and realizing that the boundary conditions (eqs . A-5 and A-6) may be 

 used to simplify equation (A-9) , the continuity equation becomes 



1^-^ gl {uch+n)} = . (A-ii) 



From the continuity equation in its basic form (eq. A-1), 

 introduction of the typical length scales, the wavelength, L, in the 

 horizontal direction and the water depth, h, in the vertical direction^ 

 shows that the order of magnitude of the vertical velocity component 

 is given by 



W = (^ U) . (A-1 2) 



Thus, for long waves, h/L << 1, vje have that W « U. This observation 

 suggests that the vertical fluid accelerations, DW/Dt, in equation 

 (A-3) may be neglected. For long waves equation (A-3) therefore 

 simplifies to a statement of hydrostatic pressure distribution, 



P = PgCn-z) , CA-13) 



where the boundary condition (eq. A-4) has been invoked. 



Introducing p as given by equation (A-13) and making use of 

 equation (A-1) the horizontal momentum equation may be written: 



DU .iU , 8U1 , 3UW. = .g in . llZ£ . (A-14) 



Dt ^8t 3x Sz -* ^ 8x Sz *■ ■' 



This equation may be integrated over the depth to yield: 



U dz + -^r^ I U^ dz -U {-^ + U -^ -W } 



St , 8x , n St n Sx n 



-h ^ -h 



By virtue of the boundary conditions (eqs. A-5 and A-6), the 

 bracketed terms vanish, and by introducing the concept of the momentum 

 coefficient. 



07 



