61 



Field Equation The field equation is mass conservation for irrotational flow, rep- 

 resented by the Laplace equation: 



Boundary Conditions The boundary conditions are the bottom boundary condi- 

 tion (BBC) for a horizontal bed, 



d(f) 

 w = -r- = at z = —h (4.3) 



dz 



the dynamic free surface boundary condition (DFSBC), 



■g- + -{u'^ + v^ + w^) + gr]-B = at z = r] (4.4) 



and the kinematic free surface boundary condition (KFSBC), 



di] drj drj 



-;r- + u^r- + v- u; = at z = ri (4.5) 



ot ox oy 



where 77 is the elevation of the water surface. 



As with the two dimensional formulation, the kinematic free surface boundary 

 condition requires the gradients of the water surface (g^, ^, and |2). Knowledge 

 of these gradients is likely to be limited. In the case of an array of water surface 

 measurements, estimates for the gradients could be computed by interpolating among 

 the measured elevations, but this would result in a low order estimate, and compound 

 the inevitable error in the measurements. In the case of subsurface measurements, 

 there are no data as to the location or the gradients of the water surface. 



To eliminate these gradients, a modified kinematic free surface boundary condition 

 is defined by differentiating the DFSBC following the motion (Longuet-Higgins 1962), 



