OF SMALL DEPTH AND WIDTH, 229 



equating now separately the coefficients of /' and /", we get 



dx* dA d$ dy 



~ * 



dx 

 The first, integrated, gives 



and the second ' 



k = -r- A 2 8v = , 



dx 



. 



<\/gy . V^r 



Hence if we neglect the superfluous constant k V^, the gene- 

 ral integral of (4) is, (v A = 



therefore, by (c'), 



and the actual velocity of the fluid particles in the direction of 

 the axis of x, is 



dx dx 



neglecting quantities which are of the order (ay) compared with 

 those retained. 



If the initial values of f and u are given, we may then deter- 

 mine /' and F', and we thus see that a single wave, like a pulse 

 of sound, divides into two, propagated in opposite directions. 

 Considering, therefore, only that which proceeds in the direc- 

 tion of x positive, we have 



