6. CONCLUSION: HULL FORM AND KELVIN- 

 WAKE FEATURES 



The classical far-rield asymptotic approximation to 

 the Kelvin wake, obtained in section 2 by applying the 

 method of stationary phase, and the low-Froude-number 

 asymptotic approximation to the far-field wave-amplitude 

 function, obtained in sections 4 and 5 by successively 

 using the Laplace method and the method of stationary 

 phase, provide a simple analytical relationship between 

 the hull shape, on one hand, and the waves it generates, 

 on the other hand. This explicit relationship between the 

 wavemaker and its waves is summarized below. 



The far-field asymptotic approximation to the 

 Kelvin wake (6) shows that at any point U,o), with | = 

 Xg/U^ « -1 and 0< o < tan" '(2"^^^) ~ 19°28', the 



wave field consists in two plane progressive waves a 



transverse wave and a divergent wave with 



wavelengths A_ and X^ propagating at angles p_ and 

 P_^ from the track of the ship, respectively. The 

 wavelengths A^ and the propagation angles p^ depend on 

 the angle from the ship track a alone, that is X^ and p^ 

 are independent of the hull shape and size, as is well 

 known. Specifically, the functions A^(o) and P^{a) are 

 defined by equations (10), (11) and (12). At a given 

 downstream distance |, the amplitudes of these waves, 

 on the other hand, are given by the product of the 

 functions A^(o), defined by equation (9), and the far- 

 field wave-amplitude function K(t) evaluated at the 

 stationary values t^(o) given by equation (7). The 

 function K(t) depends on the hull shape and the Froude 

 .number in a fairly complicated manner via an integral 

 over the mean wetted-hull surface and an integral around 

 the mean waterline, as is indicated by equation (26). A 

 low-Froude-number asymptotic approximation to these 

 integrals is obtained in sections 4 and 5. 



The analytical approximation (50) shows that for a 

 given value of t corresponding to a given value of a, as 

 is specified by equation (7), the main contributions to the 

 function K(t) stem from several particular points on the 

 mean waterline. These are the bow and the stern, on one 

 hand, and (usually but not always) one (or several) 

 point(s) of stationary phase. Indeed, the number of these 

 points of stationary phase and their position on the 

 waterhne, defined by the condition 



|dyo/dX(,| = 1/t, (68) 



depend on the shape of the waterline and the value of t. 



Let ip denote the angle between the tangent to the 

 mean waterline and the track of the ship, that is we have 



tan cp = Idy^/dx^l and < tp < n/2. (69) 



Equations (68), (69), (7) and (10) then show that the 

 stationary values t^(a) and i_(o) associated with a given 

 value of o are defined by the equation 



tp^(o) = tan-'{4o/[l±(l-8o^)'^2]}, (70) 



where o = tano. For a given waterline shape, equation 

 (70) thus defines the number of stationary points and 

 their position on the waterline corresponding to any 

 given angle a inside the Kelvin wake. In particular, 

 equation (70) yields 



< cp^ < tan-'(2'^2) ~ 54°44' < cp_ < 90° (71) 



and (p ^ ~ 2o, cp ~ n/2 -a as o -♦ 0. (72a, b) 



Points of the waterline with slope between and 54°44' 

 thus contribute mostly to the system of divergent waves 

 while waterline slopes between 54°44' and 90° mostly 

 contribute to the transverse waves. 



Equations (10), (11), (12) and (70) define the wave- 

 lengths A^, the wave-propagation angles /3^ and the 

 waterline-tangent angles cp^ corresponding to a given 

 angle from the ship track a. The functions A^(o), /3+(o) 

 and cpj.(o) are depicted in Figure 3a, where the subscripts 

 T and D are used, instead of - and + , to refer to the 

 transverse and divergent waves, respectively. The 

 foregoing relationships between a and A, p, cp may be 

 used for determining the angle from the ship track a, the 

 wavelength A and the wave-propagation angle p 

 corresponding to a given waterline-tangent angle cp. 

 Specifically, we may obtain the remarkably simple 

 relations 



a = tan"' [tancp/(2 + tan^cp)], (73) 



A/2n = sin^cp and /? = 7i/2-cp. (74a,b) 



The functions o(cp), A(cp) and /3(cp) are depicted in Figure 

 3b. Alternatively, the foregoing relationships among cp, a, 

 A and p can be represented in the form of Figures 3c and 

 3d, which depict the functions cp(/3), a{p), X{fi) and cp(A), 

 a(A). /3(A), respectively. These equivalent graphical 

 representations show that we have 



< cpp < tan" '(2'^^) £ii 54°44' < cp^ < 90°, (75a) 



< Xa/2n < 2/3 < AT/2n < 1 , (75b) 



90° ^ P^> tan -1(2"'''^)^ 35°16' > p^ > 0, (75c) 



10 



