BACKWATER FUNCTION 810 



Writing jj- = m, so that .' = H, this becomes 



HI 



n dm .[ " 3 " 1 

 11 ^T = 1 w - - -11 



/. r? ?>i -i 1 + - ^- } = 4? '* / - 

 wi 8 1 i ^^ 



Integrating this expression, 

 while 



i 7 /., 2 i \ f d m 



77 ' = '" + 1 1 - -)j^F=t + const 



f__dm_ ._!_/]_ /" + 1 1 _ ) 7 



y m a _ i - 2 / I (/// 2 -fly/ + 1) (7/ t - 1) m* + /// + 1 f '' 



\^r 



1 Jliog, ^ + + 1 -f _^__ tail - ^ :- +.11 

 2 3 (m-1 2 



The expression 



,2 



tan 



- 



, . 



(6 (m-1) V3 VF J 



is often termed the " backwater function." Writing this as </> (m), the 

 equation becomes 



> (/w) + C 



:. h = -i / + 77 1 - l ^ (m) + (7 (9) 



Thus if hi and A 2 are the depths at points distant /i and / 2 from the 

 datum, we have 



The following table gives values of </> (m) for different values of -~ 

 in the case of a dam, where /i is always greater than //. 



