98 



the free surface boundary conditions (Eqs. 5.3 and 5.4) at each of the measured nodes. 

 These are the data that would be analyzed in a practical situation. Figures 5.7, 5.9, 

 5.11, 5.13 are a comparison between the predicted and actual values for the water 

 surface and the velocities at the surface at the center of the array. The actual values 

 were computed using Fourier wave theory. The actual values would not be available 

 for comparison when analyzing field records. 



The first order computation results in errors in the free surface boundary condi- 

 tions of only order 10~^ (Fig. 5.6) as well as very accurately predicting the velocities 

 at the water surface at the center of the array (Fig. 5.7). It is a surprisingly accu- 

 rate first order solution. This is because of the local nature of the method. When 

 a single first order solution is used to capture the entire wave, the error is larger, of 

 order 10""^. It also should be noted that this first order solution is not the same as a 

 linear wave theory solution, even locally. The full nonlinear boundary conditions are 

 preserved, and the frequency and wave number are free to vary, and are not bound 

 by a dispersion relationship. For steady waves in deep water, linear wave theory is 

 fairly accurate. Linear theory is not, however, a local solution, and is not directly 

 applicable to irregular waves. 



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

 10~^, and the velocities at the surface match the Fourier 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 Fourier solution 

 well. In deep water, for waves of this height, second order is more than adequate to 

 capture the surface kinematics of this wave. Higher order solutions are likely to be 

 necessary to capture the irregularity of field records, even in deep water. 



Choice of order is dictated by the desired accuracy of the solution, and by the ease 

 of convergence to the solution. As there are more free parameters at higher order, 

 a higher order solution will always have smaller errors in the free surface boundary 

 conditions. In the case of this example, the solution converged at all orders very 

 quickly, so there is little penalty in using third or fourth order. With an irregular 

 record, in contrast, convergence can be more difficult, and it is occasionally necessary 

 to resort to lower order. 



