Singular Perturbation Problems in Ship Hydrodynamias 



As usual, we assume that there is a far-field expansion: 

 N 

 ^(x,y,z) ~ ) ^Jx,y,z), where ^n*l " ^^'''n^ ^^ € -* , 



for fixed (x,y,z). (3-4) 



and a near-field expansion: 

 N 



«Mx,y,z) ~ ) $^(x,y,z), where $„^| = o(^„) as e — 0, 



"'° for fixed (x,y/c,z/c), (3-5) 



These expansions are substituted into all of the exact conditions, 

 from which we obtain two sequences of problems which must be 

 solved sinnultaneously. 



In the far field, the first term in the expansion for (|> must 

 be just the incident uniform- streajn potential, Ux, since the body 

 vanishes as € -*" and the asymptotic representation <^ ~<|>o= Ux 

 satisfies the free- surface conditions (trivially). The second term 

 represents a line of singularities on the x axis. One rccilly ought 

 to allow the most general possible kind of singularities on this line, 

 but it is no surprise to find that just sources are sufficient at first, 

 and so we consider the special case of a line of sources on the free 

 surface. One can show that higher-order singularities could not be 

 matched to the near-field solution. Alternatively, one can construct 

 a far-field solution using Green's theorem and show that it really 

 represents just a line distribution of sources. See, for example, 

 Maruo [ 1967] . 



One can use the classical Havelock source potential to ex- 

 press the desired potential for a line of sources, but Tuck's pro- 

 cedure is more convenient in the slender-body problem: Apply a 

 double -Fourier transform operation to the Laplace equation, re- 

 ducing it to an ordinary differential equation with z as independent 

 variable: 



- (k2 + i2)<|>**(k.-«;z) +4>tz*(k,je;z) = o, 



where k and i are the transform variables, and the asterisks 

 denote the transforms. Assume for the moment that the line of 

 sources is located at z = Zq< 0. The above differential equation 

 can be solved generally, with a different solution above and below 

 z = Zq. The solution in the upper region is forced to satisfy the 

 linearized free-surface condition, the solution in the lower region 

 must Veuiish at great depths, and the two must have the discontinuity 

 at z = Zq appropriate to the source singularities. Finally, one may 



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