167 



• Dynamic free surface boundary condition (Eq. 4.4) 



• Modified kinematic free surface boundary condition (Eq. 4.6) 



There is no accommodation for an Eulerian current, although it might easily be 

 included in future work. The non-dimensional potential function, water surface, and 

 frequencies are expanded in a power series in the small parameter, (e = ka), where k 

 is a typical wave number and a is a typical amplitude of the first order component of 

 the water surface. 



6 = e<^(i' + eV''^ + e^<^*^^ + eV^'' + O(e') (A.l) 



77 = £77(1) + e'r;^^) + ^3^(3) ^ ^4^(4) ^ q^^5^ ^^2) 



u. = eul'^ + e^'^ + e^a;p^ + e^'^ + O(e^) (A.3) 



Where the c/>^"\ r/*"' are the potential function and water surface at order n, and u;)" is 

 the frequency of the ith wave at order n. The steepness of all the intersecting waves 

 is taken to be of the same order. Modern Stokes theories have found it necessary 

 to use an expansion parameter based on the physical wave height, rather than the 

 amplitude of the first order component (Fenton 1990). This is not feasible with 

 intersecting waves, as there is no clearly defined wave height. While not ideal, Stokes 

 methods based on this first order expansion parameter can be effective (Skjelbreia 

 and Hendrickson 1962), as long as they are algebraically correct and not pushed to 

 near limit waves or shallow water. 



Care must be taken when computing a given order solution when multiple parts 

 of the solution are all expanded in a perturbation series. In this case, the potential 

 function, water surface, and frequencies are expanded separately, but the potential 

 function and water surface are dependent on the full order frequency. Ideally, a 

 given order component will depend only upon parameters of the same order. In this 

 case, however, the full order frequency must first be computed, and then used as 

 the frequency at all orders. If the order of the frequency used was matched to the 

 order oi r) ov 4> being computed, the resulting solutions would be out of phase, with. 



