The second governing equation used in the refraction scheme is 

 conservation of energy. Neglecting dissipation of energy due to friction, 

 percolation, and turbulence, this equation can be expressed as 



V ME Cg) =0 (13) 



where E is the average energy per unit surface area and Cq the group velocity 

 of the wave train. Performing the operation indicated and replacing Cg by 

 its components (CgSin e) and (Cgcos e) results in the following: 



k (^ ^G ^^'^ '^'hy^^^ "^ «^ = ° (14) 



Assuming linear theory, 



£ , pgH^ (15a) 



where p is the mass density of water, g the^gravitational constant, and H 

 the wave height. Dividing the equation by ^, finite-differencing and 

 weighting the forward-differenced term as before, and solving for the wave 

 height, results in the following: 



<,] ^ I (C.cose). ■ I i^)i^\cose)._ . ( 1-2^) (Mucosa) . _.,^ 



. (T)(H\cose).^^^.,^ . £ [(Hasina). ,^^. - (Hasina) ._^^. ] 



This equation is also solved by iterative techniques and the H. . and H 

 are averaged at the conclusion of each iteration. 



Cg is determined by the linear wave theory relationship 



i,J 



CG = 2^i^iTHra^ (15; 



where h is the water depth, k the wave number, and C the wave celerity. Wave 

 height boundary conditions are input along the same boundaries as the wave 

 angles using linear theory shoaling and refraction coefficients. The e's 

 have been previously determined. In both equations (11) and (15) for a 

 variable grid system, the points (i + 1, j) and (i - 1, j) need to be 

 determined (i.e., because the y coordinates are not fixed, adjacent values 

 with the same subscripts can be farther or closer to shore, therefore 

 interpolation must be used). The actual values are found by searching the 

 (i + 1) and (i - 1) cross-shore lines, finding the adjacent values in the 

 positive and negative y-direction, and interpolating to determine the value. 



15 



