166-167] WAVES IN UNIFORM CANAL. 273 



Eliminating 77 between (4) and (5), we obtain 



The elimination of gives an equation of the same form, viz. 



d ^-ah^ (7) 



d?- g/l dx* 



The above investigation can readily be extended to the case of a uniform 

 canal of any form of section*. If the sectional area of the undisturbed fluid 

 be $, and the breadth at the free surface &, the equation of continuity is 



(iv), 



whence rj= ~^5E ....................................... ( v )&amp;gt; 



as before, provided h^=S/b, i.e. h now denotes the mean depth of the canal. 

 The dynamical equation (4) is of course unaltered, 



167. The equations (6) and (7) are of a well-known type 

 which occurs in several physical problems, e.g. the transverse 

 vibrations of strings, and the motion of sound-waves in one 

 dimension. 



To integrate them, let us write, for shortness, 



&amp;lt;f=gh .............................. (8), 



and xct = x l} x + ct = x 2 . 



In terms of x l and # 2 as independent variables, the equation (6) 

 takes the form 



o. 



The complete solution is therefore 



% = F(a;-ct)+f(a; + ct) .................. (9), 



where F,fa,rQ arbitrary functions. 



The corresponding values of the particle- velocity and of the 

 surface- elevation are given by 



|/c = - F 1 (x - ct) +/ (* + ct), } ( } 



rj/h=-F (x-ct)-f(x + ct) j ...... 



* Kelland, Trans. R. S. Edin., t. xiv. (1839). 

 L. 18 



