194 WAVES IN LIQUIDS. [CHAP. VII. 



so that the angular velocity of the element (a, 6) is 



This is greatest at the surface, and diminishes rapidly with in 

 creasing depth. 



Propagation in Two Dimensions. 



161. In the cases already considered the propagation of the 

 waves over the surface of the fluid has been supposed to take 

 place in one dimension (x) only. We will now sketch the method 

 to be pursued in treating cases where the propagation is in two 

 dimensions. 



Let the origin be taken in the undisturbed surface, the axes 

 of x and y horizontal, that of z vertical and upwards; and let 

 h be the (uniform) depth of the fluid. The velocity-potential &amp;lt; 

 must satisfy 



and the condition 



^ = 0, when z = -h .................. (36). 



Further, at the free surface we must have, making the same 

 approximations as in Art. 156 



d P_ fJ d( t&amp;gt;,r } 



dt g dz~ 



, p d6 



where - = const. - gz, 





Now (35) is satisfied by the sum of any number of terms of the 

 form 



........ .......... (38), 



