Mb EARNSHAW, ON FLUID MOTION. 231 



Similarly it may be shewn, that the density of a given part of 



B 



the converging wave varies as 6% which rapidly increases. 



33. If a plane-wave be transmitted through the medium, the 

 particles as it successively reaches them are displaced with the same 

 velocity; and there is a certain relation between the density and the 

 velocity of displacement which holds good for all plane-waves. 



For, denoting the velocity of displacement of a particle by v, and 

 the density as before, 



<p = F{fx+gy + hz - at), 

 and V- = {d,(pf + {d^fpY + {d,(pf 



= {r+g"- + Jr) {F'{fx+gy + hz-at)Y; 

 .". V = ± F' {fx + gy + h% ~ at); 

 and by referring this to a moveable origin (as in Art. 19), we have 

 v= ±F' {fx' + gy' + hz') = ±F' ip) ; 



which is independent of t, and is constant for all particles situated in 

 the same part of the wave, because for such particles p is constant. 



Again, 



a^ log^ -, = — dt<() = aF' {fx +gy + h%- at) = ± av; 



, ±i; 

 .-. p=p.e '. 



If V be reckoned positive in the direction of the wave's motion, 

 and negative when in the opposite direction, 



V 



P = p'e^ . 



Hence in plane-waves all points of equal velocity are points of 

 equal density. 



Again, when |0 is > p, v is positive, 



when p = p, V is evanescent, 

 when p is < p, v is negative. 



