A similar expression may be derived by considering the vertical 

 momentum thus leading to the momentum equations for a homogeneous, 

 imcompressible medium, 



DU . /— 5 J 



S T^ = - i^ -p{n(a + 3/u + W^) } U^ (A-68) 



Dt 3x "^ ^ ^ s 



and 



DW /—2 o 



pS p^= - |2- - Pg -plnCa + g/ U + W")} W^ , (A-69) 



where 



S = 1 + kCI - n] , CA-70) 



is expected to take on values within the range 1 <_ S <_ 1.5. 



The similarity of these governing equations and those given for 

 waves over a rough bottom (eqs. A-1 through A-3) should be noted. 

 Further development follows that given in Appendix A.l and the boundary 

 conditions to be satisfied are those previously given but now expressed 

 in terms of the seepage velocity. Hence, integration of the continuity 

 equation over depth gives: 



and with the assumption of long waves, equation (A-69) yields a 

 hydrostatic pressure distribution, 



p = pg(n + z) . (A-72) 



The horizontal momentum equation then becomes 



= JBF=-^£- («*B|ui)U , (A-73) 



where U should be interpreted as the depth averaged discharge velocity. 



a. Linearization and Solution Technique . The linearized version 

 of the governing equations may be taken as 



'^ It ^ 37 fhU) = CA-74) 



18 



