105 



In this case, the first order computation results in substantial errors in the free 

 surface boundary conditions of order 10"'^ (Fig- 5.14) and does not predict the ve- 

 locities at the water surface at the center of the array as well as it did in deep water 

 (Fig. 5.15), noticeably under-predicting the horizontal velocities. The local nature 

 of the method allows a first order solution to get fairly close, but steady waves in 

 shallow water are highly nonlinear, and a first order solution simply can not capture 

 the sharp crest. 



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

 but still of order lO"'^, and still under predicting the horizontal velocities at the 

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

 continue to decline, but the water surface velocities do not match the Fourier solution 

 visually perfectly until fifth order (Fig. 5.23). In shallow water, it may be necessary 

 to use up to fifth order to capture accurately the surface kinematics of a fairly large 

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

 and there is little penalty in using fourth or fifth order. With an irregular record, 

 convergence will be more difficult, and it may sometimes be necessary to resort to 

 lower order. 



As previously discussed, the use of higher order solutions requires that more points 

 be sampled in each window. This is not likely to be a limitation with the single wave 

 form with an array of measurements, as it is likely to be necessary to overspecify the 

 system in order to define the curvature within the window. 



5.4.2 Records of Two Intersecting Waves 



In order to develop the method using two intersecting waves, theoretical water 

 surface records were generated by Ohyama's fourth order intersecting wave theory 

 (Ohyama, Jeng, and Hsu 1995a). Ohyama's method is a Stokes-style solution for 

 irrotational intersecting waves that is accurate to fourth order, and applicable in deep 

 water (see Appendix A). As the resulting water surface from intersecting waves can 

 be quite complex, a desired wave height can not be specified. Rather, the amplitude 

 of the first order component of each of the intersecting free waves is specified. The 



