280 TIDAL WAVES. [CHAP. VIII 



c being the velocity, in the steady motion, at places where the 

 depth of the stream is uniform and equal to h. Substituting 

 for q in (1), we have 



Hence if ij/h be small, the condition for a free surface, viz. 

 p = const., is satisfied approximately, provided 



c 2 = gh, 

 which agrees with our former result. 



173. It appears from the linearity of our equations that 

 any number of independent solutions may be. superposed. For 

 example, having given a wave of any form travelling in one 

 direction, if we superpose its image in the plane x = 0, travelling 

 in the opposite direction, it is obvious that in the resulting 

 motion the horizontal velocity will vanish at the origin, and the 

 circumstances are therefore the same as if there were a fixed barrier 

 at this point. We can thus understand the reflexion of a wave at 

 a barrier ; the elevations and depressions are reflected unchanged, 

 whilst the horizontal velocity is reversed. The same results 

 follow from the formula 



% = F(ct-x)-F(ct + x) .................. (1), 



which is evidently the most general value of f subject to the 

 condition that f = for x = 0. 



We can further investigate without much difficulty the partial reflexion 

 of a wave at a point where there is an abrupt change in the section of the 

 canal. Taking the origin at the point in question, we may write, for the 

 negative side, 



and for the positive side 



