singular Perturbation Problems in Ship Hydrodynamics 



if $(x,y) were the potential function for the problem. Replace U 

 by <^o(x»0). the "local stream speed," and evaluate the condition 

 on y = 'Tq(x); then this condition transforms into the condition found 

 for $(x,y) in the low-speed problem. Thus, on a "local" scale 

 (in which a typical length is U /g) , the free-surface condition is just 

 a very ordinary condition; one cannot see that the stream velocity 

 changes slightly along the free surface, because the change occurs 

 on a scale in which a typical measurement would be a body dimension; 

 the change is very gradual. Also, the level of the undisturbed free 

 surface appears to change gradually, as given by (5-7); this change 

 also cannot be detected on the "local" scale. 



It is now clear that the two length scales are quite distinct. 

 We cannot separate the fluid-filled region into distinct parts in each 

 of which only one length scale needs to be considered. Rather, the 

 gradual changes which appear on the bod^'--size scale appear to 

 modify the short-length wave motion in the manner of a modulation. 



In trying to find a potential function which satisfies (5-23), 

 I made a nonconformal mapping: x' = x, y' = y - r|Q(x) . Then ^ 

 satisfies a complicated partial differential equation in terms of x 

 and y', but the terms in the equation can be arranged according to 

 their dependence on U, and it is found that the leading-order terms 

 are simply the terms in the Laplacian, that is, 



all other terms are higher order. In this new coordinate system, 

 the free- surface condition, (5-23), is transformed too, but again 

 the leading- order terms are just the same after the transformation 

 (but expressed as functions of x' and y')« Furthermore, the 

 boundary condition is then to be applied on y' s 0. Let us now 

 drop the primes on the new variables, for convenience. Then the 

 problem is as follows: Find a velocity potential, $(x,y), which 

 satisfies the Laplace equation in two dimensions and the free- 

 surface condition: 



4>y(x,y) +^^IU,0)%J^,0) =p'(x). 



where 



P(x) = Tio(x)(^Ox^x,0). 



In addition, the potential must satisfy a body boundary condition; 

 this has not been carefully formulated yet, and, in any case, the 

 only solution that has been produced so far is one that satisfies 

 the free- surface condition but not a body condition. There may be 



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