Understanding and Prediction of Ship Motions 



are correct, and, if they are correct, then there is no need to use Grim's more 

 complicated formulas. The questions raised here can not yet be answered. 



The first comprehensive attack on ship motions problems by slender body 

 theory was by Vossers (1962a), and I shall follow his approach in essence. 

 (However, other methods are possible. See, for example, Ursell (1962), Tuck 

 (1964).) After formulating the problem exactly, we must introduce the slender- 

 ness approximation and this proves to be a difficult task. Vossers' s analysis 

 and results are very complicated and moreover they are somewhat suspect. 

 Newman (1964) has had more success in obtaining approximations, at least for 

 the case of no forward speed, and his first numerical results are very encour- 

 aging. However, the calculations are still at a very rudimentary stage, and it is 

 too early to predict the quality of the outcome. Whether all of this effort will 

 lead to valid and useful formulas is not yet known. I leave the two following 

 authors (and their discussers) the opportunity to speculate on the future of 

 slender body theory for predicting ship motions. 



In formulating the slender body problem for ships, Vossers assumes that 

 the ratio (ship beam)/ (ship length) is a very small quantity, which we call e. 

 The purpose of his investigation is to find solutions which become more and 

 more accurate as e becomes smaller and smaller. Vossers expands various 

 q\iantities as perturbation series in powers* of e , substitutes these into the 

 various mathematical conditions of the problem, and non-dimensionalizes all 

 quantities and equations. In the last process, a number of special non-dimen- 

 sional ratios arise, and the nature of problem and solution depends on the rela- 

 tive sizes of these quantities. The important non-dimensional quantities are, 

 besides e, - 



coL/2Y, 2L reduced frequency, 



2VVgL, a forward speed parameter, proportional to (wavelength of waves 

 travelling at speed v)/(ship length), 



«2L/2g, proportional to (ship length)/ (wavelength of waves with frequency w), 



aj2B/2g, proportional to (ship beam)/ (wavelength of waves with frequency oj), 



c^v/g, a parameter for describing the pattern of radiated waves (usually 

 called "r" in the American literature), 



where 



L = ship length, 



B = ship beam, 



CO = circular frequency of exciting or motion -generated waves, and 



V = forward speed. 



^This is really not correct, and it shows the danger of loose assumptions. One 

 should, as it turns out, use double series containing factors e^cloge)". It is 

 also possible to avoid this trap altogether by assuming only that the potential 

 can be expanded: 4> = S <^^ , with 0^^j = 0(4>^). By such an approach, one must 

 determine in turn the actual order of magnitude of each term. 



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