462 Mr green, on WAVES IN A VARIABLE CANAL, &c. 



If the initial values of ^ and ti are given, we may then determine 

 f and F', and we thus see that a single wave, like a pulse of sound, 

 divides into two, propagated in opposite directions. Considering, there- 

 fore only that which proceeds in the direction of x positive, we have 



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

 and Ix to be the corresponding length of the wave, we have 



r dx 



t - / , — ■ - = a + a, 



J y/gy 



r dx Sx , 



and t — / -— =. -= = a very nearly. 



Hence the variable, length of the wave is 

 (7). ^X = a. -s/g^. 



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



t — i - -1= = const. ; 

 ■' Vgy 

 therefore 



(8). ^ = V^, 



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 ji represent the variable breadth 

 of the canal and y its depth, 



^ = height of the wave oc fi-hy-i, 

 u = actual velocity of the fluid particles ex: /3" 5 7 -5, 

 Ix = length of the wave oc 7J, 



dx / — 



and -7- = velocity of the wave's motion = vgy- 



