Waves in Flowing Water. 355 



unity, we have 



P - q =i{v*+&Hz-[.W+ff(f+i")l ■ ■ (3)- 



Now, calling the pressure at the free surface zero, we have 

 p = iga; and q = igb+j .... (4); 



v} denoting a quantity depending on wave-disturbance. 

 Hence, and by (2), 



i W c ^-g(a-b+f)+^=Q . . (5). 

 Now, put 



iJ S^=i ; and M=VD • • • • (6) - 



Thus D will denote a mean depth (intermediate between a and 

 b and approximately equal to their arithmetic mean, when their 

 difference is small in comparison with either) ; and V will 

 denote a corresponding mean velocity of flow (intermediate 

 between u and v, and approximately equal to their arithmetic 

 mean, when their difference is small in comparison with either) . 

 With this notation, (5) gives 



. w — ivf 

 b-a= ft- (7). 



1 — — 



9D 



If b— a were exactly equal to/, and if there were no beruffle- 



ment of the water beyond B, the mean level of the water 



would be the same in the entering and leaving water at great 



distances on the two sides of AB; but this is not generally 



the case, and there is a (positive or negative) rise of level, 



given by the formula 



V 2 j, ,w — w' 



j gD^ gb /ft . 



y = b-a-f= =f— .... (8). 



1 _ 



Consider now the case of no corrugation (that is to say, 

 of plane free surface and uniform flow) at great distances 

 beyond B. We have iv—w ! =zO; and therefore 



y = b-a-f= ^2- (9); 



