singular Perturbation Problems in Ship Hydrodynamics 



where C(x) is the contour around the body in the cross section at x. 

 Then clearly, 



T33^= T33= - p(ico)^ y dx J di n^^^= - (iwf J dx m(x) , 



where L is the domain of the length of the body. Similarly, we 

 obtain: 



T55 = - (ico)^ y dx x^mix) . 

 Collecting these results, we have: 



T,- = - (ico) \ dx m(x) ; 



■33 



2 f U 



.35 - ,.^) \ dxxm(x) - -^ X33 



T53 = (ico)' y^ dx xm(x) + j^ T33 ; 



(2-83) 



T„=-(ico)'y^ dxx^m(x)^(U) T33. 



III. SLENDER SHIP 



Of all the problems discussed in this paper, the slender- ship 

 problemi has led to the most important practical consequences. 

 Therefore it is not unreasonable to devote the longest chapter to the 

 problem. Even so, some aspects will not be covered; perhaps the 

 most important missing example is the case of sinkage and trinn of 

 a ship. 



In the four sections, two steady-nmotion and two unsteady- 

 motion problems are discussed. The first steady-motion problem 

 is the wave- resistance problem, that is, the problem of a ship in 

 steady forward motion on the surface of an infinite ocean. In the 

 second section, the problem treated is essentially the same, but the 

 Froude number is assumed to be related to the slenderness param- 

 eter in such a way that Froude nunnber approaches infinity as slender- 

 ness approaches zero; this rather unnatural relationship is discussed 

 at Sonne length. In the third section, I discuss in some detail the 

 problem of heave and pitch motions of a ship at zero forward speed; 

 the results are not at all surprising, but the method is quite clear 

 in this case, which helps one in approaching the final section. It is 

 concerned with the problem which is the combination of the first and 

 third problems: heave and pitch motions of a ship with forward speed. 



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