Newman and Tuck 



Kri.C'^) = -\ dk e-'^"" /3(k) coth /3(k) 



this being the kernel f;mction of Eq. (3.17). 



DISCUSSION 



H. Maruo 



National University of Yokohama 



Yokohama, Japan 



Dr. Newman and Dr. Tuck have achieved a rigorous and systematic devel- 

 opment of the slender body theory in the problem of the motion of ships among 

 waves. It must be one of the most important achievements in the theory of ship 

 motion, because it enables a consistent formulation for the damping force and 

 the added mass which cannot be realized by the thin ship theory. According to 

 the results, the effect of the free surface by which the frequency dependence ap- 

 pears, is not important unless the finite speed of advance exists. Therefore the 

 results with finite forward velocity seem to be more important. The ultimate 

 aim of the formulation is to enable the prediction of the hydrodynamic forces by 

 means of the theory. In this respect, the present analysis is not yet conclusive. 

 The reason is that the formulas for the forces and moments given by the Eqs. 

 (3.36) and (3.37) with the kernel function (3.17) are not convenient for the nu- 

 merical computation. An important thing is that the final result should be given 

 by a convergent form. However, the formulas given here involve divergent 

 integrals. In order to obtain a formula which is suitable to the computation, 

 another expression is needed. For this purpose, an expression for Green's 

 function which was obtained by Hanaoka some ten years ago is recommended. It 

 takes the following form: 



1 1 2 C ^ f" exp [-|y- y' | ^m^ + n^ - im(x-x')] 



G(x,y, z;x ,y,z) = — + -- + — dm' 



ym2 + n2 

 X {cos (nz + e) cos (nz' + e) - cos nz cos nz'} dn 



+ — exp [(z + z')(m- ajQ)Vx - |y- y' | K^ - im (x- x')] (m - c^^)'^ ■—- 



+ — exp [(z + z')(m + ojg) Vx - |y- y' | Kj + im(x- x')] (m + ajg)^ — 



J m 



(Cont.) 



162 



