302 TIDAL WAVES. [CHAP. VIII 



The dynamical equations are, in the absence of disturbing 

 forces, 



du _ dp dv _ dp 



p ~di~~dx' p di~~dy* 



where we may write 



P -Po = 9p(zo +-z), 



if Z Q denote the ordinate of the free surface in the undisturbed 

 state, and so obtain 



du_ d dv _ d 



dt~~ 9 dx> dt~ 9 dy" 



If we eliminate u and v, we find 



where c 2 = gh as before. 



In the application to simple-harmonic motion, the equations 

 are shortened if we assume a 'complex' time-factor e i(<rt+e) , and 

 reject, in the end, the imaginary parts of our expressions. This 

 is of course legitimate, so long as we have to deal solely with 

 linear equations. We have then, from (2), 



, .................. (4), 



cr dx a- dy 



whilst (3) becomes 



g + ^ + ^ =0 ..................... (5), 



da? dy 2 



where fr = a*/c? ........................... (6). 



The condition to be satisfied at a vertical bounding wall is 

 obtained at once from (4), viz. it is 



2- .............................. ox 



if Sn denote an element of the normal to the boundary. 



When the fluid is subject to small disturbing forces whose 

 variation within the limits of the depth may be neglected, the 

 equations (2) are replaced by 



du _ d dl dv _ d dl , . 



di~ ~ g dx~~fa' dt~ ~ g fy~dy" 



where H is the potential of these forces. 



