Ogilvie 



respectively, are approximately equal if co is small enough. If we 

 solve this equation on the understanding that co is very large, we 

 must keep all quantities in the exact solution. But that solution will 

 reduce numerically to the approximate solution if we evaluate it with 

 a small value of co. 



We could say that the solution obtained on the assumption 

 of high frequency becomes inconsistent if we apply it to problems at 

 low frequency, but, if the appropriate small parameter is small 

 enough, an inconsistent approximation is no worse numerically than 

 a consistent approximation. 



Once more I would warn against trying to make absolute 

 judgments of what is "small" and what is "not small," I avoid 

 careful definitions of my small parameters largely for this reason; 

 if the definition is not precise, one can never be tempted to put 

 numbers into the definition! In the problem ahead, we cannot 

 possibly judge analytically how "slender" the ship must be or how 

 high" the frequency must be for the results to have some validity. 



In all of the discussions of ship motions, I use the same 

 notation as in the study of oscillatory motion in an infinite fluid. 

 See Section 2.32. 



The ship in its mean position will be defined by the equation: 



So(x,y,z) = - z + d(x,y) = 0, (3-18) 



where d(x,y) = 0(e); the instantaneous hull position is defined by the 

 following equation: 



S(x,y,z,t) = - z + d(x,y) + i^{,t) - ^i^{t) = 0. (3-19) 



The ship is heading toward negative x (although it does not matter 

 in the zero- speed case). Upward heave and bow-up pitch are con- 

 sidered positive. 



We assume that all motions have very small amplitude. 

 Symbolically, we write that: 



^j(t) = 0(€6) as either € or 6 approaches zero. 



where e is the usual slenderness parameter, and 6 is a "motion- 

 amplitude" parameter. This convenient assumption allows us to 



748 



