The first of the governing equations used is the conservation of waves 

 equation 



31 ^^H 



t = (8) 



where VH js the_^ horizontal differential operator equal to i(3/3x) + jCa/ay) 

 in which i and j are the unit vectors in the x and y directions, respec- 

 tively, and X is the longshore direction, with positive to the right when 

 facing the water, y the offshore direction, with positive seaward, and z the 

 vertical coordinate, with positive defined as upwards. For the steady-state 

 case, equation (8) yields 



3. (k ) - L (k ) = (9) 



ax y 3y X 



where kx and ky are the wave number projections in the respective directions. 

 Defining as the angle k makes with the y-axis positive in the counter- 

 clockwise direction, the equation can be written in final form as 



^ (k cos e) = I- (k sin e) (10) 



where e = a + it (in radians). Noda (1972) and others have developed 

 numerical solutions to expanded forms of equation (10). In the present 

 study, equation (10) was initially central -differenced in the x-direction and 

 forward-differenced in the y-direction with Snell's law used to specify the 

 boundary conditions on the offshore boundary and one of the sides (i.e., the 

 side of the wave angle approach). However, a numerical problem arose. The 

 argument of the arcsine exceeded +1.0 for large Ay/AX. To overcome this 

 problem, a dissipative interface was used on the forward-difference term 

 (after Abbott, 1979). The final finite-differenced form of equation (10) is 



n+1 . -1 1 



). . = sin J i — 



1 ,J 



T(k sin e)._^^.^^ + (l-2T)(k sin e).^.,^ 



+ T(k sin 8).,^^.,^ - £ ((k cose).:^^.- (k cose)._^^ . 



where t has been taken as 0.25. The past e^^j and the present 0? j wave 

 angles are numerically averaged to give the 0i,j. Newton's method is used 

 to compute the wave number via the linear wave' theory dispersion relation. 

 In addition, numerical smoothing is used at the conclusion of the wave field 

 calculation. This approximates in an ad hoc manner diffractive effects 

 (lateral transfer of wave energy along the wave) which exist in nature but 

 have been omitted due to use of the equation for refraction (equation 8). 

 The smoothing routine is 



'i,j - 4 ^--l.j ^ 2 ^,j ^ 4 "i+l,j 



e.. . = ; e. , . + i e. . + , e.,, , (12) 



14 



