28 DYNAMICAL THEOEY OF SOUND 



the mean value of which is ^ri*bC\ Equating the mean rate of 

 decay of the energy to the mean dissipation, we get 



-ifrtC', .................. (5) 





< 6 > 



or tf=<7 <T t/T , ........................ (7) 



if T = 2a/6, as in (3). 



When there are given extraneous forces in addition to the 

 dissipative influences, the equation of energy takes the form 



& ............ (8) 



whence aq + bq + cq = Q ...................... (9) 



As in 9 we consider specially the case of a disturbing 

 force of simple-harmonic type, say 



Q = Ccospt ...................... (10) 



A particular solution of (9) is then obtained in the form 



q = Fcospt + Gsinpt, ............... (11) 



provided the constants F, G are properly chosen. The necessary 

 conditions are found on substitution to be 



-- . 



If we put 



Rcosa, pb=Rsina, ......... (13) 



we find F=cosa, = sina, ............... (14) 



c 



whence* q = -^ cos (pt a) (15) 



The values of R and a are determined by 



/nTi 



(16) 



R is to be taken positively, and a may be assumed to lie 

 between and TT. 



* A more rapid way of obtaining this solution is explained in 20. 



