 2 a 2 2 nfaf cos 2w x It 

 kr ( V cj 



n 2 W cos 2n 2 (t 



V c 



-f 2/1^2 Oi a z cos (/*! n 2 ) { t - 



2n 1 n 2 a 1 o 2 cos (^ + ^2) It JY . 



.(32) 



We thus learn that in addition to the vibrations of double 

 frequency, other simple-harmonic vibrations whose frequencies 

 are respectively the difference and the sum of the primary 

 frequencies now make their appearance. In acoustical language, 

 two simple vibrations of sufficient amplitude may give rise not 



