^»„2o2 cosh (-r— ) 



2T sinh^ (— ) 



The expressions, F]y[v' ^Dv' ^"'^ ^Lv ^^^ constant for a given set of 

 wave and pipeline conditions. 



Linear wave theory was used in the analysis because, as discussed 

 previously, there seems to be no obvious way of accurately describing 

 the lift force phenomenon mathematically using higher order theories. 

 Since the lift forces are much larger than the vertical drag or inertial 

 forces, with the drag forces being almost completely insignificant, 

 there was no point in using higher theories to express the vertical 

 components of the drag and inertial forces. 



For any vertical wave force record in which the corresponding wave 

 and pipeline conditions are known, a least squares analysis can be 

 performed on the data to determine the values of the unknown parameters 

 Cl, (j), k, Cf^^, and Cq in the vertical wave force equation. The least 

 squares analysis yields the values of these five parameters which best 

 fit the force data throughout the entire wave cycle. This is accomplished 

 by determining the values of these parameters which minimizes the sum of 

 squares of the difference between the observed force data and the corre- 

 sponding forces calculated with the mathematical model throughout a 

 complete wave cycle. 



Using the appropriate trigonometric identities. 



cos^ (6 - (J)) = 1/2 + 1/2 cos 2 (6 - (j)) 



= 1/2 + 1/2 (cos 29 cos 2(j) + sin 29 sin 24)), (AlO) 



so the lift force equation can be expressed as: 



Fl = 1/2ClPAu^j^^^[1/2cos 2{{) cos 29 + l/2sin 2(^ sin 29 + 1/2 - k] (All) 



or Fl = A^ cos 29 + B^ sin 29 + Cj^ (A12) 



where A^ = 1/4 C^ P A u^^^^ cos 2(p = 1/2 C^ ^^y cos 2(}) (A13) 



Bi = 1/4 Cl P A u^^/ sin 2(}) = 1/2 Cl Flv sin 2<^ (A14) 



127 



