twice from the ship hull. In this way, further 

 increment of 0^ makes the ray reflect from the 

 ship hull three, four ... times. However, at 

 $1 near the value of a, the ray tries to pene- 

 trate the ship hull at the starting point of 

 the bow. This cannot be allowed because this 

 kind of ray should come from outside of the 

 ship. Let the border point of 6i be 6i u . 

 This means that rays of initial value of 6 

 between 6i < 6i < 9i u reflect from the ship 

 hull. As is clear in Figures 1 through 4, 

 in general, all the rays before reflection do 

 not intersect each other. However, reflected 

 rays intersect other rays. The once reflected 

 rays intersect not only with each ray once 

 reflected from ship boundary points close to 

 each other, but also with at least one ray 

 before reflection. 



In the stationary phase, each ray has an 

 amplitude. Likewise in the ray theory, each 

 ray carries it's energy. The reflected ray may 

 have approximately the same energy as the ray 

 at Sj = Oio or 8« =. o where the amplitude 

 function of the linear theory is in general 

 more significant than amplitudes of the other 

 values of 6=>. Because the phase must be 

 approximately close to each other for the waves 

 near 6= = 0, the wave height of the once re- 

 flected ray may be close to three times that 

 of the transversal wave for 9«, = 0. When the 

 envelope of the once reflected waves is drawn, 

 the domain bounded by the ship surface and 

 the envelope, denoted by Dm, must be distinctly 

 different from other domains because in Dm 

 there are not only once reflected rays but also 

 twic or multi-reflected rays on which more than 

 three reflected rays intersect by an argument 

 similar to that used for the once reflected 

 rays. The envelope of the once reflected rays 

 behaves like the shock front which was observed 

 in Japan. -> 



In general, a line is called a caustic when 

 on one side of the line one can find a continuous 

 distribution of rays but not on the other side, 

 and along the caustic the wave slope is found 

 to be large. The wave near the ray angle 8«, = 

 35 deg is the caustic, and the wave height near 

 the caustic far downstream can be obtained by 

 an application of the Airy function in the 

 linear theory. The envelope of the once re- 

 flected rays may be a kind of caustic formed 

 by the refracted bow wave rays due to the non- 

 uniform flow perturbed by the ship. Shen, 

 Meyer, and Keller^ studied such caustics caused 

 by the sloping beach of channels and around 

 islands. Thus, the additional caustic of ship 

 waves may be called the second caustic of ship 

 waves, and should not be confused with the first 

 caustic which is the known caustic at 8„ ■ 35 

 deg. 



Second Caustic of Ship Waves of Various Ships 



For the Wigley hull, several different 

 values of parameters b and h were taken to 

 find their effect on the second caustic. In 

 addition, the effect of a bulbous bow on the 

 second caustic was considered. The most 

 distinguishable physical characteristic of the 

 second caustic is its distance from the ship 

 hull. This is related to the distance of 



reflected rays from the ship hull. If the 

 number of reflection rays increases, or if 6i 

 increases from 6j. , the maximum distance be- 

 tween the ray and the ship hull decreases. 

 The distance, before or after the ray reflection, 

 approximately behaves like a sine curve. The 

 maximum distance between the ray before the 

 first reflection and the ship hull divided 

 by the x coordinate of the point of the first 

 reflection a/xj is plotted in Figure 6 for 

 various ships. The value of a/xi for different 

 values of 9i are approximately the same for a 

 given hull and are related to the area between 

 the second caustic and the ship hull where there 

 may be breaking waves or turbulent waves. Thus, 

 if the area is large, viscous dissipation of 

 energy becomes large. Accordingly, the measured 

 momentum loss behind the ship for the breaking 

 waves' becomes large. 



The values of a/xj , increase with increas- 

 ing beam-length ratio. However, the most 

 interesting part is the effect of the bulbous 

 bow. 10 When the bulb size is increased or the 

 doublet strength is increased the curvature of 

 the ray near the bow becomes less, although the 

 streamline near the bow is such that the en- 

 trance angle is slightly large. The values of 

 a/xj decrease with increasing bulb size, and 

 eventually the ray for S„ -. cannot propagate 

 without penetrating the ship hull at the begin- 

 ning. That is, there is no reflecting ray 

 coming out of the bow with a bulb of the proper 

 size, as shown in Figure 7. 



0.2 0.4 



(h = 0.0625) 



Figure 6 - Width of the Second Caustic 

 a/xj for Wigley Hulls 



