282 TIDAL WAVES. [CHAP. VIII 



we further neglect the product of the small quantities F and 

 drj/dx, the equation reduces to 



1 dp _ drj 



jjas 'as- 



as before. The equation of horizontal motion then takes the 

 form 



where X may be regarded as a function of x and t only. The 

 equation of continuity has the same form as in Art. 166, viz. 



' *5 ........................... <*> 



Hence, on elimination of 77, 



175. The oscillations of water in a canal of uniform section, 

 closed at both ends, may, as in the corresponding problem of 

 Acoustics, be obtained by superposition of progressive waves 

 travelling in opposite directions. It is more instructive, however, 

 with a view to subsequent more difficult investigations, to treat 

 the problem as an example of the general theory sketched in 

 Art. 165. 



We have to determine so as to satisfy 



together with the terminal conditions that = for # = and 

 x = l, say. To find the free oscillations we put X = 0, and assume 

 that 



where or is to be found. On substitution we obtain 



whence, omitting the time-factor, 



,. | . (TX T-, <TX 



= A sin -- h B cos . 

 c c 



