equivalent linear elementary wave may be 

 written as 



A, (6) exp |"i k sec 6 

 ^ ° (38) 



|(x-x ) cos8 + y sine| + i k Q As] 



where xi is obtained as an intersection of the 

 tangent to the ray at °° and the x axis and 



k As = k s - k sec 8 |(x-x,) cos8 + y sinSJ 

 o o o ' 1 ' 



(39) 

 The value of 



Wave Resistance 



If all the elementary waves are assumed to 

 be propagated without reflection, the amplitude 

 function and the phase difference studied in 

 the previous sections will supply enough infor- 

 mation for the calculation of wave resistance. 

 Because at far downstream, the wave height 

 may be considered linear, the Havelock wave 

 resistance formula^ may be used for the waves 

 represented by Equations (36) through (40), 

 considering that the wave with the changed 

 phase s_, (8) has the amplitude 



= 2b 



(?, 



i Q K ) 



> ik s 2b (6) 



s, (8) = As - y l sec6 (40; 



can be obtained at any point along the ray. 

 In general, XJ is negative and S2 (8) is 

 positive meaning that the bow wave phase in the 

 ray theory is larger or more advanced than the 

 phase of the linear theory. This fact has long 

 been observed in experiments in towing tanks. 



The advancement of wave phase is computed 

 for various ships and the values of S2 and xi 

 at x = 2 are shown in Figures 8 through 10. 

 When the beam-length ratio increases and/or 

 the draft-length ratio increases, the values 

 of S2 increase and the values of x decrease 

 for all values of 8„. As compared with the 

 increment of the slopes of rays near the ship, 

 the increment of the phase angle is more 

 sensitive to the beam- and/or draft-to-length 



The most interesting phenomenon about the 

 phase difference is in regard to the bulbous 

 bow. 10 That is, the phase differences for hulls 

 with and without bulbs are almost the same even 

 with a considerably larger bulb. In the past, 

 because of the observed phase difference of 

 ship waves, the bulb was located far forward 

 to obtain good bow wave cancellation. * 

 According to the present analysis, if there is 

 no other reason, the bulb position need not be 

 far forward. Because the nonuniform flow cre- 

 ated by the ship is much more significant than 

 that of the bulb, as far as phase change is 

 concerned, both the ship bow waves (in general, 

 positive sine waves) and the bulb waves 

 (negative sine waves) propagate through the 

 same region and cancel each other. 



As for the amplitude function, although 

 it was shown by Doctors and Dagan 9 that ray 

 theory produced the best result for a two- 

 dimensional submerged body even though they 

 used a linear amplitude function, the surface 

 piercing three-dimensional case may be quite 

 different. The amplitude function is mainly 

 related to the singularity strength which sat- 

 isfies the ship hull boundary condition and 

 some improvement might result by considering the 

 sheltering effect. However, in the present 

 study, the linear amplitude function is used to 

 simplify the problem, showing the effect of 

 curved rays. 



Ill U L 



IT/, 



C = k I 



J , 



-tt/2 



ik s nl _ (6) 



'b ' "V ' 

 or in Sretten6ky's formula" 

 2, 



16 



c = 



16tt k 



j=0 



(42) 



v*m\ 



ik.s,, (dj) 



fl^J. |A, (dj) e 



^4ttj\2 

 ,k w 



(dj) 



ik s,. (6j) 



f y 



J j dxdz m exp j k d . (zd . + ix) + ik s 2 | 



= 2rr ^ 



(43) 



h-\h 



• [^ + H 1+ TTw- 2 ! 1 =sec6 



w = width of the towing tank nondimension- 

 alized by ship length L 



e . = 2 f or j S 1 



|(P b + iQ„) e 



ik s,(6) 2 



10 



