\ 





(104) 



where u and Z are reference velocity and length scales for the velocity 

 gradients as previously discussed (see eq. 86) 

 employed are given in Table 4. 



Expressions for v^ commonly 



b. Conservation Form . For the combined motion of waves and currents, 

 it is far more convenient and exact to use the conservation form of the mass 

 and motion equations. By averaging over depth and over time, in that order, 

 we define the following averaged quantities: 



udz 



-d 



vdz 



u(x,y,z,t) 



v(x,y,z,t) 



(105a) 



(105b) 



-d 



so that p and q are the discharges per unit width at any x, y, and t. Now, 

 the transport components p and q include mass transport induced by finite- 

 amplitude waves; e.g.. 



f^ 



P = 



udz + 



udz 



(106) 



-d 



and the second term on the right-hand side is not generally zero (Vreugdenhil, 

 1980). Note that generally u 7^ p/h and v ^ q/h. Some authors choose to re- 

 define U = p/h and V = q/h as "...mean horizontal velocities" (e.g., Ebersole 

 and Dalrymple, 1979, 1980). This serves no useful purpose, particularly for 

 numerical solution methods where the following physical conservation laws are 

 preferred (Vreugdenhil, 1980): 



Mass 



ill + I2. + 9^ = 



8t 9x 9y 



(107) 



Motion 



x-direction: 



l£ + 8p^/h I 9(pq/h) ^ 

 8t 3x 8y 



gh(-:^ + 



3x 9x 



'-)- 



+ -^(-x 

 p sx 



, as 



- . 1^ XX , 

 ^Bx^ - p ^-9^ ^ 



9S -, 9hT, 

 xy , 1 , Lxx 



9y '^ " P ^ 



9hT 



Lx^. 



9x 



9y 



(108) 



117 



