Newman and Tuck 

 Q5 = (-i-c^ + U :|^jxB(x) , 



These Q^ , j = 1, ... 6, are reproduced in Eq. (3.16) of the text. Note that the 

 dominant flux transfer functions are those for heave and pitch, for which Q3 and 

 Q5 are of the order of B(x), i.e., 0(e). The flux in surge is of order S'Cx), i.e., 

 O(e^), while the flux in transverse modes of sway, roll and yaw vanishes. 



APPENDIX III 



EVALUATION OF THE KERNEL FUNCTION K 

 AT FINITE SPEED 



( 1) 



Now at any finite distance from the ship (i.e., such that y^ + z^ is large 

 compared with the small lateral dimensions of the slender ship) the effect of the 

 motions of the ship for vanishing slenderness is that of a line distribution of 

 sources of strength Q^i^ per unit length. These sources are "wave sources," 

 i.e., ordinary sources modified to satisfy the linearized free surface condition 



^i!^H^"^y* = ° 



on z = . 



(A3.1) 



This potential may be obtained from well-known results on such wave sources 

 (e.g., Wehausen and Laitone, 1961). One way of writing the source potential is 

 in the form of a Fourier transform with respect to x, putting 



l,(x,y,z, t) 



( 1) 



r dk e-^*^" c/>*j^(k;y,z,t) 



•^ - CO 



Q.n(x,t) = 



dk e-^*^^ Q;i)(k;t) , 



etc., where we have for the Fourier transformed potential 0(i), 



(1) 



~ 77- ^( 1 



( 1) 



-K„'(|k| Vy^+z^) 



+ 4 (-ico - ikU)' 



J- CO Tk^T 



dk e 



iXy +zA^+X^ 



Vk2 + \2 gyk2 + \2 + (-ia;-ikU)2 



(A3. 2) 



160 



