singular Perturbation Problems in Ship Hydrodynamios 



This analysis has led to the possibility of predicting waves 

 with amplitude which is 0(e), that is, waves comparable in ampli- 

 tude to ship beam and draft. Such a possibility makes the analysis 

 worth further investigation, but it is also the cause of the major 

 difficulty, viz. , the necessity of solving a nonlinear problem in the 

 near field. 



When the above analysis was offered for publication in 1967, 

 one of the referees called attention to the fact that the conclusions 

 seemed to be quite at variance with those of Rispin [ 1966] and Wu 

 [ 1967] . Simple observation shows that, at very great distance, the 

 dominant fluid motion should be gravity- related free- surface waves, 

 whereas my high-Froude-number analysis predicts no true wave 

 motion in the far field. Actually, all aspects of the problem are in 

 complete harmony if we consider a "far-far field" in which distance 

 from the ship is 0(e"' ), that is, miuch greater than ship length. 

 The two free-surface conditions then fall into the usual linearized 

 format, and we would expect to find progressive waves in such a 

 region. 



This is quite reasonable. At very high Froude number, one 

 expects typical waves to be very long --in this case, considerably 

 longer than the ship. The appropriate distortion of coordinates is 

 an isotropic compression in scale far, far away, in contrast to our 

 usual anisotropic stretching of coordinates in the cross-plane near 

 the body. In their two-dimensional planing problems, Rispin [ 1966] 

 and Wu [ 1967] performed just such a distortion. Their problem is 

 discussed at some length later, when we come to two-dimensional 

 problems. 



The present problemi is an interesting case in which an 

 inconsistent expansion might be useful in the far field. Suppose that 

 we arbitrarily replace the free-surface condition, (3-15), by the 

 \isual moderate-speed condition, 



u\^+g<^y=0, on z = 0, 



If Froude number is indeed very high, then this condition is quite 

 equivalent to (3-15), But the potential function which satisfies this 

 condition does not represent just the simple line of dipoles implied 

 by (3-16). There will be all of the well-known extra terms involving 

 the free surface. If such an inconsistent far-field solution can be 

 matched to the near-field solution, then the waveless far-field solu- 

 tion obtained previously can be avoided. Perhaps this is worth 

 further study. 



3,3, Oscillatory Motion at Zero Speed 



A. systematic study of the zero-speed ship-motions problem 

 by means of the method of matched asymptotic expansions does not 



745 



