|,(pOv)=p.f .pof (3) 



29 . The total rate of change of momentum is the sum of the local and 

 convective changes pdQ/dt + pvdQ/dy + pQdv/dy, or pdQ/dt + pvdQ/dy + pAvdv/dy . 



30. Application of Newton's second law to the flow through the element 

 of volume can be expressed by Equation 4, 



-pffA-^ - Ti,P + T^ - psrA -g -pgAS^ 



do do , dv 



= p^ + pv-^ + pA V-^ 



^ at ^ 5y ^ dy 



Since h + Z];^ = z , the sum of the two terms -dz^/dy and -dh/dy can be replaced 

 by -dz/dy. By dividing all terms by pA, the equation simplifies to 



A dt A dy[ a) ^>= pA ^'pA ^^« ^dy 



31. Equation 5 is the mathematical expression for the conservation of 

 momentum. The bottom shear stress term Pr^/pA is commonly replaced by gS^, 

 where Sf is called the friction slope. Replacing Q by vA, and using the 

 equation of continuity to replace dA/dt by -dQ/dy + q, Equation 5 may be 

 rewritten as 



||.v|^=-.s,-.s,.^.|-.f (6) 



Equation 6 is which is the standard version of the conservation of momentum in 

 one direction. Although both forms, Equations 5 and 6, can be used for 

 numerical modeling, Equation 5 is more desirable for application to waterways 

 of irregular shape because use of the discharge Q rather than the velocity v 

 as the independent variable preserves the momentum equation in conservative 

 form. 



32. Equations 2 and 5 are known as the one -dimensional shallow-water 

 equations or the one -dimensional long-wave equations. The equations are valid 

 if the assumption of a hydrostatic pressure distribution holds. They are 



14 



