60 



water surface (77) 



MWL (z = 0) 



^?^^^^^^^^^^5^^^^^^^5^^^^^^^^ seabed 

 Figure 4.1: Coordinate system used for 3-d method 



4.1 Three Dimensional Seas 



The development of the two dimensional LFI method was closely anchored to 

 the very complete understanding of two dimensional steady waves. In contrast, the 

 understanding of three dimensional wave fields is not nearly as complete. Much of 

 the literature on three dimensional seas attempts to describe the motion through the 

 use of a directional energy spectrum. Far less attention has been directed to the 

 determination of the detailed kinematics of directional seas. 



4.2 Problem Formulation 



The governing equations for three dimensional gravity waves are a straightforward 

 extension of those in two dimensions to include the third dimension. The flow is taken 

 to be irrotational and incompressible, and thus the kinematics can be represented by 

 a potential function, (b{x,y,z,t), in a Cartesian coordinate system (Fig. 4.1), where: 



dcj) 



(4.1) 



dx dy dz 



u and V are the horizontal velocities in the x and y directions, and w is the vertical 

 velocity. 



