180 WAVES IN LIQUIDS. [CHAP. VII. 



at the beginning of Art. 147 requires then that the slope of the 

 wave-profile should be gradual, and that the length of an elevation 

 or depression should be large compared with the undisturbed depth 

 of the fluid. Waves which fulfil these conditions are called long 

 waves. The motion of waves for which the conditions are not 

 satisfied is discussed below, Art. 156. 



The second assumption of Art. 147, viz. that -~ is small, re- 



U.X 



quires, by (4), that the maximum height of the wave should be 

 small compared with the depth of the fluid. 



151. The potential energy of a wave, or system of waves, due 

 to the elevation or depression of the fluid above the mean level h 

 is, per unit breadth, gpfjydxdy, where the integration with respect 

 to y is to be taken between the limits and 77, and that with 

 respect to x over the whole length of the waves. Performing the 

 former integration, we get 



^gpjvfda; .............................. (11). 



The kinetic energy is, in the case of Art. 147, 



In a system of waves travelling in one direction only we have 



so that (11) and (12) are equal; or the total energy is half poten 

 tial, and half kinetic. 



152. If in any case of waves travelling in one direction only, 

 without change of form, we impress on the whole mass a velocity 

 equal and opposite to that of propagation, the motion becomes 

 steady, whilst the forces acting on any particle remain the same as 

 before. With the help of this artifice, the laws of wave-propa 

 gation can be investigated with great ease*. Thus, in the case of 

 Art. 147 we have, at the free surface, 



P - = const. -% q *- ff (h + v ) .................. (13), 



* See Lord Eayleigh, &quot; On Waves,&quot; Phil. May., April, 187G. 



