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the minimum mathematical requirement, there are other factors that must be taken 

 into account to assure a reasonable physical solution. The local water surface is rep- 

 resented by N nodes in each window. Defining the surface with N = J + 1 points 

 achieves a representation for the water surface of the same order as the potential 

 function being used. Care must be taken with very low order solutions; for example, 

 2 points would define the water surface to first order, but would provide no indication 

 of the curvature of the surface. As a rule of thumb, at least three points should be 

 used, regardless of order. 



Defining the water surface at J -|- 1 points provides 2(J + 1) equations {f^ and 

 f^ , at the J + 1 points). To keep the order of approximation consistent, the Bernoulli 

 equation is also applied at J+ 1 points within each window. Using J+l water surface 

 and pressure record nodes overspecifies the solution at all orders. This arrangement 

 of equations proved effective for all the examples given in this chapter on analytically 

 derived records. 



While that approach was effective on these few examples, it is important that the 

 final solution matches the measured record closely. It's possible that this suggested 

 arrangement of points would allow the optimization routine to be biased towards 

 the more numerous free surface boundary conditions, giving a solution that does not 

 match the pressure record well. The Bernoulli equation could be applied at more 

 points on the pressure record than the number of nodes defining the water surface. 

 Increasing the number of points at which the Bernoulli equation is applied will shift 

 the emphasis of the optimization to the measured record. Additional points on the 

 measured record can also be useful for accommodating measurement noise that may 

 be present in field or laboratory records. 



Order of Solution Similarly to high order steady wave theory, the order chosen 

 for the solution is influenced by a number of factors including the height of the waves, 

 the depth of the water, and the accuracy desired. As with steady wave theory, higher 

 order results in greater accuracy at the expense of computational simplicity, and is 

 necessary for larger waves and for shallow water. The following examples will help to 

 provide guidelines for the order chosen. 



