Ogilvie 



a description that becomes approximately valid (in a certain sense) 

 as the small parameter approaches zero. But we use the expansion 

 under conditions in -which the small parameter is quite finite, and 

 we just hope that the resulting error is not too big. The size of that 

 error may depend on other parameters of the problem, and we may 

 possibly reduce the error by allowing such other parameters to vary 

 simultaneously with the basic slenderness parameter. 



In the steady-motion problem that we have been considering, 

 the small parameter € could be thought of as the beam/length 

 ratio. There Is a completely different length scale In the problem, 

 namely, U /g = F L, where F Is the Froude number and L Is 

 ship length. This length Is proportional to the wavelength of a wave 

 with propagation speed equal to ship speed. When we assume that 

 F = 0(1) as e -*■ 0, we Imply that the speed Is such as to produce 

 waves which can be measured on a scale appropriate for measuring 

 ship length, and we Imply that this speed Is unrelated to slenderness. 



If we are Interested In problems of very-low-speed ships or 

 very-high-speed ships. In which the generated waves are, respectively, 

 much shorter or much longer than ship length, It Is entirely con- 

 ceivable that our severely truncated asymptotic expansions may be I 

 made even more Inaccurate by the extreme values of Froude number. * 

 We may Increase the practical accuracy by assuming, say, that 

 wavelength approaches zero or Infinity, respectively as e ~*" 0. 

 This is not to Imply that there really Is a connection between speed 

 and slenderness. It Is done only In the hope that wavelength and 

 ship length may be more accurately represented when we use the 

 theory with a finite value of c . 



Formally, the low- speed problem may be treated simply as a 

 special case of Tuck's analysis, as described in Section 3.1. One 

 finds that the appropriate far-field problem contains a rigid-wall free- 

 surface boundary condition (In the first approximation) » Thus, both 

 near- and far-field approximations are without real gravity-wave 

 effects. However, this formal approach Is quite Improper. The diffi- 

 culty Is so serious that we devote a special section later to the low- 

 speed problem. It Is perhaps the most singular of all of our singular 

 perturbation problems. The difficulty, In essence, Is that we have 

 treated all perturbation velocity components as being small compared 

 with U, and this leads to nonsense If we allow U to approach zero. 



At high speed, a slender-body theory can be developed along 

 lines paralleling Tuck's analysis. This has been done by Ogllvle 

 [ 1967] . The resulting near-field and far- field boundary -value prob- 

 lems are quite different from Tuck's however. No numerical 

 results have been obtained yet from this analysis. 



Near-field and far-field regions are defined just as In the 

 previous slender-body problem. In the far-field, the velocity- 

 potential expansion starts with the uniform-stream term, Ux, 



740 



