The numerical solution of these equations gives results comparable to 

 experiments for varying bottom topography where H/d < 0.4. 



Using the work of Mei and Le Mehaute (1966), Madsen and Mei (1969) 

 developed characteristic equations which could be solved numerically. 

 Defining 



dn dw 



- = w and ^ = a 



along the coinciding characteristics x = constant, along the two distinct 

 characteristics 



'3d 

 of - «>] = ± 



. W 9x 2 . 



and 



•H« ; £r) 



/Cd ud\ dn 5 j2 3d dw / n 1 , 3 2 d\ a 



V^- + — Jd3 + T2 d ^Td? + \ l ~ 2 d ^r)- 



(99) 



dC 2 du 

 6 dg 



aCd 2 da [ Cd _ a 3x C ' 

 = \ — D + u / u 



12 de \2 



3d 



where 



and 



d 3x /5 , „ 2 \ TnC dC (d aC 2 \ 1 rinn _ 



+ -6" ■ VI d " ° C ) a + L~ " 3" + \2 + T" ) U J W (1 ° 0) 



D = iM n .M + (M) 3 + I d 2 lil + 2d M aid (101) 



U d 9x n 3x \3x/ 3 Q 3x 3 3x 3x 2 



C = ±C (x) along g = ^J = | (t + / ^~j = constant (102) 



The Korteweg-deVries equations provide a solution for wave propagation 

 in one direction only, i.e., for an unrefracted wave. The solutions gen- 

 erated could be used to provide shoaling coefficients to obtain refracted 

 wave heights. 



Alternative methods of obtaining solutions for refracted waves in two 

 dimensions are to use the linear long-wave equations with additional terms 

 added to account for nonlinear effects, or to use solutions based on the 

 Boussinesq equations. Butler and Durham (1976) suggest a solution using 

 equations similar to those in a tidal hydraulic model. The momentum 

 equations for the tidal model are 



54 



