Current Progress in the Slender Body Theory- 

 unit lengthwise displacement of the ship, and this is clearly zero. On the other 

 hand, if j 4l then c'.^^^ does not in general vanish at zero frequency and yields 

 for oj^o the trim forces and moments on the ship in steady motion. Since in 

 evaluating added masses by (3.3) we must in any case subtract off these trim 

 forces C- j(0), the kernel 



"K - lim K" 



CO-* 



( F S ) 



may be used for all C^ ^ whenever trim forces are not required. 



At non-zero frequencies w of purely sinusoidal motion we can split the 9 

 Eqs. (3.36) into their real and imaginary parts yielding 18 added masses and 

 damping coefficients from Eqs. (3.3) and (3.4). In order to compute these 18 

 quantities we require just two functions S(x) and B(x) describing the geometry of 

 the ship and one universal kernel function K(x, w.U) which can be computed once 

 and for all. Of course the complete added masses and damping coefficients are 

 the sum of the "wall" values plus the values obtained from Eqs. (3.36), but the 

 determination of the former is, as described in Part I, a much less difficult task. 



For the lateral modes of sway, roll, and yaw, where i or j takes the values 

 2, 4 or 6, the C^^^^ vanish, so that the remaining 54 added masses and damping 

 coefficients are dominated by the "wall" values. Since the latter are frequency 

 independent, this conclusion is equivalent to the conclusion that all lateral and 

 damping coefficients are independent of frequency, to leading order in slender- 

 ness. Any frequency dependence must come from higher approximations in e. 



ACKNOWLEDGMENTS 



The authors wish to thank Mr. Werner Frank and Miss Evelyn WooUey, who 

 developed the computer programs used for the zero speed calculations. 



REFERENCES 



Newman, J. N., 1963, "Lectures on the Theory of Slender Ships," Unpublished 

 notes. Department of Naval Architecture, Berkeley, California. 



Newman, J. N., 1964, "A Slender Body Theory for Ship Oscillations in Waves," 

 Journal of Fluid Mechanics, Vol. 8, Part 4, pp. 602-18. 



Timman, R., and Newman, J. N., 1962, "The Coupled Damping Coefficients of a 

 Symmetric Ship," Journal of Ship Research," Vol. 5, No. 4, Reprinted as 

 David Taylor Model Basin Report 1672. 



Tuck, E. O., 1964, "On Line Distributions of Kelvin Sources," Journal of Ship 

 Research, Vol. 8, No. 2. 



153 



