Ogilvie 



and the oscillation problem should generally be linearized in terms of different 

 small parameters, say /? and 7, and we may neglect terms in fi'^ compared with 

 terms in /?, or terms in 7^ compared with 7, but we cannot make any arbitrary 

 assumptions about the relative size of /3 and 7. 



We assume that the pressure is given by a function which can be expressed 

 as a Taylor series about a point on s^. Let us describe the motion of the ship 

 by the two vectors: 



3 



k=l 



3 

 k = l 



^(t) specifies the linear displacement of the ship and ^(t) the angular displace- 

 ment (which is assumed to be small enough that a vector representation is ap- 

 proximately valid). The displacement from equilibrium of a point x on the hull 

 is then given by ^ + ^ x x. The pressure at a point on S can be expressed in 

 terms of the pressure (and its derivatives) at the corresponding point of s^: 



pIs = pIs + [^+^><x]-vp|s +... . 



O "" o 



In addition to expressing the pressure appropriately, we need to be able to 

 write down a set of direction cosines for calculating the effects of the pressure 

 on an element of the hull. For calculating the moments, we shall need further- 

 more a set of appropriate lever arms. 



Let us look first just at the force components, resolved along the steady 

 axes: 



Xj = j p(n-i.) dS 



We could just as well resolve forces along the unsteady (body) axes: 



x! = [ p(n -i:) dS. 



The two sets of force components are related by: 



X = X' + ^x X' , 



where X = {y.^,\^,y.^^, and so the determination of either set is sufficient. We 

 note that the factors (n . i!) have the values which we associate with the \indis- 

 turbed ship, whereas the factors (n • ip vary with time. Therefore we find it 

 somewhat easier to calculate the components x' directly. 



To the expression for Xj we add and subtract a quantity, as follows: 



70 



