where, in two dimensions 



and and i// are defined in terms of the velocity components u and w (see Figure 1) as: 



9()) _ dj) 

 ^ ~- dK " 9z 



(4) 



9(t) _ 91 

 ^ -"Ti ~+9i^ 



Boundary Conditions 



Two types of boundary conditions are required on the upper and lower surfaces (for the 

 present study, it will be assumed that the depth is uniform). The kinematic boundary 

 condition applies to both boundaries, and simply requires that the components of flow at 

 these boundaries be in accordance with the geometry and motion (if any) of the boundaries. 

 This condition can be stated as follows: 

 Bottom Boundary Condition (BBC) 



w = 0, z = -h (5) 



Kinematic Free Surface Boundary Condition (KFSBC) 



15- + u|il = w, z = n(x,t) (6) 



OL. OX 



Dynamic Free Surface Boundary Condition (DFSBC) 



The remaining free surface boundary condition, the so-called dynamic free surface 

 boundary condition (DFSBC), requires that the pressure immediately below the free surface 

 be uniform and equal to the atmospheric pressure, p . 



n+E^+A-fu^+wM ---11-= constant = Q ' , z = n (x, t) (7) 

 pg 2g ^ J g 9t ^ ' 



In the above formulation, it is tacitly assumed that surface tension effects are negligible. It is 

 customary to incorporate the atmospheric pressure term into the constant, Q', to yield a 

 new constant, Q 



