312 TIDAL WAVES. [CHAP. VIII 



This makes V^^O, so that the equation (12) of Art. 185 

 reduces to the form (1), above, and the solution is 



=AJ 8 (kr) cos sd . cos (at + e) (17), 



where A is an arbitrary constant. The boundary-condition (Art. 

 185 (14)), gives 



AkaJ s (ka)=sC, 



whence = Y, \ cos sO . cos (crt + e) (18). 



KdJ s (fCd) 



The case s = 1 is interesting as corresponding to a uniform 

 horizontal force ; and the result may be compared with that of 

 Art. 176. 



From the case s = 2 we could obtain a rough representation of 

 the semi-diurnal tide in a polar basin bounded by a small circle 

 of latitude, except that the rotation of the earth is not as yet taken 

 into account. 



We notice that the expression for the amplitude of oscillation 

 becomes infinite when J 8 (ka) = 0. This is in accordance with a 

 general principle, of which we have already had several examples ; 

 the period of the disturbing force being now equal to that of one 

 of the free modes investigated in the preceding Art. 



189. When the sheet of water is of variable depth, the 

 investigation at the beginning of Art. 185 gives, as the equation 



of continuity, 



d% _ d(hu) d(hv) .-. 



di~ ~~dx~ ~dy~ 



The dynamical equations (Art. 185 (2)) are of course unaltered. 

 Hence, eliminating f, we find, for the free oscillations, 



d /, d\ d 



If the time-factor be e i(fft+e) , we obtain 



dx \ dx) dy \ dy) g 



When h is a function of r, the distance from the origin, only, 

 this may be written 



