66 



Sn = {kx,nX + ky_ny - UJnt + On) n = [l, 2, 3] 



This form for the potential function exactly satisfies mass conservation, in the form 

 of the Laplace equation, and the bottom boundary condition for a locally horizontal 

 bottom. It allows for high order representation of each of the steady waves, as well 

 as the interaction of each wave with every other wave. The balance of the solution is 

 specified by the full three dimensional free surface boundary conditions, Eqs. 4.4 and 

 4.6. 



4.3.2 Dynamics 



The dynamics are available through the Bernoulli equation in three dimensions: 



^ + hu' + v' + w') + ^-B = (4.13) 



ot 2 p 



The dynamic pressure is the difference between the total and hydrostatic pressure 

 [Pd =P-Ph)- 



The Bernoulli Constant 



In Chapter 2, an explicit and exact expression for the Bernoulli constant, B, is 

 given for steady two dimensional waves. (Eqs. 2.14 and 2.15). A similar expression 

 can be established for unsteady three dimensional waves. 



The Bernoulli equation is applied at the bed: 



f^ + i(»f + «D + <>-n + ^ = B (4.14) 



where the subscript, b indicates the value at the bed. 



The two dimensional approach is based on analysis first presented by Longuet- 

 Higgins (1975). In that work, the analysis was applied to steady waves. With steady 

 waves, the average over either time or space of the pressure at the bed is the hydro- 

 static pressure. With unsteady waves, the pressure at the bed averaged over space 

 at any given time, or averaged over time at any given location, is not necessarily the 



