amplitude of the far-field Kelvin waves. Furthermore, the 

 numerical results obtained by Scragg and by Barnell and 

 Noblesse correspond to a particular ship form and thus 

 provide little physical insight into the origin of the 

 predicted peak in the amplitude of the far-field Kelvin 

 waves, specifically the manner in which such a peak is 

 related to the shape of the ship hull. 



A complementary analytical study of the low- 

 Froude-number limit of the Neumann-Kelvin theory for 

 an arbitrary ship form is thus presented here. This 

 asymptotic analysis of the Neumann-Kelvin theory 

 provides a simple analytical relationship between a ship- 

 hull form and its steady far-field Kelvin wake. In 

 particular, this relationship predicts the occurrence of a 

 sharp peak in the amplitude of the waves in the far-field 

 Kelvin wake at an angle, o, from the ship track that is 

 smaller than the Kelvin-cusp angle of 19°l/2 for a hull 

 form which has a small region of flare and is wall sided 

 elsewhere, if the value of the Froude number is 

 sufficiently small. A simple explicit relationship between 

 the angle, (p, between the ship track and the tangent to 

 the ship mean waterline in the region of flare and the 

 corresponding wave-peak angle a in the Kelvin wake is 

 given and depicted in Figure 3b. For instance, this figure 

 predicts the occurrence of a sharp peak in the amplitude 

 of the divergent or transverse waves at an angle a in the 

 Kelvin wake equal to 14° for a hull having a small region 

 of flare within which the waterline-tangent angle <p is 

 approximately equal to 30° or 74°, respectively. This 

 analytical result may explain the bright returns that have 

 sometimes been observed in SAR images of ship wakes 

 at angles smaller than the Kelvin-cusp angle. 



The low-Froude-number asymptotic analysis of the 

 Neumann-Kelvin theory presented in this study also 

 predicts that the wave-resistance coefficient is O(F^), 

 where F is the Froude number, for a ship form with a 

 region of flare, O(F^) for a ship form that is wall sided 

 everywhere and has either a bow or a stern (or both) that 

 are neither cusped nor round, and 0(F*) for a wall-sided 

 ship form with both bow and stern that are either cusped 

 or round. 



More precisely, the low-Froude-number asymptotic 

 approximation (50) to the far-field wave-amplitude 

 function K(t) shows that the main contributions to the 

 function K(t) stem from several particular points on the 



mean waterline. These are the bow and the stern, on one 

 hand, and (usually but not always) one (or several) 

 point(s) of stationary phase. Indeed, the number of these 

 points of stationary phase, and their position on the 

 waterline, depend on the value of t and on the shape of 

 the waterline. The first two terms in the low-Froude- 

 number asymptotic expansions for the contributions Kg g 

 of the bow and stern and the contributions K^ of the 

 points of stationary phase in equation (50) are given by 

 equations (51)-(55) and (59)-(62), respectively. The 

 second-order terms in these asymptotic expansions are 

 defined by complex expressions. However, the first-order 

 terms provide simple approximations defined explicitly in 

 terms of the geometrical characteristics of the hull and 

 the velocity components in the tangential directions Tand 

 n" X Tto the hull (see Figure 2). In particular, these low- 

 Froude-number asymptotic expansions show that the 

 contributions Kg and Kg of the bow and stern are 0(1) 

 except if the bow or stern is cusped or round, in which 

 case we have Kg g = O(F^). The contribution of a given 

 point of stationary phase is 0(1/F), and thus is 

 dominant, if the hull has flare at this point; otherwise, 

 that is if the hull is wall sided at the point of stationary 

 phase, its contribution is 0(F). The low-Froude-number 

 approximation (50) also shows that we have K(t) = 

 0(l/t') as t -• oo. In fact, this result is vaB^jkpf any 

 value of the Froude number. 



The latter result implies that the lines along which 

 the steepness of the short divergent waves in the far-field 

 Kelvin wake takes given large values, say 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 and possibly also viscosity 

 must evidently be taken into account in the vicinity of 

 the track of the ship. 



