(x.-x) y 



o ( z i +y ) r 



2 h 



3 |( Xl -x) 2 + y 2 | r(z x =h) 



h + (z -h) 



2 2 2 



(li + y ) (x^x) y Z 



1 



^2y 2 + (l-2x) (x.-x) h 



6 by' 



3h" \ (x -x) + y r(z =h) 



h +r(z =h) 2h 



5 r(z =0) r(a =h) 



2(x,-x) (l-2x) 



f d2 l 

 - 1 (z 2 r + y 2 )r 



2. Keller, J., "The Ray Theory of Ship Waves 

 and the Class of Streamlined Ships," J. of 

 Fluid Mechanics, Vol. 91, Part 3, pp. 465- 

 488 (1978). 



3. Miyata, H. , "Characteristics of Nonlinear 

 Waves in the Near-Field of Ships and Their 

 Effects on Resistance," Proc . of 13th ONR 

 Symposium on Naval Hydrodynamics (1980). 



4. Ursell, F. , "Steady Wave Patterns on a Non- 

 Uniform Steady Fluid Flow," J. of Fluid 

 Mechanics, Vol 9, pp. 333-346 (1960). 



5. Shen, M.C., R.E. Meyer and J.B. Keller, 

 "Spectra of Water Waves in Channels and 

 Around Islands," The Physics of Fluids, 

 Vol 11. No. 11, pp. 2289-2304 (1968). 



6. Yim, B., "Notes Concerning the Ray Theory 

 of Ship Waves," Proc. of the Continued 

 Workshop on Ship Wave-Resistance Comput- 

 tations, pp. 139-153 (1980). 



7. Inui, T., H. Kajitani, H. Miyata, M. 

 Tsuruoka, and A. Suzuki, "Nonlinear Proper- 

 ties of Wave Making Resistance of Wide-Beam 

 Ships," J. of the Society of Naval Arch, of 

 Japan, Vol. 146, pp. 18-26-' (Dec 1979). 



8. Lighthill, J., "Waves in Fluids," Cambridge 

 University Press, Cambridge, London pp. 332- 

 337 (1978). 



9. Doctors, L. and C. Dagan , "Comparison of 

 Nonlinear Wave-Resistance Theories for a 

 Two-Dimensional Pressure Distribution," 

 J. of Fluid Mechanics, Vol. 98, Part. 3, 

 pp. 647-672 (1980). 



10. Yim, B., "Design of Bulbous Bow by the Ray 

 Theory of Ship Waves," to be published. 



v (x.y.O) 



y 



wnere r(a,b) = r(x.=a, a =b) 



In these expressions the integrals 



are available in closed forms. 



References 



1. Inui, T. and H. Kajitani, "Study on Local 

 Non-Linear Free Surface Effects in Ship 

 Waves and Wave Resistance," Schif f stechnik, 

 Band 24, Heft 118 (Nov 1977). 



11. Yim, B., "A Simple Design Theory and Method 

 for Bulbous Bows of Ships," J. of Ship 

 Research, Vol. 18, No. 3, pp. 141-152 



(Sep 1974). 



12. Havelock, T.H., "Wave Patterns and Wave 

 Resistance," Tran . of the Inst, of Naval 

 Architects, Vol. 76, pp. 430-443 (1934). 



13. Inui, T., "Wave-Making Resistance of 

 Ships," Trans, of SNAME, Vol. 70, pp. 283- 

 313 (1962). 



14. Lamb, H., "Hydrodynamics," Dover Pub., 

 New York, N.Y., p. 395, (1945). 



15. Havelock, T.H., "The Calculation of Wave 

 Resistance," Proc. of the Royal Society 

 (London), Vol. A144, pp. 514-521 (1934). 



16. Srettensky, L.N., "On The Wave Making 

 Resistance of a Ship Moving Along in a 

 Canal," Philosophical Magazine, Vol. 22, 

 7th Series (1936). 



17. Eggers, K. , "On the Dispersion Relation 



and Exponential Variation of Wave Components 



15 



