278 TIDAL WAVES. [CHAP. VIII 



Between equations (1) and (2) we may eliminate either 77 or ; 

 the result in terms of f is the simpler, being 



(3) *- 



d* 



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 r}\li is a small quantity, the equations (2) and (3) reduce to 



#f . d f 

 and - 



The elevation 77 now satisfies the equation 



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

 ness of d^/dx 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 as 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, gpjfydxdy, where the integra 

 tion with respect to y is to be taken between the limits and ?/, 

 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, L c. 



