Furthermore, when 



>-h 



because only the transformation of coordinates 

 (x,y) to (x - Ut , y) is needed to get the un- 

 steady flow. Strictly speaking the relative 

 frequency is 



s = k u cos6 + k v sine = k U cose 



6 = 35 deg 



and the value of the integral of Equation (14) 

 for any x,y in x/|y|£-8^ can be evaluated by 

 the stationary p»hase method as the sum of two 

 waves: transverse waves (0 deg < |e| < 35 deg) 

 and divergent waves (35 deg < |e| < 90 deg). 



Except for the amplitude function exactly 

 the same result for the relation of x,y and 6 

 as in the linear theory can be obtained from 

 Equation (8) by substituting u = 1 and v = 0. 

 Thus, this means that the energy of each 

 elementary wave propagates on the uniform flow 

 along a straight line ray. The rays are con- 

 fined in x/|y| > 8^, both of the elementary 

 waves at 6 = and 9 = 90 deg correspond to the 

 ray y /x = 0, and 8 = 35 deg corresponds with 

 the ray x/|y| « 8"*. One ray between these two 

 rays corresponds to two elementary waves where 

 one is transversel and the other is divergent. 



When the velocity is not uniform due to the 

 flow perturbation caused by a ship, the ray is 

 not straight but curved near the ship as 

 shown in Figure 1. It is known that wave energy 

 flux divided by the relative frequency with 

 respect to the coordinates for which the fluid 

 velocity is zero, is conserved along the 

 curved ray tube. If the coordinates are fixed 

 in space, then the ship waves are unsteady 

 relative to the fixed coordinates and the 

 frequency is 



= U k cose 



(19) 



0.20 

 0.18 

 0.16 

 0.14 

 0.12 

 0.10 

 0.08 

 0.06 

 0.04 

 0.02 

 0.0 





/0.2265/ 



0« 



= -0.66892/ . 



/ (Geo; 0j) = 

 -0.1175:0.08 



- 



///el "JL1 



SECOND 

 -^CAUSTIC 



£ 



^n2^^5p=2 



==<~\ 



^^^ 0.14 



^^\" 



but for our discussion, the approximate value 

 serves the purpose. This means that for a 

 uniform flow the wave energy is constant 

 along the ray because for a uniform flow u = U, 

 v = 0, k = g/(U 2 cos2e), and 6 is constant 

 along the straight ray. However, the wave 

 energy is not constant along the curved ray in 

 nonuniform flow, but is dependent upon the 

 local velocity components and e because the 

 relative frequency is a function of the local 

 velocity and 6 as in Equations (5) and (19) and 

 6 changes along the curved ray. The wave 

 energy will change considerably along the 

 curved ray near the bow or stern because u and 

 v change to zero at the wave source, the 

 stagnation point, while in the far field it 

 will be constant along the ray as in linear 

 theory. Near the stagnation point, the wave 

 number increases according to Equation (5) and 

 the wave energy also increases due to the energy 

 conservation law." Thus, the wave may break near 

 the stagnation point. 



Because wave amplitude is so difficult to 

 obtain from the ray theory 2 , the linear theory 

 has been considered as an approximation. 1 A 

 good result was obtained for a two dimensional 

 flow problem. 9 However, extreme care is needed 

 in the three dimensional theory because 6 chang- 

 es by a large amount along the ray, near the 

 stagnation point, and the wave energy depends 

 upon 6. The matching amplitude function of ray 

 theory with the values from linear theory should 

 be done at the far field where 6 and the direct- 

 ion of the ray are, respecitvely, identical for 

 both cases for each elementary wave. In addi- 

 tion, the initial condition has to be taken in 

 the neighborhood, but not exactly at, the stag- 

 national point because, although the stagnation 

 point is the wave source, at the stagnation 

 point the ray equations are indeterminant , u 

 and v being zero. However, because of the large 

 change of u and v near the stagnation point, the 

 ray path is very sensitive to the initial values 

 of x, y and 6; the nearer to the stagnation 

 point, the more sensitive. Therefore, the iden- 

 tification of ray paths should be correlated 

 with the values of 6 at infinity. Then all the 

 ray paths can be properly and uniquely identified 

 by 6„. 



Since perturbations of the ship decay 

 rapidly away from the ship, 6 also rapidly 

 approaches 6oo. The relation between the initial 

 value and the value at infinity of 6 has little 

 meaning, although it was misunderstood before*'" 

 because 6 changes very rapidly near the stag- 

 nation point. 



0.2 



0.4 



0.6 



0.8 



1.0 



Figure 1 - Ray Paths for a Wigley Hull 

 (b - 0.1, h - 0.0625) 



