FAR-FIELD FEATURES OF THE KELVIN WAKE 



A. Barnell and F. Noblesse 



David W. Taylor Naval Ship Research and Develnpmeni Center 

 Belhesda. Mapland 20084 



ABSTRACT 



The classical Kelvin wake trailing a ship advancing 

 at constant speed in calm water is studied. In particular, 

 asymptotic expressions for determining the far-field wake, 

 ai large distances behind the ship, are investigated 

 numerically. This analysis, in which surface tension and 

 nonlinearities are ignored, indicates that the divergent 

 waves of a surface ship generally are infinitely steep at the 

 track of the ship, even though their amplitude vanishes 

 there; this result is theoretically possible because the 

 wavelength of the divergent waves vanishes at the track of 

 the ship. Inasmuch as infinitely-steep water waves cannot 

 exist in reality, the linear no-surface-tension analysis 

 presented in this study suggests that no divergent waves 

 can exist within a certain region in the vicinity of the 

 track of the ship, and that the Kelvin wake contains three 

 distinct regions: (i) an inner region where only transverse 

 waves can exist, (ii) an outer region where both 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, as well as 

 transverse waves, can be found. Numerical results for a 

 simple bow form show that the inner region is quite 

 narrow, and that the wavelength of the divergent waves at 

 the boundary of the inner region is of the order of 1 to 

 20 cm, depending on the speed of the ship. These results 

 appear to be consistent with the narrow V-wakes observed 

 in some SAR images of ship wakes. The analysis 

 presented in this study may thus provide a partial 

 explanation for these observations. However, the small 

 wavelengths mentioned above indicate that the divergent 

 waves in the vicinity of the track of the ship are likely to 

 be significantly altered by surface tension, which should 

 therefore be included in a more realistic analysis. 



INTRODUCTION 

 It has been observed, e.g. [1,2,3], that images of 

 ship wakes taken by the SEASAT SAR (Synthetic 

 Aperture Radar) and other airborne SAR systems 

 sometimes reveal long, narrow wakes extending back 

 several miles behind ships. According to Swanson [3], 

 "there appears a narrow, dark band along the centerline 

 of the ship wake, widening slightly with distance away 



from the ship Often accompanving these dark regions, at 

 a slightly larger angle, are bright lines, which show up 

 especially well on \-band SAR images". The apparent 

 included angle of this narrow wake is much smaller that 

 that corresponding In the Kelvin cusp line. 



Several theoretical explanations of the features 

 observed in SAR images of ship wakes have been 

 proposed [3,4,5,6). The proposed explanations include 

 interactions between the cross currents created by a ship in 

 its wake and surface graviiv waves n,6) and the occurence 

 of a sharp peak in the amplitude of the divergent waves 

 in the Kelvin wake for a ship form having a large flare 

 angle [5). However, there currently appears to be no 

 convincing electrodynamic or hydrodynamic explanation of 

 the observations. It thus seems necessary to investigate in 

 detail each one of the proposed possible explanations in 

 order to determine whether one, or possibly a combination 

 of several, of them does in fact provide a satisfactory 

 theoretical interpretation of the SAR images of ship 

 wakes. 



The classical Kelvin wake trailing a ship advancing 

 at constant speed in calm water is examined in this study. 

 Viscosity, surface tension and nonlinearities are ignored in 

 this preliminary analysis, and irrotational flow is assumed. 

 The analysis and related numerical results for a simple 

 ship bow form have led to several conclusions and 

 recommendations for further studies which are presented 

 at the end of the paper. Only two conclusions that may 

 be related to some of the features observed in SAR 

 images of ship wakes are discussed here. 



It was previously found by Scragg [5) that, for a 

 ship bow form with a large fiare angle, the zeroth-order 

 slender-ship approximation to the far-field wave-amplitude 



