function given in Noblesse [7] predicts 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 flnding of Scragg has been 

 verified in this study, as may be seen from figures 15, 16 

 and 18. Furthermore, the magnitude of the wave- 

 amplitude peak has been found to increase very rapidly as 

 the Froude number decreases below a certain threshold 

 value. This peak thus app>ears to be a large-flare low- 

 Froude-number feature. For the simple ship bow form 

 considered in this study, the peak in the amplitude of the 

 divergent waves in the Kelvin wake occurs along a line at 

 6° from the track of the ship. It may be found from 

 equation (30a) that the wavelength of the divergent waves 

 along this line varies between 0.7 m and 4.5 m for a ship 

 with speed varying between 10 knots and 25 knots, 

 respectively. 



The other conclusion of this study that may be 

 related to the narow V-wakes observed in some SAR 

 images of ship wakes is the result that the divergent waves 

 of a surface ship 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. A 

 similar result was previously obtained by Sharma [8] who 

 found that the Michell thin-ship approximation for a thin 

 and deep strut-like ship form predicted infinite slopes for 

 the divergent waves at the track of the ship. Inasmuch as 

 infinitely-steep water waves cannot exist in reality, the 

 foregoing resuh 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 "no-divergent-wave" inner region is quite narrow, as 

 may be seen from figure 20 showing the Kelvin cusp line 

 (angle ~ 19°28'), the line along which the amplitude of 

 the divergent waves exhibits a peak (angle ~ 6°), and the 

 three lines along which the steepness of the divergent 

 waves is equal to 1/20, 1/15 and 1/7 (chain line close to 

 the track of the ship). The latter three lines, along which 

 the divergent waves are steep, lie much closer to the track 



of the ship than the line along which the steepness of the 

 divergent waves exhibits a peak. Figures 21 and 22 show 

 considerable variations among the several "steep-divergent- 

 waves" lines that are represented in these figures. 

 Nevertheless, these lines may be seen to correspond to 

 values of Y/(-X) equal (o dbout 10"^ to 2.10"^. For 

 ship speeds varying between 10 knots and 25 knots, the 

 wavelength of the divergent waves corres[>onding to the 

 foregoing values of Y/( - X) may be shown to vary 

 between 0.7 cm and 4.2 cm for Y/(-X) = 10"^, and 

 2.7 cm and 17 cm for Y/(-X) = 2.10-^. These 

 wavelengths are consistent with the wavelengths of the 

 radar pulses used in SAR imaging, so that the foregoing 

 results may provide a partial hydrodynamic explanation 

 for the narrow V-wakes observed in these images. 



The tentative nature of this explanation must 

 however be stressed. Indeed, the foregoing results are 

 based on an analysis in which surface tension and 

 nonlinearities have been neglected. Inasmuch as this linear 

 no-surface-tension analysis predicts extremely short and 

 steep waves in the vicinity of the track of the ship, it is 

 evident that both surface tension and nonlinear effects are 

 liable to be significant. In particular, the short 

 wavelengths found along the steep-divergent-wave lines 

 determined in this study, and the brief description of the 

 effects of surface tension upon the Kelvin wake given in 

 Sharma [8], Lamb (9, pp. 468-470] and Wehausen and 

 Laitone (10, pp. 636-637] indicate that the system of 

 divergent waves in the vicinity of the track of the ship js 

 likely to be profoundly affected by surface tension. 

 Effects of surface tension upon the Kelvin wake will be 

 investigated in a sequel to the present study. 



APPROACH 



This study considers the steady potential flow due 

 to a ship advancing with constant speed in calm water of 

 infinite depth and lateral extent. The far-field Kelvin 

 wake, which is of primary interest here, may be 

 conveniently analyzed in terms of the nondimensional far- 

 field coordinates T = Xg/U^, velocity potential + = <J)g/U-' 

 and velocity vector V^'t' = Vy^<t>/\J, where g is the 

 gravitational acceleration and U is the speed of advance of 

 the ship, X and * represent the dimensional coordinates 

 and velocity potential, respectively, and V^^ and Vj^ are the 

 nondimensional and dimensional differential operators 

 V, = 0/3x,3/3y,a/az) and Vx = {d/dX,d/dY,d/3Z). 

 The mean free surface is taken as the plane z = 0, with 



