linear theory far downstream is reasonable and 

 likely to produce a reasonable result as in 

 the two dimensional theory.' 



The wave phase in the ray theory is ob- 

 tained, together with the ray path, indepen- 

 dent from the amplitude function. Both the 

 phase and the ray path are quite different from 

 the prediction of linear theory near the ship. 

 The rays far downstream are straight as in the 

 linear theory yet have phases different from 

 the linear theory; they are parallel to the 

 linear ray with the same wave angle but do 

 not coincide. The difference in wave phases 

 is the main factor that makes the wave resis- 

 tance different from the linear theory. Ray 

 paths and phase difference are computed for 

 various parameters of the Wigley ship, with and 

 without a bulbous bow. The ray paths for 

 different drafts and different beam-length 

 ratio of the Wigley hull are slightly different, 

 widening the wave area near the hull for the 

 wider beam and /or larger draft. However, the 

 phase difference is more sensitive to the beam- 

 length ratio and/or draft-length ratio, by 

 always advancing the linear wave phase. 



The wave resistance of the Wigley hull is 

 computed and shown to have a considerable shift 

 of phase of hump and hollow. 



By eliminating S z from these two equations he 

 obtained a dispersion relation 



The most interesting phenomenon of a ship 

 with a bulbous bow is the reduction of slopes 

 of rays and the second caustic, i.e., the 

 larger the bulb, the greater the reduction. 

 This fact was observed in the towing tank.-' 

 There exists a bulb size which totally elimi- 

 nates the reflecting rays. However, the phase 

 difference due to the bulbil ^ s very small 

 showing that the phase difference, which has 

 been observed in the towing tank, is the effect 

 of nonuniform flow caused by the main hull. 



Ray Equations 



The concept of ray theory in ship waves is 

 analogous to the concepts used in geometrical 

 optics and in geometric acoustics. A ship is 

 considered advancing with a constant velocity 

 -U which is the direction of the negative x 

 axis of a right handed rectangular coordinate 

 system O-xyz with the origin at the ship bow 

 on the mean free surface, z«0, z is positive 

 upwards . 



First the phase function s(x,y,z) is de- 

 fined so that the equation (s » constant) re- 

 represents the wave front where the value of s 

 is the optical distance from the wave source, 

 e.g., the ship bow. When Keller developed 

 his ray theory of ships by expanding boundary 

 conditions and a solution, which should satisfy 

 both the Laplace equation and the boundary 

 conditions, in a series of Froude number squares 

 F , he obtained: 



(s.. 2 + s V 



(2) 



where t is the double model potential. When 

 steady state motion is assumed with respect to 

 the moving coordinate system, - xyz, 



When the angle between the normal n to the 

 phase curve s and the 'axis is denoted by 6, 



n - icosB + jsinS (3) 



Then the wave n umb er vector is defined by 



k = k.T + k,j = si + sj 



1 



(4) 



- nk = ik cos6 + jk sine at z = 

 From equations (2) and (4) 



k "|u cos6 + v sin9J 



where 



-<i> «= u and -4 = v 

 x y 



These results are an approximation within 

 the order of f4, and the phase function and its 

 related equations are all limited to their 

 values at z = 0. Thus, from now on, unless 

 otherwise mentioned, all the physical values 

 are at z ■ 0. The ray equation of ship waves 

 is obtained from the irrotationality of the 

 wave number vector 



9k 2 3k j 

 3x 3y 



From Equations (4) through (6) , 



{2 sine (u sine - v cosB) 

 + cosB (u cosB + v sine)} — 



+ {2 cosB (-u sine + v cos6) 

 + sine (u cose + v sine)} |£ 



- 2 sine (cose |S + sine |^) 



(6) 



(7) 



(Vs) - 

 -i (S„ + V4 • 7s) 2 at z - 



(1) 



