Newman and Tuck 



Also the question arises as to how deep must the thin ship be in order to 

 avoid the three-dimensional effects that correspond to the differences (k^ - kj) 

 in the virtual mass coefficients for a body of revolution (see Lamb: "Hydro- 

 dynamics," p. 155), or to avoid the fineness ratio effects corresponding to the 

 complete elliptic integral (e) determination of the virtual mass of a thin plate 

 (see Lamb, Eq. (16), p. 154). 



REPLY TO DISCUSSION 



J. N. Newman and E. O. Tuck 



David Taylor Model Basin 



Washington, D.C. 



As Professor Maruo correctly points out, the kernel function K(x) of Eq. 

 (3.37) (which is proportional to the 4th derivative of his function K(x)) has a 

 high order singularity at x = o, so that if Eqs. (3.36) were to be used as they 

 stand to calculate the C^p\ some juggling (such as integration by parts, as 

 suggested by Professor Laitone) would be needed in order to get a finite answer. 

 However, in presenting the results in the form (3.36), we did not imply a rec- 

 ommendation that this particular form of the integrals was suitable for direct 

 computation. Just as Michell's integral can be manipulated into many different 

 forms, so also can the integrals for the transfer functions c^^^, and Professor 

 Maruo has shown an alternative form due to Hanoaka which is clearly better for 

 numerical computation than that given in (3.36), and which avoids the difficulty 

 with the singularity. In fact our initial attempts at numerical computation have 

 used precisely this form, which can be derived directly from Appendix HI by use 

 of the Fourier transform convolution theorem. The form in which we gave the 

 results in the paper was chosen for pedagogical reasons, since it illustrates 

 most clearly the simplicity of the formulas in their dependence on B(x) and S(x). 



The three-dimensional effects mentioned by Professor Laitone are of 

 smaller order of magnitude according to slender body theory than the contribu- 

 tions we calculate. As far as possible we have indicated by order of magnitude 

 statements the size of the error in each equation, but there is probably no way 

 other than comparison with experiment to test whether or not the ship is suffi- 

 ciently slender for all the neglected terms (not only those mentioned by Profes- 

 sor Laitone) to be small. 



166 



