10 



This approach can accommodate both steady and unsteady wave forms by including 

 both a set of free waves and a corresponding set of bound modes. Periodicity over 

 time and space is assumed. The record is not likely to be periodic, so the duration 

 of the record is assigned as the fundamental period, {'2tt/u), and the fundamental 

 wavelength, {2Tr/k) is found as part of the solution. The method uses a nonlinear 

 optimization to find the coefficients, An.m and Bn.m, the fundamental wavenumber, 

 kj and the Bernoulli constant that produce a solution that globally satisfies the full 

 free surface boundary conditions. 



In order to accommodate the unsteadiness of the wave profile, the evolution of 

 the wave in space must be known. This is accomplished by applying the free surface 

 boundary conditions at many locations in time and space. Since the elevation of the 

 water surface usually is measured only at a single spatial location, it is predicted at 

 the other locations as part of the solution. The potential function (Eq. 1.8) exactly 

 satisfies the bottom boundary condition and mass conservation. In order to find the 

 solution that best fits the free surface boundary conditions, the method seeks a set 

 of coefficients that minimizes the sum of squares error in both of the free surface 

 boundary conditions over a grid of nodes in time and space. 



Baldock and Swan identified a difficulty in this basic method, as the squared error 

 was equally considered at all of the nodes, most of which were at locations in which 

 the water surface elevation was also unknown. This resulted in solutions in which the 

 error in the boundary conditions was greatest at the location of the measurement. 

 The difficulty was mitigated by a weighting function, multiplying the dynamic free 

 surface boundary condition errors at the measured location by a factor of 50 in the 

 sum of squares calculation, to force the solution to match well at that point. 



Comparisons of their results with laboratory data were quite good, but the method 

 has a number of limitations. These include a huge matrix of unknown coefficients that 

 must be found simultaneously {2{MN + 1) unknowns, with typical values of M and A*" 

 of 18, for a total of 650 unknowns), and the method must solve for the water surface far 

 from the measurement location, removing the focus from the actual measured data. 

 The method makes no assumptions about the steadiness of the wave field; however, it 

 extends the horizontal bottom assumption and introduces unnecessary complication 



