et al. (1974) and Ebersole and Dalrymple (1980). However, to improve stabil- 

 ity, some of the solution techniques employed to solve these differential 

 equations were different from those used by previous investigators. The 

 models used by Noda et al. (1974) and Ebersole and Dalrymple (1980) typically 

 had stability problems when the angle of incidence of the waves was large or 

 the bathymetry was complex. The model described in this report was developed 

 also to perform calculations on a uniformly variable grid. This allowed the 

 wave calculations to be performed on the same grid as wave-induced current and 

 sediment transport calculations. 



13. In order to determine the angle of wave propagation 9 , use is made 



->■ 

 of the irrotationality of the wave number vector k : 



v x fc* = o (7) 



When vertical propagation is ignored 



9k 3k 

 x y 



3y 3x 



(8) 



where 



k = k cos 9 (9) 



k = k sin 9 (10) 



y 



k = |k | (ID 



and 9 is the angle of wave propagation. Equation 8 becomes 



Q 89 . . 39 . 1 3k ^ . . 1 3k n _. 



cos 9 -r— + sin 9 -r cos 9 — -r— + sin 9 — -r— = (12) 



3x 3y k 3y k 3x 



14. In order to conserve wave frequency when a current is present 



a) = to + k o tf (13) 



o 



where to is a constant equal to 2ir/T , T is the period of wave when 

 o ^ o o 



there is no current, to is the radian frequency as it appears to a stationary 

 observer, and V is the current velocity vector. Substituting for to , using 



12 



