166-167] WAVES IN UNIFORM CANAL. 273 



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



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



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 S, and the breadth at the free surface 6, the equation of continuity is 



whence 17 = ~h~ (v), 



as before, provided k = 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, 



o 7 /Q\ 



c = gri (o), 



and x ct == x- , x -f- ct == x% . 



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

 takes the form 



-^ ^ = 0. 



The complete solution is therefore 



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



where F, f are arbitrary functions. 



The corresponding values of the particle- velocity and of the 

 surface- elevation are given by 



?/c = - F' (x - ct) +/' (x + ct), \ (W) 



18 



v /h =-F f (x- ct) -f (x + ct) 



* Kelland, Trans. R. 8. Edin., t. xiv. (1839). 



