Ogilvie 



The nature of the free surface problem depends primarily on the length of 

 waves (of frequency w) compared with the ship dimensions. If these waves have 

 length comparable with ship length, that is, co'^B/2z - 0(e), then Vossers shows 

 that the free surface condition reduces to the rigid wall (low frequency) condi- 

 tion. If the waves are short compared with ship beam, the high frequency de- 

 generate boundary condition applies. Only if the waves are comparable in length 

 with ship beam does one obtain an interesting problem, for then the free surface 

 condition becomes (in dimensional form): 



3^cp Bcp 



and Laplace's Equation reduces from 



on X 3 , (23) 



3xj2 Bx^' Hx^ 



to , (24) 



3^cp B^cp 

 T + — ^ = . 



In words, the problem reduces to a set of two-dimensional problems; for each 

 cross section, we must find, in two dimensions only, a solution of Laplace's 

 Equation satisfying (23), the usual free surface condition, and (as Vossers shows) 

 the usual boundary condition on the body. 



This problem should be quite familiar, for it corresponds exactly to "strip 

 theory," There is no effect of forward speed and no interaction between cross 

 sections. Vossers' formulation shows clearly then that strip theory is a natural 

 consequence of assuming that the disturbance waves and ship beam are com- 

 parable in size, and accordingly it should be valid in problems of ship rolling in 

 short beam waves, for example, but not for problems of pitching and heaving in 

 waves comparable with ship length. 



It should be noted specifically that solutions which satisfy (23) and (24) are 

 functions of x^, since the body boundary condition depends on Xj. Moreover we 

 can add to such solutions any other function of x^ which we desire, without 

 violating (23), (24), or the body condition. This arbitrary additive function may 

 be interpreted as expressing the interaction between sections. But it is unknown. 

 In this formulation of the problem, we can do only as in strip theory, namely, 

 assume that there is no interaction. 



If we are interested in pitch and heave problems (and most of this paper is 

 concerned with just these problems), then we must consider wavelengths com- 

 parable with ship length, and it is apparent that we do not obtain a satisfactory 

 formulation by the above procedure. Therefore Vossers proposed a different 

 tack, viz., that we write down the solution in a general way by using Green's 

 theorem and then use the slenderness approximation to simplify the resulting 

 integral equation. In other words, we effectively establish an integral equation 

 for the solution of the problem involving a general body (not a slender body); we 



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