Joosen 



Up till ^ ;^ 2.5 the curves have a character that can be expected for three- 

 dimensional bodies. It can be compared, e.g., with the curves for a sphere cal- 

 culated by Havelock [13] . Experimental data are only available for frequency 

 parameters higher than 2.5, but there obviously the theory is not valid anymore. 



CONCLUSIONS 



It appears to be very useful, in dealing with the problem of a slender ship 

 performing oscillatory motions at different forward speeds, to express the 

 Froude number and the frequency parameter in terms of the slenderness pa- 

 rameter e. For practical purposes the range of low Froude number and high 

 frequency parameter is most interesting. 



In this paper the first order term of the velocity potential is derived for the 

 case where the Froude number is of order e^^^ and the frequency parameter is 

 of order e" ^ . 



The theory presented here can easily be extended such as to determine the 

 motion of a slender body in waves. Then a consistent pair of equations of mo- 

 tion for heave and pitch will follow. 



The analysis of section 4 is resulting in the two dimensional strip theory if 

 the slope of the bow- and stern-line is of order unity or smaller and the only 

 problem then is to solve an integral equation for each cross section separately. 

 If the slope is larger three dimensional and forward speed effects are present 

 as well. The resulting integral equation can be solved by an iteration process, 

 but an alternative method is to start the analysis from Greens' theorem instead 

 of a source distribution on the hull. 



Apart from the problem of the oscillatory motion of the ship an interesting 

 result is obtained for the steady advancing slender ship. 



For the case where the Froude number is of order e^^^ the only first order 

 contribution to the velocity potential and the wave resistance originates from 

 the source distribution on bow- and stern-line. From this fact it becomes clear 

 that it must be possible to affect the wave resistance by adding another singu- 

 larity in the bow and stern region. 



The strength of the singularity might be determined from a condition of 

 minimum wave resistance. By adding a dipole at the bow the concept of a 

 bulbous bow could be treated in the frame work of slender body theory.* 



= See comments by Laitone on paper by Newman and Tuck. 



182 



