46 



Figures 3.7 through 3.12 show the crest of the same shallow water wave shown in 

 Fig. 3.4, computed to orders J = 1 through 6. At first order, the non-dimensional 

 errors in the Boundary conditions and the Bernoulli equation are of order 10~^. These 

 are small errors, but the sharp curvature at the crest has not been captured. As 

 the order is increased, the magnitude of the errors increases. The increase in the 

 magnitude of the errors is due to the increase in the number of equations included. 

 There are three additional equations included in the optimization for each increase 

 in order (the two FSBCs at each additional water surface node, and one additional 

 pressure record node), but there is only one more parameter in the solution vector. 

 In order to best satisfy this system, the optimization finds a solution that results in 

 slightly more error in each equation. The magnitude of the errors does increase, but 

 the solution slowly converges to very precisely match the sharp crest at sixth order. 

 An asymmetric distribution of points was used in this window to accommodate the 

 symmetry about the crest. 



Figures 3.13 through 3.16 show the crest of the same deep water wave as Fig. 3.6, 

 computed to orders 1 through 4. Even at first order. Fig. 3.13, the solution is very 

 good. Deep water waves generally do not require a very high order solution, linear 

 wave theory often being reasonably adequate in these conditions. It's important to 

 keep in mind that, although this solution is first order, it is still nonlinear, having 

 found a minimum in the errors of the full nonlinear governing equations, and the 

 frequency and wave number are free to vary, not being bound by the linear dispersion 

 relationship. The local nature of the solution would allow it to change with time, 

 accommodating an irregular profile at low order better than an equally low order 

 global solution. 



In this case, first order provides an acceptable solution; however, the water surface 

 is more accurately matched as the order is increased, and the solution converged well 

 at higher order, so there is little penalty in using up to fourth order. For an irregular 

 record, higher order is more likely to be successful in matching the irregularity in the 

 record. As the examples show, deep water waves can be well represented at low order, 

 while higher order in necessary in shallow water. 



