range of values of a for which the corresponding points 

 of stationary phase are within the range of flare, the 

 contribution of these stationary-phase points is 

 dominant, of order 1/F. In this instance, the amplitude 

 of the waves in the far-field Kelvin wake is of order 

 (UVg)/(- X/L)'^^, as F - and X/L - -», for the 

 range of values of o corresponding to the region of flare 

 and of order F(U^/g)/{- X/L)''^^ for values of a outside 

 this range. If the region of flare is of small extent, and 

 the waterline-tangent angle ip does not vary widely within 

 that region, the corresponding range of values of the 

 angle a where the wave amplitude is an order of 

 magnitude larger than elsewhere is also small, and thus 

 appears as a peak for sufficiently small values of the 

 Froude number. This peak is particularly pronounced for 

 a hull with a small region of flare in the vicinity of an 

 inflexion point of the waterline. 



The low-Froude-number asymptotic analysis of the 

 Neumann-Kelvin theory presented in this study thus 

 shows that the characteristics of the far-field Kelvin wake 

 strongly depend on the shape of the ship hull, notably 

 the presence of flare and the shape of the waterline at 

 the bow and stern. This analysis also predicts that the 

 nondimensional wave-resistance coefficient R/gU L , 

 where U and L are the speed and the length of the ship 

 and e is the density of water, which is given by the 

 Havelock integral 

 nR/gU'L- = F" /" °° |K(t)|2(l -i- 1^)" '^' dt, (76) 



is O(F^) for a ship form with a region of flare, ©(F*) for 

 a ship form that is wall sided everywhere and has either 

 a bow or a stern (or both) that is neither cusped nor 

 round, and O(F^) for a wall-sided ship form with both 

 bow and stern that are either cusped or round. 



The low-Froude-number asymptotic approximation 

 (50) also shows that we have 

 K(t) = 0(l/t') as t -> °°. (77) 



This result is actually valid for any value of the Froude 

 number; indeed, the asymptotic approximation (50) is 

 valid not only in the low-Froude-number limit F -* but 

 more generally in the limit vp = (I -l-t^)'^^/F -► °°, that 

 is as F -* or/and as t -* oo_ as may be seen from tne 

 exponential functions exp (v^p^z) and E^ = 

 exp [-iv^p(x±ty)] in equation (26). 



Equations (15), (18a), (21b) and (77) then yield 



(-4)'^^s^(|,o) '%. t^'^^ '^' l/o'''^ as o - 



(78) 



for the steepness of the short divergent waves in the 

 vicinity of the track of the ship. By using equations (4) 

 and (5), which yield o ~ r)/(-4) as o -♦ 0, in equation 

 (78) we may obtain 



s + (4,n) ~ 1/f)'^^ as n -' 0. (79) 



Equation (79) thus shows that the lines along which the 

 steepness of the short divergent waves in the far-field 

 Kelvin wake takes given large values, say s^ = 1/7 and 

 1/15, are parallel to the ship track, as was found in 

 Figure 21 of Barnell and Noblesse (1986) by using the 

 Michell thin-ship approximation for a simple ship form. 

 The Neumann-Kelvin theory therefore predicts that the 

 far-field Kelvin wake contains three distinct regions: (i) a 

 narrow constant-width inner region bordering the track 

 of the ship where no divergent gravity waves can exist, 

 (ii) an outer region where the usual transverse and 

 divergent waves are present, and (iii) an intermediate 

 region at the boundary between the inner and outer 

 regions where short steep divergent waves can be found. 

 In reality, surface tension must evidently be taken into 

 account in the vicinity of the track of the ship. 



ACKNOWLEDGMENTS 



This study was supported by the Independent 

 Research program and the General Hydrodynamics 

 Research program al the David W. Taylor Naval Ship 

 Research and Development Center. I wish to thank Mr. 

 Alexander Barnell for his help in drawing Figures 1 and 

 3 and Dr. Arthur Reed for his useful suggestions. 



REFERENCES 



1. Barnell, A. and F. Noblesse (1986) "Far-Field 

 Features of the Kelvin Wake," 16th Symposium on 

 Naval Hydrodynamics, University of California, 

 Berkeley. 



2. Fu. Lee-Lueng and Benjamm Holt (1982) 

 "Seasat Views Oceans and Sea Ice with Synthetic 

 Aperture Radar," JPL Publication 81-120. 



3. McDonough, Robert N., Barry E. Raff and 

 Joyce L. Kerr (1985), "Image Formation from 

 Spaceborne Synthetic Aperture Radar Signals," Johns 

 Hopkins APL Technical Digest, Vol. 6, No. 4, pp. 

 300-312. 



4. Noblesse, F. (1983) "A Slender-Ship Theory of 

 Wave Resistance," Journal of Ship Research, Vol. 27, 

 No. 1, pp. 13-33. 



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