^1 = 1/2 ^L P ^ Vx' fl/2 - k) = C^ Fj^^ (1/2 - k). (A15) 



In an analogous manner, the vertical components of the inertial and drag 

 forces can be written as: 



(FJ = q , P (||-) cos = D- cos e • (A16) 



I V M dt max 1 >- ^ 



and (Fp) = l/2Cp pA v^^^ sin 6 | v^^^ sin e| = E^ sin 6 | sin e|, (Al?) 



where D^ = C,^ p c|^)^^^ = - C^ F^^^ (A18) 



,3v, ZTT^H ^^"^ ^— ^ 



max T smh (— j— ) 



E. = 1/2 C„ p A V V = - C^ F^ (A20) 



1 D max ' max' D Dv ^ ■' 



%ax = - -T ■ , ?^d, • f^21) 



s mh (— f — ) 



The total vertical wave force at any position G. in the wave cycle can 

 then be written as: 



F (6.) = F, + (FJ + (FJ = A, cos 26. + B, sin 26. + C, 

 V ^ 1^ L I V D'^v 1 1 1 1 1 



+ D, cos e. + E, sin 0. I sin 6. I . (A22) 

 1 1 1 1 ' 1 ' 



The parameters A^, B^, C]^, Di, and E-, are constant for any given values 

 of C^, cj), k, C^, and Cq, corresponding to the particular wave and pipe- 

 line conditions under consideration. 



The sum of squares of the differences between the observed vertical 

 forces, F (Gj^), and the corresponding calculated forces, F (6.), is 

 written as: 



128 



