ANALYTICAL APPROXIMATION FOR STEADY SHIP WAVES 

 AT LOW FROUDE NUMBERS 



by 



Francis Noblesse 



David Taylor Naval Ship R&D Center 



Bethesda, MD 20084 



ABSTRACT 



A simple analytical relationship between a ship- 

 hull form and its steady far-field Kelvin wake is obtained 

 by considering the low-Froude-number limit of the 

 Neumann-Kelvin theory. 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° 1/2 for a hull form which has a small region 

 of flare and is wall sided elsewhere, if the Froude 

 number is sufficiently small. An 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 obtained. For instance, this relationship 

 predicts the occurrence of a sharp peak in wave 

 amplitude 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 either 

 30° or 74°. This theoretical 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 

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

 ship form with both bow and stern that are either cusped 

 or round. 



1. INTRODUCTION 



It has been observed, see for instance Fu and Holt 

 (1982) and McDonough et. al. (1985), that SAR 

 (Synthetic Aperture Radar) images of ship wakes 

 sometimes reveal bright returns along rays at angles from 

 the track of the ship smaller than the Kelvin-cusp angle 

 of 19°l/2. A plausible explanation for these surprising 

 observations was proposed by Scragg (1983) who 

 considered a simple ship bow form with large flare for 

 which he found that the zeroth-order slender-ship 

 approximation to the far-field wave-amplitude function 

 given in Noblesse (1983) predicted a sharp peak in the 

 value of the amplitude of the divergent waves at an angle 

 from the track of the ship equal to approximately half 

 the bow entrance angle. This numerical result of Scragg 

 was confirmed by Barnell and Noblesse (1986) who also 

 found that the peak in the amplitude of the divergent 

 waves becomes sharper as the value of the Froude 

 number decreases, and thus suggested that the occurrence 

 of a sharp peak in the amplitude of the far-field Kelvin 

 waves was a large-flare low-Froude-number feature. 



The numerical studies of Scragg and of Barnell 

 and Noblesse are based on two simple approximations to 

 the far-field wave-amplitude function, namely the Michell 

 thin-ship approximation for which no peak was found 

 and the zeroth-order slender-ship approximation which 

 exhibited a peak as was already noted, so that it Is not 

 clear from these studies whether a more realistic 

 mathematical model for the far-field wave-amplitude 

 function, such as that provided by the Neumann-Kelvin 

 theory, would also predict the occurrence of peaks in the 



