368 Lord Rayleigh on the Momentum and 



inquire what conditions must be satisfied in order to prevent 

 the formation of a negative wave. It is clear that the answer 

 to the question whether, or not, a negative wave will be 

 generated at any point will depend upon the state of things 

 in the immediate neighbourhood of the point, and not upon 

 the state of things at a distance from it, and will therefore 

 be determined by the criterion applicable to small dis- 

 turbances. In applying this criterion we are to consider the 

 velocities and condensations not absolutely, but relatively, to 

 those prevailing in the neighbouring parts of the medium, 

 so that the form of (20) proper for the present purpose is 



du 

 whence 



V®-? < 21 >- 



•=jV©? < 22 >' 



which is the relation between u and p necessary for a 

 positive progressive wave. Equation (22) was obtained 

 analytically by Earnshaw (Phil. Trans. 1859, p. 146). 



In the case of Boyle's law, v ' {dp/ dp) is constant, and the 

 relation between velocity and density, given first, I believe, 

 by Helmholtz, is 



u = alog(p/p Q ), 



if p be the density corresponding to « = 0."" 

 In our previous notation 



dpldp=f(p) = a*+f"(p ) . (p- Po ), 



a being the velocity of infinitely small waves, equal to 

 v/{/>o)}; and by (22) 



««£=* + ^(^- 1 -)^ 2 . (23), 

 po Po\ 2a PoJ 2 



the first term giving the usual approximate formula. 



The momentum, reckoned per unit area of cross section, 



= \pu dx — pQ 1(1+ - — P°\udx. 



Introducing the value of u from (23) and assuming that 

 the mean density is unaltered by the vibrations, we get 



