176 WAVES IN LIQUIDS. [CHAP. VII. 



that the vertical motion may be altogether neglected. We shall 

 learn from our results in what cases this assumption is legitimate. 

 Let the origin be taken in the bottom of the fluid, the axis of x 

 horizontal, that of y vertical and upwards; and let us suppose 

 that the motion takes place in these two dimensions x, y. Let h 

 be the depth of the fluid in the undisturbed state, h+ 77 the ordi- 

 nate of the surface corresponding to the abscissa x, at the time t. 

 Since the vertical motion is neglected, the pressure at any point 

 (x, y) will be simply that due to the depth below the surface, viz. 



p = gp (h + rj y) + const. 

 Hence 



dp drj 



a-**!! .............................. (1) - 



which is independent of y, so that the horizontal accelerating force 

 is the same for all particles in a plane perpendicular to x. It 

 follows that all particles which once lie in such a plane always do 

 so. It is convenient, now, to follow the Lagrangian method (Art. 

 16), changing however the notation. Let x + f denote the abscissa 

 at time t of the particles whose undisturbed abscissa is x; we 

 have seen that f is in fact a function of x only. Further let the 

 independent variable x in p and 77 refer always to the same portion 

 of fluid ; in which case (1) still holds. The volume of fluid, corre 

 sponding to unit breadth originally contained between the two 

 planes x and x + dx is lidx ; at the time t + dt the same stratum 



of fluid has a thickness dx-\--~ dx, and its height is h -f rj. The 



dx 



equation of continuity therefore is 



The equation of motion of the stratum is 



d 2 % dp 



With the help of (1) and (2), this becomes 



