and a.Q = for N = 1 . The equation becomes 



N 

 (Mq+NM)zq =-NpgZQS(0) - pgA[-kTQ + S(0)] ^ sin (ka^ cos 9- - cat) 



i=l 



[16c] 



The first term on the right-hand side is just the change in buoyancy which 

 accompanies a vertical displacement of the raft. The remaining ternns 

 correspond to the vertical force obtained by integrating the dynamic pres- 

 sure due to the incident wave over the surface of the spar. This is the 

 Froude-Krylov hypothesis: To a first approximation, the presence of the 

 body does not distort the incident wave or the pressure associated with it. 



The a- equation is 



. N N 



a hAa.1 ^ sin^e. + Iq + ^ j m(z/)(z.')^ dz.' I 

 ^ i=l i=l ^ ^ 



N 

 = - PgaQ A [S(0) - kTp] y^ sin e. sin (kaQ cos 6. - cot) 



i=l 



+ NpS-|yQ- NpgaS, - NpaS2 

 N 



2 \ ' 2 



-pga^S(O) y [o-sin 9.-p sin 9. cos 9. J 



i = l 



N 

 agl ^^ J^ m(z.-')z.' dz.' + MqZq 



+ 



1 = 1 



If N > 2 , 



N N 



N 

 i=l i=l 



2J sin^e^ =^cos2 9. = 1 



19 



