DIVERQENT 



TRANSVERSE 



P,«P 



Fig. 3a - The Waterline-Tangent Angle (p, the Wave- 

 Propagation Angle p and the Wavelength A. as Functions 

 of the Angle From the Ship Track a; the Subscripts T 

 and D Refer to the Transverse and Divergent Waves, 

 Respectively. 



Fig. 3b - The Angle From the Ship Track a, the Wave- 

 Propagation Angle p and the Wavelength X as Functions 

 of the Waterline-Tangent Angle cp. 



TRANSVERSE 



DIVERGENT 



DIVERGENT 



TRANSVERSE 



Fig. 3c - The Waterline-Tangent Angle cp, the Angle 



From the Ship Track a and the Wavelength A as 



Functions of the Wave-Propagation Angle p. 



where the subscripts D and T refer to the divergent and 

 transverse waves, respectively. 



The first two terms in the low-Froude-number 

 asymptotic expansions for the contributions Kg 5 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 

 fTxTto the hull. In particular, the low-Froude-number 

 asymptotic expansions given in section 5 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 



Fig. 3d - The Waterline-Tangent Angle cp, the Angle 



From the Ship Track a and the Wave-Propagation Angle 



p as functions of the Wavelength A. 



case we have Kg 5 = 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). 



Thus, for a ship form that is everywhere wall 

 sided, the contributions of the bow and stern are 

 dominant (assuming that they are not both either round 

 or cusped) for all values of the angle a from the ship 

 track, that is everywhere in the far-field Kelvin wake. On 

 the other hand, for a hull that has flare over a portion 

 of the waterline and is wall sided elsewhere, the 

 contributions of the bow and stern are dominant, and 

 0(1), only for those angles a in the Kelvin wake for 

 which the corresponding points of stationary phase on 

 the waterline fall outside the region of flare; for the 



11 



