singular Perturbation Problems in Ship Hydrodynamics 



There are thus two difficulties: 1) The above solution is not 

 unique (a common difficulty in free- streamline problems); 2) It has 

 unacceptable behavior at infinity. 



These difficulties were resolved by Rispin [ 1966] and Wu 

 [ 1967] , who recognized that the solution of Green's problem is part 

 of a near -field (inner) expansion of the complete solution. An inner 

 expansion does not necessarily satisfy the obvious conditions at 

 infinity; it must only match some outer expansion in a proper way. 

 Rispin and Wu produced the appropriate outer expansions and showed 

 that matching does occur. The effects of gravity appear first in the 

 far field, which is hardly surprising, for two reasons: 1) Far away, 

 one expects to find gravity waves as the only disturbance. 2) The 

 divergence of the free-surface shape in Green's solution is so weak 

 that one might expect the smallest amount of gravity effect to bring 

 the free surface into the region where we expect to find it: thus, 

 the small effect of gravity eventually would have a large consequence, 

 but only far away from the planing surface. 



Rispin defines the small parameter: 



pH gi/u^= l/F^ 



where F is the usual Froude number. In the near field, the natural 

 coordinates are used, which means effectively that i is considered 

 to be 0(1). Smallness of p is achieved by allowing g "*" or 

 U-* oo. Rispin treats his small parameter properly by nondimen- 

 sionalizing everything, so that he then does not have to specify 

 whether U -* oo or g -*- . Rather than change all variables now, 

 I shall treat g as a small parameter, as in Section 3.2; the results 

 are the same as Rispin's, of course. 



In the far field, typical lengths are assumed to be 0(l/P) in 

 magnitude, or 0(l/g), in my loose notation. We could define new 

 coordinates, say, 



z = Pz; x = Px; y = Py, 



and consider that z = 0(1) as g -^ in the far field, while z = 0(1) 

 as g -* in the near field. Rather than do this, we shall just keep 

 in mind that such orders of magnitude are to be assumed. Also, we 

 note that d/dz = 0(i) In the near field and d/dz = 0(P) In the far 

 field. 



This problem Is reversed from the most common kind of 

 stretched- coordinate problem: The Inner problem Is solved by 

 natural coordinates, and the outer coordinates are compressed. 

 Note, however, that there Is no distortion of coordinates between 

 near- and far-fields. There Is just a change of scale. 



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