298 TIDAL WAVES. [CHAP. VIII 



If we now determine /(?;) so that 



(h + r,)f W = g ........................ (5), 



this may be written 



dP dP 



In the same way we find 



The condition (5) is satisfied by 



(8), 



where c = (gh)*. The arbitrary constant has been chosen so as to 

 make P and Q vanish in the parts of the canal which are free from 

 disturbance, but this is not essential. 



Substituting in (6) and (7) we find 



-u|&amp;lt;# I 



dQ 

 dx 



It appears, therefore, that dP = 0, i. e. P is constant, for a 

 geometrical point moving with the velocity 



whilst Q is constant for a point moving with the velocity 



- 



Hence any given value of P travels forwards, and any given value 

 of Q travels backwards, with the velocities given by (10) and (11) 

 respectively. The values of P and Q are determined by those 

 of 1) and u, and conversely. 



As an example, let us suppose that the initial disturbance 

 is confined to the space between x = a and x = b, so that P and Q 

 are initially zero for x &amp;lt; a and ec &amp;gt; b. The region within which P 



