Singulav Pevturhation Problems in Ship Hydrodyanamios 



the free surface. Thus, It is no longer necessary or even proper to 

 try to make the previous inner expansion satisfy the free-surface 

 conditions . 



We expect, as usual, that the first term in the expansion of 

 <j) in this new near field will be just Ux, Furthermore, we can 

 expect the next term to be rather trivial, since the second term in 

 the previous near-field expansion did actually satisfy the free- sur- 

 face condition. Using the usual arguments of slender-body theory, 

 we find in fact that the three-term expansion of <j) ^^ ^^is new field 

 is: 



<j)(x,y,z) ~ Ux + a,(x,0) + Uh,(x,0)|y| - ^ za|^^(x,0). 



0(1) 0(€) 0(€^) O(e^) 



The corresponding wave shape is found to be: 



;{x,y) ~ --^a, (x,0) 0(e) 



-■|[uh,,(x,0)|y| +-^a,^(x,0)a,^^^(x,0)] 



2 ^ 



-^[%(^'^) + U'h,'(x,0) +(|-)af^^(x,0)] 



O(e') 

 2 2 



+ . . . . 



It can be shown in straightfo rward f ashion that these results match 

 the far -field expansion as ^(y2 + z^) — oo and they match the pre- 

 vious (thin-body) near-field expansion as z -* - oo . Furthermore, 

 they satisfy the free- surface conditions without the necessity for 

 imposing unacceptable restrictions on the body shape. There is 

 just one aspect that requires special care: The free- surface condi- 

 tions cannot be satisfied on the surface z = in this near field. 

 The reason is that the first term of the t, expansion is 0(e), and 

 differentiation with respect to z is assumed to change orders of 

 magnitude by 1 /e . Thus, suppose that we want to evaluate some 

 function f(z) on z = t, in terms of its value (and values of its 

 derivatives) on z = 0, The usual procedure is to write: 



f(^) = f(0) + ;f'(0) + i ^^f"(0) +... . 



0(f) 0(e).0(f/e) 0(e2).0(f/e2) 



With our set of assumptions, this expansion is useless; we cannot 



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