176 DYNAMICAL THEORY OF SOUND 



which must take place in sound-waves propagated in actual 

 fluids, we must have recourse to accurate equations of motion. 

 On the plan of 59 we have 



and P = po/(l + A) = p /(l -f ^ (7) 



Hence, on the adiabatic hypothesis that 



P/Po ~ (p/po) 7 > (8) 



we find by elimination of p and p 



2\Y+1 



, (9) 



where c 2 = ypo/po as before. 



For illustrative purposes it is sufficient to consider the 

 isothermal case, which is derived from the above by putting 

 7=1, so that 



We have seen in 60 that on the hypothesis of infinitely 

 small vibrations there is a definite relation between particle- 

 velocity and condensation in a progressive wave. Following 

 Earnshaw, we assume (tentatively) that the same thing holds 

 in the general case, and write accordingly 



(11) 



where the form of the function is to be determined. From this 

 we deduce 



da?' 



and therefore = f ................ (IS) 



Hence (10) is satisfied provided 





