Rays of Ship Waves and Ship Boundaries 



Ship waves created by a smooth ship hull 

 propagate as regular bow and stern waves from 

 the bow and stern stagnation points to infinity 

 along rays. Because a ray carries the wave 

 energy it cannot penetrate the ship surface. If 

 the linear free-surface condition is considered 

 with the exact hull boundary condition, as has 

 been popular in recent ship wave analysis , 

 the straight rays which pass through the hull 

 boundary have to be considered. For a ray 

 theory with the exact hull boundary condition, 

 the ray is not allowed to penetrate the ship. 

 In fact, when the initial wave crest touches 

 the ship boundary it can be proved that the 

 ray of such a wave grazes along the ship 

 boundary without penetrating the ship boundary. 



In Equation (8) when the wave crest 

 touches the hull 



u cose + v sine » (20) 



because e is the angle between the normal to 

 the crest and to the*axis, and the velocity 

 normal to the ship hull is zero on the hull. 

 Equation (8) then becomes 



dy _ v 



showing that the ray touches the ship hull 

 streamline from Equation (13). 



When the wave crest is perpendicular to 

 the ship hull 



— = tan6 



(26) 



If Equation (26) Is inserted into Equation (8) 

 tanti - — (27) 



dy_ 



The ray also touches the ship initially. 

 However, Equation (9) is not compatible with 

 Equation (26) on the hull. This can be proved 

 in a similar way as follows: 



Differentiating Equation (26) with respect 

 to x along the ship hull 



dx f + 1 

 x 



Inserting Equation (26) into Equation (9) gives 



2tan9 (|H +tan eg)_ 2 (g +tan e|x) 



u (1 + — tan6) 



From Equations (25), (27), and (29), noting 



3u 3v 



When the wave crest touches the hull, 

 from Equations (11) and (13) 



(30) 



and from Equations (20) and (21) 

 tane = ^— 



(21) 



(22) 



Differentiating Equation (22) with respect to 

 x along the ship hull 



(23) 



However, inserting Equation (20) into Equation 

 (9) yields 



„ u ± v 2. 3u „ 3v 



cote -r- + t — cot e - — cote — 



d6 3x 3x 3y 3y (24) 



u(l cote) 



Differentiating Equation (21) with respect to 

 x along the hull yields 



v 3u 3y 3v 3y 



u 3y 3x 3y 3x (25) 



3 v u 3x 

 3x u u 



From Equations (20), (21), (22), and (25) 

 it can be shown that Equations (23) and (24) 

 are equivalent. That is, when the wave crest 

 touches the ship boundary, the ray equations 

 and the ship hull streamline equations are 

 equivalent. 



Equations (28) and (30) are compatible only 

 when f - , or f x - constant . 



That is, only when the ship is a flat plate 

 does the ray of the wave, whose crest is perpen- 

 dicular to the ship, follow the ship boundary. 

 This means that when the ship bow is like a 

 wedge, the ray of the bow wave, whose crest is 

 perpendicular to the wedge surface, initially 

 follows the wedge surface. 



When the ray equations are solved numer- 

 ically by the Runge-Kutta method, with initial 

 values near the bow stagnation point, the ray 

 touches and follows the ship boundary at 



e ± - -j + a (31) 



where a is the half entrance angle of the ship 

 bow and e^ denotes the initial value of 6. 



When 6^ increases from this value the ray 

 moves gradually away from the ship as shows in 

 Figures 1 through 4. The rays are curved near 

 the ship but at far downstream they are straight 

 and the ray direction becomes exactly the same 

 function of 6 as the linear theory. Thus at 

 infinity the rays are inside |dx/dy| = 8*5. 

 However, when 6^ approaches zero, the curved 

 ray near the ship gradually approaches the 

 ship, and eventually crosses the ship boundary, 

 as in Figures 1 through 4. Here the wave 

 reflection should be considered to prevent the 

 ray penetration of the ship hull. 



