37 



generated record, and the values computed from the BernouUi equation (Eq. 3.4), 

 with the parameters of the potential function generated by the initial estimate. The 

 second plot is the water surface as predicted by Fourier theory and the elevations of 

 the water surface nodes generated by the initial estimate. Note that the location of 

 the actual surface is given in the plot, but it is not available to help determine the 

 solution. These points were all generated by the method outlined in the previous 

 section, with only the pressure record as a guide. 



The third plot shows the non-dimensional errors in the objective functions: the 

 Bernoulli equation and the free surface boundary conditions (Eqs. 3.2, 3.3, and 3.4). 

 These are the errors that are minimized by the optimization to find the solution. If 

 the solution were perfect, the error in all equations throughout the window would be 

 zero. 



The initial estimates for the dynamic pressure and the water surface have order of 

 magnitude and trend agreement. The errors in the objective functions are on order 

 of .03 and show a systematic pattern, particularly in the Bernoulli equation. This 

 pattern, and the fact that the sharp crest of the wave has not been matched indicate 

 that a better solution can be found. 



The results after the nonlinear optimization are given in Fig. 3.3. At this point the 

 prediction for the dynamic pressure is essentially exact. This is virtually always the 

 case, as the pressure record is available, and the solution is optimized to that record. 

 The predictions for the water surface are also extremely close. This is an impressive 

 achievement, as location of the water surface was found only by minimizing the errors 

 in the free surface boundary conditions. The non-dimensional errors in the Bernoulli 

 equation and free surface boundary conditions are on order of .001, and show no clear 

 trend. The lack of trend indicates that the remaining error is random, and a good 

 solution has been found. The LFI method was able to capture accurately the crest 

 of a steep shallow water wave. 



Figure 3.4 shows the results of the method for the complete shallow water steady 

 wave. The LFI method finds the water surface and the kinematics on the surface 

 essentially exactly. While these results show the complete wave, each of the indicated 

 points is in the center of a separate window, and each window was computed com- 



