Ray Reflection 



There are no waves coming from places 

 other than the ship bow or the ship stern. In 

 the ray theory, the flow field perturbed by the 

 ship deflects the ray path starting from the 

 ship bow toward the ship. Thus, some rays 

 of bow waves impinge on the ship hull. 

 Because the ray theory is for small Froude 

 numbers and the wave phenomenon is considered 

 only on the free surface, it is reasonable to 

 consider only reflected waves, as in geometrical 

 optics, neglecting transmitted waves when the 

 oncoming waves impinge on the ship hull. 



Let the ray at the wave angle 8 on the 

 ship boundary (x,y) be reflected to 8 r and 

 f x = tan8 as in Figure 5. 



(32) 



Figure 5 



Angles of Waves Reflecting on 

 the Ship Hull 



Whenever the ray intersects the ship 

 boundary at (x,y) with angle 8, then the value 

 of 6 at (x,y) will be changed to 8 r = 28 - 8, 

 and x,y,8 are continuously calculated by the 

 Runge-Kutta's method. Then the ray will reflect 

 as in Figures 1 through 4. Here if 8 is zero, 

 it is easily seen in Equation (8) that the 

 impinging ray angle (9) changes to - ® f or the 

 reflected ray angle. This fact can be easily 

 shown to be true even for 8 p by just the 

 rotation of the coordinate system. 



Numerical Experiments of Ray Paths and 

 Free Surface Shock Waves 



Because the ray equations can be solved 

 only numerically, careful numerical experiments 

 with the ray equations may give valuable in- 

 formation. For simplicity, the Wigley hull 

 source distribution with 



2ir dx 



f. - 2rt (l - x 2 ) {l - (h 2 ) 



(33) 



is considered for various numbers of b and h 

 which are related to the hull beam and draft, 

 respectively. The actual hull shape corre- 



sponding to the source Equation (33), is ob- 

 tained by plotting the body streamline passing 

 through stagnation points as the solution of 



dy _ v 

 dx u 



where u and v corresponding to Equation (32) , 

 are shown in Appendix A together with u , u , 

 and v„ for the ray equation. Equations (8){ 

 (9), and (10), together with the streamline 

 equation, are solved by the Runge-Kutta method 

 with initial conditions (x,y,8) near the stag- 

 nation points with various values of 8. 



Many ray paths both reflecting and non- 

 reflecting from the surfaces of various Wigley 

 hulls are shown in Figures 1 through A. These 

 paths were computed by a high speed Burroughs 

 computer at David Taylor Naval Ship Research 

 and Development Center. The reflection condi- 

 tion is incorporated in the high-speed computa- 

 tion with a routine to find the intersection of 

 the ray and the ship boundary which is pre- 

 calculated and saved in the memory. The step 

 sizes of integrations and interpolation were 

 determined after many numerical tests, and the 

 shown results are considered to be reasonably 

 accurate. For a given initial condition, the 

 solution is stable and converges well. 



In Figures 1 through 4, some common fea- 

 tures of rays can be drawn as follows. The 

 rays in - ir/2 < 8<x> < far behind the ship 

 behave like rays of linear theory except wave 

 phases are advanced in the ray theory and those 

 near 8oo ■ are rays propagating from the ship 

 bow and reflecting from those of ship hull. 

 The rays near the ship are very different from 

 the linear theory as Inui and Kajitani^ pointed 

 out . The curved rays from the bow have a far 

 larger slope than those of linear theory and 

 the phase of each ray is considerably advanced. 

 The magnitude of the phase difference is more 

 sensitive to the beam-length ratio and the 

 draft-length ratio than the magnitude of the 

 ray slope. 



When 6„ = 35 deg, the ray will be the 

 outermost ray and the ray angle at «> will be 

 approximately tan" 8 - ^ as in the linear theory 

 and there is the corresponding initial value of 

 8 or 8i near the bow, or the origin. However, 

 the 8i which is corresponding to a single value 

 of 8o> is very sensitive to small changes of x 

 and y near the origin. At a fixed point near 

 the origin there exists a unique correspondence 

 between 6^ and 8«,. 



When from the 8^ which is corresponding to 

 8,,, = 35 deg, the initial value of 8^ Increases, 

 8oo also Increases and the ray angle decreases. 

 In general, when 8i = 0, 8<„ is still a negative 

 value. When 8^ increases further, 6«. approaches 

 zero and the ray pass is very close to the 

 stern. At this point, there exists a certain 

 increment of 8^ which makes the ray barely 

 touch the ship stern, at 8^ = B±o. When 8i 

 slightly increases from 6^ , the ray reflects 

 from the ship hull near the stern. With the 

 increment of 8^ the reflection point moves 

 toward the bow. When 8 - e^j the ray once 

 reflected touches the stern again. When 8^ 

 increases further from 8n, the ray reflects 



