MICHELL'S THEORY 

 "RAY THEORY 



0.15 



0.20 



0.25 



Figure 12 - Wave Resistance of Che 

 Wigley Ship (b = 0.2, h = 0.0625) 



Discussion 



Recently Eggers obtained a dispersion 

 relation from a low speed free-surface boun- 

 dary condition which was a slightly modified 

 version of the one Babal8 used. Eggers showed 

 that there was a small region near the stag- 

 nation point where the wave number became 

 negative; since this was not permissible for a 

 wave and it could be interpreted to mean that 

 there was no wave in the region. He suggested 

 that his form might help alleviate the sen- 

 sitivity of the initial condition to the ray 

 paths. According to the energy conservation 

 law 8 of the wave propagating through non- 

 uniform flow the wave energy flux is proportion- 

 al to the wave number along the ray tube. 

 When the wave number along a ray is considered, 

 although the Keller dispersion relation has an 

 infinite wave number only at the stagnation 

 point, the Eggers dispersion relation has an 

 infinite wave number in the flow near the bow. 

 Thus, near such a singular point or a singular 

 line, waves may break and the present theory 

 cannot be applied. When Eggers 1 equation was 

 incorporated into the ray equation in the 

 present ray computer program it was found that 

 the ray path was still very sensitive to a 

 small change of initial value (x,y,tt) and the 

 curved ray still reflected from the ship hull. 



In the experiments^ conducted at Tokyo 

 University, the second caustic can be noticed, 

 and near the second caustic the flow field is 

 violently different from the linear theory. 

 However, it seems that extreme caution is 

 needed to distinguish the second caustic from 

 the first caustic. The case of experiments- 1 

 with wedges is interesting because, according 

 to the present theory, there cannot be a second 

 caustic on the wedge although the first caustic 

 should be there. However, the wave phases of 

 the wedge should be advanced and quite sensitive 

 to the draft- and beam-length ratios due to the 

 nonuniform flow caused by the wedge. On the 

 other hand, the present theory is still not 

 exact although it takes into account the effect 

 on the propagation of water waves of nonuniform 



flow due to the double model. Thus, the non- 

 linear effect of water waves is not totally 

 analyzed here. Nevertheless, the present 

 theory supplies a great deal of hope to an 

 entirely different approach to the ship wave 

 theory - the ray theory. 



Although the strictly linear amplitude 

 function is used for simplicity in the present 

 study, a slightly improved amplitude function 

 may be easily incorporated by adding the 

 sheltering effect or other effects. However, 

 it should be noted here that, in any case, the 

 bow wave resistance and the stern wave resis- 

 tance are the same as the results without the 

 ray theory and the effect of the ray theory 

 would appear as a shift of the phase of hump 

 and hollow in the total wave resistance. There- 

 fore, the computational results would not match 

 the results of towing tank experiments unless 

 the viscous boundary layer effect on the. stern 

 wave or some other effect is considered. 



Acknowledgements 



This work was supported by the Numerical 

 Naval Hydrodynamics Program at the David W. 

 Taylor Naval Ship Research and Development 

 Center. This Program is jointly sponsored by 

 DTNSRDC and the Office of Naval Research. 



Appendix A 



For the computation of ray paths of a 

 ship, the flow velocity and its derivatives on 

 (x,y,o), u,v,u x , v ,Uy are needed. A Wigley 

 hull has the double model source distribution 



b (-2x. + 1) il - fci) 



= 0, h 



-u (x,y,o) =2 j | m |^ (^) d Xl dZj - 1 



o o 

 f h 1 



= - 2 7 / »h ( 7> d *i dz i - 1 



= 2b j f ; i^T dz, 



I J r (x 1 = 1) 1 



h i . 



J r (x, = o) 



o 



+ 4b 



+ r(z, = h) 



= 1 " 1 



-j h log j x 1 - x 



13 



