Figure 7 - Rays of the Wigley Ship (b = 0.2, h = 0.0625) with Bulb 

 (radius, r = 0.0214h, depth, Zj = 0.5h) 



These phenomena associated with the second 

 caustic are exactly the same as those of the 

 "free surface shock wave" which was observed 

 in the towing tank. 



Finally it should be pointed out that the 

 ray of the wave whose crest is perpendicular 

 to the hull surface follows exactly along the 

 flat surface as was proven by Equations (28) 

 and (30) and the following paragraphs. This 

 means that rays originating from the vertex 

 of a wedge very likely never reflect on the 

 wedge surface. This is because rays near the 

 ray which follows the ship surface did not 

 intersect each other near 6^ = -tt/2 + o, or 

 6„ - - tt/2. Rays near 6,,, '=. impinge on the 

 ship surface due to the effect of the water- 

 line curvature of the ship. Therefore, if a 

 ship has a wedge bow, the second caustic must 

 be found near or behind the shoulder. Be- 

 cause the first caustic near the bow may be 

 very prominent and both waves near the first 

 caustic and waves near the stagnation point 

 break, careful experimental analysis and more 

 theoretical study of the bow near field may be 

 needed. 



Wave Amplitude and Phase 



Because the perturbation due to a ship, 

 or both regular waves and local disturbances, 

 decays at far field, the linear theory must 

 hold in the far field. In particular, the 

 wave resistance can be calculated from the 

 energy passing through a vertical plane x= 

 constant far downstream; the linear theory 

 which is properly matched to the near field of 

 the ship will be used for calculating the wave 

 resistance. As explained in Equation (14) and 

 the following paragraphs, the expression for 

 regular waves far downstream is known to be of 

 the form of Equation (14) where the amplitude 

 function 



A(6) = P(8) + i Q(6) 



(34) 



may be taken as an approximation from the linear 

 theory but the phase difference xi must be ob- 

 tained from matching with the near field. If 

 the near field is also expressed by the linear 

 theory 



Q = "k / / 



(35) 



dxdz f e 



sec 9 



where f x is the derivative of Equation (11) with 

 respect to x. When the inner integrand of 

 Equation (35) is integrated with respect to x, 

 the value with the limit x = 1 will form stern 

 waves and the value with the limit x = will 

 form bow waves . 1 1 Then the bow waves can be 

 represented by 



tt/2 



? b " / *b (8) 6XP f 1 S l (9) ) d9 (36) 

 -tt/2 



where 



(6) ■ P v + i q. 



(37) 



When the phase s is computed from Equation 

 (10) along with the ray path from Equations 

 (8) and (9) considering that s - at the bow 

 near the origin, there are two results different 

 from those of linear theory: (1) the ray path 

 is deflected as if the elementary wave of the 

 linear theory started from (x^.O) not from the 

 origin, and (2) the phase change denoted by 

 As should be considered. That is, the 



