Note also that surge and pitch are coupled, unless P^ =.0 or unless 

 the centers of gravity and buoyancy coincide. 



The above equations of motion are not unexpected. The restoring 

 forces on the left-hand side consist of hydrostatic and inertial forces plus 

 entrained mass ternms which double the inertial force at each section. 

 This might have been deduced as a consequence of slender-body theory 

 and the fact that the entrained mass of a circular cylinder in an infinite 

 fluid is just equal to the displaced mass. In other words, the hydrodynam- 

 ic forces on the left-hand side of Equations [31] to [33] could have been 

 obtained by neglecting the presence of the free surface. Moreover, the 

 exciting forces on the right-hand side of these equations are those which 

 follow from the "Froude-Krylov" hypothesis that the pressure in the wave 

 system is not affected by the presence of the body. These results are, of 

 course, a consequence of the fact that the body is slender. 



The solutions of Equations [31] to [33] are 



1 - Xq^kh 



/ 1 - AUqI^JtL \ 



L = A sin wt I — I 



\ 1 -XKH / 



34" 



I = 2 A c 



OS cot 



^iQi -Qo(P2 + 4~^i^^^ 



2(P2 + ^y- Pi/K)- Pf 



I 



35] 



r PiQq-^Qi 1 



ijj = 2 A cos wt — 



L 2(P2 + k^ - Pj/K) - Pf J 



36] 



We note that when 



K = 



XH 



37] 



there is resonance in heave, and ■when 



K = 



Pz + kJ-ipf 



[38] 



13 



