150152.] WAVES IN CANALS. 181 



where q is the velocity. If the slope of the wave-profile be every 

 where gradual, and the depth h small compared with the length 

 of a wave, the horizontal velocity may be taken as uniform through 

 out the depth, and approximately equal to q. Hence the equation 

 of continuity is 



7j) = ch, 



c being the velocity, in the steady motion, where the depth is uni 

 form and equal to h. Substituting for q in (13), we have 



If j be small, the condition for a free surface, viz. p = const., is 

 satisfied approximately, provided 



C*=ffh, 



which is identical with equation (7). 



If we take account of the second power of j- we find that at a 



part of the stream where the average elevation is k the condition 

 for a free surface is better satisfied by 



h 



or 



The higher portions of a wave therefore advance faster than the 

 lower, so that the form of a wave continually changes as it pro 

 ceeds. Thus, in the case of a wave of elevation only, the slope 

 becomes gradually steeper in front, and more gentle behind until 

 finally the conditions on which our investigations are based foil 

 altogether to hold. The formula (14) seems due to Airy*. It is 

 otherwise obvious from (4) that the accuracy of the equation 



&amp;lt;ff_ &amp;lt;P| 

 dt* ~ c dtf 



* Encyclopaedia Metropolitan**, Vol. v., &quot;Tides and Waves,&quot; Art. 208. 



