singular Perturbation Problems in Ship Hydrodynamics 



followed by a term representing a line of singularities. The near- 

 field expansion also starts with the uniform- stream term, followed 

 by a term which satisfies the Laplace equation in two dimensions. 



The differences appear first in the boundary conditions 

 satisfied by these expansions. The proper way of setting up these 

 conditions is to nondimensionalize everything and then assume that 

 Froude number, F, is related to the slenderness parameter, e, 

 in such a way that F -♦ oo as e ~* 0. It is easier just to let the 

 gravity constant, g, approach zero in this limit. The only inter- 

 esting new case, it turns out, is: g = 0(e). We now assume this to 

 be the case. 



Since g appears only in the dynajnic free- surface boundary 

 condition, the body boundary condition will be the same as in the 

 moderate-speed problem, Eq. (3-10), and in the infinite fluid prob- 

 lem, Eq. (2-63). 



In the far field, the disturbance vanishes as e "* 0. There- 

 fore the free-surface disturbance is o(l). If we let the expansion of 

 the velocity potential, «^(x,y,z), be expressed: 



N 



4>(x,y,z) ~ Ux + ) <^^(x,y,z), for fixed (x,y,z) , 



n=l 

 the dynamic and kinematic free- surface conditions are, approxi- 

 mately: 



on z = 0. (3-14) 



0- g; + U(j),^, 



We do not know the relative orders of magnitude of t, and 4>, 

 a priori , but a study of the possibilities shows that only one combi- 

 nation is possible, namely, that £, and ^^ are the same order of 

 magnitude. Then, in the dynamic condition, the term containing 

 g is higher order than the other term, and it can be neglected in 

 the first approximation, that is, 



^1^ = 0, on z = 0, (3-14) 



which implies also that 



<^, = on z = 0. (3-15) 



Thus, the free surface acts like a pressure- relief surface, with no 



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