17 



the location of the water surface is unknown and part of the solution. When working 

 with data from a water surface probe, the location of the water surface at discrete 

 points in time is known, but the gradients in time and space are not. In order 

 to accommodate this lack of data, the kinematic free surface boundary condition 

 is transformed to eliminate the gradients of the water surface. Following Longuet- 

 Higgins (1962), the kinematic condition is subtracted from the the dynamic condition 

 differentiated following the motion: 



MKFSBC = -^(KFSBC) + -^(DFSBC) (2.6) 



Resulting in a modified kinematic free surface boundary condition: 

 d'^d) ^, du dw. 



^du dw 



+ U -TT + UW-— 



OX ox 



du 2 dw ^ 

 + uw— — h w -^— = at z = T] 

 oz oz 



This new form of the boundary condition does not include the gradients of the water 

 surface. Applying both the modified kinematic free surface boundary condition and 

 the dynamic free surface boundary condition completes the formulation. 



Observational Equations In steady wave theory, periodic boundary conditions 

 are also imposed, forcing the solution to be periodic in both space and time. For 

 irregular waves, however, the periodicity is not known. Rather, the local solution 

 is defined by a local segment of a measured record within a small window in time, 

 together with the field equation and the bottom and free surface boundary conditions. 

 In order to define a solution that fits the measured record, observational equations 

 are identified. These equations will be different depending on which quantity has 

 been measured. In the case of a water surface measurement, they are the free surface 

 boundary conditions, applied at the measured elevations at a number of points in 

 time throughout the window (Sobey 1992). In the case of a pressure measurement, 

 they are the Bernoulli equation, applied at the elevation of the measurement, and 

 also at a number of points in time within the window (see Chapter 3). 



