Understanding and Prediction of Ship Motions 



In the formula for f(Xj) we have two integrals over the length of the hull, 

 and we should distinguish between their meanings. The first integral, involving 

 the Struve and Bessel functions, represents a free surface effect. It can also be 

 looked upon as expressing an interaction between sections — an interaction caused 

 by the presence of the free surface. The second integral also represents an in- 

 teraction, but it would exist even in the absence of the free surface. We could 

 combine the latter with ^jd ^^^ ^°°^ ^^ ^® ^^^ ^^ ^^® slender body approxima- 

 tion for the three-dimensional body in the presence of a rigid wall, with the first 

 integral supplying a correction to account for the free surface effect on the 

 three-dimensional body. 



We note that the interaction, due to either term of f(Xj), involves the ship 

 geometry in a very simple way. In fact, F(xj) depends only on the ship beam at 

 section Xj, and thus f(xj) depends only on the waterplane shape. However, (p^^ 

 depends on the detailed shape of the cross section. (Fortunately, the finding of 

 02D is '^ot too difficult, since it is not really the solution of a free- surface prob- 

 lem.) Thus, the solution does depend on the hull geometry in a detailed manner, 

 but this dependence is shunted off to the mathematically easier field of problems 

 with fixed, rigid boundaries. 



We should now refer back to the earlier statements which resulted from 

 various assumptions about orders of magnitude. We have assumed here that 

 wave length is comparable with ship length. Under these conditions, Vossers 

 showed that the three-dimensional boundary value problem would reduce to a 

 set of two-dimensional problems in the cross sections, with the free surface 

 condition replaced by a rigid wall condition. This is exactly what Newman ob- 

 tains. However, we now have an explicit formula for the interaction term, i(x^). 



Perhaps it should be emphasized that (26) is valid only very near to the 

 ship hull, at distances which are of order of magnitude eL. However, this is 

 just where we need to evaluate the potential in order to find the force on the 

 ship, and we do not need to be concerned about complications far away. The 

 expression for ^ given by (26) presumably does not even satisfy the three- 

 dimensional Laplace's Equation, except in an approximate sense very near to 

 the body. Far away from the body, the potential would have quite a different 

 form from that given in (26). 



Without finding explicit formulas for the forces, we can immediately reach 

 some conclusions of importance. For the transverse oscillations, that is, yaw 

 and sway, the flux, F(xj), vanishes and so f(Xj) vanishes also. In other words, 

 the theory predicts no interaction between cross sections in these modes. If 

 this holds for the potential, it must be true for the forces too, and so the lowest- 

 order slender body theory for ship motions reduces to strip theory for the 

 transverse modes. However, to the same degree of approximation, there will 

 be non-negligible interactions between cross sections in the heave, pitch, and 

 roll modes. It is interesting to note that many years ago the same conclusions 

 were reached by Grim (1957) on the basis of physical arguments. 



In order to calculate the force and moment on the ship, we must add the 

 potential for the incident waves to the function represented in (26), and from 

 this sum we find the pressure, which we integrate over the submerged part of 



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