63 



a number of points in time within the window (Chapter 6). The method could be 

 adapted to virtually any combination of measured physical quantities by establishing 

 appropriate observational equations. In addition to the aforementioned water surface 

 and pressure measurements, these could include water surface gradient or accelera- 

 tions measured by wave buoys, subsurface velocity measurements, or any combination 

 of these. 



4.3 Formulation of the Solution in each Window 

 4.3.1 Background 



Two dimensional steady wave theory provided the inspiration for the development 

 of the two dimensional LFI method (Chapter 2). Unfortunately, the literature does 

 not provide as solid a basis for nonlinear interpretations of directional seas. There 

 has, however, been some work that can be used as a basis for a directional LFI 

 method. In an early attempt to explore the nonlinear nature of directional seas, 

 Longuet-Higgins (1962) computed the interaction of two intersecting steady waves 

 in deep water through the use of a double perturbation expansion in the steepness 

 of the waves up to third order. The result was a potential function that contained 

 terms representing the higher order interaction between the phases of the intersecting 

 waves: 



<^<" = /3(~-, S, + S2) + MZ, Si - S2) 



(4.7) 

 <^(^) = Mz, 25i + S2) + fe{z, 2,9i - S2) + friz, S^ + 2^2) + /8(^, S^ - 2S2) 



Sn = (k„ -X -U>nt + CKn) 



where n = [1,2], (/f*™' is the mth order potential function, ^i and ^2 are the phase 

 functions of the two waves, k„ is the vector wave number of wave n, x is the horizontal 

 position vector, u;„ and q„ are the angular frequency and initial phase of the nth 

 wave. The functions, fi, were determined algebraically by expanding the free surface 

 boundary conditions in a Taylor series about the mean water level, and solving for 



