114 



center of the array. The theoretical values used Ohyama's fourth order intersecting 

 wave theory. The actual values would not be available for comparison when analyzing 

 field records. 



It is clear that the first order computation results in substantial errors in the 

 free surface boundary conditions of order 10~^ (Fig. 5.28), as well as significantly 

 underestimating the horizontal velocities (Fig. 5.29). As with the steady deep water 

 wave previously discussed, the first order solution is quite reasonable. This is because 

 of the local nature of the method, and the fact that it is not a linear solution. The full 

 nonlinear boundary conditions are imposed, and the frequencies and wave numbers 

 are free to vary, and are not bound by a dispersion relationship. 



At second order, the free surface boundary condition errors are better, of order 

 10~^, and the velocities at the surface match the analytical solution visually perfectly. 

 At third and fourth order, the errors in the free surface boundary conditions continue 

 to decline, and the water surface velocities continue to match the theoretical solution 

 well. 



As with the single wave method, choice of order is dictated by the desired ac- 

 curacy of the solution, and by the ease of convergence of the optimization. For the 

 above examples, four water surface nodes (M = 4) distributed equidistantly in time 

 in each window, at each gauge, at orders 1 through 3. These values providing an over- 

 specification and sufl&cient points to define the shape of the water surface throughout 

 the window. Five points (M = 5) were used at fourth order, as four points provide 

 only 24 equations, and there are 28 parameters to be found. Five points provides 30 

 equations for a slight over specification. At fifth order, there are 38 unknowns, and 

 seven points in time (M = 7) would have to be used in each window. Sampling this 

 many points in a single window would require very closely spaced data or a wider 

 window. Another solution would be to use an array with additional measurement 

 locations. An array of four gauges, for example, would provide eight equations per 

 point in time, and would allow a fifth order solution to be computed from five points 

 (M = 5) per window. It would be advantageous to use as many gauges as possible in 

 shallow water, where higher order solutions are necessary. 



Unfortunately, the theoretical solution used for this analysis, while the best avail- 



