Equations (4), (20b) and (27) show that the asymptotic 

 expansion (23) for the free-surface elevation is valid in the 

 vicinity of the track of the ship if 

 ,3/2k(,) _► as , ^ 00. (37) 



Let us assume that we have 



|K(t)| ~ 1/t*' as t - 00. (38) 



Both conditions (37) and (36) are then satisfied if 



3/2 < n < 7/2. (39) 



Condition (37) means that the amplitudes of the 

 divergent waves in the Kelvin wake vanish as a -* 0, that 

 is as the track of the ship is approached, whereas 

 condition (36) means that the waves become infinitely 

 steep; this is theoretically possible because the wavelengths 

 of the divergent waves vanish as o -* 0, as is indicated in 

 equation (31a) and figure 7. However, infinitely-steep 

 water waves cannot exist in reality; indeed, there exists a 

 theoretical upper bound for the steepness of water waves 

 in deep water which is approximately equal to 1/7. 

 Condition (39) therefore suggests that no divergent waves 

 can exist within a certain domain in the vicinity of the 

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

 distinct regions: (i) an inner region adjacent to the track 

 of the ship where only transverse waves can exist as was 

 just noted, (ii) an outer region where both transverse and 

 divergent waves are present, and (iii) a region at the 

 boundary between the inner and outer regions where short 

 steep divergent waves, as well as transverse waves, can be 

 found. It must be kept in mind, however, that these 

 conclusions regarding the Kelvin wake are based on 

 condition (39), which was obtained on the basis of an 

 analysis in which surface-tension and nonlinearities are 

 ignored. Inasmuch as this linear no-surface-tension 

 analysis predicts short steep waves, both surface-tension 

 and nonlinear effects are liable to be significant, and these 

 effects should therefore be included in a more realistic 

 analysis. In particular, it is evident from Lamb [9, 

 pp. 468-470] and Wehausen and Laitone [10, pp. 636-637] 

 that the system of divergent waves in the immediate 

 vicinity of the track of the ship may be profoundly 

 affected by surface tension. 



y = ±(1-x)(t«n ^+ z tvi y) 



Fig. 8 — Waterlines and Framelines of the Simple Ship 

 Bow Form Considered for Numerical Applications 



A SIMPLE TEST CASE: 

 THE FAR-FIELD WAVE-AMPLITUDE FUNCTION 



The foregoing theoretical results are investigated 



numerically for the simple semi-infinite ship form studied 



previously by Scragg [5]. This ship form consists of a bow 



region, with length L, followed by a parallel body, with 



invariant framelines, extending to infinity downstream. All 



framelines, both in the bow region and downstream from 



it, are trapezoidal in shape with constant draft D. The 



waterlines are rectilinear. More precisely, the hull form is 



defined by the equations 



y = ±(tan|3-l-z tany)(l-x) 



for < X < 1 and > z > - d, (40a) 



y = ±(tan/3-l-z tany) for x < and > z > -d, (40b) 



where x,y,z and d are nondimensional in terms of the 



length of the bow region, that is we haveT=X/L and 



d = D/L. Equations (40a, b) require that the condition 



tan/3 > d tany (41) 



be satisfied. Equation (40a) shows that the entrance angle 



at the bow (x = 1, z = 0) is equal to 2p, and it may be 



seen from equations (40a, b) that y represents the fiare 



angle for x < 0. The four waterlines corresponding to 



z = 0, -d/3, -2d/3, -d and the five framelines 



corresponding to x = 1, 0.75, 0.5, 0.25, are depicted in 



figure 8 for d = 0.1, (i = 12° and y = 45°. The 



notation 



