12 



measurement, the local mean water depth is used, rather than the global still water 

 depth). This may be adapted somewhat by using a variation of Wheeler stretching: 



p^cosh{K{h + ^^)) 



Vn = TJ-j T-^^ (1.14) 



pg cosh{kn Zp) 



Nielsen also presented an alternative approach, which uses a semi-empirical trans- 

 fer function derived from Fourier wave theory to compute the water surface elevation 

 from the measured pressure and local curvature of the pressure record. While quite 

 accurate at reproducing the water surface, neither method supplies the kinematics, 

 nor do they satisfy either mass conservation and or the free surface boundary con- 

 ditions. Despite these limitations, the efficacy of these methods demonstrates the 

 potential for the local approach. 



To interpret bottom pressure measurements in the context of the kinematics while 

 preserving the full governing equations, Fenton (1986) presented a method that em- 

 ploys a local polynomial approximation to the complex potential function. In Fenton's 

 method, the potential function and water surface are represented by separate poly- 

 nomials in each small window in time: 



M 

 4>{x - ct,y) + ii^x - ct,y) = ^ ^^^(2 - cty^'^ (1.15) 



j=o ^ "^ 



M 



7j{x,t) = J2bj{x-cty (1.16) 



where z = x + iy, y = at the bed, 77 is the water surface, c is the wave celerity, 

 and the Oj and bj are real. The wave is assumed to propagate at speed, c, without 

 change in form. While steadiness is not a valid assumption in the global sense, it is 

 only applied locally, within a small window in time. 



The method solves for the coefficients, aj and bj, and the wave celerity, c, that 

 satisfy the full nonlinear free surface boundary conditions and fit the measured pres- 

 sure record. This approach provides the complete kinematics and satisfies the full 

 governing equations. Based on a polynomial variation with depth, it works well in 



