Numerical Solutions 



The ranges of integration in (31) and (32) can also be reduced to 0- v/2 by taking 

 advantage of symmetry. The number of coefficients that can be computed with 

 accuracy from (32) depends upon the number and distribution of the values of d . 

 For example, if nine values of £? in a half cycle of the oscillating integrand in 

 (32) are considered to be necessary for the numerical evaluation of the integral, 

 the series should be truncated at a^^ when intervals of 1° are used in the 

 calculations. 



From (30) we now have the parametric equations of the ship section: 



x= (A+a.) cos 6 + a, cos 36 + a^ co s 5(9 + • • • , W") 



y = (A- a,) sin 6 - a^ sin Z6 - aj sin Su - ••• • (34) 



How well a given section can be represented by these equations is shown in 

 Fig. 2. Results are shown for various combinations of mappings from the sec- 

 tion into the unit circle. It is seen that the poorest representation is obtained 

 from a direct application of the Gershgorin integral equation, with no interven- 

 ing transformations. Preliminary branch-point transformations, or a prelimi- 

 nary Joukowsky transformation for transforming the section into one of unit 

 height-to-breadth ratio, followed by the Gershgorin transformation, consider- 

 ably improve the representation. Best of all is the result obtained by the suc- 

 cession of branch-point, Joukowsky, and Gershgorin transformations. 



REFERENCES 



1. Landweber, L. and Macagno, M., "Added Masses of Two- Dimensional Forms 

 by Conformal Mapping," J. Ship Res., Vol. 11, No. 2, June 1967 



2. von Kerczek, C. and Tuck, E.G., "The Representation of Ship Hulls by Con- 

 formal Mapping Functions," David Taylor Model Basin, Hydromechanics 

 Laboratory Technical Note 58, September 1966 



3. Gaier, D., "Konstruktive Methoden der Konformen Abbildung," Springer 

 Tracts in Natural Philosophy, Vol. 3, Springer-Verlag, Berlin, 1964 



4. Becker, E.B., Wilson, H.B., and Parr, C.H., "Further Developments of Con- 

 formal Mapping Techniques," Rohm & Haas Co., Report No. S-43, July 

 1964 



5. "Advanced Mechanics of Fluids," edited by H. Rouse, Wiley-Interscience, 

 New York, 1959 



DISCUSSION 



In reply to a question by R. Barakat, the author clarified the procedure for 

 representing the entire ship hull in parametric form. If 20 sections were used, 



1626 



