PLANE WAVES 7 



The type of motion to be considered is irrotational, that is Curl F^yy, = 

 0. That is a necessary and sufficient condition for the existence of a 

 scalar velocity potential which is defined as 



dcj) dcf) d(j) 



u = — > V = — > w = — l.il 



dx dy dz 



or Fuvw = Grad 



Substitute equations 1.11 in 1.3 and multiply by dx, dy and dz 



-d<i>= --,dp,' 1.12 



dt p 



or integrating 



dt ~ Jo' 



dt J p 



Since the density changes very little, the mean density, p, may be used. 

 The J^dpo is the excess pressure, then 



^=-2 1.13 



dt p 



where p = excess pressure. 



From equations 1.9, 1.10, 1.11 and 1.13 



a/2 p \dx^ a/ dz^J 



or this may be written 



dt^ 



which is the standard D'Alembertian wave equation for </>. The velocity 

 of propagation is 



^ = c^ 1.15 



P 



For the velocity of sound in various mediums see Table 1.1. 



1.4. Plane Waves. — Assume that a progressive wave proceeds along 

 the axis of x. Then is a function of x and t only and the wave equa- 

 tion 1.14 reduces to 



^ = .^^ 1.16 



dt^ dx^ 



