Slender Body Theory for an Oscillating Ship 



Of primary interest is the leading term in the series expansion with respect to 

 e of fpj and fP2- Before starting this derivation it is necessary to introduce 

 some statements as to the order of magnitude of a, (i^ and ^^ with respect to 

 e. In this paper two cases will be considered. 



I. .= .^ ^„ =./!,, f, = %, (2-13) 



with /3j = 0(1) and ^3 = 0(i); 



II. a = e , /3o = 0(1) , ^l = O^D- (2.14) 



The first case is related to the problem of low Froude number and high fre- 

 quencies and one may expect that it corresponds with a ship moving in head 

 waves with small wave length. 



The second case deals with the problem of high Froude number and low or 

 moderate frequencies and it seems to be a good approximation for a ship moving 

 in following waves with moderate wave length. 



As far as the magnitude of the frequency parameter is concerned it is of 

 course evident that the ratio wave length to ship length is of much importance. 



The potential ^^ can be written in the form 



J-l Jc(f) 



where G. is Green's function for the free surface condition and F(^, O is the 

 source distribution to be determined from the boundary condition (2.12). 



A further notation is introduced: 



^. = cPq + cp. (2.15) 



with 



'" = f. '' I 



F(^,0 



V(^i-a'+ e2(77i-f)2+e2(^^-02 y(^i-a'+ eV^i-f)' + eHL.^ir 



and 



d: (2.16) 



'' - i '' I 



F(^, O Gi(^i,77j,^j: ^,77, Od? 



171 



(2.17) 



