Ogilvie 



It would be entirely feasible to consider the general problem 

 in which the body oscillates with the six degrees of freedom of a 

 rigid body. (We could even include more degrees of freedom by 

 allowing deformations of the body.) However, the major concepts 

 should be clear if we allow only two degrees of freedom, a) a 

 lateral translation, comparable to the heave or sway of a ship, and 

 b) a rotation, like the pitch or yaw of a ship. 



In this section, I shall depart from my usual approach and 

 first treat the problem for a perfectly general body, then Introduce 

 the slenderness property at the very end. This introduces a bit of 

 variety, but more important is the fact that some general properties 

 of the physical system can be pointed out, without any confusion 

 over the effects of assuming slenderness of the body. 



We use two coordinate systems: Oxyz is fixed in the body 

 with its origin at the center of gravity, and O'x'y'z' is an inertial 

 system which moves with the mean motion of the body center of 

 gravity. With respect to the stationary fluid at infinity, the mean 

 motion is a translation at speed U in the negative x' direction; 

 thus, in the O'x'y'z' system, there appears to be a flow past the 

 body in the positive x' direction. 



The two reference systems differ because the body oscillates 

 in the z direction, the instantaneous displacement being denoted 

 by Ijlt), and rotates about the y axis, the angular displacement 

 being denoted by ^K(t). In a more general problem, we could let 

 ^i(t)> ^2^^^' ^"-^ ^3^^) denote surge, sway, and heave (displacements 

 along the x, y, and z axes, respectively) and %^{t) , ^gCt) , and 

 |g(t) denote roll, pitch, and yaw (rotations about the x, y, and z 

 axes, respectively). It will be assumed explicitly that |j(t) is a 

 small quantity, so that squares and products can be neglected in 

 comparison with the quantity itself. Furthermore, it will be assumed 

 that Ij (t) varies sinusoidally in time and it will be represented by 

 the real part of a complex function varying as e'<^*. We shall not 

 usually bother to indicate that only the real part of a complex 

 quantity is to be implied. Thus we can write: 



ej(t) = icoe^(t), (2-68) 



The relationship between the two coordinate systems Is as 

 follows (see Figure (2-4)): 



X = x' cos ^5 - (z '-^3) sin ^5 — x' - z'^5 ; 



y = y' 5 



Z = X s 



(2-69) 



In ^5 + (z'-^J cos i.^ x'4. + z' - i, . 



722 



