»itt + g *iz + f i = ° (23) 



gh i " *it + 8 i = ° (24) 



where, for the first order perturbation solution, f- = 0, g. = and, for the nth 



order, f and e are known functions of d> . of the (n-l)th or the lower order 

 n „ °n 1 



solutions. 



Then the Lagrangian for each f. is 



f i=Ui v *ii* v *ii dzdx -ilJ 



= JJ 2 V *li* V *li dzdx -2iJ *li**litt dx 

 D (z=0)OD 



tf>, * f . dx + h * cf). . dx 

 J Y li l J t Y li 



( z =o)nD 



"/ (*li-I*2i)^2in d5 



S J 



5J. = (25) 



i 



where the solutions for 1=1, 2,...n - 1 should be used to determine the solution 



when i = n. Equation (24) gives the wave height h for each i. 



If we specify a time-dependent, free surface pressure distribution p on the 



proiection S of S to z = instead of h in S , we may use 

 F J so s t s 



so 



f l = S l = p/p ° n S j 



in Equations (23) and (24) for the first order, where p is the water density. 



11 



