This is the ray equation which can be 

 solved by the method of characteristics, and 

 is equivalent to simultaneous ordinary differ- 

 ential equations. 



dy {v- *s sine (u cos6 + v sine)} .. 

 dx {u- h cos6 (u cosB + v sinS)} 



7- -=.{2 sine (cose |H + sine |^-) 



3u 



iV> 



- 2 cose (cose -T- + sine ?r-)}/{2 sine (u sine (9) 

 dy dy 



- v cose) + cos6 (u cose + v sin6)} 



Here u cos6 + v sin 6 is the velocity component 

 normal to the phase curve of the flow relative 

 to the ship. Because the wave is stationary 

 relative to the ship, the phase velocity 

 through the water surface should be 



- u cosB - v sine 



The group velocity is one-half of the phase 

 velocity. Thus the ray direction in Equation 



(8) is along the resultant of the group velo- 

 city taken normal to the phase curve and of the 

 velocity of the basic flow as Ursell haB shown.* 

 Because the wave energy is propagated at the 

 group velocity, the ray path is interpreted as 

 the path of energy relative to the ship. This 

 can be obtained by solving Equations (8) and 



(9) with the proper initial condition. The 

 phase s can be obtained from equations (4) by 



s - f k dr 



as in potential theory; s(x,y) is a function of 

 (x,y) but is unrelated to the integration path. 

 However , 



ds « k cose dx + k sine dy 



(10) 



can be solved together with the ray equations 

 along the ray path. 



Rays of Ship Waves and Linear Theory 



To investigate the path of a ray of a 

 ship wave, a ship, represented by a double 

 model source distribution m (x,y) on 

 y ■ o, h>z>-h, is considered. Although the 

 linear relation between the source strength 

 m and the ship surface 



y - ±f(x.z) 



1 dy 



m- 2^ at 



(11) 



(12) 



the actual double model ship body streamline 

 should be obtained by solving 



through the stagnation point where u and v are 

 the velocity components of the total velocity 

 caused by the double model source distribution 

 m and the uniform flow relative to the ship. 



In the linear ship wave theory a smooth 

 source distribution produces two systems of 

 regular ship waves: the bow and the stern 

 waves starting from the bow and stern, 

 respectively, represented!' as 



tt/2 



;(x,y) •= j A(e) exp {i Sj (x,y,e)} d6 (14) 



-tt/2 



s. = k sec 8 {(x-x.) cose + y sine} (15) 

 x = the x coordinate of the bow or stern 



g ■ acceleration due to gravity 



A(6) =* amplitude function which is a 



function of source distribution m 



The regular wave c is the solution of 

 linear ship wave theory far from the ship. 1 

 Actually, it is easy to see. that the exponential 

 function satisfies both the Laplace equation 

 and the linear free-surface boundary condition 

 for any value of xi. Havelock interpreted 

 Equation (14) by discussing the integrand as 

 elementary waves, ,13 i.e., the regular waves 

 are aggregates of elementary waves starting from 

 the bow and stern of a ship. The normal 

 direction of each elementary wave crest is 

 n - i cosB + J sine. 



Because the local disturbance of the double 

 model decays rapidly away from the ship, even 

 in nonlinear theory in the far field, the 

 regular wave should be of the form of Equation 

 (14) with possibly different values of x^ and 

 A(6). When the integral of 5 is evaluated by 

 the method of the stationary phase 14 by taking 

 roots of 



If (x.y.6) = (16) 



2 tan 8 + - tane + 1-0 (17) 



y 



in general two values of e are obtained for a 

 given value of each x and y, satisfying, 



A > 8^ (18) 



dx u 



(13) 



