singular Pertuvhation Problems in Ship Hydrodynamics 



is caused by a combination of a vortex and a dipole. If the strengths 

 of the two apparent singularities were comparable, the vortex 

 behavior would dominate the dipole behavior far away, and the 

 induced velocity would diminish in proportion to 1 /z , rather than 

 1 /z , which was the case for the dipole. Thus, Fb| would be 0(e ) 

 near the free surface. In the absence of a sharp trailing edge which 

 can cause the formation of a vortex flow, the corresponding Fb| 

 would be O(e^) . This matter remains to be resolved. 



There are other interesting aspects to this problem. One 

 relates to the interpretation of the small parameter, e = t/b. In 

 defining such a dimensionless perturbation parameter, one nor- 

 mally assumes that the smallness of e can be realized physically 

 either by letting t be extremely small or by letting b be very 

 large. In the present problem, this choice is not really available 

 to us. The reason is that there is another length scale in the prob- 

 lem, namely, 1 //c = U /g , and this length scale appears generally 

 in combination with the dimension b. It has been assumed that 

 Kh - 0(1) as e — ^ 0. Therefore, if we want to consider the problem 

 of a body which is more and more deeply submerged, (b -* oo) , 

 then we must also restrict our attention to higher and higher speeds. 

 This is awkward. 



Finally, one more important aspect must be mentioned. The 

 relation between wave number, K, and forward speed, U, namely, 

 H = g/U , is based on linearized free-surface theory. In general, 

 if one seeks to find the nature of nonlinear waves which can propa- 

 gate without change of form, the wave length of those waves is not 

 related to their speed in this simple fashion. To be sure, the 

 relationship is approximately correct if the waves are not terribly 

 big in amplitude, and so one might expect that the wave length or 

 the wavenumber can be expressed as an asymptotic series in e 



/C ~ /Cq + A:, + /Cg + o . . , 



with Kq = g/U . This can indeed be done, but it turns out to be 

 much more convenient to assume that K is precisely given and then 

 to find the value of forward speed that corresponds to that wave 

 number. Thus, one expands the forward speed, U, into an asymptotic 

 expansion: 



U ~ UQ + U| + U2 + . . . . 



This procedure is discussed by Wehausen and Laitone [i960] , and ^ 

 Salvesen uses it in his hydrofoil problem. I was able to omit mention 

 of it in writing Eqs. (5-5) and (5-6) because it turns out that U| = 0, 

 and so the effect of this speed shift (or period shift) does not enter 

 the problem until the third approximation is being sought. However, 

 this is a classic example of the kind of expansion described in 



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