168 



for example, ^*^' having a different frequency to (i^'^\ It could be argued that this 

 approach would be accurate to second order, but the components would be shifted 

 out of phase as time progressed. It is more accurate to compute the results using the 

 full order frequencies at all orders of cp and 77. 



The first order solution for N intersecting waves is the familiar superposition of 

 linear waves: 



,n> sr^ ai cosh kih -\- z) . ,, , , . . ,x 



<^(i) =gy -^ ) ' sin {Kx + kyy -ut^a) A.4 



^-^ u, cosh kh 

 i=\ ^ 



N 



77'^' = y. <^j cos (kx + kyXj — ut + a) (A. 5) 



1=1 



ujf^ = y/gh tanh hhi (A.6) 



The higher order solutions are computed by applying the Laplace equation, the 

 bottom boundary condition, and the free surface boundary conditions, expanded in 

 Taylor series about the mean water level. The algebra and the resulting solutions 

 are long and extremely tedious, are presented in Ohyama et al. (1995a), and they 

 will not be repeated here. It should be noted that the frequency was only solved to 

 third order, as the fourth order frequency would require a solution to the fifth order 

 potential function, which was not presented. 



A. 3 Analytical Verification 



Ohyama et al. provided a number of comparisons with other wave theories. When 

 the method is applied for a single wave component, it represents a steady wave, and as 

 such should be expected to provide the same solution as conventional Stokes theory. 

 The authors presented a comparison with the Fenton (1985a) solution. The Fenton 

 solution is based on an expansion parameter based on the wave height, rather than 

 the first order amplitude: e = kH/2. By solving for i in terms of the e, The two 

 solutions were found to coincide exactly. 



