230 ON WAVES IN A VARIABLE CANAL, &C. 



Suppose now the value of F' (x) = 0, except from x = a to 

 x = a + a, and x to be the corresponding length of the wave, 

 we have 



[dx 

 t- -j=, = a + a, 



Wgy 



, , [ dx Sx , 



and t I r= = = a very nearly. 

 JVffy V<77 



Hence the variable length of the wave is 



............ (7). 



Lastly, for any particular phase of the wave, we have 



. dx 

 t -== = const. : 



therefore 



is the velocity with which the wave, or more strictly speaking 

 the particular phase in question, progresses. 



From (5), (6), (7), and (8) we see that if ft' represent the 

 variable breadth of the canal and 7 its depth, 



f = height of the wave @~*<y~^, 

 u actual velocity of the fluid particles 

 dx = length of the wave 7^ 



and -j- =5 velocity of the wave's motion = 



