67 



hydrostatic pressure. If, however, an average is taken over both time and horizontal 

 space, the result must be the hydrostatic pressure. 



Pb = P9(^-Zb) (4.15) 



with the double overline indicating an average over time and horizontal space. When 

 the Bernoulli equation at the bed is averaged over time and horizontal space, the 

 average time gradient of the potential function is zero, resulting in a simple expression 

 for the Bernoulli constant: 



B = lj7,+7,)+g^ (4.16) 



For the potential function given by Eq. 4.10, and z — a,t the mean water level 

 (77), B becomes 





giving a complete expression for B as a function of the parameters of the potential 

 function. 



4.4 A Local Two-Intersecting- Wave Theory 



While a large number of intersecting waves could capture a sea state of virtually 

 any complexity, it would be difficult to distinguish between the effects of each in- 

 dividual wave. It is important to remember that the familiar directional spectrum 

 description of a sea state is an integral description. While there are many different 

 frequency and direction modes represented in the spectrum, they do not necessarily 

 all play a significant rule in the kinematics the entire time. In fact, when observing 

 irregular seas, it is often the case that at any given time, there appears to be a single 

 dominant wave of a particular frequency and direction. Over time, there is a series of 

 such dominant waves, each of a different frequency and direction. The integral effect 

 of this process is a broad directional spectrum, but if time is separated into individual 



