height can be determined and is proportional to the amplitude function since 

 wave frequency is constant. 



31. Equations 27, 31, and 32 along with the dispersion relation 

 describe the combined refraction and diffraction process for linear plane 

 waves subject to the restrictions that the bottom slopes are small, wave 

 reflections are negligible, and any energy losses are very small and can be 

 neglected. These equations are assumed to be valid outside the surf zone. 

 The numerical solution scheme used to solve these equations is presented in 

 the next section. 



Wave transformation outside 



the surf zone: numerical solution 



32. The three governing equations (27, 31, and 32) are solved using 

 numerical methods. Partial derivatives within the equations are approximated 

 using finite difference operators. Finite difference solution methods require 

 the construction of a computational grid system or mesh. Solution accuracy is 

 directly related to resolution within the grid system. Discussions throughout 

 this section refer only to grid systems comprised of constant sized, rectangu- 

 lar cells. RCPWAVE is capable of computing solutions on variably sized, rec- 

 tilinear grid systems. 



33. Figure 6 shows nine rectangular cells which make up a small part of 

 a larger mesh. Each cell has a length equal to Ax in the x-direction and 



Ay in the y-direction. The maximum values of i and j are M and N , 

 respectively. All variables which vary as a function of space are defined at 

 the cell centers (see Ebersole, Cialone, and Prater (1986) for details of the 

 finite difference procedure used) . 



34. Model input includes values of the deepwater height H , direction 



6 , and period T of waves to be simulated. It also includes specification 



o '^ 



of the bottom bathymetry throughout the grid. The wave number, which is 

 related to the wave period and the local water depth through the dispersion 

 relation, is computed at every cell. It is used as an initial guess for the 

 magnitude of the wave phase function gradient. The wave celerity c and the 

 group velocity c are functions of the wave period, wave number, and water 

 depth. Therefore these variables can be calculated at each cell. 



24 



