278 TIDAL WAVES. [CHAP. VIII 



Between equations (1) and (2) we may eliminate either ?? or f ; 

 the result in terms of f is the simpler, being 



This is the general equation of ' long ' waves in a uniform canal 

 with vertical sides. 



So far the only assumption is that the vertical acceleration of 

 the particles may be neglected. If we now assume, in addition, 

 that y/h is a small quantity, the equations (2) and (3) reduce to 



-M 



The elevation rj now satisfies the equation 



This is in conformity with our previous result ; for the small- 

 ness of dQdx means that the relative displacement of any two 

 particles is never more than a minute fraction of the distance 

 between them, so that it is (to a first approximation) now 

 immaterial whether the variable x be supposed to refer to a 

 plane fixed in space, or to one moving with the fluid. 



171. The potential energy of a wave, or system of waves, 

 due to the elevation or depression of the fluid above or below the 

 mean level is, per unit breadth, gpffydxdy, where the integra- 

 tion 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. 

 Effecting the former integration, we get 



(1). 



The kinetic energy is 



(2). 



Airy, I. c. 



